THREE IRREFUTABLE ARGUMENTS AGAINST RELIGIOUS BELIEF
By Ruth Marion Graf (ruthmariongraf@gmail.com)
Copyright, Ruth Marion Graf, Jan. 15, 2016
The first has to do with the claim of religion that God created the world by
intelligent design. A sure disproof of this assertion becomes clear once we
seriously consider what we mean by intelligent design. Consider the ways a set
of objects (■■■, ■■, ■) can come about. One is certainly by
intelligent design. A person can have this specific collection of objects (■■■, ■■, ■) in mind or designed ahead of time
and make it come about by selecting three red objects, two green and one purple
from this pool of objects, ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■, to fulfill that design.
But the (■■■, ■■, ■) pattern can also come about randomly from the person picking six objects from the pool blindly and by chance coming up with (■■■, ■■, ■) though having no sense or design of it in mind to begin with. No one has ever specified that God came up with the universe, living and nonliving, in this way because random picking by definition has no sense of intelligent design in it.
A third way of obtaining (■■■, ■■, ■) also exists, the trial and error way in which this pattern is what the person has in mind or designs ahead of time, but as achieved by random picking followed by elimination of any set of six objects that isn’t (■■■, ■■, ■) until, by chance, the (■■■, ■■, ■) set does pop up and is not discarded. No theologian has ever suggested God creating things in this trial and error way has because the notion of God making something He’s designed in a way that has Him fail on His first few tries runs directly counter to the notion of God being omnipotent and never making a mistake. That is, whatever God designs to come into existence, it just does.
This distinction between the sure and direct creation of something designed and creation that comes about through chance, whether complete or partial, has parallel in what we actually see in nature. Much of nature, especially in physical, nonliving, systems, comes about randomly. This assertion is not open to debate. The energy distribution of gas molecules is random as made observably clear in the MaxwellBoltzmann energy distribution.
Wherever gaseous systems exist in the universe, they take the above form, which is not, in being random, intelligently designed by anybody, unless, of course, God thought, “I don’t care about gaseous energy distribution and how it comes out; I’ll just let it happen randomly without any design in mind for the outcome.” Living systems have this same problem of the intrusion of randomness in how they develop. For while DNA provides a blueprint or design of next generation characteristics, a design that might arguably be attributed to God’s original authorship thereby ultimately making Him the original intelligent designer of the DNA, there is significant amount of randomness in the transmission of DNA based genetic information that is centrally important to the progression of life forms in natural history. There can be no denying the genetic mutation or sexual recombination, both random dynamics, that produce novelty in life forms. This is not a matter of pitting God against evolution, for the aforementioned randomizing processes are both part of the science of genetics, denial of whose assertions is as stupid as denying that 2+3=5.
This is not to say that one cannot argue that God achieves his intelligently designed end for living creatures through random trial and error operations. Again, though, doing so has God making lots of mistakes on the way to His materially realizing the design that believers in Him believe He had in mind, errors that contradict the basic notion believers also have that God just does not make mistakes. This is not to say that this argument could possibly convince the truest of believers who discard all sensibility and logic in clinging emotionally to their belief that God is the allpowerful, unerringly wise creator of the world and everything in it. Indeed in my efforts back a halfdozen years ago to keep intelligent design out of Texas science texts I ran into no less than three fundamentalists who argued that God could make 2+3 be something other than 5 if He wished it, one of them, no less, a high school math teacher in the Lubbock, TX school district. For those raised to attend Sunday services but not quite willing to concede this level of 2+3=6 omnipotence to their God, it should yet be obvious that there is an ineradicable contradiction in God as intelligent designer of a world much of which came about and comes about irrefutably by random, inherently notdesigned, processes.
At this point, it should become clear without further arguing that science and religion are incompatible. The second nail in the coffin of God as something other than a thought and a hope in believer’s minds will reinforce this incompatibility in an even more direct and unarguable way, even for the 2+3=6 believers. Everything that happens in nature is an energy transfer of one sort or another. If you see a marble on a table first sitting still start to move, whatever the specific cause, you can be sure that energy has been transferred to the marble. There is no argument in this other than of the 2+3=6 kind. When grass grows, energy is transferred from the sun to the plant. When a wildebeest eats grass, energy is transferred from the grass to it. And when a lion devours a wildebeest, energy is transferred from the prey animal to the predator. Indeed energy transfer is so important to everything that happens that modern society depends, more or less, utterly on oil as that source of energy.
Hence, outside of the 2+3=6 school of reasoning, whatever God might make happen in the real world, as He is attributed to do by all believers, it must take the form of an energy transfer. Note that we must expand this a bit to understand energy in the most general way as matterenergy via Einstein’s famous E=mc^{2} law that is demonstrated so convincingly by atomic and nuclear explosions. Hence, if God spontaneously creates some sort of material object, which He is attributed in religious dogma to have made a lot of, that is also an energy transfer.
Now there is no problem logically in specifying God as the source of matterenergy needed to do all His creating of whatever we see transpiring in nature. The problem is with God as the storehouse of all this matterenergy because one thing that matterenergy most definitely is, except at the margins, is observable, either directly by man or with the various measuring instruments man has developed in this era of modern science and technology. But God is generally accepted without much quibbling as a spirit being that is not and cannot be observed whatever form He might take including also Allah for the Muslims or Vishnu for the Hindus.
As this argument depends on science from the starting gate, it can, of course, be argued that all the argument shows is an incompatibility between science and religion, not which one is correct but rather that both cannot be right. While that does not quite take us back all the way to the 2+3=6 school of reasoning for those who would choose science wrong and religion correct rather than science correct and religion wrong, it comes fairly close because the basic argument of energy transfer holds entirely for the computer operations we are generating and transmitting all this information on, the denial of which is evidence, if not of stupidity, minimally a lack of basic education.
Of course, if science denies religion so profoundly why don’t we hear it from the scientists? After all, the vast majority of them do not believe in God as shown in repeated polls. But that’s because modern scientists, particularly the Americans, have much less concern with truth and its dissemination than they do with position and salary, which can be seriously affected by coming out actively and publically against God. Is this saying that today’s scientists are cowards? And deserving of the endless ridicule heaped upon them and on science in the silly, mincing, respect killing treatment of scientists and science on The Big Bang Theory?
We will let that question go as we develop the third irrefutable argument against religious belief, which does not depend on a chorus of PhD scientists agreeing with it, just common sense. It comes down to what we mean by something being real. There are two kinds of things that we sense, our perceptions and the stuff of our minds, our thoughts. There is no point in arguing that our thoughts are real, at least as something we sense. But there is equally no point in arguing that perceptions and thoughts are real in the same way. Certainly my immediate perception of an ice cream cone is real in that I can eat it as opposed to my thought of an ice cream cone, its image in my mind, in my imagination, which I cannot eat. There is a difference. If I have a thought of a unicorn, I can also ask if it’s real, that is to say, real beyond its reality as a thought. Most would say, no.
This gets us directly into whether, God, which nobody has ever seen in his or her perceptions, be He God the Father, Allah, or Vishnu, is real. There is a criterion for deciding which sensations are real and which not. Other than 2+3=6 quibblers, people agree from common sense that if you perceive something, if you see it, it is real. That’s a great and sure criterion for the reality perceptions. What of things that you don’t see, but know only through thoughts or images in your mind? Is God the Father real? Now I am not denying that right of the bat, but rather trying to specify the criterion that would decide if God was real. It’s not so easy as it is for things we actually see that we say are real just by that criterion of seeing them. Let me repeat the problem again. What is the criterion for judging the reality of things we don’t ever see? Some might say that it is if we ever see them. If we see something we conjecture or think about, it’s real. But that’s just my criterion, it can be argued. And people like Joel Osteen and Deepak Chopra would with broad smiles make such a counter argument. Who are you to say that just because you can’t see something that it is not real?
The trick is that while all things we can see are real, not all the things we just know by thinking about them are real. Surely nobody says that pink elephants that live in the valleys on the other side of the moon are real. And while the West are all sure that God is real and Allah a crock, not real at all, the Muslims think Allah is as real as Swiss cheese and that God the Father is about as real as the pink elephants on the moon. This gives us a sense that for things we see in our minds eye, what people say determines what of them are real and what are not. And indeed that makes sense for the very sense of belief, as in religious belief, depends much on our believing what somebody says. So it really comes down to whether we believe somebody when they tell us that something we don’t ever see is actually real.
This gets tricky because a lot of what other people tell us has a tendency not to be real. If what the TV commercials tell us verbally and nonverbally is real, then all of us should break into an orgasm of smiles and love whenever we eat a particular breakfast cereal or brand of laxative or drive a brand new car of a certain brand, indeed any or all of them! And another kind of stuff told us incessantly is real is the gabblegobble we hear in political discourse, whose reality never quite is realized after election day, these days you might say hardly ever. Neither of these two instances of disingenuousness should be that surprising given that in both it is in the misinformation transmitter’s selfinterest, individually and/or collectively, to convince us that something that is not real is, or will be, actually real.
Does religious discourse fit into that category? Is it most basically a false expectation implanted in people in the interests of the implanter, individually and/or collectively? Well, one thing common to all religions is that the little people in the sociopolitical systems that support the religion are told by the god of the religion to obey the ruling class of the system even when it is clear to somebody in the other religion across town that this is no more than obedience to tyranny in some form, as in the murderously tyrannical regimes of Saudi Arabia and also in the slow but sure happiness destroying wage slavery we have in the West.
The problem with this is that all the bad stuff that accumulates in people stupid enough to believe that things they can never see are real makes people not only unhappy but also hyper aggressive when they have position to pass their unhappiness on to others who had nothing to do with causing it in something we’ll call redirected aggression, the worst of which comes about not just in domestic mass murder but in war, the next one likely to be nuclear and terminating, which is reason enough to found a new religion, one based on science and common sense, which I’ll continue on with as part of my earlier writings on the matter, which may be redundant in areas, all of which though is helpful rather than a hindrance for the people of the world to hook onto a new way of thinking, a new religion, that can save us from that most horrible fate.
TOTEM POLES
A New Secular Religion underpinned by
mathematically welldefined Hierarchy
that will bring True Salvation to Man
by
Saving us from Nuclear Annihilation
Hiroshima, what the ruling class and
their fundamentalist
servants have no problem risking for you, the privileged all
being homicidalsuicidal degenerates deep down and the stupider
Christians all thinking they’ll live on forever after nuclear horror in
Heaven.
By a former Lutheran Missionary
turned scientist,
Ruth Marion Graf
Hierarchy is a universal phenomenon. Totem poles, corporations, the military, religious
organizations and human societies generally, your visual field, your ideas as
in Linnaean classification and the parts of speech of language along with crystals
and quantum mechanical stability are all hierarchical. As such hierarchy enables
a unified understanding of nature  physical, biological, human and divine 
when defined quantitatively as it is with the recently discovered Mathematical
Blessed Trinity equation, called such because this diversity index equivalence ultimately
explains the mystery of the divine Blessed Trinity of God, Jesus and the Holy
Spirit clearly for the first time.
We get right down to explaining the Mathematical Blessed Trinity, (MBT) with the (4, 2, 2, 1, 1, 1, 1) number set, which in fitting the equation is identified as a hierarchical set. Not all sets that are hierarchical in the broadest sense fit the MBT. The (4, 2, 2, 1, 1, 1, 1) set as a logarithmic hierarchy does as do all logarithmic hierarchies, another example being the (9, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1) set. But there are also linear hierarchies like (7, 6, 5, 4, 3, 2, 1) not identified as a hierarchy by the MBT. And the Fibonacci series, (1, 1, 2, 3, 5, 8, 13, etc.), is a hierarchy, linear at low values, logarithmic at larger values, that also does not fit the MBT.
The value of the MBT identified hierarchies lies in such sets having welldefined properties that also apply to the nonMBT forms of hierarchy. To develop the three diversity indices of the MBT that identify a regular or logarithmic hierarchy, it is helpful to look at the number set, (4, 2, 2, 1, 1, 1, 1), first as representing an ecology of K=12 parrots from N=7 different colored species diagrammed as (■■■■, ■■, ■■, ■, ■, ■, ■). Overlooking the fact that some of these parrots will have a hard time reproducing with crossbreeding disallowed, we intuitively sense (■■■■, ■■, ■■, ■, ■, ■, ■) to be a diverse parrot ecology, one with parrots in it from many different species.
To show how the MBT identifies hierarchies it is important next to be able to quantify the diversity in (■■■■, ■■, ■■, ■, ■, ■, ■). We are fortunate to be able to do that using the Simpson’s Reciprocal Diversity Index that was introduced shortly after WWII by the British code cracker and statistician,
We introduce Simpson’s diversity index in a less arbitrary way than he did as a pure statistics function as will also give us a good look at the basic properties of a simple set like (■■■■, ■■, ■■, ■, ■, ■, ■), (4, 2, 2, 1, 1, 1, 1), whether it is taken to represent a dozen colored parrots in a jungle, a dozen colored clothespins on a clothesline or a dozen colored buttons in a button box.
Let’s do take
the (■■■■, ■■, ■■, ■, ■, ■, ■), (4, 2, 2, 1, 1, 1, 1), set to be K=12 colored clothespins in
N=7 colors so we don’t have to worry about the reproduction problem we had with
parrots. And then specify the diversity of the clothespins colorwise with Simpson’s
diversity index. Those who think the clothespins a funny illustration to use in
this day and age of electric clothes dryers should keep in mind that your
author is 74 years old and coming from way back, though still doing well as you
can see in this photo taken the year our family was so stupid as to spend all
our money trekking around the Democratic primaries to campaign for Obama’s
election.
Ruth, Champ and his Mom
The shiny fellow in the middle is my mathematically gifted grandson who in the course of our election year travels made a TV commercial for Obama in Charleston, WV, which the TV manager, unmistakably a conservative, quipped was the best commercial he’d seen for America’s Great False Hope. I should make clear that we worked as hard as we did for Obama because we are long time peace activists and hoped he’d make a big difference. That is also why I worry so much about writing this mathematical analysis of the problem in an understandable way for the fate of the world may depend on this hierarchy based unification of knowledge that predicts with mathematical accuracy that the world is heading towards nuclear annihilation unless we get rid of all the weapons first.
Look, there is no way that Russia can defeat the US in war or us them when we both have 7000 nukes locked and loaded. Yet jabs are constantly being thrown by us both in the Ukraine and Syria and a bloody nose was drawn on Russia from our Turk allies shooting down one of their warplanes. Beware the critical blow in this constant sparring that would irreversibly bring about out and out war. Both sides need to understand their mutual catastrophic vulnerability if that happens. We have to short circuit such a critical event by doing something about it ahead of time, like ending our century old communism versus capitalism conflict with a true coalition of the biggest nuclear powers so that nobody remains to fight at that level.
And once this coalition of the nuclear biggies forms, with Russia and America cooperating and no longer enemies they will have the combined power to make sure that the nations lower in this new worldwide social hierarchy get rid of all their weapons, large and small. This Nuclear Coalition will administer the death penalty to those who violate the weapons ban or the border security of the smaller city states that will arise in this new world without weapons. Other than these two death penalty laws, the city states will have complete freedom, yet competing with each other in sportsmanlike ways that obviate the bloody horror of war as we now know it.
Putin would be more likely to enlist in this joint effort than an America that has been for so long the big guy on the block and unable because of that to step back and digest the realpolitik of the situation. That’s why we may need a revolution in this country to get us to join in with the Russians to convince the rest of the world, by reason or force, to get rid of their weapons.
The easiest revolution in America would come about with the election of Donald Trump, who, whatever his shortcomings, real and exaggerated, has sense enough to see the nowin aspect of the EastWest conflict and use his noted deal making skills to end the conflict and get the world on its way to becoming one with no weapons. And if Mr. Trump needs me as his vicepresident on the third party ticket he’s going to have to run on because the Wall street oligarchy will kill him off in the primaries with the media it controls, I’m up for a fight with the unspeakably disgusting reptilians on the right and the just a bit less unspeakably disgusting Democrat politicians.
With apologies for drifting so far from the mathematics, I should make it clear that it’s not all that easy being a peace activist dedicated to the only thing worth dedicating one’s life to in this era, prevention of nuclear annihilation. Nobody pays attention whether the nuclear issue is just too big for people to get their arms around or because they believe God wouldn’t let it happen or if it did would compensate them with happiness in Heaven for eternity, a hope too ridiculously unrealistic to take the time to ridicule. Still we took that path of shouting out to people, yes, nuclear annihilation is coming unless people get together to do something about it, when Pete, my life long companion, wrote this piece back in the Cold War days with the Soviet Union.
Knickerbocker News,
Albany, NY, May 1986
Whatever you might think of the idea of a world with no weapons, espousing it requires having an independent attitude and specifically being independent of bosses and landlords and the bunch you’re confronted with when you shed those two abusers, the police. You’ll see our opinion of the cops in the story entitled, The Grand Junction Nuclear Tooth Fairy Tale, where we’ll speak of what happened to us in Grand Junction, Colorado. To prepare you for that let me now get you up to date on this Day after Christmas, 2015, as to what happened to us after we left Grand Junction the beginning of November heading for Las Vegas where Champ and his mom now live.
As cash is always tight for struggling savetheworld peace activists and knowing from experience that Las Vegas away from the high rent casinos is rough street wise, we would have preferred moving in with them, easy to do on paper as we had been separated from them for only two years.
We were half expecting the kids’ rejection, so it wasn’t the end of the world for us emotionally when they begged off. But as to where one lives in the nontourist, lowrent sections of Las Vegas, that can get kind of tricky. Let me get analytical for a moment and put this situation into broader context. Capitalism is realistically the broadest term for ownership of two different kinds of properties, real estate and standard businesses, stores or factories of some sort, both of which make a profit for their owners, though in different ways. What matters in the end analysis politically is that people’s controllers, those who have power over them, are either the boss or the landlord or both. Those who submit thoroughly to their boss as their master are generally rewarded with enough money to buy a home and be free of a landlord as overlord, a significant escape from abuse.
These career and boss devoted people are one kind of unfree person or wage slave in today’s 1984 type of hierarchical society, the boss having the home owning underling tightly on a paythemortgage leash so tightly that the careerist has all spirit of rebellion bled out of him, starting with the boss whose money hold on the careerist amounts to strangulation. And there is also another kind of wage slave, the type who doesn’t have that level of allegiance and submission to his or her boss because he rents and doesn’t lose a life’s investment of a home if he loses his job. And for his lesser obeisance, he rents from a landlord who is a second boss, and most often a classic bastard in one way or the other.
Slightly below this fellow in the social hierarchy is the fellow who doesn’t get along very well with his boss or landlord and he gets a third overlord, the police who kick his ass in from time to time within local courts and jails. And there are those who refuse to “get along” with the requisite humiliation to be endured with bosses or landlords and wind up under the control of police and the socalled justice system, whose concerns, may I digress, are much more with maintaining the rule of the privileged classes who, via the police and other institutions they control, the media and the education system and our “real” government run by our intelligence agencies plus military, keep the people they feed off of very much under control and in a vegetative existence, much like the wildebeest getting no resistance from the savannah grass they feed off of.
Well, that’s the reality of it, the big picture within which you should force yourself to fit all the individual scenes in your life. Nobody except Genghis Khan and his sort of imposing ruler have been free to do what they want since the invention of agriculture followed by technology made legalized theft of another’s sweat energy, aka wage slavery, profitable along with the superior weapons including believable mass misinformation outlets needed to keep captive the modern slave who provisions and amuses the decadent privileged with the massively expensive junk food these reptilians need to fend off the ever lingering emotional pain of exile from love, disgusting.
Freedom is a funny thing. Few get to taste it in this NaziDisneyland hierarchy of economic and police penal and amusement and suggestive information control of emotion and the behavior it directs. And those who do refuse rule by threat of punishment or delusion inspiring misinformation are usually discovered as free, captured and torn to pieces, as our family was when my brother, Don, had Pete sent to prison via the connections he had with the two million strong fundamentalist LCMS, Lutheran Church Missouri Synod, and with the connections they had, like many other powerful corporations and organizations in America do, with the FBI.
This happens every day to people who don’t have hierarchical power, though usually not as dramatically as for Pete who was sent to federal prison for three years for getting angry at a millionaire businessman who had connections with the judges in Albany, NY, played poker with them every week, he bragged, and with the local FBI. Pete got angry at Wendell Williams because he swindled us out of $70,000, that accusation hardly excessive given that Williams had to pay it back in a civil action shortly after Pete was hauled off to the torture chamber where those lucky enough to taste the joy of freedom as makes them maladjusted slaves are reshaped into good citizens that have the correct delusional in their mind of life in America as happy as the bullshit sold on ruling class controlled TV in its commercials, news and entertainments. And this torture, (I need not go into detail unless it matters to you that one of them was keeping him in a cell at fifteen below zero in his underwear for 16 hours), was done to him at age 57 as a PhD biophysicist who writes newspaper articles on saving the world from nuclear war.
Back to recent past now and Las Vegas where the hate in the noncasinotourist areas is thick on the streets. A city bus ride is like a trip on the last car in the Snowpiercer train movie, almost every bus now fitted with an armed transit cop, who, unlike his propaganda model on TV, the ever cool, ever heroic, ever mannerly cop on TV, has no problem throwing his weight around to intimidate everybody on the bus, especially the young males, a police state specialty that reminds me of the few months I spent in Franco police state Spain in the 70s where the Civil Guardia big cops did a number big time on the women. Of course it can be argued that the armed police on every bus are entirely needed these days. And that’s the truth of it as a measure of just how bad things have gotten in America, lots of unhappiness and lots of violence hanging out over the edges of that unhappiness.
For us, though, this time, the street harshness was not much of a problem because we found a place to live in close enough to the downtown casinos to get us by given our better clothes than the homeless men and enough confidence in our step to be mostly left alone on the street. Rather it was our living in a motelapartment complex called the “Siegel Suites” that is of interest because we wound up in the classic trap of those who live around the country in inexpensive motel rooms, the landlord, your motel manager. I’ll tell you a story of two of them. It’s funny, guaranteed to make you laugh, but only after I get back a bit now to the math.
I said I’d derive the diversity index as a statistical function. The K=12 object, N=subset (■■■■, ■■, ■■, ■, ■, ■, ■), (4, 2, 2, 1, 1, 1, 1), set contains x_{1}=4 red, x_{2}=2 green, x_{3}=2 purple, x_{4}=1 blue, x_{5}=1 orange, x_{6}=1 brown and x_{7}=1 black clothespin. Formally the K number of unit objects in a set as a function of the unit objects in each of the N subsets of the set, (x_{i}, i =1, 2, …N), is
1.)
The Σ symbol is a “summation” that tells you to add up whatever variable is in front of it, for K, the x_{i} number of objects in each subset. Next we want to consider the average number of clothespins in a color subset in the K=12, N=7, (■■■■, ■■, ■■, ■, ■, ■, ■), (4, 2, 2, 1, 1, 1, 1), set, µ=K/N=12/7=1.71 clothespins. Or formally and generally for any set of K unit objects divided into N subsets,
2.)
The error in the µ mean or average, the statistical error, could be the familiar standard deviation, σ, or as we prefer to use, its square, σ^{2}, called the variance, defined as
3.)
For the N=7, µ=12/7, (■■■■, ■next section ■, ■■, ■, ■, ■, ■), (4, 2, 2, 1, 1, 1, 1), set the variance is σ^{2}=1.06. Another commonly used statistical error is the relative error, r= σ/µ, the square of which, r^{2}, also called the perfect error, is useful in this analysis.
4.)
For the σ^{2}=1.06, µ=12/7,
(■■■■, ■■, ■■, ■, ■, ■, ■), (4, 2, 2, 1, 1, 1, 1), set the perfect error is r^{2}= σ^{2}/µ^{2}=.36. Next we will write
the D Simpson’s Reciprocal Diversity Index as a function of N and r^{2. }
5.)
For the N=7, r^{2}=.3611 (■■■■, ■■, ■■, ■, ■, ■, ■), (4, 2, 2, 1, 1, 1, 1), set the D diversity is D=5.14. To get a better sense of what this means, let’s calculate the D diversity next of the K=14, N=7 (■■, ■■, ■■, ■■, ■■, ■■, ■■), (2, 2, 2, 2, 2, 2, 2) set. As a balanced set, it has r^{2}=0 and, hence, D=N=7. It makes clear that for a balanced set, each subset having the same x_{i} number of unit objects, the diversity of the set is just the N number of subsets or different kinds, here colors, of objects. And that for an unbalanced set like the N=7 (■■■■, ■■, ■■, ■, ■, ■, ■), (4, 2, 2, 1, 1, 1, 1) set, the diversity index is the N number of subsets or kinds of objects reduced by the imbalance as specified by the r^{2} perfect error.
Now by what route we will not specify at the moment, the D Simpson’s Reciprocal Diversity Index is a simple function that will take us to the Mathematical Blessed Trinity via our expressing it as it is usually given in the scientific literature in terms of the same p_{i} weight fraction variable that dominates the functions in the Mathematical Blessed Trinity equation.
6.)
This p_{i} variable is just a straightforward fractional measure of the x_{i} of the set referred to as its weight fractions.
7.)
For the K=12 (■■■■, ■■, ■■, ■, ■, ■, ■), (4, 2, 2, 1, 1, 1, 1), set that has x_{1}=4, x_{2}=2, x_{3}=2, x_{4}=1, x_{5}=1, x_{6}=1 and x_{7}=1, the weight fractions are p_{1}=1/3, p_{2}=1/6, p_{3}=1/6, p_{4}=1/12, p_{5}=1/12, p_{6}=1/12 and p_{7}=1/12. Note that the p_{i} weight fractions of any number set sum to one.
8.)
Now inserting the p_{i} weight fractions of the (■■■■, ■■, ■■, ■, ■, ■, ■), (4, 2, 2, 1, 1, 1, 1), (1/3, 1/6, 1/6, 1/12, 1/12, 1/12, 1/12), set into Eq6 calculates the same D=5.14 earlier gotten from Eq5, the derivation of D in Eq5 from Eqs27 algebraically simple and straightforward for the reader. Note now, though, that Eq8 allows Eq6 to be written as something that will lead us directly to one of the God functions in the Mathematical Blessed Trinity
9.)
And this allows to write D in terms of a β (beβta) index, which will get us that much closer to the Mathematical Blessed Trinity as
10.)
And this will allow us to obtain an infinite spectrum of diversity indices as
11.)
This D_{β}
Marion Diversity Index is fascinating in including not only exact forms for
microstate temperature and entropy, β=1/2, but also the inverse sums for
series capacitance and parallel resistance and inductance in electronic
circuits. For now, we want to focus on the β=1/2 Marion Diversity Index, D_{1/2},
which we will give its own symbol to, h, because, as we will show in detail, h
as the energy diversity of a thermodynamic system is the proper exact form that
microstate entropy should be represented in.
12.)
Inserting the p_{i} weight fractions of the N=7, (■■■■, ■■, ■■, ■, ■, ■, ■), (1/3, 1/6, 1/6, 1/12, 1/12, 1/12, 1/12) into Eq12 obtains the h Marion Diversity Index as h=6. This h=6 is less of a reduction from the N=7 number of subsets parameter than D=5.14, but for both diversity indices, the diversity is the N number of subsets reduced by the imbalance in the set. While the imbalance as a reducing factor is implicit in h of Eq12, it is explic
13.)
It also can be written with a little algebraic manipulation from Eqs6&4 as
14.)
That this ξ function is also a diversity measure is clear when we evaluate it for N=7, r^{2}=.36, (■■■■, ■■, ■■, ■, ■, ■, ■), (4, 2, 2, 1, 1, 1, 1) to obtain ξ=6, the same value as for the Marion Diversity Index, h=6. Now this equivalence of ξ=h is not general for any number set but only for those with a hierarchical form like (4, 2, 2, 1, 1, 1, 1). Before we get into the nuances and ramifications of this identification algorithm, let us pick up the 3^{rd} leg of the Mathematical Blessed Trinity, a linear form of the central function for information in classical information theory, the Shannon (information) entropy,
15.)
In natural logarithm form, H is also referred to as the Shannon Diversity Index, an obviously logarithmic measure of diversity rather than linear as D, h and ξ are. But H is readily transformed into a linear diversity index as
16.)
As you might have anticipated from the above chatter, M for the (■■■■, ■■, ■■, ■, ■, ■, ■), (1/3, 1/6, 1/6, 1/12, 1/12, 1/12, 1/12) set also has the value of M=6. Hence we can write the Mathematical Blessed Trinity in short form in terms of these as h=ξ=M.
17.)
Another set that is intuitively hierarchical, at least to me, is K=108, N=40, (27, 9, 9, 9, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1). And it has the diversity equivalence of h=ξ=M=27.8. It should be pointed out that if you change any one number in this set or in the (4, 2, 2, 1, 1, 1, 1) set, the equivalence is lost. And it is also true that every h=ξ=M set is intuitively hierarchical except perfectly balanced sets like (2, 2, 2, 2, 2, 2, 2), which are h=ξ=M but only hierarchical in the sense of being a 1 level hierarchy. But none of that is convincing in itself of what is being described is a hierarchy.
Let’s give a good example that will help to take some of the mystery out of what we mean by a hierarchy, which is a very difficult concept to grasp intuitively or sketch mathematically because our minds and our mathematics are geared to linearity, 1, 2, 3, 4, simple count descriptions of what we are exposed to and a hierarchy is, indeed, nonlinear and, hence, not intuitive and not easily quantified.
And it is important to do that because hierarchy is an extremely general phenomenon in nature. Corporations in America are hierarchies as is the military. The Catholic Church and the Hindu caste system are patently hierarchical. Indeed, all organized institutions are hierarchical though most at the national level say otherwise. And though we are less interested in abstract forms of hierarchy, knowledge is inherently hierarchical as exemplified in the Linnaean Classification of living organisms and the original Roget’s Thesaurus of word forms. Scientific concepts are inherently hierarchical as is most obvious when it’s expressed in mathematical form where one or a few differential equations can describe systems nature that apply hierarchically to many subset situations. The food chain in ecologies is also patently hierarchical and is also paralleled by exploitive human hierarchies that extract labor energy for those on the upper levels of the hierarchy from those on the lower levels.
A geometric representative of the (4, 2, 2, 1, 1, 1, 1) set is very helpful for understanding it as a hierarchy in terms of its basic properties and relationships.
Figure 18. A Geometric Hierarchy
The square is 2 inches on edge with its representative (4, 2, 1, 1, 1, 1) hierarchical set consisting of N=7 parts, in measure x_{1}=4 in^{2} for the square, x_{2}=2 in^{2} for the large twotoned blue triangle, x_{3}=2 in^{2} for the large twotoned brown triangle, x_{4}=1 in^{2} for the little blue triangle, x_{5}=1 in^{2} for the little azure triangle, x_{6}=1 in^{2 }for the little brown triangle and x_{7}=1 in^{2 }for the little tan triangle. This interprets the square of Figure 18 as a hierarchy made up of the N=7 polygons below.
Figure 19. The Geometric Hierarchy of Figure 18 as a Composite of its N=7
Constituent Polygons
The N=7 polygons constitute a hierarchy of polygons at three levels, each of which level contains the same size polygon in this mathematically regular polygon which will enable us to distinguish the main features or properties of a hierarchy, which have somewhat different meaning depending on the theatre in which the hierarchy is located, of perception or cognition. The large square of x_{1}=4 in^{2} is at the 1^{st} level in the hierarchy. The pair of large, twotoned triangles of x_{2}=x_{3}=2 in^{2} are at the 2^{nd} level in the hierarchy, one twotoned blue and the other twotoned brown. And the four small triangles, brown, tan, blue and azure are at the 3^{rd} level in the hierarchy, each of an area of x_{4}=x_{5}=x_{6}=x_{7}=1 in^{2}.
Explaining the hierarchy in any detail beyond this is a waste of time because to understand a hierarchy as a set of sets, you first have to thoroughly understand the set itself, a journey we embark on next with accent on “thoroughly.” The concept of a set is all important in a unification and simplification of knowledge because the set is so general. There are 464 meanings of the word “set”, more than for any other word in the English language. This is good reason to think that a set may be the right concept and mathematical structure to base a grand unification of science on. Indeed, axiomatic set theory is already understood as the foundation of mathematics. And to the extent that everything in nature might ultimately be understood mathematically, a questionable premise, the set might provide a foundation for all human knowledge. However axiomatic set theory has a problem in that regard because its axioms, while formally defensible, are not sufficiently congruent with realistic sets of objects and events to be that set based unifying agent for a realistic scientific compendium of knowledge as a record of our experience with reality.
A suitable alternative is a set theory that includes a quantitative specification of hierarchy and a sense of the set that underlies hierarchy more in line with what we sense a set to be realistically. The importance of getting the foundation mathematics correct is that it does provide a comprehensive unified sense of nature that leads unarguably to a prediction of nuclear war unless the world is made weapons and war free first as described earlier.
To that end consider (■■■, ■■, ■) as a set of K=6 objects divided into N=3 color subsets that consist of that consist of x_{1}=3 red, x_{2}=2 green and x_{2}=1 purple objects. This set can be represented in shorthand with the letter, S, or with the number set, (3, 2, 1). For simplicity and the mathematical regularity needed for exact counting all the objects in the (■■■, ■■, ■), (3, 2, 1), set are understood to be unit objects, all the same size.
The first postulate of Totem Pole as a mathematical schema is that the objects in a set of objects are distinguished in two markedly different ways. First we distinguish every object in a set as being fundamentally different or distinct from every other object in the set. To wit, the 1^{st} red object in (■■■, ■■, ■) is fundamentally different or distinct from the 2^{nd} red object in (■■■, ■■, ■) and also from every other object in (■■■, ■■, ■) in its being in a different place in the set than all the other objects. This fundamental distinction between the objects in (■■■, ■■, ■) is specified by representing each object in the set with a different letter, namely as (abc, de, f).
And second, the expression of the set as (■■■, ■■, ■) obviously distinguishes some of the objects from others in being of a different color or contained in a different subset of the set. This is distinction of kind or categorical distinction in contrast to the fundamental distinction between objects brought up in the previous paragraph. Categorical distinction also exists between subsets, not just objects, as between the red subset, [■■■], in (■■■, ■■, ■) and the green subset, [■■].
Unfortunately, our circumstances have just taken a turn for the worst and we are, as we have been at times in the past, that story later told, unable to continue writing. Instead we’ll hitch this new front end of the information train onto what’s already been done wherein lies all of the material promised in this introduction, though not in as perfectly an orderly presentation as I started out with.
THE GRAND JUNCTION NUCLEAR TOOTHFAIRY TALE
By Ruth Marion Graf and Dr. Peter Calabria, PhD
©, Ruth Marion Graf, Nov. 7, 2015
Peter and the Wolf
You wouldn't expect it of a cop, not in this day and age of rampant cop murders of citizens and that punk faggot cop from Fox Lake who shot himself rather than go to jail because he knew what a horror and a torture it would be from inflicting it on others. But out came this federal cop from the Social Security office to spare severely pained and exhausted me from a near two mile walk to get to the nearest functioning bus stop.
He says, "You know, sir, the bus doesn't come way out to here unless you call them. I'll give 'em a call for you."
And then back he came outside again to tell me that when he called them they told him the bus doesn't come out after 1 PM, but that he told them that I was doing pretty rotten (I had mentioned to him inside the office that I'd been to the St. Mary's Hospital ER twice in the last few weeks) and that they had better come.
"Many thanks," I said, genuinely meaning it, "The pain's bad enough that I actually thought about taking a swim in the Colorado River."
"My son did that," he said, startling me though I showed it not at all partly because his remark had no emotion in it and partly because he was a cop and once you've been abused by cops you're as wary of them all as you are hateful. "Drove in, car and all," he continued in the same monotone.
"From pain?" I quickly said back, partly out of curiosity and partly because he was a cop and the situation called for some sort of response.
"Yeh, pain, he was 24." he continued in the same flat tone. Though who knows what kind of pain for I had a soninlaw whose father was a cop and who was treated badly enough in his childhood by his parents to hate them and eventually committed suicide by driving his car into a tree.
"I've got a lot of pain, too," the cop continued, "You just live with it; it's just a matter of attitude." I kind of sensed at this point he was saying that for my sake for I was so disoriented by my pain when I came into the SS office that he couldn't miss the look on my face.
"You can survive pain if you have the right attitude," he said twice more for stress, the conversation brought to an end with, "Hey, here comes your bus!"
First time I saw a cop as a human being, ever persisting negative thoughts about them as a group aside. And with that I was off to the Marillac Clinic in north Grand Junction with a nobenefits letter from Social Security in hand that I needed to get treatment for my pain that was bad enough to warrant a CScan for my head on my last visit to the ER.
Ruth and the Wolf
This is me at age 67 down in Lubbock next to my math genius grandson and his mom the year our family wasted campaigning for the Great False Hope.
Lubbock was also where Pete's TMJ problem (jaw, ear and head pain from dental misalignment) flared up a year before we got stuck in Grand Junction on our way West. Now, you would ask me from the political leanings suggested by Pete's cop story, what in God's unholy name were we doing hanging out in a place that's three times more conservative and church filled than very conservative Grand Junction. It goes way back. My father, the Rev. Arthur E. Graf, was a fundamentalist minister, who can say whether for good or evil without being accused of ideological bias. Objectively, though, and impossible to rationalize away is his nephew, my cousin, Ed Graf Jr., confessing in court below to murdering his two stepsons for insurance money by burning them to death.
Good family to run from, which I did. The fellow on the right is my older brother, Don, a senior partner in the largest law firm in West Texas who developed his aggressive style of legal maneuvering punching me, his 7 year old kid sister, on the arm and telling already frightened gullible me that a wolf was upstairs in my bedroom ready to get me when I went up to bed.
The family did not like it at all when I walked away, publicly shaming them. My mother left me and my three siblings $30,000 each when she died eight years back. Except my share was set in trust with my lawyer brother, Don, never meant to be disbursed to me except a hundred here and there as bait to keep me coming back for it. "You'll never see a penny of it unless you leave Pete," who it was I ran to when I ran from the family back then 42 years ago.
The whole inheritance thing was set up, very clever, to punish me by driving me crazy over it for the living's not easy for a 74yearold with all the typical health problems of old age. That punishment included Don's getting me locked up in Lubbock County Jail for a week for trespassing when I refused to leave his law office on a visit down from New York to get the money. Quite a well contrived revenge from the grave for my being a bad girl, pretty clever, mom.
Don't have photos of my weekly corporal punishment administering parents, burned them all. But they looked not different at all from Aunt Sue and Uncle Ed, the child killer's parents, ugly, selfrighteously cruel and perverse child abusers beneath the wolf in sheep's clothing platitudes Daddy spouted from the pulpit and Mom at the evening dinner table.
More on this story and more insulting truths on Don in a later section if you're curiosity's been aroused about this halfcentury family fight to the death and beyond. Hint as to the baddest bad feelings in recorded history: Pete backed Don down in a head to head in front of Don's trophy wife, Ruby, who a few months after that cuckolded him, right Donnie boy, and a year later divorced him.
The dentist assigned to Pete at the Lubbock Community Health Center was a Dr. K, a young lady who dressed during office hours, as I saw the one time I was her patient, in a fashion designed to show off the thighs, as they said in the old days, sufficient to get the attention and erection of as many of her male patients as she could. Though this is just one woman's opinion of another, whatever her state of mind and intentions in the clothes she wore beside the dentist chair.
This does not certify that K was a bad dentist. Pete's tooth that came to trigger his TMJ malady could have been unusually problematic, who knows, who can say, who can tell. But the pain began after she worked on it. And just got worse a few dentists later until the pain, which is sort of like having your little finger bent backwards, started driving him crazy especially in keeping him from getting any sleep for the two months before we see him at the ER and the Social Security office in Grand Junction, CO.
The Dental Clinic
The relatively young fellow who managed the Marillac Dental Clinic in Grand Junction by the name of Levi, as suggests an extreme religious upbringing, was unusual looking to be kind, top end obese and completely bald. Does such an extreme lack of attractiveness, especially in a young person, suggest an equally unattractive personality, somebody to be avoided for fear of no good coming from it? That would not be strange for attractiveness suggests somebody to be connected with that might have tangible value for you.
Functionally paranoid as such first impressions might be (consider the recent book title, Only the Paranoid Succeed), Levi turned out in the short term a lifesaver in facilitating the root canal and crown I needed to kill my near unbearable pain for a price I could actually pay, it was that inexpensive, $200 instead of $2000. Dismissing my first impression with no hesitation I told him from the heart, "Any appreciation I might express would have to be understatement." And that was not just because the price was right, but because he told me, and told Ruth separately, that they could start in a day or two. Whew! Suffering with a good attitude takes a back seat any day to getting rid of great pain. But atheist me made the mistake, seconded by Ruth, to mention to Levi that we were longtime peace activists, giving him one of our news articles from the 1980s Cold War period.
Knickerbocker News,
Albany, NY, May 1986
Given that we're all still here after 30 years, it's hard to refute the Chicken
Little label most people instantly tag the piece with, the divinely inspired
End of Timers irrelevant in the debate, their fears calmed with the hallelujah
resolution of life eternal in the afterlife.
That the end of it all does actually threaten us, maybe not in the last 30 years but sooner or later, has a strong historical, evolutionary and mathematical basis. If that sounds as trustworthy as Ben Carson's recitation of life events past, let me try to spell it out for you with as little of the math, a generally hated subject, as we can do with.
God, this is so hard to phrase without the math. Consider two distinct populations of organisms, from two different species or subspecies or lineages or, for man, cultures, living across generations in one niche or territory or environment. If the niche has lots of space and lots of resources, meaning that the two populations still have plenty of space and resources to increase their size in, there's a much less likely a chance of competition between the two relative to the case where the niche has reached its limit in terms of space and resources.
This case of where one population can grow only at the expense of the other in the niche is what, when the populations are human, prompts war between the two. To repeat, lots of room for expansion prompts peaceful coexistence and no room left prompts war.
That's what was great about America back then, lots of space and lots of resources. That's also what was great about the human race way, way back then. If early man had a problem with another group of early man, they just went their separate ways because there were a lot of separate ways available to go. And that's exactly how the planet got dispersed with our kind.
But all that changed once we filled the world up with our kind. No place to disperse to with the only alternative being your rival, your enemy, chewing you up if you don't want to fight back. And modern weapons made everything much worse. Fights between different Maori tribes in New Zealand were rough, it's true, even before the Brits got there, but horrible once the Maori picked up the white man's weapons to fight with for the best tool for mass murder is gunpowder.
When the weapons power is distributed in a sufficiently unbalanced way, one group wins out big and the rest are dead or on their knees. This is empire and it's the history of man whether one approves of it or not. It just is. Also irrefutably a part of history is that old empires fall and new one's are formed always, always by war. And no nation refrains from using a weapon in its arsenal because of fears of collateral damage, especially when they're playing from the down side of the game. Given time, the bigger competitions will arise, as we're already seeing now with nuclear biggies Russia and the US with China not exactly acting peaceably in the South China Sea.
Not stressed as much in the newspaper article was the other weapons issue, not of people being killed by them, but of being controlled by them, tyrannized, exploited, enslaved, made miserable and, as such, when the opportunity arises, more aggressive towards whomever they can release their unhappiness. And getting rid of weapons at this level gets rid of the problem. Ban all weapons use on pain of execution, just one law from the big guy, the people deciding all the rest.
It's very simple. And it's doable. Just have to get Putin together with a sane and sensible representative on our side, who seem to be in marked short supply at this time. So much so that I nominate Ruth for the job, maybe in tandem with ostensible jackass Trump, who at least has a realistic if immoderately selfish view of life. You must kill the weapons or we wind up like Hiroshima, the whole world.
This said, things turned out at the Dental Clinic not so good and that's possibly because we also called Levi's attention to the precursor of this website, which had been written in haste because of my head pain, a review of life and the world that was less polished and mannerly than what you just read:
a.) As Genghis Kahn once said: Every man wants to be the khan (the king, the top boss of it all, like he was) or be independent (no boss of any kind at all.)
b.) An interesting and quite correct observation on the part of Genghis considering that anybody within his reach (as a lot of people were back in the Mongolian empire of the 13^{th} Century) who attempted to be the Khan or to be independent and outside of his control was executed, usually by beheading.
c.) Now before we start making Genghis out to be some kind of unique devil figure in history, note that any honest history book on civilized man (and that excludes the history preached to the common folk within a nation by the rulers including to mind vulnerable children) will tell you that enslavement in one form or another has been with us since Ancient Egypt and Babylon and Sparta and Rome and in the serfdom of Medieval Christianity and British Colonialism and just last week in the 9 to 5 and beyond wage slavery all have to put up with in America.
d.) And should also tell you that top bosses (as with kings of old and _____ empowered by big money in modern times, the blank used because there are no politically correct, proper, words allowed for the billionaire wage slave controllers) would never say through the information outlets they control including religion, education and today’s quasireligion psychobabble, that people would rather be independent than be the happy wage slave suckers portrayed in the movies and media that only the millions of our billionaire rulers, may they all be given black eyes, are able to produce.
e.) And an honest telling would also make clear that life’s “personal” miseries as derive from failed and frustrated relationships (shhh, don’t tell your sisterinlaw or neighbor or coworker) are not the product of Satan or “mental illness” but of the 9 to 5 and beyond screwing people take in life as a matter of course from bosses and from authority controlled, yea owned, by money, as the police and courts certainly are (except on TV.)
f.) Really, can it be missed by any halfsensible person, the correlation between the unhappiness of the standard mass murderer and his suicidalhomicidal release from that unhappiness from his last few bloody moments? What do you mean, he was/is mentally ill and that’s why he did it? My, that’s so vague. It’s unhappiness in the heart that drives people mad, unhappiness that comes from the same place as for the rest of the suckers who can’t make the correlation between it and humiliatingly kissing the boss’s ass six ways six times a day as the years go by. Or, to push the logic of the connection one step further, let’s ask where “mental illness” comes from? Well, we just don’t know, say the welldressed shrinks on TV and their pundit and police chief sidekicks. We never have any idea what the mass murderer’s motive was, and that’s 300 such motiveunknown mass killings just in 2015. Hmmm.
g.) Though, c’mon, what’s the mass murder of a handful of people every day compared to the mass murder, legs and arms and heads blown off, of war? And does that come from the same place as domestic mass murder, from the tyranny of social control, whether from dictator setup police coercion or billionaire setup economic coercion? The boss or the authorities (in the schools now, too) too, kicks you in the ass, or threatens to day by day, but since you can’t kick the bastards back, you bullyingly and often sneakily pass your pain on to others who had nothing to do with causing your pain. And this extends to masses of people kicked in the ass passing on their hate to masses of other people, of different ethnicities, here and in other countries. Kill the bastards. That feels so good, to hate to cause others to suffer when you feel bad yourself. And it hits hard also on independent people who are often targets out of jealously.
h.) Way down in the cellar of causation, of course, it’s all Darwin’s fault, all this violent competition admixed with the worldwide economic competition that finances for the winners the best weapons, that plus the aforementioned blind hate and aggression that wells up from underlings dominated and beat up by the bosses passing their misery onto vulnerable others en masse.
i.) And there is no way to shut it down, as long as the social control font for it keeps flowing. No way to shut it down except to get rid of all the weapons in the world. And that would work because the simplest mathematics tells you that it’s absolutely impossible for a relative few people to control a whole lot of people without weapons. Which is the what the police are there for primarily, to prevent revolution, which is why it is necessary to take the weapons power away from the billionaire bosses, may their eyes be blackened, who own the police and effectively, better believe it, tell them what to do. Or is it too much to ask the question of why the billionaires NEVER go to jail.
j.) And as no small side effect of getting rid of all the weapons, you also get rid of war as we know it, up to and very importantly including mankind annihilating nuclear war, lots on exactly how to bell that cat in later sections where you’ll find that A.) It’s entirely necessary or we go up in smoke and B.) It’s quite possible to accomplish once people see clearly the 5000 years of misery up to the present that weapons have caused the human race, in war and peace.
k.) And we should also say something about the poor fellows and gal soldiers who do the billionaires’ bidding to grow and maintain the American Empire. To give your lives with arms and legs and brains blown away for the sake of maintaining the privileged lives of the decadent scum at the top? Nothing sadder than a veteran in a waiting room at a VA hospital. Really. Take a quick trip to the lunch room at the VA hospital up in the Hanover, NH, area. These poor bastards, masses of them impaired for life in irreparable ways. Even the most sensibly selfish of us can’t help feel the true horror of it. Let’s get rid of the weapons.
l.) And the rest of the a. through z. precepts of this new religion of happiness and sanity are scattered through the rest of this treatise prefaced by a bit more mathematics.
I'll get to the punch line of this not very funny joke now. After being appraised of the existence of our political awareness, not exactly what you'd say was right wing or fundamentalist, Levi tells me that they can't get to the pain problem driving me insane for a full month. Today's Nov. 6 as I'm writing this and the root canal isn't to be started until December 2, may God have mercy. So I'm just going to stick this on the front of whatever's on the website now for all those who aren't terrified of mathematics or of thinking and saying what the faggot authorities who own and run this show don't want them to know, including the low class academics in the Colorado University system including the dumb cowardly jerks at the local university here at Colorado Mesa University.
Look, when the leading candidate of the party of the rich and powerful says and worse believes that the pyramids were used to store grain as advised by the Biblical character Joseph instead of burying the pharaohs, you have to understand that the fucking sky is falling. Why, if you need a why. Because a significant fraction either believes that also or knows Uncle Potato Carson is lying and doesn't care because they believe that everybody, who's smart, lies to everybody, and that the country is a hostile place in that regard but also in the form of Orwell's take on modern life, the most perfect example of doublethink, that country is free but that we are lied to at every moment by our leaders. You don't think that the threat of nuclear war is real, assholes out there who masturbate on the toilet while watching the NFL game and drinking a beer with your free hand because you don't think much of yourself, or at least not much of yourself, and they really don't, Ruth. And that's why they are ever distracting themselves from the truth about themselves by entertaining their pained brains so as to not have to think about reality beyond the performances they have to give at work, physically and socially, their only pleasure watching the free die painfully in spirit and otherwise, the jealous faggots, all of them and including most certainly your sister, Don.
Ask yourself, who or what on earth could possibly murder by burning them to death two little kids? You don't think that's perverse. And you don’t think that somebody who would kill two babies in a pure torture way is perverse in his behavior. And if perverse in wanting to murder people for the pure pleasure of it and being quiet about that urge and those acts done is not also capable of being a smelly faggot and hiding the fact of it, Don with his observably socially dick sucking "yes" man, Kolander. They're all faggots and on the right the worst of them especially when they have power. At least I'll die with my balls still on, punks.
Those Jews with big cash at the top with tons of political power; they all look like Aunt Millie on steroids. Who’s that guy, and to his credit despised by Trump, Barry Diller. Jesus! John Paul Sarte, the world's last significant philosopher before they were collectively exterminated by the brilliantly executed near 24/7 mind cleanings of movies, media, education and the pulpit said that every person over the age of 40 is responsible for their own face.
The serfs of the medieval ages had it bad. But this modern version for it, so perfectly out of 1984 in his basics though hidden so much better that nobody even who read the book could fathom the tinseled worm infested tree they're perched in.
Every mammal past the time of maturity has sexual feelings put in them by Charles Darwin that drive them in as mammoth a way as hot pepperoni pizza being stuck under your nose makes it near impossible not to take a bite. And now the hint is that teen kids doing young adult sex in the way they are programmed for evolutionary fitness are being thought of as criminal felons liable at least according to the letter of the law, an easing of it for apparent generosity sake only done to make the law not seem as ridiculous as Uncle Ben's assertion about pyramids being used to bury grain in for God to eat to ease his constipation. How did the state get to be the complete arbiter by penal law of people's sexual behavior?
Oh, you say they're not, asshole? Well just because Uncle Ben's ding dong pulling lies are easy to see through does mean that there are people as in the media and the courts and who run businesses and social control institutions who do it a hell of a lot better than he does. And it's not that their bullshit can't be let out of the bag if they're framed scientifically, mathematically, but that all the STEM people are castrated psychologically no different than the monks who peopled the medieval universities.
The Colorado River almost seems good to think about but much more fun to think about beating the oligarchy that runs America who think their toilet paper scented with $5000 a bottle perfume is worth the pain suffered by the suckers on the bottom.
Now let’s get on to the mathematical analyses written up before we ever set foot in Grand Junction.
II: The β Diversity Index Demystification of Entropy
By Ruth Marion Graf, A. Thomas Rogovsky and Peter
Calabria, PhD
Contact: ruthmariongraf@gmail.com
No technical article on Wikipedia is more primed with “disputed” tags and the like than entropy (energy dispersal). This is a strong hint that entropy, a confusing notion for most, is also a bit of a mystery even for science despite Boltzmann’s famous entropy equation considered almost sacrosanct from inscription on his tombstone.
Boltzmann’s 1906 Tombstone
Many years after the suicide by Boltzmann prompted by initial rejection of his entropy formulation, a British statistician and WWII code cracker, Edward Hugh Simpson, steps into the picture with three numerical measures of diversity for ecological and ethnic populations.
One of his diversity functions, the Simpson Reciprocal Diversity Index, is the precursor to a new diversity expression, the β Diversity Index, which reformulates entropy in as mathematically crisp a way as Boltzmann, but is much more common sense intelligible as energy diversity. But as Simpson developed his diversity indices back in 1948 as ad hoc quantifications of population diversity of interest mainly to biologists and sociologists, there was and has been little exposure of it to the physical scientists interested in entropy. And one can be fairly certain that any physicists who have come across it in the last 65 years did not give a second thought to considering it suitable for describing entropy.
All that changes though once diversity is derived, developed and generalized as a pure mathematics function. Then Simpson’s reciprocal diversity and a companion β diversity function are readily seen to have a greater than .9995 Pearson’s correlation to Boltzmann’s S entropy, which spells out as its having as good a fit to the empirical laboratory data as the S entropy and is preferable to the S entropy from both its simplicity and clarity. There are three pure mathematics perspectives on Simpson’s reciprocal diversity as the entry point to the β Diversity Index that solves the entropy problem. We will first take up the matrix arithmetic of natural sets derivation of it.
The term natural set refers to a set of objects we would see when we
open our eyes to the world around us. The Hiroshima photo, for example,
presents a set of objects that consists of 1 atom bombed out building, 15
window frames, 956 pieces of rubble and 1 semifortunate man in the foreground
of the rubble. A natural set distinguishes the objects in a set by kind and by
the number of each kind.
For mathematical simplicity and efficiency we work with regular natural sets. An example is of this set of colored objects, (■■■, ■■, ■), represented by the number set (3, 2, 1), that consists of unit objects (all the same size) as limits the set’s elemental numerical characteristics to the K number of objects in the set, for (■■■, ■■, ■), K=6; to the N number of subsets in the set, for (■■■, ■■, ■), N=3 color subsets; and to the number of objects in each subset, for (■■■, ■■, ■), x_{1}=3 red objects, x_{2}=2 green objects and x_{3}=1 purple object.
Obvious is the use of simple arithmetic to add up the objects in the N=3 subsets of (■■■, ■■, ■) to get the K=6 total number of objects in the set: K=x_{1}+x_{2}+x_{3}=3+2+1=6. What is less obvious is that the basic arithmetic operations of addition, subtraction, multiplication and division can be done not just with single numbers but with sets of numbers. When that is done systematically with a matrix, what is developed is a general diversity measure that underpins a unified mathematical explanation for everything.
Though forms of the β Diversity
Index are salient in the addition, division and product matrices we introduce
it through the product matrix because that develops it from the
Simpson’s diversity index that science is already well familiar with. The
product matrix of the (3, 2, 1) set multiplies every number in the set with
every other number including itself.

3 
2 
1 
3 
(3)( 3) 
(3)( 2) 
(3)(1) 
2 
(2)( 3) 
(2)( 2) 
(2)(1) 
1 
(1)( 3) 
(1)( 2) 
(1)( 1) 
Figure 1. The Product Matrix of the (3, 2, 1) Set.
The sum of the products in this matrix of the K=6, (3, 2, 1), number set is K^{2}=36. Add them up to see. Keeping in mind that the (3, 2, 1) natural number set represents a set of observable objects, we readily show the origin of the K^{2}=36 parameter with a comparison matrix of all the objects in the (■■■, ■■, ■), (3, 2, 1), set.

■ 
■ 
■ 
■ 
■ 
■ 
■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■■ 
Figure 2. The Comparison Matrix of (■■■, ■■, ■)
We count the total number of pairs in this matrix of the K=6 object, (■■■, ■■, ■), (3, 2, 1), set as K^{2}=36 as we also obtained from Figure 1 as the sum of all the products in the product matrix. Out of these K^{2}=36 comparison pairs, we count Y=22 different pairs like ■■ and Γ=14 alike pairs like ■■. (Γ is the Greek letter, gamma.) The Y=22 and Γ=14 matrix variables are also obtained from the product matrix in Figure 1 with Γ=14 the sum of the products in the diagonal of the matrix (highlighted in yellow) and Y=22 as the sum of all other products in the matrix. The Γ matrix function is of sufficient importance that we specify it below for any set of K objects distributed over N subsets as
3.)
From the above for the (3, 2, 1) set, Γ=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=3^{2}+2^{2}+1^{2}=14. Important next is to see that the ratio of the K^{2}=36 pairs to the Γ=14 same color pairs is, D, the Simpson Reciprocal Diversity Index of the (■■■, ■■, ■), (3, 2, 1), set.
4.)
For the (■■■, ■■, ■), (3, 2, 1), set, this Simpson’s diversity is D=36/14=2.571. This D diversity is more familiar to ecologists and sociologists when expressed in terms of the p_{i} weight fractions of a number set, usually referred to by them as population densities.
5.)
For the (■■■, ■■, ■), (3, 2, 1), set that has x_{1}=3, x_{2}=2 and x_{3}=1, the weight fractions are: p_{1}=x_{1}/K=1/2; p_{2}=x_{2}/K=1/3; and p_{3}=x_{3}/K=1/6. Note that the weight fractions sum to unity.
6.)
Now from Eqs4&5, we express D in terms of p_{i} as
7.)
To develop the general form of the β Diversity Index we next substitute Eq6 in Eq7 to obtain
8.)
We can express this using a numerical index, β.
9.) ; β=1
And we form the β Diversity Index as a general diversity measure whose value depends on β, which has infinitely wide range as
10.)
There are diversity measures with different values for a set for β=1, the D Simpson’s reciprocal index, and for β=0, β= −1 and β=1/2. Interesting is that the three latter diversities derived from Eq10 for β=0, β= −1 and β=1/2 come about respectively from an addition matrix, a division matrix and a square root matrix and in all, as with D in Eq4, as the ratio of the sum of all terms in the matrix divided by the sum of terms in the diagonal of the matrix. Quite fascinating from a pure mathematics perspective, we do not wish to stop and considered the details of these other than for the square root matrix that generates the β=1/2 diversity in its having direct relevance to formulating the correct entropy (and temperature) measures of a thermodynamic system. And because of its importance in that regard we shall give D_{β}=D_{1/2} its own special symbol, h. Hence from Eq10, the h square root diversity index is
11.)
It is easiest to develop D_{1/2}=h using a new example set, the K=14, N=3, (■■■■■■■■■, ■■■■, ■), (9, 4, 1), set. Its p_{i} weight fractions of p_{1}=1/14, p_{2}=4/14 and p_{3}=9/14 obtain and D from Eq7 as D=2 and h from Eq11 as h=D_{1/2}=7/3=2.333. The square root matrix that develops D_{1/2}=h is a product matrix that specifies (9, 4, 1) on one axis of the matrix and the square root of the x_{i} of the set on the other axis as below.

9 
4 
1 
3 
(3)( 9) 
(3)(4) 
(3)(1) 
2 
(2)( 9) 
(2)( 4) 
(2)(1) 
1 
(1)(9) 
(1)(4) 
(1)( 1) 
Figure 12. The Square Root Matrix of the (9, 4, 1) Set.
The sum of all the products in the matrix is 84, the sum of the diagonal products, 36, and the ratio of these sums, 84/36=7/3=2.333=D_{1/2}. We need both D and h to reformulate entropy and temperature correctly but will put h on the back burner for a while to derive D next as a statistical function that is the direct entry to the derivation of entropy as energy diversity or dispersal. We start by noting that the central statistic in statistics is, but of course, the arithmetic average or mean of a number set, µ, (mu),
13.)
For the K=6, N=3, (■■■, ■■, ■), (3, 2, 1), set the mean or average number of objects in the N=3 color subsets is µ=K/N=6/3=2. We next use the µ mean as part of the definition of a primary error function in statistics, the σ^{2} variance_{, }
14.)
For the N=3, µ=2, (■■■, ■■, ■), (3, 2, 1), set the variance is σ^{2}=2/3. The square root of the σ^{2 }variance is the familiar standard deviation, σ, for the (3, 2, 1) set, σ=.816. Another commonly used statistic error is the relative error, r= σ/µ, the square of which, r^{2}, called the perfect error, is useful in this analysis.
15.)
For the N=3, σ^{2}=2/3, (■■■, ■■, ■), (3, 2, 1), set the perfect error is r^{2}=σ^{2}/µ^{2}=(2/3)/2^{2}=1/6. Algebraic manipulation of the end variance term in Eq13 with the D diversity expression of Eq4 obtains D as a statistical function.
16.)
This sees D as an important function of a natural number set in that it includes all three of the set’s basic parameters, the K total number of objects in the set, the N number of subsets in the set and the statistical error of the set as r^{2} that gives a measure of the spread of the (x_{i}) numbers in the set. Now for emphasis sake, we want to go back and repeat that D as originally conceived by Simpson as an ad hoc measure of the intuitively sensed diversity in an ecological or ethnic population was never thought of as a valuable function for physical systems by the physical science community who studied entropy. But we see in the above development of D as more of a pure mathematical structure is that D is a measure of any system in nature that can be represented as a natural set including specifically a thermodynamic system of K energy units randomly distributed over the system’s N molecules.
The development of D as a β diversity makes that all the more clear, for the β= −1 division matrix derived diversity, which we shall cover in detail at the end of this entropy presentation, shows the ratio of the µ mean to the β= −1 diversity, D_{1}, to perfectly calculate the inverse summing of series capacitance and parallel resistance and inductance of textbook circuit theory. And, hence, if that β= −1 diversity, D_{1}, shows itself to be a bona fide physical variable, so also must the β=1, D, and β=1/2, D_{1/2}=h diversities be considered in that vein able to be used to formulate and explain entropy properly.
With that encouragement for D as a valid specification of physical systems, we next turn to textbook multinomial theory to develop D as entropy where it is seen that for any distribution of K objects over N subsets,
17.)
The P_{i} term is the probability that any one of the K objects in a set will reside in the i^{th} of the N subsets of the set. Now a random distribution is an equiprobable distribution in the sense that all N subsets of a set have an equal P_{i} probability of containing one of the K objects. To clarify that, let’s look at the K=6, N=3, (■■, ■■, ■■), (2, 2, 2), set where we see that the P_{i} probability that any object chosen at random will be red, green or blue is P=1/3=1/N. Hence substituting P_{i}=1/N in Eq9, we see that the σ^{2 }variance of a random distribution of K objects over N containers is
18.)
Before we proceed too far, too fast, let us make it clear that this applies to a thermodynamic system in its being understood, as it is by all scientists without argument, as a random or equiprobable distribution of K discrete energy units over the N molecules of the system. And let us also make it clear just what the σ^{2} variance in Eq18 is referring to in a random distribution. We’ll do that by illustration with a minithermodynamic system or random distribution of K=4 energy units over N=3 molecules. The variance of this random K=4 over N=3 distribution from Eq18 is σ^{2}=(4/3)(2/3)=8/9. To understand where this σ^{2}=8/9 variance comes from consider next the “states” of this system that specify the various ways the K=4 energy units can be distributed over the N=3 molecules given in the table below along with the σ^{2} variance of each of the states as calculated from Eq14.
State 
σ^{2} Variance 
(4, 0, 0) 
32/9 
(3, 1, 0) 
14/9 
(2, 2, 0) 
8/9 
(2, 1, 1) 
2/9 
Table 19. The States of the K=4 over N=3 Random Distribution and their
Variances.
Now we will take the average of the state variances as weighted by the number of ways that each of the states can come about, aka, the permutations of each state. These are derived from textbook combinatorial statistics that specify the total number of permutations in the system as N^{K}=3^{4}=81 and the number of permutations per state as listed below.
State 
# of Permutations 
σ^{2} Variance 
(4, 0, 0) 
3 
32/9 
(3, 1, 0) 
24 
14/9 
(2, 2, 0) 
18 
8/9 
(2, 1, 1) 
36 
2/9 
Table 20. The States of the K=4 over N=3 Random Distribution and their Number
of Permutations
The various states and their permutations, whose relative values give an
indication of the relative time the system is in that particular state, come
about by the random molecular collisions and energy transfers that take place
most easily pictured as coming about for a system of gas molecules in a
container of fixed volume. Now the variances of the states are averaged by
weighting them with the fractional measure of the permutations of each
state.
State 
# of Permutations 
Weight 
σ^{2} Variance 
Weighted Variances 
(4, 0, 0) 
3 
3/81=1/27 
32/9 
32/243 
(3, 1, 0) 
24 
24/81=8/27 
14/9 
112/243 
(2, 2, 0) 
18 
18/81=6/27 
8/9 
48/243 
(2, 1, 1) 
36 
36/81=12/27 
2/9 
24/243 




Sum =216/243=8/9= σ^{2}_{AV} 
Table 21. The States of the K=4 over N=3 Random Distribution and Pertinent
Measures to Determine Their Average Variance.
Hence the σ^{2} of Eq18 of the random K over N distribution is the
average variance of the distribution, σ^{2}_{AV}, and
should be specified as such.
22.)
Eq18 also obtains the average r^{2} perfect error for a random distribution from Eqs15 as
23.)
For the K=4 over N=3 random distribution, r^{2}_{AV}=1/2. The
above also obtains the D diversity for a random distribution of K objects over
N subsets from Eq16 as
24.)
For our K=4 over N=3 random distribution, D_{AV}=(4)(3)/6=2. Eq24
understands D_{AV} as a weighted average of the D diversities of all
the states of a random distribution and, thence, as the diversity of a
thermodynamic system as a whole. Also the D_{AV} diversity provides a
clear picture of the thermodynamic system in the form of one of the states of
the random distribution, (2, 2, 0), that has the very same variance, σ^{2}=8/9,
the same perfect error, r^{2}= σ^{2}/µ^{2}=(8/9)/(4/3)^{2}=1/2,
and the same diversity, D=2, as the σ^{2}_{AV}=8/9, r^{2}_{AV}=1/2
and D_{AV}=2 of the system as a whole. This (2, 2, 0) state is called
the Representative State of the system.
Next let’s look at another random distribution, K=12 over N=6, to reinforce this understanding of the Representative State. Its energy diversity is from Eq24, D_{AV}=(12)(6)/(12+6−1)=4.235. The state of the system that has this diversity as calculated from Eq4 is the number set, (4, 3, 2, 2, 1, 0), which is the Representative State of the system for that reason. If the Representative State truly represents the thermodynamic system as a whole, it should have the characteristics of the system as a whole, one of which should be the MaxwellBoltzmann energy distribution of a thermodynamic system.
Figure 25. The MaxwellBoltzmann Energy Distribution
To see if it does we plot of the number of energy units on a molecule vs. the number of its molecules that have that energy for the (4, 3, 2, 2, 1, 0) set.
Figure
26. Number of Energy Units per Molecule vs. the Number of Molecules Which
Have That Energy for the Average Configuration of the K=12 over N=6
Distribution
Calling this a
MaxwellBoltzmann distribution may seem a bit of a stretch, though the plot could
be (generously) characterized as a very simple, choppy form of a
MaxwellBoltzmann. What we need to make the point are larger K and N
distributions like the K=36 energy unit over N=10 molecules random
distribution. Its D_{AV} diversity is from Eq24, D_{AV}=8. The
K=36, N=10 state that has this diversity is (1, 2, 2, 3, 3, 3, 4, 5, 6, 7), and
as such is the Representative State of the system. A plot of the energy
distribution of this set is
Figure
27. Number of Energy Units per Molecule vs. the Number of Molecules Which
Have That Energy for the Average Configuration of the K=36 over N=10
Distribution
This curve was greeted without prompting as “It’s an obvious protoMaxwellBoltzmann,” by the author of a primary graduate text on the thermodynamics of surfaces, John Hudson, Prof. Emeritus of Materials Engineering, Rensselaer Polytechnic Institute. The larger the K and N of a random distribution, the better the fit to the MaxwellBoltzmann distribution of Figure 25 of a realistic thermodynamic systems that has extremely large K and N parameters. So next we’ll look at the K=40 energy unit over N=15 molecule distribution, whose diversity is from Eq24, D_{AV}=11.11. A Microsoft Excel search program finds four states that have this diversity including (0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 6) whose number of permutations is the most of the four and is the most representative state from that perspective. Its plot of energy distribution is
Figure
28. Number of Energy Units per Molecule vs. the Number of Molecules Which
Have That Energy for the Average Configuration of the K=40 over N=15
Distribution
This is a not very symmetric MaxwellBoltzmann, but we are getting there. Next we will look at the K=145 energy unit over N=30 molecule distribution whose D_{AV} diversity is from Eq24, D_{AV}=25. There are nine K=145, N=30 number sets with this diversity including (0, 0, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 7, 7, 8, 8, 9, 10), which has the most permutations of the nine as the most representative state and whose plot of energy distribution is
Figure
29. Number of Energy Units per Molecule vs. the Number of Molecules Which
Have That Energy for the Average Configuration of the K=145 over N=30
Distribution
At this point we are beginning to consider K and N values of random
distributions large enough to display a fairly good resemblance to the standard
MaxwellBoltzmann distribution of Figure 25. And as we progressively increase the K and N values of
distributions, the above plots would more and more approach the shape of the
realistic MaxwellBoltzmann distribution of Figure 25. This suggests that this
Representative State approach we have taken is reasonable and that the D_{AV}
diversity is a meaningful variable of a thermodynamic system. In what way?
The entropy of a thermodynamic system is currently accepted to be Boltzmann’s equation inscribed on his tombstone, which is in modern terminology,
30.) S=k_{B}lnΩ
It easy to show that the D_{AV} diversity of a random distribution of Eq24 and Boltzmann’s S entropy of Eq30 are, for large K and N random distributions that mimic realistic thermodynamic systems, nearly directly proportional and, hence, can be substituted for each other mathematically. To demonstrate this we need not explore the meaning of the Ω (capital omega) variable in Boltzmann’s entropy formulation but only to render the Ω and lnΩ variables as functions of K and N (k_{B} is a constant) to show the extremely high correlation that exists between D_{AV} and lnΩ, hence, between energy diversity and entropy. This is easily done with a standard formula for Ω found in any advanced physics textbook.
31.)
And lnΩ is
32.)
For large K over N equiprobable distributions it is easiest to calculate lnΩ with Stirling’s Approximation, which approximates the natural logarithm (ln) of the factorial of any number, n, as
33.)
Stirling’s Approximation works exceedingly well for large n values as with ln(170!) =706.5731 being very closely approximated as 706.5726 by the Stirling’s. The Stirling’s Approximation form of the lnΩ expression of Eq20 is
34.)
We use this
formula to compare the lnΩ of a list of randomly chosen large K over N
equiprobable distributions to their D_{AV} diversity measure of Eq24.
K 
N 
lnΩ 
D_{AV} 
145 
30 
75.71 
25 
500 
90 
246.86 
76.4 
800 
180 
462.07 
147.09 
1200 
300 
745.12 
240.16 
1800 
500 
1151.2 
381.13 
2000 
800 
1673.9 
571.63 
3000 
900 
2100.88 
692.49 
Table 35. The lnΩ and D_{AV} of Large K over N Distributions
The Pierson’s
correlation coefficient for the D_{AV} and lnΩ of these
distributions is .9995 as indicates a near perfect direct proportionality
between the two as can be appreciated visually from the near straight line of
the scatter plot of these D_{AV} versus lnΩ values.
Figure 36. A plot of the D_{AV} versus lnΩ data in Table 35.
This high .9995 correlation between lnΩ and D_{AV} becomes greater
the larger the K and N values of K>N distributions surveyed. For values of K
on the order of EXP20 as found in realistic thermodynamic systems, the
correlation for K>N distributions becomes.9999999 indicating a near perfect
direct proportionality between lnΩ and D_{AV}^{. }As the
Boltzmann S=k_{B}lnΩ entropy is judged to be correct ultimately by
its fit to empirical laboratory data, given the very high correlation of D_{AV}
to S, the diversity entropy formulation must also be judged correct from that
empirical perspective.
But that is not the end of the story because the D_{1/2}=h diversity of Eq11, of a random distribution, h_{AV}, also has an exceedingly high correlation with lnΩ and, as will be shown from its perfectly fitting the temperature measure of a thermodynamic system, is the correct β Diversity Index for entropy. Showing the correlation of h_{AV} to lnΩ is not as straightforward as it was for D_{AV}, however, because h_{AV} is not expressible as a simple function of K and N as was D_{AV }in Eq24 as D_{AV}=KN/(K+N−1). There is a remedy for that, though, because h_{AV} is the h diversity of the representative state of a thermodynamic system, a number of which we developed for the random distributions of Figures 2629. We obtain h_{AV} for the K over N distributions of Figures 2629 from the x_{i }and p_{i} values of the Representative States from Eq11. Below are listing their h_{AV} values along with the lnΩ values of those representative states from Eq32. And we also include their D_{AV} diversities from Eq24 for comparison sake.
Figure 
K 
N 
lnW 
D_{AV} 
h_{AV} 
26 
12 
6 
8.73 
4.24 
4.57 
27 
36 
10 
18.3 
8 
8.85 
28 
45 
15 
26.1 
11.11 
12.33 
29 
145 
30 
75.88 
25 
26.49 
Table 37. The lnΩ, D_{AV} and h_{AV} of the
Representative States of Figures 2225.
The Pearson’s correlation coefficient for the lnΩ and h_{AV} values in the above is .995. And for the lnΩ and D_{AV} values it is .997. Note that though this lnΩ and D_{AV} correlation of .997 is high, it is less than the .9995 correlation between lnΩ and D_{AV} from Table 35 for larger K and N distributions. This is attributed to the Pearson’s correlation coefficient between diversity and Boltzmann’s S entropy being a function of the magnitude of the K and N of the random distributions tested, those of the distributions in Figures 2629 used in Table 37 being significantly smaller than the K and N of the distributions in Table 35. Hence the Pearson’s correlation between lnΩ and h_{AV }of .995 for the K over N distributions in Table 37, being little different than the .997 correlation between lnΩ and D_{AV} for these distributions, implies that the lnΩ and h_{AV }correlation is, as was the .9995 between lnΩ and D_{AV }for larger K and N distributions, sufficiently great to have h_{AV} also accepted as a diversity candidate for entropy from its high correlation with the S=k_{B}lnΩ Boltzmann entropy.
To choose between the two turns out to be little problem as it is easily shown that h_{AV} derives the true microstate temperature of a thermodynamic system. This analysis will also make clear why the diversity function is so important for mathematics and mathematics based science. To that end consider the K=6, N=3, (■■, ■■, ■■), (2, 2, 2), set to be a balanced set given that all of its x_{i} have the same value of 2. Its D diversity is seen from Eq4 to be D=N=3. In general the D diversity and N number of subsets are one and the same for a balanced set.
Now recall that we devised the (■■■, ■■, ■), (3, 2, 1), set as a regular set made up of unit objects (all the same size) with the original rationale that it was mathematically easy to work with. But we could have said, as we will now, that the reason for using a regular set was that when things counted are the same size, the count of them is exact. But when things counted are not the same size, the count of them is inexact. This becomes intuitively clear if we vary the sizes of the objects in (■■■, ■■, ■) to be, say, (■■■, ■■, ■). Now while it is not forbidden to say there are K=6 objects in (■■■, ■■, ■), that count of them is inexact for we would never use this K=6 parameter to develop the mathematics of the set without taking into consideration the size variations of the objects in the set. The dictum of things counted needing to be the same size is why standard measures as for weight like the pound are all the same size. If they were not, one pound having a slightly different magnitude than another pound would make commercial transactions involving them be greatly open to question.
In the same way the N number of subsets in an unbalanced set, like N=3 for (■■■, ■■, ■), (3, 2, 1) is inexact because the N subsets are of different sizes, not all containing the same number of unit objects as was the case for the (■■, ■■, ■■), (2, 2, 2), set whose N=3 parameter is exact because the subsets are all the same size in having the same number of unit objects in each. Note that N being inexact for an unbalanced set like (■■■, ■■, ■), (3, 2, 1) strongly suggests that µ=K/N is also inexact or somehow in error for an unbalanced set. And that should not be surprising given that the µ=K/N mean is invariably accompanied by a measure of statistical error like the σ^{2} variance or the σ standard deviation or the r relative error or r^{2} perfect error, all of which are zero for balanced sets with that the absence of statistical error implying the absence of inexactness as a roundabout rationale for balanced sets (whose subsets are all the same size) being exact.
Now to make the point that the β Diversity Indices of a set are exact both for balanced and unbalanced sets. That is clear in a most intuitive way for D itself given its D=N/(1+r^{2}) specification from Eq16 in which the inexactness in N can be understood to be offset by the inclusion of the r^{2} statistical error within the D=N/(1+r^{2}) function for D. More generally, though, it is clear that all β diversity indices are exact because they are functions entirely of p_{i} as we see in Eq10. And p_{i}=x_{i}/K is exact because as we see for (■■■, ■■, ■), (3, 2, 1), the x_{i} of it as x_{1}=3, x_{2}=2 and x_{3}=1 are exact because the objects counted in each subset are all the same unit size and similarly for its K=6 parameter, exact because all objects in the set are unit objects.
The above makes it clear that D in particular and D_{β} in general are surrogates, exact surrogates, for inexact N. This is very important for the mathematical sciences, particularly thermodynamics, our mathematical science of immediate interest. For it question science’s acceptance of the N number of molecules in a thermodynamic system as exact. This discrepancy is profound, for however much we may use moderately inexact measures in practical situations (as in counting 12 oranges at the grocery store not all perfectly the same size as 12), nature does not operate on inexact counts and our use of inexact counts as being approximately correct should be understood as not obtaining the most exact and clear understanding of nature.
More to the immediate point, much as we can use D as an exact surrogate for the inexact N of an unbalanced set, we ask if we there is an exact average that is a surrogate or standin for inexact µ=K/N. One is obtained by replacing the inexact N in µ=K/N with exact D to form a biased average, φ, (phi),
38.)
The φ=K/D biased average is an exact average in being a function of K, which is exact, and of D, which is also exact as has been made clear earlier. The K=12, N=3, µ=K/N=4, D=2.323, (■■■■■■, ■■■■■, ■), (6, 5, 1), set has a biased average of φ=K/D=12/2.323=5.166. This is greater than this set’s arithmetic average of µ=K/N=4 from the φ biased average being weighted or biased towards the larger x_{i} values in (6, 5, 1). To intuitively appreciate the bias in the φ biased average towards the larger x_{i} values in an unbalanced set, we express φ from Eqs38,4&5 as
39.)
This shows the φ biased average to consist of the sum of “slices” of the x_{i}
of a set, slices that are p_{i} in thickness, as biases the φ
average towards the larger x_{i} subsets in weighting them with their
correspondingly larger p_{i} weight fraction measures. We can also
develop a square root biased average, ψ, (psi), also exact, as the
ratio of K to h, which from Eq11 obtains ψ as
40.)
The numerator in the rightmost term shows the ψ square root biased average to be the sum of “slices” of the x_{i} of a set of thickness p_{i}^{1/2}, which biases the average towards the larger x_{i} in the set in their having larger p_{i}^{1/2}. The ∑p_{i}^{1/2} term in the denominator of the rightmost term, it should be explained, is a normalizing function required to make the p_{i}^{1/2 }“slices” in the numerator sum to one, this summing to one of fractional “slices” being necessary for the construction of any kind of an average of a number set.
Having developed all this somewhat digressive, somewhat tedious material let’s show now how it formulates temperature correctly. The simplest form of the microstate temperature of a thermodynamic system is currently taken in the standard physics rubric to be the arithmetic average energy per molecule of the system, µ=K/N. But a flag is immediately raised that there might be something wrong with this specification of temperature because, as we have made clear, the µ=K/N arithmetic average of any unbalanced set is inexact and a thermodynamic system is decidedly an unbalanced set in its N molecules having different energies as is made unarguably clear from the MaxwellBoltzmann energy distribution of Figure 25, which has been empirically proven to be correct.
That says what the temperature isn’t. As to the function that properly represents temperature we must consider how it is actually measured, physically, with a thermometer. Each of the N molecules in the thermodynamic system collides with the thermometer to contribute to its temperature measure in direct proportion to its frequency of collision with the thermometer. And that is equal to the velocity of the molecule, which itself is directly proportional to the square root of the x_{i} number of energy units a molecule has. Because of this, the slower moving molecules with the smaller energies in the MaxwellBoltzmann energy distribution of Figure 25 collide with the thermometer less frequently and have their energies recorded or sensed by the thermometer in its cumulative compilation of the temperature measure less frequently, which records their energies with smaller p_{i}^{1/2} slices of the actual values of their energies. This is in contrast to the faster moving, higher energy molecules that collide with the thermometer more frequently and have their energies recorded as larger p_{i}^{1/2} slices of their actual energy. This biases the temperature measure towards the energy of the higher energy, faster moving, molecules as properly understands temperature as the square root biased average of molecular energy, ψ=K/h, rather than the simple, but inexact, µ=K/N arithmetic average of the molecular energy.
There is one more factor that must be added to this argument to make it precisely correct. As a thermodynamic system cycles through its many states over time with the average of the characteristics of those states being the characteristics of the representative state of the system and of the system as a whole, the temperature of the system is the ψ of the representative state or ψ_{AV}. And by logical extension from Eq40, the correct diversity measure for entropy is therefore
41.)
So, however much Boltzmann is revered, and rightly so, for his breakthrough efforts to understand entropy microscopically, its specification in terms of h_{AV} energy diversity, which is impossible to distinguish from Boltzmann’s S entropy quantitatively given the exceedingly high correlation between them, finally makes sense out of entropy as a physical quantity as energy diversity or energy dispersal.
Was Boltzmann, therefore, wrong? Unfortunately, especially given that he gave his life for science, literally, he was. Some who have reviewed this material came to the conclusion that though the analysis cannot be argued with, the diversity take on entropy provides a parallel understanding of entropy, not an overthrow of Boltzmann physics. But it is patently impossible for both Boltzmann and The β Diversity Index Demystification of Entropy to be correct because the two are mutually contradictory in their axioms. Boltzmann statistical mechanics starts off with the premise that the K energy units are indistinguishable and therefore subject to indistinguishable object combinatorial statistics while the β diversity prescription considers the K energy units to be distinguishable and subject to distinguishable object combinatorial statistics as the analysis of the K=4 energy units over N=3 molecules in Eq18 to Eq22 makes clear.
Hence they cannot both be right. Assuming Boltzmann wrong, let’s now consider just how this thoroughly canonized saint of science screwed up with the problem of distinction. There are two kinds of distinction in nature, categorical and fundamental. In the (■■■, ■■, ■), (3, 2, 1), set, it is obvious that a red object, ■, is categorically distinct, a different kind of object, than a green object, ■. That’s why they are put in different subsets in the set. But any two red objects, like the first and second red objects in (■■■, ■■, ■) are also distinct, or distinguishable from each other, in residing in different places. This is called fundamental distinction.
One can develop a quite extensive subscience of distinction and the lack of it. But to cut to the chase in regards to our focus on thermodynamic systems, not that even when all the molecules of a thermodynamic system are of the same kind and have no categorical distinction, as with all being Helium molecules, they are still distinct from each other, fundamentally distinct. And the easiest badge of that is that they are all in different places at any one moment in time. Indeed, Boltzmann statistical mechanics does not dispute this and accepts the idea that all N molecules in a thermodynamic system are distinguishable or distinct.
But the K energy units that reside in or on those molecules, which are in different places, are also in different places in being contained in or on molecules in different places. Thus at least as between energy units residing on different molecules, they are certainly distinguishable (however much Boltzmann’s unpolished intuitions about distinction may have told him different.) And one can also make the argument that during a collision between two molecules when there is a transfer of some of the energy units of a molecule to another molecule, certainly ne distinguishes between the energy units that left the parent molecule and the ones that remained.
Rather what happened is that Boltzmann’s sense of indistinguishable led to the use of indistinguishable combinatorial statistics, which produced an lnΩ based entropy, which never has made any intuitive sense to anybody over the last hundred years, which happened to have a happenstance very high numerical correlation to diversity, which Boltzmann didn’t have a prayer of using in his analyses, because it wasn’t invented until a full halfcentury after his death. We leave the details of a diversity centered thermodynamics to professionals in this area who are intelligent enough not to continue to favor Boltzmann and though he were an infallible true saint of some church of thermodynamics. And move now to the exposition of the full range of the β Diversity Index in the next part of this treatise.
III: The β Diversity Index Demystification of
Nature,
Physical, Biological, Human and Perverse
By “perverse nature” we mean not only out of step with biology homosexual inclinations, but much more importantly the root cause of it beyond the left wing myth of genetics in the exploitive subjugation of people by the ruling classes up the top of nations including our own same sex relations heavy America. The problem with explaining social control and its unpleasant side effects which go way beyond the relatively trivial LGBT issue to the perverse aggression misdirected to innocent victims by unhappy midlevel wage slaves is that this fact is kept fairly well hidden by the black propaganda that shouts from ruling class misinformation outlets 24/7. For as Machiavelli philosophized back a half a millennium ago, the best way to keep the workplace serfs out of the rebellion their situation logically deserves is to never stop preaching the right wing the right wing myth that the peasants’ suffering on earth will be fully rewarded by an eternity, no less, of happiness, ecstatic no less, after they’ve gone to the grave.
Because the religious and ideological bunk is so ingrained in people’s heads from their preschool days to their death beds and because the harm done is so horrible up to and including man’s collective misery being the cause not just of our daily mass murders but also pointless vanitydriven wars like Bush’s slaughter in Iraq and potentially even soon to come nuclear war, the only way to can explain what’s going on in the world and with your life is to scientifically sketch out the truth in precise, logical, impossible to spin mathematical language .
Hiroshima, what this thesis proves the world is heading towards worldwide if we don’t get rid of the weapons.
So please bear up with the technical dryness of the exposition that immediately follows until we have enough ground to stand on quantitatively to spell the reality of life out in ordinary language in some of the later sections. Let’s start now.
We have already talked about the β=1 and β=1/2 diversities at length along with their associated matrices. The β=0 diversity sheds little light on nature – physical, biological, human or perverse – but tells a good deal about the cohesiveness of the mathematics that properly explains it. The β=0 diversity index is from Eq10
42.)
Notice how the β Diversity Index, D_{β}, “grows” for the N=3, (9, 4, 1), set from D=D_{1}=2 to D_{1/2}=2.333 to D_{0}=N=3 as we go from β=1 to β=1/2 to β=0, the spread of the numbers in the set being taken less and less into account in the specification of the diversity of the set as β gets smaller and smaller. This makes it clear that all of the β diversities are, indeed, measures of diversity given the intuitive origin of the notion of diversity. The derivation of the D_{0}=N diversity, which doesn’t take into account the spread of the numbers in a number set at all, from its associated matrix, the addition matrix, drawn out for the K=6, N=3, (3, 2, 1) set below, is also striking.

9 
4 
1 
9 
9+9 
9+4 
9+1 
4 
4+9 
4+4 
4+1 
1 
1+9 
1+4 
1+1 
Figure 43. The Addition Matrix of the (9, 4, 1) Set.
The sum of all the terms in the addition matrix is 2KN=84 and the sum of the diagonal terms, 2K=28. So as we saw with the β=1 and β=1/2 diversities, the β=0 diversity, D_{β}=D_{0}=N=3 is also the ratio of the total sum, 2KN, divided by the diagonal sum, 2K, as D_{0}=2KN/2K=N. Is there a contradiction here, though, given N as a β diversity measure, which we said were all exact, and N for an unbalanced set, which we said were all inexact? No, the D_{β}=D_{0}=N diversity measure is exact because the p_{i}^{0}=1 terms that directly generate N in Eq42 neutralize the spread or dispersion of the p_{i}=x_{i}/K weight fractions and the x_{i} they are functions of thus treating even an unbalanced like (9, 4, 1) as though it were balanced, showing internal consistency in the mathematics of The β Diversity Index Demystification.
We next consider the β= –1 diversity, which is important to our demystification of physical nature to be given its own symbol, M. For the K=14, (9, 4, 1), set, whose p_{i} are p_{1}=9/14, p_{2}=4/14 and p_{3}=1/14, D_{1}=M is
44.)
Note first how the diversity index
continues to increase in value for a given set as β decreases in value
even into negative numbers. And next note how M is derived from the division
matrix shown below of (9, 4, 1) as the ratio of the matrix sum of terms to the
diagonal sum of terms as it did for β=1, β=1/2 and β=0.

9 
4 
1 
9 
9/9 
9/4 
9/1 
4 
4/9 
4/4 
4/1 
1 
1/9 
1/4 
1/1 
Figure 45. The Division Matrix of the (9, 4, 1) Set.
The sum of all the terms in the division matrix is Я=343/18=19.0556. (Я is the Russian letter, ya.) And the sum of diagonal terms is N=3, with the ratio of these sums being Я/N=6.352, the β= –1 diversity. Hence, we generalize for the division matrix of all natural number sets,
46.)
Again we see that the N term in Я/N that M is a function of and might be thought to be inexact from our earlier arguments on N, thus making M inexact, is not exact as a count of the unit ratios in the diagonal, which shows the β diversity index, M=D_{1},_{ }to be exact as we generalized earlier for all β diversity indices. Again this shows the internal consistency of these mathematics.
More meaningful as regards science proper and the solving its “mysteries” is that the ratio of the µ mean of K=14, N=3, (9, 4, 1), which is µ=K/N=4.667, to the M=D_{1}=6.352 diversity is the inverse sum of the x_{i} of the set as we would find if the x_{i} were the values of capacitors in series or resistors or inductors in parallel.
47.)
For the K=14, N=3, M=6.352, (9, 4, 1) set, the inverse sum is K/NM=.7347. To understand more clearly what is being talked about here as regards capacitors in series, let’s look at a diagram for the RC circuit, an electronic resistancecapacitance circuit.
Figure 48. An RC Circuit
The capacitor on the right, with capacitance, C, is connected up with the battery on the left, with voltage, V_{S}, so as to fill the capacitor with electronic charges. If N=3 capacitors are used in the circuit together they can be arranged in two ways. One way is “in parallel” as shown below,
Figure 49. Capacitors in Parallel
In this parallel arrangement, the capacitance, C_{P}, of a set of N=3 capacitors, (9, 4, 1), as C_{1}=9, C_{2}=4 and C_{3}=1 is just their simple arithmetic sum.
49.) C_{S }= C_{1} + C_{2} + C_{3}
For our (9, 4, 1) set of capacitors, hence, the three behave as one capacitor of value C_{S }= C_{1} + C_{2} + C_{3}=14. This is intuitively very reasonable as the three capacitors act like three containers to be filled with charges whose amounts sum together little differently than the number of peaches you might fill up three wicker baskets of sizes (9, 4, 1) with. The other way the capacitors can be hooked up is “in series” as shown below,
Figure 50. Capacitors in Series
Now the three behave as one capacitor of value, C_{S},
51.)
For the (9, 4, 1) set of capacitors, C_{S}=.7347. Hence
52.)
There are a couple of important conclusions that can be drawn. The first is that this clarifies for the first time in science what the inverse sum of C_{S} means intuitively from C_{S}=K/NM. If we look at K divided by N by itself, it is obviously the µ=K/N mean or arithmetic average. And if we look at K divided by M by itself, it is, perhaps less obviously, a β diversity based biased average entirely akin to the φ=K/D biased average of Eq38 and the ψ=K/h square root biased average of Eq40. Hence we can understand this new representation of electronic circuit inverse summing as a double average that first averages the K=C_{1}+C_{2}+C_{3} simple sum of the capacitors over the N number of them and average that again over their M diversity, more to be said on this later.
The second is that inverse summing as derived from the pure mathematics of the β diversity index and the number set division matrix is a general mathematical structure, not one necessarily derived from or restricted to electronic circuits. That this is empirically the case is obvious when we take a look at the mammalian circulatory system to see the blood vessels fanning out, so much like parallel resistors in an electronic circuits, in order to optimally minimize the resistance to blood flow in mammals including man. And this suggests that inverse summing as fits circuit elements can also be used to explain some of the many mysteries of human relationships, for people tend to bond or connect with each other very often in obligate fashion, like capacitors in series, and also in optional fashion, like capacitors in parallel.
To explain relationships and indeed behavior in general in these very precise and clarifying mathematical ways, we must take the view that not only is inverse summing an elementary mathematical function that can be applied generally, but also that the elemental Kirchhoff’s Laws of electronic circuits (to be reviewed shortly) can be applied generally to nature. Before we begin that analysis, though, we want to mop up two “side show” aspects of the matrix and β diversity mathematics we have been dealing with, for they are both powerful instruments for science whether used in conjunction with each other or separately.
We have considered now the matrix arithmetic of sets
as it applies to addition, division and multiplication, the latter also in
square root form. Is there also a meaningful subtraction matrix for a set? Our
first attempt at forming one, as for the (3, 2, 1) set would look like this.

3 
2 
1 
3 
(3−3) 
(3−2) 
(3−1) 
2 
(2−3) 
(2−2) 
(2−1) 
1 
(1−3) 
(1−2) 
(1−1) 
Figure 53. The Simplest Possible Subtraction Matrix of the (3, 2, 1) Set.
A quick glance at the diagonal terms shows them all to be zero. And the sum of all the terms in the matrix is also zero, which effectively renders the subtraction matrix in this form without little ostensible value. We can rectify this problem, though, by squaring the differences, which will give a measure of their magnitude without zeroing out their sum.

3 
2 
1 
3 
(3−3)^{2} 
(3−2)^{2} 
(3−1)^{2} 
2 
(2−3)^{2} 
(2−2)^{2} 
(2−1)^{2} 
1 
(1−3)^{2} 
(1−2)^{2} 
(1−1)^{2} 
Figure 54. A Better Subtraction Matrix for the (3, 2, 1) Set.
Doing this doesn’t change the zero values of the diagonal differences but it does prevent the zeroing out of the sum of the matrix differences, which we see adding up to λ=12, (λ, the Greek letter, lambda.) The average of all N^{2}=9 squared differences in the matrix is λ/N^{2}=12/9=4/3. Note that this is exactly twice the σ^{2}=2/3 variance of the (3, 2, 1) set. And generally for any number set,
55.)
This λ/N^{2 }function as the average of the λ sum of matrix terms per its N^{2} terms, though twice the σ^{2} variance, provides as good a measure of the spread of the numbers in a number set as does the σ^{2} variance, which should be understood as an inherently arbitrary specification of the spread, as indeed are all of the statistical error functions of the science of statistics. From this perspective the spread of numbers quantified in terms of the 1/N^{2} average of the λ matrix squared differences is as intuitively sensible, if not more so, than the 1/N average of the deviation of the x_{i} numbers in a set from the µ mean in the textbook definition of the σ^{2 }variance of Eq14. And from this perspective one could theoretically revamp the entirety of statistics using the λ/N^{2 }doubled measure of the σ^{2} variance from the subtraction matrix as the foundation statistical error of such a new statistics. It is much easier, of course, to retain statistics in the form it is presently in, of course, though recognizing that Eq55 as λ/2N^{2}=σ^{2 }is valid alternative derivation of the σ^{2} variance that has some advantages.
One of these comes from averaging the λ sum of squared differences terms, not over an N^{2} number of terms that includes the zero value difference terms in the diagonal (which can be thought of, hence, as nonexistent) but over the N (N−1) nonzero differences in the matrix. Doing that derives not the σ^{2} population variance, but the S^{2} sample variance.
56.)
This provides a simple and direct explanation for the statistical science’s differentiation between the population standard deviation, σ, and the sample standard deviation, S, which if not as much of a mystery for science as entropy, has confused countless students and a goodly number of science professionals over the years.
And further validating the meaningfulness of this natural set matrix mathematics is comparison matrix of Figure 2 that derives from the product matrix, the quite illuminating as regards the foundation operation of the human mind of distinguishing or making a distinction shall take up in detail in a later section.
And before we get to an understanding of human behavior in terms of Kirchhoff’s Law circuits, we also want to make a few quick observations on the relationship between diversity and information, for however much Claude Shannon’s information theory (arrived 1947) has provided some interesting insights on information in the way we usually understand the word, it has failed, despite many attempts especially in its early days, to provide anything approaching a comprehensive understanding of information as the biological kingdom’s premier information processing machine, the human mind, uses it. As a brief preview of this we want to cite two logarithmic information functions that are closely associated with diversity.
One of these is the Renyi (information) entropy, simply just logD where D is our β=1 diversity of Eqs4,7&8, which does double duty in the science literature as both a central function for information in information theory and as the Renyi Diversity Index. And the other is the Shannon (information) entropy, which also does double duty as the central function for information in information theory in log_{2} form as the Shannon Diversity Index in natural logarithm form. While it does not have a perfect analog in the β diversity pantheon, it is very close in value for most sets and equal in value for some to logh, the h square root diversity of Eq11.
In a later section we will extend this synonymy of information and diversity to include an understanding, through the comparison matrix of Figure 2, of the (nonlogarithmic) β diversity functions and its derivatives as information, which provides a quite beautiful, mathematically precise specification of all of our human emotions including fear, excitement, relief, disappointment, hunger, sex and anger, the latter in conjunction with the mathematics of natural selection we also develop giving a perfectly clear understanding of violent behavior and why the only thing that can keep it from bringing about the greatest mass murder of all time as nuclear attack is for us to get rid of all the weapons in the world before the weapons get rid of all of us.
Now doubling back to the RC circuit of Figure 48, we will first explain its dynamic behavior in terms of Kirchhoff’s Law and then use that to explain goal directed human behavior.
Figure 48. An RC Circuit
The capacitor on the left, C, connected up the battery on the right, V_{S}, fills the capacitor with charges. What is relevant to human behavior is that activity directed to achieving a goal, a task, can be configured conceptually as filling up a container with some kind of valuable, needed, objects as with filling a bushel basket with apples at harvest time or, indeed, filling up one’s stomach or other storage places like the liver and fatty tissue with food. While that is a simple ordinary language intro to this analysis, it’s nuances and ramifications cannot be understood with recourse to the RC form Kirchhoff’s law spelled out in the language of differential calculus as
57.)
We see that the rate of filling up of the capacitor with q charges, dq/dt, from a batter with voltage, V_{S}, is inversely proportional to the resistance, R, and the capacitance, C. The greater the R resistance, the longer it takes to fill it; the greater the C capacity of the capacitor, the longer it takes to fill it. Applied to human behavior as the task of filling a container, the bigger it is, the larger its capacity, analogous to C, the longer it takes to fill it; and the greater the environmental resistance, how difficult the task is, analogous to the R resistance, the longer it takes to fill it. Capacitance in series represents tasks that require more than one person to do them. And …
[Painful Illnesses from the natural breakdown of the body as occurs once you get substantially into your 70s regardless of what the smiling sucker ads on TV for the purgatory of retirement say has put a halt to our writing. Before we quit, though, we’ll write up the a. to z. precepts as in a scientific bible for what life actually is about along with what’s best to do about it. And following that is a ton more mathematics that underpins these precepts done in years past. For that reason, there are breaks here and there in in the topics covered and in the equation numbers, though the material is yet well ordered and well written enough to be read quite easily, the newness of some of the mathematics aside. And there are also a few extended sections written in ordinary language mixed in with tit that help to fill in the blanks in the a. to z. precepts that follow.]
a.) As Genghis Kahn once said: Every man wants to be the khan (the king, the top boss of it all, like he was) or be independent (no boss of any kind at all.)
b.) An interesting and quite correct observation on the part of Genghis considering that anybody within his reach (as a lot of people were back in the Mongolian empire of the 13^{th} Century) who attempted to be the Khan or to be independent and outside of his control was executed, usually by beheading.
c.) Now before we start making Genghis out to be some kind of unique devil figure in history, note that any honest history book on civilized man (and that excludes the history preached to the common folk within a nation by the rulers including to mind vulnerable children) will tell you that enslavement in one form or another has been with us since Ancient Egypt and Babylon and Sparta and Rome and in the serfdom of Medieval Christianity and British Colonialism and just last week in the 9 to 5 and beyond wage slavery all have to put up with in America.
d.) And should also tell you that top bosses (as with kings of old and _____ empowered by big money in modern times, the blank used because there are no politically correct, proper, words allowed for the billionaire wage slave controllers) would never say through the information outlets they control including religion, education and today’s quasireligion psychobabble, that people would rather be independent than be the happy wage slave suckers portrayed in the movies and media that only the millions of our billionaire rulers, may they all be given black eyes, are able to produce.
e.) And an honest telling would also make clear that life’s “personal” miseries as derive from failed and frustrated relationships (shhh, don’t tell your sisterinlaw or neighbor or coworker) are not the product of Satan or “mental illness” but of the 9 to 5 and beyond screwing people take in life as a matter of course from bosses and from authority controlled, yea owned, by money, as the police and courts certainly are (except on TV.)
f.) Really, can it be missed by any halfsensible person, the correlation between the unhappiness of the standard mass murderer and his suicidalhomicidal release from that unhappiness from his last few bloody moments? What do you mean, he was/is mentally ill and that’s why he did it? My, that’s so vague. It’s unhappiness in the heart that drives people mad, unhappiness that comes from the same place as for the rest of the suckers who can’t make the correlation between it and humiliatingly kissing the boss’s ass six ways six times a day as the years go by. Or, to push the logic of the connection one step further, let’s ask where “mental illness” comes from? Well, we just don’t know, say the welldressed shrinks on TV and their pundit and police chief sidekicks. We never have any idea what the mass murderer’s motive was, and that’s 300 such motiveunknown mass killings just in 2015. Hmmm.
g.) Though, c’mon, what’s the mass murder of a handful of people every day compared to the mass murder, legs and arms and heads blown off, of war? And does that come from the same place as domestic mass murder, from the tyranny of social control, whether from dictator setup police coercion or billionaire setup economic coercion? The boss or the authorities (in the schools now, too) too, kicks you in the ass, or threatens to day by day, but since you can’t kick the bastards back, you bullyingly and often sneakily pass your pain on to others who had nothing to do with causing your pain. And this extends to masses of people kicked in the ass passing on their hate to masses of other people, of different ethnicities, here and in other countries. Kill the bastards. That feels so good, to hate to cause others to suffer when you feel bad yourself. And it hits hard also on independent people who are often targets out of jealously.
h.) Way down in the cellar of causation, of course, it’s all Darwin’s fault, all this violent competition admixed with the worldwide economic competition that finances for the winners the best weapons, that plus the aforementioned blind hate and aggression that wells up from underlings dominated and beat up by the bosses passing their misery onto vulnerable others en masse.
i.) And there is no way to shut it down, as long as the social control font for it keeps flowing. No way to shut it down except to get rid of all the weapons in the world. And that would work because the simplest mathematics tells you that it’s absolutely impossible for a relative few people to control a whole lot of people without weapons. Which is the what the police are there for primarily, to prevent revolution, which is why it is necessary to take the weapons power away from the billionaire bosses, may their eyes be blackened, who own the police and effectively, better believe it, tell them what to do. Or is it too much to ask the question of why the billionaires NEVER go to jail.
j.) And as no small side effect of getting rid of all the weapons, you also get rid of war as we know it, up to and very importantly including mankind annihilating nuclear war, lots on exactly how to bell that cat in later sections where you’ll find that A.) It’s entirely necessary or we go up in smoke and B.) It’s quite possible to accomplish once people see clearly the 5000 years of misery up to the present that weapons have caused the human race, in war and peace.
k.) And we should also say something about the poor fellows and gal soldiers who do the billionaires’ bidding to grow and maintain the American Empire. To give your lives with arms and legs and brains blown away for the sake of maintaining the privileged lives of the decadent scum at the top? Nothing sadder than a veteran in a waiting room at a VA hospital. Really. Take a quick trip to the lunch room at the VA hospital up in the Hanover, NH, area. These poor bastards, masses of them impaired for life in irreparable ways. Even the most sensibly selfish of us can’t help feel the true horror of it. Let’s get rid of the weapons.
l.) And the rest of the a. through z. precepts of this new religion of happiness and sanity are scattered through the rest of this treatise prefaced by a bit more mathematics.
All processes in nature can be unified in our understanding of them in terms of their common property of moving towards some quantifiable end point. An example of this unifying generalization is found in the thermostatic heating of a room initially at θ=32^{o}F to the θ_{S}=72^{o}F temperature set on the room’s thermostat. The Ԑ (epsilon) error function in this classic negative feedback control system is the difference between the actual room temperature, θ, and the end point temperature the process is moving towards, θ_{S}.
901.) Ԑ = θ_{S }− θ
When an Ԑ error is sensed by the system, Ԑ=θ_{S}−θ=40^{o}F, a furnace automatically turns on to heat the room to the θ=θ_{S} set point to eliminate the error, Ԑ=(θ_{S}−θ)=0, at which point the furnace turns off. From a purely mathematical perspective Ԑ can alternatively be specified as a negative quantity
902.) Ԑ = θ_{ }– θ_{S }
It is clear that the negatively signed error, Ԑ = θ_{ }– θ_{S }= −40^{o}F in this example, is also eliminated when the room temperature reaches the set point temperature on the thermostat, θ= θ_{S}, to zero out the error, Ԑ=θ_{ }– θ_{S}=0. While textbook feedback control theory specifies the sign of the Ԑ error as positive, its negative specification has intuitive advantage in couching an error as a deficit. All processes in nature are unified in behaving in some way as a cybernetic or negative feedback control system that proceeds to some end point by eliminating some form of Ԑ= θ−θ_{S} error. This unification clarifies a spectrum of otherwise confusing phenomena that range from thermodynamic entropy to human emotion and behavior.
The Cybernetic Age was born from mathematical theories of negative feedback control developed at MIT in the 1940s. This research soon led to the building of the forerunner machines of modern day computers. The cybernetic science developed though these efforts also led to a clear understanding of homeostasis or negative feedback control in biological systems. One such homeostatic system warms a person up when cold by the negative feedback control process of shivering that comes about from the brain sensing a θ skin temperature that deviates from a genetically inborn θ_{S} temperature set point for the body, the difference as an Ԑ=θ− θ_{S} error being eliminated by the muscle movement of shivering that generates an increase in body temperature. This is textbook physiological negative feedback control.
A third way to get warm when one feels cold is by warming behavior like walking into a warmer room or putting on more clothes or building a fire in a fireplace. Warming behavior starts with feeling the unpleasant sensation of cold as an emotional measure of Ԑ=θ−θ_{S} error that is zeroed out or eliminated by such behaviors. To explain all of the human emotions in a precise mathematical way as elements in a negative feedback control of behavior we start with behavior directed not to achieving the pleasurable end point goal of attaining warmth but to the broader pleasurable goal of obtaining money. This mathematical formulation of the human emotions develops a math based cognitive science that explains precisely how the mind works while avoiding the psychobabble vagueness of ideologically corrupt, pseudoscience standard psychology.
In doing so A Theory of Epsilon will enable us to address the troublesome and contentious problems of the day with the same assurance one has in solving technical problems using the mathematical sciences of Newtonian mechanics and electronic circuit theory. While many, disliking mathematical analysis, would prefer to discuss contentious issues in the same entertaining and easy to follow format television employs there are great advantages in using logical mathematical analysis because the unambiguous meaning of mathematical symbols makes impossible the spin enabled by the ambiguities of ordinary language that tolerates lies as patent as “You are Now Entering the No Spin Zone” that should provoke ridicule at the level of outing Bill O’Reilly as an even worse predatory closet faggot than fellow Republican spin masters Ted Haggard, Dennis Hastert and Karl Rove. That should clarify which side of the issues we are not on.
I should also make clear that the primary problem we have been concerned with over the last forty years of developing A Theory of Epsilon is the threat of nuclear annihilation.
To that end A Theory of Epsilon directs itself to explaining human nature, especially our violent emotions, well enough to encourage readers to actively participate in a worldwide political movement to change our planet into A World with No Weapons. This not only gets rid of the misery and horror of war but also the near equally miserable unhappiness that results from exploitive control ultimately sustained by the ruling class in such societies including capitalism through their military and police armed with weapons.
To explain such treasonous attitudes with unarguable mathematical precision consider next a specific behavior that has as its end point goal getting money through a behavior that provides a V dollar payoff gotten with probability, Z. This approach to understanding human emotion and behavior cybernetically that is examined in great detail in Sections 913 starting with Eq84 is sketched out here in a simpler way with a game of chance that uses one die.
If you roll a 3 on the die, which has a probability of Z=1/6, you win V=$60. The mathematical expectation, E, which is your average payoff if you play the game repeatedly, is
903.) E = ZV
Specifically the expectation for this game is E=ZV=(1/6)($60)=$10. If you play six times, for example, on average you win the V=$60 prize one time in six for an average payoff of E=$10 per play. In this game the negative expression of Ԑ error of Eq902 takes the form of
904.) Ԑ = θ – θ_{S }= ZV –V
Here we see the V dollar prize as the end point goal of the process in parallel to θ_{S} for the heating system and the ZV expectation as where you are prior to rolling the die to achieve the V prize in parallel to the θ temperature as where the heating system is at prior to achieving the θ_{S} set point temperature from the furnace heating the room. The Ԑ= ZV –V error can also be expressed in terms of U=1–Z, where U is the probability of failing to win the prize when you toss the die or the improbability or uncertainty in winning, as
905.) Ԑ = ZV –V = –UV
This Ԑ= –UV error is sensed as a neural signal from the brain as displeasure, specifically of the anxiety felt in the uncertainty about getting the V dollars. The greater the U uncertainty and the greater the V dollars one is uncertain about getting, the greater the displeasure in the Ԑ= –UV anxiety. In synchrony with a person instinctively wanting to eliminate unpleasant feelings, they thusly act to eliminate the error that is associated with the displeasure of anxiety. That is why displeasure evolved, to motivate behavior directed to eliminating the Ԑ error in the feedback loop associated with attaining a goal, here obtaining money. The full set of emotions associated with this behavior is brought out by next spelling out the E=ZV expectation of Eq903 from Eq905.
906.) E = ZV = V –UV
Much as –UV specifies an unpleasant anticipatory emotion via its negative sign, so does V in the above specify from its implicit positive sign a pleasant anticipatory emotion, that of anticipating the pleasure of obtaining V dollars or pleasure in one’s contemplation or wish or desire for money. The words one uses for the pleasant anticipation of obtaining V dollars is secondary to its primary symbol representation as V. And similarly with the word we gave to the –UV symbol, anxiety, which can also be called in ordinary language anxiousness or worry or concern or fear. Indeed there is so much latitude in which word or words in ordinary language we might call –UV that we will give it the technical name of meaningful uncertainty meaning U uncertainty associated with the meaningful item of money or V dollars.
This makes it clear that the E=ZV is a measure of the pleasure of one’s hopes of getting V dollars tempered or reduced by one’s unpleasant feelings of anxiousness or doubt about actually getting the money as would be universally felt by anyone playing this game, the marginal effect of the player’s wealth on the pleasure experienced in anticipating V=$60 notwithstanding.
The ZV, V and –UV emotions of hope, desire and anxiety are all anticipatory feelings experienced prior to doing the behavior of tossing the die. There is also a fascinating set of mathematically welldefined emotions that arise after one tosses the die. These depend, of course, on whether or not the toss is successful. If it is one feels pleasure as joy in getting or realizing the money we will label as R=V, the R symbol specifying a pleasure that comes from an actual realization rather than expectation. The amount of pleasure in getting V dollars, of course, depends on how big the V prize is. The bigger V is, the greater the pleasure, taken for simplicity to be a linear relationship and further a measure that ignores the wealth of the player, which without question has a marginalizing effect on the intensity of the pleasure experienced.
There is also an additional pleasure from winning, the excitement of winning that depends on the intensity of the –UV anxiousness from its negation from winning which we’ll represent with the letter, T.
907.) T = – (–UV) = UV
The greater the uncertainty, U, and the amount money one is uncertain about getting, V, the greater the UV excitement in winning it. If one plays a variation of this dice game with the V=$60 win coming about if one rolls a 1, 2, 3, 4 or 5 with probability Z=5/6 and with uncertainty U=1–Z=1/6, as one pretty much expects to win, though there is R=V joy in winning the V=$60 in either case, as there is much less meaningful uncertainty or anxiousness beforehand, there is less of a T=UV thrill or excitement than in the Z=1/6 game won only with the roll of a 3. Specifically the easier game to win at elicits a thrill of T=UV=$10 measurable as the pleasure of getting an extra $10; and the harder game, more of a thrill as T=UV=$50.
The T symbol stands for transition emotion, which categorizes excitement as a transition emotion in contrast to an E expectation and an R realized emotion, the other two broad categories of emotion humans experience in their goal directed behaviors. We can also define the T emotion in a more general way as
908.) T = R – E
We will refer to this function as The Law of Emotion. For a winning toss of the die with E=ZV expectation, the realized emotion is R=V and, hence, as derives UV thrill or excitement in a way different than Eq907,
909.) T = R – E = V – ZV = (1 – Z)V = UV
The Law of Emotion of Eq908 also generates a T transition emotion for when one does not throw a winning number. In that case, as no money is realized, a realized emotion is lacking, R=0, and the T transition emotion produced is from Eq908
910.) T = R – E = 0 – ZV = –ZV
This T= –ZV emotion is the disappointment felt when one fails to win the V dollar prize, an unpleasant feeling as denoted by its prefatory minus sign and one greater in intensity the greater the E=ZV expectation of winning felt beforehand. To wit as is universal among humans, a low expectation of winning carries with it little disappointment when you lose.
People don’t just behave to get good things like money; they also act to avoid losing good things they already have like money. This is illustrated with the same one die game of chance where you have to roll, say, the 3 to avoid losing v=$60 (lower case v.) The probability of failing to roll a 3 is U=5/6 so the expectation of incurring the v=$60 penalty is
911.) E= –Uv
The endpoint goal of throwing the die is to lose 0 dollars, which allows us to specify the Ԑ error that exists prior to the throw as
912.) Ԑ = E= 0 – Uv
The Ԑ error is eliminated or reduced to zero, Ԑ = 0, when the 3 is thrown and no money is lost, R=0, as the U uncertainty of avoiding the loss goes to U=0. The Ԑ error of Ԑ=E=0– Uv to be eliminated is associated with a feeling of displeasure, the fear of losing v dollars. As one acts to eliminate the displeasure by this or that behavior, here to toss the die to roll a 3, one also is behaving to eliminate the error in keeping with the generalization that all dynamic systems direct themselves to some quantifiable end point aiming to eliminating the Ԑ error in the system.
Two other expectation emotions besides E= –Uv fearful expectation that are familiarly associated with avoiding a loss become salient once Eq911 is expanded via U=1–Z to
913.) Ԑ = E= –Uv = –(1–Z)v = –v + Zv
In the above the –v term is the anticipation of incurring the penalty, dread of it we might say, a distinctly unpleasant feeling as its minus sign implies. And +Zv is the hope one has of avoiding this penalty, the emotional measure of the security one feels in this situation, a pleasant feeling as implied by its prefatory positive sign. This makes it clear that the E= –Uv emotion of fear of incurring the v penalty is a tempered sense of what we are calling one’s –v dread of the penalty tempered by one’s sense of +Zv security or hope that the penalty will be avoided by rolling the 3. The many nuances of these three emotions of fear, dread and security are detailed in Section 9 starting at Eq120. 3The T=R–E Law of Emotion of Eq908 also applies to behaviors directed to the goal of avoiding a loss. When one is successful in that effort by rolling the 3, no money is taken and R=0. With that and E= –Uv the T transition emotion for a successful throw of the die is
914.) T = R – E = 0 –(–Uv)= Uv
The positively signed T=Uv transition emotion is the pleasant emotion of relief felt when one avoids losing something of value as in incurring the v dollar penalty in this game. The intensity of the pleasure of T=Uv relief is greater the greater the v penalty that might be incurred and the greater the U probability of incurring it. The emotion realized when the penalty is incurred is R= –v, the grief or sadness felt when you lose something of value like money. Again with E= –Uv as one’s fearful expectation,
915.) T = R – E = –v –(–Uv)= –v +Uv = –v(1–U) = –Zv
The T= –Zv emotion is the dismay felt in losing v dollars whose displeasure is greater the greater one’s zV hopes of avoiding the loss. This T= –Zv dismay felt from Zv hopes of avoiding the penalty being dashed is above and beyond the –v grief felt in losing the money. We can lump the positive pleasant T transition emotions of UV excitement and Uv relief together as elation and the negative unpleasant ones of –ZV disappointment and –Zv dismay as depression. The function or purpose of the T transition emotions in man’s emotional machinery along with some of their other important nuances and ramifications are developed in detail in Sections 912 starting at Eq84 in a fashion spectacular enough to derive the Economics 101 Law of Supply and Demand from the T=R–E Law of Emotion of Eq907. Then in Section 13 starting at Eq235 we substitute the penalty of losing one’s v*=1 life for losing v dollars to derive the feeling of cold we have that threatens one’s life as the emotion couched error whose displeasure motivates us to warming activity to save our life or survive.
Having in the above given the necessary directions for tying that ribbon of getting money and getting warmth together, we want to double back now to the thermostatic heater to develop systems engineering’s basic expression for 1^{st} order negative feedback that will further show the generality of cybernetic control in nature’s processes and then lead to a clearer understanding of our emotions (as with emotional energy and how it is gained and lost) and of the evolutionary processes that created them over time. To do that we will use mathematics a bit more advanced than the simple algebra we have tried hard to stick with to make the entry to this introductory section as easy to follow as possible.
A special kind of thermostatic heater is a proportional heater, called that because its rate of θ temperature increase from the furnace, dθ/dt in calculus terms, equal to the rate at which the Ԑ=θ_{S}−θ error is eliminated, is directly proportional to the Ԑ error as
916.) −dԐ/dt = −d(θ_{S}−θ)/dt = d(θ−θ_{S})/dt = dθ/dt = k(θ_{S}−θ)
In the above k is a constant of proportionality. The proportional heating system raises the θ room temperature to the θ_{S} temperature set on the thermostat to eliminate the Ԑ=θ_{S}−θ error according to the solution to Eq916 presented in chart form below.
Figure 917. θ Temp (in green) & Ԑ=θ_{S}−θ Error (in blue) over Time for Proportional Heating
The horizontal axis is time. And the numbers on the vertical axis represent the
θ room temperature as it increases over time on the green curve, keeping
the numbers simple, from θ=0^{0}C to a θ=θ_{S}=5^{o}C
thermostatic set point. And the descending blue curve in the graph represents
the elimination of the Ԑ=θ_{S}−θ error, which as
it approaches closely 0^{0}C is eliminated as automatically shuts off
the furnace that’s has been heating the room up. Eq216 is the textbook function
for 1^{st} order negative feedback control. It is very general
in nature as we will see next.
The RC circuit diagrammed below operates in basically the same way as the thermostatic heater.
Figure 918. An RC (resistancecapacitance) Electronic Circuit
Electric charge in coulombs, q, flows
from the battery on the left with a voltage of V_{S}=2.5 volts to the
capacitor on the right with a capacitance of C=2 farads. The maximum number of
charges the capacitor can hold, the RC circuit’s end point, is q_{max}=CV_{S}=5
coulombs. With its textbook Kirchhoff’s Law representation in which R and C are
constants we see that the current or rate of charging up, dq/dt, is directly
proportional to an effective Ԑ error for
the circuit, (q_{max}−q).
919.)
−dԐ/dt = −d(q_{max}−q)/dt = dq/dt = k(q_{max}−q)=
(1/RC)(q_{max}−q)
This is in perfect parallel to the error eliminating 1^{st} order negative feedback control function of Eq916. Indeed the temperature graph of Figure 917 perfectly fits the behavior of the RC circuit with the number 5 on the vertical axis representing the maximum number of coulombs of charge on the capacitor, q_{max}=5. This identifies the RC circuit as negative feedback control, albeit “passive” feedback control as the systems engineering textbooks categorize it. This distinction between active and passive feedback control in terms of the source of the end points, θ_{S} for the thermostatic heater deriving from a person’s wishes and the q_{max} end point being a fixed part of the RC circuit, is a minor difference relative to the cybernetic properties that the RC circuit and the proportional heater have in common.
Without getting into the technical details, a discharging RC circuit is also an instance of passive 1^{st} order feedback control as is an LR (inductanceresistance) circuit both for the growth and decay of current. And the function for another basic electronic circuit, an LC (inductancecapacitance) circuit, perfectly fits albeit passively textbook 2^{nd} order feedback control in which the quantifiable end point of a process is not a single value but an average of a repeated set of harmonically oscillating values.
Now without expanding this list of electronic circuits in detail we can say that every circuit is an instance of passive feedback control in its being directed towards some quantifiable end point in a way that eliminates some Ԑ error. This makes it clear that there is another broad category of cybernetic processes beyond manmade automatic machines and biological and behavioral homeostasis seen in electronic circuits. This cybernetic interpretation of electronic circuits, as we shall see later in this introductory section, allows emotion to be most fully described by circuit functions and most basically by the Maxwell’s Equations for electromagnetism that underpin circuits.
Ultimately we will show that all processes are essentially cybernetic, minimally in terms of their attempting to move towards some quantifiable end point by eliminating some form of Ԑ error. Seeing nature in this unified cybernetic way is extremely helpful in explaining how the more complex processes in nature operate. In that regard we note a variation of 2^{nd} order feedback control that has as its end point the average of a statistical distribution of values. A clear example of this is the average of the number of Heads that appear as the end point of the repeated flipping of N=6 coins, namely 3 Heads and 3 Tails. Such stochastic 2^{nd} order feedback control is also manifest in thermodynamic processes, which have as their end point an average of the energy diversity of the system as its entropy, a variation of a Simpson’s Diversity Index developed in Section 5 starting at Eq31 as an intuitively sensible entropy replacement for Boltzmann’s century old intuitively incomprehensible entropy formulation. As such thermodynamic processes provide yet another class of processes that can be understood as essentially cybernetic in nature.
We can also add to this list of processes governed by negative feedback control, Newtonian mechanics. All applications of Newton’s laws of motion can be derived from circuit theory itself understood as cybernetic as made clear above. While this interpretation of Newton called “kinetic theory” is less in fashion today than it was fifty years ago when it was first developed, the mathematical parallels of mechanical to electronic circuit processes are eminently clear. For example, the transfer of the kinetic energy of a body moving over a surface with friction is governed by a differential equation identical in form to that for the RC circuit in Eq919. Newtonian kinematics interpreted cybernetically also can provide a mathematical understanding of “behavioral driving forces” and “emotional energy.”
There are limits to any explanation of human emotion and behavior in cybernetic terms without first understanding their origin in evolution, the topic we will take up next. Doing so will also introduce us to the above mentioned Simpson’s Diversity Index that is also necessary for a proper microstate understanding of thermodynamics. And explaining evolution as a cybernetic process will also clarify why money is meaningful and where the possession of it gets its pleasure from.
The two main components of biological evolution are the natural selection of populations by competition and the coming on the scene of new populations from variation to join in the competition. An equation for natural selection useful for pointing out its cybernetic properties was first worked out by the classical population biologists of the 1920s, R.A. Fisher, J.B.S. Haldane and Sewell Wright. This equation is also derived by us in a more direct way in Section 14 at Eq269, the critical term in it for determining which of two rival populations wins out in evolutionary competition being
920.) F_{1 }= g_{1}−g_{2}= (b_{1}−d_{1}) − (b_{2}−d_{2}) =b_{1}−d_{1}−b_{2}+d_{2}
F_{1} is the competitive fitness of population #1 of two populations that occupy the same niche and compete for the resources and space in it needed for a population to persist from generation to generation and avoid dying out or going extinct in the niche. The b and d terms in F_{1} are birth and death rates, b_{1} and d_{1} of population #1 and b_{2} and d_{2} of population #2. And the g=b−d terms in it are the growth rates of the two competing populations. When the g_{1}=b_{1}−d_{1} growth rate of population #1 is greater than the g_{2}=b_{2}−d_{2} growth rate of population #2, hence F_{1}>0, population #1 in blue below flourishes to eventually occupy the entirely of the niche while population #2 in red below goes extinct in the niche.
Figure 921. Competitive Population Growth or Natural Selection
Essential to this process is the limit to the number of organisms of either population the niche can hold called the carrying capacity, K. In the example competition in Figure 921, K=100 organisms, the maximum the niche can hold. There is no arguing with this dynamic for it is purely mathematical in form and valid for any two distinguishable populations of objects located in the same niche or container of limited capacity, even if the objects are not living organisms as with the red and green populations of M&M’s seen in the basket below.
Figure 922. Natural Selection of Competing M&M Populations
I will load more M&M’s into the basket right up to the very tippity top,
but many more red than green, which specifies the “birth rate” of red M&M’s
in the basket to be greater than the “birth rate” of the greenies. Then after
stirring all the M&Ms in the basket around to mix them up well I’ll scoop
out all the M&M’s at the top until I get the basket back to the same level
started with as seen above. This random removal of M&M’s produces an equal
“death rate” for the two kinds and given the superior “birth rate” of the reds,
a superior “growth rate” for the red M&M’s. If I do this repeatedly as
mimics biological reproduction generation after generation in time all that’s
left is the basket are the red M&Ms. Try it yourself to see! Even the
legion of double talking assholes on Fox News doing this experiment will see
that the red M&M’s have been naturally selected and the green one’s “gone
extinct” because of their different growth rates. Note that there is no
intelligent design of this outcome of which color of M&M’s survives in the
basket, just relentless application of the differential growth rate dynamic of
natural selection.
The general validity of the natural selection dynamic made clear, let’s return now to the fitness function of Eq920 as F_{1 }= b_{1}−d_{1}−b_{2}+d_{2} applied to living populations to make clear the nuts and bolts of how the population with the greater growth rate succeeds in persisting over evolutionary time. Unarguably the population that has the greatest g growth rate and, hence, positive F fitness, say population #1 with F_{1}>0, wins out. But how do the members of population #1 behave so as to make F_{1}>0? To maximize the likelihood of F>0 for any population, its members should behave in a way that maximizes F. And for the members of population #1 this means behaving so as to optimize the variables in F_{1 }= b_{1}−d_{1}−b_{2}+d_{2} by maximizing b_{1} and d_{2} and minimizing d_{1} and b_{2}.
An organism behaves to minimize its population’s d_{1} death rate by trying to stay alive as long as it can. That’s why being in the cold, feeling cold, is an error for the organism felt as something unpleasant to get rid of and avoid. The value of staying warm then, from the F fitness function, is that it keeps one alive. This makes it clear that one source of the pleasure in and value of money is in its enabling survival behaviors including its enabling one to purchase food and shelter as directly affects the life span of the organism and by extension the minimization of its population. Note that “errors” in survival, like lacking food as causes hunger, are generally unpleasant as prompts activity directed to eliminate the error and its genetically associated displeasure.
The F_{1 }= b_{1}−d_{1}−b_{2}+d_{2} fitness function is also maximized when the b_{1} birth rate of population #1 and d_{2} death rate of rival population #2 are maximized by behaviors of the members of population #1.These evolutionary drives driven by our emotions of sex and violence are difficult to discuss because sexual and violent behavior is very much affected beyond the instinctive emotions that influence it by significant cultural restrictions on sexual and aggressive behavior. It is obvious from the simple algebra of the F fitness function and from the M&M experiment that having a high birth rate helps make for a successful population. The intense pleasure of sex that drives this is the origin of male lasciviousness, the curbing of which via the sexual mores of a culture, though, also contributes to the survival of a population without our getting into the details of the mechanics of it. And the pleasure of murdering a rival for resources, space and mates derives from the same source of optimizing evolutionary fitness whatever the cultural sculpting of it that glorifies violence under some circumstances and condemns and punishes it most severely under others. The main body of the text in Section 14 gets into these matters in greater detail. The point is that the F fitness function directs us by pleasure and pain to survive, to reproduce and to compete, sometimes mortally.
The naturalness of violence especially in men, in a way the main topic in this thesis, is further clarified by explaining natural selection in evolution as a feedback control process that moves inexorably towards a quantifiable end point while eliminating a welldefined Ԑ error. Understanding evolution in that way will make super clear how the cybernetic characteristics of natural selection get so easily confused with the cybernetic characteristics of cognitive selection so as to attribute evolution to the intelligent design of a super powerful person, be it our God the Father or the equally delusional Allah of the Muslim humanoids. This is very important to make clear for the retaining of an unseen god in one’s thoughts as an agent that makes real things happen magically is totally destructive of the sensible thinking (based on your senses) needed to solve the terrible problems that confront mankind individually and collectively as include the unhappiness of wage enslavement and the possibility of nuclear annihilation.
Natural selection is passive feedback control that aims at an end point of ecological uniformity or zero diversity for taxa competing for the resources of a common niche. To see this requires the introduction of a mathematical function for diversity called the Simpson’s Reciprocal Diversity Index.
923.)
Consider a niche occupied by with N=3 competing populations that has a K=1000 carrying capacity. Its population #1 has x_{1}=200 members, population #2, x_{2}=300 members and population #3, x_{3}=500 members. Each population can also be specified quantitatively in terms of its population density,
924.) p _{i}= x_{i}/K
Population #1 has a population density of p_{1}=200/1000=.2; population #2 of p_{2}=300/1000=.3; and population #3 of p_{3}=500/1000=.5. The diversity of this niche is thus from Eq923
925.)
The densities of N balanced populations, all with the same x_{i} size, are p_{i}=1/N. And their diversity with 1/N substituted for p_{i} in Eq923 is
926.) D = N; balanced
Hence the Simpson’s Diversity Index is just the N number of populations reduced by the imbalance in their sizes as it is for our example set of N=3 population that has a diversity index of D=2.63. When these N=3 populations have different growth rates of, say, g_{1}=1.6, g_{2}=1.4 and g_{3}=1.2, they inherently compete over time until population #1 with the largest g growth rate wins out in the niche according to a function derivable from Eqs262&263 in Section 14 of
927.)
With x_{i0} as the initial sizes of the N=3 populations given above, the competition proceeds as below with population #1 in blue, population #2 in green and #3 in green.
Figure 928. Natural Selection in an N=3 Population Competition
The abscissa of the graph is time in years and the ordinate, the x_{i}
size of the populations. What we see is that population #1 comes to occupy the
entire niche and that the other two populations die out in time. The N=1
population that triumphs in the competition has from Eq926 a diversity index of
D=1, no diversity at all. One can also develop the D diversity in the niche
from Eqs923,924&927 as a function of time as graphed below, which also
shows the end point diversity to be D=1.
Figure 929. Diversity vs. Time for the Natural Selection in Figure 928
The D=1 diversity is the end point or set point diversity of this natural selection process, best written with the S subscript as D_{S}=1. This D_{S}=1 is the end point of every natural selection process that occurs with the Ԑ error eliminated passively in this passive feedback control process of natural selection being
930.) Ԑ = D_{S }–D =1 − D
Ԑ is eliminated, made to be Ԑ=(1−D)=0, by generation after generation repetitive processes of competitive survival, reproduction and combat won by the population with the greatest g growth rate. This was also made clear earlier for the N=2 case in Figure 921 where again only N=1 population with diversity D=N=1 occupies the niche in the end. This tells us that natural selection can be added to our growing list of processes that operate via movement towards a quantifiable end point and the elimination of some form of Ԑ error as we progressively show that every process in nature can be understood as cybernetic and unified from that common feature.
We can further unify processes by identifying all change other than purely mechanical or kinematic not just as cybernetic but also evolutionary. The logic of it is very direct. Everything that exists had to come into existence at some rate, its effective birth; and, except for an eternally stable item, go out of existence at some effective death rate. Religions do this and nations do it and biological species do it and cultures do it and molecular species do it and businesses do it. Do what? Compete actively or passively with parallel forms to continue to exist at the expense of their rivals.
If this notion of general evolution seems excessive outside of biological and cultural evolution consider the evolution of competing molecular populations in the in vitro transformation of amorphous calcium phosphate (ACP) to crystalline calcium phosphate or hydroxyapatite (HA) as mimics bone and tooth maturation in animals including man. This is described in detail in Section 18 where it is shown that both the ACP and the HA have a precipitation constant, b, that specifies the rate they come into solid phase existence and a dissolution constant, d, for the rate they go out of existence as solids. In this transformation the HA molecular species, which has the larger growth rate constant, precipitation minus dissolution, increases its number of crystalline molecules autocatalytically to eventually become the only calcium phosphate moiety in the mix while the ACP with the lower growth rate constant goes completely out of existence.
This section is worth reading because it shows evolution to be a very general process spelled out with the simplest mathematics and that efforts to deny its reality by the right wing is stupidity at a level even beyond denying global warming in the face of endless floods, fires and tornados. It is also worth reading for its clarity on the recommendation of a leading American mathematician, Dennis Sullivan, the National Medal of Science winner in 2004.
Mon, 27 Feb 2006 23:50: why not publish the part that explains Posner's data in terms of the logistical equation first... then do some of the rest next... etc... then as your acceptance takes hold do the more radical parts...as it is you may be preempting any real success by indulging your own deeply felt philosophy... by the way your explanations in the first parts were very clear....you may want to read how Einstein in similar and simple layman's terms dispelled the notion of absolute time in the 1905 paper....and how he did it without being untoward...
good luck
dennis Sullivan
If not entirely in response to Sullivan’s admonitions we tried over the last
decade to become less “untoward” towards some of the bonehead academics
we’ve come across while trying to educate them with math as correct
and clear as 2+3=5. But it’s not that easy to be nice to dumb jackasses posing
as scientists. Most recently I tried to explain to the science faculty of
Colorado State University at Pueblo (CSUP), that all of nature  physical,
biological and human  can be understood in a mathematically unified way in
terms of the Ԑ error function of feedback control systems. But this CSUP
crew led by Frank Zizza, Chairman of the Math/Physics Dept. there, is a perfect
example of the worst that university teachers have become in America, most of
them memorizers incapable of understanding anything other than what they can
regurgitate from the textbooks they ground their way through in school to get their
degrees and positions.
Worse than that as regards the welfare of the students at CSUP as I make clear at the end of Section 17 one of these I got into an email exchange with is a strong bet to be your typical position empowered dominant closet fag predator. This crew epitomizes an American academia that truly stinks in caring more about paychecks, position and power over students than scientific truth. It’s bad enough that our American media has evolved into hard science deniers, Fox News leading the way. But that the science profession as a whole just sits there with its mouth shut so passive at this critical time for America looking for all practical purposes like the bunch of laughable beanie babies in The Big Bang Theory is utterly unforgivable.
Professionals ignoring innovative science that clarifies the cause of the violent troubles plaguing America today puts them at the level of the monks in medieval universities who rejected Copernican science that contradicted the dogma and ideology of that day to please the ruling class of the day. For this modern era is no freer than any of the serf and slave based societies seen in human history including our professional class equivalent of the castrated obedient scribes that served the emperors of China and the pharaohs of Egypt for thousands of years. It’s right out of the South Korean masterpiece, Snowpiercer.
Well that’s enough of a smoke break from the math writing. It really does helps the rebel in you. That’s why for the last 50 years since the late 60s rebellion (which was much more against the warmongering capitalist system than for same sex marriage) they locked up people who smoked weed, locked them up in a cage where as you know if you’ve ever done time, even if just for stupid stuff, they torture you until the rebel in you is killed and you become a “normal” wage slave who lives off delusions like a gloriously happy life after death as compensation for the pain they put you through in life before you die. Gives you that occasional boost you need to temper the downside of the obligate paranoia needed to keep you aware enough of actual reality to keep on plotting against the regime and to keep on avoiding their agents. Who are many, anybody with police power, including in many venues as the experienced rebel knows, managers of the nostar motels we flit around in, like the managers of the Santa Fe Inn in Pueblo, CO, whose gross obesity and ugly faces betray their ugly predatory inclinations. Like they say in the Matrix movie, when you see such effective police agents don’t hang around too long.
Fuck the Orwellian ruse of ISIS as every American’s worst enemy. Our hearts and spirits are murdered everyday by smiling sadists right here at home like the boss who never has to tell you he has Trump’s “You’re Fired” hanging over your head to kick your ass in daily with the threat. Speaking of whom, somebody should tell Boss Trump that he won’t win in the third party bid he’ll have to create without picking up more than a few progressive votes. And that the best way to do that is to use his quick if avaricious mind to understand that we have to get to A World with No Weapons or we all lose big. It’s a long shot tricky maneuver given that Trump’s not exactly Mother Theresa. But with nuclear annihilation as the worst thing that can happen to us and not at all that unlikely, if the light clicks on in this smart moneymaker’s head, the NoWeapons Party ticket for 2016 of Graf and Trump might could make it to the White House. Who better to make a deal with Putin to set aside nuclear weapons once and for all? I used to think Elizabeth Warren might do the trick as you’ll read in material I penned earlier. But looks like the next woman president might best be me if the science community and Trump wake up on time.
Well, enough bullshit. I’m going to end this intermission from mathematical analysis with the Orwellian bit I started the blog with first but didn’t put on because I thought the truth of it too threatening to the butchers up at the top who might find grounds to put 74 year old grandmother me into the torture pit. As Orwell’s protagonist resurrected below might say, kids, be willing to die to defend your happiness. It’s all you have in your brief existence on the planet that’s worth anything. And do read on past my Orwell blurb attempt at creative writing and the first bunch of math sections to see how the nonviolent revolution I am proposing can actually be pulled off.
A Geometry of Time: 2084
By Pete Peterson, PhD
Love was in the air or, I should say, on the air. Love was in the fast foods and JELLO commercials that beckoned you to eat them. Love was in the cars they beckoned you to be deliriously happy in. Love was in the pretend smiles of the women in the ads and talk shows and in the interminable giggling of the fauxhappy journalists who spun the morning news. Love was even in the laxative they said, with inappropriate musical accompaniment, was better than the other laxatives and in the toilet paper they promised would make you swoon with joy when you used it.
But beyond the applauded samesex and other substitute relationships of the 3^{rd} Millennium, love was not to be found, not on a springtime walk or in any place real except the minds of a few young men in 2084 America who had managed to escape the systematic psychological castration used to tamp down rebellion against this Disneyland Hell whose control of information was so complete that the adult humanoids who watched TV endlessly in their off hours to forget the pain of their enslavement actually believed the silly lies broadcast 24/7 that the miserable lives they led as wage slaves were happy, free and fair.
How to draw a true picture from this black hole of misinformation was in Dr. Peterson’s thoughts every moment as he struggled to avoid losing his own manhood and happiness. For though it was possible to run from the shackles of the regime for a short time, without any real intent or tangible plan to destroy the clockwork predation, it was impossible to avert succumbing to the unavoidable stream of institutionally applied slaps that eventually killed everyman’s vigor. Especially with the lock up in a cage reserved for the most resistant young men where pressure could be applied through torture disguised as needed correction for a criminal attitude that threatened, as blared 24/7 on TV, the security and happiness of all of society. From his training as a scientist who had been groomed for the junk food pampered enslavement of working in the technological sector, Dr. Peterson worked instead in hiding on an information weapon to kill the regime, a treatise that exposed the horror he called A Geometry of Time.
Extending technical work started 70 years earlier in A Theory of Epsilon Dr. Peterson came to mathematically explain the ways in which people in this America of 2084 were so tightly controlled cognitively as to turn them into humanoids with thinking completely rewired and stripped down from human instinct to what was efficient for a workplace machine part. Peterson was also sure it would explain the daily cluster of mass murders that young men caught up in the suicidehomicide provoking hell of the regime’s castration program were the perpetrators of. His approach was based on the passive feedback control of an RC circuit that he was well familiar with. Peterson thought hard about the Kirchoff’s Law expression for it we first brought up in Eq919,
919.) dq/dt = (1/RC)(q_{max}−q)
He saw that if you divide both sides of the equation by q_{max} you got
930.) dp/dt = (1/RC)(1 – p)
In the above p is the fractional or percent measure of the extent to which the capacitor in the RC circuit has been filled with charge, p = q/q_{max}. It can also be a similar percent measure of completion for any goal directed behavior a person may engage in as such making 1−p what’s left to do to achieve the goal, in essence a very general form of the Ԑ error of negative feedback control. In this representation, the C capacitance is understandable as a measure of “the size” of the task or goal aimed at: the bigger is C, the longer it takes to finish the task. And R is understood as the “resistance” to getting the task finished as derives from all the other factors that slow down finishing the task and getting p to p=1=100% and the error to Ԑ=1−p=0.
So eliminating the Ԑ=1−p error or what you have left to do is a simple and inarguable way of describing how one attains a goal. This generalizes all goal directed behavior as cybernetic in the sense that doing them eliminates or reduces the Ԑ=1−p error. The shape of the time course of an activity that expressly follows Eq930 is that of the green curve of Figure 917. Making the dp/dt rate of getting a job done a perfect function of the Ԑ=1−p error or what’s left to be done is, of course, a mathematical simplification or idealization in deriving it from the RC circuit equation. But the broader point to be made is that every goal directed behavior we do is directed to eliminating a Ԑ=1−p error from the person continuously working to complete the Ԑ=1−p remainder of what’s left to be done. And as we shall show later this is the case for every conceivable human behavior, goal directed or not.
This understanding of human behavior as cybernetic implies that our behavior functions automatically or in a machine like way much as does, by definition, every piece of feedback control machinery developed like the RADAR guided antiaircraft guns developed for use against the Luftwaffe in WWII that fired automatically. How is our behavior controlled by our nervous system automatically? Our central nervous system or CNS, which includes the brain and the network of peripheral nerves that flow into and out of it, operates generally in a negative feedback control way. But rather than give a minitreatise on all aspects of biological homeostasis, we just want here to stress how the CNS works automatically in machinelike fashion.
What you see, hear, smell, taste and feel, your basic senses, all derive from measurable physical properties as with visible electromagnetic waves of variable frequency and intensity for sight and waves of air molecules of variable frequency and intensity for sound. One property of your nervous system that affects the information you get about the objects and events in your environment is the intensity of the incoming (afferent) sensory signals. Very low intensity inputs, we humans just don’t sense. Nerve impulses for them just don’t make it all the way to the brain. Understanding this properly will make it clear that we do not decide what to do on the basis of “free will” but rather act automatically like a machine.
At night time visual energy from objects has a much lower intensity and very low intensity objects are just not seen. The threshold of energy needed to register sensory data in an organism’s brain is different for different species, an owl, for example, seeing objects at night that a human doesn’t see. Shortly we’ll mathematically explain how the CNS disregards insignificant energy input for the right wing taffy pullers out there skilled enough in rhetoric to successfully argue that a fish’s ass is twice as morally powerful as the hypotenuse of an oblique triangle. But let me begin with a textbook illustration of a nerve impulse. Bear up with the neurophysiology lecture for a moment that will be easy to follow because I’ll be talking only about one small part of a nerve impulse.
Figure 932. Diagram of a Nerve Impulse
The “threshold of excitation” tag tells us that only nerve impulses with energy
above a certain level discharge sufficient to get the neural signal all the way
to the brain. You don’t sense the insignificant sensory inputs because they
never make it to your brain. This evolutionary design of the CNS makes a great
deal of sense pragmatically because little objects that reflect few light rays
from the sun to your eyes generally speaking can’t do you much good or much
harm, so why even see them or notice them?
What does get to the brain then goes on a roller coaster ride of information processing (details given later) that often causes the brain eventually to send neural signals in the opposite (efferent) direction, out to your muscles, which gets you to move about and speak. Not all of the efferent nerve impulses make it to the muscles. The insignificant one’s below threshold die out and cause nothing to happen.
This sense of neural significance versus
insignificance is important to understanding how and why we do what we
do. It shows up at higher, conscious levels in a way that can be spelled out
mathematically. Information as processed by the human mind has measure in the D
diversity index of Eq923 interpreted more broadly than ecological diversity as
the number of significant subsets in a set. This is covered in
great detail in Section 3 towards Eq17. That the human mind routinely operates
on significance factors is made clear with the three sets of colored objects
shown below, each of which has K=21 objects in N=3 colors.
Sets of K=21 Objects 
Number Set Values 
D, Eqs923&924 
Rounded to 
(■■■■■■■, ■■■■■■■, ■■■■■■■) 
x_{1}=7, x_{2}=7, x_{3}=7 
D=3 
D=3 
(■■■■■■, ■■■■■■, ■■■■■■■■■) 
x_{1}=6, x_{2}=6, x_{3} =9 
D= 2.88 
D=3 
(■■■■■■■■■■, ■■■■■■■■■■, ■) 
x_{1}=10, x_{2}=10, x_{3}=1 
D=2.19 
D=2 
Table 934. Sets of K=21 Objects in N=3 Colors and Their D Diversity Indices
The N=3 color set, (■■■■■■■■■■, ■■■■■■■■■■, ■),
(10, 10, 1), has a diversity index of D=2.19, which rounded off to D=2 implies
D=2 significant subsets. And that further implies that the one
object purple subset is insignificant in its being the smallest subset
that contributes only token diversity to the set. In contrast the D=3, (■■■■■■■, ■■■■■■■, ■■■■■■■), (7, 7, 7), set has 3 significant subsets, red,
green and purple as does the (■■■■■■, ■■■■■■, ■■■■■■■■■), (6, 6, 9), set when the D=2.88 diversity of the set
is rounded off to D=3. A more intuitive sense of the mind’s automatic
evaluation of significance and insignificance is had by manifesting the K=21,
N=3, colored object sets in Table 25 as K=21 threads in N=3 colors in a swath
of plaid cloth.


(10, 10, 1), D≈2 
(7, 7, 7), D=3 
(6, 6, 9), D≈3 

A woman who owns a plaid skirt with the (10, 10, 1), D≈2, pattern on the left, as I do, would spontaneously describe it as her red and green plaid skirt, omitting reference to the low density insignificant threads of purple in the plaid. She would make this description automatically without conscious consideration because the human mind just does automatically disregard the insignificant. And it does this, indeed, not just in its peripheral if any visual sense of the insignificant but also in our automatically not verbalizing the insignificant. This verbalization of only the significant colors in the plaid should not be surprising given that the word “significant” has as its root, “sign” meaning “word”, which suggests that what is sensed by the mind as significant is signified or verbalized, while what is insignificant in not being sensed or noticed isn’t assigned a word in discourse or in thought. Another example would be of minorities with few people in them like “South Sea Islanders” being omitted as separate groups on a census.
Insignificance is not only disregarded by the CNS when afferent or sensory but also when it is efferent. Only when neural signals sent by the brain to the muscles are over threshold do they reach our muscles to cause behavior. The neural impulses, which can be complex in origin, must be over threshold to get us to act. Playing the dice game in A Theory of Epsilon for a penny rather than V=$60, for example, though there is some prize, it is so insignificant that it is not acted on whatever the specifics of the neural mechanism that causes that inaction. It is enough to understand the CNS controlled behavioral system being as automatic as the propagation of nerve impulses as automatic, propagating those over threshold but dying out for those that are subthreshold or insignificant.
In the debate over “free will” this makes it clear that while “decision” is important, it is not determining. While a million Americans decide to go on a diet every day, more than half over the age of 25 are obese and that is because once the neural impulses go over threshold, the displeasure of not eating and anticipated pleasure of shoving that ice cream or pepperoni pizza in your mouth, you do it as cybernetic machine like as nerve impulse propagation generally, as 100 million pathologically obese Americans will attest to. The psychobabblers, even the fat ones, will argue that you can “decide” otherwise. But motivation for successful dieting comes in the form of having strong competing goals to binge eating, like very much wanting to look good at the beach, stress on “very much.”
One concludes from the above that people are basically automatic machines (made of biological cells rather than metal parts) designed by evolution to optimize the F fitness function as their set point in order to succeed in persisting over time from generation to generation. This is not to say that humans have no “control” over what they do. But that is only by understanding the causes of the overthreshold neural impulses that bring about behavior.
And that is exactly what Dr. Peterson said in his A Geometry of Time to the young people of 2084 and the handful of adults yet salvageable. He first explained how the human machine works instinctively and then how the ruling class takes advantage of the natural mechanisms to control the subjugated population’s behavior much to their detriment and sorrow as you can read about in detail in all that follows starting with the problems of this present time in history, 2015, that must be solved.
UNKILLING THE MESSENGER: THE BALTIMORE RIOTS AND BEYOND
The obvious message in the Baltimore riot photo is that these people are angry
about the police murder of Freddie Gray. It’s not just the fellow smashing the
window whose anger is released in this act of destruction. It’s the people
behind him, too, along with the thousand others who helped tear Baltimore
apart. And not just because of the killing of this one man but because they
all, as with millions of others across America, black and white, have been the
target of police abuse in one form or another. That’s the message the media
incessantly kill, that the police are predators on nonprofessional people
effectively making America with its highest lockup rate in the world and in all
of human history a police state.
Oh, no its not, say the journalist, politician and clergy mouthpieces of the ruling class who control the media message. The people who trashed Baltimore are thugs, bad people who used Freddy Gray’s death in police custody as an excuse to steal and destroy property for no sensibly justifiable reason. These are two very different messages. It matters a lot which of them is understood to correctly represent reality. It matters a lot which one people believe is true.
This is Hiroshima, 1945, right after the bomb fell. The utter destruction of
that city many years before the Baltimore riots came from the same emotion, the
violence from man’s evolutionary heritage intensified by the unhappiness caused
by civilization’s inbuilt restrictions and exploitation. Hiroshima’s can happen
again and on a scale that terminates the human experiment. For that reason it
is important to get the message right. What is the true nature of man,
especially as it concerns his [propensity towards violence? What is his
predictable fate? Only in getting the message straight can we find the way to
prevent more Baltimore’s, more Sandy Hook mass murders, more police murders, by
and of them, and more Hiroshima’s.
We are going to show which message is correct using science. We’re going to prove it with a mathematical argument that is logical and trustworthy and starts off in a somewhat different place than cybernetics. Now there are some out there who think God is above mathematics in being able to make 2+3 be equal to something other than 5 if He wishes. Such people believe whatever anyone in authority tells them, be it God, the media or the police. The violence in the world that threatens to end it will not be quelled with the help of such stupid people who are beyond sensible arguments that even an elevenyearold can understand. Indeed, let me introduce you to this mathematical take on violence and what to do about it with the help of one such elevenyearold.
One day in 2008 down in Acapulco, Mexico, while in retreat there from the Bush regime of 20002008, my grandson, a dropout in attitude but with a high speed curious mind, asked me: How did they measure distance before rulers and such were invented, Gram? They measured distance in feet with their actual feet instead of with one foot rulers, I answered. If you wanted to know how long a road was in ancient times you walked the distance off, one foot pressed in front of the other, while counting the number of feet you walked off.
Champ, the nickname we gave the boy at birth in hopes that fate would be kinder to him than most, quickly replied: But that’s not very exact because unless you had one guy measuring the distance of all the roads in an ancient kingdom with just his feet, the difference in the length of the feet of the different people who might be doing this measuring would lead to inaccuracy in the distances measured.
Smart kid, a bit lazy like I said, but with a very quick mind. For that is very true and, as we shall see, the basis of an elementary correction in mathematics that configures science differently. That includes not only revising thermodynamics to clarify the centuries old mystery of entropy but also the human sciences to mathematically clarify the emotion of anger and how to tamp down violence and the horror it causes domestically and in war. To see the unarguable connection between extreme violence and tyranny and the inexactness in the count of things unequal in size you have to do the homework needed to follow the mathematical argument.
A count of things unequal in size, of people’s feet with different lengths or of molecules with different speeds, is inaccurate or inexact. Early humans took care of the inexact distance problem when they replaced people’s anatomical feet as measures of distance with foot long rulers, all of whose feet, unlike people’s feet, are the same size and whose count is, hence, exact. The problem of inexactness in counting things unequal in size goes beyond the distance measure problem, though, and is troublesome even today for a proper understanding of physical nature and of human nature.
Count the number of objects in (■■■■). There are 4, of course. Now count the number of objects in (■■■■). There are also 4, you answer quickly. But is that count of 4 for (■■■■) an exact count? No more than a count of the number of feet an ancient road is long is exact when the feet walked off are of different sizes. This problem of inexactness is why the concept of entropy has remained so difficult to understand for so long. And it’s also why the psuedoscience of psychology is as vague in explaining human nature with its elusive psychobabble as Christian dogma is in explaining the birth of God Jesus with the Virgin Mary’s magical, biologically impossible, nosex pregnancy.
Perhaps it was God, the Father of Jesus, who slipped it in with Mary so fast asleep she didn’t notice? Or perhaps the Devil did the dirty work with Mary not as asleep as she pretended to be?
Excuse the blasphemy. The point I’m trying to make with this joke on the Pope and His followers is that supervague notions like God’s ability to shit without needing to wipe His Ass, Satan’s evil and the ever mysteriously caused disease of “mental illness” don’t give rational cause and effect explanations for the mass dysfunctional behavior, unhappiness and violence we see in modern times. True believers in psychology should question its validity as an authoritative science right from the getgo in psychology giving a pass to all the nutty superstitions of religion as acceptable thinking for welladjusted people. Psychology’s failure to label praying (talking to a personality nobody can see and asking a favor of it) as delusional, even if emotionally comforting, is quite contrary to how all the other sciences, the ones that make our computers and fly us to the moon, operate. Can people be so blind as to not see that psychology acts consistently like religion, as a moral code whose central preachment is that disobedience to and revolution against the authority of the ruling class is wrong?
Holding the misleading notions of religion and clinical psychology in one’s mind as firm truth makes impossible a causal explanation of the bad things that happen in life, like today’s epidemic unhappiness from failure in love, and, thus, impossible to correct. Resolving the error in counting inexactness gets science spruced up enough to properly explain not only complex physical phenomena like entropy but also today’s epidemic unhappiness and the violence it brings about and what can be done to eliminate the worst of both the unhappiness and the violence. If this sounds like bullshit at least as doubtful as Mike Huckabee’s promise that an electric bass playing fundamentalist minister in the White House would bring God’s favor upon America, you have the read the God damned mathematics. Before Newton’s mathematical theory of gravitation was accepted, everybody in medieval Europe, rich and poor, educated and peasant, believed as firmly as the sun rising in the East each morning that angels commanded by God pushed the planets around in their observable orbits.
The problem with resolving today’s contentious issues with mathematics, though, is that the scientists who do understand mathematics fluently are effectively paid off in pay and status to avoid speaking out on matters that cast doubt on the benevolence of the ruling class and that the little people who take the greatest hit from their lowly position on the totem pole see mathematical language as readable as Transylvanian Bulgarian. All use basic math and trust it without question when it comes to adding up the groceries at the checkout counter and computing the dent the grocery bill makes in one’s weekly paycheck. But the article about whether Kim or Taylor’s cute ass is the cuter is invariably read with more interest than a mathematical explanation of life’s pains, even if the latter can clearly explain their cause and what can be done about them. To get around such math phobia I’ll start off in this new area of mathematics minus the equations that give the math haters indigestion. That should give the math compromised the gist of the argument if not the proof while also encouraging the more educated readers to go on to the Section 1 where we begin the mathematical analysis needed to nail the truth down in an unarguable way.
Ancient peoples had to develop standard measure devices like twelve inch foot rulers or they would not have been able to make the impressive things they came to construct in centuries past like the pyramids.
If the building blocks of a pyramid aren’t made exact in size with standard measure, they won’t fit together well enough to make a pyramid. Another thing needed was a lot of people, all guys back then, to do the backbreaking work to make the pyramids the pharaoh used to glorify himself. Was it slaves who made the pyramids? Whatever the spin put on working class life in ancient Egypt, one thing for sure was that there was a lot of slavery around back then, in Egypt and Babylon and Persia and then Rome and Greece and then in the medieval Christian Europe that fueled itself on the labor of its jillions of serfs and slaves and then in the precivilwar Southern states of America.
Not only are people made miserable being under the thumb of a slavemaster mentality ruling class, but also, hard truth be told, women, of which I am one, are not impressed by men who are slaves and have the welladjusted slave mentality. I’m not saying women lack admiration for slaves who have the balls to resist their enslavement by rebellion or flight. Three large cheers for any Django Unchained type who retaliates on his masters with guns blazing. But run of the mill male slaves that suck off their master to avoid a whipping or get a meal, women do not turn on to or fall in love with. With sincere apologies to the sensibilities of my suffering black brethren, Thomas Jefferson’s black slave mistress, Sally Hemings, much preferred screwing President Jefferson by whom she had six kids than the niggers that Jefferson owned as slaves who picked his cotton and cleaned his toilets.
Excuse the word nigger that I use, as has Pres. Obama, to hammer home an important point, namely that slavery, in any and all of its forms, be it plantation or modern wage slavery, makes a nigger out of a guy, black or white, as makes a mess out of love. It doesn’t matter if you’re only the boss’s slave for just 40 hours a week. Women just aren’t attracted to men made jerks out of in this way. Not that a woman doesn’t give such men a try, for the female mind says instinctively in the absence of an emotionally healthy fellow, maybe this one will do. But in the end, after a couple of years and a couple of kids, the smell that lingers from hubby’s cleaning the boss’s toilet during the work hours, despite the gewgaws his salary might provide, eventually overwhelms the perfume of love. The initial glow of love she felt turns to contempt and eventually that overpowering urge to reject him unless the prospect of living alone in relative poverty comes to tolerate his noxious smell as the lesser of evils.
Now not only does slavery, in whatever Christmas paper the slave culture wraps it in, destroy love, but breaking up makes the male slave feel much like a dumb dog out in a pounding rain just happy just to have a bone thrown to him and not very prone to rebel against the subservient state that caused the breakup to begin with. Conversely, few things make a man feel more vigorous and strong than the love of a woman and prone as such to resist the slave state of existence.
The frustration and loss of love, then, is an effective emotional crippling of men that kills the rebel in them. The providence of this for the ruling class, so happy to have slaves who lack the rebel element in them, has made true love taboo. Sexually successful men are much more likely to rebel against control, which slants the culture morally sharply against their kind. On the one hand sexually dominant men are displayed to the girls in the media and in other ruling class controlled information outlets as devils with cruel and abusive horns. The word is out: stay away from this kind of men; they’re going to do nothing but hurt you, rape you, use you, abandon you, make you miserable. And girls who disregard these warnings are made to feel like inferior, stupid fools. What you want, they are told 24/7, is a nice soft squishy slave type guy who will commit to bringing home the bacon and to not fussing too much when you agree to have sex with him only once every two weeks with him on the bottom and your toy up his ass. It’s hard for the girl to avoid rejecting healthy men and choosing a jerk when every movie, sitcom and news article tells the girls to beware the aggressive male who might have any vigor in him to make things click.
And in recent times the “men are bad” campaign that Christianity has fostered for the last millennia has been ramped up by laws across America that make a lingering glance by a man attracted to a pretty girl liable to be taken as “sexual harassment” of some kid that could destroy his life as a “sex offender”, a fate not much worse than being crucified slowly. This broad antilove strategy imposed by the ruling class through its religious, psychology and politician mouthpieces also turns out to be hell for women in the end. Yes, you get to be a Queen in your teen years and early twenties, the slim girl with a practiced smile who tells the guys to get down on their knees for a sniff of the backside of her jeans. But in the later years it all comes back to haunt the women when they wind up with lives devoid of any intimacy and anybody to love them. Except for the company of other women, which, whatever its glorification these days, has great limits as you can see from the vast ocean of unhappy unattractive women over the age of 30, professional actress bitches on TV and in the movies, who are full of crap in everything they say and do and fake smile about, notwithstanding.
To sum up, wage slavery, along with its associated social controls and restrictions, is the reason for the pained epidemic of frustrated and broken love relationships. Failed lovers make better slaves. And, not incidentally my dears, women wage slaves make better whores for the bosses at work who bleed the last juice from the women in the labor force who lack strong loving men to protect them from asshole bosses emotionally and otherwise.
And, wait a minute, that’s not the end of it. For what comes ultimately from the unhappiness of slavery and the failed love it causes in modern civilization is the channeling of that unhappiness into aggression towards others. For passing on one’s unhappiness to others, so often individuals who had nothing to do with causing the unhappiness, is a main way of mitigating your unhappiness, indeed, often the only way in highly controlled societies that don’t let you punch the abusive boss who caused the unhappiness in the face.
Such aggression towards innocent victims to tamp down unhappiness caused by an institutionally protected tyrant is termed redirected aggression. It is so common in life that we take it for granted. And beyond the petty meanness it causes in day to day existence it is the real reason for the mass murders endlessly seen in the news that are persistently touted as having no discernable motive. We just have no idea, say the TV journalists, why the guy killed his wife and his neighbors or the kids at the elementary school or his boss and coworkers. The possibility that the fellow was unhappy enough to crazily kill for the aforesaid reasons of the pain of being a jackass wage slave is never considered though his unhappiness is utterly obvious if one even lightly scratches below the surface of the story. God forbid that his unhappiness and violence might come from the life he’s forced to live in today’s hyper1984 societies. The fellow who mass murdered the 150 people by driving the airplane into the Alps? The scary looking Adam Lanza who murdered all those second graders at Sandy Hook Elementary? These and the rest of their kind are just the extreme tip of the iceberg of unhappy individuals sideeffect manufactured by our highly ordered society. Killing others makes them feel better than doing nothing at all. It reduces the unhappiness in them that comes from the humiliations and restraints of wage slavery and its effect on the possibility of them finding and keeping love.
Redirected aggression emanates not just from an unhappy individual but also from an unhappy group of people. Hate and violence fostered by the unhappiness of civilized life spills across national borders as a significant cause of war. Nothing better than having a foreign enemy with different values and beliefs to hate and kill to distract people from the unhappiness caused them by the rule they live under in their own countries. And these dysfunctional emotions of redirected aggression that make nations prone to war can cause the very worst for mankind when nuclear weapons are in the arsenals of aggressor nations. Doesn’t it make sense that a person as enthralled with mass murder as Adolph Hitler was had to be an unhappy man to begin with, a nut in that sense? And if Hitler had nukes to use, need we question whether or not he would have used them? Can’t we see the dangers in the endless bloodletting in the Middle East and in the Ukraine as to where these might escalate once real weapons of mass destruction come into play? Are people so pacified in their thinking not to see the ease with which a nuclear player backed up against the wall and facing impending defeat by a hated enemy might use his nuclear weapons?
There is only one sensible solution. The people have to get rid of the weapons before the weapons get rid of the people. Two profoundly good things come out of this. For one, it’s very hard for one person or one group of people to control others significantly without weapons. The ruling class in America ultimately controls the subservient population with police who enforce laws written by legislatures owned through campaign contributions by the wealthy ruling class. The police ultimately keep the people in line with weapons. Get rid of the weapons and you get rid of the tyranny. And, of course, getting rid of the weapons also gets rid of the horrors of war.
Now you have the gist of the main idea minus the mathematical proof of it. You say you don’t like this picture as too strong a condemnation of the free and fair America you love so much? Or you don’t like the solution of eliminating the weapons as too idealistic or against citizens’ constitutional right to own weapons and use them against bad people who deserve to be shot dead? Or you think this attack on the religious ideas that blur the reality is sinful or on psychology crazy? Well, that’s what the mathematics is for. It makes perfect sense out of these controversial issues to anybody who believes that 2+3=5 provides truth that can’t be denied.
As to actually achieving A World with No Weapons to solve the problem of violence, it’s just a matter of convincing Russia, the world’s other dominant nuclear power, that any war between us two big guys on the block is the end of the game for both of us and for all of us. This is not impossible, for the Russians have produced some of modern history’s top mathematicians and trust in science, whatever our endless vilification of Putin, at least as much as evolution denying America does. Then working with Russia, our two nations can convince the rest of the world, which we two dominate weapons wise, to give up their weapons, or be destroyed by our two nation coalition if they should refuse to give up their weapons. Other nuances of this utterly indispensible Utopia, achievable with great effort and a small miracle or two, are spelled out at the end of the nonmathematical Section 15. And the mathematics that shows how indispensible A World with o Weapons is for mankind’s avoidance of extinction is also indispensible, for people are not about to drop whatever they have felt is so important in their lives to dedicate themselves to a movement for a mass weapons ban without some form of tangible proof that it absolutely must be done to avoid the mass death of themselves and their kids and grandkids.
We begin such mathematical proof with a discipline called information theory. It will take us back in a formal way to our original the problem of inexactness in the counting of things unequal in size and help to develop a solution to that technical problem. Information theory is the science of the digital, synthetic, information that computers run on as distinct from the meaningful information that the mind runs on. This inability of information theory to develop mathematical specifications of meaningful information is a major shortcoming of it as is made clear in a June, 1995, Scientific American article, From Complexity to Perplexity:
Created by Claude Shannon in 1948, information theory provided a way to quantify the information content in a message. The hypothesis still serves as the theoretical foundation for information coding, compression, encryption and other aspects of information processing. Efforts to apply information theory to other fields ranging from physics and biology to psychology and even the arts have generally failed – in large part because the theory cannot address the issue of meaning.
It is most important to solve this problem because the greatest impediment to mankind avoiding an annihilating nuclear war is our failure to correctly understand the mind’s information processing operations which include man’s propensity for violence that is so dangerous in this era of nuclear weapons. The solution to this problem shown by the mathematics to be A World with No Weapons must begin in the United States and must have political muscle to succeed. To that end we strongly urge support for Elizabeth Warren in 2016 as the only candidate who is honest and caring enough about people to lean in this direction.
She stands in contrast to Hillary Clinton who merely as an astonishingly good actress pretends to care about people. Further it is important to understand that beyond her genius as a politician during campaign time, Hillary is as stupid a thinker as she is disingenuous. Certainly hubby Bill Clinton gets the Blue Ribbon as the smoothest liar American politics has ever seen. Do you think that this smooth talker’s wife can possibly have a different mindset after four decades of mutual plotting and scheming to hold the public stage?
And it is so important to have an intelligent person in the White House, for the repeated warnings of Vladimir Putin to use nuclear weapons if push comes to shove must be taken seriously in the absence of rapprochement with America. Hillary Clinton calling recently for more military assistance for Ukraine, (4/17/15), makes clear what a stupid she is in this most important issue of our time.
A sensible peace with Russia and
movement towards A World with No Weapons will not come about under Hillary.
That is for sure. Or under any of the Republican war hawks who might be
elected. That pointedly includes current Republican frontrunner, Jeb Bush,
whose blood relationship to the pair of assholes in the Bush monarchical
dynasty who contrived the War in Iraq strongly suggests a continuation of
maximum bloodshed in the Middle East with all that portends for eventual world
destruction, anybody who believes Jeb’s campaign denials of it understood to be
even stupider than Jeb looks. Once you understand the stakes, given Putin’s
determination to never back down to the US, Elizabeth Warren is the only sane
choice for president.
Those who wish to help start up a movement towards A World with No Weapons with
a donation of $20 can do so by clicking here. In asking for it we’re either
Bernie Madoff in some very mathematically elaborate sheep’s clothing or we’re
the real thing. Read the mathematical analysis that follows before you decide
which. It provides a most beautiful mathematical explanation of the human
emotions and how man’s violent ones can be tamed only by getting rid of weapons
that make aggression so horribly bloody. And for the resolute nonmath readers,
to highlight the underlying social causes of violence, there’s my personal
story in Section 15, “Revolution in the Garden of Eden”, that tells of
child murders by my cousin, Ed Graf, and my brother, Don Graf, both of them
members of the fundamentalist LCMS Christian sect I managed to escape from many
years ago.
That’s cousin, Ed Graf, on the left pleading guilty in court to burning his
stepson’s to death to get insurance money, and lawyer brother, Don Graf, on the
right. And Section 16, “Waiting for the Bomb”, also provides some
nonmathematical material for the reader who wants to pass on the technical
analysis that begins below.
1. Some Basics of Information Theory
The central structure in information theory is the information channel or message channel diagrammed below.
The set of messages that can be sent through a particular message channel have mathematical representation in terms of their relative probabilities of being sent. Consider a message set consisting of N=4 color messages, red, green, purple or black, that derive from a person blindly picking one object from a set of K=12 objects, (■■■■■, ■■■, ■■■, ■), which consists of x_{1}=5 red, x_{2}=3 green, x_{3}=3 purple and x_{4}=1 black object. The probability of the color picked being red and of a message about it being sent to some receiver is p_{1}=x_{1}/K=5/12; that of green, p_{2}=x_{2}/K=3/12=1/4; of purple, p_{3}=x_{3}/K=1/4; and of black, p_{4}=x_{4}/K=1/12. The message set of N=4 color is specified in terms of these probabilities as [p_{i}] = [p_{1}, p_{2}, p_{3}, p_{4}] = [5/12, 3/12, 3/12, 1/12].
The amount of
information in a message is a function of the probabilities of the N messages,
p_{i}, i=1,2,…N. There are two main functions used for information in
information theory. The one used most often is the Shannon (information)
entropy.
1.)
The other important information expression in information theory, which is closely related functionally to the Shannon entropy, is the Renyi entropy, in logarithm to the base 2 form,
2.)
The key to understanding information as we ordinarily understand that word as information that has meaning for us entails understanding the Renyi entropy rather than the Shannon entropy as the primary function for information as follows. We note from the blurb on it in Wikipedia that the Renyi entropy is important in ecology and statistics as an index or measure of diversity. That is not surprising given that the nonlogarithmic part of the Renyi entropy is another longtime used measure of diversity in ecology and sociology, the Simpson’s Reciprocal Diversity Index,
3.)
This allows us to specify the Renyi entropy, R, in terms of the Simpson’s diversity index, D, as
4.) R=log_{2}D
An equiprobable message set for color would derive from a random pick of an object from a balanced set of objects like (■■■, ■■■, ■■■, ■■■), whose N=4 colors have equal probabilities of being picked and sent a message out about of p_{1}= p_{2}= p_{3}= p_{4}=1/N=1/4. Substitution of p_{i}=1/N in Eq3 obtains a simplified D diversity index expression for the balanced case of
5.) D = N, balanced
This is intuitively sensible as indicating that the diversity of a set of objects is measured by the N number of different kinds of objects, in (■■■, ■■■, ■■■, ■■■) as D=N=4 differently colored kinds of objects. Note how this also simplifies the Renyi (information) entropy as
6.) R=log_{2}N
Clearly R is a function of the number of color messages derived from random picking from the (3, 3, 3, 3) set of objects, (■■■, ■■■, ■■■, ■■■), namely, R=2 bits. In contrast the D diversity index of the (5, 3, 3, 1) unbalanced set of objects, (■■■■■, ■■■, ■■■, ■), and the message set derived from it is from Eq3 and its [p_{i}] = [5/12, 3/12, 3/12, 1/12], D=3.273, with a Renyi entropy from Eq4 of R=log_{2}(2.323)=1.71. Now we are not at the moment interested in the meaning of R, but rather that R is for all sets a function of the D diversity of the message set. This includes R=log_{2}N for the equiprobable or balanced case in which N is equal to D from Eq5 and can be considered a diversity measure for the balanced case also. Hence the R Renyi entropy as information is a function of the D diversity of the message set. This gives diversity a central role in information. Why is this? It has to do with the problem of exactness in counting things.
2. Counting
The simplest items in mathematics are the counting numbers: 1, 2, 3, 4, and so on. But counting isn’t as simple as it seems. Count the number of objects in (■■■■). You count 4 objects here, of course. Now count the number of objects in (■■■■). It is also 4 one says at a glance. But is that count of 4 an exact count?
There is something not quite right with counting the unequal sized objects in (■■■■) as 4. Counting 4 objects in (■■■■) should be understood to be inexact as follows. You remember the grade school caveat against adding things together that are different in kind like adding 2 galaxies and 2 kittens together. This caveat also holds for adding things or counting things that are different in size. Consider (■■■■) as pumpkins of sizes (5, 3, 3, 1) in pounds. Is the count of them of 4 pumpkins exact? A grocer selling the pumpkins would think not, which is why pumpkins are sold not by the pumpkin, but by the pound, all of which pounds being exactly the same in size. Four pounds is an exact enumeration of pounds because all pounds are the same size in weight while four pumpkins is an inexact count of pumpkins when the pumpkins counted are not the same size. This requirement of sameness in size for a count of things to be exact applies to all standard measure whether pounds, fluid ounces or inches. That is why standard measure underpins all commercial transactions unless things bought and sold are the same size, like large eggs, which are sold by a straightforward count of them, as by the dozen.
We make this point of inexactness in a count of things not the same size in a more rigorous way by next considering our set of K=12 unit objects, all the same size, (■■■■■, ■■■, ■■■, ■), divided into N=4 color subsets that are not the same size in having different numbers of unit objects in some of them. The K=12 count of all the objects is exact because the objects are “unit objects” all the same size. But the N=4 count of the subsets, on the other hand, is inexact because the subsets are not the same size in having a different number of unit objects in some of them. To make it analytically clear that there is some sort of error in counting the number of subsets in (■■■■■, ■■■, ■■■, ■) as N=4, we first formally specify the set as consisting of x_{1}=5 red, x_{2}=3 green, x_{3}=3 purple and x_{4}=1 black object or in shorthand the (5, 3, 3, 1) natural number set. The sum of the objects in each of the N=4 subsets is the K=12 total number of objects in the set, or generally for any natural number set,
7.)
For the (■■■■■, ■■■, ■■■, ■) set, the total number of objects is K = x_{1}+ x_{2}+ x_{3}+ x_{4 }= 5+3+3+1 =12. Now it is a simple matter to show that the N=4 count of the unequal sized subsets in (■■■■■, ■■■, ■■■, ■), (5, 3, 3, 1), is inexact or in error with statistical analysis. The basic statistic of a set of numbers like (5, 3, 3, 1), here representing (■■■■■, ■■■, ■■■, ■), is the mean or arithmetic average, µ, (mu).
8.)
For the K=12, N=4, set, (5, 3, 3, 1), the arithmetic average is µ=K/N=12/4=3. That the µ arithmetic average is inexact is well made in 47 chapters of myriad examples in the modern classic, The Flaw of Averages by Sam Savage of Stanford. A more immediate register of the inexactness or error in the µ arithmetic average comes from noting that it is always associated with a statistical error measure, explicitly or implicitly, the most common of which is the standard deviation, σ, (sigma),
9.)
For the N=4, µ=3, (5, 3, 3, 1) set, the standard deviation is
10.)
Another commonly used statistical error is the relative error or percent error, r,
11.)
For the µ=3, σ=1.414, (5, 3, 3, 1), set, the relative error is r=σ/µ=1.414/3=.471=47.1%. The statistical error in the µ=K/N=3 arithmetic average of (5, 3, 3, 1), whether expressed as σ=1.414 or r=47.1%, implies a counting error in the N number of subsets parameter in µ=K/N. The K=12 count of the unit objects in (■■■■■, ■■■, ■■■, ■) in µ=K/N is exact because its K=12 unit objects are the same size. Hence the statistical error or inexactness in µ=K/N must arise from the inexactness in the N=4 count of the unequal sized subsets in (■■■■■, ■■■, ■■■, ■).
To further make the point of the straight N count of unequal sized subsets being inexact via the statistical error associated with it, we look at the µ, σ and r of the K=4 object, N=4 subset, “balanced” set of objects, (■■■, ■■■, ■■■, ■■■), (3, 3, 3, 3), all of whose subsets are the same size, x_{1}=x_{2}=x_{3}=x_{4}=3. This set also has a µ=K/N=12/4=3 arithmetic average, but from Eq9, it has no statistical error, σ=r=0, which logically implies from what we just said above that there is no error or inaccuracy in µ=K/N for it and, hence, no error or inaccuracy in the K or in the N variables of µ=K/N. And this fits perfectly with our understanding of an exact count coming about when things counted, including the N=4 subsets in (■■■, ■■■, ■■■, ■■■), (3, 3, 3, 3), are all the same size.
Now while the N=4 count of the number of subsets in (■■■■■, ■■■, ■■■, ■) is inexact, its diversity index of D=3.273 is an exact quantification of the subsets in the set. And this holds generally for any set, balanced or unbalanced. To see this, let’s formally define the p_{i} of a set in terms of the K and x_{i} of a set as
12.)
The two variables that p_{i} is a function of are exact, both the K count of the total number of same size objects in a set and the x_{i} number of (same sized) unit objects in each subset. From this perspective, D in being entirely a function of the p_{i} of a set in Eq3 is exact. Another way of appreciating the exactness in the D diversity index comes from understanding D as a statistical function. To do that we first express the σ standard deviation of Eq3 as its square, σ^{2}, called the the variance statistical error.
13.)
And then we solve this for the summation term to obtain
14.)
Now from Eq12, we express the D diversity index as
15.)
And lastly inserting the summation term in Eq14 into D of Eq15 obtains D also via Eqs8&11 as
16.)
This derives an exact D quantification of the subset constituents of a set as a function of their inexact N count effectively made exact by the inclusion of the r relative error measure of the inexactness in N in the function. This understands D as an exact correlate or substitute for inexact N that can be used in place of N as an exact quantification of the constituent subsets of an unbalanced set. And it further understands diversity, in its sense as an exact quantification of balanced or unbalanced subset constituents of a set, as what information is in the most general sense of the word. Let us back up a bit to make what we mean clear here. Earlier we made note that the R Renyi entropy is used in the scientific literature as a measure of diversity, a logarithmic measure of diversity. Now let’s also note that the Shannon information entropy has also been used in the scientific literature of the last 60 years as a measure of ecological and sociological diversity as called the Shannon Diversity Index, it’s configuring as a natural logarithm rather than a base 2 logarithm being irrelevant to the synonymy of information and diversity in the Shannon entropy as well as the Renyi entropy. Furthermore both the logarithmic diversities, Shannon and Renyi, and the linear diversity, D, are exact functions as defined above.
This strongly suggests that the D diversity index is information also. Later we shall prove this rigorously with a bit signal encoding recipe interpretation of D and with a Gödel based rebuttal of the Khinchin argument in its derivation of the Shannon entropy as the only correct form for information (along with the Renyi entropy generalization of the Shannon entropy). These tedious technical proofs, though, are delayed for the moment as too much of a digression from showing rather and first how the D diversity index is readily understandable intuitively as meaningful information.
3. Diversity as a Measure of Meaningful Information
Information as processed by the human mind has measure in the D diversity index
interpreted as the number of significant subsets in a set. The
exercise that follows will explain how the mind intuitively distinguishes what
is significant in its sense and memory of things from what is insignificant. We
illustrate with an item in the recent news about the makeup of the K=53 man
Ferguson Police Dept. at the time of the protest over the death of Mike Brown,
namely x_{1}=50 White officers and x_{2}=3 Black officers. Few
have a problem intuitively understanding the Black contingent of the Ferguson
P.D. to be insignificant (quantitatively) even without any mathematical
analysis. But D diversity index interpreted as the number of significant
subsets in a set makes the understanding of insignificance mathematically
precise.
The Ferguson P.D. as the number set, (50, 3), has from Eq15 a diversity of D=1.12, which rounded off to the nearest integer as D=1 implies that there is only 1 significant subset or subgroup in the department. Were the force made up in a more diverse way of, say, x_{1}=28 Caucasians and x_{2}=25 Blacks, the diversity of its (28, 25) representative number set of D=1.994 rounded off to D=2 would indicate that both subgroups were (quantitatively) significant. Returning to the actual (50, 3) makeup calculated to have a rounded diversity measure of D=1 significant subset, the x_{1}=50 preponderance of the White contingent suggests that it is the significant subgroup and, hence, that the x_{2}=3 Black officer subgroup is insignificant as can also be interpreted as its contributing only token diversity to the police force.
Before we continue this analysis, given the contentiousness of this issue, it should be made clear that considering the police as the enemy of a hoped for genuinely free and fair society is a mistake. Police are strictly the hired hands of the ruling class business and political leaders of communities that range in size from the small city of Ferguson to the entire USA. Police do not make policy. They simply execute it and do so on threat, like the rest of Americans who work jobs, of being fired and having their lives ruined if they fail to comply with the directives of the upper echelon in the American social hierarchy. There is no good cop, bad cop dichotomy, therefore, only a good leader versus bad leader differentiation. And this current crop of leaders in America are as disgustingly predatory, uncaring of the little people and deceitful as any ruling oligarchy you’ll read about in history. If you want change, that’s where you have to look for change, in the people at the top, not the cops who are the ruling class’s well controlled ultimate instrument of coercive control of the people.
That important political digression aside, let’s now continue the mathematical analysis of significance versus insignificance by showing how to assign a significance index to each one of the constituent subsets of a set. We will use the K=12, N=3, (■■■■■■, ■■■■■, ■), (6, 5, 1), x_{1}=6, x_{2}=5, x_{3}=1, set to introduce significance indices. We calculate from Eq15 a D=2.323 diversity index for this set, which rounded off to D=2 suggests 2 significant subsets, the red and the green, with the purple subset that is represented by only x_{3}=1 object in it understood as insignificant. To specify these attributions of significance and insignificance to each subset in a more direct way, we define next the root mean square average, aka the rms average, of a number set, ξ, (xi) as
17.)
The rms average squared, ξ^{2}, is
18.)
The ξ rms average of the K=12, N=3, µ=K/N=4, (■■■■■■, ■■■■■, ■), (6, 5, 1), unbalanced set is ξ =4.546 with ξ^{2}=62/3=20.667. And the rms average of the K=12, N=3, µ=K/N=4 balanced set, (■■■■, ■■■■, ■■■■), (4, 4, 4), which we will use for comparison sake, is from the above ξ=µ=4 with ξ^{2} =µ^{2} =16. Next note from Eqs13,8&18 that the D diversity index can be expressed as
19.)
We define the significance index of the i^{th} subset of a set as s_{i}, i=1, 2,…N,
20.)
This obtains the D diversity as the sum of its s_{i} significance indices as
21.)
For sets, balanced and unbalanced, that have N=3 subsets containing a number of objects in each of x_{1}, x_{2} and x_{3},
22.) D = s_{1} + s_{2} + s_{3}
For the N=3, (■■■■, ■■■■, ■■■■), (4, 4, 4), set, x_{1}=4, x_{2}=4 and x_{3}=4, the D=N=3 diversity index of this balanced set from Eq5 alternatively computed from the above is
23.) D = s_{1} + s_{2} +s_{3 }= 1 + 1 + 1 = 3 = N
What D=3=1+1+1 tells us is that all N=3 subsets in having significance indices of s_{1}=s_{2}=s_{3 }= 1 are significant. Now consider the unbalanced (■■■■■■, ■■■■■, ■) set, whose subset values of x_{1}=6, x_{2}=5 and x_{3}=1 develop its significance indices from Eqs22&23 as
24.) D = s_{1} + s_{2} +s_{3 }= 1.161 + .968 + .194 = 2.323
What D= 1.161 + .968 + .194 indicates is that the x_{1}=6 red objects subset in having a significance index of s_{1}=1.161 rounding off to s_{1}=1 is significant; that the x_{5}=5 green objects subset in having a significance index of s_{2}=.968 rounding off to s_{2}=1 is significant; and that the x_{3}=1 purple object subset in having a significance index of s_{3}=.194 rounded to s_{3}=0 is insignificant. This analysis applied to the Ferguson P.D. situation has us interpret from the s_{1}=1.056 index evaluated from the above for the x_{1}=50 White cops and rounded to unity that they are (quantitatively) significant and from the s_{2}=.056 rounded off to s_{2}=0 for the x_{2}=3 Black cops specifies them as (quantitatively) insignificant.
That the human mind genuinely operates with these
significance functions, or some neurobiological facsimile of them, is made
clear in the next illustration of significance and insignificance of the three
sets of colored objects shown below, each of which has K=21 objects in it in
N=3 colors.
Sets of K=21 Objects 
Number Set Values 
D, Eq15 
Rounded to 
Significance Indices, Eq21 
(■■■■■■■, ■■■■■■■, ■■■■■■■) 
x_{1}=7, x_{2}=7, x_{3}=7 
D=3 
D=3 
s_{1}=1, s_{2}=1, s_{3}=1 
(■■■■■■, ■■■■■■, ■■■■■■■■■) 
x_{1}=6, x_{2}=6, x_{3} =9 
D= 2.88 
D=3 
s_{1}=.824, s_{2}=.824, s_{3}=1.24 
(■■■■■■■■■■, ■■■■■■■■■■, ■) 
x_{1}=10, x_{2}=10, x_{3}=1 
D=2.19 
D=2 
s_{1}=1.04, s_{2}=1.04, s_{3}=.104 
Table 25. Sets of K=21 Objects in N=3 Colors and Their D Diversity and s Significance Indices
The N=3 set, (■■■■■■■■■■, ■■■■■■■■■■, ■), (10, 10, 1), has a diversity index of D=2.19, which rounded off to D=2 implies D=2 significant subsets, the red and the green from their s_{1}=s_{2}=1.04 significance indices. And that also implies that the one object, purple subset is insignificant as reinforced by its s_{3}=.104 significance index as might also be interpreted as the purple set contributing only token diversity to the set. In contrast, the D=3, (■■■■■■■, ■■■■■■■, ■■■■■■■), (7, 7, 7), set has 3 significant subsets, red, green and purple, s_{1}=1, s_{2}=1, s_{3}=1; as does the (■■■■■■, ■■■■■■, ■■■■■■■■■), (6, 6, 9), set whose D=2.88 diversity rounds off to D=3 with s_{1}=.824, s_{2}=.824, s_{3}=1.24.
One gets the most intuitive sense of the mind’s automatic or subconscious evaluation of significance and insignificance by manifesting the K=21, N=3, colored object sets in Table 25 as K=21 threads in N=3 colors in a swath of plaid cloth.


(10, 10, 1), D≈2 
(7, 7, 7), D=3 
(6, 6, 9), D≈3 

A woman who owns a plaid skirt with the (10, 10, 1), D≈2, pattern on the left, as I do, would spontaneously describe it as her red and green plaid skirt, omitting reference to the low density and, hence, relatively insignificant threads of purple in the plaid. She would make this description automatically without any conscious calculation because the human mind just does automatically disregard the insignificant and, indeed, not just in its visual sense of it but also in its not verbalizing things sensed as insignificant. This verbalization of only the significant colors in the plaid swath of red and green should not be surprising given that the word “significant” has as its root, “sign” meaning “word”, which suggests that what is sensed by the mind as significant is signified or verbalized while what is sensed as insignificant and little or not at all sensed or noticed, isn’t signified or assigned a word in discourse or in one’s thoughts.
Our sensory perceptions are generally quite automatically affected by the magnitude of the sensory input, small inputs being insignificant and disregarded in our sense of them as with our lack of sensing or tasting salt added to a stew when it is just the slightest pinch of salt. This sense of the possibility of insignificant ingredients in a recipe when added in the smallest yet nonzero amounts is what gave me the conceptual germ of A Theory of Epsilon.
Quantitative significance is not just a characteristic of the size or quantity of things but also of the frequency of our observation of things and events. Consider a game where you guess the color of an object picked blindly from a bag of objects, (■■■■■■■■■■, ■■■■■■■■■■, ■). Assume that you don’t at all know the makeup of the objects in the bag ahead of time as you go about guessing and observing which colors get picked and in with what frequency. Then your sense of what colors are significant or insignificant comes only from the frequency the colors are picked from the bag (picked with replacement). And over time, as you see purple picked so infrequently that the purple color will come to seem insignificant in your mind as a possible pick and to also be disregarded as a color you might think likely to be picked. The human mind’s operating automatically to disregard the insignificant is an important factor for behavior because we generally think, talk about, pay attention to and act on what we consider significant while automatically disregarding the insignificant in our thoughts, conversations and behaviors.
From a sociopolitical perspective the development of one’s sense of the significance of some things and insignificance of others via media exposure a central aspect of propaganda and mind control because issues and opinions frequently disseminated through mass media and other ruling class information outlets are subconsciously taken to be significant to some degree and tend, as such, to take up much of one’s thoughts, conversations and behavioral considerations in contrast to issues, observations and opinions infrequently brought up or not at all broadcast, which become regarded, as such, as insignificant and effectively paid little regard if any at all.
In this way personally immaterial sporting events and entertainments along with political opinions with minimal factual bases come to be subconsciously thought of as significant, crowding out issues and interpretations of news events that are genuinely meaningful for people’s individual welfare, but in being shown infrequently or not at all, such as the daily abuse in workday life people take from all powerful bosses, become insignificant in discourse and in thought for people and in their intentions for future action. This does not come about by chance for people drugged with such an endless stream of misinformation tend to stay in line. To hear a warning to avoid such brain washing set to music, take a few minutes break from the mathematical analysis to listen to Curse That TV Set.
An obvious example of propaganda via frequent repeat of a message is seen in Republican talking point strategy. This political party instrument of the ruling class repeats things as nonsensical as the sky is green and the trees are blue through media controlled by ruling class TV station and newspaper ownership and through advertising with such frequency that the drugged population out there in the audience come to think that such opinions have significance. Hey, maybe there is something to the idea of a green sky and of no evolution and of no climate change and of working people not always living fearfully on the edge of homelessness while the privileged amuse themselves with expensive trivialities purchased at the expense of the misery of the working class.
A classic illustration of political disingenuousness via a distortion of significance is found in the Republicans getting the public to support the war in Iraq in 2003 by describing our invading force as a “coalition.” It consisted approximately of K=163,700 soldiers from N=32 nations distributed set wise as (145,000, 5000, 2000, 2000, 1000, 1000, 1000, 1000, 500, 500, 500, 500, 500, 500, 500, 200, 200, 200, 200, 100, 100, 100, 100, 100, 50, 50, 50, 50, 50, 50, 50, 50). The invading force set’s number of significant members calculates from Eq15 as D=1.26, which rounded off to D≈1 specifies 1 significant nation in the socalled coalition, the United States, distinctly at odds with the general sense of a coalition as a plural entity rather than a collection of subordinates dominated by on nation, here the modern American empire.
The cleverness of calling it a coalition along with the endlessly repeated WMD talking point as rationalizations for entering this costly and unnecessary bloody war were clear enough to be recognized by the astute back then as raw political hokum without need for the D diversity index to clarify N−1=31 nations in the “coalition” as insignificant, though using D as a measure of the number of significant nations allows us to call out the deceitfulness of the politicians and the media that supported them with mathematical precision.
Manipulation of the intuitive D diversity based operation of the mind to assess significance is a cornerstone foundation of propaganda. It works by repetition of mistruth as evident both in patently totalitarian societies, organized religions and in capitalist pseudodemocracies where almost all politicians are controlled by the big money of corporations and Wall St., an obvious and highly meaningful fact that is made to be insignificant in the minds of people in its seldom being publically voiced. And those who do bring such facts to light, whether about the harsh realities of war or the institutional corruptions and misery of a highly ordered peace, are made to seem significantly bad.
A case in point is the documentary film maker, Michael Moore, whose primary sin is feeling disgust in both areas and making it known. His calling out Bush’s vanity driven Iraq War that, for no good national purpose and supported by endlessly repeated lies, unnecessarily took the lives of 5000 young American soldiers, crippled 30,000 more and destroyed well a million Iraqi lives, not to speak of the instability it caused that brought cutthroat ISIS to power in the Middle East, all these hard facts made to seem insignificant. Indeed, Moore’s insightful and prophetic castigation of the war at the Oscar nominations in 2003 and later in his documentary, Fahrenheit 9/11, brought about an active encouragement by right wing snake, Glenn Beck, held up in the media as a paragon of virtue, to go out and literally murder Michael Moore.
Those reading this and in despair about finding anything that can be done about it can be given one word of good advice that fits well with their desperation: run. To retain some modicum of selfrespect and the possibility of squeezing any real happiness out of life: run. That’s the best you can do in the short term to avoid the powerful if well hidden agencies of control in modern America. One thing that you can, and must, avoid is the TV set. This is one very important route to maintaining a sane view of life. It is admittedly a limited cure for the bigger problem. To solve that you have to fight back, not just run. And you do that by supporting and placing your hopes in the only true solution to this mess of human existence, helping to bring about A World with No Weapons, not just for bypassing nuclear Armageddon but also in a weapons ban restoring a true balance of power to life, the details to be talked more about in later sections.
To get to A World with No Weapons requires political power, it must be pointed out. Anarchy, everybody doing their own thing, is great if you can get it. But you can’t get it in the real world where the rules forbid it and punish those who break the rules. You have to first get to A World with No Weapons, which has to start with somebody in the White House here in America who actively carries that destiny on their shoulders. As I’m the only suitable candidate for that in 2016, encourage me by dropping a line to ruthmariongraf@gmail.com and sending a $20 donation to join the movement for A World with No Weapons and for the democratic revolution needed to make it happen by clicking here.
4. A Biased Average
The sense of a Utopia where there is no war or tyranny still must seem to most farfetched. We need more precise mathematical argument to make clear that it’s the only direction we can go in to get out of the hell of debilitating control in our lives and to avoid nuclear annihilation. To that end we next want to consider an adjunct function to the exact specification of the inexact N subsets in an unbalanced set, namely an exact average of the number of objects in each subset. As we made clear earlier, the µ=K/N arithmetic average number of objects per subset in an unbalanced set of K objects distributed over N subsets is inexact. Much as we form the arithmetic average as the ratio of the K objects in a set to the N inexact number of subsets in the set as µ=K/N, we can form an exact average as the ratio of the K objects to the D exact quantification of the subsets as K/D, an exact if biased average, to which we give the symbol, φ, (phi).
27.)
The φ=K/D biased average is an exact average in being a function of K, which is exact, and of D, which is also exact as was made clear earlier. The K=12, N=4, µ=K/N=4, D=2.323, (■■■■■■, ■■■■■, ■), (6, 5, 1) set has a biased average of φ=K/D=12/2.323=5.166. This is greater than this set’s arithmetic average of µ=K/N=4 from the φ biased average being weighted or biased towards the larger x_{i} values in (6, 5, 1). To see the details of the bias in the φ biased average of an unbalanced set towards the larger x_{i} values in the set, let’s develop the µ=K/N arithmetic average in an unusual way from Eqs7&8 as
28.)
This understands the µ mean or arithmetic average as the sum of “slices” of the x_{i} of a set of thickness 1/N. We can develop a function for the φ biased average with a parallel form from Eqs27,13,8&12 as
29.)
This shows φ to be the sum of “slices” of the x_{i} of a set that are p_{i} in thickness to bias the φ average towards the larger x_{i} subsets weighting them with their correspondingly larger p_{i} weight fraction measures. Much as the human mind’s sense of significance is affected in a biased way by the diversity of what it senses, so also is its sense of the average of the constituent subsets of set affected in a biased way towards the greater x_{i} of a set as representing the size of the objects in a set and/or the frequency with which they are sensed. A familiar example is our sense that a dinosaur is generally speaking very, very big. This comes about as a biased average of dinosaur sizes both in terms of the larger ones biasing our sense of the average towards the large sized dinosaurs and also from the fact that people see images of dinosaurs that are very large much more frequently than they see medium sized dinosaurs or small ones. Note also that the diversity index, D, can be understood as a function of the φ average when Eq27 is solved for D as
30.)
This expression for D tells us that the K total number of something in a set divided by its φ biased average is its D diversity. This relationship allows us to corroborate this analysis of significance and insignificance for the mind by showing it for physical systems where the concepts of significance and insignificance have measurable, empirical, reality. Specifically we will do it for a thermodynamic system of K energy units distributed over N molecules with a highlight on the concept of entropy, the tight argument presented reinforcing the D diversity based understanding of the highly contentious issue of brain washing propaganda.
5. Entropy
Entropy is a somewhat mysterious concept. Feeling cold in wintertime and hot in summertime comes about from the 2^{nd} Law of Thermodynamics said to be caused by an increase in entropy. But what it is exactly that’s increasing in these processes has been a confusion for science for the last two centuries since the French engineer, Sadi Carnot, first became aware of processes described in terms of entropy. The equation for entropy in terms of measurable quantities is the Clausius macroscopic formulation of entropy
31.)
Even though this differential equation tells us that entropy, S, is dimensionally, energy, Q, divided by temperature, T, it still leaves us with a confused mysterious sense of entropy because we lack an intuitive sense of what energy divided by temperature might be. Some light is thrown on the problem by next noting that (absolute) temperature is explained in the standard rubric of physical chemistry as being directly proportional to the µ=K/N arithmetic average of the K energy of a thermodynamic system per its N molecules. But this immediately raises a red flag because we made it very clear earlier that the µ arithmetic average is inexact for an unbalanced set and because the N molecules of a thermodynamic system are an unbalanced set of constituents of a thermodynamic system from the energy units of the system being distributed over them in a skewed or unbalanced way from the empirical MaxwellBoltzmann energy distribution.
Figure 32. The MaxwellBoltzmann Energy Distribution
The inexactness in the µ=K/N average energy function that derives from the
inexactness in the N number of molecules parameter suggests that it is an error
in physical science assuming that systems in nature necessarily operate in an
exact way. Or alternatively we might say that this arithmetic average
specification of temperature is a poor one in being inexact and is perhaps the
reason why the S entropy is so poorly understood and mysterious as a physical
quantity.
Following this line of reasoning tell us that temperature might be better understood as a biased average of energy per molecule. That supposition provides us with a very clear and physically sensible interpretation of entropy for as we see from Eqs30&31, K total energy divided by φ as a biased average energy would be the D diversity of the system of molecules which in perfectly fitting dimensionally Q energy divided by T temperature as S entropy pegs S entropy as the energy diversity of the system. As this quite fits the intuitive or qualitative sense of entropy as energy dispersal, just another word for energy diversity, (see entropy as energy dispersal in Wikipedia), it is worth tracking it down further and provide hard core evidence to show if this is truly the case. Doing so will also increase our confidence of the D diversity as a measure of significance and insignificance in cognitive systems and underpinning of a mathematical specification of the human emotions. And it will also provide us with a mathematical template for the generalizations of ideas and thoughts that the human mind also operates on. The evidence we will provide shows a mathematically perfect fit of energy diversity to the Boltzmann formulation of entropy, understood in science to be the most basic expression for entropy as honored by its inscription of Boltzmann’s tombstone in the terminology of 100 years ago as
33.) S = klogW
Ludwig Boltzmann’s 1906 Tombstone
Diversity is a property not only of a set of K objects divided into N color categories, but also of K candies divided between N children and K discrete or whole numbered energy units divided between N molecules. The distribution of K=4 candies to N=2 children takes the form of three natural number sets: (2, 2) for both children getting 2 of the K=4 candies with a diversity from Eqs5&8 of D=2; (4, 0) for one child getting all 4 of the K=4 candies and the other child none with a diversity from Eq5 of D=1; and (3, 1) for one child getting 3 of the K=4 candies and the other child, 1, with a diversity from Eq5 of D=1.6. The (2, 2), (3, 1) and (4, 0) manifestations of the random distribution are also referred to as the configurations of the distribution. The distribution of K=4 energy units over N=2 molecules has the same diversity values as the distribution of K=4 candies over N=2 children: for (2, 2), both molecules having 2 of the K=4 energy units, a diversity of D=2; for (4, 0), one molecule having all 4 of the K=4 energy units and the other molecule none, a diversity of D=1; and for (3, 1), one molecule having 3 of the K=4 energy units and the other molecule, 1, a diversity of D=1.6.
The candies over kids distribution is easiest to picture and follow, so we begin with it. The random or equiprobable distribution of candies to children as might come from grandma tossing K=4 candies of different color, (■■■■), blindly over her shoulder to her N=2 grandkids, Jack and Jill, has a number of ways of occurring, ω,
34.) ω = N^{K}
ω (small case omega) is referred to as a combinatorial statistic. Specifically for K=4 candies distributed randomly to N=2 children, the ω number of ways that can occur is
35.) ω = N^{K} =2^{4}= 16
These ω =16 ways are, with Jack’s candies set to the right of the comma and Jill’s candies to the left,
36.)
(■■■■, 0); (■■■, ■); (■■■, ■); (■■■, ■); (■■■, ■); (■■, ■■); (■■, ■■);
(■■, ■■)
(■■, ■■);
(■■, ■■); (■■,
■■); (■■■, ■); (■■■, ■); (■■■, ■); (■■■, ■); (0, ■■■■)
The probability of each of these permutations or ways or microstates of the random distribution is the same,
37.) 1/ω=1/16
If grandma did the tossing of the K=4 candies to the N=2 kids 16 times, on average, Line16 would come about though not necessarily in the sequence depicted. It is possible to compute the average diversity of this random distribution. Here we see that the probability of a (4, 0) permutation is 2/16=1/8; of a (3, 1) configuration, 8/16=1/2; and of a (2. 2) permutation, 6/16=3/8. It is a simple matter to compute the σ^{2 }variances of these permutations from Eq11: for (4, 0), σ^{2}=4; for (3, 1), σ^{2}=1; and for (2, 2), σ^{2}=0. Note that (4, 0), (3, 1) and (2, 2) are also referred to as the configurations of the distribution. The average variance of the ω = 16 permutations, also understandable as the probability weighted average variance of the configurations, is
38.)
The average variance, σ^{2}_{AV}, enables us to calculate the average diversity of the random distribution, D_{AV}, from Eq16 with σ^{2}_{AV} replacing σ^{2} and D_{AV} replacing D.
39.)
Understanding the arithmetic average of the number of energy units per molecule for the K=4 energy unit over N=2 molecule distribution to be µ=K/N=4/2=2, the parameters of σ^{2}_{AV}=1 and N=2 have us calculate the average diversity of the random distribution, D_{AV}, as
40.)
This dynamic plays out as above  it must be emphasized  even if the candies are all of the same kind, say K=4 red candies, (■■■■). This comes about because the candies, even though all of the same kind, are fundamentally different candies. Let’s back up a minute to explore this in greater depth. The (■■■■) candies are said to be categorically distinct or distinct in kind. But we don’t just distinguish things as being different kinds, as between a red candy, ■, and a green candy, ■. We also distinguish between two of the same kind of thing, as between two red candies, ■■, which though they are categorically indistinguishable or the same kind of thing, are yet distinguishable fundamentally. If you are holding one of these red candies in your hand and the other is on the kitchen table, you definitely distinguish between the two.
This is called fundamental distinction. It is different than categorical distinction, but yet a distinction between things people make as intuitively as they distinguish between different kinds of things. To show the fundamental distinction between K=4 red candies, (■■■■), we can represent them each with a different letter as (abcd). With the fundamental distinction so marked, the number of ways or different permutations of K=4 red candies, (abcd), that come about from their random distribution to N=2 kids is also calculated as ω= N^{K} =2^{4}= 16 of Eq35, those permutations being
41.)
(abcd, 0); (abc, d); (abd, c); (adc, b); (bcd, a}; (ab, cd); (ac, bd); (ad, bc)
(bd, ac); (bc, ad); (cd, ab); (a, bcd); (b, acd}; (c, abd}; (d, abc}; (0,
abcd)
Note that everything we said for the random distribution of (■■■■) in Eq37 to Eq40 applies also to the random distribution of (■■■■) as is readily understood once we delineate the fundamental distinctions in (■■■■) as (abcd). Now determining the average diversity, D_{AV}, for random distributions gets a bit tedious as the K and N of random distribution get large, indeed, practically impossible for very large K and N values. Fortunately we can develop a shortcut formula for the D_{AV} average diversity of any K energy unit over N molecule random distribution from a shortcut formula for σ^{2}_{AV} that already exists in standard multinomial distribution theory. In general for any multinomial distribution of K objects over N containers,
42.)
For an equiprobable multinomial distribution, the P_{i} term is P_{i}= 1/N, a relationship that tells us that each of the N containers in a K over N distribution has an equal, 1/N, probability of getting any one of the K objects distributed to it. This P_{i}=1/N probability for an equiprobable distribution is the P=1/N=1/2 probability of each of grandma’s N=2 kids having an equal, 50%, chance of getting any one candy blindly tossed by grandma. The P_{i} =1/N probability for a random distribution greatly simplifies the multinomial variance expression of Eq24 for the equiprobable case to
43.)
As things turn out this variance of an equiprobable multinomial distribution is the average variance of an equiprobable distribution, σ^{2}_{AV}, of Eq38 we developed for the K=4 over N=2 random distribution. Hence we can write Eq43 as
44.)
That the variance of an equiprobable multinomial distribution is, indeed, the average variance, σ^{2}_{AV}, is demonstrated by calculating the σ^{2}_{AV}=1 average variance of the K=4 over N=2 distribution in Eq41 from the above as
45.)
Eq44 can now be used to generate a shortcut formula for the average diversity, D_{AV}, by substituting its σ^{2}_{AV} into Eq39 to obtain
46.)
And we can further demonstrate the validity of the above shortcut formula for D_{AV} by calculating the D_{AV}=1.6 average diversity of the K=4 over N=2 distribution obtained in Eq43 with it.
47.)
These conclusions also hold for a system of K=4 energy units distributed equiprobably over N=2 gas molecules flying about in a container of fixed volume. The equiprobable or random distribution results from collisions between the N=2 molecules that result in random energy transfers of the energy units between molecules. In that case, Line41 represents the microstate permutations that arise on average from the collisions, though not necessarily in that sequence. The average variance, σ^{2}_{AV}, of the microstate permutations and their average diversity, D_{AV}, is the same as for the random distribution of K=4 candies between N=2 children.
With this picture of a thermodynamic system as our template, we can now confirm the dimensional analysis that suggested from the Clausius macroscopic formulation of entropy that entropy is basically energy diversity or energy dispersal. This is done specifically by showing that the average diversity, D_{AV}, has near perfect direct proportionality to the expression for microstate entropy Boltzmann developed that is expressed in modern terminology as
48.) S=k_{B}lnΩ
To demonstrate this we need not explain the meaning of the Ω (capital omega) variable in Boltzmann’s S entropy, held off until later, nor the k_{B} term in the function, a constant, but only show the exceedingly high correlation coefficient between D_{AV} and lnΩ. That is easy to demonstrate because both D_{AV} and Ω are functions solely of the K number of energy units and N molecules in a thermodynamic system, D_{AV} as seen in Eq46 and Ω from a standard formula in mathematical physics.
49.)
And the lnΩ as a function of K and N is
50.)
For large K over N equiprobable distributions it is easiest to calculate lnΩ using Stirling’s Approximation, which approximates the natural logarithm (ln) of the factorial of any number, n, as
51.)
Stirling’s approximation works very well for large n values. For example, ln(170!) =706.5731 is very closely approximated as 706.5726. The Stirling’s approximation form of the lnΩ expression of Eq50 is
52.)
We can use this formula to compare the lnΩ of randomly chosen large K over N equiprobable distributions to their D_{AV} average diversity of Eq46.
K 
N 
lnΩ, Eq52 
D_{AV}, Eq46 
145 
30 
75.71 
25 
500 
90 
246.86 
76.4 
800 
180 
462.07 
147.09 
1200 
300 
745.12 
240.16 
1800 
500 
1151.2 
381.13 
2000 
800 
1673.9 
571.63 
3000 
900 
2100.88 
692.49 
Table 53. The lnΩ and D_{AV} of Large K over N Distributions
The Pierson’s correlation coefficient for the D_{AV} and lnΩ of these distributions is .9995, which indicates a very close direct proportionality between the two as can be appreciated visually from the near straight line of the scatter plot of these D_{AV} versus lnΩ values.
Figure 54. A plot of the D_{AV} versus lnΩ data in Table 33
This high .9995 correlation between lnΩ and D_{AV} becomes greater yet the larger the K and N values of K>N distributions surveyed. For values of K on the order of EXP20 the correlation for K>N distributions is .9999999 indicating effectively a perfect direct proportionality between lnΩ and D_{AV} as fits very large, thermodynamically realistic, K over N equiprobable distributions. As the Boltzmann S=k_{B}lnΩ entropy is judged to be correct ultimately by its fit to laboratory data, given the near perfect correlation of the D_{AV} to it, this diversity entropy formulation must also be correct from that purely empirical perspective. This correlation of diversity entropy to the Boltzmann microstate formulation of entropy powerfully reinforces the dimensional analysis of entropy as energy diversity done from the Clausius macroscopic formulation of entropy.
It must be emphasized, though, that the two microstate formulations of entropy, diversity and Boltzmann, cannot both be correct even though both mathematically fit the data because the assumptions that underpin the two formulations are absolutely mutually contradictory. This requires some explaining. The ω = N^{K} number of ways combinatorial statistic of Eq34 implies from the 16 microstate permutations of Line21 for the K=4 over N=2 distribution that the energy units are all fundamentally distinguishable from each other. Understanding the random distribution in this way is what made possible the foregoing derivation of the average variance, σ^{2}_{AV}, and the average diversity, D_{AV}.
A quite different combinatorial statistic exists for enumerating the number of observably different ways that K categorically indistinguishable objects can be arranged in N containers. It is the Ω variable that we have already seen from Eq49 that sits in Boltzmann’s S=k_{B}lnΩ entropy.
49.)
Irrespective of Boltzmann’s use of it in his entropy equation, Ω can specify the number of ways that K=4 red candies, (■■■■), which are categorically indistinguishable even if as we made clear they are fundamentally distinguishable, can be arranged over N=2 containers or N=2 children. From Eq49 that is 5 ways
55.)
These Ω=5 ways are as we see below with Jack’s candies to the right of the comma and Jill’s to the left.
55a.) (■■■■, 0); (■■■, ■); (■■, ■■); (■, ■■■); (0, ■■■■)
However the Ω=5 value this has absolutely no meaning as regards the random distribution of K energy units over N molecules because such a distribution is necessarily governed from elementary probability theory by the ω = N^{K} combinatorial statistic of Eq14 that implicitly assumes that the energy units, though they are categorically indistinguishable, are fundamentally distinguishable. This perspective is bolstered by the energy units residing in distinguishable molecules, which themselves reside in different places in space. This suggests that Boltzmann researched a number of mathematical functions associated with the distribution of energy units over molecules until he came to one, lnΩ, which fit the data. In theoretical physics such a fit of a mathematical hypothesis to empirical data is generally taken as strong proof that the hypothesis is correct. In this case, though, it turns out that lnΩ is little more than a fluke fit to another function, D_{AV}, which not only also fits the data, but also makes physical sense out of entropy as energy diversity or energy dispersal One adds that neither Ω nor lnΩ make any sense out of entropy as a physical quantity, the reason for entropy’s mysteriousness over the last century.
One readily absolves Boltzmann for this error given that mathematical formulations for diversity did not come into existence until near half a century after his death and we emphasize that without Boltzmann’s breakthrough efforts my clarification of entropy as diversity would have been impossible. The complete acceptance of Boltzmann’s notions for the last hundred years from their perfect fit to data makes his ideas very difficult to overthrow for Boltzmann is as much a revered “saint” of physical science as Newton or Maxwell or Einstein. The task of rectifying our understanding of entropy as energy diversity would be much easier, for that reason, if both interpretations in their both fitting the data empirically, could be accepted. However the two assumptions of energy unit distinguishability and energy unit indistinguishability are totally incompatible and only one can be accepted. Hence Boltzmann is overthrown rather than just refined, difficult to accept for physical scientists who have embraced him as correct in the most foundational way over the last century.
This impediment to a correction of Boltzmann’s error, thus, asks for as much supporting evidence for the diversity entropy proposition as can be mustered. This is doubly important for not only does diversity explain entropy correctly and clearly for the first time in science, but also understanding diversity as a measure of entropy also very much makes clear the underpinning of meaningful information with diversity. That includes not only showing the concept of significance as a marker for what is meaningful in a very firm way in physical systems, but also uncovers a welldefined mathematical structure for the generalizations that the human mind operates on verbally called compressed representation.
A very strong supporting argument for diversity based entropy shows that my diversity based statistical mechanics much better explains the MaxwellBoltzmann energy distribution than Boltzmann statistical mechanics does.
Figure 32.
To show it we next introduce a new structure in mathematics called the Average
Configuration of a random distribution. The configurations of the K=4 over
N=2 distribution are listed below with their variances and diversity indices.
Configuration 
Microstates 
Variance, σ^{2} 
Diversity, D 
(4, 0) 
[4, 0], [0,4] 
4 
1 
(3, 1) 
[3, 1], [1, 3] 
1 
1.6 
(2, 2) 
[2, 2] 
0 
2 
Table 56. The Variance, σ^{2}, and Diversity, D, of the Configurations of the K=4 over N=2 Distribution
Recall now the average variance of σ^{2}_{AV}=1 of the K=4 over N=2 distribution from Eq38&45 and its average diversity of D_{AV}=1.6 from Eqs40&47. We see in the above table that the same values of a σ^{2}=1 variance and a D=1.6 diversity are seen for the (3, 1) configuration. On that basis the (3, 1) configuration is understood to be a compressed representation of all of the distribution’s configurations of (4, 0), (3, 1) and (2, 2) and as such is called the Average Configuration of the distribution. The Average Configuration is one configuration that represents all the configurations of a random distribution in compressed form much like the µ arithmetic average is one number that represents all the numbers in a number set in compressed form as, for example, the μ=K/N=4 arithmetic average does for all the numbers in the K=24, N=6, (6, 4, 2, 1, 5, 6), number set.
A configuration includes all of the permutations describable with the same number set, much as the (4, 0) configuration of the K=4 over N=2 distribution includes the permutations, (abcd, 0) and (0, abcd). Hence the Average Configuration should be understood as a compressed representation not only of all of a distributions configurations but also of all ω=N^{K }of its permutations as develop physically over time as the system’s microstates, each of which exists at any one moment in time. This exceedingly clear microstate picture of a thermodynamic system is worth taking a moment or two to sketch out. Recall the ω=16 permutations or microstates in Line21 for the K=4, N=2 distribution.
41.)
(abcd, 0); (abc, d); (abd, c); (adc, b); (bcd, a}; (ab, cd); (ac, bd); (ad, bc)
(bd, ac); (bc, ad); (cd, ab); (a, bcd); (b, acd}; (c, abd}; (d, abc}; (0,
abcd)
These should be understood as appearing in this proportion on average though not necessarily in this sequence over 16 moments of time as coming about from the random molecular collisions and transfers of energy in a thermodynamic system. As such, the (3, 1) Average Configuration represents the state of the system as measured over an extended period of time. Now if this microstate picture of a thermodynamic system is correct, the MaxwellBoltzmann energy distribution should be the average energy distribution of all the microstate configurations as manifest in the energy distribution of the Average Configuration.
The K=4 energy units over N=2 molecules distribution has too few K energy units and N molecules for its Average Configuration of (3, 1) to show any resemblance to the MaxwellBoltzmann energy distribution of Figure 32. Rather, we need random distributions with higher K and N values. And we will look at some starting with the K=12 energy units over N=6 molecule distribution. To find its Average Configuration we first calculate from Eq44 the σ^{2}_{AV} average variance of this distribution to be
57.)
The Average Configuration of the K=12 over N=6 distribution is a configuration that has this variance of σ^{2}_{AV} =1.667. The easiest way to find the Average Configuration is with a Microsoft Excel program that generates all the configurations of this distribution and then locates the one/s that has the same variance of σ^{2}=σ^{2}_{AV}=1.667. It turns out to be the (4, 3, 2, 2, 1, 0) configuration, taken to be the Average Configuration on the basis of its having as its variance, σ^{2}_{AV}=1.667. A plot of the number of energy units on a molecule vs. the number of its molecules that have that energy for this Average Configuration of (4, 3, 2, 2, 1, 0) is shown below.
Figure
58. Number of Energy Units per Molecule vs. the Number of Molecules Which
Have That Energy for the Average Configuration of the K=12 over N=6
Distribution
Seeing this distribution as the MaxwellBoltzmann energy distribution of Figure 32 is a bit of a stretch, though it might be characterized generously as an extremely simple choppy form of a MaxwellBoltzmann. Next let’s consider a larger K over N distribution, one of K=36 energy unit over N=10 molecules. Its σ^{2}_{AV} is from Eq44, σ^{2}_{AV}=3.24. The Microsoft Excel program runs through the configurations of this distribution to find one whose σ^{2} variance has the same value as σ^{2}_{AV} =3.24, namely, (1, 2, 2, 3, 3, 3, 4, 5, 6, 7). A plot of the energy distribution of this Average Configuration is
Figure
59. Number of Energy Units per Molecule vs. the Number of Molecules Which
Have That Energy for the Average Configuration of the K=36 over N=10
Distribution
This curve was greeted without prompting by Dr. John Hudson, Professor Emeritus of Materials Engineering at Rensselaer Polytechnic Institute and author of the graduate text, Thermodynamics of Surfaces, with, “It’s an obvious protoMaxwellBoltzmann.” Next we look at the K=40 energy unit over N=15 molecule distribution, whose σ^{2}_{AV} average variance is from Eq44, σ^{2}_{AV} =2.489. The Microsoft Excel program finds four configurations that have this diversity including (0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 6), which is an Average Configuration of the distribution on the basis of its σ^{2}=2.489 variance. A plot of its energy distribution is
Figure
60. Number of Energy Units per Molecule vs. the Number of Molecules Which
Have That Energy for the Average Configuration of the K=40 over N=15
Distribution
And next we look at the K=145 energy unit over N=30 molecule distribution whose average diversity is from Eq44, σ^{2}_{AV} =4.672. There are nine configurations with a σ^{2 }=4,672 including this natural number set of (0, 0, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 7, 7, 8, 8, 9, 10), which is an Average Configuration of the distribution on that basis. A plot of its energy distribution is
Figure
61. Number of Energy Units per Molecule vs. the Number of Molecules Which
Have That Energy for the Average Configuration of the K=145 over N=30
Distribution
At this level we are considering K and N values large enough to display a
moderately good resemblance to the classical MaxwellBoltzmann distribution of
Figure 32.
Figure 32.
As we progressively increase the K and N values of distributions, the plot of the energy per molecule versus the number of molecules that have that energy more and more approaches and eventually perfectly fits the shape of the above realistic MaxwellBoltzmann distribution. Now it should be clear that this unorthodox development of the MaxwellBoltzmann distribution as a property of the Average Configuration comes directly from the mathematics of a random distribution of distinguishable energy units and does not require any additional assumptions, which the Boltzmann derivation of the MaxwellBoltzmann decidedly does. As only one theory, mine or Boltzmann’s, can be correct because of the mutually contradicting assumptions for the two theories of distinguishable versus indistinguishable energy units, there is a strong argument in favor of mine from the Occam’s razor principle that is used generally in science to decide between two competing explanations on the basis of which has the fewest assumptions, in this case the diversity based explanation.
The other reinforcing argument for diversity based entropy comes about from the use of a diversity index slightly different than the D diversity index, one whose biased average perfectly fits microstate temperature and so makes the dimensional argument with the Clausius formulation of entropy air tight. We began our consideration of diversity based entropy with the D diversity index because of the mathematical regularities it has that made it easy to work with, but now that we have developed the basic concepts of diversity based entropy from D, we will switch our focus to Square Root Diversity Index, h.
62.)
This h diversity index not only provides a precise dimensional argument for diversity based entropy with the Clausius macroscopic entropy formulation, but also has, like D, a very high Pearson’s correlation to the Boltzmann microscopic entropy. As such it is the proper diversity underpinning of entropy. The p_{i }in h are the weight fraction measures of the x_{i} number of objects in each subset. The K=12 object, N=4 color, (■■■■■, ■■■, ■■■, ■), (5, 3, 3, 1), set, x_{1}=5, x_{2}=3, x_{3}=3 and x_{4}=1 has p_{i}=x_{i}/K weight fractions of p_{1}=x_{1}/K=5/12, p_{2}=x_{2}/K=3/12=1/4; p_{3}=1/4 and p_{4 }=1/12. This makes for an h square root diversity index of the set of
63.)
Note that the h diversity index is exact in being solely a function of p_{i}, which we made clear earlier is exact. Note also that the h=3.464 of the (■■■■■, ■■■, ■■■, ■), (5, 3, 3, 1), set compares well to its D=3.273 index in being, as is D, a reduction from the N=4 of this unbalanced set, though not quite as much as D=3.273 is. The two diversity indices, D and h, have comparable measures for the sets below with h=N=D for balanced sets and h < N for unbalanced sets as is D<N.
Set of Unit Objects 
Subset Values 
D, Eq3 
h, Eq62 
(■■■, ■■■, ■■■, ■■■) 
x_{1}=x_{2}=_{ }x_{3}=x_{4}=3 
4 
4 
(■■■■■, ■■■, ■■■, ■) 
x_{1}=5, x_{2}=_{ }x_{3}=3, x_{4}=1 
3.273 
3.468 
(■■■■■■, ■■■■■■) 
x_{1}=x_{2}=6 
2 
2 
(■■■■, ■■■■, ■■■■) 
x_{1}=x_{2}=_{ }x_{3}=4 
3 
3 
(■■■■■■, ■■■■■■, ■■■■■■■■■) 
x_{1}=x_{2}=6, _{ }x_{3}=9 
2.882 
2.941 
(■■■■■■, ■■■■■, ■) 
x_{1}=6, x_{2}=5,_{ }x_{3}=1 
2.323 
2.538 
(■■■■■■■■■■, ■■■■■■■■■■, ■) 
x_{1}=x_{2}=10,_{ }x_{3}=1 
2.194 
2.394 
Table 64. Various Sets and Their D and h Diversity Indices
Earlier back in Eqs2729 we developed an exact biased average for the D diversity of φ=K/D. We can also develop an exact biased average for the h square root diversity of Eq42 that we’ll call the Square Root Biased Average. In parallel to K/N=µ and K/D=φ, we define the square root biased average as K/h and give it the symbol, ψ, (psi).
65.)
The ψ=K/h biased average is an exact average in being a function of K, which is exact, and of h, which is also exact. The K=12, N=3, µ=K/N=4, h=2.538, (■■■■■■, ■■■■■, ■), (6, 5, 1), set has a square root biased average of ψ=K/h=12/2.538=4.72, greater than the arithmetic average of this set, µ=4, in being biased towards the larger x_{i} values in the (6, 5, 1) set. We detail the basis of this bias in the ψ average towards the larger x_{i} in a set by expressing ψ from Eqs3,8&62 as
66.)
The numerator in the rightmost term shows the ψ square root biased average to be the sum of “slices” of the x_{i} of a set of thickness p_{i}^{1/2}, which biases the average towards the larger x_{i} in the set in their having larger p_{i}^{1/2}. The ∑p_{i}^{1/2} term in the denominator of the end fraction is a normalizing function used to make all the p_{i}^{1/2 }“slices” in the numerator sum to one, this summing to one of the fractional “slices” being necessary for the construction of any kind of an average of a number set. We next invert Eq60 to express the h square root diversity index as a function of the φ square root biased average as
67.)
Now let’s understand ψ of Eqs65&66 as the microstate temperature of a thermodynamic system of K energy units distributed over N gas molecules. As we said earlier back after Eq31, the microstate temperature is currently understood in the standard physics rubric to be the arithmetic average energy per molecule, µ=K/N. And we made clear that what is wrong with that is that the K energy units of the system being distributed over the N molecules in an unbalanced way as seen in the MaxwellBoltzmann energy distribution of Figure 32 tells us that the this µ=K/N arithmetic energy average is an inexact specification because the N number of molecules parameter in µ=K/N is inexact. Earlier we suggested a replacement of inexact µ=K/N with the exact biased average, φ=K/D and we saw how doing that quickly developed a dimensional argument from (normalized, absolute) temperature being the φ=K/D exact biased energy average and S entropy being dimensionally from the dS=dQ/T Clausius entropy expression of S entropy deriving from the division of energy by temperature.
We can make the same dimensional argument for the square root biased average, ψ, as temperature and it is a substantially better one based on how temperature is actually measured physically with a thermometer. Each of the N molecules in the thermodynamic system collides with the thermometer to contribute to its temperature measure in direct proportion to its frequency of collision with the thermometer, which is equal to the velocity of the molecule, which itself is directly proportional to the square root of the x_{i} number of energy units a molecule has. Because of this, the slower moving molecules with the smaller energies in the MaxwellBoltzmann energy distribution of Figure 32 collide with the thermometer less frequently and have their energies recorded or sensed by the thermometer in its compilation of the temperature measure less frequently to record or sense smaller p_{i}^{1/2} slices of their energies. And conversely, the faster moving, higher energy molecules, which collide with the thermometer more frequently have their energies recorded or sensed as larger p_{i}^{1/2} slices of their energy thus biasing the temperature measure towards the energy of the higher energy molecules in an unarguable way. This determines the molecular energy average to be the square root biased average energy per molecule, ψ=K/h.
This first order model is incomplete, however, because it does not take into account the fact that the h energy diversity of a thermodynamic system along with its ψ=K/h biased energy average is changing at every moment in time from one microstate permutation to another as the system’s molecules collide with and transfer energy between themselves, thus continuously altering the distribution of the K energy units over the system’s N molecules. Hence the ψ and h parameters that form the basis respectively of the system’s temperature and entropy must be their average measures, h_{AV} and ψ_{AV}, which are properties also of the system’s Average Configuration that represents the system as a whole. This makes it clear by extension from Eq66 that the K total number of energy units divided by the ψ_{AV} square root biased energy average as temperature is the h_{AV} square root diversity of the system
68.)
Now it is only one quick stop at the Clausius macroscopic entropy formulation, dS=dQ/T, to see from a dimensional argument that h_{AV} diversity is the measure of the entropy of the system. As S entropy is dimensionally Q energy divided by T temperature and K energy divided by ψ_{AV} temperature is h_{AV} diversity, h_{AV} diversity must be dimensionally entropy.
To conclude that h_{AV} is the proper function for entropy based on its dimensional fit to the Clausius macroscopic entropy, we must also show that h_{AV} has a high correlation to Boltzmann’s S=k_{B}lnΩ entropy (or more simply, to lnΩ.) To demonstrate this, though, is not as straightforward as it was for D_{AV} because h_{AV} is not a simple function of the K energy units and N molecules of a thermodynamic system as D_{AV }was back in Eq46 as D_{AV}=KN/(K+N−1). There is a remedy for this problem, though. Because h_{AV} is the h diversity the Average Configuration much as D_{AV} was the D diversity index of the Average Configuration, we can obtain h_{AV} for the K over N distributions for which we have a specific Average Configuration and its x_{i }and p_{i} values. Those are the K over N distributions of Figures 5861. Below we list their h_{AV} values as calculated from Eq62 alongside the lnΩ values of those Average Configurations as calculated from Eq50. And we also include their D_{AV} diversity indices from Eq46 for comparison sake.
Figure 
K 
N 
lnW 
D_{AV} 
h_{AV} 
36 
12 
6 
8.73 
4.24 
4.57 
37 
36 
10 
18.3 
8 
8.85 
38 
45 
15 
26.1 
11.11 
12.33 
39 
145 
30 
75.88 
25 
26.49 
Table 69 The lnΩ, D_{AV} and h_{AV} of the Distributions in Figures 7376
The Pearson’s correlation coefficient between the lnΩ and h_{AV} values of the above is .995. And between lnΩ and D_{AV} it is .997. Note that though this lnΩ and D_{AV} Pearson’s correlation of .997 is high, it is less than the .9995 correlation between lnΩ and D_{AV} seen in Table 53 for larger K and N distributions. This is attributed to the Pearson’s correlation coefficient being a function of the magnitude of the K and N parameters of the random distributions, those of the distributions in Figures 5861 used in Table 69 being substantially smaller than the K and N of the distributions in Table 53. Hence the Pearson’s correlation between lnΩ and h_{AV }of .995 for the K over N distributions in Table 69 being little different than the .997 correlation between lnΩ and D_{AV} implies that the lnΩ and h_{AV }correlation is also, as was the .9995 between lnΩ and D_{AV }for larger K and N distributions, understood to be sufficiently great to have h_{AV} accepted as a candidate for entropy from its high correlation with the S=k_{B}lnΩ Boltzmann microstate entropy.
Now we will show how the h_{AV} diversity index replaces S in the 2^{nd} Law of Thermodynamics. The standard form of the 2^{nd} Law is
70.) ΔS > 0
This says that entropy, S, always increases in an irreversible or spontaneous process. One such process the 2^{nd} Law applies to is thermal equilibration. In it two bodies at different temperatures both go to some intermediate temperature upon thermal contact. While the mathematics of this is unarguable when entropy is expressed in dS=dQ/T form, any sense of the increase in entropy microscopically or molecularly that can be gleaned from representing entropy with the Boltzmann S entropy is unintuitive and conceptually mysterious. And that, we posit, is not because entropy is inherently difficult to understand or mysterious, but because S=k_{B}lnΩ is incorrect and must be replaced with h_{AV} diversity to make any sense out of the process intuitively.
To show this we will demonstrate the h_{AV} diversity based entropy increasing in a thermal equilibration between two “minithermodynamic” subsystems. Subsystem A has K_{A}=12 energy units distributed randomly over N_{A}=3 molecules. And subsystem B has K_{B}=84 energy units distributed randomly over N_{B} =3 molecules. These two subsystems are initially isolated out of thermal contact with each other. From Eq44 the average variance of the K_{A}=12 energy units over N_{A}=3 molecules subsystem is σ^{2}_{AV }=2.667, which calculates an Average Configuration for the subsystem, which has that variance, of (6, 4, 2) along with a normalized microstate temperature of the subsystem from Eq66 of ψ_{AV}(A)=4.353. And the K_{A}=84 energy units over N_{B}=3 molecules subsystem B has from its Eq44 average variance of σ^{2}_{AV}=18.667 an Average Configuration of (34, 26, 24) with a normalized microstate temperature from Eq66 of ψ_{AV}(B)_{ }=28.328.
Upon thermal contact the system, now comprised of the two subsystems as one whole system, consists of N=N_{A}+N_{B}=6 molecules over which are distributed K=K_{A}+K_{B}=96 energy units. At the first moment of contact, we represent the whole system as a composite of their separate Average Configurations, to wit as (6, 4, 2, 34, 26, 24). At this first moment there is no ψ_{AV} temperature of the composite system because it is not in thermal equilibrium. But it can be understood to have a square root diversity index of h_{AV}=4.394 from Eq62. This specifying of the composite system not in equilibrium as h_{AV}, as an average diversity, is awkward in (6, 4, 2, 34, 26, 24) being made up of the average h diversities of the two subsystems. But the meaning of h_{AV} is clear here despite the (6, 4, 2, 34, 26, 24) set being made up of the h_{AV} of the subsystems.
After molecular collisions sufficient to bring about a random distribution of the K=96 energy units over the N=6 molecules, the Average Configuration as obtained from the σ^{2}_{AV }=13.333 average variance of Eq44 is (11, 14, 15, 16, 17, 23). It has a square root diversity index from Eq62 of h_{AV}=5.85 and a normalized microstate temperature from Eq66 of ψ_{AV}=16.409.
Note that the usual computation of temperature of the whole system from the 1^{st} Law of Thermodynamics, an energy conservation law, suggests a temperature that is the simple average of the temperature of the two subsystem’s, which would be of the ψ_{AV }normalized microstate temperatures, (4.353 + 28.328)/2 =16.341. The discrepancy between this value of 16.341and ψ_{AV}=16.409 calculated from Eqs62&66 is not a violation of energy conservation because temperature from our unorthodox diversity based perspective is understood as an average molecular energy biased toward the higher energy molecules.
What is important to demonstrate here is that the h_{AV} energy diversity or energy dispersal understood as entropy increases upon thermal contact from an initial value of h_{AV}=4.39 for (6, 4, 2, 34, 26, 24) to a final value of h_{AV}=5.85 for (11, 14, 15, 16, 17, 23). The change in h_{AV} energy diversity is, hence,
71.) Δh_{AV}=5.85 – 4.39 = +1.46
This fits the increase in entropy for thermal equilibration demanded by the 2^{nd} Law of Thermodynamics with entropy now expressed as h_{AV} energy diversity.
72.) Δh_{AV}> 0
There are two things that are different about this unorthodox manifestation of the 2^{nd} Law entropy increase for thermal equilibration. The first is that the entropy increase expressed in terms of Δh_{AV }=1.46 is measured as a change in the whole system of N=6 molecules. And the second is that what is happening physically in the thermal equilibration process is very clear intuitively when the entropy increase is understood as an increase in energy diversity or dispersal. Indeed, nothing could be clearer intuitively especially by comparison to the standard take on entropy increase as an increase in the Ω microstates of the system, which makes zero sense out of entropy as a physical quantity. This diversity based entropy change quantitatively fits the sense of entropy as energy dispersal, (See Wikipedia), which though taken by most scientists to be the qualitatively sensible interpretation of entropy, has never been given a firm quantitative basis until now.
There are other major improvements in thermodynamics that come about from this diversity based statistical mechanics as in a clearer understanding of free energy and of a real gas law in terms of diversity. We will not detail these and other improvements in thermodynamic theory that a diversity based entropy brings about, leaving that to specialists in the field who have sense enough to expand on our seminal work in the detail it warrants.
6. Entropy and Information
Two concepts that derive from the development of diversity based entropy have relevance to the information processing operations of the mind. The first is that the h diversity index of Eq62, like the D diversity index of Eq3, is understandable as a measure of the number of significant subsets in a set. In the (■■■■■■■■■■, ■■■■■■■■■■, ■), (10, 10, 1), set of Table 25 and Figure 26 it was seen that the D=2.19 diversity index of the set could be interpreted, when rounded off, as the set having D=2 significant subsets, the x_{1}=x_{2}=10 red and the green subsets, with the x_{3}=1 purple subset understood as insignificant. This interpretation of the D=2.19 measure was reinforced with significance indices for it from Eq21 of s_{1}=1.04≈1 for the red subset, s_{2}=1.04≈1 for the green subset and s_{3}=.104≈0 for the purple subset. The h diversity index of this set, h=2.394 from Eq62, rounded off to h=2, can also be understood as specifying 2 significant subsets in the set.
This has us interpret h_{AV} diversity based entropy as the number of energetically significant molecules in a thermodynamic system, that is, in its Average Configuration or equivalently as the average number of them in the system. For example, consider the K=36 energy units over N=10 molecules random distribution of Figure 59 as its (1, 2, 2, 3, 3, 3, 4, 5, 6, 7) Average Configuration and its h_{AV} from Eq62 of h=8.853≈9. This is readily interpreted as the system having 9 energetically significant molecules, the molecule with just 1 energy unit being insignificant energetically. This sense of the energetic significance versus insignificance of molecules goes a long way to understanding temperature specified as ψ_{AV}=K/h_{AV} as coming about from molecular collision with the thermometer that is biased towards the faster moving, more energetically significant molecules. That, in turn, makes clear that the “reality” of a thermodynamic system as manifest in its most basic property of temperature as ψ_{AV}=K/h_{AV} depends not just on the molecular energies of the system but also on how the molecular energies are tallied in the temperature measure in a biased way.
This also makes clear by parallel the human mind’s appreciation of significance in its sensory operations as measured by the D diversity index. Ultimately this parallel derives from the commonality between thermodynamic and sensory systems in the measures of both being affected by the magnitude of their subset constituents, be it of the energy of molecules measured in a biased way by a thermometer or of the size of color subsets sensed by the CNS, the central nervous system.
And this helps make clear that our perceptions and the thoughts we developed from them as used to guide our behaviors and communication with others depend not just on what it is that objectively exists out there but also on how they are sensed or measured with our CNS sensing apparatus. Hence understanding entropy as diversity in physical systems as diversity strongly validates the reasonableness of the concept of significance in our sensory apparatus, which tells us that reality for people is what is sensed or measured rather than some totally objective phenomena that lies outside our senses. From a purely epistemological perspective, then, this greatly calls into question transcendental notions like gods and angels and devils that have absolutely no basis in anybody’s sense of them as these items claimed to exist somehow have never been sensed. This is very important to developing a clear picture of what is meaningful in life for plaguing our sense of life with nonsensed imaginations muddies the picture critically.
Significance is one major determinant of what makes information meaningful. Later we will make it clear that the other determinant of the meaningfulness of things is the association of emotion with them major, something diversity will also present a beautiful picture of by enabling a representation of our basic emotions of fear, hope, excitement, relief, sex, love, warmth, anger and the like with great mathematical precision in a later section.
The other concept basic to the mind’s information processing introduced through diversity based entropy is compressed representation. The µ mean is the most familiar and commonly used compressed representation. Specifying the number of objects in a subset of the K=12, N=4 set of (■■■■■, ■■■, ■■■, ■), (5, 3, 3, 1), as its µ=K/N=12/4=3 arithmetic average compresses the number of numbers needed to describe the set from N=4 of them, (5, 3,3, 1) to the one number of µ=3. While that constitutes efficiency in description in using less information to describe the set, note that the loss of some information in this compression generates error as seen in the statistical error associated with the arithmetic average of an unbalanced set. This compressed representation of the arithmetic average is also inexact as we made clear earlier.
Another compressed representation of the (■■■■■, ■■■, ■■■, ■), (5, 3, 3, 1) is its D=3.273 diversity index. Again, like the arithmetic average, this is a 1 number representation of an N=4 number, number set and an efficient compression in that regard. It also leaves out information about the set but less so than the set’s µ=3 arithmetic average because D includes a measure of the distribution of the K objects over the N=4 subsets as is clear from D=N/(1+r^{2}) of Eq16 inclusion of r^{2} as a measure of set distribution. And D=3.273 is also an exact compression of the set as is the h=3.468 square root diversity index of the set.
The h_{AV} average diversity or entropy of a thermodynamic system is a higher level compressed representation in representing the h diversity compressed reductions of all of the microstate permutation of the system as their average. And the ψ_{AV}=K/h_{AV} average square root biased average is another higher level compressed representation as the average over time of the square root biased average energy per molecule that changes over time from continuous collisions between molecules.
We can appreciate these mathematically formulated compressed representations as one category of the generalizations we make about the world around us, quantitative generalizations. The usefulness of these compressed representations or quantitative generalizations is obvious enough that we need not enumerate them.
These quantitative compressed representations also shed great light on the nonmathematical generalizations we humans use as compressed representations that range from the common nouns and verbs we use to represent objects and actions with to our generalized knowledge of complex processes of all sorts. In the simpler case of common nouns as compressed representations, note that the word “dog” conjures up a picture in the mind of a person who hears the word that is an average or morph of all the dogs a person has come across including in picture books and movies seen. The mind compresses everything it comes into sensory contact with in its memory of those things. Such compressed information from the past is used in the interplay of emotions and of thought manifest as generalization, which very much affects both the things we decide to do and our communications with others.
This section has been a relatively qualitative discussion of two important information concepts gleaned from our development of diversity based entropy: significance and compressed representation. In the next section we will use the diversity concept to more formally revise information theory.
7. Revising Information Theory
To do that, let’s start back with the central function for information in information theory, the Shannon (information) entropy of Eq1.
1.)
Information theory was developed in 1948 by Claude Shannon to characterize messages sent from a source to a destination. Consider (■■■, ■■■, ■■■, ■■■) as a set of K=12 colored buttons in a bag in N=4 colors. I’m going to pick one of the buttons blindly and then send a message of the color picked to some destination. The probability of any color of the N=4 colors being picked is the p_{i} weight fraction of the color, for all the colors in this case,
73.) p_{i }= 1/N = 1/4.
So there’s a p_{1}=1/4 probability of my sending a message saying “I picked red.” And a p_{2}=1/4 probability of my message saying, “I picked green,” and so on. Plugging these p_{i}=1/4 probabilities into messy Eq71 obtains the amount of information in the color message sent as
74.)
This tells us that there’s H=2 bits of information in a message sent. What does that mean? The most basic interpretation of the H=2 bits is as the number of binary digits, 0s or 1s, minimally needed to encode the color messages gotten from (■■■, ■■■, ■■■, ■■■) in bit signal form, namely as [00, 01, 10, 11]. Red might be encoded as 00, green as 01, and so on. Then when the receiver of the message gets 00 sent, he decodes it back to red. The H=2 bits measure is considered to be the amount of information in a message as the number of bit symbols in each bit signal. This bit signal information is the synthetic or digital information that computers run on. There is a simpler form of the Shannon information of Eq1 used for balanced or equiprobable sets like (■■■, ■■■, ■■■, ■■■). Because the p_{i} probabilities are all the same for a balanced set as p_{i}=1/N, substituting 1/N for p_{i} in Eq1 derives the simpler form for H of
75.) H= log_{2}N
This equation gets us the same H=2 bits result for messages sourced from (■■■, ■■■, ■■■, ■■■) as did Eq74, but in a simpler way as H= log_{2}N = log_{2}4 = 2 bits. Now let’s also use Eq75 to calculate the amount of information in a message that derives from a random pick of one of K=16 buttons in N=8 colors, (■■, ■■, ■■, ■■, ■■, ■■, ■■, ■■). Because this set is balanced, the probability of picking a particular color and sending a message about it is the same for all N=8 colors, p_{i}=1/N=1/8. And the amount of information in a color message from this set can be calculated from the simple, equiprobable, form of the Shannon entropy of Eq75 as H= log_{2}N= log_{2}8= 3 bits. This has us encode messages from N=8 color (■■, ■■, ■■, ■■, ■■, ■■, ■■, ■■) with the N=8 bit signals, [000, 010, 100, 001, 110, 101, 011, 111]. Each bit signal has H=log_{2}8=3 bits in it understood as the amount of information in a color message derived from this set.
Now we want to make the case that information and diversity are synonymous with each being a measure of the other. We already tried to make that case before by suggesting that the D diversity interpreted as the number of significant subsets in a set was an instance of meaningful information. The synonymy of diversity and information can also be demonstrated in a more technical way. First of all it is well known that the H Shannon information entropy expressed in natural log terms is the Shannon Diversity Index used over the last 60 years in the scientific literature as a measure of ecological and sociological diversity. Paralleling Eq1 for the Shannon information entropy is the Shannon Diversity Index of
76.)
And for a balanced set, paralleling Eq75, the Shannon Diversity index is H = lnN. The difference between the Shannon entropy as information and the Shannon entropy as diversity is merely the difference in logarithm base, the direct proportionality between the two telling us from measure theory in mathematics that what one function is the measure of, the other must also be a measure of, here both of information and diversity. Another conceptual equivalence between diversity and information that derives from classical information theory comes from Renyi entropy, R, which is taken in information theory to be information in being a parent function or generalization of the Shannon information entropy. Its connection to diversity lies in its being the logarithm of the D Simpson’s Reciprocal Diversity Index as we saw earlier in Eqs24.
4.) R = logD
This also strongly suggests a synonymy between diversity and information. The above two diversityinformation associations suggest two kinds of diversity indices that further imply two kinds of information functions. The two kinds of diversity indices are the logarithmic kind, as with H and R; and the linear kind, as with D. To better understand the two kinds of information that the two kinds of diversity indices, logarithmic and linear, imply we next develop the D (linear) diversity index as a bit encoding recipe that parallels H as the bit encoding recipe we introduced it as. The sociopolitical implications of this somewhat tedious exercise are profound and make the following technical considerations worth our time and effort.
Recall the H=2 bits for (■■■, ■■■, ■■■, ■■■) that specify for its N=4 color messages an encoding of N=4 bit signals, [00, 01, 10, 11], each consisting of H=2 bits or binary digits. We can also use the D=4 diversity index as a bit encoding recipe. The D=4 diversity index of this set translated as the number of bits in a bit signal obtains N=4 bit signals for the N=4 colors of the set of [0001, 0011, 0111, 1111], each of which consists of D=4 bits. And for the N=8 color set of (■■, ■■, ■■, ■■, ■■, ■■, ■■, ■■) whose H=3 bits measure encoded it as [000, 001, 010, 100, 110, 101, 011, 111], the D=8 diversity index used as a coding recipe encodes it as [00000001, 00000011, 00000111, 00001111, 00011111, 00111111, 01111111, 11111111], with each bit signal consisting of D=8 bits. Note in both D encodings that only one permutation of a given combination of 1s and 0s can be used. This restricts us writing the 20s and 61s combination of bits in only one permutation of it, as for example as 01111011 or as 00111111, but not both. And also note that the all 0s bit signal is disallowed in this D encoding recipe.
Anyone familiar with information theory will immediately note that the D bit recipe is inefficient as a practical coding scheme in its requiring significantly more bit symbols for a message than the H Shannon entropy coding recipe. This is not surprising since Claude Shannon devised his H entropy initially strictly as an efficient coding recipe for generating the minimum number of bit symbols needed to encode a message in bit signal form. The D diversity index as a coding recipe fails miserably at that task of bit symbol minimization. But we have developed it not trying to engineer a practical coding system but rather to show how D can be understood in parallel to H as an information function in being understandable as a bit coding recipe, its efficiency for message transmission being quite beside the point.
We show D to be an information function for a very familiar kind of information, quantitative information, by next looking carefully at the details of the difference between the H and D bit encodings. Recall the (■■■, ■■■, ■■■, ■■■) set, whose N=4 colors are encoded in H encoding with [00, 01, 10, 11] and in D encoding with [0001, 0011, 0111, 1111]. Now look closely to see that these are two very different ways of encoding the N=4 distinguishable color messages derived from (■■■, ■■■, ■■■, ■■■) with N=4 distinguishable bit signals. What is special about the D bit encoding of (■■■, ■■■, ■■■, ■■■) with [0001, 0011, 0111, 1111] is that all of these N=4 bit signals are quantitatively distinguishable from each other with each bit signal having a different number of 0s and 1s in it than the others.
This is not the case for the H encoding of (■■■, ■■■, ■■■, ■■■) with [00, 01, 10, 11]. For with them it is seen that the 01 and 10 signals have the same number of 0s and 1s in them and, hence, are not quantitatively distinct from each other. Rather the distinction between 01 and 10 is positional distinction from the 0 and 1 bit signals being in different positions in 01 and 10. So 01 and 10, we could say, are qualitatively distinct rather than quantitatively distinct.
This quantitative versus qualitative distinction for D and H encoding is even more clear for the N=8 set, (■■, ■■, ■■, ■■, ■■, ■■, ■■, ■■), and its D=8 bit encoding of it as [00000001, 00000011, 00000111, 00001111, 00011111, 00111111, 01111111, 11111111]. For there we see that every one of the N=8 bit signals is quantitatively distinguished from every other bit signal in each having a different number of 0s and 1s in them. This quantitatively distinguishable bit encoding with D contrasts to the H=3 bit encoding of that color message set as [000, 001, 010, 100, 110, 101, 011, 111] in which we see that the 001, 010 and 100 signals are not quantitatively distinguished, each of them having 20s and 11, but rather distinguished entirely by the positions of the 1 and 0 bits in them. And that positional or qualitative distinction is also seen between the 011, 101 and 110 signals of its H bit encoding, all of which are quantitatively the same rather than quantitatively distinct.
The qualitative versus quantitative H versus D encodings corresponds to our everyday sense of information as being either of two broad kinds, qualitative or quantitative. When I tell you General George Washington worked his Virginia planation with slaves rather than hired help, that’s qualitative information for you. But when I tell you that Washington owned 123 slaves at the time of his death, that’s quantitative information. With our example set of (■■■, ■■■, ■■■, ■■■) we see that the color subsets are all qualitatively distinct from each other as is well represented with their [00, 01, 10, 11] H bit encoding. It is also clear, though, that there are N=D=4 color subsets, which is well denoted with [0001, 0011, 0111, 1111], which distinguishes them as the 1^{st} color, the 2^{nd} color, the 3^{rd} color and the 4^{th} color, which effectively counts the number of colors.
This qualitative versus quantitative differentiation explains why the H, the qualitative coding recipe is logarithmic and why the D quantitative coding recipe is linear. H is logarithmic because it is a coding recipe for information communicated from one person to another. The human mind distinguishes intuitively between the positions of things as between 20s and 11 arranged as 001 or 010 in different positions. This property of mind allows us to represent distinguishable messages sent from one person to another, like ■ and ■, encoded with signals distinguished via positional or qualitative distinction like 001 and 010. Because the N number of distinguishable messages that can be constructed from H variously permuted, variously positioned, bit symbols is determined by N=2^{H}, a power function, the information in one of those messages specified as the H number of bits in each bit signal is inherently logarithmic via the inversion of N=2^{H} as H=log_{2}N.
Compare this to D=4 encoding of the N=4 colors in (■■■, ■■■, ■■■, ■■■) as [0001, 0011, 0111, 1111]. This D encoding recipe encodes the colors via the number of 1s in the bit signals or effectively with ordinal numbers that encode the colors as the 1^{st} color, the 2^{nd} color, the 3^{rd} color and the 4^{th} color, which is most basically just a count of the number of distinguishable colors, clearly quantitative information about them. D is linear because counting is inherently linear as with 1, 2, 3, 4 and so on. While much information transmitted or communicated from one person to another is qualitative in form as suits practical efficiency, information sourced directly from nature is quantitative in form when it is a precise description of nature as every practitioner of physical sciences understands. As such an encoding of such quantitative information from nature should be most basically linear in form rather than logarithmic as is the h diversity encoding of the number of energetically significant molecules in a thermodynamic system.
The above development of quantitative versus qualitative information provides the broadest understanding of information as diversity. That includes logarithmic diversity for information communicated from person to person as the H Shannon information entropy also understandable as the Shannon Diversity Index provides; and linear diversity, linear in form as D or h. Take careful note that quantitative descriptions of items can also be represented and communicated via positional distinctions as seen in the Arabic numerals that write thirteen as 13 rather than 1111111111111 for efficiency sake, with 13 distinct position wise from thirtyone as 31. But that should not take away from the reality of the elemental linear nature of counting and, hence, of science’s distinguishing things quantitatively in the linear rather than logarithmic form the D and h diversity indices have.
This argument that information can be logarithmic or linear in form runs sharply counter to the notion by many information theorists that that the only proper form of information is from the Khinchin derivation of it, the Shannon information entropy, which is logarithmic. That this rigid perspective, which makes impossible an understanding of meaningful information, is nonsense is easy to show in the Khinchin derivation of information as the Shannon entropy defines information to begin with, in a narrow way that obviates our familiar sense of what information is, and then proceeds to derive information as it specified information must be in its axiom set. This approach is a prime example of the value of Gödel’s incompleteness theorem, which considers all axiomatic schemata invalid for that very reason of any conclusions one wishes to attain being achievable via a biased selection of the axiomatic underpinning of the argument to fit the conclusions.
Recognizing this and allowing the D diversity index to be understood as a measure of quantitative information allows us to understand one cornerstone of meaningful information to be information that is judged significant by the human mind rather than insignificant. In a nutshell: significant information is meaningful information. And D diversity also allows is to develop the other cornerstone of meaningful information and that is information associated with emotion, the central topic of the section following this next one, which deals with significance as it affects people.
8. The Significance of Individuals
Much of what people do and think is affected by their sense of themselves as being significant or insignificant individuals. This is important for our understanding corruption in social institutions as deriving from an individual’s selfinterest being greater than his or her commitment to the institution. That is, we can attribute corruption to the great need of an individual to feel significant rather than insignificant, which is generally very difficult to achieve in a hierarchically ordered, exploitive, society. This drive is a determinant not only in the Wall St. corruption that jiggled the mortgage market and caused the 2008 recession that near destroyed many American families, but also in the unspoken judicial, political and medical corruption that abounds in today’s America. And such corruption extends to an academic community whose selfinterest in status and position tends to trump considerations of truth. This condemnation of academia is so harsh and so difficult to make stick that we approach this problem of people’s motives to cheat with mathematical analysis.
Social dominance, malevolent and benevolent, is a universal reality however much the concept is unspoken because it abrades on the hope and promise of freedom and fairness. In general the dominant individual is thought to be significant by himself and others and the subordinate person, insignificant. Dominance in a two person relationship, the simplest to analyze, depends ultimately on the relative probabilities of success in competition between the two whether the competition be explicit as in contact sports or implicit as in the track and field, “I can do this better than you can” type. The characteristics of explicit competition are clearer and, hence, is easiest to consider first.
Consider the random selection of an object from the K=10, N=2 color set, (■■■■■, ■■■■■), x_{i}=5 and x_{2}=5, p_{1}=1/2 and p_{2}=1/2, where the p_{1} and p_{2 }weight fractions of the set are the probabilities respectively of red and green being picked. If red is picked, the player assigned red gets $100 from the player assigned green, and vice versa. And in contrast to this balanced game, consider it played with the (■■■■■■, ■■■■) set, x_{i}=6 and x_{2}=4, p_{1}=6/10 and p_{2}=4/10, in which the red player has an edge over the green player of
77.) Δp = p_{1}−p_{2 }=6/10 – 4/10 = 2/10 = 20% =.2
This Δp=.2 measure is also understandable as the vulnerability of the green player. Assume now that this “unfair game must be played by the green player unless he or she opts out of playing by paying the red player $25. Of course, it makes little sense financially for the green player to pay $25 to avoid the game because the average loss is less than that as Δp=.2($100)=$20 per game. Now let’s change the game further to one where the color is picked randomly from (■■■■■■■■■, ■), x_{i}=9 and x_{2}=1, p_{1}=9/10 and p_{2}=1/10. The edge for the red player has increased now to
78.) Δp = p_{1}−p_{2 }=9/10 – 1/10 = 8/10 = 80% =.8
In this game it makes great sense for the green player to pay $25 to avoid the game because the average loss is greater than $25 at Δp=.8($100)=$80 per game. This doesn’t completely obviate the possibility that the green player might get lucky if he plays and win, thereby getting $100 instead of losing the $25 by forfeit without any attempt to win. But generally speaking in any kind of competition that has this form of two players having very different probabilities of winning, past some critical Δp value the additional cost for losing after playing the game and trying to win gets the inferior player to capitulate to the superior player and accept the lesser penalty without a fight.
If a game with lopsided probabilities were played frequently as part of the ongoing relationship between the two players, this level of control and exploitation would instinctively make the inferior player feel insignificant. This emotional outcome of the lopsided relationship has firm mathematical expression in the D diversity index as expressed as D=N/(1+r^{2}) in Eq16 when its r^{2} relative error term is given in terms of the weight fraction probabilities as
79.)
This expression for r^{2} is easily derived
from previous functions we’ve considered but is just simply demonstrated here
with an example for expediency sake. Consider the K=9, N=3 natural number set,
(4, 4, 1), which from Eqs2,3&4 has a relative error of r=.471 and r^{2}=.222=2/9.
With p_{i} = (4/9, 4/9, 1/9), the r^{2} statistical error of
(4, 4, 1) is calculated from Eq79 as
80.)
This demonstrates the validity of Eq79 for r^{2}. For an N=2 set of relative probabilities of success of two persons in a competition of p_{1} and p_{2}, the r^{2} term is obtained from Eq79 as
81.)
This develops the number of significant people in the N=2 person relationship from D=N/(1+r^{2}) as
82.)
When there is perfect balance in the competition and, hence, in the relationship, p_{1}=p_{2}1/2 and r^{2}= (Δp)^{2}=0, as brings about D=2, which indicates that there are 2 significant people in the relationship. In such both persons tend to think of themselves and of the other person as significant, a positive or pleasant feeling both in terms of what you think of yourself and what the other person thinks of you. In competitions between N=2 persons where the probabilities of winning are p_{1}=.9 and p_{2}=.1, the D diversity is
83.)
Rounding off D=1.22 to D=1 indicates that there is but 1 significant person in the relationship, the person with the large p_{1}=.9 probability of winning, the other person, the one with the slight p_{2}=.1 chance of winning, being insignificant in the relationship. That is, the mathematically insignificant person feels insignificant and is also thought to be such by the dominant person. There are other factors in a relationship that mitigate the displeasure of being the insignificant partner in a relationship, it must be stressed. A child inherently thinks of its adult parent as the “big person”, the significant one, and is yet quite happy with a parent who gives love as makes up for this lesser and unequal role in the relationship. But this changes necessarily as the child matures and seeks to develop its own sense of significance. Indeed this mathematical specification of personal significance is a manifestation of a person’s “ego” or sense of self as propounded by Sigmund Freud. The mathematics of the factors of caring for another less able individual, as a child is relative to an adult parent objectively, must wait until we develop functions for the emotions in later sections, some of which provide balance for the generally unpleasant feelings of being insignificant or inferior.
Until then we will understand insignificance as a generally unpleasant enough feeling that it causes people to prefer to be significant as an equal or as dominant rather than to be insignificant. Let us repeat for emphasis that there are other factors involved than just competition in relationships and we will get to those in due time. In the meantime we will generalize that people prefer having power than not having power. Nobody who plays an explicit competitive game prefers anything other than winning for this reason.
Tom Brady of football fame from his success in the game feels significant and is thought of as significant, while the fellow on the losing end in games and in scrimmages who soon gets cut from the roster feels insignificant and is thought of as such by others in and out of his profession, (indeed, as insignificant, usually not thought of at all.) We need not dwell at length on the obvious rewards of being and feeling significant. Hence, the question to be asked is whether Tom Brady would cheat to win and be significant and enjoy the rewards of being significant? Would Tom deflate the footballs to be significant? This is not asking the question of whether or not he actually did. We know Tom Brady personally and can vouch for the fact that he’s not that kind of a guy. But maybe somebody other than Tom Brady would deflate the footballs to achieve Brady’s significance. That’s the point. Would somebody do that or something like that if he or she could get away with it?
The payoff for winning, for having a high p probability of winning, for having the edge in competition and being significant, is so high and the cost of having a low p probability of winning, of being vulnerable to loss and being insignificant, is so great emotionally, that in many situations, people who can get away with corruption, with cheating on the rules, just do it. Moreover it is also the case that such corrupt behavior is never openly confessed to or in any way revealed because doing so kills the “getting away with it” factor in making a deal with the devil.
For that reason, there’s lots of corruption and not much open talking about it. Really does any even halfintelligent person think that no Wall Streeter going to jail, not even for a day, after six years of investigation of this multibillion dollar scam by the Justice Dept. is anything but a manifestation of consummate business, judicial and political corruption? Even though nobody with a public voice ever says, in contradiction to the Orwellian doublespeak reasons why there was no prosecution of this grand theft by Wall Street banks on tens of millions of Americans, that our social system is thoroughly corrupt, is it not apparent on its face?
Indeed capitalism is corrupt. All civilized social systems are inherently corrupt. The drive to be on top, to be significant, to avoid insignificance, is a powerful incentive to break any code of fairness held up as the moral norm in a society. Even heads of state in a military dictatorship never tell their underling citizens that the game is corrupt and unfair. Be sure that Hosni Mubarak of Egypt, 30 years the military dictator of that vassal state of the USA, a fellow who scuttled billions for himself and his family through systematic corruption, never went on Egyptian TV before the Arab Spring to tell the Egyptian peasants that he was robbing and fucking them blind.
All power systems work this way including our capitalism, which never buys two hours of airtime on TV to proclaim to the American people that it is robbing them blind. The only difference between military dictatorship and capitalism is how the edge is obtained, not whether it exists and is used for the benefit and privilege of those who have the edge. Buying and selling is inherently corrupt and deceitful. The game is to buy cheap and sell dear. The seller is always out to tell the buyer anything he has to in order to extract the maximum amount of money out of the buyer. In the small, as hagglers over price in a New Delhi marketplace know, it is part of the game of the seller to lie in order to get the maximum cash out of the buyer. Institutional corruption comes in when the political system and the judicial system joins in the game against the rules in order to tolerate the inherent corruption and deceit of the marketplace.
But these instances of corruption get way ahead of the game and require much more analysis for their picture to be drawn our fully. What we want to talk about in this section in detail, rather, is the corruption in the marketplace of analytical ideas, that makes it impossible in the end to honestly talk about and uncover the corruption in the broader society that will soon be taking us all to nuclear hell because of the stupidity and lack of foresight from leaders who attained their power through cleverness in becoming socially significant by maximum capability at corruption and deceit rather than really being smart. Beyond the exceptions like Bill Gates, who is too much of a coward and a pussy to enter the tussle of the political and economic arenas where real people suffer daily, the entrepreneur is nothing but a clever thief.
In academics, as in all other professions, there is a hierarchy of power and as in all other hierarchies, it is corrupt as long as it can get away with being corrupt as requires centrally hiding the corruption. As in all areas of corruption, people maintain their p probability of being successful by helping to enable other people’s p probability of being successful. That is, in colloquial language, you scratch my back and I scratch yours. The temptation to do this in obviation of the rules is very strong because the penalty for not doing it is frequently that one winds up with little p probability of winning in competition and of being insignificant, while the prize for “playing the game” is attaining some significance, which feels much better than being insignificant.
In academics, just as in every profession, some people do the hiring and firing, the awarding of cash grants to do research and the editing of journals where research papers are published as enable position and grant support. Most often the people on the top of this game are “experts” in the field. Or better said they are recognized as experts by others in the field, which gets you right back to the “you scratch my back, and I’ll scratch yours” game. Of course, playing that game with the devil is lost if one advertises the fact that one is an academic entrepreneur. One most amusing instance of it in recent times was the Complexity Theory bullshit engineered by Stuart Kauffman whose debunking was the primary focus of the Scientific American article by John Horgan, From Complexity to Perplexity, I brought up at the beginning of A Theory of Epsilon.
Unlike my quip about Tom Brady, we actually do very much know Stuart Kauffman personally. And it is interesting that long before we read Horgan’s article ridiculing complexity theory, we had a sense after an hour’s long lunch with Kauffman from the obvious lack of clarity in his doubletalk mathematics that he was as slick a charlatan as Bernie Madoff. For what Kaufmann did was to spin a mathematical argument so complex about complexity that it was near impossible to take apart, for a while anyway. Before it was taken apart publically and Kaufmann chased off to Canada to talk his nonsense to the more naïve Canuck academics he enjoyed great significance in association with the Santa Fe Institute and even wound up a McArthur grant winner.
The corruption in academics is not as obvious in most cases as Kaufmann’s carny game. In our case at hand it has consisted of “experts” in the field of thermodynamics being unwilling to admit that the material they claim to be experts at is incorrect. Though we have run into a good number of them over the years in various universities, the attitude of Bill Poirier stands out. Note his comments to us about our revision of microstate entropy.
In short, though the Gibbs and Boltzmann Shannonlike formulations of entropy have their limitations/issues, there is nothing really mathematically "wrong" about themthey are what they claim to be, within wellknown caveats. Conversely, this is not to say that your approach is "wrong" or otherwise without value; as I said in an earlier email, there may well be more than one useful quantity associated with the same general concept. But I would be wary of making claims that classical entropy is "fundamentally incorrect", and that your approach "provides the only correct understanding of microstate entropy."
Poirier is alluding to both the BoltzmannShannon take on entropy and our meaningful information derivation of it being mathematically equivalent and both correct in that regard. Yet despite the “limitations/issues” with the standard formulation that Poirier cites, which have interminably confused students and professionals alike for the last 100 years, he still favors the standard, perplexing take on entropy unable to shake the inferior explanation he has grown to accept over the years despite its obvious shortcomings. Proof that Poirier is dead wrong lies in his insistence in Chapter 10 his recent book that entropy can be explained from information theory despite the general understanding in the scientific community conveyed in the above Scientific American quote that information theory, as it presently stands, cannot be applied to explaining physical systems. Our extension and elaboration of information theory to include meaningful or significant information as it is found both in physical and in human nature makes the intimate association between entropy and information clear enough that even a high school chemistry student can understand entropy now.
We also mention lightly the pettiness and stupidity of today’s scientists in clinging to orthodoxy for the sake of retaining the crown of “expert” and the position, status and pay that go along with it. A perfect example of this is of our front page poster boy, Bill Poirier of Texas Tech in Lubbock. Perusal of Chapter 10 in his recent book, A Conceptual Guide to Thermodynamics shows a frivolous notion of an information theory interpretation of thermodynamic entropy. It can be judged on its merit by anybody who takes the time to pick up the book and read that chapter in it. His frivolous interpretation of entropy as “the amount of information you don’t know about the thermodynamic system” should be damned because it totally misunderstands the mathematical similarities between information and entropy that scientists have been aware of for the last 65 years. The reason for the similar form of the two is that both are inexact measures of sets of things that are generally unbalanced and whose exactness mathematically and as a clear correct understanding of them is provided by the same functional replacement for the N number of constituents in a set, namely diversity be it D or h. He personally should also be damned for not being willing to budge an inch for fear of making a fool of himself in his hypothesis being shown to be wrong, this after a many email exchange with him that laid out the foregoing in a series of first drafts of this material.
He is hardly the only one out there who thinks this way, science having fallen into the same state as all other endeavors in modern day mathematics where people learn to feather their own nests at the expense of the broader needs of society. If such is the case in the natural sciences, that much more is it prevalent in the human sciences, which are thoroughly adulterated by ideology to ascribe people’s unhappiness to the bugaboo of mental illness which is almost as vague, spirit like and intangible as Satan as the cause of evil and unhappiness. Clinical psychology never prescribes rebellion against unfair authority and its humiliations as a remedy for the unhappiness caused by it, but rather “adjustment” to the pain of it and that by any means which includes developing a chronic dependency on psychotropic drugs and delusional belief in religious superstition. God, Heaven, after life and the devil are quite alright with the pseudoscience of clinical psychology.
From a purely logical perspective, one should have great doubt about a supposed science that purports to understand abnormal emotion without giving any clear sense of the normal human emotions, as we will do starting in the next section as the foundation of a new set of mathematics based human sciences.
9. The Mathematics of Human Emotion
We do not wish to throw the baby out with the bathwater in our revision and expansion of information theory in Section 6. It is in no way a denial of all of its basic principles, a primary one of which we form the foundation of our specification of all of the human emotions in mathematical form. That principle I am talking about is information theory’s alternative interpretation of the H Shannon entropy of Eq71 as the amount of uncertainty that getting a message resolves upon being received. Uncertainty and information are closely related in information coming about as the resolution of uncertainty. If you have no idea of the way Company XYZ you hold stock in is going and I tell you from what my cousin, the president of the company, told me that they are contemplating bankruptcy in two weeks, that message is information for you because you had uncertainty about the company’s situation to begin with. But if I tell you that Osama bin Laden was the mastermind of 9/11, something you certainly knew beforehand, that message would not be information for you because you had no uncertainty about that.
In a more mathematically treatable way, if you are playing a game where you must guess which of N=4 colors I’ll pick from the set of K=8 colored buttons, (■■, ■■, ■■, ■■), inherently you have uncertainty about what the color is. Keep in mind from our earlier considerations the H=2 bits amount of information associated with this set. That value of H=2 is a measure of the amount of uncertainty you have as the number of yesno binary questions one needs to ask about the colors in (■■, ■■, ■■, ■■) to determine which color I picked. By a yesno binary question is meant one that is answered with a “yes” or a “no” and, as binary, cuts the number of possible color answers in half.
One might ask of (■■, ■■, ■■, ■■), “Is the color picked a dark color?” meaning either purple or black? Whatever the answer, a “yes” or a “no”, the number of possible colors picked is cut in half. Assume the answer to the question was “no”, then the next question asked might be, “Is the color green?” If the answer to that next question is also “no”, by process of elimination the color I picked was red. It took H=2 such questions to find that out. So the amount of uncertainty about which color I picked is understood to be H=2. And the amount of information gotten from receiving a message about the color picked is H=2 bits understood as the amount of uncertainty felt beforehand.
Let’s play that game with (■■, ■■, ■■, ■■, ■■, ■■, ■■, ■■) now whose H=3 bits Shannon entropy is the amount of uncertainty you feel about which color I picked from that set of buttons because it takes H=3 yesno binary questions to determine the color. The first question for (■■, ■■, ■■, ■■, ■■, ■■, ■■, ■■) might be, “Is the color a light color?” meaning red, green, aqua or orange. When “no” is the answer, it halves the field of colors picked to (■■, ■■, ■■, ■■). And two more yesno binary questions will then reveal the color picked. The amount of uncertainty for the color picked from (■■, ■■, ■■, ■■, ■■, ■■, ■■, ■■) is then its H Shannon entropy of H=3 bits interpreted as 3 binary questions. And the amount of information you would get if I sent you a message about which color was picked would be H=3 bits of information as the resolution of the H=3 bits of uncertainty felt beforehand.
That information is affected by emotion is obvious from the sense of information underpinned by uncertainty, something people generally feel as an unpleasant emotion. Moreover when uncertainty is resolved, by whatever means, a person tends to feel something akin to relief or elation, a generally pleasant emotion. Now while it is true that the H Shannon entropy provides some measure of uncertainty as discussed above, the human mind really doesn’t work on logarithmic measures for the most part. We tend rather to evaluate uncertainty probabilistically. Let’s go back to guessing the color picked from the N=4 color set, (■■, ■■, ■■, ■■).
The probability of guessing correctly, which we’ll give the symbol, Z, to, is
84.)
And the probability of failing to make the correct guess, understood as the uncertainty in guessing, is
85.)
Now let’s recall the D diversity of a balanced set from Eq4 to be D=N. This allows us to understand the U uncertainty as
86.)
Now let’s make a table of sets of buttons that have more and more D diversity and list the U uncertainty in guessing the color picked from them.
Sets of Colored Buttons 
D=N 
U=(D–1)/D 
(■■, ■■) 
2 
1/2=.5 
(■■, ■■, ■■) 
3 
2/3=.667 
(■■, ■■, ■■, ■■) 
4 
3/4=.75 
(■■, ■■, ■■, ■■, ■■) 
5 
4/5=.8 
(■■, ■■, ■■, ■■, ■■, ■■) 
6 
5/6=.833 
(■■, ■■, ■■, ■■, ■■, ■■, ■■) 
7 
6/7=.857 
(■■, ■■, ■■, ■■, ■■, ■■, ■■, ■■) 
8 
7/8=.875 
Figure 87. Various Sets and Their D and U Values
Very obviously the U uncertainty is an increasing function of D diversity. That
is, as D increases, U increases. More formally U is an effectively continuous
monotonically increasing function of D. This, from measure theory in
mathematics, tells us that whatever D is a measure of, U is a measure of.
Earlier we made it clear that D diversity was a measure of information. And
that would also make the U uncertainty measure understandable as information,
which fits with the classical information theory take on information as the
resolution of uncertainty.
This gives us two ways to specify the resolution of uncertainty being information, one as H in a logarithmic way, the other as U in a linear probabilistic way. The breakthrough in psychology that finally makes sense out of human nature is to understand emotion as meaningful information. And to do that we need to specify the uncertainty that precedes information in terms of probability, U, not only because the human mind is geared to sensing uncertainty as probability rather than in bits and bytes, but also because doing so, using U, allows us to connect it up with something meaningful, and that meaningful something is money.
Specifically, configuring uncertainty and information in terms of U probability connects uncertainty up with that meaningful item of money through a game of chance designed to have a cash penalty imposed on you if you fail to win at it. It is a color guessing game that uses the N=3 color set of colored buttons of (■■, ■■, ■■). If you fail to guess the color I pick, you pay a penalty of v=$120. The probability of guessing correctly is
88.)
And the probability of failing to guess correctly is
89.)
Now the product of the penalty, v, and the uncertainty, U, which is the probability of paying the penalty, is called the expected value of the game
90.) E= –Uv
Putting in the values, U=2/3 and v=$120, we calculate the E expected value or expectation to be
91.) E= –Uv= –(2/3)($120)= –$80
The negative sign specifies E= −$80 as a loss of money, the average loss incurred if you are forced to play this game repeatedly. If you play the game three times, for example, on average you will roll a lucky number and escape the v=$120 penalty one time out of three; and you will fail to roll a lucky number and pay the v=$120 penalty two times out of three as adds up to a $240 loss that averaging out over three games is an E= −$240/3= −$80 loss per game.
The E= –Uv term that is the product of the U uncertainty and the v penalty of money, a meaningful item, can also be understood as meaningful uncertainty. A more familiar expression for this E= –Uv meaningful uncertainty is the fear you have of losing money when you are made to play this game. That E= –Uv is a fitting equation for such fear is clear from three perspectives. The first is that your fear of losing money is a function of the U uncertainty or your probability of failing to guess the color. If we change the game to my randomly picking a colored button from the N=8 color set of buttons of (■■, ■■, ■■, ■■, ■■, ■■, ■■, ■■), then the probability of a successful guess goes down to
92.) Z = 1/N =1/8 = .125
And the probability of failing to guess correctly and of your having to pay the v=$120 penalty goes up to
93.) U=1–Z=7/8=.875
And the expected value translated as the amount of fear you have in having to play this game is
94.) E= –Uv= –(7/8)($120)= –$105
That fear feels unpleasant is manifest in the negative sign of the E= –Uv expectation. And you see that this function also fits the natural sense of fear that would be felt including as a measure of the displeasure in it if we change the v penalty. If we increase it to v=$360, the displeasure of fear felt for this game played with (■■, ■■, ■■, ■■, ■■, ■■, ■■, ■■) goes up to
95.) E= –Uv= –(7/8)($360)= –$315
We have introduced emotion now in a very straightforward way in terms of understanding meaningful uncertainty as fear. Next now consider what happens if there is a third person involved in this game who sees which color I picked and tells it to you on the quiet. Then you can use it as your guess and avoid paying the penalty. To go along with the basic algorithm that information is the resolution of uncertainty in information theory, we’ll understand the amount of information you got in the color told you, meaningful information from its resolving your meaningful uncertainty, to have the same measure as the meaningful uncertainty, –Uv, except we’ll get rid of the – minus sign understanding the removal of the meaningful uncertainty to be specified as
96.) T= –(–Uv) = Uv
We’ll explain where the T symbol comes from later on, understanding it now to represent the amount of meaningful information you got from the message told you about color. Now intuitively, you are going to feel an emotion of relief in getting this meaningful information. And the T=Uv function is a very good measure of the amount of relief, how pleasant it is in intensity. For the greater the v penalty, the greater the relief you feel in avoiding it. And the greater the U uncertainty, the greater the relief also. And the implicit (+) positive sign of T=Uv= +Uv, is a reasonable marker for the positive feeling or pleasure you get in relief, that as opposed to the E= –Uv fear, which is unpleasant as it (–) minus sign denotes.
Of course, the amount of fear you feel in expectation of paying the v penalty and the amount of relief you feel in avoiding the penalty are both dependent not only on the U probability of paying the penalty and the v amount of the penalty, but also on how much money, how much wealth, you already have. A millionaire doesn’t really care about losing $80 or get that much pleasure of relief in avoiding the loss as compared to a person who had but $5 in their purse or bank account. This marginality aspect that affects the emotions involved, we’ll obviate by making everybody who plays this game and in every game played have the same amount of wealth.
With that we have the basics down: getting rid of E= –Uv meaningful uncertainty by some activity, here guessing a color randomly chosen, generates T=Uv meaningful information. And this also introduces two primary emotions people feel, fear as E= –Uv and relief as T=Uv. The color guessing game was fine for an introduction to emotion, but next we want to develop the basic emotions, and there are a few more of them, in a more general way. Specifically we want to do it for all goal directed behaviors.
And to do that we are going to switch the game to a dice game called Lucky Numbers. It will develop mathematical functions for a fuller spectrum of our most basic emotions like hope, anxiety, excitement, disappointment, fear, relief, dismay, relief, joy and depression, which we’ll refer to as our operational emotions. And then later we’ll modify the game played to develop functions for our visceral emotions like sex, anger, hunger and the taste pleasures of eating.
We’ll start off playing this Lucky Numbers dice game for a prize, one of V=$120. The lucky numbers in the game are the 2, 3, 4, 10, 11 and 12. If you roll any one of them you win a prize of V=$120. The individual probabilities of rolling the numbers 2 through 12 on a pair of dice are:
97.) p2=1/36; p3=2/36; p4=3/36; p5=4/36; p6=5/36; p7=6/36; p8=5/36; p9=4/36; p10=3/36; p11=2/36; p12=1/36
And the probability, Z, of rolling one of these lucky numbers, 2, 3, 4, 10, 11 and 12, is just the sum of their individual probabilities.
98.) Z = 1/36 + 2/36 + 3/36 + 3/36 + 2/36 + 1/36 =12/36 =1/3
This obtains the probability of rolling a number other than one of these lucky numbers of
105.) U=1− Z=2/3
(Note: equation numbers 99104 are not used.) This U=2/3 is the improbability or uncertainty in success of rolling a 2, 3, 4, 10, 11 or 12 lucky number. The amount of money one can expect to win on average in this V=$120 prize game is
106.) E = ZV = (1/3)($120) = $40
E is the expected value of the game, the average amount won per game played. If you played this dice game three times you could expect to win V=$120, on average, one play in three for an average payoff of E=$40 per game played. Eq105 enables us to write the expected value of E=ZV in Eq106 as
107.) E = ZV = (1−U)V= V −UV
The E expected value has three component terms in the above, E=ZV, V and – UV. To understand E=ZV and V in Eq107 in terms of the pleasure associated with them we need to fast forward for the moment to the successful outcome of playing this game of winning the V=$120 prize. We label the prize money gotten or realized with the letter R, hence, R=V=$120. This distinguishes it from the V=$120 in E=V−UV of Eq107, which is most broadly an expectation or anticipation of getting money that is quite different than actually getting or realizing money.
And assumed is that getting money is pleasurable with the intensity of the pleasure greater the more money gotten. Consider a spectrum of prizes offered that can be won by a player. Then R=V=$120 is understood to be more pleasant than R=V=$12 and both less pleasant than R=V=$1200. This assumption is reasonable in being universal in people old enough and sane enough to appreciate money. The pleasure of the R=V emotion of winning is referred to variously as joy, delight or elation.
For simplicity sake we will take R=V=$120 to provide ten times more pleasure than R=V=$12 and R=V=$1200 to provide ten times more pleasure than R=V=$120. So we will understand the pleasure experienced in getting R=V dollars to be a simple linear function of V. This simplifies the relationships derived for the mathematics of human emotion. One could also assume that the pleasure involved in getting money is marginal, that the more money one gets, the less pleasure felt per unit of money gotten. We could also develop a mathematics of human emotion with functions that model this assumption of marginality, but in the end, the cornerstone relationships of the emotion mathematics derived would be essentially the same as with the linear model, but the computations involved significantly more difficult to develop and to follow.
It is also accepted that the pleasure in getting a certain amount of money is a function of how much money the receiver of some R+V amount already has in her purse or in the bank. Clearly getting R=V=$12 means a lot more and provides more pleasure to a homeless woman with $2 in her purse and no money in the bank than it does to someone like Bill Gates. This is just another manifestation of marginality that we can also omit from consideration by assuming that all recipients of R=V dollars have the same amount of money already in their possession.
The V term in E= V− UV of Eq107 differs from an R=V realization of money in its being the anticipated goal of playing this prize awarding Lucky Numbers game. The V dollar prize in E=V− UV is what the player wants. It is his desire, his wish, his goal in the game, to obtain the V=$120 prize. There is a pleasure in the V wish or desire for obtaining the V dollar prize. Again we will understand the intensity of that pleasure to be directly proportional to or a linear function of V.
We will also understand the pleasure in anticipating V dollars to be equal to the pleasure in realizing R=V dollars. At first this seems incorrect. Surely, one would think, people enjoy greater pleasure in getting R=V dollars than in expecting to get V dollars. That confusion, though, is cleared up by understanding the –UV term in E=V−UV of Eq107 as a measure of the anxiousness or anxiety felt about getting the V dollar prize. The greater the U uncertainty in success, the greater the anxiety in expecting it as also inflated by the V size of the prize expected. That is, the greater the V size of the dollar prize desired or wished for, the greater the −UV anxiousness about getting it. The negative sign in –UV is understood as indicating that the emotion of anxiousness is unpleasant, which is in experience universal for people.
Note then that the –UV anxiousness reduces the V pleasure of anticipating the prize in E=V−UV of Eq107. This understands the E expected value as a measure of the realistic hope or hopes a person has in getting the as a reduction of the wish for the V prize via the –UV anxiousness the player has about succeeding. That is our realistic hopes take into account both the desire or wish for the V prize and the U probability of not getting it. Indeed, when that U improbability or uncertainty of success is not taken into account, we call it wishful thinking.
Very often, and especially in a game of chance like the prize awarding Lucky Numbers game, there is always some U uncertainty in expectation of the prize. Hence anticipation of the prize in terms of the E=V−UV measure of realistic hope for it is very often less intense pleasure wise than the R=V pleasure of actually realizing the prize. But that is not always the case as is clear when a person anticipates a paycheck at the end of the week with absolute surety, Z=1, and no uncertainty, U=1−Z=0. In that case E=V−UV=V, and experientially there is no significant difference between surely expecting to get the R=V money on the day before pay day and actually getting it on pay day, E=V=R=V.
Backing up a bit we see that our hopes are a function of what we hope for, V dollars in this case, and our sense of the likelihood or probability of getting it, Z in this case. The greater the V prize desired and the Z probability supposed of getting it, the “higher” our hopes and greater the pleasure in the E=ZV expectation. Note that we use the word “supposed” in association with Z and the pleasure incumbent in our E=ZV hopes. In this Lucky Numbers game, it is taken that the supposed probability is the true probability of success in rolling a winning lucky number. But generally speaking people may have false hopes, excessive hopes, which actually do feel more pleasant in anticipation of success than if a lesser, more realistic, probability were supposed. Indeed much of the pleasure in believing in religion and the reward of a happy after life derives from a delusional high hope of its actually happening, the reality of the outcome irrelevant to the true believer’s pleasure in anticipating it.
Backing up again we also should understand that the –UV anxiousness felt also goes in ordinary language by other names like anxiety or fear or concern or worry about getting money wished for. For that reason we also give –UV a technical name, that of meaningful uncertainty as uncertainty, U, made meaningful by its association with V dollars in –UV, money generally being a meaningful or valuable item for people.
Next we want to state a general function for all the emotions involved in this prize awarding Lucky Numbers game, The Law of Emotion. To do that we have to add one more elemental function to the mix. It is what is realized when a lucky number is not rolled. Nothing is gotten or realized as expressed by R=0. The elemental emotions we have considered up to this point now allow us to write the Law of Emotion as
108.) T = R − E
We are already familiar with two of the three functions in The Law of Emotion. E is the expectation of winning a V dollar prize and R the realization or outcome of the attempt to win by throwing the dice, R=V for a successful attempt and R=0 for an unsuccessful one. The T term is now introduced as a transition emotion that comes about as a combination of what was expected, E, and what was actually realized, R. In a failed attempt where R=0, the transition emotion develops from T=R−E, The Law of Emotion, as
109.) T = R −E = 0 −ZV = −ZV
This T= − ZV transition emotion is the disappointment felt when one’s hopes of winning the V prize, E=ZV, are dashed or negated by failure to throw a lucky number. Disappointment is specified as unpleasant from the minus sign in T= − ZV and its displeasure is seen to be greater, the greater is the V size of the prize hoped for but not won and the greater the Z probability the player felt he had to win. In the game for a V=$120 prize that can be won with probability of Z=1/3, the intensity of the disappointment is
110.) T = −ZV = −(1/3)($120) = −$40
The T= −$40 cash value of the emotion of disappointment indicates that the intensity of the displeasure in it is equal in magnitude, if not in all its nuances, to losing $40. The T= − ZV disappointment over failing to win a larger, V=$1200, prize hoped for, is greater as
111.) T= − ZV= − (1/3)($1200)= − $400
Note that though the realized emotion, R=0, produces no feeling, pleasant or unpleasant in itself, from failure to achieve the goal of obtaining the V dollar prize in the game, failure does produce displeasure in the form of the T= − ZV transition emotion. This transition emotion and the three more basic transition emotions we will consider have a specific function in the emotional machinery of the mind that we will consider in depth once we have generated those three T emotions from The Law of Emotion.
We call attention to the universal emotional experience of T= − ZV disappointment being greater the more V dollars one hoped to get but didn’t. The T= − ZV disappointment is also great when the Z probability of winning is great. Consider this Lucky Numbers dice game where every number except snake eyes, the 3 through12, is a lucky number that wins the V=$120 prize. These lucky numbers have a high probability of Z=35/36 of being tossed, so the hopes of winning are great as
112.) E = ZV = (35/36)($120)= $116.67
And we see that the disappointment from failure when the ZV hopes are dashed or negated to –ZV by rolling the losing 2 is also great as
113.) T= −ZV = − (35/36)($120)= − $116.67
Compare to T= − ZV = −$40 in Eq10 played for the same V=$120 prize, but when the probability of success was only Z=1/3. This fits the universal emotional experience of people feeling great disappointment when they have a high expectation of success and then fail. And at the other end of the spectrum, as also predicted by T= −ZV, people feel much less disappointment when they have a very low Z expectation of success to begin with. As an example, consider the T=−ZV disappointment in this dice game when to win you must roll the low Z=1/36, probability snake eyes, the 2, as the only lucky number to win with. Then the disappointment is much less as
114.) T= ZV= − (1/36)($120)= −$3.33
Now let’s consider the T transition emotion that arises when one does win the V dollar prize with a successful toss of the dice. With hopeful expectation as E=ZV and realized emotion as R=V, the T transition emotion is from the Law of Emotion of Eq108, T=R−E, via the U=1−Z relationship in Eq105,
115.) T = R−E = V −ZV = (1− Z)V = UV
The T= UV transition emotion is the thrill or excitement of winning a V dollar prize under uncertainty. It is a pleasant feeling as denoted by the implied positive sign of UV with the pleasure in the thrill greater the greater is the V size of the prize and the greater is the U uncertainty of winning it felt beforehand. When one is absolutely sure of getting V dollars with no uncertainty, U=0, as in getting a weekly paycheck, while there is still the R=V pleasure of delight in getting the money, the thrill of winning money under uncertainty is lacking. That is, with uncertainty present, U>0, there is an additional thrill or excitement in winning money as in winning the lottery or winning a jackpot in Las Vegas or winning a V=$120 prize in the Lucky Number dice game. In the latter case, with an uncertainty of U=2/3 from Eq105, the intensity of the excitement in winning the V=$120 prize is from Eq115
116.) T=UV=(2/3)($120)=$80
That this additional pleasure of T=UV excitement in obtaining V dollars over and above the R=V delight in getting money depends on feeling U uncertainty prior to rolling the dice is made clearer if we look at an attempt to win V=$120 by rolling the dice in a game where only tossing snake eyes, the 2 on the dice, with probability Z=1/36 and uncertainty U=35/36, wins the prize. In that case, if you do win, as with winning in any game of chance where the odds are very much against you, the uncertainty very great, there’s that much more of a thrill or feeling of excitement in the win.
117.) T= UV= (35/36)($120)= $116.67
By comparison consider a game that awards the V=$120 prize for rolling any number 3 through 12 with Z=35/36 probability of winning and low uncertainty of U=1−Z=1/36 as makes the player near sure he is going to win the money. While there is still the R=V=$120 delight in getting the money upon rolling one of these many lucky numbers, there is much less thrill because like getting a paycheck, the player was almost completely sure of getting the money in this Z=35/36 dice game to begin with.
118.) T=UV=(1/36)($120)=$3.33
This relationship between the uncertainty one has about getting something of value and the excitement felt when one does get it is clear in the thrill children feel in unwrapping their presents on Christmas morning. The children’s uncertainty about what they’re going to get in the wrapped presents is what makes them feel that thrill in opening them up. This excitement is an additional pleasure for them on top of the pleasure realized from the gift itself. That special thrill in opening the presents under the Christmas tree is not being felt when the youngsters know ahead of time what’s in the Christmas presents and feel no uncertainty about it.
As is predicted by T=UV, it is seen to be universal for people that winning a V=$1200 prize in a game of chance is more thrilling than winning a V=$120 prize when the U uncertainty (or probability of not winning) is the same in both cases. And we get a fuller picture yet of the T=UV thrill of winning under uncertainty from the T=R−E Law of Emotion of Eq108 when the E expectation term in it is expressed from Eq107 as E=V− UV.
119.) T = R− E =V−(V−UV) = − (−UV)=UV
This derivation of T=UV as the negation –UV anxiousness, T= − (− UV) =UV, derived for the Lucky Numbers dice game is the basis of excitement coming about generally by the negation or elimination of anxiousness via a successful outcome. Adventure movies generate their excitement or thrills for the audience in just that way by being loaded with anxiousness or dramatic tension at the beginning of a drama from the hero’s meaningfully uncertain situation, which the audience feels vicariously. When the hero’s uncertain situation is resolved by success towards the end of the movie, it vicariously brings about thrills and pleasurable excitement for the audience that empathizes with the hero by negating or eliminating the anxiousness they felt about his or her situation to begin with. Though the emotions felt by the audience are vicarious, the essence of the dynamic is essentially the same as spelled out in Eq119.
We have in the above explained excitement as resulting from an outcome of goal directed behavior of success. People are also generally aware of excitement as a feeling that prefaces success. That is also very easy to explain mathematically, as we will in Section 8, but only after a proper workup that makes its understanding instantly simple and clear.
THE OTHER broad category of goal directed behavior that people engage in is to try to avoid losing something of value, like money. This category is well illustrated with the v= S120 dollar penalty game we introduced earlier in the color guessing game. The player is forced to play this game and the penalty can be avoided with the Z=1/3 probability roll of a 2, 3, 4, 10, 11 or 12 lucky number. The probability of not rolling one of these lucky numbers as results in paying the v=$120 penalty is U= 1− Z =2/3. And the expected value as Uv=$80 is given below in more proper form with a negative sign as
120.) E= U(−v)= −Uv= −(2/3)($120)= −$80
The negative sign on –v makes clear that the v dollar value represents a loss of dollars for the player. The E= −Uv= −$80 expected value of this game is the average penalty paid if one were forced to play this game repeatedly. It tells us that if you played three of these penalty games, on average, you will fail to roll a 2, 3, 4, 10, 11 or 12 lucky number two times out of three to pay the v= −$120 penalty for a total of $240 as averages out over the three games to a penalty per game of E= − $80.
E= –Uv is a measure of the fearful expectation or fear of incurring the penalty. The negative sign prefix of E= −Uv indicates that this fear is an unpleasant emotion with the intensity of the E= −Uv displeasure of the fear greater the greater the U probability of incurring the v penalty and the greater the size of the v penalty, as fits universal emotional experience.
The −Uv fear goes by a number of other names in ordinary language including worry, distress, apprehension and concern. This plethora of names for E= –Uv fear has us give it the technical name also of meaningful uncertainty as puts –Uv fear, as an anticipation of the possibility of losing dollars, in the same general category as −UV anxiety, as an anticipation of the possibility of failing to win V dollars that are hoped for. That both –Uv fear and –UV anxiety are classified together as forms of meaningful uncertainty should not be surprising given that they are very often referred to with the same names of fear, anxiety, concern, worry, distress, apprehension, trepidation, nervousness and so on. Note that we refer in this treatise to –Uv as fear and –UV as anxiety to distinguish between the two however the words are often used interchangeably in ordinary language. We will have more to say about the naming of emotions shortly after we develop a more complete list of them.
Next we consider the realized emotions of the penalty game. The first is the realized emotion that comes about when the v penalty is realized from the player failing to roll one of the 2, 3, 4, 10, 11 or 12 lucky numbers, R= −v. This unpleasant emotion is one of the grief or sadness or depression felt from losing money. Again there are many names for it in ordinary language. And when the outcome is of a successful toss of a lucky number the realized emotion is given as R=0 because as no money changes hands when the player is spared the penalty, there is no emotion that comes from the outcome, per se.
That is not to say that there is no emotion felt from avoiding the penalty, but it is a T transition emotion derived from the T= R−E Law of emotion of Eq8 rather than as a form of R realized emotion. When the lucky number is rolled the fearful expectation of E= −Uv is not realized, R=0, and the T transition emotion is from the T=R−E Law of Emotion of Eq108,
121.) T = R−E = 0 − (−Uv) = Uv
This T=Uv measures the intensity of the relief felt from escaping the v dollar penalty when you roll one of the lucky numbers. The positive sign of T=Uv specifies relief as a pleasant emotion with its pleasure greater, the greater is the v loss avoided and the greater is the U improbability of avoiding the loss. The T=Uv relief felt when a 2, 3, 4, 10, 11 or 12 lucky number is tossed in the v=$120 penalty game with uncertainty U=2/3
122.) T= Uv= (2/3)($120) =$80
To make clear how dependent the intensity of Uv relief is dependent on the U uncertainty, note that if one plays a v=$120 penalty game where rolling only the 2 avoids the penalty, with uncertainty U=35/36, there is greater relief in successful avoidance of the penalty by rolling the lucky number because you felt prior to the throw that most likely you would lose.
123.) T=Uv=(35/36)($120)=$116.67
This increase in relief with avoidance of a penalty under greater uncertainty is universal. But if you play a v=$120 penalty game that avoids the penalty with any number 3 through 12, with uncertainty of only U=1/36, there is much less sense of relief because you felt pretty sure you were going to avoid the penalty, with high probability of Z=35/36, to begin with.
124.) T=Uv=(1/36)($120)=$3.33
Note also that the larger the v penalty at risk, the more intense the relief felt in avoiding it as with a v=$1200 penalty in the game where only rolling the 2 lucky number game with uncertainty, U=35/36, escaped the penalty.
125.) T=Uv=(35/36)($1200)=$1166.67
Compare to the relief of T=$116.67 in Eq123 when the penalty was only v=$120. The universal fit of mathematically derived Uv relief to the actual emotional experience of feeling relief is remarkable. We also use the Law of Emotion of Eq108 of T=R−E to obtain the T transition emotion felt when the player does incur the v penalty by failing to roll a lucky number. In that case, with expectation of E=−Uv and a realized emotion of R= − v, the T transition emotion is via Z=1− U
126.) T = R − E = −v − (−Uv) = −v+ Uv= −v(1−U)= −Zv
This T= − Zv transition emotion is the dismay or shock felt when a lucky number is not rolled and the v penalty is incurred. The T= −Zv emotion of dismay is an unpleasant feeling as implied by its negative sign and one greater in displeasure, the greater the Z probability of escape one supposed before rolling. Its value for the 2, 3, 4, 10, 11 or 12 lucky number v=$120 penalty game with Z=1/3 is
127.) T= − Zv = − (1/3)($120)= − $40
But if you have a very small Z probability of avoiding a v=$120 dollar loss as in the dice game where only rolling the 2 as the lucky number provides escape from the v penalty to probability, Z=1/36, there is little − Zv dismay when you fail to roll that lucky number and must pay the penalty because you had such a high sense of E= −Uv with U=35/36=.9667 surety that you’d have to pay the penalty to begin with.
128.) T= − Zv = − (1/36)($120)= − $3.33
One develops a more intuitive feeling for dismay by expressing the E= −Uv fearful expectation via U=1− Z as
129.) E= −Uv = −(1–Z)v = −v + Zv
The − v term in Eq129 is the anticipation of incurring the entire v penalty, which we will call one’s dread of the penalty for want of a better word. The displeasure in the dread of paying the v penalty is marked by the negative sign of − v with the intensity of its displeasure greater, the greater the v dollar penalty that is dreaded. Were the penalty raised to −v= − $1200, the dread and its displeasure would be proportionately greater than the –v= −$120 penalty. This −v dread in E= − v + Zv of Eq129 is partially offset by the +Zv term in it as the (pleasant) hope one has that one will escape the penalty by rolling a lucky number.
This Zv term is understandable emotionally as the sense of security one has that one will avoid the penalty, the greater the Z probability of escaping the penalty in +Zv and the greater the v penalty one is protected from by Z, the greater the sense of security one has when one is forced to play the penalty game that one will be able escape the penalty. The combination of unpleasant –v dread and pleasurable +Zv security produces the realistic fear or fearful expectation of incurring the penalty, E= −v + Zv = −Uv, of Eq129.
Expressing the E expectation as in Eq129 adds an important nuance to the derivation of dismay from the T=R−E Law of Emotion of Eq108.
130.) T = R –E = −v −(−v + Zv)= −(Zv)= −Zv
This understands T= –Zv dismay as coming about from the dashing or negation of one’s Zv hopes or expectation of avoiding the v penalty by failure to roll a lucky number. The low dismay that results from failure preceded by low Zv expectation is why some people subconsciously develop a strategy of low expectations in life to avoid the unpleasant feeling of dismay if they do fail. This contrasts to the considerable T= − Zv dismay or shock felt in the v=$120 penalty where the lucky numbers on the dice that are needed to avoid the penalty are the 3 through 12 whose probability of being rolled is Z=35/36.
131.) T= − Zv = − (35/36)($120)= − $166.67
In short the dismay in this case is high because of the high Zv expectation of not paying the penalty to begin with. Great dismay from failure preceded by a high Z=35/36 probability of escaping failure is also felt and referred to as shock, familiarly as a person’s surprise at failure when what was expected from the preceding high probability was success. Unpleasant unexpected surprise specified here as great −Zv dismay is also the fundamental basis of horror.
The above development of the E fearful expectation as E=− Uv = − v + Zv gives us functions for three more elementary emotions: the − v dread of incurring a penalty; the Zv sense of security one feels in the possibility of escaping the penalty; and the probability tempered E= −Uv fear of incurring a penalty. These add as expectations to the V desire of getting a V prize, the –UV anxiousness about getting it and the E=ZV probability tempered hopes of getting a prize considered earlier to give a complete set of our basic anticipatory emotions.
The −Uv, ZV, V, −v, Zv and –UV symbols are the best representations of our anticipatory emotions rather than the more familiar names for them in ordinary language respectively of fear, hope, desire, dread, security and anxiety. Ludwig Wittgenstein, regarded by many as the greatest philosopher of the 20^{th} Century, made the point well in his masterwork, Philosophical Investigations, of the inadequacy of ordinary language to describe our mental states. Words for externally observable things like a “wallet” are clear in meaning when spoken from one person to another because if any confusion arises in discourse, one can always point to a wallet that both the speaker and the listener can see. “Oh, that’s what you mean by a wallet.” But with emotions, however, as nobody feels the emotions of another person, the words we use for an emotion have no common sensory referent one can point to in order to clarify its meaning.
The mathematical symbolwords of −Uv, ZV, V, −v, Zv and –UV, on the other hand, are at least clear in meaning because they have countable referents of money as V and v and numerical probabilities of Z and U as components. And the fit of these therefore mathematically welldefined wordsymbols to emotional experience, pleasant and unpleasant, is universal. That is, all people feel these −Uv, ZV, V, −v, Zv and –UV anticipatory feelings in the same way when playing the V prize and v penalty Lucky Number games assuming all have the same quantitative sense of dollars and of probability. Hence quibbling over the “correct” names to call −Uv, ZV, V, −v, Zv and –UV or any of the other mathematical symbols we will develop for the emotions is not a valid criticism of this analysis.
Our expectations determine our behavioral selections, what we choose or decide to try to do. The basic rules are simple.
Rule #1. If we have a choice between entertaining a hopeful expectation as with E=ZV of the V prize awarding Lucky Numbers game and a fearful expectation as with E= −Uv of the v penalty assessing Lucky numbers game; we act on the behavior that generates hope rather than fear. This is so intuitively obvious that it is almost not worth stating at all other than for the sake of completeness. We can understand Rule #1 as sensible from the standpoint of a V dollar gain being preferred to a –v dollar loss; or, hedonistically, from the pleasure felt in ZV hopes triumphing cognitively over the displeasure of –Uv fear.
Rule #2. If we have a choice between two hopeful expectations, E_{1}=Z_{1}V_{1} and E_{2}=Z_{2}V_{2} with E_{1}>E_{2}, we choose E_{1} whether E_{1}>E_{2} comes about via Z_{1}>Z_{2} or V_{1}>V_{2} or both. As an example, one would choose to play the standard Z=1/3, V=$120 prize game with E=$40, than a V=$120 game with just 2, 3 and 4 as the lucky numbers, Z=1/6 and E=$20. We may attribute the underlying cause of greater hopeful expectation triumphing cognitively over less hopeful expectation to the anticipated average gain in E_{1} being better than in E_{2}; or, hedonistically, to their being greater pleasure in entertaining E_{1}=Z_{1}V_{1 }than in E_{2}=Z_{2}V_{2}. _{. }
Rule #3. If we have a choice between two v penalty games, one with fearful expectation, E_{1}= –U_{1}v_{1}, and the other with E_{2}= –U_{2}v_{2}, one of which games we must play, we choose the game with the smaller expectation (in absolute terms.) Or more exactly, if E_{1}>E_{2 }numerically, we choose to play the E_{1} game. To clear up any confusion, as between the games in Eqs4&5, we choose to play the E_{1}=–80 game, E_{1}>E_{2}, rather than the E= –$240 game if we have to play one of them. This comes under the colloquial heading of “choosing the lesser of two evils”, also known as a Hobson’s choice.
The nuances and extensions of these three rules are many. The main point is that they show the primary function of our expectations, hopeful and fearful, to be to determine the choices we make. The next section explains the function of the transitional emotions of excitement, relief, disappointment and dismay in our emotional machinery. And then we go on to show how the Law of Emotion derives the Law of Supply and Demand in the most elementary way, something that even the most ardent capitalist hater of our revolutionary ideas cannot deny.
10. The Function of the Transition Emotions
We continue with our systematic explanation of our emotional machinery by explaining the purpose and function of the transition emotions of T= −ZV disappointment of Eq109, T=UV excitement of Eq115, T=Uv relief of Eq121 and T= −Zv dismay of Eq125. Recall that they all come about from the T=R−E Law of Emotion of Eq108. In it the E expected value depends in a very direct way on the Z and U probabilities: of the E=ZV=V−UV hopeful expectation in the V prize game; and of the E= −Uv= −v+ Zv fearful expectation in the v penalty game.
In our analysis up to this point, the
player’s sense of the values of the Z and U probabilities were taken directly
and correctly from the mathematics of throwing dice. But that need not be the
case. A player may suppose any probabilities of success or failure,
which affects the player’s E expectations, and in turn, affects from the
T=R−E Law of Emotion, the intensities of the T transition emotions from
T=R−E the player experiences upon success or failure.
As an example of a player supposing incorrect values of Z and U, consider in
the V=$120 prize game where rolling a lucky number of 2, 3, 4, 10, 11
or 12 has an actual probability of Z=1/3 that a naïve player supposes it is
Z’=1/2 for whatever reason. This distorts the hopeful expectation from the
player thinking she will win half the time instead of just 1 time in 3 from the
correct expected value of E=ZV=(1/3)($120)=$40 of Eq107 to
191.) E’=Z’V=(1/2)($120)=$60
(Note: Equation numbers 132190 are not used.) The player has higher hopes of winning than she should and though that cannot affect the actual (average) R outcomes or realizations it does from the T=R−E Law of Emotion of Eq108 affect the T transition emotions that arise. To show this let’s assume the game is played three times as results in the average winloss record of winning 1 time in 3 with R realizations of (0, 0, $120). And we’ll also assume that the player sticks to her incorrect probability suppositions for all three games played. The transition emotion felt after the first failed attempt of a realization of R=0, labelled T’, is
192.) T’=R−E’=0−Z’V= −Z’V= −$60
This T’= −Z’V= −$60 emotion is of disappointment in greater intensity than the disappointment of T= −$40 of Eq10 felt when the correct Z=1/3 probability is supposed. This is because the naïve player thought she had a greater possibility of winning. The 2^{nd} game played is also an R=0 failure and again a T’= −$60 disappointment is felt. On the 3^{rd} play, though, as fits the average % of games won a lucky number is rolled for R=V=$120 and the thrill of winning with E’=Z’V=$60 is from the law of emotion as T’=R−E’
193.) T’=R−E’=V−Z’V=$120−$60=$60.
This a smaller excitement than the E=ZV=$80 of Eq16 that would have been felt had the player supposed the correct probability of winning of Z=1/3. The player, hence, feels greater disappointment and less excitement over the three games, the sum of the T’ emotions experienced being
194.) ∑T’_{ }= −$60 −$60 +$60 = −$60
And the average of these T’ transition emotions per game is
195.) ∑T’/3 = T’_{AV}= −$60/3= −$20
Now, though the player retained her incorrect suppositions of probability for the three games, failure to meet her expectations over the three games manifest as an overall unpleasant set of transition emotions of ∑T’_{ }= −$60 and T’_{AV}= −$20 per game lowers her hopeful expectation in the next game she plays and, as we will show below, to the correct E=$40 per game.
Her emotional machinery does this with a T=R−E Law of Emotion inversion that understands T for a game as the T’_{AV} average of prior games, E as E’, the incorrectly supposed expectation and R as what is realized cognitively from T’_{AV} and E’, which is a revised or new expectation, E_{NEW}. Hence, not T=R−E, but
196.) T_{AV} =E– E’
Or solving for E_{NEW}, we arrive at the Law of Emotion Inversion,
197.) E_{NEW} = E’ + T_{AV}
For the example case developed above, this obtains an E_{NEW} expectation of
198.) E_{NEW }= $60 −$20 = $40
Now this revised E_{NEW}=$40 is just the E=ZV=$40 of Eq110 that arises from the correct Z=1/3 probability. So we see that the function of the transition emotions is to correct errors in expectation, and to do it using the E_{NEW} = E’ + T_{AV} variation of the general T=R−E Law of Emotion of our emotional machinery. If this seems too beautifully precise and simple a way for out emotional machinery to act, let’s try another example.
This will be of a fellow who has no confidence at all that he can win at any game, Mr. Unlucky. His sense of probability is hence, Z’=0 and of expectation, E’=Z’V=0. Again we will consider a three game play that realizes R outcomes of the actual average of (0, 0, $120). From the Law of emotion as T’=R−E’, we see that his first two games result in –Z’V disappointments of
199.) T’=R−E’=0−Z’V= −Z’V=0
He has no disappointment in the losses because he had absolutely no hopes of a win to begin with. The excitement of winning on the 3^{rd} game, though, is, from R=V=$120, great, as
200.) T’=R−E’=$120−0=$120
Note that this is an excitement greater than the T=$80 of Eq116 he would have felt had he supposed correctly a probability of winning of Z=1/3 and an expectation of E=ZV=$40. Now we see that the sum of his T’ transition emotions felt are
201.) ∑T’_{ }= 0 + 0 + $120 = $120
And the average of these T’ transition emotions per game is
202.) ∑T’/3 = T’_{AV}= $120/3 = $60
And from the Law of Emotion Inversion of Eq197 we obtain the correct expectation felt in the next play of the game of
203.) E_{NEW} = E’ + T_{AV}= 0 + $60 = $60
From the two above examples we see, as fits universal emotional experience, that preponderant disappointment in a goal directed behavior reduces subsequent hopeful expectation or confidence in that behavior and that preponderant excitement from winning increases subsequent confidence. The fit of function to experience is unarguable, quite remarkable, and makes clear that the function of the transition emotions is to keep one’s expectations in line with one’s reality of outcomes. This is reinforced all the more if one repeats the above exercise starting with the correct supposition of E=$40. In this case over the play of three games that realizes outcomes (0, 0, $120), the (correct) transition emotions felt of disappointment and excitement are (−$40, −$40, $80), which sum to 0 as produces no change in expectation from the Law of Emotion Inversion of E_{NEW} = E’ + T_{AV}.
This Law also works in a numerically exact way for the v penalty Lucky Numbers game to show that preponderant relief in repeated play of a penalty game results in subsequent decreased E= −Uv fear of losing; and that preponderant dismay results in a subsequent increase in E= −Uv fearful expectation; as universally fits emotional experience.
While this analysis cannot without neurobiochemical assay say absolutely that the mind uses this exact functional algorithm to keep our expectations in line with the reality of actual experience, the fit of the equations to experience in the broad ways cited above and the exactness of the corrective dynamic they bring about, especially as based on a variation of the Law of Emotion as seen in Eq197 makes clear that the mind’s neurobiochemistry and neurophysiology must operate as controlled by these functions in some way.
The universality of the fit of the equations for the emotions and of the Laws of Emotion of Eqs108&197 that control the relationships between these basic emotions is very important, for it counters any facile rebuttal of this understanding on the basis of the human emotions not being susceptible to empirical verification. Rather this mathematical explication of the emotions is effectively empirical in being universal.
Our specification of disappointment as the −ZV negation or dashing of ZV hopes, for example, is universal in that all human beings feel disappointment when they fail to achieve a desired goal. Indeed, all of the emotional specifications and dynamic relationships we have considered are universal. Such universal agreement is the fundamental factor in all empirical validation. When ten researchers all read the same data off a laboratory instrument, that data is taken to be empirically valid because all ten agree on what they see. This criterion for empirical validity of universal agreement extends to the emotions of the dice game laid out as what all people would feel if they played it. To deny the validity of the above interlocking, experience reflecting, quantitatively precise emotion specifications and relationships on the basis of an abstract principle of absence of empirical verification is to fail to understand the underlying basis of empirical validity in universality.
11. Emotions of Partial Success
To further provide observable, empirical proof of the emotion mathematics, we will next consider the emotions that arise from partial success. To that end we alter the Lucky Number V prize awarding game to one where you must roll a lucky number of 2, 3, 4, 10, 11 or12 not once but three times to win the prize, one of V=$2700. The excitement gotten from the partial success of rolling the 1^{st} lucky number of the three needed to win the V prize has observable reinforcement in parallel games of chance seen on television. The three rolls of the dice taken to roll three lucky numbers and win the V=$2700 prize may be with three pair of dice rolled simultaneously or with one pair of dice rolled three times in succession. The probability of rolling a 2, 3, 4, 10, 11 or12 lucky number on any one roll of dice is from Eq2, Z=1/3. Hence for the 1^{st} roll or with the 1^{st} pair of dice, Z_{1}=Z=1/3; for the 2^{nd} roll or pair of dice, Z_{2}=Z=1/3; and for the 3^{rd} roll or pair of dice, Z_{3}=Z=1/3.
204.) Z_{1}=Z_{2}=Z_{3}=Z=1/3
And the U uncertainties for each toss are
205.) U_{1}=(1− Z_{1})=_{ }U_{2}=(1− Z_{2})_{ }=U_{3}=(1− Z_{3})=(1−Z)=2/3
The probability of rolling a lucky number of 2, 3, 4, 10, 11 or12 on all three rolls is the product of the Z_{1}, Z_{2} and Z_{3} probabilities, which is given the symbol, z, (lower case z).
206.) z = Z_{1}Z_{2}Z_{3 }= Z^{3 }= (1/3)^{3 }= 1/27
And the improbability or uncertainty of making a successful triplet roll
successfully is
207.) u=1–z = 26/27
The expected value of this triplet roll game to win the V=$2700 is in parallel to Eq6,
208.) E = zV =V−uv=Z_{1}Z_{2}Z_{3}V_{ }= (1/27)($2700) = $100
This is also a measure of the intensity of the player’s pleasant hopes of winning the game. The displeasure of disappointment from failure to make a successful triplet roll is from the T=R−E Law of Emotion of Eq108 with R=0 in parallel to Eq109,
209.) T=R−E = 0−zV= −zV= −(1/27)($2700)= −$100
And the pleasure of the excitement or thrill in making the triplet roll, R=V=$2700, is, in parallel to Eq115
210.) T=R−E = V−zV=(1−z)V= uV= −(−uV)=(26/27)($2700)=$2600
Next we derive the emotion felt from rolling a lucky number on the 1^{st} throw of three sequential throws on one pair of dice. After a 1^{st} toss that does roll a lucky number, the probability of winning the V=$2700 prize by tossing lucky numbers on the next two rolls increases to
211.) Z_{2}Z_{3 }=(1/3)(1/3) = 1/9
And the hopeful expectation of making the triplet roll after a lucky number is rolled on the 1^{st} toss is, increases from the original E=Z_{1}Z_{2}Z_{3}=$100 to
212.) E_{1 }= Z_{2}Z_{3}V = (1/9)($2700) = $300
Next we want to ask what is realized when the 1^{st} toss is successful. It is not R=V=$2700, for the V prize is not awarded for just getting the 1^{st} lucky number. And it is not R=0, what is realized when the player fails to make the triplet roll and win the V=$2700 prize, for rolling the 1^{st} lucky number successfully quite keeps him on track to roll the next two numbers successfully and win the V=$2700 prize. Rather what is realized when the 1^{st} roll is of a lucky number is the E_{1}=$300 increased expectation in Eq212. This understanding of the increased E_{1}=$300 expectation as what is realized has us specify E_{1} as a realization with the R symbol as
213.) E_{1}=R_{1}=Z_{2}Z_{3}V = $300
Now we use the Law of Emotion of T= R−E, of Eq108 to obtain the T transition emotion that arises from a successful 1^{st} toss. This specifies the T term in T=R−E as T_{1}; the R term in it as R_{1}=Z_{2}Z_{3}V from Eq213; and the E term in T= R−E as the expectation had prior to the 1^{st} toss being made, E=zV=Z_{1}Z_{2}Z_{3}V of Eq208. And with U_{1}=(1−Z_{1}) from Eq205 we obtain T_{1} as
214.) T_{1 }= R_{1}–E = E_{1}– E= Z_{2}Z_{3}V –Z_{1}Z_{2}Z_{3}V = (1−Z_{1})Z_{2}Z_{3}V =U_{1}Z_{2}Z_{3}V =(2/3)(1/3)(1/3)($2700) =$200
Now we can ask what kind of emotion this T_{1}=U_{1}Z_{2}Z_{3}V transition emotion is. We answer that by noting that the T=uV excitement of Eq210 from making the triplet toss and winning the V=$2700 prize can be written, given R=V for success, as
215.) T=uV=uR
And we also see that we can substitute Z_{2}Z_{3}V=R_{1}=E_{1} from Eq113 into the T_{1}=U_{1}Z_{2}Z_{3}V term in Eq219 to obtain T_{1} as
216.) T_{1 }=E_{1}−E=U_{1}Z_{2}Z_{3}V =U_{1}E_{1}=U_{1}R_{1}
The parallel of this T_{1}=U_{1}R_{1} to the T=uR excitement of Eq215 identifies T_{1}=U_{1}R_{1} as the excitement experienced from rolling the 1^{st} lucky number, excitement that is felt even though no money is awarded for rolling just the 1^{st} lucky number. Note that the intensity of this partial success excitement of T_{1}=$200 of Eq214 is much less than the T=$2600 excitement of Eq210 that comes about from making the triplet roll and actually getting the V=$2700 prize.
This development of partial success excitement from the T=R−E Law of Emotion is borne out from observation of situations that go beyond the Lucky Numbers dice game. Excitement from partial success is routinely observed on TV game shows like The Price is Right where a contestant is observed to get visibly excited about entry into the Showcase Showdown at the end of the show, which offers a large prize, by first getting the highest number on the spinoff wheel, which offers no prize in itself. This and other observed examples of the partial success excitement on TV games shows and the like derived as above from the Law of Emotion is a form of empirical validation of the law, even if not a perfectly quantitative validation.
We can further validate the Law of Emotion with this partial success analysis as follows. We understood that what is realized from getting the 1^{st} lucky number is an increase in expectation from the original E=zV=$100 of Eq108 to E_{1}=Z_{2}Z_{3}V=$300 in Eq212. Now we ask what is realized in rolling the 2^{nd} lucky number after the 1^{st} lucky number is gotten. It is a greater expectation yet of making the full triplet roll and winning the V=$2700 prize,
217.) R_{2}= E_{2 }=Z_{3}V=(1/3)($2700)=$900
The transition emotion that comes from rolling a 2^{nd} lucky number after having gotten the 1^{st} lucky number is specified as T_{2 }and from the Law of Emotion, T=R−E, with T as T_{2}, R as R_{2}=Z_{3}V from the above and E as the expectation felt after the 1^{st} lucky number was gotten as E_{1}=Z_{2}Z_{3}V in Eq213, is
218.)
T_{2} = R_{2}−E_{1 }= E_{2}−E_{1}=Z_{3}V−
Z_{2}Z_{3}V =(1−Z_{2})Z_{3}V = U_{2}Z_{3}V
= (2/3)(1/3)($2700) = $600
Expressing T_{2 }= U_{2}Z_{3}V via Z_{3}V=R_{2}
of Eq217 as T_{2}=U_{2}R_{2} makes it clear from its
parallel form to the excitement of T=uR of Eq215 that T_{2}=U_{2}R_{2}
is the excitement felt when the 2^{nd} lucky number is guessed after
the 1^{st} lucky number has been rolled.
And we can also use the Law of
Emotion, T=R−E, of Eq108 to derive the excitement felt in getting the 3^{rd}
lucky number after getting the first two, which wins the V=$2700 prize. What is
realized in that case is the R=V=$2700 prize. Given the expectation that
precedes getting the 3^{rd} lucky number of E_{2}=Z_{3}V
from Eq217, the Law of Emotion, T=R−E, obtains a T_{3 }transition
emotion of
219.) T_{3 }= R−E_{2 }= V –Z_{3}V
= (1−Z_{3})V = U_{3}V = (2/3)($2700) = $1800
Now expressing T_{3 }=U_{3}V from R=V as T_{3}=U_{3}R and noting its parallel form to T=uR excitement of Eq215 identifies T_{3 }= U_{3}R as the excitement of rolling the 3^{nd} lucky number after the first two have already been rolled as obtains the V=$2700 prize. Note that the intensity of the T_{3}=$1800 excitement from rolling the 3^{rd} lucky number and getting the V=$2700 prize is significantly more pleasurable than the T_{1}=$200 and T_{2}=$600 excitements for the antecedent partial successes.
Such significantly greater excitement in actually winning the prize than from achieving prefatory partial successes is what is observed in game shows like “The Price is Right” where actually winning the Showcase Showdown has the winner jumping up and down and running around screaming and showing more excitement than the excitement felt and shown from the prefatory partial success of getting into the Showcase Showdown by getting the highest number on the spinning wheel. Again this fit of observed excitement in the approximate relative amounts suggested in the above analysis with the Law of Emotion constitutes an empirical, if not perfectly quantitative, validation of the Law of Emotion.
Also note that the Law of Emotion, T=R−E, of Eq108 is further validated from the three partial excitements, T_{1}, T_{2} and T_{3} of Eqs214,218&219 summing to the T=uV=$2600 excitement of Eq210 that arises from making the triplet roll in one fell swoop as you might from throwing three pair of dice simultaneously.
220.) T_{1 }+ T_{2 }+ T_{3 }= $200 + $600 + $1800 = $2600 = T = uV
The internal consistency in this equivalence is another validation of the T=R–E Law of emotion. It is also revealing and further validating of the Law of Emotion to calculate what happens when you roll the first two lucky numbers successfully but then miss on the 3^{rd} roll and fail to get the V=$2700 prize, R=0. To evaluate the T_{3} transition emotion that arises from that we simply use the T_{3}=R−E_{2} form of the Law of Emotion in Eq119 that applies after the first two lucky numbers are gotten, but with the R realized emotion as R=0 from failure to obtain the V=$2700 prize.
221.) T_{3 }= R−E_{2 }= 0 – Z_{3}V = – Z_{3}V = −(1/3)($2700)= −$900
This T_{3 }= −Z_{3}V= −$900 is the measure of disappointment felt in failing to make the triplet roll after getting the first two lucky numbers and after experiencing the prefatory partial success excitements in getting them. Note that this T_{3}= −Z_{3}V= −$900 disappointment is significantly greater than the T= −zV= −$100 disappointment of Eq209 that arises from failure to roll the lucky numbers in one fell swoop. And note that the −$800 increase relative to the T= −$100 disappointment in the T_{3}= −$900 disappointment felt after partial success is exactly equal to T_{1}+T_{2}=$200+$600=$800 sum of the two partial success excitements of Eqs114&119. This understands the additional −$800 displeasure of disappointment from failure in the 3^{rd} roll to rescind or negate the prefatory $800 pleasure of excitement that was followed by ultimate failure. This fits the universal emotional experience of an increased let down or disappointment when initial partial success is not followed up by ultimate success in achieving a goal as the letdown felt when one counts their chickens before they hatch and they then do not hatch.
The sequential scenarios that end in success in Eq220 and in ultimate failure in Eq221 universally fit emotional experience and as such are a convincing validation of the Law of Emotion, T=R−E, of Eq108. The linear sums and differences of the transition emotions in these two instances also importantly show that understanding our emotions to reinforce each other positively and negatively in a linear fashion via simple addition and subtraction of the emotion intensity values provides an excellent modeling of our emotional processes regardless of the factor of marginality that affects the linear aspects of emotional intensity.
In a slight digression, everybody knows that we don’t just feel excitement from winning in a game as T=UV but also in anticipation of a win. The above analysis can be used to derive this sense of prefatory excitement felt by people in anticipation of success. To do that consider the emotional state of a person to whom the opportunity to play this triplet V=$2700 prize game is denied or not offered. For that person the expectation of winning is zero, specified as E_{0}=0. It is only when the game is offered and available to play that there is any expectation of winning, namely or E=zV=$100 of Eq208. Much as we saw that sequential increases in expectation produced a T transition emotion of excitement in Eqs214,218&219, so should we also see that this increase in expectation from E_{0}=0 to E=zV=$100 from being offered the game produces from the T=R−E Law of emotion as it did for other increases in expectation a feeling of excitement. Specifically, with T as T_{0} and E as E_{0}=0, the original expectation prior to the game being offered, and R=E=R_{0} as the expectation realized once the game is offered, the T=R−E Law of emotion generates the excitement felt as
222.) T_{0}=E−E_{0}=zV−0=zV
As E=R_{0}=zV and as the probability of success prior to game availability is Z=0 and of failure, U=1, then E=R_{0}=zV can also be understood as
223.) T_{0}= E=R_{0}=zV=UzV=UR_{0}
Again by parallel to T=uR excitement of Eq215, T_{0}=UR_{0} is excitement, the excitement of getting to play the game to begin with. Note that its value is equal to the expectation or the player’s hopes of winning. Next we see in a successful game that expectation increases as each lucky number in the triplet is tossed. As they are, excitement is also felt as we saw in Eqs214,218&219. The difference between the increasing expectations and the excitement that accompanies their experience is that the excitement is cumulative, it adds to prior excitement or builds with progressive success. This very much fits excitement building in a sequential composite effort to achieve a goal. And it explains the origin of the prefatory excitement that, again, is a universal in emotional experience. All of the transitional emotions, whether excitement, disappointment, relief or dismay, can be shown to have this sense of existence prefatory to the final R outcome of success or failure and in the same entirely exact mathematical way.
We next consider the transition emotions of partial success in the v penalty Lucky Numbers game. Consider a game one is forced to play that exacts a penalty of v=$2700 unless the player rolls three of the lucky numbers of 2, 3, 4, 10, 11 or12. In parallel to the E= −Uv expectation of Eq120 and with u=26/27 from Eq207 as the improbability of rolling three lucky numbers, the fearful expectation of incurring the v=$2700 penalty is
224.) E= −uv= −(1−Z_{1}Z_{2}Z_{3})v= −(26/27)($2700)= −$2600
The Law of Emotion, T=R−E, of Eq108 generates a uv emotion of relief from avoiding the v penalty, R=0, when one successfully rolls a lucky number on three pair of dice simultaneously as
225.) T=R−E=0−(−uv)=uv=$2600
With the game played with three sequential rolls on one pair of dice, the increased expectation of avoiding the v=$2700 penalty after rolling a lucky number on the 1^{st} toss of the dice, E_{1}, understood as what is realized from the toss, R_{1}, is, with the improbability of escaping the penalty then as 1−Z_{2}Z_{3}
226.) E_{1}=R_{1}= −(1−Z_{2}Z_{3})v= −(1−(1/9))$2700= −(8/9)($2700)= −$2400
Hence the transition emotion, T_{1}, is, via the T=R−E Law of Emotion expressed as T_{1}=R_{1}−E, and with R_{1}=E_{1} from the above
227.) T_{1}=R_{1}−E= E_{1}−E=−(1−Z_{2}Z_{3})v−[−(1−Z_{1}Z_{2}Z_{3})]=(1−Z_{1})Z_{2}Z_{3}v=U_{1}Z_{2}Z_{3}v=$200
Now recall the parallel forms of the T=UV excitement of Eq115 and the T=Uv relief of Eq121. This understands T_{1}=U_{1}Z_{2}Z_{3}v, in parallel to the partial excitement of U_{1}Z_{2}Z_{3}V of Eq214 for the V prize game, to be the partial relief felt upon rolling the 1^{st} lucky number in the v penalty game. The rest of the analysis for the triplet v penalty game then perfectly parallels that for the triplet V prize game except that the partial emotions felt in sequentially rolling the 1^{st}, 2^{nd} and 3^{rd} lucky numbers are those of relief in escaping a v=$2700 loss of money rather than excitement gotten from a V=$2700 gain of money. The universal fit of this v penalty game analysis to universal emotional experience with sequential behaviors whose goal is escape from a penalty validates the law. And further validating the Law of Emotion and its underlying mathematics is its next deriving the Law of Supply and Demand.
12. The Law of Supply and Demand
The Law of Supply and Demand of Economics 101 states that the price of a commodity is an increasing function of the demand for it and a decreasing function of the supply of it. An alternative expression of the Law of Supply and Demand determines the price as an increasing function of the demand for the commodity and of its scarcity as the inverse of its supply or availability.
Now let’s return to the triplet lucky number V=$2700 prize game with the 1^{st} lucky number in the triplet sequence understood as a commodity that can be purchased. This assumes the existence of an agent who runs the dice game and pays off the V prize money and who will give this commodity of the 1^{st} lucky number to the player for a price. Question: what would be the fair price of the 1^{st} lucky number?
As having the 1^{st} lucky number changes the probability of winning the V=$2700 from z=Z_{1}Z_{2}Z_{3}=1/27 in Eq106 to Z_{2}Z_{3}=1/9 in Eq211, it is certainly a valuable commodity for the player. But exactly what is its value, what is the fair price of it? It is the difference between the E_{1}=$300 average payoff of Eq213 expected when the 1^{st} lucky number has been gotten and the E=$100 average payoff of Eq108 expected prior to any of the three lucky numbers being attained. Or given the symbol, W_{1}, the fair price for the 1^{st} lucky number is
228.) W_{1 }= E_{1}−E
This W_{1} fair price is a function of a number of variables associated with the E_{1}−E term in Eqs214&216.
229.) W_{1 }= E_{1}−E = Z_{2}Z_{3}V – zV = U_{1}Z_{2}Z_{3}V = T_{1} = U_{1}E_{1 }=$200
This W_{1}=$200 is the fair price for the 1^{st} lucky number in the latter increasing the average payoff from E=$100 to E_{1}=$300. From the perspective of economic optimization the player as buyer would want to pay as little as possible for the 1^{st} lucky number and the agent as seller would want to charge as much as possible for it. But W_{1}=$200 is the fair price of the 1^{st} lucky number from that price paid by the player effectively maintaining the initial average payoff of E=$100 for the player.
The fair price expressed in Eq229 as W_{1}=U_{1}E_{1 }is a primitive form of the Law of Supply and Demand given in terms of the emotions that people feel that control the price they’ll pay for a commodity. W_{1}=U_{1}E_{1} is an increasing function of the scarcity of the 1^{st} lucky number as the uncertainty in rolling it on the dice, U_{1}=2/3, and is an increasing function of the demand for the 1^{st} lucky number as the E_{1}=Z_{2}Z_{3}V=$300 average payoff it provides with its value as such understood as the underlying determinant of the demand for it. This derivation from the emotion mathematics of the Law of Supply and Demand, a firm empirical law of economics universally accepted as correct, is a powerful validation of it.
There are a number of important nuances in this formulation of the Law of Supply and Demand. Note the equivalence in Eq229 of the W_{1}=$200 fair price for the 1^{st} lucky number and the T_{1}=W_{1}=$200 pleasurable excitement gotten from rolling the 1^{st} lucky number on the dice. This equivalence of T_{1} excitement and W_{1} price suggests that the price paid for a commodity is a measure of the pleasurable excitement the commodity generates for the buyer. This fits economic reality quite well as seen from TV commercials for automobiles and vacations and foods that hawk these products by depicting them as exciting.
And, further, the value of the 1^{st} lucky number can be calculated not just in terms of the W_{1} amount of money one would spend for it but also in terms of the time spent in acquiring the money needed to purchase the 1^{st} lucky number. Given that the time taken to obtain money, risk based investment aside, is directly proportional to the money earned as in a dollars per hour wage, W_{1} is understandable as a measure of the amount of time spent to get the 1^{st} lucky number. This has the W_{1}=U_{1}E_{1}=T_{1} Law of Supply and Demand showing that people spend their time to obtain commodities that pleasurably excite them, whether as the time spent working to get the money to spend on the commodity or as time spent directly to obtain pleasurable excitement like watching the Super Bowl for some.
We can also derive a parallel primitive Law of Supply and Demand from the v penalty game that requires the toss of three lucky numbers to avoid the penalty. The W_{1} fair price of the 1^{st} lucky number is again W_{1}=E_{1}−E, but with E_{1}−E from Eq227 in the v penalty as
230.) W_{1 }= E_{1}−E = U_{1}Z_{2}Z_{3}v = T_{1} = $200
This form of the primitive Law of Supply and demand tells us that people also spend their money for commodities that provide T_{1}=U_{1}Z_{2}Z_{3}v=W_{1} relief. This is in addition to commodities that provide T_{1}=U_{1}Z_{2}Z_{3}V=W_{1} excitement as seen in Eq229. The two forms of the Law of Supply and Demand of Eqs229&230 provide a strong empirical validation of the Law of Emotion of Eq108 that underpin them from the observed fact that people do spend their money and their time to obtain commodities, goods and services, that provide relief and excitement. This is readily seen in the complete spectrum of TV ads, all of whose products are pitched in ads as providing relief, as with insurance and antacids and other medicines, or excitement, as with exciting cars, foods and vacations.
Next we want to express the primitive Law of Supply and Demand in as simple a form as possible. We do this as preface to our deriving in the next section simple functions for our visceral emotions like the pleasures of feeling warm and of eating food. In Eq229 we saw the equivalence of the W_{1} fair price with T_{1} partial success excitement, W_{1}=T_{1}. This implies that the simplest form of T excitement we have seen in Eq115 as T=UV for the one number Lucky Number game should also be a measure of the fair price, W, one would pay for this one lucky number.
231.) W=T=UV
Now in recalling Eq116 we see the value of the T excitement in getting the V=$120 prize to be T=$80, which allows us to express the fair price of being given the lucky number that gets the V=$120 prize as
232.) W=T=UV= $80
At first this may seem odd. One may ask what sense there is in paying $80 to win the V=$120 prize. The point is rather that W=$80 is the fair price that you would pay. Consider what happens if you do this for three games, with the total price paid being 3($80)=$240. This wins the player $120 in each game for a total of 3($120)=$360 for the three games. The net winnings for the three games are, thus, $360−$240=$120. And this is what is won on average in three games played strictly from the throw of the dice with no lucky number purchased. That is V=$120 is won one time out of three. Hence W=T=$80 is, indeed, the fair price of the lucky number. And W=T=UV is a most simple form of the Law of Supply and Demand with U as the uncertainty in rolling the lucky number as a measure of its scarcity and V as the cash value of the prize as a measure of the demand for it.
Next we want to write this most simple form of the Law of Supply and Demand with a slight algebraic manipulation as
233.) W= UV= −(−UV)
This tells us that people spend W dollars or spend equivalent time both to obtain UV excitement and to negate or eliminate –UV anxiety as very much fits universal emotional experience. And without our going through the details of its derivation or explanation we can write an equivalent simple Law of Supply and Demand pricing law based on T=uV relief of Eq121 with W=T assumed from earlier considerations as
234.) W= Uv= −(−Uv)
This tells us that people also spend W dollars or spend equivalent time both to obtain Uv relief and to negate or eliminate their –Uv fears, again as very much fits universal emotional experience. To sum up for emphasis, this mathematics derives people spending their money and time, being motivated to do that, both in the pursuit of the pleasures of excitement and relief and in the avoidance of the displeasures of anxiousness or anxiety and fear. Understanding behavior to be motivated by the pursuit of pleasure and the avoidance of displeasure is the essence of hedonism. It should be made clear that this sense of hedonism is not an encouragement for people to seek pleasure and avoid displeasure, but rather a conclusion drawn from the foregoing mathematical analysis that people just do behave so as to achieve pleasure and avoid displeasure as the essence of human nature. To generalize hedonism you need, of course, to also take into consideration the visceral emotions that motivate our behavior at the most basic levels like hunger and feeling cold and the pleasures of eating and warmth along with the pleasures and displeasures of social and sexual behavior, which we will begin explaining mathematically in the next section.
13. Survival Emotions
Many of our most basic emotions are associated with surviving or staying alive. We derive the pleasant and unpleasant emotions that drive survival behavior from the primitive Law of Supply and Demand in the form of Eq234 of W=Uv= −(−Uv). We do that by applying it not to avoiding the loss of v dollars but to avoiding the loss of one’s v*=1 life. That is, the penalty for failing at a survival behavior like getting food to eat or air to breathe is the loss of one’s own v*=1 life rather than the loss of v dollars. The other terms in W=Uv= −(−Uv) are also asterisked in using it to explain survival behavior to show that they are all associated with avoiding the loss of one’s V*=1 life rather than the loss of v dollars.
235.) W*=T*= U*v*= −(−U*v*)
It is best to introduce this function with specific survival behaviors and save the generalizations of what these variables mean until after we do that. Let’s start with the survival behavior of breathing air whose emotional properties are cut and dried. Consider Eq235 for a situation where air to breathe is lacking whether from a person being underwater and drowning or having a critical asthmatic attack or having a pillow placed forcibly over his face or being water boarded. From Eq235 understood as the Law of Supply and Demand, U* is a measure of the scarcity of air as the uncertainty or improbability of getting air. We can assign a very high value to it in this case of suffocation of, say, U*=.999, also interpretable as the high probability of losing one’s v*=1 life under these circumstances.
The T*=U*v* transition emotion in Eq235 experienced when a behavior is done to obtain air under this U*=.999 circumstance is, in parallel to T=Uv relief of Eq21, the very pleasurable relief felt in getting air to breathe when one is suffocating. While not all have had the experience of suffocation followed by escape those who have will attest to the great intensity of the pleasurable relief felt. One measure of this great relief is from Eq235 evaluated for the v*=1 life saved and its prior U*=.999 scarcity of air or uncertainty in getting it as
236.) T*=U*v*=(.999)(1) =.999
This .999 fractional measure of relief very close to unity, 1 or 100%, is a good way of indicating an intensely pleasurable level of relief. We can also specify the relief in dollar terms as we did in the Lucky Number games by putting a cash value or price on one’s v*=1 life, the one that one doesn’t want to lose. One measure might be if one was alone in the world, all the money one had, let’s say v*=$100,000. That calculates a cash value for the T*=U*v* relief of
237.) T*=W*= –(–U*v*)=U*v*=(.999)$100,000=$99,900
This effectively says that one would pay a price of W*=$99,900 or pretty much all of one’s money to escape terminal suffocation, which is true of all with the above assumption of nobody else to worry except the pathological. The –U*v* term in Eqs235&237 that is negated or resolved by the behavior of escaping suffocation to T*= –(–U*v*)=U*v* relief is a measure of the fear instinctively felt upon suffocation, parallel to the E= −Uv fear in Eq120 of losing money in the Lucky Numbers v penalty game.
The W*=T* equivalence of Eq235 also makes clear that the W*=T*=U*v function that governs the emotional dynamic operates as the Law of Supply and Demand with the demand for some commodity, be it goods or service, object or behavior, that provides escape from suffocation and preservation of one’s v*=1 life measured by the instinctively great value a person places on his or her life; and with the supply of what is needed to preserve that life measured inversely by the scarcity of air to breathe or uncertainty in getting it as U*.
The fact that we can so simply derive the emotions of breathing under suffocation, the panic fear it causes and the great relief experienced in escape from the suffocation, is a remarkable validation of Eq235, W*=T*_{}=U*v*= −(− U*v*), and of its derivation from the cash based Lucky Numbers game. It gives confidence that this mathematical understanding of man’s emotional machinery can impact the central problem for mankind of unhappiness from e