A Slot Machine Treatise
on Emotion and Politics

Disappointment is the emotion felt when hopes are dashed a slot machine tells us mathematically. Indeed slot machine play reveals all the basic human emotions in a precise mathematical way. The importance of this for politics lies in the unhappiness caused by financial stress, workplace servitude, police brutality, legal restriction and personal failure from such control being based on the emotions of disappointment, dismay, anxiety, fear, sadness and depression.  

The properties of these other of our emotions are hard to study scientifically because people make their feelings known to others in only a sketchy way and when they do, often not honestly particularly in this era of rampant paranoia and mistrust. Slot machines solve this problem of revealing the emotions analytically by correlating what slots players feel with the exact quantifiable properties of the probability of winning at slots play and the cash payoffs.

This providing a precise mathematical specification of the emotions felt by a slots player including his or her expectation or hopes of winning money, anxiety at the thought of losing, the excitement felt in winning and the disappointment, dismay and depression felt in losing. One then extrapolates the slot machine based equations for the basic emotions to all feelings people can have with mathematical functions for evolution derived from the early 1920s work of the classical population biologists, R.A. Fischer, J.B.S. Haldane and Sewell Wright.  

What is striking beyond deriving precise functions for the emotions, a first in science in itself, is their perfect fit to the mathematics of evolution, a fit of emotion to evolution that mathematically proves the mind was designed by evolution rather than, in incorrect inverse fashion, the products of evolution being intelligently design by a mind belonging to some superhuman deity. Just as Newton’s mathematically firm Law of Gravity disproved the childish medieval belief of angels steering the planets around in their orbits, so these mathematically firm laws of emotion developed from slots playing disprove the equally childish belief of the intelligent design of the living world by a mythical spirit being.  

This will hardly come as a surprise to biologists, who don’t need a mathematical proof of already quite obvious Darwinian evolution, but is sure to upset science rejecting right wing politicians presuming they have yet to cross the threshold of denying even 2+3=5 mathematics. One might expect, though, on Fox News soon the female expert in a low cut dress making it clear that, of course, God can make 3+2 be something other than 5 if He wants to. This treatise makes clear that evolution denying Republican would be a most dangerous choice for president in lacking the logical thinking needed to steer America away from the cliff edge economic, social and military challenges it faces today.

This is not to tout Obama for reelection we should make clear, for we are equally disdainful of this corporate get-along disguised as a man of the people. After initially working hard for his election in 2008 because like many we were overwhelmed by his practiced oratorical skills, we quickly came to see him as the standard politician he is after getting close up to his campaign in the West Virginia primary as is made clear in the newspaper article we wrote against him in 2008 in the San Francisco Chronicle.

As an alternative to both mainstream candidates our family of scientists makes available mathematician, Ruth Calabria, for president. She’s an excellent choice both in being a woman, exactly what’s needed as a change from the hyper-aggressive male leadership that has dominated government and messed it up so badly recently, and in her sparkling intelligence as co-creator of this mathematics of the emotions. Our teen grandson, Thomas, urges all his age to work for his grandmother’s election so they can have a life to live when they grow up. We also hope Jon Huntsman, the only Republican with any sane thinking, comes on board to form a bipartisan third party ticket that has some real chance of winning the White House in 2012.

 

As to how we came across our slot machine analysis of the emotions, five of us came to Las Vegas in 2010, not to gamble, but to help Harry Reid get reelected by sending out a quarter million emails for him over Tea Party wingnut, Sharon Angle. But while in town it was impossible not to dabble in the penny slots that ultimately led to our developing mathematical functions for the human emotions. From that perspective one might give the casinos some credit in passively assisting in this scientific breakthrough in their providing the slot machines that made it possible. But looking at the “gaming industry” from a more meaningful perspective shows the casinos to effectively tamper with elementary probability theory in its fleecing people out of their money in a con game that’s a microcosm of the broader ripoff that all of America has come to be.  

Let me explain. In probability theory, the probabilities of the possible outcomes must add to one. If flipping a bent coin whose probability of landing on Heads is 6/10, the probability of Tails must be 4/10 in order to have the probabilities of the two outcomes, Heads and Tails, add up to one, (6/10) + (4/10)=1.

This gives a ballpark sense of where one stands when one gambles in a casino. They have a good chance of winning your money and you have a poor chance of winning theirs. Indeed Laplace’s Law of Large Numbers tells you that unless you are the proverbial one in a million, you must lose your money over time with people losing it often in a way that shocks the loser who was made to feel beforehand by every piece of information, tacit and subtle, the casinos can drive into the gambler’s emotion dominated mind, that he or she actually had a reasonable chance of winning. Just how shocking losing all the money in the wallet or purse and bank account is, is made clear by these stories not displayed on CBS or Google.

·  Man commits suicide in fall from Binion’s in downtown Las Vegas
·  Man commits suicide by leaping off Cosmopolitan in Vegas
·  Man commits suicide by leaping off Caesars Palace parking garage in Vegas
·  Man commits suicide by jumping off 11th floor of El Cortez hotel in Vegas
·  Man dies in fall from Venetian hotel parking garage in Las Vegas

Essentially what casinos do is wiggle around the laws of probability by making the gambler feel with a variety of bells and whistles distractions that he or she has a good chance of winning as would make the 4/10 real probability of the bent coin landing on Tails be equally in the range of 6/10 so as to make the probabilities of the casino winning, 6/10, and of the gambler winning, 6/10, add up to more than one, (6/10) + (6/10) >1. This is quite impossible according to the firm mathematics of it whatever the gambler may be made to feel emotionally.       

This contradiction of probability theory comes from stratagems that jigger the player’s emotions towards confidence in winning when the player should logically be wary that he or she is about to be fleeced. The emotional jiggering specifically consists of the casinos pumping the emotion of excitement into the brain of the player from loud exciting music and bright flashing lights and free liquor and the grandiose architecture of the casinos, excitement we shall show mathematically that automatically engenders confidence in a person. This infusion of excitement into the Las Vegas tourist also derives from the exciting entertainments of all kinds found on every corner of the city directed to keeping the spirits elevated and emotionally numb of the fleecing about to happen, a fleecing manifest in the billion dollars that flows from the players to the casino owners each month, every month, as sure as the sun coming up in the morning.   

Particularly disgusting in this carnival con game twisting of the emotions and the player’s real chances of winning are the smiley faced casino employees who repeatedly wish the player “good luck” when they all know perfectly well that just about everybody goes home a loser. This extends all the way to the poker games where the skim of the pot makes sure that almost all players will be losers over time, not to speak of the double dealing so easy to do for a professional Vegas poker dealer and so lucrative.

If there is any silver lining to this dark cloud of the casino ripoff, it is that its obviousness makes it the perfect microcosm for revealing how the broader society of ripoff America works. As in Las Vegas, while it is not completely impossible to win in the casino of American life, the probability of being happy is exceedingly small for in this hierarchical workplace slave colony where the bosses run the games and the ruling class are the casino owners, whatever the American Dream delusion fed the young and the distracting excitements of movies, sports and computer games that emotionally jigger the true low probability of finding happiness, the real chance of it is very low with the rampant broken families and relationships and 25% of the nation living on pills and mass murders and suicides that manifest the unhappiness attributed in the media to mental illness or to the devil or to your own fault, in short, to anything but the socio-political system that is the cause of the unhappiness. Much as casino employees consider their patrons as an alien species, the suckers, though it is not broadcast openly, our ruling class feels the same way about the 99% of us under their control who support their lives of absurdly luxurious decadence.

All this is hard for the emotion dominated human mind to fathom because the unhappiness caused by socio-political control is actually reduced by these entertainments and by culturally approved delusions, however ridiculous, like being happy after death and by drugs, legal and illegal, and by the intake of food beyond physiological need. Indeed, the rampant obesity of America is nothing but a manifestation of the sadness and frustrations of people being partially soothed by the non-stop compulsive stuffing of their mouths with food.

One such artificial lessening of unhappiness that is particularly troubling and dangerous is people sloughing off their bad feelings on others in aggressive ways. The violent tendency of unhappy people to scapegoat and treat vulnerable innocents badly is the cause today not only of the petty meanness that dominates our daily lives as the pain laid on workers by unhappy bosses, on students by unhappy teachers, on children by unhappy parents, especially corporal punishing conservative parents, and from bullying generally, but also of the mass murders that have become so common these days that we accept the daily reports of them as a normal part of American life.

That unhappiness provokes an instinctive aggressive response is clear in the justifiably angry street rebellions seen in so many nations of the world these days, including in Russia with its demonstrations against the Putin dictatorship and in America with the Occupy movement. It is when such tyranny provoked resistance is bottled up by fear of punishment - as it is in most people most of the time - that the aggressive emotions worked up by abusive social control become violently misdirected to people who had nothing to do with the abuse to begin with. And the worst of that goes beyond the cruelties inflicted within a nation to the misdirected aggression done to people in other cultures that is the cause of unnecessary war that serves little purpose other than to release the pent up anger of a nation’s unhappy people. Indeed the street rebellions seen in so many countries these days is a dark harbinger of the aggressions within unhappy nations spilling over to violence between nations as war eventually on a grand global scale.   

This should be particular distressing to people at this time in man’s history because the escalation of the tempest of Mideast wars to the next level with the invasion of Iran and such is sure to bring nuclear weapons into play soon enough as will ultimately spell the end of all of us unless great changes come about in the way people think and behave. Unfortunately the reality of the high probability of nuclear destruction, which would be a warning to us all that something dramatic must be done to avoid it, as with the probability of bad outcomes in casino gambling, is swept under the rug with emotional distractions and misinformation to hide the great need for the true revolutionary change needed to save the world. Our family’s concern with this problem especially as it impacts the children in the world who don’t deserve to be incinerated and radiated to death because of the stupidity and callousness of the world’s ruling class starting with this newspaper article written by us during the Cold War.

Knickerbocker News, Albany, NY, May, 1986

The article speaks for itself, though to fully appreciate the high probability of the nuclear nightmare being realized requires that one read the mathematics that follows. It starts with the mathematics of human emotion and then of evolution followed by the correct formulation and clarification of the century old mystery of entropy that demonstrates the power of the mathematics we use to revise the currently ideologically corrupt pseudo-science of psychology. And that is topped off with a mathematical unification of emotion, evolution and entropy and all knowledge using the 1949 Simpson’s Diversity Indices rediscovered by us to provides a new foundation for mathematics and for science generally.  

Those for whom the mere thought of mathematics produces fear and nausea but who still want to get some feel for our politics can link to songs written back 25 years ago that I played to put my son, Benito, to sleep with when he was a baby, Songs from a Rebels Past. The reader is warned, though, that this mathematical scientist makes no claim to being a professional musician and that the songs are no substitute for the mathematic analysis that follows.

Copyright: Feb. 7, 2012, Peter & Ruth Calabria & Family
Contact: petercalabria@matrix-evolutions.com

Those who want to encourage our disarmament, scientific and political efforts with a contribution can do so the old fashioned way by mailing a check or money order to Ruth at:

Ruth Calabria
P.O. Box 3035
Albany, NY 12203

An Introductory Sketch of the Mathematics of Human Emotion

Disappointment is the emotion felt when hopes are dashed. The greater your hopes or expectations, the greater your disappointment when your hopes are dashed. It’s easy to mathematically characterize the hopes or expectation people have when they play a slot machine because the mathematical expectation is numerically just the average payoff per play as the product of the probability of getting a winning set of symbols on the slot machine times the payoff it gives. So it’s also easy to mathematically characterize the disappointment felt when expectations fail, which is just the negation or negative of the mathematical expectation, and from there it’s also a simple matter to mathematically characterize the excitement felt in winning and all the other basic human emotions people feel not only when they play a slot machine but in all other circumstances, too.  

Thus we begin the much needed complete restructuring of mankind’s thinking by developing a mathematical science of emotion and behavior whose validity is proven with slot machine play. What do we mean by proven? Newton’s Law of Gravity is proven by its fit to data on the movements of the planets in the nighttime sky and the movements of objects falling to the ground here on Earth, data that everybody agrees on. In the same way the Law of Emotion we will develop is proven by its fit to the emotions felt by slots players, emotions that all slots players agree they feel.

The slot machine we use to illustrate this perfect fit of mathematics to emotional reality is a super simple slot machine that has only three symbols, a red seven, 7, a green seven, 7, and a purple seven, 7. These three symbols appear on a three reel pay line. The 27 ways the “sevens” can appear on the three reels are listed below.

The 27 Ways of Three “sevens” on a 3-Reel Pay Line of a Slot Machine

Only the 3 triplet matches of [7, 7, 7], [7, 7, 7], and [7, 7, 7] give a payoff, one of V=$270 on this simplified slot machine that is different from the usual casino slot machine in that no money is required to play it. As each color “seven” has an equal chance of appearing, each of the 27 ways has an equal chance of appearing, so the probability of getting a triplet match and winning the V=$270 payoff is Z=3/27=1/9. The mathematical expectation in this slots game is the probability, Z, multiplied by the payoff, V, or

E=ZV

For this game with probability, Z=1/9, and payoff, V=$270, 

E=ZV=1/9($270)=$30

Mathematical expectation has two interpretations. One is the average payoff per slots game played. If you play 9 times, on-average you get the V=$270 payoff 1 time in 9 or E=$30 per play. The other interpretation of E=ZV is its emotional interpretation of the hopes you feel of winning the V=$270 payoff. The greater the V payoff the machine pays off for the winning triplet match, the higher your hopes and the greater your pleasure in anticipating the V payoff. That is, it’s more pleasant hoping for a V=$27,000 payoff than a V=$270 payoff. And the greater the Z probability you feel you have of winning, the higher your hopes and the pleasure you feel in anticipating winning. 

Now consider how you feel when you get only one shot at getting a V=$270 payoff and you lose and your hopes are dashed. What you feel then is disappointment. Not surprising your feeling of hopes dashed is represented by a negation of the E=ZV hopes you had or 

T= −ZV

The negative sign in T= −ZV disappointment says that it is an unpleasant feeling. This T= −ZV function for disappointment quite fits how people actually feel in telling us that the bigger the V payoff expected, the more disappointed a person is when he fails to get it and that the more sure a person is of getting the V payoff, the bigger the Z probability of getting it, the more disappointed he is if he fails to get it. It is when a person feels almost completely sure he will get a payoff, say Z=.99, that he feels the most disappointed when he fails to get it. So the T= −ZV representation of disappointment is a perfect fit of how people actually feel when they fail to get V dollars they expected.

Now let’s ask how a person feels when he or she does get the [7, 7, 7], [7, 7, 7], and [7, 7, 7] triplet match and win the V payoff. They feel a thrill or rush of excitement, the bigger the V payoff, the bigger the excitement. Also the greater the uncertainty about winning felt ahead of time affects the amount of excitement felt, the greater the uncertainty, the greater the excitement. The uncertainty over winning, given the symbol, U, is just the converse of the Z probability of winning,

U=1−Z

If you have a Z=1/9 chance of winning, then you have a U=1−Z=8/9 chance of not winning, which your sense of it is your uncertainty. To see better that the amount of excitement felt when you win money depends on the amount of uncertainty you feel beforehand, consider feeling absolutely sure of getting a $1,000 weekly pay check with no uncertainty. While you still feel pleasure in getting a $1000 paycheck when there is no uncertainty, it is not an exciting thrill like winning $1,000 on a spin of the roulette wheel where there is uncertainty. So the amount of excitement felt is a function of how big the V payoff is and how much U uncertainty you have beforehand or 

T=UV

The intensity of excitement for the Z=1/9, U=1−Z=8/9, V=$270 slots game is T=UV=$240. Note that T=UV is a positive function, which tells us mathematically that excitement as a pleasant feeling, much as –ZV disappointment as a negative function tells us that it is an unpleasant feeling. The excitement you feel in winning money is a feeling over and above the most basic pleasure of getting (or realizing) money. We’ll symbolize that basic pleasure as

R=V

The more V dollars are gotten or realized, the greater the pleasure of delight or elation in getting them. That is independent of the excitement of getting the money, which is an additional feeling that is a function of your uncertainty beforehand. Also independent of the disappointment you feel when you don’t win, we can write out how you feel from getting or realizing no money, which in itself is no feeling at all or 

R=0

So in quick review when you have E=ZV expectation and get nothing or R=0 you feel T=−ZV disappointment. And when you do get the triplet match and get the payoff of R=V you feel T=UV excitement. Now if you look carefully at these functions we have specified for the initial emotion of expectation, E=ZV; for the realization emotions of R=0 or R=V; and for what we will call the transition emotions of T= −ZV disappointment when you get nothing, R=0, and T=UV excitement when you get the payoff, R=V, you see a general pattern for the T transition emotions deriving from the E expectation emotion and the R realization emotion of 

T = R−E

This is the Law of Emotion. The T= −ZV disappointment comes about from it when you have E=ZV expectation and you fail to get the payoff, R=0.

T = R−E = 0 –ZV = −ZV

And T=UV excitement comes about from the Law of Emotion when you have E=ZV expectation and succeed in getting the payoff, R=V.

T = V−ZV = (1−Z)V = UV

This makes it clear that T=R−E is a general Law of Emotion because it works in both cases, whether you win or lose. Now we can make the proof of the Law of Emotion even stronger by using it to generate the excitement a slots player feels when the first two symbols on the pay line come up matching as [7, 7] or [7, 7] or [7, 7]. When the doublet match shows it tells the slots player that he is part on his way to getting the triplet match of [7, 7, 7] or [7, 7, 7] or [7, 7, 7] and winning the V payoff, which makes him feel some excitement. 

Here’s how we develop that from the T=R−E Law of Emotion. Note that there are 9 doublet matches in the 27 ways the “sevens” can show on the pay line. 

The 27 Ways Three “sevens” Can Show on a 3-Reel Pay Line of a Slot Machine

That tells us that the probability of getting the doublet match is

Z_A=9/27=1/3

And the probability of getting the right color “seven” on the third reel to match the doublet match and make a winning triplet match is

Z_B=1/3

Since you need both the doublet match and the third “seven” to match it to get a winning triplet match, it should not be surprising that the product of these two component probabilities, Z_AZ_B, is the probability of getting the triplet match we saw earlier of Z=1/9.

Z = Z_AZ_B = (1/3)(1/3) = 1/9

Now we want to ask what it is that is realized when we get the doublet match of [7, 7] or [7, 7] or [7, 7]. It is not R=V, for we do not get a V payoff by just getting the doublet match. And it’s not R=0, which is what is realized when the slots player loses. Rather what is realized when the doublet match appears is an increased expectation of getting the triplet match, an expectation greater than the original expectation of

E = ZV = (1/9)($270) = $30

The increased expectation of winning the V=$270 payoff once you have the doublet match is given the symbol E_A and it is just the product of the V payoff and the Z_B=1/3 probability of getting the “seven” on the third reel to match the doublet match.

E_A=Z_BV=(1/3)(270)=$90

As this E_A expectation is what is realized, we can write it as a realization using the R symbol as R_A.

R_A=E_A=Z_BV

Now we can use the Law of Emotion, T=R-E, to obtain what emotion is felt when we realize the R_A=E_A=Z_AV increased expectation. We just use the Law of Emotion, T=R−E, expressed in modified form for the emotion felt when the doublet match is achieved as T_A=R_A−E 

T_A= R_A-E = Z_BV−ZV = Z_AV-Z_AZ_BV = (1−Z_A)Z_BV = U_AZ_BV = (2/3)(1/3)V=(2/9)($270)=$60

To find out what kind of emotion T_A=U_AZ_BV is, let us now recall that 

Z_BV=R_A

This allows us to write T_A=U_AZ_BV as

T_A=U_AZ_BV=U_AR_A

Now what kind of emotion might T_A=U_AR_A be? To find out, let us recall that the excitement of getting the triplet match and the V payoff, R=V, was T=UV. The R=V relationship allows us to write the T=UV excitement as

T=UV=UR­

Now if T=UR excitement in getting the triplet match is the product of the U uncertainty of getting the triplet match and the R realization of the triplet match, so T_A=U_AR_A emotion of getting the doublet match, which is the product of the U_A uncertainty of getting the doublet match and the R_A realization of the doublet match must be the excitement of getting the doublet match. Note that the T_A=U_AR_A=$60 excitement from getting the doublet match is much less than the T=UV=$240 excitement from getting the triplet match as fits the actual feelings of slots players. This precise derivation of the excitement felt by a slots player upon getting the doublet match on a pay line is sure proof of the validity of the Law of Emotion, T=R−E, an enormous breakthrough in precisely describing the workings of the emotional machinery of the human mind, a first for science.

As impressive as the above proof of The Law of Emotion is, even more validating of it is the formal continuation of the mathematics of the emotions below that derives the universally accepted Law of Supply and Demand of economics and explains in detail why the poker room boss killed himself.  Or proceed to the fairy tale of the The Wizard of Odds if you wish and toggle back and forth between it and the math below to fill in the big picture. 

A More Formal and Complete Mathematics of Human Emotion

We develop mathematical functions for the basic human emotions in a more formal way using the same simple slot machine game. All of what we said above is repeated, though in a more detailed way, and then extended to explain with mathematical precision all the things you always wanted to know about that are systematically obfuscated by the media, religious dogma, political ideology and ideologically corrupt pseudo-science psychology. 

If you know anything about slot machines you know that they are made up of a number of reels marked with various symbols like “lucky sevens” that pay off when the symbols come up matching. Slot machines provide an excellent way to mathematically develop the emotions because those felt by slot machine players such as hope, disappointment, anxiety and excitement are functions of numerically exact probabilities and payoffs.

The slot machine we use is a simple one consisting of three reels and three symbols that show on the pay line, a red seven, 7, a green seven, 7; and a purple seven, 7. A list of all 27 configurations of the three “sevens” that can show on the three reels is:

Table 1. The 27 Configurations of a 3-Reel, 3-Icon Slot Machine

Only the 3 triplet matches of [7, 7, 7], [7, 7, 7], and [7, 7, 7] give a payoff, one of V=$270 in this slots game that is different than the usual casinos slots in that no money is required to play. As the 3 “sevens” are stipulated to equiprobable chance of 1/3 of appearing, each of the 27 configurations has an equiprobable chance of 1 in 27 of appearing, which makes the probability of one of the 3 winning triplet matches appearing Z=3/27=1/9. The mathematical expectation of winning in this slots game is the probability, Z, multiplied by the payoff, V.

2.)          

The objective interpretation of the mathematical expectation is as the average payoff per game. For our slot machine with Z=1/9 and V=$270. 

3.)          

This tells you that if you play repeatedly, you will get the V=$270 payoff 1 time in 9 for an average payoff of E=$30 per play. A subjective interpretation of this E=ZV=$30 expectation is as a measure of the emotion of your hopes of getting the V=$270 payoff, which can be understood as the emotional equivalent to what one would feel if they were to receive $30 with a 100% surety. The expectation of getting money is universally a pleasant one. If the V payoff is increased to V=$2700, the pleasure of the expectation increases measurably to

4.)          

And if we altered the game in some unspecified way to increase the Z probability of winning to Z=2/3 the pleasure in the expectation of winning would increase, measurably for the V=$270 payoff to

5.)          

Assumed in the above is that the player supposes his chances of winning to be the actual Z probabilities. Now consider how you would feel if you played one time for the V=$2700 payoff with the probability of winning of Z=1/9 and you didn’t win. You would feel disappointed. And if you were playing for the smaller payoff of V=$270, you would feel less disappointed. And if you were playing for an even smaller payoff of V=$27 and didn’t get it, you would feel even less disappointed yet. This suggests that disappointment is an increasing function of the size of the V payoff expected.

Now note that if you played for the V=$2700 payoff in the Z=2/3 game of Eq4 you would feel more disappointed if you lost than if you played for the V=$2700 in the Z=1/9 game of Eq3 and lost because you had less expectation of winning to begin with, in the latter case. And if you played for the V=$2700 payoff in a game that had a miniscule probability of winning of Z=1/900, you would feel little disappointed because you felt you really had no chance of winning to begin with. This suggests that disappointment is an increasing function of the Z probability of winning you suppose to begin with.

The increase in disappointment as an increasing function of the V payoff expected and of the Z probability of getting it has us understand disappointment as the product of V and Z or ZV. That this ZV representation of disappointment, which is the same function as the ZV=E expectation or hopes of Eq2, is reasonable given disappointment being generally understood as hopes dashed. But as ZV disappointment is an unpleasant feeling in contrast to E=ZV hope being a pleasant one, we differentiate between the two ZV measures by placing a negative sign in front of it as −ZV when it represents disappointment.

Disappointment is the emotion felt when one fails to win. When one succeeds at winning the V dollar payoff by getting the triplet match of [7, 7, 7] or [7, 7, 7] or [7, 7, 7], the emotion felt is the thrill or excitement of winning, a pleasant feeling. The greater the V payoff, the greater is the excitement, winning V=$2700 being more exciting than winning V=$270. The excitement of winning in a game of chance is also a function of the uncertainty of winning defined as the probability of not winning or

6.)          

When the probability of winning is Z=1/9, the uncertainty is U=1−Z=8/9. That the excitement in winning is an increasing function of the U uncertainty felt beforehand is easy to demonstrate. If you are absolutely sure of getting V=$270 dollars with Z=1 and no uncertainty or U=1−Z=0, while there is still pleasure in getting the V=$270 dollars as might be felt in getting a sure V=$270 weekly paycheck, there is no excitement in contrast to winning V=$270 in a game of chance where there is uncertainty. And if on the other hand, the Z probability of winning V=$270 is very low as with Z=1/900 and, hence, the uncertainty very great as U=1−Z=899/900, there is great excitement in winning. This understands the thrill or excitement in winning to be an increasing function of V and U as UV, positively signed to indicate that UV excitement is a pleasant emotion.

A familiar example of the thrill of getting something of value as a function of one’s uncertainty beforehand is of Christmas presents for kids. Their uncertainty of not knowing what’s in the carefully wrapped packages under the Christmas tree is what makes for the excitement of opening them on Christmas morning, this pleasure of excitement being an additional pleasure for them on top of the pleasure in receiving the gift itself, for the thrill on Christmas Day of opening the presents is not felt if the youngster knows ahead of time what the present is and has no uncertainty about it.

In brief review we have developed three very basic emotions so far: of E=ZV as your expectation or hopes of getting something of value; of −ZV as your disappointment when you don’t get what you hopes for and your hopes are dashed; and of UV as the excitement you feel when you do get what you hoped for.

Note that getting something of value produces a pleasant feeling independent of the pleasure of excitement that derives from any U uncertainty about getting it felt beforehand. Paychecks gotten with no uncertainty and presents received whose value and contents you have no uncertainty about beforehand are still pleasant to receive. In the slots game, the feeling that comes from getting the V payoff independent of the excitement in it is a function of the size of the V payoff. The bigger the V payoff, the greater the pleasure of elation or joy in getting it as symbolized by R=V with the R symbol specifying the emotion felt as a realized emotion, one that arises from actually getting something of value in contrast to the expectation emotion of E=ZV that arises from anticipating something of value. Note that when no payoff is gotten or realized, there is no realized emotion, which is specified as R=0.

There is a simple functional relationship between the expectation emotion, E=ZV, the realized emotion, R, and disappointment and excitement categorized together as transition emotion, T.

7.)          

With the expectation as E=ZV, when no money is won and the realized emotion is R=0, the Emotion Equation of Eq7 specifies the T transition emotion to be the –ZV disappointment felt when no money is won.

8.)          

And with the expectation as E=ZV, when V dollars are won and the realized emotion is R=V, the Emotion Equation of Eq7 via Eq6 specifies the T transition emotion to be the UV excitement that is felt when money is won.

9.)          

The Emotions of Partial Success

The validity of the Emotion Equation of Eq7 is reinforced by two empirical verifications. One is its derivation of the excitement felt by slots players and contestants on game shows like “The Price is Right” from initial partial success that yields no prize in itself and the other lies in this phenomenon of partial success deriving the universally accepted, empirically verified Law of Supply and Demand of pricing for free market economies.

To win in slots play by getting the triplet matches of [7, 7, 7] or [7, 7, 7] or [7, 7, 7] one must first obtain doublet matches of [7, 7] or [7, 7] or [7, 7]. When the doublet match appears on the pay line, an initial excitement is felt by a slots player who is observing them. To derive this subcomponent excitement from Eq7, T=R−E, note that there are 9 configurations out of the 27 in Table 1 that have the doublet match of [7, 7] or [7, 7] or [7, 7]. Hence the probability of a doublet matching is Z_A=9/27=1/3, the “A” subscript in Z_A denoting it as the probability of getting the “A” subcomponent of the triplet match, which is the doublet match. And the uncertainty of getting the initial doublet match is, in parallel to Eq6, U_A=1−Z_A=2/3.

The probability of getting the triplet match of [7, 7, 7] or [7, 7, 7] or [7, 7, 7] following the doublet match of [7, 7] or [7, 7] or [7, 7] appearing is Z_B=1/3, the “B” subscript of Z_B denoting it as the probability of getting the “B” subcomponent of the triplet match, which is a matching “seven” to the initial doublet match. This enhanced Z_B=1/3 probability of success once the doublet match has been achieved and its enhanced E_A=Z_BV=(1/3)($270)=$90 expectation of winning relative respectively to the initial Z=1/9 probability and the E=ZV=$30 expectation understands the E_A=Z_BV enhanced expectation from the doublet match as what is realized, which allows us to specify E_A=Z_BV as a realized emotion given the symbol, R_A.

10.)       

This specification of E_A as a realized emotion, R_A, in turn allows us to apply the Emotion Equation of Eq7 of T=R−E to generate the transition emotion felt when the doublet match is achieved with T_A for T and R_A=Z_BV for R as

11.)       

To get a clearer sense of what this T_A transition emotion of partial success is, we specify the Z=1/9 probability of getting the triplet match of [7, 7, 7] or [7, 7, 7] or [7, 7, 7] as the product of the Z_A=1/3 probability of getting the initial doublet match of [7, 7] or [7, 7] or [7, 7] and of the Z_B=1/3 probability of getting the third “seven” to match the doublet match and make it a triplet match.

12.)       

This Z=Z_AZ_B relationship evaluates T_A in Eq11 with V=$270 as

13.)       

To see what emotion this is, let’s return to Eq9 for T=UV excitement in getting the winning triplet match to express the V term in it as V=R, a substitution we can make because R=V when there is a win.

14.)       

Now from Eqs10&13 we see the T_A transition emotion of partial success expressed in a parallel way as

15.)       

The parallel form of the T_A=U_AR_A partial success emotion in Eq15 to the T=UR complete success emotion of excitement in Eq14 indicates T_A=U_AR_A to be the excitement felt in achieving the partial success of getting the initial doublet match. Note that the intensity of the doublet match excitement of T_A=$60 in Eq15 is significantly less than the T=$240 excitement of getting the triplet match that wins the V=$270 payoff as quite fits what is universally actually felt by a slots player.

We can also calculate the increment in excitement felt when the 3^rd color appears to match an initial doublet match to give the triplet match and the R= V=$270 payoff. In evaluating this we note that the expectation prior to the 3^rd matching “seven” appearing is from Eq10, E_A= Z_BV, and that the realized outcome in getting the payoff is R=V. These parameters obtain the transition emotional for getting the 3^rd matching “seven” after getting the initial doublet match via the Emotion Equation of Eq7, T=R−E, with T as T_B and E as E_A=Z_BV and R=V as

16.)       

Now we see from Eqs15&16 that the two partial emotions of excitement, T_A=$60 of excitement from getting the doublet match and T_B=$180 of excitement in getting the 3^rd matching “seven” after the doublet match, sum to the T=$240 excitement in Eq14 of getting the matching triplet from scratch.

17.)       

The partial excitements of T_A=$60 and T_B=$180 adding up in this linear way to obtain the T=$240 excitement of winning the payoff from scratch gives further confidence in the Emotion Equation on which this analysis is based. These successive partial excitements are not only experienced universally in slot machine play but also on game shows like “The Price is Right.” There we see initial partial success that provides no payoff in itself eliciting excitement in a contestant as when she or he gains entry to the Grand Prize game at the end of the show by getting the highest number under $1.00 on the Price is Right spinning wheel. And we see a much greater excitement upon actual success of winning the Grand Prize as displayed by the winning contestant jumping up and down and running around on the stage screaming excitedly. Observation of this two part excitement in slots games and on game shows like “The Price is Right” predicted by the Emotion Equation of Eq7 constitutes a form of empirical validation of it.

The validity of the Emotion Equation of Eq7 is also shown in its deriving the Law of Supply and Demand of commodity pricing in the most primitive way in terms of the emotional sense that people have of supply and demand. The commodity we want to price is of the doublet match in the slots game, which we will assume our fictional casino would sell to you.

To determine the price of the doublet match, note that when [7, 7] or [7, 7] or [7, 7] is provided in every game, the average payoff is from Eq10,

18.)       

But when you start from scratch to get the triplet match of [7, 7, 7] or [7, 7, 7] or [7, 7, 7], the average payoff is from Eq2,

19.)       

So the fair value  to pay for the doublet match is what you would pay that keeps the average payoff the same, which is just the arithmetic difference between the above two average payoffs of E_A=Z_AV and E=ZV specified as W_A, which as calculated from Eqs10&13 is

20.)       

This is not to say that the player wouldn’t want to pay less than this fair price of W_A=$60 or that the casino wouldn’t want to get paid more than W_A=$60, but that W_A=$60 is the fair price for the doublet match in it keeping the average payoff per game at E=$30.

The W_A=U_AE_A fair price relationship is the most primitive form of the Law of Supply and Demand of pricing. In standard economic theory the Law of Supply and Demand specifies the price of a commodity to be an increasing function of the demand for it and a decreasing function of its supply. Or put in another way, it specifies the price of a commodity to be an increasing function of the demand for it and of its scarcity, which is just an inverse measure of its supply.

The buyer’s emotional sense of the scarcity of a commodity is the uncertainty the buyer feels in obtaining it. The more scarce the commodity, the more uncertainty the buyer feels about obtaining it, with the uncertainty in obtaining the doublet match being the U_A=1−Z_A=2/3 uncertainty measure term in W_A=U_AE_A in Eq20.

And the primitive emotional sense of the demand the buyer has for a commodity derives from its value for the buyer once it is obtained. For the doublet match, its value is the E_A=Z_BV=$90 expectation of getting the V=$270 payoff that it provides, which is the other term in the W_A=U_AE_A=U_AZ_BV fair price of Eq20.

Hence the W_A=U_AE_A fair price relationship in Eq20 is in its most primitive formulation from the most elementary human emotions the most basic form of the Law of Supply and Demand. This derivation of the Law of Supply and Demand, a universal empirically observed law of free market pricing, is an empirical validation of the Emotion Equation of Eq7 and the emotion functions that underpin it.

Note also the equivalence in Eq20 of the W_A=$60 fair price for the doublet match to the T_A=W_A=$60 excitement in getting the doublet match. This suggests that the cash value of a commodity is a function of the excitement gotten from it, the greater the excitement pleasure provided, the greater the price people are willing to pay for it. This quite perfectly fits common sense economic reality as, for example, in TV ads that incessantly tout the exciting nature of the product advertized. The applications of this primitive Law of Supply and Demand of Eq20 are so broad that we will later use it as a mathematical function for a female’s attractiveness to a male and of the excitement it generates for him.

The above verifications of the Emotion Equation from partial success and the Law of Supply and Demand suggest it to be a very powerful and fundamental equation in science in its telling us with such great mathematical preciseness how the human mind operates emotionally. Specifically it tells us that the mind has three broad categories of emotion, realized emotion felt upon actual realization of an outcome, expectation emotion felt upon anticipation of an outcome and transition emotion, which is a function of realized and expectation emotion as in Eq7. No understanding of human emotion and the behavior controlled by it can be correct that is not based on the Emotion Equation, which calls the discipline of psychology as it currently stands seriously into question. To show this in fuller detail we continue to sketch out our picture of the mind’s emotional machinery as follows.

A More Complete List of Anticipatory Emotions

We can express the E=ZV expectation of Eq2 as a function of the U=1−Z uncertainty of Eq6 as

21.)       

This alternative expression of one’s E=ZV hopes as E=V–UV splits one’s hopes of winning into two component emotions felt more or less simultaneously. One is the V in E=V−UV that is the desire for V that anticipates getting it whose pleasure is greater the greater the V payoff anticipated or daydreamed about. And the other is the –UV in E=V−UV emotion that reduces the V pleasure of desiring it as the –UV anxiety or worry that one will not get the desired V dollar payoff. A more precise technical name for the –UV feeling is as meaningful uncertainty, that is, uncertainty, U, made meaningful by its functional association with V dollars in −UV, money being an intuitively meaningful object for all normal adults.

The negative sign of −UV anxiety labels it as an unpleasant feeling with the intensity of its displeasure greater the greater the V payoff one is anxious about getting and the greater the U uncertainty felt about getting it. Expressing the E=ZV expectation as a function of the –UV anxiety as E=V−UV derives the T=UV excitement in Eq9 of getting the R=V payoff from T=R−E of Eq7 as

22.)       

This derivation of T=UV excitement or thrill makes it clear that the UV thrill of success comes about as the resolution of antecedent –UV anxiety or meaningful uncertainty. This is how adventure movies generate their thrills by being front loaded with meaningful uncertainty or dramatic tension over the hero’s uncertain situation, which when the resolved successfully brings about thrills for the audience.

Rearranging Eq21 introduces two other familiar emotions, anticipated excitement and anticipated disappointment.

23.)       

The form of the UV term again understands it as excitement from Eq9, but as anticipated excitement because it is felt prior to the outcome being realized. And the form of the –ZV term understands it as disappointment from Eq8, but as anticipated disappointment because it is felt prior to any outcome being realized. We see then that the transition emotions of –ZV disappointment and UV excitement also occur mathematically as in reality in anticipated form.

The relationship of UV=V−ZV in Eq23 of a small amount of –ZV anticipated disappointment being associated with a great amount of UV anticipated excitement is seen in lottery play where the chances of winning, Z, are felt to be very small, Z≈0. This generates little anticipated –ZV disappointment over the thought of losing as one’s chances were so small to begin with and great UV anticipatory excitement about the thought of winning. And on the other hand when one is nearly completely sure, Z≈1, of getting V dollars, there is minimal anticipatory excitement, UV≈0, while the thought of not getting the nearly sure V dollars produces considerable –ZV anticipatory disappointment because one’s ZV hopes were so high to begin with.

This increases our list of anticipatory emotions to include E=ZV expectation or hopes, V desire, −UV anxiety, UV anticipated excitement and –ZV anticipated disappointment. Note that UV=V−ZV anticipated excitement will be shown in a later analysis to be the command signal for activity done by free will.

Emotions Felt Over Losing What You Already Have

Our expectations concerning money don’t just arise from the possibility of getting money but also from the possibility of losing it. We analyze that with a slots game that gives no V payoff but rather penalizes a player v dollars when he or she fails to achieve the triplet match of [7, 7, 7] or [7, 7, 7] or [7, 7, 7]. The probability of failing to achieve it the U uncertainty of Eq6 and, hence, the expectation of incurring the −v dollar penalty in this game is

24.)       

The negative sign of –Uv characterizes it as an unpleasant emotion, that of the fear or worry of losing v dollars, which has intensity greater the greater the amount of v dollars worried about and the greater the U uncertainty or probability of incurring the loss. As with the –UV anxiety over not getting dollars hoped for, the –Uv fear of losing dollars you already have is technically referred to meaningful uncertainty.

Two outcomes realized in this game of trying to achieve the triplet match to avoid the v penalty. Either you lose v dollars if you miss the triplet match, R= −v, which produces the realized emotion of grief or depression over incurring a loss. Or you avoid losing the v dollars by getting the triplet match, R=0. This produces no emotion in itself, but does bring about a transition emotion from Eq7 of

25.)       

The Uv transition emotion is the relief felt in avoiding a feared loss, the positive sign of Uv indicating relief to be a pleasant feeling and one of intensity greater the greater is the v loss avoided and the greater is the U uncertainty felt about avoiding the penalty beforehand. If one is very sure of avoiding a v loss, Z≈1 and U=1−Z≈0, there is little T=Uv relief in avoiding it because one was sure the loss would not occur to begin with. Though, as the T=Uv function also makes clear, if the v amount of money one fears losing is enormous, this v component of T=Uv makes for considerable relief in avoiding loss of the money even when U, the possibility of losing it, is small.

Expressing the E= −Uv expectation of loss of Eq24 in an alternative way from Eq6 is the most intuitive way of understanding the feelings that come about when the penalty is incurred.

26.)       

This expression of E= −Uv fearful expectation as E= −v + Zv shows it to take an alternative form as two more or less simultaneous feelings, one of the unpleasant anticipation of incurring the entirety of the −v dollar penalty we will call the dread of loss reduced in its displeasure by the positively signed Zv pleasant feeling of the security or hope we have of a Z possibility of not losing the v dollars. If the triplet match is not achieved and the R= −v penalty is incurred, from Eq7 we see the T transition emotion to be 

27.)       

It is the sense of dismay or shock felt when the v dollars are lost. This –Zv dismay, an unpleasant feeling from its negative sign, is greater the greater was the Zv security felt that the money won’t be lost in the first place. If you had little Z hope of avoiding the v loss to begin with, Z≈0, while you yet feel the displeasure of grief or sadness from incurring the R= −v penalty, your low expectation, Z≈0, of avoiding the loss to begin with keeps the –Zv dismay over losing it small, except to the extent that the –v loss itself in the T= −Zv dismay may be very large.

The marginality of such a v dollar loss, its amount relative to one’s total wealth, is an important variable affecting the emotions experienced. But that complicating factor, which we set aside here, does not affect the validity of this analysis, which can be understood to assume an equal baseline of total wealth and equal marginality for all calculations. This disregarding of marginality as affecting the emotions involved also holds for the V dollar prize slots game we first considered. Marginality will be clearly explained mathematically in terms of the Simpson’s diversities once we have developed them in a formal way.

For now we want to complete this introduction to emotions associated with potential loss by considering its anticipatory transition emotions. Solving Eq26 for –Zv obtains

28.)          

The form of the –Zv term understands it as the dismay or shock of losing from Eq27, but as anticipated dismay because it is felt prior to the outcome being realized. And the form of the Uv term in the above understands it as relief from Eq25, but as anticipated relief because it is felt prior to any outcome being realized. We see than that the transition emotions of –Zv dismay and –Uv relief also occur mathematically as in reality in anticipated form. The relationship derivable from Eq28 of a small amount of –Zv anticipated dismay in losing being associated with a great amount of Uv anticipated relief were the penalty avoided is seen when there is Z≈0 little chance sensed of avoiding the –v penalty.

We have at this point mathematically characterized the basic emotions in a systematic way as consisting of three types.

·         The basic expectation emotions, E, are a function of a V payoff or −v penalty and of the probabilities of incurring them, respectively Z or U. These emotions:

o    Feel good as hope in the case of a positive expectation, E=ZV

o    Feel bad as fear in the case of a negative expectation, E= –Uv

·         The basic realization emotions, R:

o    Feel good as elation, joy or delight in the case of a positive outcome, R=V

o    Feel neutral in the case of a null outcome, R=0

o    Feel bad as depression, sadness or grief in the case of a negative outcome, R= –v

·         The basic transition emotions, T, are functions of the R realization and E expectation emotions as their difference, T=R−E, of Eq7. These emotions:

o    Feel good when positively signed as with excitement, T=UV, and relief, T=Uv

o    Feel bad when negatively signed as with disappointment, T= −ZV, and dismay, T= −Zv

·         An extended list of expectation emotions also includes those that:

o    Feel good as the anticipated V as desire

o    Feel good as the anticipated transition emotions of UV excitement and Uv relief

o    Feel bad as the anticipated −v as dread

o    Feel bad as the anticipated transition emotions of –ZV disappointment and –Zv dismay

These basic human emotions are centrally important phenomena because as we shall see very soon they control behavior. The emotions should not be disregarded on whatever philosophical grounds just because they are difficult to characterize in not being externally observable, the primary criterion for accepting data as empirically trustworthy. To do that is truly to throw out the baby with the bathwater. To minimize any out of hand dismissal of this analysis of emotion from the empirical criterion, note that we have considered only emotional experience that is universal as with the feeling of −ZV disappointment from ZV hopes dashed, which is inarguably universal for all people.

This approach taken here, hence, is that assertions that derive from universal agreement are as valid as externally observable data because the fundamental criterion for the trustworthiness of such data is the universal agreement as to what the data is. When ten laboratory workers accept the data they are reading off the same laboratory instrument as valid, they do so from the primary criterion that all of them are sensing the exact same data. It is this universal agreement on what is sensed that is the underlying basis of empirical trustworthiness.

Our characterization of the basic emotions is trustworthy because the emotions that machine players feel under the circumstances cited is universal for all slots players, its extension to parallel general circumstances or obtaining or retaining something of value to be made clear after we mathematically develop the evolutionary basis of all of the human emotions. This validation of the emotion functions based on universality is further strengthened by the derivation of the T_A=U_AE_A excitement of partial success in Eq13 and by the primitive Law of Supply and Demand obtained in Eq20.

Partial Success in Avoiding a Loss

We will next derive partial success emotions and The Law of Supply and Demand for the slots game played to avoid a v penalty, which will further reinforce the validity of the emotion analysis and add a few significant nuances to it. We begin by reviewing the probability of getting the doublet match of [7, 7] or [7, 7] or [7, 7] as Z_A=1/3 and of the probability of matching that doublet with the same color to get the triplet match of [7, 7, 7] or [7, 7, 7] or [7, 7, 7] as Z_B=1/3, with the overall probability of getting the triplet match that now avoids the v penalty being the same as in Eq12.

29.)       

The relevant uncertainties of failing to get the doublet and triplet matches are

30.)       

The expectation of incurring the v= −$270 penalty is from Eq24

31.)       

And the relief felt in getting the triplet match and avoiding the –v penalty is from Eq25 and Eq6

32.)       

The above T=v−Zv expression of relief specifies it as an increase in security, getting the triplet match increasing the security for avoiding the −v loss from Zv to v with the latter v term understandable as complete surety or security that the v penalty will not be incurred once the triplet match is gotten. Next we see that achieving the doublet match decreases the probability of incurring the –v loss from U=1/9 to U_B=2/3 in Eq30, which decreases the E_A expectation of incurring the –v loss from E= −Uv to

33.)       

The E_A=R_­A equivalence in the above understands the E_A= −$180 reduced fearful expectation of a –v loss to be what is realized by achieving the doublet match. And the transition emotion, T_A, that arises from this R_A realized emotion taken relative to the E expectation comes about from Eq7, T=R−E, with T as T_A, R as R_A= −U_Bv in Eq33 and E = −Uv in Eq24.

34.)

Much as T=v−ZV represents relief from the initial Zv security increasing to v security in avoiding the –v loss when the triplet match is achieved, so T_A=Z_Bv−Zv represents the relief felt in partial success from the security Zv security increasing to Z_Bv when the doublet match is achieved.

We can also calculate the amount of relief felt from getting the triplet match after getting the doublet match. To evaluating this we note that the fearful expectation after getting the doublet match is from Eq33, E_A= −U_Bv= −$180 and that the realized outcome in avoiding the loss is R=0. These parameters obtain the transition emotion for this from Eq7, T=R−E, with T as T_B, R=0 and E_A= −U_Bv from Eq33.

35.)       

Again we see the transition emotion that arises to be an increase in security, in this case of Z_Bv as the probability of avoiding the –v penalty once the doublet match has been achieved to v as complete surety or security of avoiding the loss when the triplet match has been gotten. And see from Eqs34&35 that the two partial relief emotion of T_A=$60 and T_B=$180 sum to the T=$240 relief in Eq32 of getting the triplet match in one fell swoop to avoid the –v penalty.

36.)       

The partial relief emotions of T_A and T_B adding up in this linear way to obtain the T total relief in getting the triplet match give further confidence in the validity of the Emotion Equation of Eq7 on which this analysis is based.

Now let’s look at a failed effort to get the triplet match after getting the doublet match in this penalty slots game to further validate the Emotion Equation. If one fails to get the triplet match by whatever route, the transition emotion is the T= −Zv dismay of Eq27, which calculated for the v=$270 penalty with Z=1/9, is

37.)       

Now consider if the doublet match is initially gotten to produce the T_A=$60 partial relief of Eq34, but is followed by failure to achieve the triplet match and a realization of the R= −v= −$270 penalty. That elicits a transition emotion from Eq7, T=R−E, with T as T_B, R= −v= −$270 and E=E_A= −U_Bv = −$180 from Eq33.

38.)       

This T_B= −$90 of dismay from failing to achieve the triplet match after getting the doublet match sums with the T_A=$60 worth of pleasant relief of Eq34 to produce the overall dismay of Eq37 of T= −$30.

39.)       

This relationship of Eq39 represents the universal emotional experience of a person who experiences some initial relief in potentially avoiding a bad outcome but then fails ultimately to avoid the failure as deems the initial T_A=$60 relief of Eq34 to have been premature by producing a greater “let down” or exaggerated dismay of T_B= −$90 once the final failure is realized. This further validates the Emotion Equation of Eq7 on which this analysis is based.

Now let’s derive the fair price of the initial doublet match in this penalty slots game assuming that it can be purchased. The average loss per game once the doublet match has been achieved is E_A= −$180 of Eq33 and prior to spinning the reels it is E= −$240 of Eq31. This understands the W_A fair price of the doublet match to be the difference in these average losses per game, which is evaluated from Eq34 to be

40.)       

This is essentially the same primitive Law of Supply and Demand derived in Eq20, but with the W_A fair price of the doublet match equal to the T_A=W_A=U_AZ_Bv partial relief of getting the doublet match of Eq34. This expands what we said from the Eq20 Law of Supply and Demand that people spend their money to obtain excitement to their also spending it to obtain relief, which is a powerful objective validation of this emotion analysis as people do, indeed, spend their money on commodities that provide either excitement or relief.

From the fact that money is generally earned as an hourly wage or a weekly or monthly salary, that is, in direct proportion to time spent working for it, this further indicates that people spend their time on obtaining the pleasures of excitement and relief and, hence, also on avoiding the respective associated anxiety over getting what they want and the fear of losing what they already have. The mathematical derivation from this primitive Law of Supply and Demand of people spending their time to attain pleasure and avoid displeasure makes it clear that behavior is motivated by attaining pleasure and avoiding displeasure or by hedonism.

Note that this is not a philosophical encouragement that people should act so as to achieve pleasure and avoid displeasure but a analytical conclusion that people do act to achieve pleasure and avoid displeasure, something we shall show in a more generally way once we have developed natural selection mathematically and shown its functions to derive the emotion functions we have generated from slot machine playing

A Further Exploration of Transition Emotion

Next we examine the T transition emotions in greater detail to see what their basic function is. Recall the T_A=$60 partial success excitement in Eq13 in the slots game with the V=$270 payoff that came from getting the doublet match of [7, 7] or [7, 7] or [7, 7]. We can show from Eqs10&13 that this T_A=$60 excitement comes about as the increase in the expectation of getting the triplet match of [7, 7, 7] or [7, 7, 7] or [7, 7, 7], specifically an increase from E=ZV=$30 to the E_A=Z_BV=$90 expectation of getting the triplet match once the doublet match has been achieved.

41.)       

This expression of the T_A partial excitement as T_A=E_A−E suggests that T_A transition emotion can be generally derived as a change in expectation.

42.)       

Consider the case of the initial doublet match of [7, 7] or [7, 7] or [7, 7] not coming about. Then the expectation of getting a winning triplet match is E_A=0 because it’s impossible. This E_A=0 is a decrease in expectation from the initial E=ZV=$30 expectation. Expressing the T_A transition emotion as the ΔE difference in these two expectations derives the transition emotion of −ZV disappointment from failure to win in a different way than we did earlier, now as a change in expectation.

43.)       

Eqs41&43 deriving excitement and disappointment as the T_A=ΔE change in expectation of Eq42 strongly suggest that this sense of transition emotion is general. Deriving the T_A transition emotions of relief and dismay from ΔE, which we shall do shortly, will make clear that the T_A=ΔE is, indeed, general. And we can also understand the Emotion Equation of Eq7, T=R−E, as a ΔE change in expectation by taking the R realization emotion to be a sure expectation of the final outcome of an R=V payoff or an R= −v penalty or an R=0 null event. This general understanding of transition emotion as a change in expectation can be written as below with the understanding that T can take the form also of T_A and T_B.

44.)       

Now let’s use this simple but powerful generalization to develop what the function of T transition emotion is in the emotional machinery of the mind. The T_A=$60 excitement of Eq41 felt when the doublet match is achieved is felt by slots players as part of their E_A=Z_BV=$90 increased expectation of getting the winning triplet match as can be derived from Eq42.

45.)       

This is a somewhat different specification of the E_A=$90 expectation than we see from Eqs10&13 as

46.)       

In Eq46 the E_A expectation is given as the V payoff multiplied by the Z_B probability of getting it while in Eq45 it is given as the original E=ZV=$30 expectation augmented by the T_A=$60 excitement. The equivalence of these two forms of expectation fits people’s two ways of sensing expectation as seen in the two synonymous expressions of expectation as “looking forward to something”, on the one hand, and “getting excited about it” on the other.

There is a substantive if subtle difference between these two senses of expectation easy to see if we stop the slots game after the doublet match shows and not spin the 3^rd reel for a couple of days. By that time the T_A=$60 excitement from getting the doublet match dies down and the E_A=$90 expectation of getting the last “seven” to match the doublet and yield the triplet match is entirely in the E_A=Z_BV=$90 “looking forward to something” form rather than in the E_A=E+T_A=$90 “getting excited about it” form.

The E_A=E+T_A=$30+$60=$90 excitement laden expectation of getting the triplet match upon getting the doublet match is termed novel expectation in including the T_A=$60 excitement felt right after the doublet match is obtained that boosts the initial E=ZV=$30 expectation up to E_A=$90. This E_A=E+T_A=$90 novel expectation converts over time to what we will call seasoned expectation, E_A=Z_BV=$90, which has the same value as E_A=E+T_A=$90 and the same intensity of pleasure, but which does not include the T_A=U_AZ_BV=$60 component of patent excitement in it. In its dissipating away over time, the T_A=$60 excitement functions to boost the E=$30 initial expectation to the E_A=$90 seasoned expectation. This explains the function of T_A excitement, which derives from success under uncertainty, to increase the expectation of or confidence in success.

There are myriad examples of this in real life. One is of a teenager learning to drive a car who has considerable uncertainty to begin with, the resolution of which uncertainty by successful efforts makes learning to drive very exciting. Upon repeated success in excitedly handling the uncertain situations that face a new driver, the excitement of the learning process dissipates and produces an increased expectation or confidence in successfully driving a car.

The function of all the transition emotions of excitement, disappointment, relief and dismay is to alter subsequent expectation. The E_A=E+T_A relationship of Eq45 is entirely general. Excitement from success increases subsequent expectation; disappointment from failure reduces expectation; relief from success reduces subsequent fearful expectation; and dismay increases fearful expectation. Let’s look at the transition emotions other than excitement to show how they alter subsequent expectation in greater detail.

When the 1^st two colors that show in the slots game with a V=$270 prize don’t match, the T_A transition emotion is seen from Eq43 to be disappointment of intensity

47.)       

That T_A= −$30 feeling of disappointment in failing to get the doublet match of [7, 7] or [7, 7] or [7, 7] brings about the player’s E_A=0 expectation of getting [7, 7, 7] or [7, 7, 7] or [7, 7, 7] via its reducing the initial E=ZV=$30 expectation to zero as

48.)       

This novel expectation of E_A=E+T_A patently includes the T_A= −ZV= −$30 disappointment as a modifier of the initial E=ZV=$30 expectation. This sense of the E_A expectation changes when we wait for a couple of days time before spinning the 3^rd reel, at which time the T_A= −ZV disappointment dissipates while transforming the initial E=ZV=$30 expectation to the seasoned expectation of E_A=0. This changing or transforming of expectation should be considered to be the primary purpose of the –ZV transition emotion in the emotional machinery of the human mind.

Now let’s consider the slots game that imposes a v= $270 penalty unless the triplet match of [7, 7, 7] or [7, 7, 7] or [7, 7, 7] is achieved. The fearful expectation of incurring the v=$270 penalty is from Eq31, E= −Uv= −$240. If the doublet match of [7, 7] or [7, 7] or [7, 7] comes up, then this initial expectation of E= −$240 reduces to the expectation of E_A= −U_Bv= −$180 of Eq33 with the T_A transition emotion of partial relief from seeing the doublet match in Eq34 being the ΔE difference in the expectations.

49.)       

This T_A= $60 relief operates on the initial E= −$240 fearful expectation to reduce it to

50.)       

The first felt novel expectation of E_A=E+T_A= −$180 consists of the original E= −Uv= −$240 expectation reduced by the T_A= $60 relief from getting the doublet match. If we wait for two days before spinning the 3^rd reel, the T_A= U_AZ_Bv=$60 relief of Eq34 dissipates and what is then felt emotionally in anticipation of getting the 3^rd matching color is the seasoned expectation of E_A= −U_Bv= −$180 of Eq33, which should be understood to have been developed by the T_A=$60 of relief operating on the E= −Uv= −$240 initial fearful expectation to alter it over time to the lesser E_A= −U_Bv= −$180 seasoned expectation.

When the 1^st two colors don’t match in the penalty slots game, the T_A transition emotion is the difference between two expectations, T_A=ΔE=E_A−E of Eq42 with E_A= −U_Bv= −v, which indicates a sure, U_B=1, incurring of the v penalty because the triplet match of [7, 7, 7] or [7, 7, 7] or [7, 7, 7] can’t possibly be achieved.

51.)       

This T_A= −$30 dismay felt when the first two colors don’t match sums with the initial E= −Uv= −$240 expectation of loss to make for a sure, U_B=1, E_A= −U_Bv= −v= −$270 expectation of incurring the v penalty as we see from Eq51 solved for E_A.

52.)       

The novel expectation of E_A=E+T_A consists of the initial E= −Uv= −$240 fear of incurring the penalty increased by the T_A= −Zv= −$30 dismay of not achieving the doublet match to the E_A= −$270 sure expectation of incurring the penalty. If we wait for some time before spinning the 3^rd reel, the T_A= −Zv= −$30 feeling of dismay of not achieving the doublet match fades and what is felt emotionally is the seasoned expectation of a E_A= −v= −$270 sure loss.

Now we have shown how the transition emotions affect subsequent expectation. Excitement from success under uncertainty increases E expectation and reduces the –UV anxiety in E=V−UV. Disappointment from failure decreases E expectation and from E=V−UV increases −UV anxiety. Relief from successful avoidance of loss decreases E fearful expectation and from E= −v+Zv increases the Zv sense of security. And dismay or shock from unexpected loss increases E fearful expectation and from E= −v+Zv destroys the Zv sense of security. This give a clear mathematical picture of the function of the transition emotions in the mind’s emotional machinery mind, its fit to universal emotional experience lending great credence to the validity of the Emotion Equation, T=R−E, of Eq7 that underpins this analysis and to its alternative T=ΔE form of Eq44

This emotional dynamic we have developed tell us that our emotions work in a cyclical way. E expectations lead to actions, even if just pressing a slot machine button, which then lead to outcomes and their R realized emotions and to T transition emotions that arise from R realization emotions taken relative to antecedent E expectations and equivalently as a ΔE change in expectation as in T_A=E_A−E=ΔE of Eq42. This is one pillar of the mind’s emotional operations, this development of transition emotion such as T_A as T_A=E_A−E=ΔE of Eq42.

A second pillar of our emotional machinery is of the transition emotions operating on initial expectation to bring about subsequent expectation as spelled out in Eq42 solved for E_A for a particular case in Eq45 as

53.)       

These two pillars of our emotional machinery, the development of the transition emotions as a ΔE difference in expectation and of the transition emotions affecting subsequent expectation in the above way, both derive from the general T=ΔE difference in expectation relationship of Eq44.

Behavioral Selection

The third pillar of our emotional machinery that completes it is the mind’s selection of which behavior to try, which is also based on the ΔE difference in expectation. More specifically the mind selects which behavior to try from a menu of available behaviors on the basis of which one has the greatest expectation. The function that determines behavioral selection is τ, (tau),

54.)       

The τ term specifies another category of emotion, the selection emotion. It is felt by a person when the expectations of two possible behaviors are compared to see which is greater. We will illustrate how τ comes about with the slots game that imposes a penalty of v=$270 for failure to get the triplet match of [7, 7, 7] or [7, 7, 7] or [7, 7, 7]. The two games chosen between will be the standard game that has an expectation of incurring the −v= −$270 penalty of E= −Uv= −$240 and the doublet match primed game whose expectation of incurring the penalty is E_A= −U_Bv= −$180 of Eq50. The selection emotion felt for choosing the doublet match primed game that has the E_A expectation of winning, τ_A, comes about by using the E_A term as the minuend in the ΔE difference. 

55.)       

What is this selection emotion of τ_A? It is seen as a reduction in meaningful uncertainty from –Uv to –U_Bv in the above.  Now let’s go back and look at T=0−(−Uv)=Uv relief in Eq25. Its form is as a reduction in meaningful uncertainty from –Uv meaningful uncertainty to zero meaningful uncertainty when the triplet is actually achieved, Z=1 and U=1−Z=0. Though the sense of the τ= –U_Bv–(−Uv) selection emotion as deriving from a comparison of two behaviors chosen between is quite different than the T=0−(−Uv) transition emotion that arises when a penalty avoiding behavior is successful, both being reductions in meaningful uncertainty peg τ_A as the relief felt in selecting the “lesser of two evils” doublet primed penalty incurring game to play is obvious.

This classic Hobson’s choice can be looked at in reverse to see what emotion would be felt were the E= −$240 primed game chosen. 

56.)       

The form of τ is seen to be an increase in meaningful uncertainty from –U_Bv to –Uv. To see what emotion this represents, let’s look again at Eq27 for –Zv dismay written with Z=1−U from Eq6 as 

57.)       

This understands the –Zv dismay as an increase in meaningful uncertainty from –Uv to –v, the latter –v term being –Uv meaningful uncertainty at a maximum with U=1. This understands the τ= −Uv−(−U_Bv) increase in meaningful uncertainty of Eq56 to represent the dismay felt in considering the “greater of the two evils” choice to do. As people always take the Hobson’s choice of the “lesser of two evils” choice, it is implied that people do behaviors that have positive, pleasant, expectations and avoid those with negative, unpleasant, expectations.

This principle of selection can be stated as follows: if τ>0 for the behavior and emotionally pleasant, do it; if τ<0 for the behavior and emotionally unpleasant, don’t do it. This holds, of course, at the most elementary level when the choice is between playing a game and not playing it. Intuitively we understand a person to always play the no-cost V payoff slots game because of its positive, pleasant, expectation and per play average ZV payoff.

This intuitive choice can be brought under the τ selection principle by considering the choice of playing the V payoff game to have an expectation of E₁=ZV and the choice of not playing to have the expectation of E₂=0. Playing the V payoff game then produces from Eq54

58.)       

This choice of playing the game is understood to be taken because τ₁=ZV>0, which implies pleasant positive expectation or hope of winning the V payoff. On the other hand the choice of not playing the game has a selection emotion of

59.)       

This choice of not playing the game is understood to be taken because τ₂= −ZV<0, which implies anticipated disappointment because one cannot win the V payoff if one doesn’t play. And the τ selection emotion for avoiding the –v penalty in the penalty incurring slots game also fits our intuitive sense of that being the proper choice to make when the E₁= −Uv expectation playing of the game is contrasted with the E₂=0 of avoiding it to develop the τ₂ emotion for avoiding the game as 

60.)       

So the penalty imposing slots game is avoided because the τ<0 emotion of relief in doing so is positive and pleasant. Choosing between two V payoff paying slots games also follows the τ selection emotion function. Specifically consider selecting between two slots game that both reward a V payoff for getting [7, 7, 7] or [7, 7, 7] or [7, 7, 7]. The two games chosen between are the standard game on one slot machine that has a payoff of V₁=$270 with probability, Z=1/9, and an expectation and average payoff of E₁=ZV₁=$30; and the doublet match primed game on another machine that has a payoff of V₂=$60 with probability, Z_B=1/3, and an expectation and average payoff of E₂=Z_BV₂=$20. The selection emotion experienced in choosing the standard game whose E₁ expectation would be listed as the minuend in the τ=ΔE expectation difference function of Eq54 is

61.)            

The positive sign of τ₁, that is, τ₁>0, and its development as a ΔE>0 that is a function of a V payoff suggests it to be the pleasurable excitement of selecting the higher average payoff. But if one chooses the doublet match primed game as makes E₂=Z_BV₂=$20 the minuend in Eq54, the τ selection emotion is

62.)       

The negative sign of τ₂, that is, τ₂<0, dictates avoidance of the doublet match primed game with the poor V₂=$60 payoff. Such avoidance can be attributed equivalently to the lesser average payoff of the inferior game or from the unpleasant feeling implied in playing it from its τ₂<0. This understands displeasure is as a neurological signal to avoid whatever is associated with it and pleasure as a neurological signal that motivates doing whatever behavior is associated with it, which fits in perfectly with the hedonistic basis of behavior developed earlier after Eq40.

The τ=ΔE selection operation completes the cycle of the emotional workings of the mind. We have seen now that expectations compete until the one selected motivates a behavior whose realization emotions referenced to the expectations develop transition emotions that in turn shape subsequent expectations that are the factors which act to select the next behavior will be selected with all components in this cycle based on the ΔE distinction between expectations in Eqs44,53&54. It is fascinating to see the vast seeming complexity of our emotional machinery working on these simple three pillars, all of which derive from the ΔE distinction between expectations. 

This suggests that the human mind is an organ of distinguishing between things.  A clear sense of why the mind evolved that way and how this analysis of emotion derived from slot machine play can be extended to all behaviors requires the explanation of natural selection coming shortly. But before we do that mathematically we want to consider a few other kinds of emotion driven selections that can be demonstrated with the slot machine games.

The first focus is on the emotions associated with aggression. Let’s return to the −v dollar penalty incurring slots game and imagine the existence of agent who forces player to play this game is the one who takes the money away from the player when he or she fails to get the triplet match. Recall from Eq24 the fearful expectation of incurring the –v penalty.

24.)       

In the slots game with a penalty of –v= −$270 and a probability of incurring it of U=8/9, the player can avoid the average loss of E= −Uv= −$240 per game by not playing the game at all with one way to avoid it if he or she cannot simply walk away from it being to destroy the agent who forces him to play by killing or violently disabling the agent. Aggressive destruction of the agent results in an R=0 outcome (no loss) that produces from Eq7, T=R−E, the emotion of

63.)       

This T=Uv emotion felt from successful aggression as obtains an R=0 outcome is felt as the relief we earlier derived in Eq25 with intensity proportional to the size of the −v loss avoided by the aggression and the U probability of the −v penalty being imposed. Aggression can be employed not just to avoid having v dollars being taken away by an agent to some U probability, but also to eliminate an agent who is in whatever way is responsible for the U probability of a person failing to get the V dollars in the payoff rewarding slots game.

The transition emotion felt in that case is in parallel to Eq9, T=UV, the excitement or thrill of receiving V dollars under U uncertainty, but this time by destroying an agent who inhibits success to probability U. Note that the pleasant emotions of Uv relief and UV excitement that derive from successful aggression have overtones particular to aggression. Also note that a person acting destructively towards another to obtain or retain something of value, whether money or food, is seen empirically in today’s drought ravaged Africa where instances of people aggressively fighting over food have been broadcast. 

Now let us understand that not all behaviors have expectations that are either purely positive as E=ZV or purely negative as E= −Uv. Indeed most realistic situations have mixed emotions that are positive and negative. One such situation is the gamble, which we can explore with a slot machine game where achieving the triplet match to probability Z=1/9 obtains a payoff of V=$270 with expectation E=ZV=$30 and failing to achieve it to probability U=1−Z=8/9 incurs a penalty of v= $18 with expectation of E= −Uv= −$16. In such a case the E expectation is a compound expectation that includes both the positive and negative expectations as

64.)       

For the specific case cited above, we see that

65.)       

As the E=$14 expectation is positive or overall hopeful, this gamble is taken. Were the penalty v= $45 and the expectation of loss E= −Uv= −$40, the compound expectation would be

66.)       

As the E= −$10 expectation is negative or overall apprehensive or fearful, this gamble would not be taken. Now consider two gambles selected between, one with expectation E₁=Z₁V₁ and the other with E₂=Z₂V₂. A comparison of the two for possible selection generates a τ selection emotion from Eq54 of

67.)                

If τ₁>0, the E₁ gamble is selected and if τ₁<0, the E₁ gamble is not selected, but rather the E₂ gamble. This form of the classic gamble specified in Eq64 basis the expectation of losing money on the U uncertainty with U=1−Z from Eq3. In casino slot machine gaming, one always “loses” money up front as the money required to play the slots game with a V payoff. We specify this gamble with a fixed cost to play as

68.)       

Were the cost to play v=$40 and the positive expectation ZV=$30, the compound expectation would be E=$30−$40= −$10, which motivates not playing this game. All realistic casino slots games have negative expectation for the player that indicate they shouldn’t be played. And indeed they wouldn’t be if the player’s supposition of the expectation was the actual negative expectation, though there are other emotional factors that work to bring about play rather than avoidance as will be explained later.

There are a few other general forms of compound expectation. An important one is when a behavior with a negative expectation is followed by one that has a positive expectation with magnitude greater than the negative expectation, the compound behavior of sacrifice spelled out functionally as

69.)       

The lower case u is used for the probability of incurring a penalty is u≠1−Z as fits the scenario of player having to first “endure” playing a −v penalty incurring game to gain access next to a V payoff rewarding game. When E>0, the sacrifice is worth undertaking and when E<0, it is not.

Another common compound behavior is of initially positive pleasant behavior deterred by punishment or negative unpleasant consequences. It takes the functional form of Eq69 written in reverse as

70.)       

It is modeled with the slot machine game in terms of Z probability of a V payoff game necessarily followed with a game a u probability of incurring a –v penalty game when E<0 deters the initial positive behavior. In both sacrifice and punishment the choice to make a sacrifice when E>0 and to avoid a game you will be punished for when E<0 is understandable in terms of E expectation interpreted objectively as average penalties and payoffs or as displeasure and pleasure. When the pleasure of eventual success is greater than the displeasure of the initial sacrifice and, hence, E>0, the sacrifice is worth making. When the displeasure of subsequent punishment is greater than the pleasure of the initial behavior, E<0, the punishment deters it.

Another important compound expectation situation that can be represented with Eq69 is restorative behavior. This occurs when an unavoidable threat of loss, −uv, or actual loss, u=1, happens unavoidably as from a –v penalty incurring slots game forced upon a player who then seeks to find a V payoff rewarding game to play that makes up for the initial losing one. Many behaviors in life are like this. Over time if we don’t eat, the blood sugar in our blood stream depletes along with stored reserves as takes away something of value from us, which eventually causes us emotionally to eat food as increases blood sugar and stored energy reserves to make up for this depletion. Similarly we lose valuable heat energy is we stay out in the cold too long and/or too exposed, which causes us emotionally to mover ourselves to a warmer place to have the heat loss made up by a heat intake from warming ourselves, the details of both eating and warming to be explained in the context of restorative slot machine playing after we have taken up natural selection. 

Eq70 also explains a class of misguided restorative behaviors called misdirected behaviors which occur when loss in one area of activity cannot be made up for by the proper restorative behavior and instead is made up for with a substitute behavior that while providing pleasure to mitigate displeasure to some degree does not in hard fact make up for loss. A loss of a loved person in a relationship, for example, which produces great displeasure but cannot be properly restored may be made up for by overeating which produces some pleasure, that of food taste, to help assuage the displeasure of the broken heart. To develop the full panoply of misdirected behaviors, which are many and very important in modern times, requires the mathematical consideration of evolution we will undertake shortly.

Emotions that operate sequentially over time are complicated by the phenomenon of recency or how long ago a past emotion that compounds with a present emotion was experienced because the intensity of a past emotion generally fades over time. Much as we ignored the complicating factor of marginality in developing the basic picture of the mind’s emotional machinery, so do we also ignore the recency factor, though marginality and recency are easily taken into account once the fundamental functions for the emotions have been developed.

To repeat importantly, all of the above selection algorithms based on ΔE expectation differences can be understood in an objective way by considering expectation in mathematically firm terms as the average payoff or penalty incurred. But selection can and should also be understood in terms of expectation understood as the pleasant and unpleasant emotion that instinctively cause people choose to do without their consciously calculating average gain or loss. This perspective fits perfectly with our consideration of behavior as fundamentally hedonistic or pleasure and displeasure driven via Eq40. Much of selection based on pleasure and displeasure that we have developed is found in the Behaviorism of B.F. Skinner, indeed in a splendid and highly elaborated form, but with Skinner’s sense of positive and negative reinforcement that motivates behavior specified in this work as pleasure and displeasure via the emotion functions we developed for them.

Evolution with Emphasis on Natural Selection

We spell out natural selection in evolution mathematically because a precise explanation of evolution is a necessity in the anti-science climate that exists in America at this time. Our form of it is entirely new and not seen in print anywhere, but the conclusions reached in this treatment are but a special case of the rubric of population biology developed in the 1920s by master biologists, Sir R.A Fisher, Sewell Wright and J.B.S. Haldane, whose work has been thoroughly accepted over the years as standard biology and validated empirically by much wet-lab work. We begin with the differential equation for exponential growth.

71.)       

It should be made clear that this starting point for our developing a mathematical function for natural selection is entirely ideologically neutral for when g is the annual interest rate on a savings rate, Eq71 represents the exponential growth of money in a savings account that is compounded daily. For biological population growth g is the growth rate of a population of x organisms as a function of its birth and death rates.

72.)       

More specifically the growth rate, g, is the difference between the annual birth rate per number of organisms, b, and the annual death rate, d.

 73.)      

A differential equation tells a story in mathematical language, here that the rate of growth, dx/dt, is higher the greater the x number of organisms reproducing in the population, the greater the b birth rate and the smaller the d death rate. The time equation for exponential population growth is

74.)       

The x₀ term is an initial population of organisms at some arbitrary time, t=0. There are a number of important nuances to this classical expression of exponential population growth. For one, it generates fractional numbers of organisms. But that is not a problem as fractional x values are just rounded off to whole numbers of organisms. For another, the growth specified is essentially asexual as in bacterial growth. But that in no way interferes with our applying it eventually to human population growth. And also Eq74 is mathematically at odds with realistic growth other than for bacteria because it implicitly specifies newborn organisms as ready to reproduce the moment they are born as is the case for bacteria. Fortuitously this is a fine fit to human reproduction when birth is understood as the birth or coming to be of a sexually mature organism as humans are at adolescence. Hence the birth rate, b, is that of biologically sexually mature adolescent organisms.

Evolution, the great gift of Charles Darwin to the world, consists of two parts. Organisms produce a number of offspring, some of them different than others. This is the principle of variation. Most simply, variation in the anatomy, physiology, biochemistry and neurology of organisms makes for variation in the capacity to survive and reproduce. When one genetically related group competes with another for the same resources in a niche, the population whose individuals are more capable at surviving and reproducing persist from generation to generation while their less competent competitors die out over time or become extinct in the niche. This is the principle of natural selection. Together variation and natural selection bring about biological evolution. We are focusing on the natural selection aspect of biological evolution here, though we will include the variation component in our considerations when needed.

Also note that Darwin wrote his masterwork on evolution back in the mid-19th century and understood evolutionary competition to take place between populations as species. In agreement with modern theory as clarified by the master evolutionary biologist, Richard Dawkins, we take competing populations to be lineages of genetically related groups. This readily has one understand the competitions of natural selection we are about to delineate to take place between different human lineages including broad genetically distinct ethnic groups that are also bound as cultural lineages by common cultural information.

Returning how to Eq74 we illustrate its exponential growth with a population that starts out with x₀=2 individuals and has an annual growth rate of g = b – d = 2 organisms as graphed below.

Figure 75. Exponential Growth

Now consider exponential growth for two populations that grow apart from each other and don’t compete in any way for resources and which have an unlimited amount of resources and space to grow in. It is specified by the exponential growth functions we have already considered with growth terms subscripted for the two populations as

76.)       

77.)       

78.)       

79.)       

The time solutions to the differential equations of Eqs77&78 take the form of Eq74.

80.)       

81.)       

The x₁₀ and x₂₀ terms are the initial sizes of the #1 and #2 populations respectively. Now what we are going to do is count the x₁ and x₂ sizes of the two independently growing populations together as

82.)       

While there may seem to be no particular reason to do this, it is a mathematical operation that certainly can be done as can the x₁ size of the #1 population and x₂ size of the #2 population be specified relative to the x₁+x₂ combined populations sizes.

83.)       

84.)       

Real populations, though, don’t grow exponentially without limit because the biosphere has limited resources for growth. And real populations generally don’t exist without rivals that compete with them for whatever limited resources are available. To mirror this reality of limited competitive growth mathematically, we now take these two independently growing populations and put them together in a common niche, one that has a limit to the total number of organisms its territory and resources can support called the carrying capacity of the niche, K. The K carrying capacity limits the x₁+x₂ combined size of the two populations to be equal to or less than K.

85.)       

Of interest to us is the case of two populations growing together in a common niche with carrying capacity K whose combined x₁+x₂ population sum is at the K carrying capacity or maximum combined population size.

86.)       

This boundary condition renders Eqs83 as

87.)       

88.)       

And the boundary condition of Eq86 renders Eq84 as

89.)       

90.)       

The boundary condition of Eq86 also provides the initial condition of combined population sizes of

91.)       

This allows us to specify x₂₀ as

92.)       

And the above obtains Eq88 as

93.)       

Now dividing the numerator and denominator of the above by the exponential term in the numerator,, obtains x₁ as

94.)       

95.)       

And for the x₂ population we obtain by similar algebraic operations

96.)       

97.)       

A plot of Eqs95&97 for an example niche with a carrying capacity of K=100 and populations with x₁₀=1, x₂₀=99, g₁=2 and g₂=1 obtains the curve shown below with x₁ in blue and x₂ in red.

Figure 98. Evolutionary Competitive Growth or Natural Selection

The #1 population in blue, which has the higher growth rate, g₁ =2, survives from generation to generation in the niche while the population in red, which has the smaller growth rate, g₂ =1, dies out or goes extinct in the niche. We can simplify the functions of Eqs95&97 which Figure 98 is based on by defining g₁−g₂ as the fitness of the #1 population, F₁,

99.)       

This expresses Eq95 more succinctly as

100.)                

The easiest way to now obtain the differential equation form of the above algebraic time function is by reference to the well known logistic equation, also called the Verhulst equation after its originator, the Belgian mathematician, Pierre Verhulst, who introduced it to science in 1838. Also called the limited growth equation of a single population, it is derivable from Eq95 in our terminology with g₂=0, that is, with the growth rate of any competing population nonexistent or zero, as

101.)       

This is a rather beautiful derivation of the classic logistic equation from our more general function for natural selection of Eqs97&100. There are over 8 million references to the logistic equation on Google, with the differential form of the above expression of it in our terminology as

102.)     

The similarity in form between Eq100 and Eq101, indeed with F₁ in the former substituted with g₁ in the latter, makes it clear that the differential equation for Eq100 must be

103.)     

This can be simplified further from Eq86, which understands x₂=K−x₁, as

104.)     

And similar considerations for Eq97 develop a succinct differential equation for the competitive growth of the #2 population as

105.)     

106.)     

The foregoing argument makes clear that a positive fitness for the #1 population, F₁>0, as derives from its having a higher growth rate, g₁ >g₂, destines the #1 population to be the “fit” population selected for in the niche to survive over time, while the “unfit” #2 population with negative fitness, F₂=<0, as derives from its smaller growth rate, g₂ <g₁, is selected against and goes extinct in the niche. Eqs99-106 mathematically tells us that natural selection selects the population whose members are superior in survival activities as results in a lower death rate and in reproductive activities as results in a higher birth rate. This is a perfect fit to Darwinian natural selection described in non-mathematical language as by the Harvard grandmaster evolutionist, Ernst Mayr.

“.....it must be pointed out that two kinds of qualities are at a premium in selection. What Darwin called natural selection refers to any attribute that favors survival, such as better use of resources, a better adaptation to weather and climate, superior resistance to diseases, and a greater ability to escape enemies. However, an individual may make a higher genetic contribution to the next generation not by having superior survival attributes but merely by being more successful in reproduction.” Mayr, Ernst, One Long Argument: Charles Darwin and Modern Evolutionary Thought, Harvard Univ. Press, 1991, p.88).

Behavioral Ramifications of Evolution

It is clear from the foregoing analysis as depicted in Figure 98 that successful lineages have a fitness, F₁=b₁−d₁−b₂+d₂ of Eq99, that is positive, F₁>0. The optimal mathematical strategy for attaining an F₁>0 positive fitness and avoiding extinction is for the individuals of a successful lineage to behave so as to maximize their individual fitness, specifically: by maximizing their individual birth rate as will maximize the b₁ birth rate of their lineage; by minimizing their individual death rate (or maximizing their life span) as will minimize the d₁ death rate of their lineage; and by maximizing the d₂ death of the rival population. And that is what human beings have been evolutionarily programmed to do via their emotions: to try to avoid death at every turn; for adult human males to want to mate at every turn, at least emotionally, whatever the restraints of their culture and circumstances; and to try to win in battle against rival groups as we see in the endless parade of wars in history.

 

We specify a function for individual fitness, Ψ, in terms of the F₁ fitness of the #1 lineage as

107.)     

This understands the F₁ fitness of the lineage as the average of the Ψ individual fitness of the m members of the lineage.  We specify the individual fitness of the j=1 member of the #1 lineage as Ψ₁₁ and express it in terms of his individual birth rate, β₁₁ and his individual death rate, δ₁₁, as

 

108.)     

This individual in maximizing his β₁₁ birth rate, minimizing his δ₁₁ death rate and maximizing the d₂ death rate of his rivals maximizes his Ψ₁₁ individual fitness as develops a maximization of the F₁ fitness of his lineage. We are less concerned with his minimizing his rival’s b₂ birth rate.  The individual growth rate is γ₁₁,

 

109.)     

  Hence the Ψ₁₁ fitness of an individual is

 

110.)     

So individual fitness, Ψ₁₁, is personal growth, γ₁₁, relative to the growth rate of the rival lineage, g₂, which is an average of the growth rates of all the individual rivals. The individual’s behaviors are programmed by his emotions to increase, retain or restore his Ψ individual fitness. Note that we use the masculine pronoun and not the feminine so as to focus our attention on male behavior because natural selection works through adult males rather than females because of their differential growth rates. That is, almost all mammalian females mate, but generally speaking, less than half of the males. While the male competition that makes for an uneven distribution of females over males the rule for all mammals is less seen in civilized societies, especially the monogamous ones, the underlying instinctive emotions which developed genetically prior to the advent of cultural, often religious, restrictions on reproduction, aim towards his maximization of the male β₁₁ birth rate. This is not an ideological position but hard mathematics. 

 

Sex pleasure motivates sexual activity without which the β₁₁ individual birth rate drops to zero, sharply reducing the Ψ₁₁ individual fitness from Eq108. A healthy male’s sexual arousal at just about any female with her clothes off and often with her clothes on is a clear indication of the primitive emotions made to drive him to maximum reproduction when circumstances make it possible. For humans, reproductive activity includes not only mating but also raising babies to sexual maturity at adolescence as satisfies the assumption in the exponential growth equation of Eq71 that an organism is able to reproduce from the moment of birth. Taking the arrival of an adolescent as birth, the birth of a sexually mature organism, satisfies this tacit proviso in the natural selection equations of Eqs104&105. And note that it is the pleasure of love, of romantic and family love, that motivates the relationship between mother and father and between parent and child that are needed to successfully raise an infant to adolescence. The importance of male-female love as makes for two parents to raise a child rather than one is not an ideological position but hard mathematics given that two persons working cooperatively generally have a higher probability per unit time of achieving a goal than does one person .

 

Food taste pleasure motivates eating without which the δ₁₁ individual death rate goes up sharply also reducing the Ψ₁₁ individual fitness. Hunger displeasure also motivates eating to retain Ψ₁₁ individual fitness in a backwards logical way by motivating avoidance of not eating, much as the displeasure of losing money motivates avoidance of the penalty incurring slots game. This parallel strongly suggests that money, which we all use to buy our Ψ fitness retaining food with as motivated by our getting food pleasure and/or avoiding hunger displeasure, derives its value in bringing about an increase and/or retention of fitness.

 

The pleasures of glory in winning in mortal combat motivate behaviors directed to winning as do the fearful displeasures of losing in competition motivate avoidance of losing, which may take the form of winning in or avoiding mortal combat completely. Note that not does killing the enemy increase his d₂ death rate and the aggressor’s Ψ₁₁ fitness from Eq108 but also driving one’s rivals out of the competitive niche in territorial battles that that “kill them off” numerically within the niche as also satisfies the emotional drive dictated in Eq108 to maximize the Ψ₁₁ individual fitness and, indeed, the F₁=b₁−d₁−b₂+d₂ fitness of the #1 lineage. Money is also used to buy weapons, offensive and defensive. 

 

The pleasures of food and sex and success in battle, all of which are purchasable or enabled by money, have the same functional role as V in E=ZV expectation. Much as we have an E=ZV expectation of the pleasure of obtaining V dollars diluted by the Z probability of attaining the V dollars, so we have an E=ZV expectation of the V like pleasure of eating food diluted with the Z like probability of getting the food, which is sometimes low as with the probability of purchasing and eating filet mignon steak for the poor. Unlike slot machine playing, though, the fitness and emotion value of the food is not precisely quantified as V is in the slot machine games nor is the probability of getting it quantified precisely like the Z probabilities in our slot machine games, which is why we introduced the emotions with the quantitative V and Z characterized slot machine games.

 

The emotional machinery described in Eqs2-69 for getting and losing money, though, yet applies to getting and retaining any object that has biological (evolutionary fitness) value including information on how to get it. When there is some uncertainty in getting food and one finally finds or catches it as is the case in hunter-gatherer societies, there is a genuine excitement or thrill in getting the food. In my younger days a severe case of asthma drove Ruth and I high up in the mountains of Northern California for four years. Our last three months there before I returned to civilization to finish my PhD in biophysics, we ran out of money and had to survive on wild apples and berries that genuinely provoked excitement when found especially when we had to look through Siskiyou National Forest for a new apple tree or berry bush with fingers crossed that we would find one.

 

Our analysis of the basic emotions done with slot machines holds for getting and retaining anything of value, not just money.  When a person loses a loved one whether through death or rejection, in another example, one feels grief, dismay and shock the same as from losing money as described in Eq27 and indeed in proportion to the value of the loved one and in reference to shock, in proportion to the (Z) surety one had beforehand that the loved one would not be lost, that is to the (U) uncertainty or unexpectedness of the loss. 

 

Selection also derives in the same way in non-money behaviors from τ=ΔE of Eq54 but with the expectations not specified numerically as E=ZV or E= −Uv but as the relative pleasures or displeasures in the possible choices. For example if one is hungry and cold at the same time one instinctively chooses to find food or to find warmth depending on which (E= −Uv) expectation displeasure is the greatest.

 

Understanding the E expectation as deriving in its V/v term from the Ψ fitness value of the things money buys makes clear that the pleasure of having and displeasure of not having are a measure of the fitness value of the things as diluted by the probability term in E, be it Z or U of getting or losing these things. Ultimately, an expectation represents a probability of increasing or maintaining fitness. Indeed, we can understand expectation most clearly as a creation of natural selection by understanding fitness as a measure of the probabilities of birth and death. A woman having an individual β₁₁ birth rate of 2 child in 8 years translates to her having or having had an annual probability of giving birth of 1/4=.25. And the b₁ birth rate of the #1 population can also be understood as a probability of birth year. 

 

Once the value of V money is clarified as deriving from money’s ability to increase or retain Ψ fitness, the translation of birth and death rates to probabilities understands E=ZV expectation as the probability of getting something that increases or retains the probability of avoiding death (surviving), of giving birth (to an adolescent) or of successful combat against rivals. Or more succinctly yet, expressing fitness as competitive growth, Ψ₁₁=γ₁₁−g₂ in Eq110, understands E=ZV expectation and all the emotions that arise from it as the probability of acquiring something that increases or retains the probability of competitive growth whether via differential survival, reproduction and/or success in combat. 

 

This firm mathematical rooting of expectation and its related emotions in the growth probabilities of natural selection make it inarguably clear, as common sense also tells the scientifically literate, that the emotional operation of the human mind via our emotions derived from the evolutionary processes that shaped the mind. This mathematical correlation of mental attributes and evolutionary characteristics, on the one hand, reinforces the validity of this mathematically precise clarification of the human emotions, difficult to understand because another’s emotions cannot be sensed (other than when openly displayed). And on the other hand it provides a powerful reinforcement of evolution as the creator of man, which despite the powerful circumstantial evidence for it is still difficult to otherwise prove because man’s evolution happened in a past that cannot be replayed to be observed.

 

This origination of our emotional machinery in evolution from this powerful mathematical argument is so tight that to argue it is equivalent to arguing the Law of Gravity. It finally and firmly explains for the first time precisely how the mind works. And it also inarguably makes clear that the mind of man and man himself is a product of evolution. If God did create man, a moronically childish supposition to begin with, clearly He did so in a very unoriginal way by perfectly imitating the Darwinian evolution pattern for it. Those who are so attached emotionally to the notion of a spirit mind that intelligently designed everything as to deny this argument are stupid at a level so infantile that they deserve all the ridicule that can be heaped on them.

 

At this point we have fleshed out the basics of human emotion, the behavior motivated by it and their evolutionary basis. There are a great many topics one can go to next because this foundation underpins every facet of human nature and how it has played itself out over the years including how civilization has affected it. We will focus on those topics relevant to how people’s emotions, thoughts and actions are shaped today, especially those that affect their happiness or lack of it. 

Romantic love plays a special role in happiness because of its relevance for keeping two parents together in raising children that are genetically both of theirs. The outcome of the actual genetic parents of children staying together to raise them is important both because of the strong instincts that control behavior towards one’s own children and as stated earlier two competent cooperating persons can generally do a better job at anything than one. As pleasure is a measure of the Ψ fitness of an outcome, so the deep and profound pleasure of obtaining and retaining romantic love is a measure of its ensuring fitness or evolutionary success by producing vigorous competent young adults to continue the lineage from generation to generation. 

Paramount in forming and retaining this relationship is the man’s attraction to the female. We begin with a mathematical function for attraction generally. Our attraction to something derives from the value of the thing we are attracted to. We have already derived a function for value in the Law of Supply and Demand of Eq20 that specifies the monetary and emotional value of a thing. This is in keeping with the universal intuitive sense of the value of a thing in emotional and monetary terms being a measure of how much we are attracted to it or want to have it, a new (expensive) Mercedes, for example, being more exciting and attractive for the automobile aficionado than an old (inexpensive) and unexciting Ford.

20.)       

The T_A excitement value and W_A dollar cost of a thing are obvious measures of how attracted people are to it. For men as this applies to female attractiveness, the excitement she generates and the money or time a man would be willing to spend to mate with her are obvious measures of her attractiveness. The V term in Eq20 when W_A=T_A measures female attractiveness is not a number of dollars possible to get as in the slot machine game but the number of (eventually mature) offspring the man can produce and raise with the female. The U_A term is his uncertainty in acquiring her. And the Z_B term is the probability or ability of the female once acquired to bear infants and do her part in raising them to maturity.

The most obvious factor in Eq20 for a woman’s attractiveness and the excitement she generates is the V number of children she can bear, which is a simple biological function of her age. The older the women, the fewer V number of children she can produce for the male who acquires her and as is predicted from Eq20, the less attractive she is as a function of her age. Who would deny that the average 16 year old girl is prettier and instinctually more attractive than the average 36 year old female other than a 36 year old female who has acquired the attractive nuances of a 16 year old via beauty aids for an older woman, all of which do or are purported to make her look younger, and more attractive in doing so. And what is all the extreme dieting to mega-thinness about than women trying to imitate the body shape of a 14 year old. 

The two probability functions in W_A=T_A=U_AZ_BV of the U_A uncertainty of the male getting the female and of her Z_B probability of bearing a child and raising it with the male’s assistance to maturity are also important factors in the female’s attractiveness and excitement for the male. The U_A uncertainty is, one the one hand, the actual scarcity of females for males, for if there is only one pre-menopausal female in the environment of competing males as indicates a scarcity of them as causes an uncertainty in each of the many males, the female is definitely sensed as more attractive than would otherwise be the case. This uncertainty as a cause attraction also comes about by the female indicating “maybe” to the male as part of her instinctive courtship behavior, this hard to get attitude or ploy causing uncertainty in the male and a greater evaluation of her value and greater attractiveness to her than would otherwise be the case, easy women being generally less attractive to a man.

The Z_B probability of the female to have and cooperatively with the male raise an infant to sexual maturity is a product of two primary component probabilities. One derives from the probability of the male connecting with the female to begin with and mated and then his retaining her for without the love bond, cooperative parenting is pretty much impossible. And the other characteristic needed is the female as an individual’s competence in mothering tasks required to successfully raise the child to maturity.

The complications of civilization especially in its modern form aside for the moment, the probability of a man and a woman producing a child and then staying together as mother and father includes both the probability of the male mating with the female and then hanging on to her for the duration of the child’s development to adolescence. And much of that has to do with the relative rank of the male and the female, which is often described in terms of dominance. And sexual dominance between human males and females is much affected by their living in civilized societies. 

So to make clear what it is that affects a woman’s attractiveness as it relates to her probability of mating and staying with the male without which cooperative parenting is not possible, we have to explain both rank and civilization, which while these will be digressions from the topic of female attractiveness, considering rank and civilization are also important for understanding topics in human nature that go beyond a male’s attraction to the female. Explaining this mathematically will get us into some important considerations that are secrets females generally keep from males. 

To understand rank in its basics, let us consider a V prize slots competition between two players. One of the players called Player plays the slots game that has a probability of winning a V=$270 prize of Z=1/9 and an expectation or average payoff is E=ZV=$30 per game. And the other player called Competitor plays a slots game with the initial doublet match provided where the probability of winning a V=$270 prize is Z_B=1/3 and the expectation or average payoff is E_A=Z_BV=$90

We make this into a competition by having Player pay Competitor whenever Competitor wins the V=$270 prize and having Competitor pay Player V=$270 whenever Player wins. This has Competitor winning ΔE=E−E_A=$60 per game on average and Player losing ΔE=E−E_A= −$60 per game. Assuming that this game is played once a day it is clear that Player would be better off not playing this game at all because of the average loss per day of $60 caused here by the difference in the two players’ probabilities of winning of ΔZ=Z−Z_B =2/9=.222.

Now let us stipulate an alternative for Player that would allow him to contribute $15 per day to Competitor’s bank account and skip playing the competitive slots game that is unprofitable to him to the tune of costing Player $60 per day. This sort of compromise payment, which we will call the contribution, happens frequently in real life situations, one individual submitting to another without competing and contributing something to avoid being beaten in competition and incurring an even higher penalty. This is the origin of one person respecting a competitor’s ΔZ edge in the probability of winning and of taking the Hobson’s choice of Eq55 here of choosing a lesser penalty without a fight over competition fairly sure of producing a greater penalty.

The Competitor also gains in this compromise for a lesser payoff in that there is no risk at all of losing for Competitor when Player just concedes the lesser payoff and that no time is taken up in competing, time that competitor can then spend on other profitable activities. When such a relationship of Player surrendering and contributing to Competitor is continuous and stable over time, Competitor has rank or dominance Player.

The degree of rank or dominance is most simply when both are playing for the same V prize the ΔZ difference in their probabilities of winning the prize competed for, in the above case ΔZ=2/9. Whether contributing to the dominant without a fight is better than trying to win to avoid any penalty depends much on the ΔZ differential probabilities of the two competitors winning. If ΔZ is large as indicates that the higher Z competitor is going to win most of the time, it is better to not compete and take a smaller loss. This in effect is what a poker player does when he thinks the ΔZ difference in the chance of winning is significant and folds his cards to accept the smaller loss of what he already has in the pot rather than putting more money up for grabs that he thinks he has significantly less chance of winning than the other guy.

In physical competition the ΔZ difference in the likelihood of the two competitors winning is also the primary factor in whether one person will submit to another without actually getting into a fight that can cost more than just submitting without a fight. In the absence of a hippopotamus police force or a lion police force, the males of these species basically overpower the female via their significant ΔZ edge in winning and the females sooner or later give into sex as an effective surrender to the male as part of the courtship ritual. In the primitive absence of a police force and a law against rape or any level of forcible sex the average sized male would easily overpower the average sized female.

One reason the primitive human female resists to begin with is that a male who can beat her is stronger than a male who cannot beat her and is preferable as a mate given the protective value of the male in potentially lethal situations the two of them and their future family may have and give the pre-birth control focus of females for the last half million years on raising kids and the protective and provisioning focus of the larger and more aggressive partner of the pair, the male. While things are not now at all like they were in primitive times, the instinctive, inborn emotional machinery is little changed from a half million years ago.

The signal that the aggressor male and eventual dominant or person of higher rank is the right guy for her, the female tends to turn on to sexual pleasure when dominated especially after a vigorous courtship battle. As I say in the song on YouTube, ”What Girls Want”, they aren’t going to tell you because the drive of the female to resist dominance is as natural and powerful as a true test of the value of the male during courtship as her neural signal to give up the fight once the sexual feeling comes upon her. If this is not reality then human courtship rituals are a sharp disjunct from those of almost all other mammals, which would be doubly odd given the sexual dimorphism in size that exists between male in female in almost all mammalian species including man.

Rather it is reality and to get back to our original point, a primary element in a woman’s attractiveness is her relative submissiveness or ability to be dominated, femininity, a term that has strong overtones of submissiveness being understood universally to be a central factor in a female’s attractiveness. Being under the male also allows the female to focus on and specialize in mothering while the male specializes in protecting the mother child relationship from outside threats, a job that is so consuming of time and effort and focus that women who take it on, whatever its touted merits, takes away focus and time from her tasks as mother. Pretty or attractive women who this argument goes make better, more Ψ fit, mothers tend to be attractive because of this higher fitness value that women able to be dominated by males have.

The lack of ability in the modern male to sexually excite the modern female is the primary reason that heterosexual relationships are so difficult to form and maintain for without the truly large voluptuous pleasure of sex to share with all of its love overtones, the love bond between the male and female is limited to totally impossible, for the female attempted to be mated by a less than vigorous male is humiliated by her instinctive disrespect for guys who aren’t “cool” and instinctively rejects them unless cultural factors brought to bear make for her holding her nose and accept the males sperm despite her nausea over the fact. 

The cause of modern man’s low testosterone titer and related personality problems derives from the nature of servitude in civilized societies that dates all the way back to man’s discovery of agriculture. Prior to agriculture, the vanquished population in a war over territory and/or resources who were not able to run away was killed by the victors. Such mass butchery of the male losers in battle is referred to more than a few times in the Bible for such maximizes the d₂ death rate of rivals in F₁=b₁−d₁−b₂+d₂ of Eq99 along with the capture and taking away of the women as additional wives or concubines as maximizes the b₁ birth rate in F₁ of the conquerors. Agriculture enables great economic exploitation from slavery to decrease the d₁ death rate and increase the F₁=b₁−d₁−b₂+d₂ F₁ fitness of the conquerors more than it is increased by increasing the d₂ death rate of the vanquished potential slaves.

Great dominance is a central factor in slavery, the ΔZ differential probability of winning in a fight between master and slave being very great, indeed, ΔZ≈1. This causes male slaves not to resist their masters and to contribute their labor rather than incur the worse penalty of punishment for resisting their master/s. It is this psychological castration of civilized males by the ruling classes of civilizations, which actually does lower testosterone significantly even if not as much as physical castration, that makes the typical civilized male a bad lay, a poor lover and husband and a poor father for a woman to have to protect her and her children, which she, of course, is unable to do from the ruling class who will use her children to be the next generation of slaves. This is why most modern males are emotionally rejected sooner or later by females today despite the fact (another secret) that females may fake it in a variety of culturally sanctioned ways to “keep the marriage together.”

The case, of course, can be argued that although all of the old Middle East empires of Babylon and Egypt and Persia were slave societies followed by the slave based Western Hellenic (Greek) and Roman empires followed by the atrociously exploitive serf societies of the Middle Ages that slavery stopped short with democratic capitalism. While it is easy to show from logical argument that this public assertion as with a free and fair America is nothing but Machiavellian deception, it is better first to explain the nature of mass media propaganda mathematically to show how the obvious pain of corporate ruling class intrusion, control and abuse in our workday and personal lives is brushed under the rug publically and, hence, in the minds of people who suffer and are so brainwashed as to not understand the central cause of their unhappiness to be their servile working class exploitation by the capital class of the corporate wealthy. 

To explain the cornerstone strategy of propaganda we introduce Simpson’s Reciprocal Diversity Index written in its most immediately digestible form as  

111.)     

The meaning of D as the number of significant subsets is developed from the sets of objects in the table below that consist of K=21 objects divided into n=3 subsets distinguished by color. 

Table 112. Sets of K=21 Objects Divided into n=3 Subsets and Their D Significance Values

113.)     

114.)     

115.)     

The D=2.19 of the (10, 10, 1)↔(■■■■■■■■■■, ■■■■■■■■■■, ■) set, which rounds off to D≈2 is interpreted as there being 2 significant subsets in the set, the (■■■■■■■■■■) red subset and the (■■■■■■■■■■) green subset, with the (■) purple subset with only one object in it being insignificant. The (7, 7, 7)↔(■■■■■■■, ■■■■■■■, ■■■■■■■) set with D=3 is understood to have 3 significant subsets, the red, the green and the purple as is the (6, 6, 9)↔(■■■■■■, ■■■■■■, ■■■■■■■■■) set whose D=2.88 rounds of to D≈3. We can better feel for the mind’s intuitive sense of significance by representing the objects in the above sets as interwoven lines.

 

If one had a plaid skirt with the (10, 10, 1), D≈2, pattern, one would describe it as a red and green plaid, automatically and subconsciously omitting reference to the one thread of purple in it. So what the D≈2 specification of the plaid tells us is that there are 2 significant colors in it, the red and the green with the single purple thread being insignificant. The other two plaids would intuitively be described as red, green and purple plaids as suits their D=3 and D≈3 specification. 

That the insignificance of the purple thread indicated by the D≈2 diversity measure would manifest itself linguistically in its being disregarded in the description of the plaid as a red and green plaid should not be surprising for the word “significant” has as its primary root, “sign”, which means “word”, which tells us that what is significant is given a word or is verbalized or mentioned while what is insignificant doesn’t get a word or isn’t verbalized or mentioned. The mind’s differentiation of the significant from the insignificant and of its disregarding the insignificant in mention is an important factor for behavior for you pay attention to, think about, talk about, give a word to and act on what you sense is significant but disregard the insignificant in your thinking and behaving.

Politicians and other propagandists often speak in a way that makes the realistically significant seem insignificant and the insignificant seem significant. As to the latter, consider the fact that it was repeated often and touted as significant by the Bush administration that the invading force in the Iraq War was a coalition. But in consisting approximately of n=32 nations with numbers of soldiers contributed of (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), this set’s number of significant contributors calculated from Eq111 is D=1.26≈1, the one significant nation in the coalition being the United States. This tells us either that the so-called coalition was not a significant coalition or that it was not a coalition of significant nations, the political deceit in calling it a coalition obvious enough to be recognized as deceit by the astute via common sense without the need for this mathematic analysis based on Simpson’s Reciprocal Diversity Index, though using the D measure allows us to call down the liars unequivocally in mathematical terms because of another way that D of Eq111 measures significance.

Also affecting the acceptance of political misinformation, no matter how absurd, is how often it is repeated. Let us reconsider the (10, 10, 1)↔(■■■■■■■■■■, ■■■■■■■■■■, ■) set as representing events that happen over time. The red event, ■, happens with a relative frequency of p₁=10/21=.476, the green event, ■, of p₂=10/21=.476 and the purple event, ■, of p₃=1/21=.048. Much as the purple set of x₃=1 purple object in (10, 10, 1)↔(■■■■■■■■■■, ■■■■■■■■■■, ■) was earlier considered insignificant on the basis of the size or magnitude of the purple set at a given time, so also is the purple unit object when considered an event that happens in time considered to be insignificant relative to the red, ■, and green events, ■, which happen with a higher frequency.

Now let us consider an n=2 set of objects, (20, 1)↔(■■■■■■■■■■■■■■■■■■■■, ■). The D measure of the number of significant subsets is from Eq111, D=1.099≈1, which suggests D=1 significant event, the red event, ■, with the purple event, ■, being insignificant. When these two events are people taking opposite sides of an argument, the red event, ■,      When p₁=.952 is the frequency of occurrence of the red event, ■, which happens much more frequently than p₂=.048 of the purple event, ■, the red event, ■, is seen as significant and the purple event, ■, as insignificant. When these events are mutually contradictory, as with the red event, ■, being affirmation of a socio-political position and the purple event, ■, it denial, what is considered to be significant and true is what is repeated with the greatest frequency.

Now we see that a large frequency repeat of the same kind of event makes for a significant event in people’s minds relative to events seldom experienced or not at all. That is sensible from the realistic point of view that things we do often as with the people we interact with are significant, our parents, children, siblings, neighbors, salespersons at a store we frequent and co-workers and bosses are all significant as are the venues in which these people are located. And from an analytical perspective, frequency as a marker for significance has meaning in that the relative frequency an event in the past, relative to other events in the past, projected to future repeats of the same kind of event is mathematically the relative probability of that event happening, relative to the other events.

As probability, it will be recalled, it is as Z part of the E=ZV mathematical expectation, which was the determinant of selection in Eq54 via the τ=ΔE selection emotion.  Hence much we would discard the lower Z probability activity in τ=ΔE=E₁−E₂ when V is the same in both, τ= E₁−E₂ = Z₁E−Z₂E, so also does the mind automatically disfavor or discard the relative low frequency event as insignificant relative to the higher frequency event.

This is also the foundation of discarding the insignificant as it was introduced in terms of a low time stationary characteristic like the size or number of objects seen at any moment. Why? It is because large things or groups of things tend be very good for a person’s Ψ fitness or very bad, that is, important in value as measured by V dollars or the V number or size of anything observed that has any potential of increasing or decreasing Ψ fitness, and as a large thing, a chance of affecting fitness in a large or significant way.

What we have considered in this forgoing topic of significance for whatever has great or potential high value (V) and great or potentially high (Z) probability of occurrence is just an interesting interpretation by the mind of the more general earlier rule of Eq54 of the mind selecting for the activity or event with the highest E=ZV expectation and disregarding or not selecting the low expectation event with low Z and V as make up E=ZV.

Hence, this is but the instinctive intuitive manifestation of the τ=ΔE selection rule that we can also can and do operate on in a more emotional way when we actively feel the pleasures and pains associated with competing possibilities and choose on that basis and further yet when we operate from the τ selection function in a totally calculating way when we are able to specify the Z and V terms in E=ZV accurately and evaluate E=ZV as the average return per unit game played (in slots play) or per unit effort made more generally. This confluence of two relationships come at from two quite different directions is a powerful proof of the mathematics of the human emotions we have developed.

There are two reinforcements of earlier conclusions things to note from this. The first is that as we give words to what is significant as manifesting a high relative expectation of Ψ fitness optimality, it is obvious that we give words to those things that have high fitness value, language now being an obvious adaptive characteristic developed by evolution, which further bolsters the general argument that the machinery of the human mind is an evolved structure, and in a very mathematically detailed way, which has no need for a god as an intermediary than our solar system needed angels informed by Newton’s F=Gm₁m₂/r² Law of Gravity to be able to push the planets about in their orbits obeying that law.

Also what this tells us is that the mind is a very mechanistic piece of neural anatomy fixed in its aim for optimal fitness in well specified ways no different than the heart or liver or human hand that cannot be contravened by the magical undefined operator of “free will.”  We work according to a set of rules underpinned by genetic information from our distant past and experiential information from our life’s past ultimately gotten from an environment that lies outside our control and by mental mechanisms whose hardware information we also have no control over. Hence life and its outcomes, whatever our relatively narrow input into those outcomes, is most basically a matter of fate, a fate determined not only by the natural laws we are spelling out in this dissertation that underpin our emotions and the decision making they bring about, their purpose, but also by our environment, which is what we wish to consider next.

Let us first go back to Eq54 to expand it and understand the E=ZV expectation activity chosen as competing against not just one other E=ZV expectation but numerous of them. In that case by extension from the τ selection law, we choose to do the activity that has the highest expectation associated with it. The big question is to how an individual’s mind gets to know or learn the values of Z and V in the set of E=ZV alternatives he can choose from. As we already have made semi-clear, those expectation values, especially the V aspect of the E=ZV and E= −Uv arise primarily from instinct for the pleasurable feeling we get from eating food is inborn as is the unpleasant feeling we get from being exposed to frigid temperatures.

Their values are, more specifically, by a testing of objects in an environment and activities able to be done in an environment. If there is a match of an object tested by experience and the template for it indicating its fitness value in mind, a V function, as with fruit, which keeps us alive and surviving, our neural anatomy makes it taste good, which if it has a reasonable Z probability of being gotten, gives it a high E=ZV expectation, which makes us select the activity of getting the piece of fruit and eating it.

The probabilities associated with such expectations, on the other hand, are obtained by experience as learning. If there are two ways of getting a peach, either by throwing a rock at a peach tree repeatedly which has a lower likelihood of success per unit time than climbing a ladder and picking the peach, we choose the latter activity over the rock throwing one to get a peach. In this way genetic and experiential information combine together to develop our expectations of success in various activities.

Another way we get information on the Z and V aspects of the E=ZV expectations of activities we can take up is by information from others. This is the basis of the great adaptive power of the culture for human beings. Often the testing necessary to determine the V value of an object and/or the Z probability of a particular activity achieving the object of desire is a lengthy process or dangerous for the individual. What saves time and the penalty of risky enterprises is information from another person. For humans so much of this kind of learning takes place in childhood and originates from parents or adults in loco parentis, who, indeed, determine so very much of a child’s expectations, both in terms of what the child can make happen and what can happen to the child from outside agencies. Indeed, this is a great part of growing up, being taught how to optimize Ψ fitness or in colloquial terms, to be successful in life.

Such information or advice from parent figures to children is readily accepted by the children. There are two factors in the value of this secondary information for its receiver. The Z likelihood and V values taught must be accurate reflections of reality, that is, not in error. And besides error, the other factor that kills the value of secondary information is when it is intentionally inaccurate and intended to make the individual do something that is not in his Ψ fitness interests but in those in some manner of the deceiver. In both cases the secondary information, (general information, suggestion, advice and command), should be rejected by the individual as much as he or she would reject a freezing temperature situation.

Deceitful information is not a problem, for the most part, in the most primitive social groups that consist of clans of genetically closely related individuals up to the second and third cousin level, who also have a reason for acting co-operatively from the fruits of group activity being shared by all to assist each in optimizing their individual Ψ fitness, as with hunting groups sharing the meat from the kill. But there was a dramatic shift for mankind in a number of very important ways that occurred when agriculture was discovered and came to supersede the hunter-gatherer phase of man’s evolution, which our basic emotional machinery developed in and for. 

The nature and nuances of the lies of propaganda are difficult to understand without considering the mammoth impact that man’s discovery of agriculture had on fitness priorities and human behavior. The upside of agriculture is readily understood for those primitive human cultures that left behind the hunter gatherer way of life and adopted farming as their primary way of surviving for these evolved culturally to completely dominate the earth both over all other animal species and over the hunter-gatherer primitives they left behind. How agriculture did this is seen in an analytical way from an expansion of the fitness function of Eq99. 

99.)                                      

It is obvious that those organisms that have proficient survival behaviors which get them to live long as to minimize their lineage’s d₁ death rate are more likely to have a positive F₁ fitness and to survive from generation to generation, that is, to avoid extinction. Avoiding extinction from a positive fitness, F₁>0, also includes in order to maximize the b₁ birth rate of adolescents being reproductively successful, which includes for the male not only getting the female but also raising a family successfully. And as the math shows clearly in Eq198, one maximizes the F₁ fitness by maximizing the d₂ death rate of one’s rivals for space and resources in the niche. This can be done mathematically either by killing them, the evolutionary causation of the endless parade of wars we see in human history, or by territorial battles which accomplishes the same end of reducing the numbers of rivals in the competitive niche.

The importance of staying alive by having a steady food supply as minimizes the d₁ death rate and helps maximize the F₁ fitness in Eq99 was not lost on those who took up farming as a way of life and discarded the primitive hunter-gatherer mode of survival.  But there was a price to pay for humankind adopting agriculture, not only for the hunter gatherers who were outcompeted by the civilized agriculturalists that had more time and energy to devote to achieve the combat skills and weaponry needed to best their rivals (as the British-American settlers did against the Indians in America and the British-Australian settlers against the aborigines in Australia), but also in the slavery made possible and lucrative by agriculture.

Prior to agriculture based civilization, the losers in a war over a territory and its resources were killed by the victors if not fortunate enough to flee to new territory, activity that surely was a large part of human beings coming to survive in just about every kind of environment on the planet. This statement of the mass butchery of the losers referred to more than a few times in the Bible should be amended, of course, to refer more specifically to the male losers who failed to escape for the women of the vanquished in war were invariably taken as additional wives or concubines for the conquers as satisfied the drive to maximize the b₁ birth rate of the conquerors as was also seen for our Biblical predecessors.

But the murder of vanquished males came to an end with the invention and refinements of agriculture that enabled a profitable keeping of the losers in war captive to use as labor in the agricultural fields, for owning and using slaves thusly maximizes F₁ fitness by decreasing the d₁ death rate of the conquering population much more than the killing of the vanquished increases F₁ by increasing the d₂ death rate of rivals in Eq99. 

The F₁ economic fitness of owning slaves has the slave owners develop optimal slave systems to maximize that fitness. Such optimization takes on a number of forms. One of them is allowing slaves as much freedom as possible commensurate with the slaves conditioned obedience to their enslavement. In most slave systems in all civilizations, the descendents of captured slaves are given a less constrained status after a few generations of patent ball and chain servitude. This makes the slaves more valuable economically not just from the lesser need and expense of having them constrained, but also because for many occupations which slaves may take up, less constraint translates to higher productivity.

Indeed, some slaves are given slave driver occupations as is economically optimal for the slave owners, a common practice seen in the pre-bellum plantation slavery of the American South. Such is also the totally extensive practice in hierarchical form in modern cash controlled slavery where the primary coercion is from the withholding of money rather than from the whip of the archetypal slave driver or billy club of the police of a military dictatorship police state and the control of the slave colony as a mega-unit is hierarchical with bosses controlled by upper level bosses controlled by higher level bosses yet, usually referred to in nicer terms as management, higher level management and top management, as though being managed via the loss of job or promotion at your head and the humiliation that goes with it for those still young and intact enough to feel the sting of humiliation or being supervised by your supervisor was better than being bossed by a boss or certainly better than being beaten up every day by a slave driving bastard as alien to your self-interests unless they are co-incident with his or hers as an alien predator from another planet.         

Children of slaves who are raised to respect the wishes of slave drivers or managers and the rules of slavery are obvious candidates for lesser constraint status and no small effort is made in this area to raise slaves “properly” much as like raising farm livestock with great care for optimal end results. Along these lines, part of raising such slave children is the inculcation in them of the notion that the children are not slaves, a notion that has them behave in adulthood in a less rebellious, less resistant manner, for if there is no real slavery than there is nothing to revolt against or resist, slave masters then taking on the form of “authority” like family authority whose judgment of matter and wishes should be respected because in the end they are in the best interests of the slaves, both in the adult and childhood developmental stage.

All of this is by way of explaining how such propaganda is inculcated in the minds of the slaves. Much of such slave ideology in all civilizations, especially those whose culture predates the scientific revolution, entails superstition that promises rewards of one form or another for slaves after they die if they are obedient and follow the rules when they are alive. However much the notion of this is ludicrous to any sensible human being, subscription to it is almost total sure if the brainwashing begins at a very young age, is done as much as possible directly by the slave parents of such slave children and is reinforced both for children and for adult slaves by the entire society agreeing that the information provided, no matter how intuitively farfetched, is valid and truthful.

This takes us back to the D diversity of Eq111 that specifies quantitative significance. How often or frequently one encounters objects and situations is as important in determining their significance for the human mind as how physical large or numerous the objects associated with situations encountered are. If an n=2 set of specifications are mutually contradictory, one being the opposite of the other as for example the dichotomy that giant spiders exist on the other side of the mountain or do not exist, in the absence of actual observation, if 99 people say they do exist and only 1 in a 100 that they don’t, the information from the former is taken to be significant and is accepted, unless of course the reliability of the word of the 99 is seriously questioned.

This is how propaganda generally works. This mode of brainwashing by extensively repeating falsehood is especially effective on children who instinctively are very unlikely to question the reliability of adult information or the veracity of adults who are closely related or very familiar to them. It also works astonishingly well on adults especially when the information is pleasurably received or is entertaining.  And the essence of it is giving people misinformation as to the Z probability and V or –v values of activities especially to youngsters who are very vulnerable in the mind as to whatever is told them. That is, the “this is bad and that is good” information of standard culture as seen on TV, preached from the pulpit or psychologist’s chair is for the most part and taught in school’s is as bogus as the sense from TV and movies that jip joint Las Vegas is an exciting place where you can go to win lots of money and find love.

Yesterday morning (Sat., Jan. 9), I walked past the parking garage of the California Casino off Fremont St. where a losing patron, more than 80% are the first time around with a higher percent for returnees, jumped off its four floor to his death after losing all his money. They not put that in the news and, truth be told absolutely it happens all the time. That’s why none of the high rise hotel windows don’t open, to prevent people jumping out of buildings after they lost all their money from false expectations put in their mind’s by the standard culture about casinos. America is the same game, all advertizing and bait and switch when life wears on and one’s expectations aren’t fulfilled, can’t possibly be because they were bogus misdirection by all manner of fictional (sitcoms and movies) and so-called factual programming misdirection.

The information given by the corporate owned and run media is information in their interests and not yours. Basically people are coerced by their financial situation and not infrequently by fear of police to do what the ruling class has set up the rules to be (in their favor), which makes people unhappy, the only way to get around that being to overthrow the cruel jackass bastards, and the only way to do that in a social system where they have all the police power being to elect my wife, Ruth, as president on the Occupy the Ballot ticket, possibly along with Jon Huntsman, the only Republican with any sanity on the issues, because the ticket has to be bipartisan to beat down deep corporate suck, Obama. The effort has to be serious to get somebody elected who will put all the robber barons in jail and fine them their billions, and that requires a bipartisan ticket and a vice-presidential candidate who even with a few warts is electable. Huntsman has a good mind and Trump, whatever his merchant’s perspective and downside of having been born also with a silver spoon in his mouth, has a basically good heart.

There is much more that can be said and derived from this mathematics including a revision of Boltzmann statistical entropy in terms of diversity, but for practical purposes, most of what needs to be said to save the world from its personal unhappiness and that collectively causing the next world war (nuclear) and the end of us all and our precious children and grandchildren, the above says all that needs be said. We need revolution now, the only change that can make the real difference of getting rid of the jerk bastards above us who are the cause of all our misery and confusion. 

Now let’s take a sharp turn to get into the mathematics we have been using in greater detail.

Figure 117. Boltzmann’s Tombstone

Now that we have the gist of the centrally meaningful idea of this work that abusive control is at the root of unhappiness and violence, let’s double back to explaining it again but in more thorough detail with mathematics so powerful that it also resolves the century old mystery of entropy by correcting the misleading formulation of it derived by Ludwig Boltzmann and inscribed on his tombstone back in 1906. It does this by specifying entropy as energy diversity. Its connection to the mathematics we have already worked with lies in the Z probability of the E=ZV expectation of Eq2 and the D significance function of Eq111 also deriving from diversity measures that were originally developed back in 1949 by Hugh Simpson, a British statistician.

The Simpson’s Diversity Indices are found extensively in the ecological and sociological literature of the last 60 years. We give the symbol, Z, to the first Simpson’s Diversity Index that Simpson developed.

118.)     

We introduce Z with an ecology of parrots that has K=12 parrots in it divided into n=3 species, red, green and purple feathered, diagrammed as the set of objects, (■■■■, ■■■■, ■■■■). This parrot ecology can be specified with the number set, (4, 4, 4), which denotes x₁=4 red parrots, x₂=4 green parrots and x₃=4 purple parrots. The fractional measures of each species or their population densities are p₁=x₁/K=1/3, p₂= x₂/K=1/3 and p₃= x₃/K=1/3. These are the p_i terms in the diversity index of Eq118.

119.)      

The diversity Z for the n=3 species, (4, 4, 4)↔( ■■■■, ■■■■, ■■■■) parrot ecology that has p₁=1/3, p₂=1/3 and p₃=1/3 is

120.)     

By contrast the diversity, Z, for an n=2 species, (6, 6)↔(■■■■■■, ■■■■■■) parrot ecology with x₁=6 red parrots and x₂=6 green parrots and population densities of p₁=1/2 and p₂=1/2 is

121.)     

As the n=3 species, Z=.333, (4, 4, 4)↔(■■■■, ■■■■, ■■■■), ecology is by every sense of diversity more diverse than the n=2 species, Z=.5, (6, 6)↔(■■■■■■, ■■■■■■), ecology, it is clear that Simpson’s Diversity Index, Z, of Eq118 is a reciprocal measure of diversity, the smaller the value of Z, the greater the diversity of the ecology. This is borne out for the K=12 parrot, n=3 species, (7, 2, 2)↔(■■■■■■■, ■■■, ■■), ecology that has x₁=7 red parrots, x₂=3 green parrots and x₃=2 purple parrots with population densities p₁=7/12, p₂=3/12 and p₃=2/12 and diversity

122.)     

Its Z=.431 diversity reciprocally indicates that the n=3, (7, 2, 2)↔(■■■■■■■, ■■■, ■■), ecology is more diverse than the n=2, Z=.5, (6, 6)↔(■■■■■■, ■■■■■■) ecology because it has more different species than (6, 6), but is less diverse than the n=3 species, Z=.333, (4, 4, 4)↔(■■■■, ■■■■, ■■■■) ecology because though both the (4, 4, 4) and (7, 3, 2) ecologies have n=3 species, the (7, 3, 2) ecology divides up its n=3 species in an unbalanced way in contrast to the balanced n-3, (4, 4, 4) ecology.    

Simpson developed his Z measure of diversity in probability terms as follows. If you randomly select a parrot from the n=3 species, (■■■■, ■■■■, ■■■■), ecology the population density, p₁=1/3, of the red species is the probability that the parrot selected will be a red parrot and p₁²=(1/3)²=1/9 is the probability you will select (with replacement) a red parrot twice. Similarly p₂²=1/9 is the probability of selecting a green parrot twice and p₃²=1/9, that of selecting a purple parrot twice. So the probability of selecting parrots twice from the same species is the sum of p₁² + p₂² + p₃² or ∑p_i²=Z of Eq118: Z=1/3=.333 for n=3, (4, 4, 4) from Eq120; Z=1/2=.5 for n=2, (6, 6) from Eq121; and Z=.431 for n=3, (7, 3, 2) from Eq122.

Note from Eqs120&121 that for balanced n-species ecologies where the number of organisms in each species is the same,

123.)     

This is the basis for our using the Z symbol as probability in the E=ZV expression for mathematical expectation of Eq2 as applied to the slot machine game where the probabilities of the red seven, 7, the green seven, 7, and the purple seven, 7, were balanced or equiprobable. Later we will show in detail how unbalanced or non-equiprobable selection as develops the emotions for many of people’s realistic behaviors uses Z of Eq118 as probability or a diversity function derived from it as the general case. We hold off on it for now because the mathematics of it is best introduced by our first using the diversity measure to solve the entropy problem in science. The point to be made at the moment is that the general case of probability for emotion derives from Simpson’s Diversity Indices that we have rediscovered to be centrally useful for examining and solving a number of very difficult and up to now unsolved problems in science.

To do that we must understand the concept of diversity to applied to any set of objects that can be grouped into n subsets. That includes, for example, K=12 buttons in a button box that can be divided into n=3 color categories as (4, 4, 4)↔(■■■■, ■■■■, ■■■■). And such a representation can be extended to any set of K=12 things distinguished from each other in some fashion as (4, 4, 4)↔(■■■■, ■■■■, ■■■■) with the colors shown here in the subsets put there merely as markers of subset distinction as might distinguish 4 Indians, 4 Irishmen and 4 Kenyans or 4 Chevrolets, 4 Fords and 4 Audis. In this generalization of diversity, the p_i=x_i/K variable of Eq119 is called the weight fraction of the objects in the n different subsets of the set.

Next note that the D measure of the number of significant subsets in Eq111 is seen via Eq119 to be the inverse of Z diversity

124.)     

In fact, D is also a Simpson’s diversity found extensively in the ecological and sociological literature of the last 60 years called Simpson’s Reciprocal Diversity Index. It’s values for our example ecologies are: for Z=1/3, (4, 4, 4)↔(■■■■, ■■■■, ■■■■), D=3; for Z=1/2, (6, 6)↔(■■■■■■, ■■■■■■) , D=2; and for Z=.431, (7, 2, 2)↔(■■■■■■■, ■■■, ■■), D=3.232. Note that the D Simpson’s diversity is an increasing measure of diversity as better fits intuition. Note also as applied to the example parrot ecologies this would indicate respectively D=3, D=2 and D=2.232≈2 significant species. This makes it clear how Simpson’s diversities are also measures of probability and significance. We shall see next how they also provide the proper understanding of thermodynamic entropy. This will give us greater confidence in them as new, very primitive mathematical functions capable of spelling out all the human sciences mathematically. It will also tell us exactly what a word is, very important because ultimately words control behavior and their outcomes, happy and unhappy, in terms of a representative configuration of a random distribution, a mathematically specified space-time generalization that is structurally identical to and explains the origin of the meaning of common nouns and verbs and, indeed, all words.         

To understand the verbal underpinning of our thinking and behavior requires that we understand entropy properly, difficult because it has been from its introduction to science 200 years ago an intuitively confusing and mysterious concept, even over the last hundred years since the Austrian physicist, Ludwig Boltzmann, provided the microstate mathematical expression for it inscribed on his tombstone in Figure 117, S=klogW.  Students who’ve studied entropy think it is difficult to understand because it is somehow an inherently complicated phenomenon. A Professor Wunderlich I had back in graduate school in an Advanced Chemical Thermodynamics class told us students not to worry about making sense out of entropy because nobody under the age of 40 understands it. But the real reason entropy is confusing is because Boltzmann’s expression of it written below in modern notation is flat out wrong despite its fit to thermodynamic data and is totally misleading in that regard as to what entropy actually is as a physical quantity..

125.)     

 

We will explain the symbols in Boltzmann’s entropy as we prove it to be incorrect and replace it with a mathematical formulation for energy diversity, which is what entropy actually is.  Bypassing the confusing gibble gabble of the popinjays who currently rule in the academic kingdom of confused Boltzmann statistical mechanics, a thermodynamic system is most simply a random or equiprobable distribution of K whole numbered, discrete, energy units over n molecules. It is no different in essence than a more tangible system we shall examine first of the repeated random distribution of K candy bars by a grandfather to his n grandchildren. Specifically we will mathematically consider the distribution of K=4 candy bars by grandpa to his n=2 grandchildren, Jack and Jill. This is done by him tossing the candy bars blindly over his shoulder to them. Such a distribution is said to be equiprobable because the probabilities of each of the n=2 children getting a candy bar blindly tossed are equal, P=1/n=1/2 for both the grandkids.  

 

The K=4 candy bars we have grandfather toss are different, distinguishable, brands, a Snickers, S, a Hershey’s, H, a Butterfinger, B, and a Tootsie Roll, T. We could use the same kind of candy bar for all K=4 of them and still get the same results, but using different brands most clearly distinguishes between the Ω=n^K=2⁴=16 permutations of candy bar allocations marked out in {braces} below. Note that the candy bars Jack gets in the random tossing are listed to the left of the comma in the {braces} and Jill’s candy bars to the right of the comma.

Table 126.The Ω=16 Permutations and W=5 States of the Random Distribution of K=4 Candy Bars to n=2 Children

The Ω=16 permutations are grouped into W=5 states that specify in [brackets] how many candy bars Jack and Jill receive followed by the number of permutations per state. And from the understanding that all permutations in a random distribution are equiprobable as Ω=1/16, the probabilities of the states are listed. And beneath that is the number set notation of each state, with x₁ being the number of candy bars Jack has in a given state and x₂, the number Jill has.

If grandfather performs this random tossing of the K=4 candy bars 16 times, on average each of the Ω=16 permutations will come about once. The W count of the number of states of a random distribution, here the W=5 number of states for the K=4 candy bar distribution over n=2 children, is the W term in Boltzmann’s entropy of Eq125 for the random distribution of K energy units over n molecules (with S, the entropy in that function and k_B, a constant called Boltzmann’s constant.) And the probability of a state for the candy bar distribution as seen in Table 126 is the number of permutations of that state divided by the Ω=16 total number of permutations.  

Some states have more diversity of distribution than other states as to how the K=4 candy bars are divided over n=2 kids, which parallels the division of K organisms over n species in an ecology. The [2, 2] state that has both kids getting the same number of candy bars in a K=4 toss has more diversity of distribution than the [4, 0] and [0, 4] states in which one kid gets all the candy bars tossed and the other gets none. The diversity of a state can be specified with the Z diversity index written from Eq124 as

127.)     

This specifies the Z diversity of the [3, 1] state, for example, as 

128.)     

The Z diversity of each state of the K=4 candy bars over n=2 children distribution is 

Table 129.Diversity, Z, of the States of the K=4 over n=2 Distribution

We see again that Simpson’s Z diversity index is an inverse function of diversity in having a larger Z value for less diverse states like the [0, 4] state of Z=1 and a smaller Z value for states with greater diversity like the [2, 2] state of Z=1/2. One can form an average of the Z diversities of all W=5 states in the K=4 over n=2 distribution that is a probability weighted average, one that weights the contribution of the diversity of each state by the probability of the state given in Table 126. 

Table 130.The Average Z Diversity of the States of the K=4 over n=2 Distribution

The average diversity of the states of a random distribution is written as Z_RC, here Z_RC=5/8. The RC notation of the subscript in the Z_RC average state diversity stands for Representative Configuration, as will be clarified later. We can also specify the Simpson’s Reciprocal Diversity Index, D, of Eq124 for each state.

124.)     

The D diversities of the states are just the reciprocals of their Z diversities listed in Table 129.

        

Table 131.Diversity, D, of the States of the K=4 over n=2 Distribution

Note as clarified earlier that Simpson’s D diversity is an increasing function of diversity and, as such, a preferred measure of diversity for physical systems. The average state reciprocal diversity, D_RC, is just the reciprocal of the Z_RC=5/8 average state diversity, D_RC=1/Z_AC=8/5=1.6. Shortly we will derive a shortcut formula for the average state reciprocal diversity, D_RC, as a function of the K and n parameters of a random distribution, but for the moment we will just state the shortcut function.   

132.)     

You can test the correctness of this formula by using it to obtain the D_RC=1.6 average state diversity of the K=4 over n=2 distribution.  

133.)     

We will derive this shortcut formula from scratch shortly but for now just want to use it to cut to the chase in order to show the error in the century old Boltzmann’s entropy equation, S=k_BlnW, of Eq125. The k_B term in S=k_BlnW is a constant, Boltzmann’s constant, and not of immediate concern to us. The W term in it is the number of states of a K over n random distribution, which can also be obtained from a shortcut formula known generally to mathematical physics.

134.)     

It calculates the W=5 states of K=4 candy bar over n=2 kids random distribution listed in Table 126 as it would the W=5 states of a K=4 energy unit over n=2 molecule random distribution as  

135.)     

Boltzmann specified a logarithm function of the W number of states, lnW, as the central variable of entropy in Eq125 because lnW fit the empirical thermodynamic data.

136.)     

The thermodynamic data’s good fit to Boltzmann’s lnW entropy of Eq125 made for the function’s eventual acceptance by the science community of the turn of the 20^th Century. But this was not before initially rejecting it because of its vague representation of entropy as a physical quantity, which caused so much distress in Boltzmann that he hung himself back in 1906, which was likely a powerful reason for reconsideration and eventual acceptance culminating in its being honored by being placed on a new tombstone for Boltzmann inscribed with S=klogW, Eq125 in its older notation. The lack of any intuitive sense it gave entropy, though, produced much confusion over the last century because the lnW logarithm function of the W number of states of a distribution makes no tangible sense as a physical property of a random distribution.

The D_RC diversity of a random distribution of Eq132 makes plenty of sense intuitively on the other hand and, as we will show next, there is a very high numerical correlation between lnW of Eq136 and the average state diversity, D_RC of Eq132, that makes them interchangeable with each other and impossible to tell apart as to which is the proper codification of the thermodynamic data.  We will show this high correlation using a series of candy bar random distributions over a fixed number of n=40 kids with the K number of candy bars distributed ranging from K=40 to K=91. 

Table 137.The lnW and D_RC of K over n=40 Distributions with K ranging from K=40 to K=91.

The Pearson correlation coefficient between lnW and D_RC in the above of .999 is manifest visually in the near straight line of the scatter plot of lnW against D_RC.

 

.
Figure 138.The Scatter Plot of lnW versus D_RC for the K=40 to K=91 Candy Bar Allocation to n=40 Kids.

 

Any way one might chose to compare lnW and D_RC for a distribution including via random computer generated values of K and n with K>n consistently produces a high correlation in the range of .99. This high correlation should not be surprising given from Eqs132&136 that the D_RC average diversity of the states of a K over n random distribution increase in tandem with the W number of states and lnW, both being functions of the K and n parameters of the distribution.

The high .999 correlation between lnW and D_RC makes clear that whatever fit the lnW based Boltzmann entropy of Eq125 has to empirical thermodynamic data, so also must the D_RC diversity have that fit within an acceptably small limit of error. As both functions fit the data well, both can be used to represent entropy on that basis. But the D_RC diversity based entropy or diventropy is vastly superior to Boltzmann’s lnW based entropy because it makes clear intuitive sense out of entropy as energy diversity in contrast to lnW based entropy which makes little intuitive sense as a physical quantity especially given that the lnW logarithm of the W number of states of a random distribution has no tangible meaning even from a pure mathematics perspective as a property of a random distribution, whether one of candy bars over kids or energy units over molecules.

Beyond D_AC diventropy being superior to lnW Boltzmann entropy in making better intuitive sense, it should importantly also be pointed out that Boltzmann’s lnW entropy is patently incorrect because of Boltzmann’s axiom of equiprobability of the W states which he had to assume in order to derive his S=k_BlnW entropy for it is mathematical poppycock, entirely impossible mathematics, as seen in the candy bar over kids random distribution of Table 126 whose W states are not equiprobable. Indeed, equiprobable W states for any random distribution that can be conjured up are impossible mathematically, as we shall show alter in detail. Hence of the two tightly correlated entropies based respectively of D_RC and lnW, the former must be correct.  

As to why Boltzmann choose the lnW based entropy formulation, it must be understood that it was the only function available mathematically to fit the thermodynamic data back at the turn of the 20^th Century because Simpson’s D diversity, the correct form of entropy that lnW so highly correlates to, was not developed by Simpson and available to science until 1949, 46 years after Boltzmann’s death, and even then as a measure of biodiversity that lacked ready dissemination to the physical science community whose aegis was statistical mechanics and entropy.

To bolster the above assertion that Boltzmann lnW entropy is dead wrong we now formally develop the D_RC=nK/(K+n−1) function assumed in Eq132. An important property of every number set and of every state of a K over n distribution considered as a number set is the μ mean of a state.  

139.)     

All W=5 states of the K=4 candy bar over n=2 children distribution listed in Table 126, [4, 0], [3, 1], [2, 2], [1, 3] and [0, 4], have a mean value of μ=K/n=4/2=2 candy bars per child. Also important is the statistical error function, the variance of a number set.  

140.)     

The variance is of the [4, 0] state is σ²=4. The variance, σ², is the square of the more familiar statistical error, the standard deviation, σ, (sigma), which for the [4, 0] state is σ=2. The σ² variances of all W=5 states of the K=4 over n=2 distribution are

 

Table 141.The σ² Variance of the States of the K=4 over n=2 Distribution

One can also form an average of the variances of these states, a probability weighted average that weights the contribution of the variance of each state by the probability of occurrence of these states as listed in Table 126.

Table 142.The Average State Variance for the K=4 over n=2 Distribution

The average of the state variances, written as σ²_RC, is σ²_RC=1. Now to understand better what this σ²_RC average variance is and means, let us look up the textbook formula for the variance of a multinomial distribution in Wikipedia expressed in our notation as

143.)     

The P_i term is the probability that any one of K objects distributed over n containers will be allocated to the i^th container. For an equiprobable distribution, this probability is P_i= 1/n, every one of the n containers having an equal 1/n chance of receiving the object in a random or equiprobable distribution. We saw this earlier for the K=4 candy bar over n=2 kids equiprobable distribution where the probability for each child getting any candy bar tossed is P=1/n=1/2. This P_i =1/n probability for the equiprobable case simplifies the above distribution variance expression to 

144.)     

 

For the K=4 over n=2 equiprobable distribution, this computes as  

145.)          

Note the equivalence of this value of 1 for the distribution variance of the K=4 over n=2 distribution and its σ²_RC=1 average state variance of Table 142, which tells us for this distribution that the distribution variance and average state variance are one and the same and that both can be denoted with the σ²_RC symbol and expressed from Eq144 as

146.)     

Ceaseless observation of every K over n random distribution shows the above to be entirely general. Next we develop the D diversity of Eq124 as a statistical function of a number set by first solving Eq140 of the variance for the rightmost summation term in it.  

147.)     

Then substituting this summation into Eq124 obtains D as

148.)         

The above logically implies that the average state diversity, D_RC, is a function of the average state variance, σ²_RC, as

149.)         

This logical implication is demonstrated with the K=4, n=2, μ=K/n=4/2=2 distribution by computing its previously developed D_RC=1.6 value from the σ²_RC=1 average state variance of Table 142.  

150.)          

And by substituting σ²_RC from Eq146 into Eq149 we obtain the D_RC expression introduced without proof of Eq132.    

151.)             

To review briefly, recall that earlier in Table 137 we showed a high .999 correlation between D_RC and lnW, which made it clear that given the correctness of Boltzmann’s lnW entropy based on its fit to thermodynamic data, so also must D_RC be considered correct on that basis. And we further argued that D_RC was the proper representation of entropy because Boltzmann’s lnW based S entropy was disqualified because the assumption Boltzmann needed of the equiprobability of the states of a distribution contradicted the basic mathematics of random multinomial distributions.

Also it should be pointed out that in contrast to the conceptual fuzziness of Boltzmann’s lnW based entropy as a physical quantity, the diversity of any allocation of K objects, be they organisms or discrete energy units or otherwise, over n subsets or containers, be they species or molecules or otherwise, is an undeniably firm intuitive concept and for physical allocations, physical quantity. That is, the energy diversity of a random distribution of energy units over molecules certainly exists and a very likely candidate for its measure is the average energy diversity, D_RC, and our understanding of this energy diversity as entropy is very attractive given the very high .999 correlation of D_RC with lnW displayed from Table 137 in Figure 138 and the fit of the lnW entropy with thermodyanmic data, which extends to the D_RC diversity entropy or diventropy because of its high correlation with Boltzmann’s S=k_BlnW entropy.       

An additional helpful argument regarding the Maxwell-Boltzmann energy distribution that supports diversity entropy requires the introduction of a mathematical structure for multinomial distributions called a configuration. It is simply the collection of states that have the same number set. For example, the states of [0, 4] and [4, 0] are essentially the same number set, (4, 0), given the name, configuration. Note that we write a configuration in parenthesis, (4, 0), rather than in the bracket form we used for the [4, 0] and [0, 4] states that comprise the (4, 0) configuration. The (4, 0), (3, 1) and (2, 2) configurations of the K=4 over n=2 distribution and their component states are

Table 152.The Configurations of the K=4 over n=2 Distribution and Their States

A quick glance back to Table 141 makes clear that a configuration has the same σ² variance and same D diversity as the states that comprise it.

Table 153.Variance, σ², and Diversity, D, of the Configurations of the K=4 over n=2 Distribution

Now see that the distribution variance of σ²_RC=1 and diversity of D_RC=1.6 of the K=4 over n=2 distribution are the same respectively as the σ²=1 and D=1.6 of the (3, 1) configuration. On that basis of the (3, 1) configuration is understood to represent the entire K=12 over n=3 distribution in a succinct way as the representative configuration of the distribution and for that reason we also will specify its variance and diversity respectively as σ²_RC=1 and D_RC=1.6

The representative configuration of a random distribution represents the entire distribution comprised of all its configurations with a single configuration much like the μ mean of a number set represents the entire number set with a single number. Consider the K=12, n=3, (6, 4, 2), number set whose single number mean of μ=K/n=4 represents the many numbers in the (6, 4, 2) set and as such the entire number set in a succinctly and simple way. The (3, 1) representative configuration of the K=4 over n=2 distribution parallels this as a single configuration that represents the many configurations of the distribution and, hence, the entire distribution in a succinct and simple way. 

This has us understand σ²_RC as the average state variance of a distribution, the distribution variance and the variance of the representative configuration of the distribution, the value of the variances of all these mathematical structures being the same have. This is why we use the RC subscript in σ²_RC for these variance measures. And we also understand D_RC to be equivalently the average state diversity, the distribution diversity and as the diversity of the representative configuration.

To further clarify the meaning and nuances of the average configuration in order to explain entropy and ultimately to explain what words are, it is helpful to develop another example random distribution, one of K=12 candy bars over n=3 children. The nineteen configurations of the K=12 over n=3 distribution are listed below along with the number of states in a configuration, the number of permutations in a configuration, the probability of occurrence of the configuration, its D diversity, σ² variance, and the σ²_RC average state variance of the distribution.

Table 154.Properties of the K=12 over n=3 Random Distribution

The W=91 states of the distribution is obtainable from Eq134 as

155.)     

The # of permutations of each configuration, ω, (omega), is a combinatorial statistic available in any standard elementary probability theory text whose details we will not review here.  

156.)     

It obtains the ω=49,896 permutations of (5, 5, 2) configuration, for example, as

157.)     

And the total number of permutations, Ω, is also a standard combinatorial statistic, Ω=n^K, which evaluates for the K=12 over n=3 random distribution as

158.)     

The probability of occurrence of a configuration is its ω number of permutations divided by the total number of permutations in the distribution, Ω=n^K=3¹²=531,441.The probability of occurrence of the (5, 5, 2) configuration is, hence, ω/Ω=(49,896)/(531,441)=.094. The D diversities of the configurations are calculated from Eq124 and their σ² variances from Eq140. And the last column of Table 154 calculates the σ²_RC average configuration variance of σ²_RC=2.667 as the probability weighted average of the σ² configuration variances, which can also be calculated independently from Eq146 as

159.)     

And from that and the μ=K/n=4 of the distribution we can calculate the distribution diversity, D_RC, from Eq151 as 

160.)     

Now importantly note that the distribution variance of σ²_RC=2.667 is the variance of the (6, 4, 2) configuration and that the distribution diversity of D_RC=2.571 is the diversity of the (6, 4, 2) configuration, which understands the (6, 4, 2) configuration to be the representative configuration of the K=12 over n=3 distribution. Ceaseless observation of K over n random distributions shows the above procedures for obtaining the representative configuration of a random distribution to hold for all K over n random distributions. This will enable us next to show that the energy distribution of the representative configuration is the empirical Maxwell-Boltzmann distribution of a realistic thermodynamic system shown below, which will prove the representative configuration to be the correct representation of a thermodynamic system, which will prove that diversity based diventropy is the correct formulation of entropy and not Boltzmann’s S=k_BlnW entropy. 

Figure 161.The Classic Maxwell-Boltzmann Energy Distribution

 

While we developed the representative configuration for random distributions of K candy bars over n children, it is also valid for thermodynamic random distributions of K discrete energy units over n molecules because the latter is a perfect mathematical model of the latter. The random or equiprobable distribution of candy bars over children comes about from grandfather tossing the candy bars to the children blindly over his shoulder, which when done repetitively perfectly maps the random or equiprobable distribution of energy units over molecules that comes about from repetitive molecular collision and energy transfer between molecules when the thermodynamic distribution is pictured in the simplest way as that of a system of gas molecules moving about and repetitively colliding in a fixed volume.

A K=4 energy unit over n=2 molecule system and K=12 over n=3 system have too few energy units and molecules for their representative configurations, respectively (3, 1) and (6, 4, 2), to show any semblance to the Maxwell-Boltzmann distribution of Figure 161. We need to go to distributions with higher K and n values starting with the K=12 energy units over n=6 molecule distribution. To find its representative configuration we first calculate the σ²_RC distribution variance from Eq146.

162.)     

The representative configuration of the K=12 over n=6 distribution will have a variance with the same value as this distribution variance of σ²_RC=1.67. An easy way to locate the representative configuration is with a Microsoft Excel program that runs through all configurations of this distribution to find one whose variance has the same value as σ²_AC=1.67. It is the (4, 3, 2, 2, 1, 0) configuration, which is the representative configuration on that basis. A plot the (4, 3, 2, 2, 1, 0) representative configuration’s number of energy units on a molecule versus the number of molecules that have that energy is shown below.

 

Figure 163.Number of Energy Units per Molecule vs. the Number of Molecules Which Have That Energy for the RC of the K=12 over n=6 Distribution

Seeing this distribution as a Maxwell-Boltzmann energy distribution is a bit of a stretch, though it might be intuitively characterized as a very simple, choppy one. So let’s next consider a larger K over n distribution next, of K=36 energy unit over n=10 molecules. Its σ²_RC distribution variance is from Eq146.

164.)       

Our Microsoft Excel program runs through the configurations of this distribution to find one that has the σ²_RC =3.24 variance, namely, (1, 2, 2, 3, 3, 3, 4, 5, 6, 7). A plot of the energy distribution of this RC is

 

Figure 165.Number of Energy Units per Molecule vs. the Number of Molecules Which Have That Energy for the RC 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 proto-Maxwell-Boltzmann.” And next we look at the K=40 energy unit over n=15 molecule distribution, whose σ²_AC distribution variance is from Eq146

166.)     

Our Microsoft Excel program finds four configurations that have this variance including (0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 6), which is a RC of the distribution on the basis of its σ²=2.489 number set variance. A plot of its energy distribution is

 

Figure 167 Number of Energy Units per Molecule vs. the Number of Molecules Which Have That Energy for the RC of the K=40 over n=15 Distribution

 

And next we will look at the K=145 energy unit over n=30 molecule distribution whose variance is from Eq146.

168.)     

There are nine configurations with this σ²_RC =4.672 variance 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 is the RC on that basis. A plot of its energy distribution is

 

Figure 169.Number of Energy Units per Molecule vs. the Number of Molecules Which Have That Energy for the RC of the K=145 over n=30 Distribution

 

As we increase the K and n of the random distribution examples, their distribution of energy versus the number of molecules with that energy more and more takes the shape of the realistic Maxwell-Boltzmann distribution of Figure 161. There is a practical limit on increasing the K and n of a random distribution because of the prohibitively high numbers of configurations produced that must be sifted through to find representative configurations, but higher K and n distributions such as for K=10,000 over n=1000 molecule systems are an exceedingly close match to the realistic Maxwell-Boltzmann curve of Figure 161.

This derivation of the Maxwell-Boltzmann energy distribution from the RC, representative configuration, concept not only adds to the argument that diventropy is the proper form of entropy rather than Boltzmann’s entropy. It also provides an intuitively sensible picture of a thermodynamic system in terms of the RC allowing us to picture the K=12 energy units over n=3 molecule system as consisting of Ω=n^K=3¹²=531,441 permutations of energy distribution that develop over 531,441 time units from 531,441 repetitive random collisions, each time unit as the time between a collision assumed to be the same, with energy transfers occurring in each collision to produce all configurations of the system in numbers on average equal to the number of permutations per configuration and with the distribution represented by the (6, 4, 2) RC in its having the physical properties that the system’s configurations have on average. 

Though one might think that the foregoing has provided enough proof to select diversity entropy over Boltzmann entropy as the correct formulation and understanding of entropy, ideas firmly held in science for a century die hard. So let’s return to the consideration that powerfully disqualifies Boltzmann statistical mechanics and that is Boltzmann’s premier axiom that the W states of the random distribution are equiprobable. As said earlier, this is patently not the case mathematically for the simple K=4 over n=2 random distribution as shown in Table 126 where the probabilities of the W=5 states are shown to be 1/16, 1/4, 3/8, 1/4 and 1/16, decidedly not equiprobable.

This lack of equiprobability for the W states of a random distribution is seen again for the K=12 over n=3 distribution in Table where the probability of any one of the W=91 states is just the probability of the configuration it belongs to in the 4^th column of the table divided by the number of states of the configuration in the 2^nd column. If all these probabilities are computed, it is seen that the W states are not equiprobable. .

This lack of equiprobability for the W states of a random K over n distribution is a general mathematical fact that affects even the distribution of outcomes for the throwing of dice. The number of permutations that can arise from tossing K=2 dice whose outcomes are distributed over the n=6 sides of a die is Ω=n^K=6²=36, namely: {1,1}, {1,2}, {2,1}, {1,3}, {3,1}, {2,2}, {1,4}, {4,1}, {2,3}, {3,2}, {1,5}, {5,1}, {2,4}, {4,2}, {3,3}, {1,6}, {6,1}, {2,5}, {5,2}, {3,4}, {4,3}, {2,6}, {6,2}, {3,5}, {5,3}, {4,4}, {3,6}, {6,3}, {4,5}, {5,4}, {4,6}, {6,4}, {5,5}, {5,6}, {6,5} and {6,6}

There are W = (K+n−1)!/K!(n−1)! = 7!/2!5! = 21 states of these Ω=36 permutations: [1,1], [1,2], [1,3], [2,2], [4,1], [3,2], [5,1], [4,2], [3,3], [6,1], [5,2], [4,3], [6,2], [5,3], [4,4], [6,3], [5,4], [6,4], [5,5], [6,5] and [6,6]

While the Ω=36 {permutations} are equiprobable, each having a probability of 1/Ω=1/36, the W=21 states of the dice are decidedly not equiprobable, the probability of the “doubles” state like [1, 1] and [2, 2] being 1/36, but of the “non-doubles” state like [1, 2] and [2, 5] being 1/18.  

Indeed, the only K over n random distribution that is equiprobable is the trivial K=1 over n distribution. Hence the error in Boltzmann assumption for the general K>1 case makes it clear that the axiom on which his statistical mechanics and lnW entropy formulation is based must be incorrect. This completely invalidates Boltzmann’s S entropy and by default prescribes diventropy, which has a .999 correlation to lnW entropy, as the correct form for entropy.

A rebuttal that Boltzmann’s energy units are indistinguishable and obviate the above considerations as such are specious for it matters not at all in the candy bar thought experiment whether the candy bars are distinguishably different brands or all the same and indistinguishable for the mathematics is the same in both cases though easier to see when we make the candy bars distinguishable.    

As to why Boltzmann, a master mathematical physicist, missed the correct form is easy to understand for Simpson’s Diversity Indices weren’t developed until near a half century after Boltzmann’s death. Hence with his S=k_BlnW function fitting the data and being the only function that did, it was accepted as correct even if conceptually confusing. Telling, though, is that Boltzmann’s S entropy is not that distant from the concept of diversity. Indeed as part of standard physical chemistry theory today it is understood that Boltzmann’s S entropy derives from a parent function called the Gibb’s entropy.

170.)     

Without getting into the details that can be looked up on Wikipedia, when this function is summed over a distributions W states assumed to be equiprobable, which we showed above is a mathematically ridiculous assumption, the function turns into Boltzmann’s entropy, S=k_BlnW. What is interesting is that a half century later, just about the time that Hugh Simpson introduced the Simpson’s Diversity Indices Claude Shannon introduced the world to the Shannon Diversity Index, in natural logarithm form

171.)        

This function has been used also along with Simpson’s diversities as a standard measure of ecological and sociological diversity. It is impossible not to see the near identity of it to the Gibb’s entropy form that derives Boltzmann’s S entropy, though Boltzmann had no idea that its form was that of a diversity measure, the Shannon diversity, that wasn’t developed until long after his death. While the details are a bit complicated as to how Boltzmann’s S entropy gets entangled with Shannon diversity and have to be gradually sifted in as we proceed, most simply we see that the similarity in form of entropy in Eq170 and diversity in Eq171 makes it clear that diversity was in Boltzmann’s picture of entropy, even if in an incorrect way with the wrong diversity function and even if Boltzmann was not aware that the foundation function for his entropy took the form of diversity.        

Boltzmann’s famous S entropy equation was accepted as correct anyway, though, despite its theoretical thorns because the numbers generated by them fit the data from their fluke, high correlation fit to the diversity function that would not appear on the scene for nearly a half century later. This in no way disparages Ludwig Boltzmann, who was so dedicated in his life to unearthing nature’s truth that he hung himself in 1906 suddenly while on vacation with his wife and daughter because the equation for entropy he had worked on for a lifetime was rejected by his scientific peers. After his suicide and very likely because of it, a second look was taken at his S entropy and honored shortly after by inscription on his tombstone. One might conclude that were it not for Boltzmann surrendering his life for scientific truth, his equation would not have been known today to make possible our clarifying entropy as diversity and using that discovery to help ferociously urge the world to disarm in order to avoid a nuclear homicidal suicide for man that is otherwise sure to come.

We want to point out now, briefly with the intention of more detailed follow up later, that a word like a common noun is basically a representative configuration or RC. We introduce this idea now because doing so will also sharpen the picture of the thermodynamic system we have been considering and how Boltzmann made his error. The RC for energy or candy bar distributions we’ve considered are really averages over two dimensions, of space and of time. Let’s take a close look at the D diversity measure of Eq148 as applied to the diversity of the permutation of a state and its configuration as it the permutation exists at one particular moment in time.   

148.)     

Note the inclusion of two central statistics functions, the μ mean or arithmetic average and the σ² variance statistical error that explicitly or implicitly accompanies every μ mean, often when explicitly as the σ standard deviation. This should make it clear that the D diversity is a kind of averaged or statistical measure of a distribution. Then to obtain the D_RC diversity of the representative configuration and the distribution as a whole, we average the D diversities of the permutations that appear over time weighted by their probabilities or how often they appear relative to each other over time. Hence the D_RC of the RC is most basically a two dimensional averaging, over space at any moment in time to obtain D, and then these D diversities over time to obtain D_RC. Or to put it another way, the D_RC is in mathematical language a lumped or generalized measure of the x_i amounts in each of the n containers of a system further lumped or averaged or generalized over time.          

In the same way, the meaning of the word fish is a rough average of every instance of a fish that exists at any moment in time with the average values at all the moments in time averaged over time. Indeed every generalized word or common noun or verb we use is an averaging of every instance of it in every place at any moment with this at time average further averaged to include all instances at every moment in time, a two dimensional, space-time, averaging entirely like the RC of a multinomial distribution. Characteristics of an RC tell us much about words and how unintended confusions and intended deceptions can readily arise from them, as we intend to show in detail later.    

Now we return to more formal technical topics because we have not yet fully solved the problem of entropy. There is yet another diversity function besides D_RC of Eq148 that has a high correlation with Boltzmann S entropy and must also be considered as a candidate for replacing Boltzmann’s entropy along with D_RC. As it derives directly from the temperature of a thermodynamic system, it turns out to be the correct form of diversity based entropy.  To develop it we need to express the μ=K/n mean of Eq139 in a way that requires us to first express K, the sum of the x_i of a natural number set, formally as

172.)     

For the (7, 3, 2) number set that has x₁=7, x₂=3 and x₃=2, K= x₁+x₂+x₃=12. Eq172 has us define μ of Eq139 in a not usual way as

173.)     

The (1/n) term in the summation can be understood as the average weight fraction of a number set, that is, as the average of a number set’s p_i=x_i/K weight fractions of Eq119. As an example, note that the average of the p_i weight fractions of the (7, 3, 2) number set, p₁=7/12, p₂=3/12 and p₃=2/12, is their sum, which is 1, divided by the number of p_i weight fractions, which is n=3. So the average weight fraction, given the symbol,, is for this set, =1/n=1/3. Or generally for any number set, its average weight fraction, ,

174.)     

This expresses the μ mean in Eq173 as

175.)      

This can be interpreted as saying that the μ mean is the sum of “slices” of x_i of “thickness”, the average weight fraction. This form of μ computes the μ=K/n=12/3=4 mean of the K=12, n=3, (7, 3, 2), number set as

176.)          

All of this is preface to our defining a new average of a number set called the biased average, φ, (phi). In parallel to the μ arithmetic average as the ratio of K to n as μ=K/n, the ratio of K to the D diversity of a number set is φ, the biased average.   

177.)           

For K=12, (7, 3, 2), which has Z=62/144 from Eq122 and, hence, D=1/Z=144/62=2.323, φ=K/D=5.167. Now let’s specify φ in a way parallel to μ in Eq173 as a function of p_i weight fractions via Eq124 as

178.)            

This has us interpreting the φ biased average as the sum of “slices” of x_i of “thickness” p_i, that is, of thickness of the actual weight fractions of each x_i rather than of the  average weight fraction as was the case with μ in Eq173. The above form for φ obtains the φ=5.167 biased average of (7, 3, 2) as

179.)           

This development of φ is an introduction to another unusual average, the square root biased average, ψ, (psi), which is important for correctly formulating temperature and entropy. The ψ average has the form of the sum of slices of the x_i of a set of thickness, p_i^1/2, the square root of the weight fractions of a number set.

180.)              

A question mark is placed in this introductory definition of the ψ square root biased average to indicate that there is something not quite right with the Ψ function as it stands. What isn’t right is that the p_i^1/2 weightings don’t add up to 1 as they must to form an average of the x_i of a set. This problem can be illustrated with the (6, 5, 1) set, which while its p_i weight fractions of (1/2, 5/12, 1/12) do add up to 1, the p_i^1/2 square roots of these weight fractions, (.7071, .6454, .2887), don’t add up to 1. Rather their sum is .7071+.6454+.2887=1.6412. Because they don’t add to 1 they can’t be used to weight the x_i in forming a weighted average of them.

The difficulty is readily resolved, though, by normalizing the p_i^1/2 to get them to add up to 1 by dividing each of them, (.7071, .6454, .2887), by their 1.6412 sum. This obtains normalized p_i^1/2 of (.4308, .3933, .1759), which do add up to 1 and, hence, can properly weight the x_i of the (6, 5, 1) set to form its ψ square root biased average as

181.)            ψ = (.4308)(6) +(.3933)(5) + (.1759)(1)= 4.727

The general form of the normalizing factor of the sum of p_i^1/2 is written as

182.)         

It corrects the ψ{?} questionable function of Eq180 by dividing its p_i to normalize them and obtain the ψ square root biased average properly as

183.)           

We can also express ψ in an alternative way via the p_i=x_i/K weight fraction relationship as 

184.)            

This form evaluates the ψ=4.727 of the K=12, (6, 5, 1), set, which has p_i weight fractions of (1/2, 5/12, 1/12), as

185.)             

A calculation of the φ biased average for K=12, n=3, μ=4, (6, 5, 1), from Eq178 shows it to have φ=5.167, an increase over the μ=4 arithmetic average. And we see from the above that the Ψ=4.727 square root biased average for (6, 5, 1) is also increased from the μ=4 mean or arithmetic average, but less of an increase than its φ=5.167.    

What justifies this somewhat tedious derivation of the ψ square root biased average is that ψ properly specifies the microstate absolute temperature of a thermodynamic system. In standard physico-chemical theory temperature is taken to be the average kinetic energy, which is directly proportional to the μ=K/n average energy per molecule in our model. But a μ=K/n temperature measure must be seriously questioned from the perspective of the physical reality of how temperature is actually physically measured.

When the x_i of a number set are the energy units of the i^th of n gas molecules of a thermodynamic system moving at velocities proportional to the square root of the x­_i number of energy units on them, the molecules necessarily collide with the thermometer that measures the temperature of the system as the average of molecular kinetic energies that are recorded at a frequency equal to those velocities. Hence the smaller energies of the slower moving molecules in the Maxwell-Boltzmann energy distribution of Figure 69 collide with the thermometer less often to be recorded less often than the higher energies of the faster moving molecules.

This necessarily produces a temperature measure of an average of molecular energies weighted toward the higher energies because of greater velocities of the higher energy molecules that cause their higher frequency of collision with the thermometer. The velocities of the molecules are directly proportional to the square root of the x_i energy of the molecules. Hence the average molecular energy, which is temperature, is a square root of x_i energy weighted average, which is the ψ square root average energy per molecule of the representative configuration of the thermodynamic distribution, labeled ψ_RC, which is the temperature of the thermodynamic distribution as a whole.

186.)               

A rebutting argument that collision velocities average out over time to produce temperature as the simple μ=K/n arithmetic average is facetious for while velocities do average out for any given molecule, the molecules that have more energy and a higher velocity at any moment in time do collide with the thermometer more often to have their energies recorded more often as necessarily weights the average energy as temperature towards the higher energies as specified by ψ_AC. Now recall from Eq177 that the φ biased average derived from the D diversity as φ=K/D, which allows the D diversity to be specified as

187.)               

Much as the D diversity can be understood as deriving from the φ biased as above, so also can one understand there to be a diversity measure that derives in a parallel way to the ψ square root biased average as 

188.)                 

We will call G the square root diversity. We can also express G solely as a function of the p_i of a set via Eqs188&184 as  

189.)            

The complex makeup of G as a function of the p_i weight fractions in the above makes it impossible to obtain a G_RC square root diversity of the RC of a thermodynamic distribution as a simple function of K and n as we did for D_RC as D_RC=Kn/(K+n−1) in Eq151. Hence we cannot compare G_RC to lnW in the same simple and direct way as we did D_AC to lnW in Table 137. There are two approximate methods we can use to compare G_AC to lnW, though.

The first employs computer generated random allocations of K objects over n containers. When the K and n parameters of such are large enough, the law of large numbers makes their σ² variance very close to the σ²_RC variance of the RC. Specifically we consider allocations of K objects ranging from K=40000 to K=91000 over n=4000 containers whose σ² variances vary from the distribution variance, σ²_RC, by no more than 2 parts per 1000. The G square root diversity calculated from Eq189 from the p_i of the computer randomized distributions are used as approximations of G_RC to be compared to the lnW of the allocations as computed from Eq136.