A WORLD WITH NO WEAPONS:
A REVISION OF ENTROPY EXPLAINS THOUGHT AND EMOTION
AND HOW A UNIVERSAL WEAPONS BAN WOULD ELIMINATE
NOT ONLY WAR AND MASS MURDER BUT ALSO THE GLOOM
OF POLITICAL AND ECONOMIC SUBJUGATION
By
Ruth and Peter Calabria and Angel Thomas Rogovsky
©, September, 1, 2014, Ruth Calabria
Contact:
ruthcalabria@matrixevolutions.com
ENTROPY
Ludwig
Boltzmann’s formulation of entropy honored by inscription on his 1906 tombstone
is considered one of the most important equations in science.
But whatever its merits it has confused students and professors alike over the last 100 years in its failing to make any intuitive sense out of entropy as a physical quantity. We will show that entropy is better formulated as energy dispersal or diversity with a variation of the Simpson’s Reciprocal Diversity Index unavailable to Boltzmann in its not being introduced to science by the British statistician, Edward Simpson until 40 years after Boltzmann’s death. This diversity based formulation of entropy has a .999 numerical correlation to Boltzmann’s entropy, S=k_{B}lnW in modern notation, and as such is empirically indistinguishable from it while providing a perfectly clear and understandable sense of entropy as a measure of energy dispersal or diversity.
The upshot of this revolutionary appreciation of molecular systems goes beyond resolving the confusions associated with entropy to introducing a number of concepts that also revolutionize cognitive science and the human sciences generally in mathematically clarifying for the first time in science the nature of thoughts and emotions. Of special interest in this breakthrough is a clear explication in mathematical terms of anger and the destructive behavior it generates in terms of its origin as part of evolutionary fitness neurological programming. Also made clear is the impossibility of eradicating it as is readily observed in daily news bulletins on murder and war. All that can be done in this regard is to lessen the outcomes of aggression by eliminating the weapons that so easily make for the deaths, gore and crippling generally associated with weapons enabled violence.
Also
the elimination of weapons from everybody including the police agencies that
guarantee ruling class subjugation of the little people makes for the true
balance of power between individuals needed to end that level of social control
and the gloom that flows from it, unhappiness that is a primary initiator of
today’s epidemic violence in the world, the worst of which manifests itself
collectively in war. This mathematical treatise also makes inarguably clear
that mankind’s fate is one of extinction if we do not avoid nuclear war with a
worldwide weapons ban. Our group first made the call for such back during the
time of the cold war between the United States and the Soviet Union.
Knickerbocker
News, Albany, NY, May 1986
Little argument is needed today in light of the proxy battle in the Ukraine
between nuclear armed America and Russia, the endless fighting in the Middle
East and the persistent rise of nuclear armed China as a military power of the
relevance of the article’s central concern for the need for worldwide
disarmament. While the goal may seem impossible at first thought, it is not
with America leading the way to a worldwide weapons ban with the carrot of
peace on earth for all and the stick of punishment with its military might for
those nations unable to see the collective good of laying down their arms. In
effect this is a call for a war against weapons, to be achieved by any and all
means possible with the details of the new social structure for the world,
ordered but much more free than now, that will result to be considered in
detail in a later section.
Unfortunately, though, most people are locked in their highly controlled lives and after the hopes of youth have faded are unable to care about their own fate or the fate of others beyond the delusions of a happy retirement and afterlife, two of the funnier jokes that have ever been told. For that reason we spell out the argument for A World with No Weapons with firm mathematical reasoning in hopes of waking up a blind population of workers and soldiers before it is too late to avoid nuclear annihilation.
To that end I also make myself available for president in the 2016 election as a one issue candidate focused on avoiding nuclear war through the eventual elimination of all weapons worldwide. If a mathematically minded grandmother who has never held political office seems farfetched as president, certainly I can do no worse in solving America’s problems, great and small, than the lot of Wall St. controlled crippled roosters currently in office. Those who wish to encourage our disarmament efforts and my candidacy should click here. Now on to the entropy mathematics followed with a personal tale of illustrates life’s hidden violence on children.
2. Diversity, Distinction and Entropy
Diversity
is a measure of distribution in physical, biological and human systems. It is
most basically a property of any set of objects divided into subsets be it a
set of animal organisms divided into species, a set of people divided into
ethnic groups or a set of buttons in a button box divided by color. The (4, 4,
4), (■■■■, ■■■■, ■■■■) set of K=12 buttons divided into
N=3 color subsets, x_{1}=4 red, x_{2}=4 green and x_{3}=4
purple, is intuitively more diverse than a set of red and green buttons, (6,
6), (■■■■■■, ■■■■■■), in N=2 colors and both of these
less diverse than the (2, 2, 2, 2, 2, 2) N=6 color set of buttons, (■■, ■■, ■■, ■■, ■■, ■■). One can give this intuitive sense
of diversity for these sets of buttons quantitative measure with a function for
diversity devised by the British statistician Edward Simpson, back in 1948
called the Simpson’s Reciprocal Diversity Index, which we will give the symbol, D,
to and write as
1.)
The K term is the total number of objects in a set, K=12 for all three of the above sets of buttons. And the x_{i} term in Eq1 is the number of objects in each of the subsets of a set. For (■■■■, ■■■■, ■■■■) this is x_{1}=4 red, x_{2}=4 green and x_{3}=4 purple buttons, which can be written in shorthand notation as (4, 4, 4). The diversity in the (4, 4, 4) set is from Eq1
2.)
We see that the D=3 diversity index of this set is equal to the N=3 different kinds of buttons in the set. If we applied Eq1 to the K=12 button, N=6 subset, (2, 2, 2, 2, 2, 2) set of (■■, ■■, ■■, ■■, ■■, ■■), we would calculate its diversity to be D=N=6. And for the K=12 button, N=2 subset, (6, 6) set, (■■■■■■, ■■■■■■), we calculate the diversity to be D=N=2.
The D diversity measures for these three sets fit our intuitive sense of their diversity. The D=N equivalence for these sets holds because they are balanced, the number of buttons in each subset of (4, 4, 4), (6, 6) and (2, 2, 2, 2, 2, 2) being the same.
3.) D = N (balanced)
From Eq1 we can also calculate the D diversity of an unbalanced set like (6, 5, 1), (■■■■■■, ■■■■■, ■), which has K=12 buttons divided into N=3 color subsets as x_{1}=6 red, x_{2}=5 green and x_{3}=1 purple.
4.)
For this and all unbalanced sets, we see that the D diversity index is less than the N number of subsets,
5.) D < N (unbalanced)
We might
understand the D<N in Eq4 for the unbalanced, N=3, (6, 5, 1) set by
considering the x_{3}=1 object purple subset in (■■■■■■, ■■■■■, ■) to contribute only token diversity.
Next we want to understand the D diversity index as a statistical function. To do that, we next
specify the mean or arithmetic average, μ, (mu) of a set of
objects as.
6.)
The average number of buttons in the N=3 subsets of the (6, 5, 1), K=12, N=3, (■■■■■■, ■■■■■, ■) set is μ=K/N=12/3=4. The mean of a set is often accompanied in statistics by a measure of statistical error. A very commonly used statistical error is the standard deviation, σ. It is the square root of the statistical error most useful to us at the moment of the variance, σ^{2}.
7.)
The variance of the N=3, µ=4, (6, 5, 1) set is
8.)
The variance is a measure of the amount of imbalance in a set as is its standard deviation, for this set, σ=2.16. Now consider another K=12, N=3 set, the (10, 1, 1), (■■■■■■■■■■, ■, ■). It is also unbalanced and intuitively more unbalanced than (6, 5, 2), (■■■■■■, ■■■■■, ■). If the σ^{2} variance is truly a valid measure of the imbalance of a set, we’d expect the variance of (10, 1, 1), (■■■■■■■■■■, ■, ■) to be greater than the σ^{2}=4.67 of the (6, 5, 2), (■■■■■■, ■■■■■, ■), set. And so it is as we see from Eq5 for it, σ^{2}=18.
Another commonly used error function is the relative error, r, or the σ standard deviation divided by the μ mean.
9.)
And a form of the relative error is its square, r^{2}, that we’ll call the perfect error.
10.)
For the σ^{2}=4.67, µ=4, (6, 5, 2), (■■■■■■, ■■■■■, ■) set, r^{2}=4.67/16=.29 and for the σ^{2}=18, µ=4, (10, 1, 1), (■■■■■■■■■■, ■, ■) set, r^{2}=18/16=1.125. From this we see that r^{2} is also a measure of the imbalance of a set. That sense of r^{2} as a measure of imbalance is further reinforced when we calculate the r^{2} perfect error for the balanced, (4, 4, 4), (■■■■, ■■■■, ■■■■), set from Eqs6,7&10 to be r^{2}=0 as denotes zero or no imbalance in the set.
We now see from the diversity index of (4, 4, 4) in Eq2, D=3, and of (6, 5, 1) in Eq3, D=2.32 and of (10, 1, 1) from Eq1, D=1.41, and from the perfect errors of these N=3 subset sets respectively of r^{2}=0, r^{2}=.29 and r^{2}1.125 that their D diversity indices is inversely proportional to their r^{2} perfect errors. This inverse relationship can be developed analytically by first solving the σ^{2} variance of Eq7 for the summation term in it.
11.)
Then one inserts this summation into Eq1 to obtain D via µ of Eq6 and r^{2} of Eq10 as
12.)
We will use this specification of the D Simpsons Reciprocal Diversity Index as a statistical function in our diversity based formulation of entropy. We also want to explore diversity further as a function of distinction for our derivation of diversity based entropy and also of human emotion. To do that we note one minor discrepancy in the D Simpson’s Reciprocal Diversity Index as a measure of diversity lies in its evaluating a completely uniform set like the N=1 color (■■■■■■■■■■■■), which has in its total sameness of color of all objects zero or (0) diversity, as having via Eq1 a diversity index of D=1. This is easily rectified by specifying a new diversity index we’ll call the Exact Diversity Index.
13.) L = D ‒ 1
Number
information does calculate the intuitive zero diversity of the uniform set, (■■■■■■■■■■■■), as L=D‒1=0. In reducing the
diversity indices of the other sets we have been considered by 1, it also, as
with D, gives a quantitative measure of the relative diversities of the sets
we’ve been considering.
Set of Objects 
Number Set 
D, Simpson’s Diversity Index 
L, Exact Diversity Index 
(■■, ■■, ■■, ■■, ■■, ■■) 
(2, 2, 2, 2, 2, 2) 
6 
5 
(■■■■, ■■■■, ■■■■) 
(4, 4, 4) 
3 
2 
(■■■■■■, ■■■■■, ■) 
(6, 5, 1) 
2.32 
1.32 
(■■■■■■, ■■■■■■) 
(6, 6) 
2 
1 
(■■■■■■■■■■, ■, ■) 
(10, 1, 1) 
1.41 
.41 
(■■■■■■■■■■■■) 
(12) 
1 
0 
Table 14. Some Sets and Their D and L Diversity Indices
The Exact Diversity Index, L, has other advantages over D in having a
foundation in the primitive mental operation of our distinguishing or making a distinction
between things. In the N=6 color subset (■■,
■■, ■■, ■■, ■■, ■■), every subset is distinct from 5
other subsets and this subset distinction is specified in its Exact Diversity
Index of L=5. And in the N=3 subset (■■■■,
■■■■, ■■■■), every subset is distinguished
from 2 other subsets and that is specified by its L=2 diversity. And for the
N=2 subset (■■■■■■, ■■■■■■), each of the N=2 subsets in the
set is distinguished from one (1) other subset and its Exact Diversity Index is
L=1. And we see from the table that the L=0 measure for the uniform, N=1 subset
(■■■■■■■■■■■■) set specifies that the red subset
in the set is distinct from no or 0 other subset.
Hence L is
a measure of the diversity of a set and of subset distinction. It
is also a measure of the distinction we intuitively make between
the objects in a set. In the (2, 2, 2), (■■,
■■, ■■), we automatically distinguish an
individual red object, ■,
from an individual green object, ■,
and both from an individual purple object, ■. We can quantify these object distinctions we make by
comparing the objects in (■■, ■■, ■■) to each other systematically in a
matrix.

■ 
■ 
■ 
■ 
■ 
■ 
■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■■ 
Figure 15. The Comparison Matrix of (■■, ■■, ■■)
The K=6 set of objects,
(■■, ■■, ■■), has K^{2}=36 comparison
pairs in its matrix. Y=24 of them are distinctions or mismatches
in color and ε=12 of them are what we’ll call samenesses or
matches in color. This tells for this matrix and for the comparison matrix of
any set of objects that
15a.) Y + ε = K^{2}
Now from Eqs1&13 we calculate L=2 for (■■, ■■, ■■), which is the number of distinctions that any one subset in (■■, ■■, ■■) has from the other subsets in the set. We also see that L can be calculated as the ratio of the Y distinctions in the matrix of Figure 15 to the ε (epsilon) samenesses in the matrix.
15b.)
Hence L is
not only a direct measure of the subset distinctions of a set but
also a simple function of its Y object distinctions. In that
sense both L and D=L+1 are measures of diversity and distinction. a set. For
this (2, 2, 2), (■■,
■■, ■■) set, L=Y/ε=24/12=2. L in
Eq15b is also valid for unbalanced sets like K=6, N=3, (3, 2, 1), (■■■, ■■, ■). From Eqs1&13, we calculate this set having a
D=2.57 Simpson’s Diversity Index and an L=1.57 Exact Diversity Index. Now let’s
look at its comparison matrix.

■ 
■ 
■ 
■ 
■ 
■ 
■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■■ 
Figure 16. The Comparison Matrix of (■■■, ■■, ■)
We count out of K^{2}=36 comparison pairs, Y=22 distinctions and
ε=14 samenesses as calculates the Exact Diversity Index as
L=Y/ε=22/14=11/7=1.57. This makes it clear that L is an elemental
diversity index in deriving most basically from the mind’s primitive senses of
distinction and sameness.
The matrix
underpinning of diversity in terms of distinction and sameness is the
foundation of mathematical formulations of the basic human emotions of hope,
fear, anxiousness, excitement, relief, dismay, depression, hunger, anger, sex
and love among others. Take another look at the comparison matrix of (■■, ■■, ■■) in Figure 15 to see how this is
developed.

■ 
■ 
■ 
■ 
■ 
■ 
■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■ 
■■ 
■■ 
■■ 
■■ 
■■ 
■■ 
Figure 15. The Comparison Matrix of (■■, ■■, ■■)
Consider the (■■, ■■, ■■) set of objects as a bag of K=6
colored buttons from which one is picked blindly and randomly. A game can be
played in which a person makes a guess at which color will be picked. Assumed
is that she knows the (■■,
■■, ■■) contents of the bag as will tell
her that each color has an equal probability of being picked. Because of that
she guesses each of the N=3 colors with the same probability or frequency. As
such the (■■, ■■, ■■) set you see on the vertical axis
of the matrix is readily understood as an average of the color guesses she
would make over six guesses. And the (■■,
■■, ■■) set you see on the horizontal axis
of the matrix in Figure 15 is the average of the picks from the bag for any six
picks.
This interprets the K^{2}=36 pairs in the matrix of Figure 15 as K^{2}=36 guesspick pairs with the Y=24 distinctions between color guessed and color picked such as ■■ understood as wrong guesses and the ε=12 samenesses between the color guessed and color picked understood as correct guesses. Assumed is that every wrong guess feels unpleasant and every correct guess pleasant, much as the way guessing does when one plays along at home with the Jeopardy game on TV or any other guessing game.
It is clear that the ratio of Y wrong guesses relative to the K^{2} total number of guesses is, when projected to future guessing, the probability of failure in guessing correctly as a measure of the uncertainty in guessing correctly.
16a)
This ratio that we give the symbol, U, to is a measure of the displeasure of the uncertainty in guessing as a lumped measure of the displeasures of the Y individual incorrect guesses in the matrix. And the ratio of the average ε correct guesses to the K^{2} total guesses is, projected to future guessing, the probability of success.
16b.)
It is measure of the pleasure of the expectation or hope of success in guessing as a lumped measure of the pleasure in the ε individual correct guesses in the matrix. Details aside for now as will be taken up later, this distinction and sameness based matrix foundation of the pleasures and displeasures of correct and incorrect attempts spells out as we shall see later all the human emotions in a firm mathematical way.
This is a very important breakthrough for the human emotions have been without question the most difficult phenomena in nature to give clear explanations for over the last five millennia in which knowledge has set down in written form. One the one hand, we have the devil vs. angel theory of the emotions in which characters like Satan are responsible for the feeling or temptation to sex. And on the other hand, we have today the mental illness theory of unpleasant emotion, which is almost as foggy as the supernatural good and bad spirits explanation of human feeling.
We hold off
on jumping immediately into this most meaningful of topics of our emotions,
reserving it for a later section, because the feelings aroused by talking about
our feelings can be so intense and potentially unpleasant when they run against
the rationalizations and delusions people have in this area that it is
preferable to proceed first to explaining the concept of entropy in terms of
diversity and its sister concept of distinction. This will also develop a firm
mathematical formulation of thoughts or ideas that in conjunction with the
mathematics of emotion will give a clear and precise explanation of how the
human mind operates to control our choices of the behaviors we do and try in
life, most important for understanding violent behavior.
For now let’s consider entropy in depth, its problems and their resolution. The
accepted fundamental expression for entropy is the Boltzmann S entropy, in
modern terminology,
17.) S=k_{B}lnW
The problem with it is that it doesn’t explain entropy as a physical quantity
however the mathematical fit of it to the empirical data. It is easy to show
that the Boltzmann S entropy is a fundamentally flawed formulation right off the
bat because one of its main axioms, namely that the discrete or whole numbered
energy units of a thermodynamic molecular system are indistinguishable,
is flat out incorrect.
To make clear how that assumption of Boltzmann statistical mechanics cannot possibly be the case, we need to go back to the sets of colored buttons and explain first the difference between categorical distinction and fundamental distinction. The Y=24 distinctions in the matrix of Figure 15 for (■■, ■■, ■■) are clearly the distinctions we make between the buttons in (■■, ■■, ■■) on the basis of their color differences. Recall also the L=2 Exact Diversity Index (■■, ■■, ■■) understood as each subset in the set’s number of distinctions from the other subsets in the set, here L=2 of them. Both the object distinction and the subset distinction on the basis of color are a form of categorical distinction with the different colors in the set understood as the categories the buttons are grouped into.
But those are not the only distinctions we make for this set because also every button in the set is also fundamentally distinct from every other button. Take a close look at the x_{1}=2 red buttons in (■■, ■■, ■■). Those two red buttons are also distinguishable or distinct from each other, the 1^{st} red button in (■■, ■■, ■■) being a different object than the 2^{nd} red button. Certainly they are different in being in different places in (■■, ■■, ■■) as seen by us. And if we think of an actual K=6 buttons in a button box represented by the (■■, ■■, ■■) diagram, the two red buttons would certainly in different places, which understands them as fundamentally distinct buttons even if they are the same color and categorically the same as each other in that way. Indeed all K=6 buttons in (■■, ■■, ■■) and in the button box are fundamentally distinct from each other in existing in different places.
We can represent this hard fact that all K=6 buttons are fundamentally distinct by giving each their own distinct letter as (ab, cd, ef). This representation takes into account both the categorical distinctions of color and the fundamental distinction that exists between all the objects. Now let’s apply this general understanding of distinction to the problem of entropy.
As said above, Boltzmann’s development of entropy assumes in the distribution of K energy units over N molecules that the K energy units are indistinguishable from each other. The consequence of that attribution to the energy units as indistinguishable is purely mathematical for Boltzmann in developing from textbook combinatorial statistics for indistinguishable objects the Boltzmann S=k_{B}lnW entropy.
An explanation of his S entropy term by term is not necessary for the moment, only to point up that it must be incorrect because the axiom of indistinguishable energy units on which it is based is seen to be patently false once one clarifies distinction properly as we have done above. In a thermodynamic system of N gas molecules all of the same kind, say all of He, (Helium), the molecules are yet fundamentally distinct from each other. Each molecule is materially distinct from every other and at any moment in time has a particular place it is located in. This fundamental distinction between molecules is another of the basic axioms of Boltzmann statistical mechanics and one we are also in agreement with from our understanding of fundamental distinction being defined in terms of entities existing in different places.
Now understand that the K energy units of a gas thermodynamic system are distributed over its N molecules such that each molecule has a packet of energy units contained in or on it. For example, were the system to consist of N=3 molecules and K=12 energy units, the latter might be distributed at a given moment in time as x_{1}=6 energy units on the 1^{st} molecule, x_{2}=4 on the 2^{nd} molecule and x_{3}=2 on the 3^{rd} molecule. It is obvious that the packet of energy units on each molecule also occupies the space that the molecule occupies, which clearly shows the packets of energy units of each molecule occupying different places in space. Clearly the energy units of a given molecule must be distinguishable or distinct place wise from those of the other molecules, fundamentally distinct on that basis of occupying different places. It is therefore unarguably incorrect to consider all the energy units of a thermodynamic system to be indistinguishable from each other.
Whatever rebuttal might be laid on this argument, minimally it puts Boltzmann’s axiom on indistinguishable energy units and his S entropy formula of Eq17 developed from it seriously in doubt. A formula for entropy based on the diversity or dispersal of energy units over molecules is readily developed to correct Eq17 as based on textbook combinatorial statistics and multinomial theory. The reason why Boltzmann’s famous equation has been accepted over the years even if incorrect is because it numerically fits observed thermodynamic data. But that, it shall be shown, is the result of a fluke mathematical coincidence of the lnW term in Boltzmann’s S=k_{B}lnW entropy and the average energy diversity of a thermodynamic system.
Jumping ahead for the moment to the conclusions of the derivation of diversity based entropy, we will develop the lnW in Boltzmann’s S=k_{B}lnW entropy as a function of the K energy units and N molecules of a thermodynamic system via Stirlings Approximation as
17a.)
And we will develop the average energy diversity of a thermodynamic system as
18.)
It is then a simple matter to show that the two above functions have a Pearson’s correlation coefficient minimally of .999, more so for the high K and N values of realistic thermodynamic systems, which makes the two functions of lnW and D_{AV} near identical and effectively quantitatively indistinguishable. Anybody with computer skills can test that correlation for themselves. The math comes out the same no matter the details of the testing process to show a near perfect correlation approaching unity for large K and N systems.
It would have been very difficult if not impossible for Boltzmann to correct his error by using mathematical diversity because functions for diversity, a population biology concept for the most part, didn’t come along until 1948 long after Boltzmann held a pen in his hand. And when they did, the scientific journals in which mathematical diversity was disseminated were not the ones read by the physicists who studied entropy. From that perspective, Boltzmann’s error being accepted by the scientific community as correct is understandable given that the correct explanation was unable to be formulated prior to the advent of mathematical diversity after WWII and, indeed, until now from our studied elaboration and application of the Simpson’s diversity.
That begins with bypassing the conceptual and mathematical mess erroneously accepted as valid by the popinjays who currently rule in the kingdom of statistical mechanics to understand a thermodynamic system in the most simple way as an equiprobable or random distribution of K discrete (or whole numbered) energy units over N molecules. This microstate picture of a thermodynamic system is intuitively spelled out well with an equiprobable or random distribution of K candy bars to N children by their playful grandfather.
Specifically let’s consider the random distribution of K=4 candy bars to N=2 grandchildren, Jack and Jill, as done by grandpa tossing the candy bars blindly over his shoulder to them. Such a distribution is equiprobable because each of the N=2 children has an equal, P=1/N=1/2, probability of getting a candy bar on any given toss by grandpa.
For ease in explanation we’ll have the K=4 candy bars be different brands: a
Snickers, S; a Hershey bar, H; a Butterfinger, B; and a Tootsie Roll, T. We
could as well use the same brand for all K=4 of the candy bars and get the same
mathematical results, but using different brands makes the argument easier to
follow. There are Ω=N^{K}=2^{4}=16 permutations
of candy bar distributions possible in this action as are listed in {braces}
below with the candy bars Jack gets listed to the left of the comma in the
{brace} and the candy bars that Jill gets, to the right of the comma.
Ω=16 permutations 
{SHBT, 0} 
{SHB, T} 
{SH, BT} 
{T, SHB} 
{0, SHBT} 


{SHT, B} 
{SB, HT} 
{B, SHT} 



{SBT, H} 
{ST, BH} 
{H, SBT} 



{BTH, S} 
{HB, ST} 
{S, BTH} 




{HT, SB} 





{BT, SH} 


States 
[4, 0] 
[3, 1] 
[2, 2] 
[1, 3] 
[0, 4] 
Permutations per state 
1 
4 
6 
4 
1 
Probability of a state=permutations per state/Ω 
1/16 
4/16=1/4 
6/16=3/8 
4/16=1/4 
1/16 
State number set notation 
x_{1}=4, x_{2}=0 
x_{1}=3, x_{2}=1 
x_{1}=2, x_{2}=2 
x_{1}=1, x_{2}=3 
x_{1}=0, x_{2}=4 
Table 19.The Ω=16 Permutations and other Properties of the Random Distribution of K=4 Candy Bars to N=2 Children
All Ω=16 permutations are equiprobable, the probability of each one coming
about by grandpa’s random toss being 1/Ω=1/16. To make clear what we mean
by equiprobable permutations, if grandfather repeats his random tossing of the
K=4 candy bars to the N=2 grandkids 16 times, on average each of the Ω=16
permutations of distribution listed in the table will appear once. The Ω=N^{K}=16
permutations are grouped into W=5 states, [4, 0], [3, 1], [2, 2],
[1, 3] and [0, 4], each state having a specified number of permutations in it.
The [1, 3] state consists of 4 permutations, for example, as tells us that
there are 4 ways that Jack can get 1 candy bar and Jill, 3. And below that in
the table, the probability of each state coming about is listed. For example,
the probability of each child getting 2 of the K=4 candy bars, the [2, 2]
state, is 3/8=.375. And below that in the table is listed the number set
notation of each state, x_{1} being the number of candy bars Jack gets
and x_{2},_{ }the number that Jill gets for that state.
The W number of states for a random distribution of K=4 candy bars to N=2 kids is W=5, as is counted in the table. There is a shortcut formula that in textbook mathematical physics.
20.)
Note how this formula calculates the W=5 number states of the K=4 over N =2 distribution seen in Table 19.
21.)
The W term is the W in Boltzmann’s S=k_{B}lnW entropy of Eq17, the W=5 in Table 19 also being understandable as the number of states for an equiprobable distribution of K=4 energy units over N=2 molecules. The k_{B} in S=k_{B}lnW is a constant, Boltzmann’s constant, which does not impact the argument.
It should be obvious that some of the W=5 states in the K=4 over N=2 candy bars are more diverse in them than others. The [2, 2] state that has both children getting the same number of candy bars in a K=4 toss, for example, has more diversity as the balanced state of the distribution than the [1, 3] and [3, 1] unbalanced states of this K=4 over N=2 distribution.
The σ^{2} variance of a state can be calculated from Eq7. And the D diversity of a state can be calculated from the variance and the µ=K/N=2 mean of the distribution from Eq12. For example, the variance of the [3, 1} state of x_{1}=3 and x_{2}=1 is σ^{2 }=1 and the diversity of that state
22.)
The variance and diversity of the W=5 states of the distribution are from Eqs7&12
State 
Variance, σ^{2} 
Diversity, D 
[4, 0] 
4 
1 
[3, 1] 
1 
1.6 
[2, 2] 
0 
2 
[1, 3] 
1 
1.6 
[0, 4] 
4 
1 
Table 23. The Variance, σ^{2},
and Diversity, D, of the W=5 States of the K=4 over N=2 Distribution
One obtains
the average of the σ^{2} variances of the W=5 states
in the K=4 over N=2 distribution as a probability weighted average
of the variances of the states that weights the variance of each state by the
probability of that state occurring as listed in Table 19.
State 
Variance, σ^{2} 
Probability of State 
Probability Weighted Variance 
[4, 0] 
4 
1/16 
(4)(1/16)=1/4 
[3, 1] 
1 
Ľ 
(1)(1/4)=1/4 
[2, 2] 
0 
3/8 
(0)(3/8)=0 
[1, 3] 
1 
Ľ 
(1)(1/4)=1/4 
[0. 4] 
4 
1/16 
(4)(1/16)=1/4 



Average Variance=σ^{2}_{AV}=4/4=1 
Table 24. The Average Variance, σ^{2}_{AV}, of the W=5 States of the K=4 over N=2 Distribution
Note that the average variance is specified as σ^{2}_{AV},
which for the K=4 over N=2 equiprobable distribution is σ^{2}_{AV}=1.
Now let’s modify D in Eq12 as a function of σ^{2} to an average
diversity, D_{AV}, as a function of the average variance, σ^{2}_{AV}.
25.)
This has us calculate the average diversity, D_{AV}, of the K=4 over N=2 distribution from its σ^{2}_{AV}=1 average variance as
26.)
To show that Boltzmann’s S=k_{B}lnW entropy has a very high correlation to the D_{AV} average diversity of an equiprobable distribution, we need to calculate the D_{AV} and σ^{2}_{AV} of distributions with larger K and N values than our teeny tiny K=4 and N=2 distribution. This gets complicated for calculating σ^{2}_{AV} by probability weighting as we did for K=4 and N=2 distribution in Table 24. Fortunately we easily develop a shortcut formula for σ^{2}_{AV }from a textbook formula for the variance of a multinomial distribution in Wikipedia. For the general case the formula is
27.)
This simplifies greatly for the equiprobable case where the P_{i} term in the above is P_{i}= 1/N, which tells us that each the N containers for K random or equiprobable distributed items has an equal, 1/N, chance of getting them whether they are equiprobable distributed candy bars or equiprobable distributed energy units. We saw this P_{i}=1/N probability for the K=4 candy bar over N=2 children equiprobable distribution to be P=1/N=1/2. This P_{i} =1/N probability for the equiprobable case simplifies the variance formula of Eq27 via substituting 1/N for P_{i} to
28.)
This variance of an equiprobable multinomial distribution is perfectly the same as the average variance of an equiprobable distribution, σ^{2}_{AV}, first developed in Table 24. Hence we can express the above variance formula as
29.)
It is a simple matter to demonstrate the correctness of Eq29 for the K=4 over N=2 distribution by calculating its σ^{2}_{AV}=1 average variance of Table 24 from Eq29 as
30.)
And now from Eqs25&29 we derive the simple formula for the average diversity, D_{AV}, we first introduced without derivation in Eq18.
31.)
We demonstrate its correctness by using this formula to obtain the D_{AV}=1.6 average diversity of the K=4 over N=2 distribution of Eq26 as
32.)
Next we want to develop an easy to use lnW in Boltzmann’s S=k_{B}lnW for large K over N distributions to see the extent of its correlation to D_{AV} of Eq31. The lnW term is from W of Eq20
33.)
A formula that provides ease in calculating lnW for large K and N values is Stirling’s Approximation, which approximates the ln (natural logarithm) of the factorial of any number, n, as
34.)
This approximates lnW of Eq33 as the function introduced earlier of Eq17a
17a.)
To
demonstrate the accuracy of Stirling’s Approximation we note that it computes
lnW for a K=145 over N=30 random distribution as lnW≈75.71 as compares
well to the exact lnW=75.88 value of lnW calculated exactly from Eq33. The
larger the K and N values, the better the approximation. Realistic
thermodynamic systems of K energy unit over N molecule distributions have very
large K and N so we compare the lnW of large K and N as computed from Eq17a to
their D_{AV} average diversity of Eq31.
K 
N 
lnW 
D_{AV} 
145 
30 
75.71 
25 
500 
90 
246.86 
76.4 
800 
180 
462.07 
147.09 
1200 
300 
745.12 
240.16 
1800 
500 
1151.2 
381.13 
2000 
800 
1673.9 
571.63 
3000 
900 
2100.88 
692.49 
Table 35. The lnW and D_{AV}
of Various K over N Distributions
The
Pierson’s correlation coefficient between D_{AV} and lnW for these
large K over N random distributions is .9995, close to a near perfect 100%
correlation as appreciated visually with the near perfectly straight line
scatter plot below of D_{AV} versus lnW in Table 35.
Figure 36. A plot of the D_{AV} versus lnW data in
Table 35
The .9995 correlation between lnW and D_{AV} (and increasingly greater the larger the K and N of a distribution, K>N), tells us that Boltzmann’s S=k_{B}lnW entropy can be replaced by D_{AV} because the fit of D_{AV} to empirical thermodynamic data must be as good as the lnW based Boltzmann’s S=k_{B}lnW entropy. In conjunction with the incorrectness of Boltzmann’s axiom of indistinguishable energy units, this tells us that entropy should be specified as energy diversity or dispersal as the spread of K energy units over N molecules, which understands and explains entropy in a much more intuitively clear way as a physical quantity than Boltzmann’s lnW based S=k_{B}lnW entropy whose lnW term has absolutely no meaning as a physical quantity. The quantitative fit of this D_{AV} energy diversity characterization of a thermodynamic system to the qualitative description encouraged in the Wikipedia article Entropy (energy dispersal) is quite remarkable and a conclusively strong argument in favor of it given its .9995 correlation to it.
As Boltzmann and his work are highly regarded in science, right up there with Isaac Newton, James Clerk Maxwell and Albert Einstein, it does not hurt to bolster the argument that Boltzmann was wrong and that energy diversity is the proper understanding of entropy in any way we can. We do that next with an argument that starts with the introduction of a property of a distribution called a configuration.
A
configuration is well defined as the
group of all states in a distribution that have the same number set representation. For example, the states of [0, 4]
and [4, 0] of the K=4 over N=2 distribution have the same number set, (4, 0),
understood as a configuration of that K=4 over N=2 distribution. Note that we
write a configuration in parenthesis, (4, 0), in contrast to the brackets we
used for the [4, 0] and [0, 4] states of the (4, 0) configuration. We see that
the K=4 over N=2 equiprobable distribution has three configurations, (4, 0),
(3, 1) and (2, 2), which the W=5 states of the distribution of Figure 19 belong
to as
The 3 configurations of the K=4 over N=2 Distribution 
(4, 0) 
(3, 1) 
(2, 2) 
The W=5 states of the K=4 over N=2 Distribution 
[4, 0] 
[3, 1] 
[2, 2] 
[0, 4] 
[1, 3] 

Table 37. The Configurations of the
K=4 over N=2 Distribution and Their States
A quick
look back to Table 23 makes it clear that a configuration has the same σ^{2}
variance and same D diversity as the states that comprise the configuration.
Configuration 
States 
Variance, σ^{2} 
Diversity, D 
(4, 0) 
[4, 0], [0. 4] 
4 
1 
(3, 1) 
[3, 1], [1, 3] 
1 
1.6 
(2, 2) 
[2, 2] 
0 
2 
Table 38. The Variance, σ^{2}, and Diversity, D, of the Configurations of the K=4 over N=2 Distribution
Now note carefully in the table that the average variance of σ^{2}_{AV}=1 of the distribution of Eq30 and average diversity of D_{AV}=1.6 of Eq32 of the K=4 over N=2 distribution are the same respectively as the σ^{2}=1 variance and D=1.6 diversity of the (3, 1) configuration as seen in Table 28. On that basis the (3, 1) configuration is a condensed representation of the entire set of the three configurations that make up the K=4 over N=2 distribution as what we call the Average Configuration of the distribution. That is, much as the µ mean is a condensed representation of a set of N numbers as with K=24, N=6, (6, 4, 2, 1, 5, 6), being represented in condensed or reduced form by its μ=K/N=4 mean, so the Average Configuration of a random distribution is a condensed representation of the entire distribution comprised of its manifold configurations as with the (3, 1) Average Configuration representing the entire K=4 over N=2 distribution of its many configurations of (4, 0), (3, 1) and (2, 2).
Both kinds of condensed representations, the µ mean and the Average Configuration, leave out information from their respective larger set of numerical descriptions. The mean leaves out the (x_{i}) of the number set it represents in condensed form and the Average Configuration leaves out the specific configurations that make up the equiprobable distribution it represents in condensed form. There are other important pure mathematics condensed representations we shall present later when we make the point that the human mind operates primarily on condensed representations or generalizations associated with emotions to direct or control our behavior.
The Average Configuration is a condensation representation of an equiprobable distribution because it has the same values of σ^{2} variance and D energy diversity as the σ^{2}_{AV }variance and D_{AV} diversity respectively of the entire distribution. If the Average Configuration is truly representative of the distribution as a whole, it should exhibit the salient properties of the distribution, which include not only its σ^{2}_{AV} variance and its D_{AV} diversity but also the MaxwellBoltzmann energy distribution that is an observed property of every realistic thermodynamic distribution.
Figure 39. The MaxwellBoltzmann Energy Distribution
Hence showing that the Average Configuration of a K over N distribution
exhibits the MaxwellBoltzmann distribution should go a long way in supporting
diversity as the correct formulation and understanding of entropy rather than
the Boltzmann’s S=k_{B}lnW formulation of it. While we developed the Average
Configuration for the random distribution of K candy bars over N children it is
also valid for the random distribution of K discrete energy units over N
molecules of a thermodynamic system.
The equiprobable distribution of K candy bars over N children from grandfather tossing the candy bars to the children blindly over his shoulder when done repeatedly is a perfect mathematical model for the equiprobable distribution of K energy units over N molecules that comes from repeated collisions between molecules and from the random energy transfers those collisions bring about for the simplest thermodynamic system of gas molecules moving about and repetitively colliding in a container of fixed volume.
A K=4 energy units over N=2 molecule distribution has too few K energy units and N molecules for its Average Configuration of (3, 1) to show any resemblance to the MaxwellBoltzmann distribution of Figure 39. We need equiprobable distributions with higher K and N values to show it starting with a K=12 energy units over N=6 molecule distribution. To find its Average Configuration we first calculate the σ^{2}_{AV} distribution variance from Eq29.
40.)
The Average Configuration of the K=12 over N=6 distribution will have a σ^{2} variance with the same value as the distribution variance of σ^{2}_{AV}=1.667. The easiest way to find Average Configuration of a distribution is with a Microsoft Excel program that generates all the configurations of this distribution and their variances to find one whose σ^{2} has the same value as σ^{2}_{AC}=1.67. It is the (4, 3, 2, 2, 1, 0) configuration, the Average Configuration of the distribution on the basis of its having a variance of σ^{2}=1.667. A plot of the number of energy units on a molecule for this Average Configuration of (4, 3, 2, 2, 1, 0) vs. the number of its molecules that have that energy is shown below.
Figure 41. Number of Energy Units
per Molecule vs. the Number of Molecules Which
Have That Energy for the
Average Configuration of the K=12 over N=6 Distribution
Seeing this distribution as the MaxwellBoltzmann energy distribution of Figure 39 is a bit of a stretch, though it might generously be characterized as a very simple, choppy MaxwellBoltzmann distribution. Next let’s consider a larger K over N distribution, one of K=36 energy unit over N=10 molecules. Its σ^{2}_{AV} distribution variance is from Eq29, σ^{2}_{AV}=3.24. The Microsoft Excel program runs through the configurations of this distribution to find one whose σ^{2} variance has same value as the σ^{2}_{AV}^{ }=3.24 distribution variance, namely, (1, 2, 2, 3, 3, 3, 4, 5, 6, 7). A plot of the energy distribution of this Average Configuration is
Figure 42. Number of Energy Units
per Molecule vs. the Number of Molecules Which
Have That Energy for the
Average Configuration of the K=36 over N=10 Distribution
This curve was greeted without prompting by Dr. John Hudson, Professor Emeritus of Materials Engineering at Rensselaer Polytechnic Institute in Troy, NY and author of the graduate text, Thermodynamics of Surfaces, with, “It’s an obvious protoMaxwellBoltzmann.” Next we look at the K=40 energy unit over N=15 molecule distribution, whose σ^{2}_{AV} distribution variance is from Eq29, σ^{2}_{AV}=2.489. 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 an Average Configuration of the distribution on the basis of its σ^{2}=2.489 variance. A plot of its energy distribution is
Figure 43. Number of Energy Units
per Molecule vs. the Number of Molecules Which
Have That Energy for the
Average Configuration of the K=40 over N=15 Distribution
And next we look at the K=145 energy unit over N=30 molecule distribution whose variance is from Eq29, σ^{2}_{AV}=4.672. There are nine configurations with this σ^{2}_{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 44. Number of Energy Units
per Molecule vs. the Number of Molecules Which
Have That Energy for the
Average Configuration of the K=145 over N=30 Distribution
All of the other configurations of this distribution bear similar resemblance
to the MaxwellBoltzmann of Figure 39. As we progressively increase the K and N
of our example equiprobable distributions we see that the plot of their energy
per molecule versus the number of molecules with that energy progressively more
and more approach and eventually fit the shape of the realistic
MaxwellBoltzmann distribution of Figure 39.
This generation of the empirical MaxwellBoltzmann energy distribution from the Average Configuration not only supports diversity formulated entropy as the proper form of entropy but also provides an intuitively sensible microstate picture of a thermodynamic system. Specifically it depicts a K energy unit over N molecule system as consisting from an extrapolation of the picture presented in Table 19 of Ω=N^{K} energy distribution permutations that develop from Ω=N^{K} sequential collisions between molecules over time. The random energy transfers from the collisions produce on average all of the various Ω=N^{K} equiprobable permutations from the Ω=N^{K} sequential collisions, those permutations represented by the Average Configuration that consists of the average properties of the system that include its average variance, its average energy diversity and its MaxwellBoltzmann energy distribution.
This easy to intuitively digest picture of a thermodynamic system developed in terms of the Average Configuration in conjunction with the .9995 correlation of energy diversity and Boltzmann’s S entropy and the great doubt cast on Boltzmann’s indistinguishable energy unit axiom from a clarification of fundamental distinction overthrows Boltzmann statistical mechanics. This is not to disparage Boltzmann’s genius, for it is a simple matter to understand and “forgive” his error. Boltzmann proved his S=k_{B}lnW entropy equation with its quantitative fit to the thermodynamic data. But this fit to data only came about by a fluke or chance correlation to mathematical diversity, something Boltzmann couldn’t possibly have known about because the diversity function wasn’t developed by Simpson and available to science until 1948, long after Boltzmann died. Credit for genius where credit is due, though, for the correct specification of entropy given here as energy diversity a century after Boltzmann would have been quite impossible we feel without Boltzmann developing his S formula first.
To make this story of hit and miss scientific discovery more interesting yet, we now clarify that the above understanding of entropy as energy diversity is not the complete story. To solve the problem of entropy completely, yet another diversity function that has a very high correlation with Boltzmann S=k_{B}lnW entropy must be considered. To develop that diversity function we begin by expressing the K number of objects in number set with N subsets in a formal way as
45.)
For the (6, 5, 1) number set, for example, which has x_{1}=6, x_{2}=5 and x_{3}=1, we see K= x_{1}+x_{2}+x_{3}=6+5+1=12. Eq45 allows us to define the μ mean of Eq6 as
46.)
Next we want to express 1/N in terms of what we are calling the weight fraction of a number set. It is just a fractional measure of the x_{i} of a number set as the ratio of the x_{i} number of objects in each of the N subsets to the K total number of objects in the set.
47)
For the K=12, N=3, (6, 5, 1), number set that has x_{1}=6, x_{2}=5 and x_{3}=1, the weight fractions are p_{1}=x_{1}/K=6/12=1/2, p_{2}=x_{2}/K=5/12 and p_{3}=x_{3}/K=1/12. We can write these p_{i}^{ }weight fractions of (6, 5, 1) in shorthand form as (1/2, 5/12, 1/12). Note that the p_{i} weight fractions of a number set necessarily sum to one.
48.)
The weight fractions of (6, 5, 1) as (6/12, 5/12, 1/12) sum to one. The (1/N) term in the rightmost summation of Eq46 can now be understood as the average weight fraction of a number set. We take the average of a number set’s_{ }weight fractions by adding up its p_{i} to one (1) and then dividing that sum by the number of p_{i} in the set, which is N. For example, the weight fractions of the N=3, (6, 5, 1), set are p_{1}=6/12, p_{2}=5/12 and p_{3}=1/12, which sum to 1. Dividing this sum of 1 by N obtains 1/N. So the average weight fraction, which is given the symbol,, is for any number set,
49.)
This allows us to express the μ mean in Eq46 in terms of the average weight fraction,, as
50.)
This interprets the μ mean of a number set as the sum of “slices” of its x_{i} with each “slice” the “thickness” of, the average weight fraction. This form for μ computes the μ=K/N=12/3=4 mean of the K=12, N=3, (6, 5, 1), number set as
51.)
All of the above is spelled out as preface to our defining a new kind of average of a number set called the biased average, φ, (phi). In parallel to the μ arithmetic average of a number set being the ratio of K to N as μ=K/N, we specify the φ biased average as the ratio of K to the D diversity of a number set as
52.)
For the K=12, (6, 5, 1), set, which has D=2.323 from Eq4, φ=K/D=5.167. Note that this φ=5.167 biased average of (6, 5, 1) is greater than its μ=4 mean or arithmetic average. Next let’s express the D Simpson’s Reciprocal Diversity Index of Eq1 via Eq47 as
53.)
This has us specify φ=K/D of Eq52 in a way parallel to μ in Eq50 as a function of the p_{i} weight fractions of a number set of Eq47 as
54.)
This has us interpret the φ biased average as the sum of “slices” of x_{i} with each “slice” the “thickness” of p_{i}, that is, of the thickness of the actual p_{i} weight fraction rather than of the =1/N average weight fraction, as was the case for μ in Eq50. The above form for φ in obtains the φ=5.167 biased average of (6, 5, 1) as
55.)
This makes it clear how the φ biased average is biased in its exaggerating the contribution of the larger subsets in a set to the set’s φ biased average. With regard to our perfecting our formulation of diversity based entropy, the φ biased average is just an introduction to another diversity related biased average called the square root biased average, ψ, (psi), which will be shown to be the proper foundation of microstate entropy and temperature.
In parallel to the φ biased average as the sum of slices of the x_{i} of a set of thickness p_{i} in Eq54, the ψ square root biased average is 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. So in parallel to Eq54 for the φ biased average, we write ψ as
56.)
We place a question mark on this introductory definition of the ψ square root biased average to indicate that there is something not quite right with it 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 any kind of an average of the x_{i }of a set, that proviso being in the intrinsic nature of what an average is. This problem is 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 an average of them.
This problem is readily resolved, though, by normalizing the p_{i}^{1/2} to get them to add up to 1 as is done 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
57.) ψ = (.4308)(6) +(.3933)(5) + (.1759)(1)= 4.727
The sum of p_{i}^{1/2} function that divides the p_{i}^{1/2} to normalize them is expressible as
58.)
This revises the ψ{?} questionable function for ψ in Eq56 by dividing the p_{i}^{1/2} in the numerator to obtain the ψ square root biased average correctly as
59.)
Next note that we can express ψ in an alternative way via the p_{i}=x_{i}/K weight fraction relationship of Eq47 as
60.)
The φ=K/D biased average as the ratio of the K number of objects in a set to the set’s D diversity index implies that the D diversity index can be understood as the ratio of K to the φ biased average.
61.)
In parallel we define a diversity index that is the ratio of K to the ψ square root biased average and that we’ll call the Square Root Diversity Index. From Eq60
62.)
Next we show that the average G diversity of a thermodynamic distribution, G_{AV}, also has a high correlation to the lnW term in Boltzmann’s S=k_{B}lnW entropy and as such that it along with D_{AV} is a candidate to replace Boltzmann’s S=k_{B}lnW as the correct diversity based function for entropy. This is a somewhat tricky correlation to compute, however, because the form of G in Eq62 is not amenable to developing a simple function for its average value, G_{AV}, as we did for the average diversity, D_{AV},_{ }in Eq31 as D_{AV}=KN/(K+N−1).
But
G_{AV} is the G diversity index of the Average Configuration much as
was D_{AV} was the D diversity index of the Average Configuration.
Hence we can obtain G_{AV} for the K over N distributions for which we
have the specific number sets of the Average Configurations, the K=4 over N=2
distribution and those in Figures 4144. Below we list those K over N
equiprobable distributions along with their lnW values from Eq33 and the D and
G of these distributions’ Average Configurations, the D_{AV} and G_{AV}
respectively from Eq31 and Eq62.
K 
N 
lnW 
D_{AV} 
G_{AV} 
4 
2 
1.61 
1.6 
1.76 
12 
6 
8.73 
4.24 
4.57 
36 
10 
18.3 
8 
8.85 
45 
15 
26.1 
11.11 
12.33 
145 
30 
75.88 
25 
26.49 
Table 63. The lnW, D_{AV} and G_{AV} of Distributions in Figures 101107
The correlation between the lnW and D_{AV} of the above distributions
is .996. Though quite high, this is less than the .9995 correlation between lnW
and D_{AV} that we saw in Table 35 for high value K over N
distributions. What is noteworthy is that the correlation between lnW and G_{AV
}for
the K over N distributions in Table 63 is also quite high as .994, very little different than
the .996 correlation between lnW and D_{AV} for these distributions.
As the correlation of D_{AV} with lnW went from .996 for the low value K over N distributions to .9995 for the high value K and N distributions back in Table 35, so we can assume reasonably from the closeness in the lnW correlations for D_{AV} and G_{AV} of .996 and .994 respectively that for high value K and N distributions the correlation of G_{AV} to lnW would also be in the range of .999 or near 100%. Thus we see that either diversity function, D_{AV} or G_{AV}, can replace Boltzmann’s S=k_{B}lnW entropy from both having a high correlation to lnW. What will have us choose G_{AV} over D_{AV} starts with specifying these two average diversity functions by extension from Eqs61&62 as
64.)
65.)
In standard theory, via the equipartition theorem, temperature is a simple function of the average kinetic energy, μ=K/N, where K is total number of energy units and N the number of molecules. But a μ=K/N temperature specification must be seriously questioned from the perspective of the reality of how temperature is actually measured physically with a thermometer.
Let us understand the K energy units of a thermodynamic system of N gas molecules to be distributed over them with x_{i} energy units, i=1,2,…N, for each of the N molecules that move about in a container of fixed volume. Because the molecular energy units are divided equally from the equipartition theorem over kinetic, rotational and vibrational energy, the velocity of the molecules as a function of the kinetic energy is proportional to the square root of the x_{i} number of energy units on each molecule. This has each of the molecules collide with the thermometer that measures the temperature of the system at a frequency equal to the molecular velocity, which is proportional to the square root of the x_{i} number of energy units on the molecule. Hence the smaller energies of the slower moving molecules in the MaxwellBoltzmann energy distribution of Figure 39 collide with the thermometer less frequently and are, hence, recorded less frequently as part of the temperature than the higher energies of the faster moving molecules that collide with the thermometer and are recorded more frequently.
This necessarily develops temperature as an average of molecular energies weighted toward the higher energies of the faster moving molecules because of their greater velocities that cause them to have a higher frequency of collision with the thermometer. As the velocities of the molecules are directly proportional to the square root of the x_{i} energy of the molecules, the average molecular energy, which is temperature, is the square root of the x_{i} energy weighted average, which is the square root ψ average energy per molecule of the thermodynamic distribution, ψ_{AV}. From this we see that the diversity of the distribution measured in terms of the ψ_{AV} temperature is G_{AV}=K/ψ_{AV} rather than D_{AV}=K/φ_{AV}, which has us choose the G_{AV} diversity index of the distribution as the proper correction of Boltzmann’s S entropy.
3. My Brother, Don
The data needed for an
analysis of entropy is a simple matter of the cookbook laboratory numbers that
Boltzmann nested his S entropy in which we piggybacked on by showing the near
perfect correlational equivalence of diversity based entropy to the S entropy.
But the data needed for an analysis of a society's emotional reality is
entirely another thing in an informational environment saturated with
propaganda that tinsels police and economic subjugation with endless free and
fair sloganeering further confused with people's emotionally soothing
rationalizations that their lives are other than those of servile cowardly
jackasses. A true story indicative of what life is actually about, obedience on
threat of punishment, follows.
I was born four months before America entered WWII in with the last wave of
Christian girls whose minds were controlled as tightly and painfully as the
foot bound women of imperial China. My father, the Rev. Arthur Graf, was a
fundamentalist minister with rural parishes in Cullman, AL, where I was born,
to Serbin, TX, winding up as a reward for his success as a pastor at Lutheran
seminary where he taught Stewardship, a fancy name for how to get the cash out
of parishioners' wallets and purses.
My mother was a classic bulldog faced nut out of Stephen King's Carrie who believed
Jesus talked to her personally every day, that the fossils in Dinosaur National
Monument in Utah were plaster fakes buried secretly in the ground by people who
hated God and that the best way to raise children, including this little girl,
was with a weekly britches pulled down whipping. If she didn't get off sexually
with this game for she had a way of twisting truth in all other matters, I
wouldn't believe it.
Fear ruled my life, fear of punishment for taking a cookie without permission,
fear of the dark, fear of dogs, fear of a full moon that stretched into my
early thirties when I was finally able to escape from this idiotic horror so
hidden by my father's Joel Osteen smiling sermons. Some of the worst of it was
my role as fodder for my brother, Don, two years older than me. He was the
recipient of the same sort of corporal punishment until he became my mother's
toad and henchman. My hearing her spank him brought on tears for him, a waste
of emotional energy in that my mother's iron rule could never be softened with
tears and in Don's passing on a good amount of the pain she gave him to his
younger sister, me. That was quite acceptable in those days when southern
Cristian women only came in two varieties, the crazy beasts who ruled in
adulthood with the rod and the pretty pastries of children the monsters
devoured, some of whom were the seeds of the next generation of monster mothers
and some of whom were destined to the worst horrors of misery and madness in
their own adulthood.
I was lucky. I was not so destroyed as to be unable to hate my mother. I was
yet aware enough to hate coming home to her from school every day. Also what
was left of the dumbbell in me she created was pampered enough on the margins
to make me a pretty if awkward young girl, for the minister's daughter is a
public figure and if thought pretty a considerable status symbol, meaningful
for stewardship and rising in the pastoral ranks.
All that mattered to this little girl growing up was the thought and hope of
love. The most daring books in our home library on the shelf in the living room
were Zane Grey novels. My imagination translated the best of them into heroes
scooping me up on their horses and taking me far away from my home situation
while squirting me in my preteen private parts, front and back, with hot sauce
of unknown composition.
My attitude towards men was also shaped, no doubt, by my brother, Don, who
sustained himself by terrifying me with daily punches on my arm and tales told
me of a wolf on the prowl upstairs near my bedroom who was always about to bite
me to pieces, my mind so dumbed down from constant disapproval and punishment
that I actually believed these stories that he delighted in frightening me
with. I was the model he practiced on in learning to control and humiliate
people as a lawyer in later life.
My young romances were typical failures. The boy I came to love most my parents
hated and never stopped talking down. Unfortunately the poor fellow, only
seventeen like me, lacked the vigor and toughness of a Zane Grey hero even if
his fondling was enough to kindle a strong flame of desire and affection for
him. It takes more weapons to be the knight in shining armor that rescues a
damsel in as much distress as I was in than any seventeen year old kid could
possibly have had.
My tears from the inevitable breakup were doubly painful with my mother
reveling in soothing me over what I took to be a personal failure on top of the
loss of love. And soon I found myself connected with a seminary student in my father's
class as superficially charming as my father, himself, the standard minister
type.
After two years of college at age twenty I married this guy, Len Schoppa. The
error in it was marked by my brother, Don, not attending his sister's wedding
because he needed to study for an exam in law school, beyond an insult of major
proportions in any family, a fairy tale level omen of bad things to come on
this most important day in a woman's life.
To speak of myself as gullible as Len and I headed off to Japan as Lutheran
missionaries is as much an understatement as calling a blind person gullible. I
came equipped only with an ingrained sense of duties to be performed, cook and
wash the dishes after meals and prepare the Sunday communion wafers. And a few
primitive feelings that escaped my mother's guillotine like my continued
feeling of longing for love and sex. And the endless subtle misery of a
loveless arranged marriage that manifest itself in the daily migraine headaches
that I'd had with me since early grade school.
Can you imagine the preposterousness of living your life with the goal of
converting the Japanese to Christianity? For my husband it was all dominance
games on the young Japanese men who came to our mission church in search of
escape from the empty life that awaited that generation of losers to America in
WWII; and to make contact with the pretty young wife of the pastor whose vacant
personality fit so well the docility expected culturally of Japanese women.
Many fell in love with me, no awareness of what was going on my part as window
dressing attraction for them out of Tennessee William's masterpiece Suddenly
Last Summer with Len reveling over them in being the one to have the woman they
were falling in love with. And down went all these poor bastards to him, one of
them eventually committing suicide.
I dare not talk in any depth about my relationship to the three kids I bore for
this common predator, they at the same time being the only love I had ever had
in my life, especially for the first born; or the unbearable pain I felt in
seeing the terrible job I had done as a mother as was so clearly marked by the
absence of any spark in their eyes as they approached adolescence. Makes you
wish you were dead. If I could kill myself and offer up the pain of a torturous
exit to assuage what I was incapable of giving them that added to their primary
hell of being under my minister missionary husband, I'd take the knife to my
throat without hesitation. As bad as what you become in life, worse is what you
pass on to others intended or not especially to the innocents.
In America as a pastor's wife in idiot form rather like Sandy Dennis in Afraid
of Virginia Wolf I would have been totally devoured by the older women in a
congregation. But in Japan I was semiworshipped by a vast gaggle of Japanese
men who extended beyond our mission church young guys to the classes of college
boys I taught English to at Hokkaido University and then to the Japanese public
at large as a commercial model on Japanese TV. One of our social contacts
through the mission church was a television producer who signed me up to pitch
Japanese bean soup on TV. For six years I was on TV all over Japan in this
guise and stopped by strangers on the street and asked, "Aren't you the Koiten
Soup Girl?"
My next would be Zane Grey hero was a college boy, a ski bum type, who took the
missionary's wife bait Len dangled in front of all the kids off to bed. This
was on church related ski trips Len didn't come on because he didn't ski. It
was love as close as I'd ever been to it, a great relief from the emotionally
empty love life I had in my parent arranged marriage to the missionary.
Physical love when you want it is Heaven and when you don't and have to do it,
worse than Hell.
Perhaps affairs like this are easy for the smart women on the Unhappy
Housewives of New York TV show to have without blowing up, but in a crowd of 30
LCMS (Lutheran Church Missouri Synod) missionary couples in Japan at the time,
once the gossip sparked it spread in a flash right back to the Rev. Leonard
Schoppa. The climax of it in our confrontation had for me surprising twists and
turns. I didn't hesitate to confess. How could I have said, "No". I
was too dumb to tell a good lie and why would I have wanted to hide it from him.
What surprised me was his falling to the floor when I said, yes, I did it, and
writhing on the rug like a big piece of bacon frying in an fry pan turned up
too high; and while twisting all about confessing to have had sex with farm
animals of all kinds when he was young, sheep, pigs and even the large dog his
parents had called, "Lassie." What that had to do with my having had
an affair the last six months with one of our converts just could not register
in my head and somehow got me to think that the rumor that he had had sex with
his retarded cousin Larry that a few of the good old boys in Harrold, Texas,
had joked about must have been true.
And once you know that, parallax with pastor personalities makes it clear that
they're all closet fags, all the fundamentalist ministers, from Len to Ted
Haggard to my own father, for who would possibly marry a woman as ugly and
bearish as my mother if he had any normal feelings about women. I'd even go as
far as to say that all fundamentalist conservative men are queer as the
comedian Joel McHale so wittily brought out at the White House correspondents
dinner this year for who looks prissier and weirder than Ted Cruz and Rand Paul
and Karl Rove and Limbaugh and my dippy brother, Don, especially after his
wife, Ruby, divorced him, and I'm sure for good reason for what woman divorces
a wealthy lawyer especially in a town as crazily religious as Lubbock, TX.
The headline of Missionary's Wife Has Affair with College Boy Parishioner
quickly spread beyond just our Lutheran circle to all of the Christian
missionaries in Japan and shortly within a year or so brought about the recall
of 29 of our 30 LCMS missionaries back to the States. The scandal hit home
stateside, too, for my father was way up there in the LCMS church hierarchy as
a candidate for Bishop of the Texas District at just about this same time, not
to speak of half my close male relatives being ministers of teachers in the
LCMS. So I was not exactly welcomed back with smiles and flowers after Len and
I were thrown out of Japan as the first of the 29 to be sent home to America.
Rather the word was put out by my immediate family, Don included, so directly
affected by the scandal of it all that I was mentally ill, for why else would a
girl from such a good Christian family do something so dirty and sinful and to
such a wonderful fellow as all ministers are painted to be, especially your
soninlaw. Mentally ill, though, was not how I felt. Rather I felt scared to
see my whole family siding with the snake, indeed that they were all snakes,
and snakes with an eye on biting me as punishment for my sin and to get me back
with Len, the thought of whom at this point, animalfucker and God knows what
else, made me feel like vomiting. Ted Haggard's wife may have remained loyal to
her homosexual condemning fundamentalist minister husband after it turned out
he took it up the ass every Tuesday from a male prostitute he paid with
collection box money, but she was heavily invested in the bigger game and about
as much in need of love from anybody as Donald Trump's third wife.
I should insert in the story at this point, probably a little late because I'm
not much of a creative writer, that in the two years leading up to the affair
in Japan, that I found Len's touch so much like smelling horse manure that I
literally slept on the couch every night. I was lucky to have an excuse for
doing this in Junko.
At Len's insistence, I am sure in retrospect to make us look like the Holy
family to the Japanese, we adopted a beautiful baby girl, Junko Hirota, later
June, the product of a young prostitute from Yokohama and a Norwegian seaman,
strikingly adorable with this mix of Asiatic and Nordic features. In a way she
saved my life, the something about her that was not a product of the snake and
his snaky mission church set up. And June was also an excuse for my sleeping on
the couch, to be close to her to make sure she didn't cry at night. He knew it
was shit but he bought it anyway because all that mattered to the snake was
appearances in life.
Anyway, whatever hell was threatened me if I didn't go back with Len, it was
impossible, like my cutting off my own little finger with a butter knife. So I
ran away and they all ran after me, Len, the family and a couple of dozen
minister friends of my father who harassed me morning, noon and night on the
phone and ringing the bell at the front door. I ran away, I should make clear,
in my mind, for I couldn't leave my kids behind and actually run away.
Frightened and with no real solution to my ever increasing problems, I ran away
in my fantasy thinking.
And it came true. In the guise of another guy appearing on the scene just in
the nick of time, Pete, one of my coauthors in this attempt to save the world
from its epidemic social misery and the nuclear war it will bring soon to wipe
us out with if we're not successful in getting A World with No Weapons.
Back then around 1970 you didn't just up and get a divorce if you wanted one,
at least not if you were a good Christian woman. At least I didn't, coming from
where I was coming from in life. I insisted to Len upon our being booted out of
Japan that we go to Berkeley where I'd read in an issue of International Time
Magazine that things were happening, things that gave hope, just what I needed
personally at this time of despair in my life.
Len enrolled at this school, really a Presbyterian seminary, in San Anselmo,
north of San Francisco in Marin County, to get a degree in something called
pastoral counseling so he could become a marriage counselor, LOL, or a drug
counselor. We lived in student housing on campus, barely speaking to each
other. At this point I am going slowly mad, like locked up in a cage. I avoid
the other minister student's wives, endlessly smiling for no good reason, ugly
and as sweetly phony as food bank artificially sweetened soda pop. It was not
what I came to the Bay Area for. A great relief it was to go 40 miles away for
a weekend of environmental exploration with my oldest boy's seventh grade
class. It is an especially great relief because I am due on Monday to go with
Len to see two psychiatrists who are teachers of his as some kind of marriage
therapy Len has set up to patch us up. Like a stuffed doll with a broken arm I
had agreed to this, perhaps as evidence of how utterly stupid I was back then.
The collection of people who were out at this Youth Hostel we'd be staying in
at the Point Reyes National Seashore included not only all the other kids in
Lenny's class but also genuine users of a youth hostel, many of the guys at
this time with long hair and the girls with torn jeans and flowers in their
hair, the kind that favored organically produced cheese. They were mostly sweet
kind of looking people, except for one who wasn't particularly sweet looking.
Pete, as I found out later, was coming from upstate New York, a dropout from
graduate school at Rensselaer Polytechnic one credit shy of a PhD in
biophysics. He's my entropy writer, not just very smart, but very tough too.
That's how he looked. But that's not quite the right description for how he
looked more was not afraid of anybody, that type of person, confident in a
maximum way.
Later he would tell me that on first sight that he thought I looked like a
model in Woman's Day magazine, which wasn't far from the truth as I had been a
TV model in Japan for the previous six years who directed herself to selling
soup to Japanese moms and families.
We talked for six hours that night, Friday night, his eyes never leaving mine.
He said the selfhelp psychology book I had brought with me was bullshit, that
they all are. When I told him about my husband and going to a therapy session
that Monday with Len's two psychiatrist professors teachers, he said don't go,
it's a trap, two psychiatrists can commit you.
The next morning at breakfast in the communal kitchen of the youth hostel, he
got talking with two Australian fellows who were arguing that you had to
compromise in life to survive and were a fool not to. Pete subtly made clear
that he thought that was cowardly if you compromised on matters that were
important to you. Both the Australians were big guys. When it became clear that
their differences with Pete were irreconcilable and the cross remarks were
bordering on insult, Pete raised his eyebrow and lowered his tone just a bit
and stopped smiling and they both more or less ran out of the kitchen. He was
not somebody who made you afraid of him, but it was also clear that he would
not back down in a fight for honor, kind of like the heroes in the Zane Grey
novels.
We separated during an environmental tour of the seashore and later that
afternoon when we met again I opened up to him. When he asked why I was so sad,
I said, "Look at my son, look at his eyes." To me, anyone could see
he hadn't turned out so well, not very confident with people in some way. It
killed me.
Pete talked to reassure me saying he didn't look that bad, looks better than a
lot of the other kids. But I knew he was being generous to make me feel better.
The conversation went on and on again Saturday night too touching a lot on
politics for he was heavy into the radical antiestablishment politics of the
sixties.
We school people were all due to leave the next morning on Sunday. At some
point during our last exchange, he touched my upper arm, squeezing it in a firm
way as I was about to go, something I could feel down to my knees. As we were
about to get into our blue Toyota, I suddenly asked him, stupidly in
retrospect, if he wanted to come over to the house and have dinner with the
family. Given my situation with Len, I don't know why those words came out of
my mouth. I suppose I wanted to see him again, but didn't know how to say it in
a socially acceptable way.
He smiled and shook his head and said, "Three doesn't work." And we
parted. That night after we got back I told Len I wasn't going to the therapy
session he'd set up. And the next morning after Len went off to classes for the
day I called the youth hostel and told Pete I wanted to make the 40 mile drive
and come out to see him and talk some more.
He was very forward, aggressive at the level of putting his hands down my jeans
without saying a word the minute I got out there and we were alone. The thought
that came into my head at the moment was that he was some sort of a sex maniac
that you hear about and that women are told, of course, to avoid. As it turned
out I suppose he was sort of a sex maniac, but it was something deeply
pleasurable enough that you can't help want to do again once you've done it
once. A little more aggressive and forceful than you might think a honeymoon
should be. But like the best pepperoni pizza you ever ate even if it was kind
of shoved down your throat a bit to begin with, once you've tried it, it's hard
to not want another slice. And he quite felt the same way, maybe even doubly
from the second and third slices he wanted right away.
I stayed overnight and by the time morning came and I knew I had to get back to
the kids and Len  this was in the days before cell phones  he was telling me
that he had never seen a girl as beautiful as I looked, not in a movie, not in
a magazine and not in real life, not ever. As I've been with him 41 years now,
I know he meant it, though some credit to him because all that physical
attention does make a girl feel really good and I suppose as a result look very
good. He also said the words that first intimate day, "I'd die for you.
I'd kill for you." He was as they say in that song by the Cars, "just
what I needed" as things would turn out.
Whatever the bullshit they say about making a commitment, Darwin says it all
and much better than God or Freud. When the sex clicks in that super pleasure
way, you say hello to each other forever. And when it doesn't, as I'd also
found out, and it feels sour, there's no future in it or in whoever the guy is.
Japan was great teen sex. This was a pleasure leash around your hips and your
brain you didn't get away from because you just didn't want to get away from
it. Either the guy's got the testosterone without needing a prescription for it
or he don't. There's no love in America today, all divorces and breakups and
loneliness because all the guys but the bravest who resist critical compromise
whatever the cost and risk have been gelded.
Len knew what was up the minute I got back. "I can tell by your eyes."
Pete said to tell him the minute I got back to get out of the house. He refused
at first until I told him angrily I'd run screaming out onto the campus if he
didn't. It helps to be furious at critical moments. This pious fraud jerk
minister dog I'd had the misfortune to live with for the previous ten years
came back the next day, though, and tried to rape me. I ran from the apartment
with bruises on my arms. Pete was furious when he heard about it when I
hitchhiked out to the youth hostel the following day after Len took the car
keys away from me. "I'll kill the bastard." He didn't have to wait
long. Len drove out to the youth hostel to ask questions and confront him a
couple of days later.
Pete's funniest war story to hear was how he backed down the leader of a Puerto
Rican gang of ten he had with him at the time in this neighborhood on East 11th
St. he lived in in Manhattan. By the time he left New York City to come West he
had acquired four bullet holes in him and a half dozen knife scars and said he
never lost in a fight, even against the gun. I've heard the fight with Len many
times over the years but basically Pete said that he could easily have torn
Len's eyes out of his head and felt like doing it he was so angry but didn't
because he knew that would go over the line and surely get him locked up.
He didn't have to do that, though, because whatever the details of their fight,
Len got the point and was quite scared enough of Pete to never come over and
bother me again. But that was hardly the end of the pain he caused for quickly
following my filing the papers for divorce he got visitation rights and it was
swiftly understood that he enjoyed coming over to bother me with the courts
backing him up, something 50 million women in the same situation I am sure have
experienced. It was obvious in my case because Len never cared anything about
the kids any more than he did about me. We were all window dressing for the
creep. But there's nothing you can do about it without legal repercussions.
Even Pete swallowed his urge to crack his skull open.
All of the legal leveraging was calculated to get me back to Len, not to
produce a livable divorce. Len made no bones about it. Neither did my parents
or my brother who called and talked to me like I was a disobedient eight year
old to get me back to Len to avoid the scandal on themselves. As time passed it
became clear that Len's various legal maneuvers were engineered by my brother,
Don, we always thought because Len's California lawyer was a cheapo prematurely
balding geeky type who mostly wanted me to like him anytime we had contact or
discussions with each other.
Part of the endless harassment to get me to leave the "evil Pete" and
go back to Len included not only near daily phone calls and house calls from
near a dozen LCMS ministers but a ring at the door bell, unannounced, visit
from my mother who flew up from Texas with a large roast beef in tow.
Fortunately Pete was right there in the living room two feet from the front
door when she suddenly arrived. The interaction between the three of us was
short and to the point.
Mildred Graf, my mother, whom Pete described after the fight as looking
remarkably like a "basilisk", a mythical lizardlike monster,
threatened us both with punishment from God, repeating to Pete a few times what
she had told me when I was young that God spoke to her directly on a daily
basis. What Pete said God to do, shouted back in her face, is what you might
imagine a long haired politically radical physically confident lover fed up
with the bullshit that had been raining down on me would say, that God and she
could both go fuck themselves and for her to get the hell out of the house.
When she hesitated he nearly pushed her out and to make his point, our point,
he tossed her roast beef in the garbage can that was sitting outside not far
from the front door.
Near the minute she was gone, he said, "Seems to me more like a squabble
with a dyke over a girlfriend. Your mother really is weird. No wonder you hated
her so much as a child." My memory of some of her more invasive, hygienic,
punishments made that picture of my mother a fairly accurate one. She was
disgusting on top of being overbearing and always in need of being in control.
I'm sure though I don't know how I'd prove it that maternal rape of children
has to be the commonest and most hidden crime in America. I'm sure that kind of
abusiveness is centrally responsible for the plight of so many kids in America
in terms of the unhappy faces that abound outside of the cereal commercials on TV
with bright happy kids in them that gives a totally wrong impression of the
reality of what you see if you visit a middle school. Check out the unhappy
real faces of real kids these days. One thing for sure is that the Columbine
and Virginia Tech and Newtown mass murders were all perpetrated by unhappy kids
who were only the tip of the iceberg, the far end of the statistical curve of
unhappiness that comes from absent and predatory mothers. I am sure fathers
too, but whatever the current psychobabble bullshit on parental equality
related to insuring that capitalism has an ample female labor force, mothers in
an especially big way because we still are what we are instinctively in that
area. Pity the children.
When my mother saw how forceful Pete was, also making clear to her during her
visit that my kids liked and respected him, they changed gears with the custody
strategy of who'd get the kids. First Len said explicitly that, of course, I'd
get the kids, the idea being from my cynical translation of the situation that
he'd get to keep his feet in the door forever with visitation and that
eventually Pete and I would dissolve away. But after my mother's visit, the
legal papers changed abruptly for Len asking for custody of the three of our
biological kids, something I am sure my brother, Don, had a hand in as he was
my mother's lackey in all matters and the change in custody strategy happened
near immediately after her visit.
Then the strategy was to take the kids away to break my heart, which it did.
There really was no contest for once the kids chose Len to a great extent from
the suggestion of the grandparents on both sides, there was nothing I could do.
Their tone became: we're going with daddy; and you should go back with him,
too.
This thing of losing custody of your children is played out so commonly in our
mass media as to be a thing of like choosing this or that cut of meat at the
grocery store. But it's damn not. It killed me. Almost. Now truly turning into
the crazy person they said I was to begin with because of the kids leaving, I
still refused to go back. That wasn't going to work, fuck you all and your
horrible games, I thought.
Soon the kids went with him and Pete and I bought a $200 trailer to live in
with June, whom I still had custody of. They left her behind to keep up Len's
connection to me, for the theme was endlessly, come back. Len still had
visitation rights with June, every other weekend, and those comings and goings
were so painful. Too sad to tell, on the third or fourth occasion of one of
these visitations, he brought her back but she wouldn't talk anymore. Wouldn't
talk and wouldn't smile and wouldn't do anything but crawl around on the floor
making sounds like a kitty cat. Whatever had been June was dead.
After a half hour of this horrifying nightmare in the living room of the
trailer, I called Len on the phone and screamed out, "What did you do to
her!?" Only to hear him say back in an obviously emotionally fake and
contrived manner, "What did you do to her." This doubled the
scariness of what they did to her, whatever it was, by making it clear that
whatever they had done, they wanted to use it on us to destroy me by destroying
her and somehow blaming it on us, which made clear that they had intentionally
done something to destroy this poor little three year old.
What did we do? We ran, picking up stakes with the trailer and driving off the
next day, screw the legal agreements as to visitation. I'd rather be locked up
for violating court ordered visitation than ever let him get his hands on her.
7oon we crossed from California into Oregon, leaving the state upping the
potential charges to the felony level. We didn't care. Threatening letters from
Len and his lawyer and the authorities came to our Post Office Box over the California
border. We didn't care. We worried constantly that they'd track us down, every
sight of a car in Oregon with California or Texas plates producing a jolt. Pete
said if he ever came across Len, he'd kill him. And he would have. I was so sad
and crazy after that, I don't know how we made it. Pete never quit. All the
love available after that went to June. Spoiled her to get her to smile and
keep her looking the most beautiful child in the world, whatever it cost in
time and energy.
Pet never quit. He really was a real revolutionary, true and blue, to the death
he vowed long before he met me. I should make that clear and why. When he came
of age, his thesis advisor, name of Posner, a big man in science, stole his
research. Pete said at first he couldn't believe it. Stole it and published it
without Pete's name on it. Then he told Pete, partly because Pete was and
looked like a 60s rebel at this time, practically just told him out and out
that he wouldn't sign Pete's PhD thesis unless Pete kissed his ass.
There was no uncertainty about what was going on. It was just a pure power
play, teaching Pete who was boss, just a kind of rape. So what did he do? He
told Posner and the rest of his thesis committee who were too cowardly to
challenge big time Posner, to all go fuck themselves. And from that, he said, a
genuine miracle, an unexpected miracle happened. Said he was reborn as a young
god, a whole new level of confidence in his life. He joked that his sex life,
which wasn't the worst before that, took off to new levels, that women started
near fighting to see who could sit on his lap in watering places in New York
City. On his way from New York to California not long before we met he said
he'd had sex with three different women on the Greyhound bus ride cross
country. That it was a new life impossible to turn back from even though he'd
lost his PhD as the price he paid. (Got it back ten years later when his
research was validated by a team in Czechoslovakia.)
Anyway he was a fighter and we fought hard to bring June back to life. She
hardly spoke a word over the next three years. But what she did do was draw a
lot, an amazingly gifted artist even though so little. And when she was about
six years old, she started drawing odd pictures about strange looking creatures,
people that Pete thought might have hurt her back then because the pictures had
this dark look to them. And Pete had also taken a special course in Montessori
method of teaching reading to deaf children that he used for June until he
gradually got her talking again.
Not only did it seem a miracle in itself but also it came to explain what had
been done to her by them. I should point out that June never used a pillow when
she went to bed. She didn't like pillows. Eventually she told us that they had beaten
her up because she wouldn't be quiet in church where they took her on
visitation. They took her home after church and beat her up. And then, horror
of horrors revealed, they put a pillow over her face and partially smothered
her and told her if she ever told anybody, they'd smother her, put that kind of
fear in her. I'm not exaggerating in this.
She also talked about things done to her that you'd deem sexual. But Pete never
took it too seriously because once you start thinking and talking in that way,
nobody would believe you. It was horrible enough that they beat her dumb
without accusing them of anything more than that. I should answer the question
now of how June turned out. Our other coauthor, Angel Thomas Rogovsky, is her
17 year old son, and he is a wonder to behold as well as a special kind of
genius.
Like I said, Pete eventually got his PhD from Rensselaer in 1980 after the
other thesis committee members threw Posner off of it. And they also gave him a
faculty position in the Dept. of Biomedical Engineering. This new found status
7 years after we met and the worst had gone down enabled us to make a visit to
my family in Texas so I could have some contact with my other three kids again.
I'd have to write an entirely separate novel about this, but what has immediate
relevance to the point I'm trying to get across is that we were invited to
brother, Don's house in Lubbock. To make sense of what happened, I need to
briefly fast forward to tell you that the following year Len tried to get custody
of June and failed, the judge not only giving us custody but actually nixing
visitation by Len. That was in 1981.
In the visit to my brother, Don's, house in 1980, it came out over breakfast
that Don accused us of running off with June and violating Len's visitation
rights. As he talked this way it became very clear that he actually was very
much involved with Len's legal strategy. But what was particularly revealing
was when he accused us of crossing state lines and committing a felony that he
implied he would try to get us prosecuted on. When Pete said, "How do you
know? You weren't there," Don said, yes he was, that he had come up from
Texas that weekend of that last visitation.
And at that point I knew that this punk rat murdering bastard had been in on
June's beating and had probably been the one to suggest it in a cold
calculating way to begin with because he knew from his same torture of me when
I was little what the result would be. As June escaped and recovered over time
enough to have a happy life as one of the finest mothers this world has ever
seen, her son solid proof of that, I can tell this story without tears in my
eyes and I do tell it to make the point that much happens that is hidden,
things that destroy people's lives and happiness.
In the following sections, I'll prove that likelihood with mathematical
analysis starting with our mathematics of the human emotions that will
eventually make clear in association with a mathematical explication of natural
selection how weapons overwhelm people and cause their freedom in life to be
taken away. And how that unhappiness in life from the loss of freedom is passed
on to innocent victims as a way of releasing it, much like my brother, Don,
passed it on to me and June and whomever he could in life as a result of the
great control my mother imposed on him, which he did not rebel against like I
did. Here’s a couple of photos.
This is me, Ruth, with my genius grandson, Thomas, and his mom, June, down in Acapulco shortly before we returned to campaign for Obama in 2008. We were sold a brave Obama fighting for the people and spent $5000 to help get him elected.


CBS photo of the kids at the Las Vegas Occupy March four disappointing Obama years later before the Occupy movement was destroyed by 6000 slamtotheground wristbreaking arrests. To support our movement to end the NSA run police state and to disarm the world by electing me president (or by radical protest if needed) click here. . 
4. The Mathematics of Emotion
The notion of condensed representation developed in the entropy section is essential to understanding our basic emotions of desire, hope, anxiousness, disappointment, excitement, fear, dismay, security, depression and relief in a firm mathematical way and to provide an analytical foundation for what we are calling our visceral emotions like hunger, anger, sex and love. .
In
a game called Lucky Numbers we’ll use to develop the basic human
emotions, the lucky numbers are 4, 7 and 10. If you roll one of them you
win V=$12. We use this chart from Encyclopedia Britannica to evaluate the
probabilities of rolling these lucky numbers.
There are 36 ways a pair of dice can fall with 3 of those 36 ways resulting in a 4 with probability of 3/36=1/12; 6 ways out of 36 for a 7 with probability of 6/36=2/12; and 3 ways out of 36 for a 10 with probability of 3/36=1/12. Hence the probability, Z, of rolling a lucky number of a 4, 7 or 10 is the sum of their individual probabilities as
66.) Z=1/12 +2/12 +1/12 =4/12=1/3
This calculates the probability of rolling a number other than a 4, 7 or 10 lucky number as
67.) U=1− Z=2/3
U is also understood as the improbability or uncertainty in rolling a 4, 7 or 10. The amount of money one can anticipate winning on average in this V=$12 prize game is
68.) E=ZV=(1/3)$12=$4
E is the expected value of the game, the average amount you win per game. If you play the dice game repeatedly you can expect to win V=$12 on average or one time in three for an average payoff of E=$4 per game.
The E expected value is understandable in the broadest way the mind works as a condensed representation. Consider that you play the lucky numbers game twelve times with the following dollar payoffs laid out in number set form as ($12, 0, 0, 0, 0, $12, $12, 0, $12, 0, 0, 0). The average payout per game has been arranged here to be $4, that is to say, the E=$4 expected value.
This way of understanding the E expected value as the average or condensed representation of past experiences projected to future play is valuable because it is the way that the mind generally operates to consider future events in terms of similar events in the past condensed in the amount of information by taking their average values. The mind does this in many ways, a most general example being the way we develop expectation from the words we use.
The word dog, for example, conjures up a picture of what to expect when one encounters a dog that is the average of all the dogs that person has ever come across including in books and in the movies. This dependence on expectation from past experience is clear in the slightly different sense Mexicans have of a dog when they use their word for it, perro, for Mexican dogs tend much more to run wild and be a bit snarly as one sees on Mexican beaches and less beloved as pets than they are in the United States. Their expectation of what will greet a person if they’re told a dog will be coming into their life is different for Mexicans because the past experiences on which they base in average form the condensed representation word for a dog is different.
With the Lucky Numbers game we have taken a shortcut of developing the E expected value not from past experiences as is the general case for the human mind, but from mathematical formulae in elementary probability theory. But our most basic sense of expectation via condensed representations of past experiences should be remembered at all times especially when we are considering the emotions that derive from the E expected value. This is in line with most people not reckoning future possibility mathematically but from past experience represented in a reduced or condensed way as includes the condensed representations of others told to them. We will get into the ramifications and nuances of condensed representations, thoughts or ideas, and how they’re used after we develop the human emotions most efficiently and clearly with mathematical analysis.
The U=2/3 uncertainty in Eq67 tells you that on average you will fail to win two games out of three with the U=1−Z function enabling us to write the expected value, E, of Eq68 as
69.) E=ZV=(1−U)V=V−UV
The E expected value written as E=V− UV has two components, V and – UV. The V term in E=V− UV is the amount of money one anticipates or desires or wishes to win as a measure of the pleasure felt in desiring or wishing to get V dollars, the goal of the behavior of rolling the dice. This pleasure in desire is greater the greater the V dollar prize one desires to win. If we raise the prize to V=$120, the pleasure in its anticipation is greater.
This V pleasure in anticipating V dollars is accompanied in E=V−UV by a −UV term that is a measure of the displeasure of anticipating failure to win the V dollars. It goes variously under the everyday language name of anxiousness or anxiety or fear or concern or worry about winning and we’ll also give it a technical name of meaningful uncertainty, uncertainty, U, made meaningful by association with V dollars in –UV, money being a generally meaningful or valuable thing for all sane adults
That this feeling of anxiousness over winning is unpleasant is indicated by the minus sign of –UV with its displeasure greater the greater the U uncertainty in winning and the greater the V number of dollars one is anxious or uncertain about winning. In the roll of the 4, 7 or 10 for V=$12 with uncertainty of U=2/3, the displeasure intensity of the –UV anxiousness is measurable in dollar terms as
70.) −UV= − (2/3)($12)= −$8
This –UV= −$8 translates in everyday language to the anticipation of not getting the V=$12 one wishes for, which reduces the pleasure in the V=$12 wish to
70a.) E=V−UV=$12−$8=$4
Note that the –UV anxiousness felt is more unpleasant when a larger prize of V=$120 is offered and desired.
71.) −UV= −(2/3)($120)= −$80
And also note that the −UV feeling of anxiousness is greater when the U uncertainty of winning is greater as when only a roll of the 4 with probability of Z=1/12, and uncertainty of U=1− Z=11/12 is the lucky number that wins the V=$120 prize.
72.) −UV= − (11/12)($120)= −$110
This appreciation of −UV understands E=ZV=V−UV as a reduction in one’s pleasant anticipatory desire or wish to win V=$120 by a arithmetic reduction in V=$120 desire by the –UV anxiousness or difficulty in winning, which also results in a Z fractional reduction of V in one’s probability tempered E=ZV hopes of getting the V=$12 prize.
The pure V wish to get the cash prize is, by itself in isolation from the −UV difficulty or uncertainty or improbability of getting the V dollars, wishful thinking, the V term in E=V−UV when the –UV term of realistic difficulty in getting the V prize is ignored, the measure of the pleasure felt in pure wishful thinking.
Young children’s expectations for noncash desirables are dominated by simple desire and wishing as represented by V independent of worry or anxiety as represented by –UV in E=V−UV because parents provide what children desire or wish for with no or minimal uncertainty felt by the child. This sense of expectation in children of E=V, expectation as wish sure to be granted with no –UV meaningful uncertainty gradually develops to E=ZV=V−UV as the child grows more independent and matures and becomes aware of the difficulty or uncertainty in getting what it wishes for when it has to help get what it wishes for.
That is, a child’s existence is generally one of wishful thinking that is successful via the child’s wishes being granted by the parent/s with no or minimal uncertainty for the child. Such wishful thinking of a child takes on its familiar negative overtone in an adult who disregards the –UV meaningful uncertainty that is a realistic part of getting what you want or wish when you have to get it for yourself.
The other general category of behavior is activity directed to the goal of avoiding losing something of value. It is crisply illustrated with a dice game that one is forced to play that exacts a v=$12 penalty (lower case v) if you don’t roll a 4, 7 or 10 lucky number. The probability of incurring the penalty by failing to roll a 4, 7 or 10 or the uncertainty in avoiding the penalty is U= (1− Z)=2/3. The E expected value of paying the v=$12 penalty is, hence,
73.) E= −Uv= −(2/3)($12)= −$8
This is the average penalty you would pay when playing the game repeatedly. It tells us, for example, that if you play three of these penalty games, on average, you will escape the v=$12 penalty one time out of three by rolling the 4, 7 or 10 and you will pay the v=$12 penalty two times out of three, $24 in total for an on average penalty per game of E= − $8. The U=1− Z uncertainty of Eq67 allows us to write E in Eq73 in an alternative way as
74.) E= −Uv = −(1–Z)v = −v + Zv
The − v term in Eq74 is the anticipation of the entire –v penalty, which we will call dread of the penalty for want of a better common word for it. The displeasure in dread of paying the entire penalty is specified in the negative sign of − v with the intensity of the displeasure of this dread greater, the greater the v dollar penalty. Were the penalty raised to −v= − $120, the dread and its displeasure is greater than that for the –v= −$12 penalty.
The −v dread in E= − v + Zv of Eq74 is reduced by the +Zv term in it as the pleasant hopes one has that one will escape the penalty by rolling a 4, 7 or 10 to probability Z=1/3. This Zv term is also understandable as the security one has that one will avoid the penalty, the greater the Z chance of escaping the penalty in +Zv and the greater the v penalty, the greater the pleasant feeling of security when one is forced to play the penalty game.
The Zv hopes in E= − Uv = − v + Zv of avoiding the –v penalty reduces the displeasure of the − v dread of the penalty to bring about the –Uv probability tempered fearful expectation of incurring the penalty. This − Uv fear of something unpleasant happening goes by a number of other names in everyday language including anxiety, worry, distress and concern, which has us label –Uv fear as we did −UV anxiousness as a form of meaningful uncertainty. Note for E= −Uv fear that one’s unpleasant feeling of fear is greater the greater the U probability of the loss happening and the greater the v penalty imposed as fits universal emotional experience.
The above gives us functions for three more basic emotions: the − v dread of incurring a v penalty; the Zv security one feels of escaping the penalty; and the probability tempered E= −Uv fear of incurring the penalty. These add to the V desire of getting a V prize, the –UV anxiousness about getting it and the Z probability tempered hopes of getting a prize to give a complete set of our general anticipatory emotions.
It should be emphasized that −UV, ZV, V, −Uv, Zv and –v are the correct names for our anticipatory emotions rather than the more familiar words for them of anxiety, hope, desire, fear, security and dread and so on. Ludwig Wittgenstein, regarded by many as the greatest philosopher of the 20^{th} Century, made the point well in his masterwork, Philosophical Investigations, of the inadequacy of everyday language to describe the mental states of ourselves and others. Words for externally observable things like “wallet” are clear in meaning because if any confusion in meaning arises, one can point to a wallet to make clear what is meant by the word. With emotions, however, nobody feels the emotions of another person, so the familiar words we use to describe the emotions have no referent one can point at to clarify its meaning. You can’t point to somebody’s anxiety, hope, desire, fear, security or dread to make clear the meaning of these words other than in an indirect, imprecise way from how people may show their emotions, which is highly error prone as regards how they actually feel from intention and/or incompleteness.
The words in mathematical language of −UV, ZV, V, −Uv, Zv and –v, on the other hand, are perfectly clear in meaning because their symbols have specific countable referents of money gained or lost and the pleasures and displeasures generated and of probability. And the fit of these mathematical words to emotional experience is universal. That is, all people feel these −UV, ZV, V, −Uv, Zv and –v anticipatory feelings when playing the V prize and v penalty dice games. Hence quibbling over the “correct” everyday language names to call −UV, ZV, V, −Uv, Zv and –v is not a valid criticism of this analysis because the root of the problem of understanding emotion is to begin with the inadequacy of everyday language to describe emotion.
Mathematical language is superior to everyday language not only in the unequivocal meaning of its number and symbol words, something the fellow up there in the No Spin Zone at Fear and Balanced Fox News can appreciate, but also in the trustworthiness of the logical relationships of mathematics and the conclusions they reach as seen in Euclidean Geometry and Newtonian Mechanics. This logic and clarity of mathematics in geometry and the physical sciences points up the value of putting the human sciences on a mathematical foundation. Indeed, this is entirely necessary given the inability of current psychology to provide any clear understanding of normal emotion and but a poor one for the abnormal emotions. And that makes understanding socioeconomic control as the cause of our unpleasant emotions of unhappiness, dubbed “mental illness” in the ideologically corrupt pseudoscience of clinical psychology, impossible, a problem that Wishful Thinking rectifies mathematically by the end of this analysis.
So far we have considered just anticipatory emotions, the feelings felt prior to rolling dice to win money or avoid losing it. Next we will consider the emotions felt after the dice are thrown and a lucky number is seen or not seen on them. In the prize game one wins V dollars by rolling a lucky number to feel a pleasant emotion we’ll call delight or most generally elation, an “up” feeling specified mathematically as R=V. The greater the V amount of dollars won, the greater the R=V delight. The R symbol is used to specify the R=V delight in winning V dollars as a realized emotion, one that comes about from something that actually happens or is realized rather than from one expecting something to happen as was the case with the anticipatory emotions we just considered.
For the v penalty game, the realized emotion felt when the penalty is incurred or realized from failure to roll a lucky numbers is R= − v. This is an unpleasant feeling of sadness or most generally of depression, a “down” feeling that is more unpleasant the greater the v money loss. Note that depression here is not defined as a disease or mental illness but rather as an unpleasant emotion whose origin lies clearly in losing something of value. It is also possible in the V prize game to fail to win the money. In that case, as nothing is realized, no money changing hands, there is no realized emotion, which is specified by R=0. And it is possible in the v penalty game to not lose any money, which also has no realized emotion as specified by R=0.
This is not to say there are no emotional consequences from the R=0 failing to win the V prize or the R=0 escaping the v penalty. Specifically they constitute a third class called transition emotion, symbol, T, that is neither E expectation nor R realized emotion but the arithmetic difference between R and E.
75.) T = R−E
This equation is called the Law of Emotion because it holds generally for E=ZV hopeful expectation and for E= −Uv fearful expectation and for all manner of realized outcomes of R=V, R= −v and R=0. With hopeful expectation of winning a V prize as E=ZV, when a lucky number is not rolled, the V prize not won and the realized emotion R=0, the T transition emotion is
76.) T = R −E = 0 −ZV = −ZV
This T= − ZV emotion is the disappointment felt when one’s hopes of winning the V prize, E=ZV, are dashed or negated. Disappointment is specified as unpleasant from the minus sign in T= − ZV and its displeasure is seen to be greater, the greater is the V size of the prize not won and the greater the Z probability the player felt he had to win it. In the dice game for the V=$12 prize won by rolling a 4, 7 or 10 with probability Z=1/3, the intensity of the disappointment is
77.) T = −ZV = −(1/3)($12) = −$4
The T= −$4 cash value of the disappointment indicates that the intensity of the displeasure in it is equal in magnitude if not in all its nuances to losing $4. The T= − ZV disappointment over failing to win a larger, V=$120, prize, is greater as
78.) T= − ZV= − (1/3)($120)= − $40
Eqs77&78 fit with the universal emotional experience of disappointment being greater the more you hoped you’d get but didn’t. The T= − ZV disappointment is also great when the Z probability you felt of winning the V prize is great. Consider a dice game where every number except snake eyes, the 3 through12, is a lucky number that wins the V=$120 prize. These lucky numbers have a high probability of Z=35/36 of being tossed, so the hopes of winning are great as
79.) E = ZV = (35/36)($120)= $116.67
And the disappointment from failure when the ZV hopes are dashed or negated to –ZV is also great as
79a.) T= −ZV = − (35/36)($120)= − $116.67
This fits the universal emotional experience of people feeling great disappointment when they have a high expectation of success and then fail. And at the other end of the spectrum people feel much less disappointment when they have a very low expectation of success to begin with. As an example, consider the T= − ZV disappointment in a dice game where to win you must roll snake eyes, the 2, as the only lucky number. With such a low probability of winning felt of Z=1/36, the disappointment is much less as
80.) T= ZV= − (1/36)($120)= −$3.33
This amount of disappointment is much less than for the dice games of Eqs78&79a because of the low expectation of winning of E=ZV=(1/36)($120)=$3.33 to begin with. We see in the broader picture that though there is no realized emotion when one fails to win in a V prize game, R=0, there is still felt the T= − ZV transition emotion of disappointment.
Now let’s consider the T transition emotion that arises when one does win the V dollar prize with a successful toss of the dice. With hopeful expectation as E=ZV and realized emotion as R=V, the T transition emotion is from the Law of Emotion of Eq75, T=R−E, via U=1−Z,
81.) T = R−E = V −ZV = (1− Z)V = UV
This T= UV transition emotion is the thrill or excitement of winning a V prize under uncertainty. It is a pleasant feeling as denoted by the positive sign of UV with its pleasure greater, the greater is the V size of the prize and the greater the U uncertainty of winning it felt beforehand. When one is absolutely sure of getting V dollars as in a weekly paycheck with no uncertainty, U=0, there is still the R=V delight in getting the money. But with uncertainty present, U>0, there is an additional thrill or excitement in winning the money as in winning the lottery or winning a jackpot in Las Vegas or winning V=$120 in the dice game by rolling a lucky number of 4, 7 or 10. In the latter case with an uncertainty of U=2/3 from Eq67, the intensity of the excitement in winning is from Eq81
82.) T=UV=(2/3)($120)=$80
That this additional pleasure of excitement in getting V dollars over and above R=V depends on prior U uncertainty is made clearer if we look at trying to win V=$120 by rolling the dice in a game where rolling only the 2 with probability Z=1/36 and uncertainty U=35/36 wins the prize. Here if you do win, as with winning in any chance game when the odds are very much against you, there’s that much more of a thrill or feeling of excitement in the win.
83.) T=UV=(35/36)($120)=$116.67
By comparison consider a game that awards V=$120 for rolling any number 3 through 12 with Z=35/36 probability of winning and low uncertainty of U=1−Z=1/36 as makes the player quite expect to win the money. While there is still the R=V=$120 delight in getting the money, there is much less thrill because like getting a paycheck, the player was almost completely sure of getting the money in this particular Z=35/36 dice game.
84.) T=UV=(1/36)($120)=$3.33
This relationship between the uncertainty of getting something of value and the excitement or thrill felt when you do get it is clear in the thrill children feel in unwrapping their presents on Christmas morning. The children’s uncertainty about knowing what they’re going to get in the wrapped presents is what makes them feel that thrill in opening them up. This excitement is an additional pleasure on top of the pleasure from the gift itself, that special thrill in opening the presents under the Christmas tree not being felt if the youngsters know ahead of time what’s in their wrapped packages and have no uncertainty about it.
It is also universal that winning a V=$120 prize is more thrilling than winning a V=$12 prize when the U uncertainty in winning is the same in both cases. We see the UV excitement in the V=$12, 4, 7 or 10 lucky number game to be significantly less than for the V=$120 prize in Eq82 as
85.) T=UV=(2/3)($12)=$8
We get a fuller picture of our emotional machinery by deriving the T=UV thrill of a win from the T=R−E Law of Emotion of Eq75 with the E=ZV expectation expressed from Eq69 as E=ZV=(1− U)V=V− UV.
86.) T = R− E =V−(V−UV) = − (−UV)=UV
This derivation of T=UV as T= − (− UV)=UV sees the T=UV excitement as the negation or elimination of − UV anxiousness via a successful outcome. Adventure movies generate their excitement or thrills for the audience in just that way by being loaded with anxiousness or dramatic tension from the hero’s initial and/or continuously meaningfully uncertain situation, which the audience feels vicariously. When the hero’s meaningful uncertain situation is resolved by success towards the end of the movie, it vicariously brings about thrills and pleasurable excitement for the audience by negating or eliminating the anxiousness they felt about the hero’s situation. Though the emotions felt by the audience are vicarious, the essence of the dynamic is essentially the same as is spelled out above in Eq86.
Now let’s consider the T transition emotions felt in the v penalty exacting game. With a fearful expectation of E= − Uv, when the v penalty is avoided by rolling the 4, 7 or 10, the realized emotion is R=0 and the T transition emotion from the Law of Emotion of Eq75 is
87.) T = R−E = 0 − (−Uv) = Uv
This T=Uv emotion is the relief gotten from escaping the v dollar penalty when you roll one of the lucky numbers. The positive sign of T=Uv specifies relief as a pleasant emotion with the pleasure of relief greater, the greater is the v loss avoided and the greater the U improbability of avoiding the loss. The T=Uv relief felt when a 4, 7 or 10 lucky number is tossed in the v=$120 penalty game is with U=2/3 from Eq66
88.) T=Uv=(2/3)($120)=$80
But if one plays a v=$120 penalty game where rolling only the 2 with uncertainty U=35/36 avoids the penalty, there is greater relief in rolling the 2 and avoiding the loss because you felt prior to the throw with U=35/36 that most likely you would lose.
89.) T=Uv=(35/36)($120)=$116.67
But if you play a v=$120 penalty game that avoids the penalty with any number 3 through 12 and an uncertainty of only U=1/36 of avoiding the penalty, there is much less sense of relief because you felt pretty sure you were going to avoid the penalty to begin with.
90.) T=Uv=(1/36)($120)=$3.33
Note also that the larger the v penalty, the more relief there is in avoiding it as with a v=$1200 penalty, the relief in escaping it the 2 lucky number game with U=35/36 being that much greater than for the v=$120 penalty game of Eq88.
91.) T=Uv=(35/36)($1200)=$1166.67
The universal fit of mathematically derived Uv relief to the actual emotional experience of felling relief is remarkable. Lastly we use the Law of Emotion of Eq75 of T=R−E to obtain the T transition emotion felt when the player does incur the v penalty by failing to roll a lucky number. In that case, with expectation of E= − Uv and a realized emotion of R= − v, the T transition emotion is via Z=1− U
92.) T = R − E = −v − (−Uv) = −v+ Uv= −v(1−U)= −Zv
This T= − Zv transition emotion is the dismay or shock felt when a lucky number is not rolled and the v penalty is incurred. The T= − Zv emotion of dismay is an unpleasant feeling as implied by its negative sign and one greater in displeasure, the greater the Z probability of escape one supposed before rolling. Its value for the 4, 7 and 10 lucky number v=$120 penalty game with Z=1/3 from Eq66 is
93.) T= − Zv = − (1/3)($120)= − $40
But if you have a small Z probability of avoiding a v=$120 dollar loss as in the dice game with only rolling the 2 providing escape to probability, Z=1/36, there is little of this − Zv dismay when you fail because you had such low Zv hope of escape to begin with.
94.) T= − Zv = − (1/36)($120)= − $3.33
This low dismay from failure given low expectation is why many people develop low expectations in life to avoid the unpleasant feeling of dismay if they do fail. This contrasts to the considerable T= − Zv dismay or shock felt in v=$120 penalty game where the numbers on the dice of 3 through 12 are the lucky numbers that avoid the v penalty to probability, Z=35/36. Then if you roll the 2, the only losing number, and must pay the penalty, the intensity of the dismay when you lose is great because you did not expect to lose to begin with.
95.) T= − Zv = − (35/36)($120)= − $166.67
Great dismay from a high Z=35/36 probability felt of escaping the v penalty is felt as shock from a person’s surprise at failure when what was expected from the high Z=35/36 probability was success in avoiding the penalty.
In brief review we now list the primary emotions that people experience in goal directed behavior demonstrated with a dice game whose goals are getting money and avoiding losing it: ZV, V, −UV, −Uv, −v, +Zv, −ZV, −Zv, UV, Uv, R=V, R= −v. We differentiate between V and R=V with the former understood as the pleasant wish or desire for V dollars, an expectation emotion, and the latter as the pleasant emotion of actually getting V dollars; and similarly between –v and R= −v. Whatever everyday language names we want to assign to these emotions, they are a complete set of what we will call the operational emotions. They will be augmented to a complete set of all the human emotions after we later include the visceral emotions, pleasant and unpleasant, like hunger, food taste, cold, warmth, anger, glory, sex and love.
For
now, though, we want to explain the purpose of the T transition emotions of
disappointment, excitement, relief and dismay in our emotional machinery.
Recall that they derive from the T=R−E Law of Emotion and the E expected
value function in it, which depends in a very direct way on the Z and U
probabilities whether as E=ZV in the V prize game or E= −Uv in the v
penalty game. In our analysis up to this point, the player’s sense of the values of the Z and U probabilities were
understood to be calculated correctly from the mathematics of throwing dice.
But that need not be the case. A player may suppose any probabilities
for success and failure, which will affect his or her E expectations, be they
hopes or fears, and in turn from the T=R−E Law of Emotion, the T
transition emotions experienced upon success or failure.
Let’s illustrate this with a Christian girl who makes a bet with her
nonbelieving fatherinlaw that she will play the v=$120 penalty game with
4, 7 and 10 as the lucky numbers three times and win it each time to
avoid the penalty. If she succeeds he promises to go to church every Sunday for
the rest of his life and if she fails she must pay him the penalties incurred.
The daughterinlaw with God on her side for sure success supposes a
probability of Z=1 for each roll instead of the actual Z=1/3 and a zero
probability of failure, U=1−Z=0, instead of U=2/3 of Eq67. Her
supposition of Z=1 and U=0 generate a fear of losing of
96.) E= −Uv= −(0)$120=0
This total optimism is at variance with the correct fear of losing she should have of
97.) E= −Uv= −(2/3)$120= −$80
As luck would have it, the first time the daughterinlaw rolls she does get one of the lucky numbers and escapes the v=$120 penalty. This gives her no sense of T=Uv relief because her perfect faith told her she couldn’t lose. This fits mathematically with her Z=1, U=0, supposition of the probabilities involved. She doesn’t feel the T=Uv=$80 relief of Eq88 that derives from the correct Z=1/3, U=2/3, probabilities, but rather feels no relief as derives from her Z=1, U=0, supposition as
98.) T=Uv=(0)$120=0
On the second roll, however, she fails to roll a lucky number and must pay the professor the v=$120 penalty. Her dismay as the −Zv in Eq92 is great.
99.) T= −Zv= −(1)$120= −$120
Note again that this emotion of dismay felt by her depends on her supposition of Z=1 and U=0 probabilities rather than on the correct Z=1/3, U=2/3, value. Her faith in God is not broken by this outcome, however, for she assumes the Devil must have intervened in some way and that God will surely guide the dice in her favor on the third roll so she can minimize the penalty money she must pay to her now smiling fatherinlaw.
On the third roll, the girl again misses a lucky number to get the on average outcome, which while not a surety is the most probable outcome. This ups the penalties she must pay to the professor by another v=$120 while generating another dose of the T= −$120 dismay of Eq99 in her. At this point her two heavy doses of T= −$120 dismay begin to emotionally question her faith in God, at least in His Divine power to overrule elementary probability theory. What she does not realize is that the T transition emotions she felt in her dice rolling are working on her mind subconsciously to alter her previous expectation based on the Z=1 and U=0 incorrect suppositions she was making about probability. This works on a formula that is a simple variation, as we shall show later, of the T=R−E Law of Emotion.
100.) E_{NEW} = E_{OLD }+ T
The daughterinlaw’s original or old expectation was from Eq96, E_{OLD}=0. The T transition emotion in the formula is the average of the T transition emotions felt, the T=Uv=0 relief felt once of Eq98 and the T= −Zv= −$120 dismay of Eq99 felt twice.
101.) T= (0 −$120 −$120)/3= −$240/3= −$80
Now inserting into Eq100 from Eq96, E_{OLD}=0 as her initial fear of losing and from Eq101, T= −$80 will generate her new fear of losing of
102.) E_{NEW} = E_{OLD }+ T= 0 −$80 = −$80
This is the correct expectation seen in Eq97 evaluated from the correct Z=1/3 and U=2/3 probability values that she should have had to begin with. This tells us that dismay from failure, T= −Zv, increases fearful expectation in subsequent play, E= –Uv. And from E= −Uv= −v+Zv of Eq74 we see that –Zv dismay from failure also decreases the Zv feeling of security in subsequent play.
Note that for the average T transition emotion to be T= −$80 in this example in a predictable way would have taken a much larger number of repeated rolls. This would have made the mathematics and the story significantly more tedious to tell, which is why we made the outcomes of the daughterinlaw’s three rolls artificially fit the average.
That clarified, it should be obvious that the daughterinlaw is engaging in wishful thinking in basing her E= −Uv=0 expectation of paying the penalty on U=0, no uncertainty of losing given the divine intervention of God. Let’s explain this by reference to wishful thinking we have already considered in the V prize game where we made clear that it entails ignoring the realistic –UV meaningful uncertainty in the E=V−UV expectation. Now let’s write the E= –Uv fearful expectation in the v penalty game a little differently as
103.) E= 0 −Uv.
This stipulates 0 or no penalty as the wish or desire in this game, paying no penalty. Wishful thinking occurs in the v penalty game by thinking that will happen for sure as by ignoring the realistic –Uv uncertainty stipulated in elementary probability theory as the Christian girl did by supposing no uncertainty, U=0, which renders the fearful expectation in Eq38 as E=0.
To demonstrate the generality of the E_{NEW }= E_{OLD }+T formula of Eq100 let’s turn now to another anecdote, this one about the husband of the Christian girl, the professor’s son. Junior is a total pessimist in situations that in any way involve risk or chance. Delusional in the opposite way of his born again wife, Junior turned out to be a psychological eunuch who submits to even unfair authority obediently to please his wife. Personality issues aside, when Junior plays the dice game that awards a prize of V=$120 for rolling the 4, 7 or 10 lucky number, he fearfully supposes the probability of winning to be less than the realistic Z=1/3 probability that generates an expectation of
104.) E=ZV=(1/3)$120=$40
Specifically he supposes that the probability of winning is only Z=1/12, uncertainty U=1−Z=11/12, which generates a hopeful expectation in him of only
105.) E=ZV=(1/12)$120=$10
Now we will have Junior play twelve of these games. From the realistic probability of winning of Z=1/3 we will take it that he succeeds in four of the twelve games, winning the V prize, and fails to win eight times. Assuming he retains his deflated probability suppositions for the entire 12 games, the excitement he feels in each of the four times he wins is from T=UV of Eq16 with his supposition of uncertainty of U=11/12,
106.) T=UV=(11/12($120)=$110
And the disappointment he experiences each of the eight times he doesn’t win is from T= −ZV of Eq11 with his supposition of Z=1/12,
107.) T= −ZV= −(1/12)$120= −$10
His average T transition emotion per game for the twelve games is, thus,
108.) T= [4($110) −8($10)]/12=$360/12=$30
His average transition emotion is T=$30 of excitement. After twelve games and these emotional outcomes Junior comes to feel that his chances of winning in a game are greater than he initially supposed because of the predominance of the T=UV=$110 excitements over the T= −$10 disappointments. Specifically his mind operates according to Eq100 to increase his E=ZV expectation in subsequent play via E_{OLD}=$10 from Eq40 and T=$30 in Eq108 to
109.) E_{NEW} = E_{OLD }+ T= $10 + $30 = $40
We see that Eq109 generates the correct E=$40 expectation of Eq104 from trial and error experience. Generally speaking predominant T=UV excitement increases E=ZV hopes and from E=V−UV decreases –UV anxiousness about winning. To see the general pattern, recall that predominant T= –Zv dismay from failure increased one’s E= −Uv fear of losing in the penalty game and from E= −v+Zv decreased one’s Zv security. This tells us from Eq105 without our having to going through mathematical examples of it that predominant T=Uv relief from success in the v penalty game decreases one’s E= −Uv fear and increases one’s Zv security in subsequent plays; and that predominant T= −ZV disappointment from failure in the V prize game decreases one’s E=ZV hopes and increases one’s –UV anxiousness.
This makes clear the function of the T transition emotions, namely to bring one’s E expectations realistically in line with actual experience. This seems unnecessary in these dice games where simple mathematical calculations give immediate correct knowledge of the Z and U probabilities without the need for prior experience. But man’s mind did not evolve to play dice games but rather to survive, reproduce and compete in an often uncertain environment where probabilities of success in such activities were centrally important for success in life. Under those circumstances a priori exact values for the probabilities of success are seldom known and can only be supposed and shaped by actual trial and error experience with cultural transmission of the probability values from other individual’s experiences also taken into account.
The importance of having realistic expectations that reflect experience lies in man making decisions on what to do on the basis of his expectations. If a person has the opportunity to play a prize awarding dice game for V=$120 either with 4, 7 and 10 as the lucky numbers (Z=1/3 and E=$40) or with 4 and 7 as the lucky numbers, (Z=1/4 and E=$30), he or she chooses the E=$40 game because in its having a higher average payoff, there is greater pleasure in its expectation. Or similarly if a person must choose between playing one of two v penalty games, one with an expectation of E= −$60 and the other E= −$80, he chooses the former in this Hobson’s choice as “the lesser of two evils” because in incurring a lesser penalty its expectation is less unpleasant. Clearly, having unrealistic expectations that don’t fit actual Z and U probability values make one choose badly with consequences of less pleasure and more displeasure than could have been had.
Now we have given a thumb nail sketch of how the mind works in terms of three classes of emotions, E expectations, R realized emotions and T transition emotions, and how they relate to each other through the formulas of Eqs75&100. Its simplicity, mathematical clarity and generality strongly suggest that is truly how the mind works. Calling it a “theory of the mind” as implies the possibility of competing theories is as misleading as calling the Law of Gravity a theory of how our solar system works rather than a clear and correct description of it. Whatever may seem missing in the larger picture of how the mind works will be filled in.
We also want to make clear that this mathematical explication of the emotions is effectively empirical in being universal. Our specification of disappointment as the −ZV negation or dashing of ZV hopes, for example, is universal in that all human beings feel disappointment when they fail to achieve a desired goal. Such universal agreement is the fundamental factor in empirical validation. When ten researchers all read the same data off a laboratory instrument, that data is taken to be empirically valid because all ten agree on what they see. This criterion for empirical validity of universal agreement extends to the emotions of the dice game laid out as what all people would feel if they played it.
5. The Law of Supply and Demand
Though this is somewhat of a veering off from our main line of thought, as the
Law of Supply and Demand is the foundation of all free market economics denied
by nobody sane, deriving it from the Law of Emotion proves the controversial
conclusions of this analysis. We begin our derivation with first explaining
the emotions people feel from partial success. To do that, we alter our dice
game to one where you must roll a lucky number of 4, 7 or 10 not once but
three times to win the prize, one of V=$2700. These three rolls may be
with three pair of dice rolled simultaneously or with one pair of dice tossed
three times in succession. As the probability of rolling a lucky number in any
one of the three pairs or three rolls is from Eq1, Z=1/3, so the probability of
rolling a lucky number on the 1^{st} pair of dice or the 1^{st}
roll is Z_{1}=Z=1/3; on the 2^{nd} pair of dice or roll, Z_{2}=Z=1/3;
and on the 3^{rd} pair of dice or roll, Z_{3}=Z=1/3.
110.) Z_{1}=Z_{2}=Z_{3}=Z=1/3
And the uncertainties in each toss are
110a.) U_{1}=U_{2}=U_{3}=(1−Z)=2/3
Hence the probability of rolling a lucky number of 4, 7 or 10 on all three rolls is the product of the Z_{1}, Z_{2} and Z_{3} probabilities, which is given the symbol, z, (lower case z).
111.) z = Z_{1}Z_{2}Z_{3 }= Z^{3 }=(1/3)^{3 }= 1/27
And the improbability or uncertainty of making the triple roll successfully is
112.) u=1–z = 26/27
The expected value of this game as a measure of the player’s hopes of winning the V=$2700 is
113.) E = zV =V−uv=Z_{1}Z_{2}Z_{3}V_{ }= (1/27)($2700) = $100
The displeasure of disappointment from failure to make the triplet roll is from the T=R−E Law of Emotion of Eq75 with R=0
114.) T=R−E = 0−zV= −zV= −(1/27)($2700)= −$100
And the pleasure of the excitement or thrill in making the triplet roll with R=V=$2700 is
115.) T=R−E = V−zV=(1−z)V= uV= −(−uV)=(26/27)($2700)=$2600
Next we derive the emotion felt from rolling a lucky number on a 1^{st} throw of three sequential throws. After a 1^{st} toss that rolls a lucky number, the probability of winning the V=$2700 prize by tossing lucky numbers on the next two rolls is
116.) Z_{2}Z_{3 }=(1/3)(1/3) = 1/9
And the hopeful expectation of making the triple roll after a lucky number is rolled on the 1^{st} toss is, as increased from E=Z_{1}Z_{2}Z_{3}=$100
117.) E_{1 }= Z_{2}Z_{3}V = (1/9)($2700) = $300
Next we want to ask what is realized when the 1^{st} toss is successful. It is not R=V=$2700, for the V prize is not awarded for just getting the first lucky number. And it is not R=0, what is realized when the player has failed to make the triplet roll and win the V=$2700 prize. Rather what is realized when the 1^{st} roll is of a lucky number is the E_{1}=$300 increased expectation in Eq117. This understanding of the E_{1}=$300 expectation as what is realized has us specify E_{1} understood as a realization in terms of the R symbol as
118.) E_{1}=R_{1}=Z_{2}Z_{3}V = $300
Now we use the Law of Emotion of T= R−E, of Eq75 to obtain the T transition emotion that arises from a successful 1^{st} toss. This understands the T term in T=R−E as T_{1}; the R term in it as R_{1}=Z_{2}Z_{3}V from Eq118; and the E term in it as the expectation had prior to the 1^{st} toss being made, E=Z_{1}Z_{2}Z_{3}V of Eq113. And with U_{1}=(1−Z_{1}) from Eq110a we obtain T_{1} as
119.) T_{1 }= R_{1}–E = E_{1}– E= Z_{2}Z_{3}V –Z_{1}Z_{2}Z_{3}V = (1−Z_{1})Z_{2}Z_{3}V =U_{1}Z_{2}Z_{3}V =(2/3)(1/3)(1/3)($2700) =$200
Now we ask what kind of emotion this T_{1}=U_{1}Z_{2}Z_{3}V transition emotion is. We answer that question by noting that the T=uV excitement of Eq115 from making the triplet toss and winning the V=$2700 prize can be written, given R=V, as
120.) T=uV=uR
And we see that we can write Z_{2}Z_{3}V=R_{1}=E_{1} from Eq103 in the T_{1}=U_{1}Z_{2}Z_{3}V transition emotion of partial success in Eq119 as
121.) T_{1 }=E_{1}−E=U_{1}Z_{2}Z_{3}V =U_{1}E_{1}=U_{1}R_{1}
The parallel in form of T_{1}=U_{1}R_{1} to the T=uR excitement of Eq120 identifies T_{1} as excitement, the excitement felt in rolling the 1^{st} lucky number, excitement that is felt even though no money is awarded for just rolling the 1^{st} lucky number. Note that the intensity of this partial success excitement of T_{1}=$200 of Eq119 is much less than the T=$2600 excitement of Eq50 that accrues from making the triplet roll and getting the V=$2700 prize.
The above development of excitement from partial success from the T=R−E Law of Emotion is borne out from observation of situations that go beyond the dice game. Excitement from partial success, indeed, is routinely observed on TV game shows like The Price is Right where a contestant is seen to get visibly excited about entry into the Showcase Showdown at the end of the show, which offers a large prize, by getting the highest number on the spinoff wheel first, which offers no prize in itself. This and other observed examples of the partial success excitement we just derived from the Law of Emotion is a form of empirical if not perfectly measurable empirical validation of The Law of Emotion.
We further validate the Law of Emotion by deriving from it the excitement felt in rolling the 2^{nd} lucky number in the triplet roll once the 1^{st} lucky number has been rolled. We saw that what is realized from getting the 1^{st} lucky number is an increase in the expectation of winning the R=V prize from E=zV=$100 in Eq113 to E_{1}=Z_{2}Z_{3}V=$300 in Eq118. What is realized from rolling the 2^{nd} lucky number after the 1^{st} is a greater expectation yet of making the full triplet roll and winning the V=$2700 prize,
122.) R_{2}= E_{2 }=Z_{3}V=(1/3)($2700)=$900
The transition emotion that comes from rolling 2^{nd} lucky number after having gotten the 1^{st} is specified as T_{2 }and from the Law of Emotion, T=R−E, with T as T_{2}, R as R_{2}=Z_{3}V in the above and E as E_{1}=Z_{2}Z_{3}V in Eq118, the expectation prior to the 2^{nd} lucky number being rolled, is
123.) T_{2} = R_{2}−E_{1
}= E_{2}−E_{1}=Z_{3}V− Z_{2}Z_{3}V
=(1−Z_{2})Z_{3}V = U_{2}Z_{3}V =
(2/3)(1/3)($2700) = $600
Expressing T_{2 }= U_{2}Z_{3}V via Z_{3}V=R_{2}
of Eq57 as T_{2}=U_{2}R_{2} makes it clear from its
parallel form to the excitement of T=uR of Eq120 that T_{2}=U_{2}R_{2}
is the excitement felt when the 2^{nd} lucky number is guessed after
the 1^{st} lucky number has been rolled.
And we can also use the Law of
Emotion, T=R−E, of Eq75 to derive the excitement felt in getting the 3^{rd}
lucky number after getting the first two to win the V=$2700 prize. What is
realized in that case is finally the V prize, R=V. Given the expectation
that precedes getting the 3^{rd} lucky number of E_{2}=Z_{3}V
from Eq122, the Law of Emotion, T=R−E, obtains a T_{3 }transition
emotion of
124.) T_{3 }= R−E_{2 }= V –Z_{3}V
= (1−Z_{3})V = U_{3}V = (2/3)($2700) = $1800
Expressing T_{3 }= U_{3}V from R=V as T_{3}=U_{3}R and its parallel form to T=uR excitement of Eq120 identifies T_{3 }= U_{3}R as the excitement of rolling 3^{nd} lucky number after the first two are rolled to obtain the V=$2700 prize. Note that the intensity of the T_{3}=$1800 excitement from rolling the 3^{rd} lucky number and getting the V=$2700 prize is significantly more pleasurable than the T_{1}=$200 and T_{2}=$600 excitements for the antecedent partial successes.
Such significantly greater excitement in actually winning the prize than from achieving prefatory partial successes is what is observed in game shows like “The Price is Right” where actually winning the Showcase Showdown has the winner jumping up and down and running around screaming showing much more excitement than the excitement had from the prefatory partial success of getting into the Showcase Showdown by getting the highest number on the spinning wheel. Again this fit of observed excitement in the approximate relative amounts suggested by the above derivations from the Law of Emotion constitutes an empirical if not perfectly measurable empirical validation of the Law of Emotion.
Also note that the Law of Emotion, T=R−E, of Eq75 is further validated from the three partial excitements, T_{1}, T_{2} and T_{3} of Eqs119,123&124 summing to the T=uV=$2600 excitement of Eq115 gotten from making the triplet roll in one fell swoop as you might from throwing three pair of dice simultaneously.
125.) T_{1 }+ T_{2 }+ T_{3 }= $200 + $600 + $1800 = $2600 = T = uV
It is also instructive to calculate what happens when you roll the first two lucky numbers but then miss on the 3^{rd} roll and fail to get the V=$2700 prize, R=0. To evaluate the T_{3} transition emotion that arises from that we simply use the T_{3}=R−E_{2} form of the Law of Emotion in Eq124 that applies after the first two lucky numbers are gotten, but with the R realized emotion as R=0 from failure to obtain the V=$2700 prize.
126.) T_{3 }= R−E_{2 }= 0 – Z_{3}V = – Z_{3}V = −(1/3)($2700)= −$900
This T_{3 }= −Z_{3}V= −$900 is the measure of disappointment felt in failing to make the triplet roll after getting the first two lucky numbers and experiencing the prefatory partial success excitements in doing so. Note that this T_{3}= −Z_{3}V= −$900 disappointment is a significantly greater disappointment than the T=−zV=−$100 disappointment of Eq114 that comes from failure to roll the lucky numbers in one fell swoop.
Note that the −$800 (negative) increase in the T_{3}= −$900 disappointment relative to the T= −$100 disappointment felt without partial success is equal to T_{1}+T_{2}=$200+$600=$800 sum of the two partial success excitements of Eqs119&124 gotten from rolling the first lucky numbers before failing in the 3^{rd} roll. This understands the additionally unpleasant −$800 disappointment from failure in the 3^{rd} roll rescinding or negating the prefatory $800 pleasant excitements that were followed by ultimate failure. This fits universal emotional experience of an increased let down or disappointment when initial partial success is not followed by ultimate success in achieving a goal as the letdown disappointment gotten when one counts their chickens before they hatch and they fail to hatch.
The sequential scenarios that end in success in Eq125 and in ultimate failure in Eq126 universally fit people’s emotional experience and as such are a convincing validation of the Law of Emotion, T=R−E, of Eq75. The linear sums and differences of the transition emotions in these two instances also importantly show that our emotions reinforce each other positively and negatively in a linear fashion via simple addition and subtraction of the emotion intensity values.
One can do the same analysis for a v penalty game. Consider a dice game one is forced to play that exacts a penalty of v=$2700 unless the player rolls three lucky numbers, a lucky number being a 4, 7 or 10, whether simultaneously with three pair of dice or sequentially on one pair of dice. In parallel to the E expectation of Eq73 with u=26/27 in Eq112 as the improbability of rolling the three lucky numbers, the fearful expectation of incurring the v=$2700 penalty is
127.) E= −uv= −(1−Z_{1}Z_{2}Z_{3})v= −(26/27)($2700)= −$2600
The Law of Emotion, T=R−E, from Eq75 generates a uv emotion of relief in avoiding the v penalty, R=0, when one successfully rolls the lucky numbers on three pair of dice simultaneously as
128.) T=R−E=0−(−uv)=uv=$2600
With this game played with three sequential rolls of the dice, the increased expectation of avoiding the v=$2700 penalty after rolling a lucky number on the 1^{st} toss of the dice, E_{1}, understood as what is realized from the toss, R_{1}, with the improbability of escaping the penalty then as 1−Z_{2}Z_{3}, is
129.) E_{1}=R_{1}= −(1−Z_{2}Z_{3})v= −(1−(1/9))$2700= −(8/9)($2700)= −$2400
Hence the transition emotion, T_{1}, is via the Law of Emotion, T=R−E, expressed as T_{1}=R_{1}−E, is
130.) T_{1}=R_{1}−E= E_{1}−E=−(1−Z_{2}Z_{3})v−[−(1−Z_{1}Z_{2}Z_{3})]=(1−Z_{1})Z_{2}Z_{3}v=U_{1}Z_{2}Z_{3}v=$200
T_{1}=U_{1}Z_{2}Z_{3}v is understood in parallel to the partial excitement of U_{1}Z_{2}Z_{3}V of Eq119 as the feeling of partial relief gotten from rolling the 1^{st} lucky number. The rest of the analysis for the triplet v penalty perfectly parallels that for the triplet V prize game, except that the partial emotions felt are those of relief in escaping a money loss rather than excitement in achieving a money gain.
The fit of the analysis based on the Law of Emotion to universal emotional experience validates it and the underlying Lucky Numbers mathematics. Further validating them next is the Law of Emotion deriving the universally accepted Law of Supply and Demand.
The Law of Supply and Demand of Economics 101 states that the price of a commodity in a free market economy is an increasing function of the demand for it and a decreasing function of the availability or supply of the commodity. An alternative equivalent expression of it determines the price as an increasing function of the demand for and the scarcity of the commodity, the latter as the inverse of the commodity’s supply or availability.
Now let’s consider the triplet lucky number V=$2700 prize game with the 1^{st} lucky number in the triplet sequence as a commodity that can be purchased. This assumes, of course, that some agent who runs the game and pays off the prize money exists to sell such a commodity to the player. What is the fair price of the 1^{st} lucky number, that is, of the pair of dice being placed on the table for the player with a 4, 7 or 10 showing on it that counts in getting the V=$2700 prize?
As this changes the probability of winning the V=$2700 from z=Z_{1}Z_{2}Z_{3}=1/27 in Eq111 to Z_{2}Z_{3}=1/9 in Eq116, it is certainly a valuable commodity for the player, but what is the fair price of it? It is an amount of money that maintains the E=$100 average payoff in Eq106 of the game played when all three of the lucky numbers must be gotten by random throws of the dice.
Once the 1^{st} lucky number is obtained by purchase from the agent, the average payoff for the player increases from E=$100 of Eq113 to E_{1}=$300 of Eq118. Hence the fair price for the 1^{st} lucky number, W_{1}, must be such that its subtraction from the player’s improved E_{1}=$300 average payoff by purchase must be equal to the original E=$100 average payoff.
131.) E_{1}−W_{1}= E
Solving for W_{1} obtains the fair price as
132.) W_{1 }= E_{1}−E
This W_{1} fair price is shown to be a function of variables we have already encountered from the algebraic manipulation of E_{1}−E done back in Eqs119&121 as
133.) W_{1 }= E_{1}−E = Z_{2}Z_{3}V – zV = U_{1}Z_{2}Z_{3}V = T_{1} = U_{1}E_{1 }=$200
The W_{1}=$200 price for the 1^{st} lucky number that increases the average payoff to E_{1}=$300 is equal to the original average payoff of E=$100, which is what understands W_{1} as its fair price. From the perspective of economic optimization the player as buyer would want to pay as little as possible for it and the agent as seller would want to charge as much as possible for it. But W_{1}=$200 is its fair value or fair price in maintaining the original average payoff or expected value of E=$100.
The W_{1}=U_{1}E_{1} fair price formula of Eq133 is a primitive form of the Law of Supply and Demand in its specifying it in terms of the emotions people feel that control the price they’ll pay for a commodity. W_{1}=U_{1}E_{1} is the Law of Supply and Demand for price understood as an increasing function of scarcity and demand with the scarcity of the 1^{st} lucky number as the uncertainty in rolling it on the dice, U_{1}=2/3, and the demand for the 1^{st} lucky number in terms of its value to the player as the E_{1}=Z_{2}Z_{3}V=$300 average payoff it provides, the greater the value of a commodity, the greater the demand for it being an intuitively reasonable assumption. This derivation from the Law of Emotion of the Law of Supply and Demand, a firm empirical law of economics universally accepted as correct, is a powerful validation of the Law of Emotion and the mathematics of Lucky Numbers on which it is based.
There are a number of fascinating nuances in this understanding. Note the equivalence in Eq133 of the W_{1}=$200 fair price for the 1^{st} lucky number and the T_{1}=W_{1}=$200 pleasurable excitement in rolling the 1^{st} lucky number on the dice. This equivalence of T_{1} excitement and W_{1} price tells us that the price paid for a commodity is a measure of the pleasurable excitement the commodity generates for the buyer. This fits economic reality quite well as seen from TV commercials for automobiles and travel ads and foods that hawk them by depicting them as exciting.
The value of the 1^{st} lucky number can be calculated not just in terms of the W_{1} money one would spend for it but also in terms of the time spent in acquiring the money needed to purchase it. That is, W_{1} is understood as a measure of the amount of time spent to get the 1^{st} lucky number when time is taken to be directly proportional to money, as is certainly the case for most people in the dollars per hour wage or per month salary they get their money from.
This extends the W_{1}=U_{1}E_{1}=T_{1} Law of Supply and Demand to showing that people spend their time to obtain commodities that pleasurably excite them, whether as the time spent working to get the money to be spent on the commodity or as time spent directly to get the commodity like time spent growing a pleasant tasting food like strawberries in one’s backyard.
We can also derive a parallel primitive Law of Supply and Demand from the v penalty game requiring the toss of three lucky numbers. The W_{1} fair price of the 1^{st} lucky number is again W_{1}=E_{1}−E, but with E_{1}−E from Eq130 as
134.) W_{1 }= E_{1}−E = U_{1}Z_{2}Z_{3}v = T_{1} = $200
This form of the primitive Law of Supply and demand tells us that people also spend their money for commodities that provide T_{1}=U_{1}Z_{2}Z_{3}v=W_{1} relief in addition to commodities that provide T_{1}=U_{1}Z_{2}Z_{3}V=W_{1} excitement as seen in Eq119. The two forms of the Law of Supply and Demand of Eqs133&134 provide a strong empirical validation of the Law of Emotion of Eq75 that derived them from the observed fact that people do spend their money and time to get things that provide relief and excitement as is readily seen in the complete spectrum of TV ads, all of whose products are pitched in ads as providing either relief, as with antacids and other medicine and insurance, or excitement, as with exciting cars, foods and vacations.
Next we want to express the primitive Laws of Supply and Demand we see in Eqs133&134 as simply as possible. In both we note the equivalence of the W_{1} fair price with T_{1} partial success excitement or relief. This implies that the simplest forms of T excitement and relief in Eqs115,128,81&87 should also be a measure of the fair price, W, of what is achieved that is generating this excitement and relief. Hence we write from Eq115
135.) W=T=uV= −(−uV)
This specifies W as the price of all three lucky numbers needed to win a V prize. And we write from Eq128
136.) W=T=uv= −(−uv)
This
specifies W as the price of all three lucky numbers needed to avoid a v
penalty. And we write from Eq81
137.) W=T=UV= −(−UV)
This specifies W as the price of one lucky number needed to win the V prize. And we write from Eq87
138.) W=T=Uv= −(−Uv)
This specifies W as the price of one lucky number needed to avoid the v penalty.
Now while we have supersimplified the pricing Law of Supply and Demand in form, the intuitive meaning of it seems blurred for what is the fair price of a commodity used to avoid a v penalty than perhaps as intuition suggests the cost of the penalty. Regardless of this objection reasonably raised, these simple form are instrumental to deriving our visceral emotions of hunger, anger, pain, sex and love in an upcoming section. So we will now demonstrate that they are correct functions for fair price on the basis of the earlier argument that the W fair price of a lucky number is equal to the increase in average payoff generated by getting that lucky number.
We’ll show that with the simplest of Eqs135138, namely Eq135, generating a V=$120 prize from the rolling of one lucky number. From E=ZV with Z=1/3 the average payoff is
139.) E=ZV=(1/3)$120=$40
We also see from Eq82 that the excitement gotten is
82.) T=UV=$80
This from Eq135 tells us that the fair price for the lucky number is W=T=$80. If we pay this each time for three games the total price paid is $240. This has us win $120 each game for a total of $360 for the tree games. The net winnings are thus $360−$240=$120, which is what is won on average in three games if the game is played strictly from the throw of the dice with no lucky numbers purchased. Hence W=T=$80 is, indeed, the fair price of the lucky number and W=T=UV is a most simple form of the law of supply and demand with U as the uncertainty or scarcity of the lucky number and V its value as a measure of the demand for it. This should also make clear that without our having to go through the details of it, all four of Eqs135138 are valid forms of the Law of Supply and Demand. We will use these simple forms shortly to develop the visceral emotions of hunger, anger, pain, sex and love.
More immediately Eqs135138 tell us in a clear way that people not only spend money and time to attain the pleasures of relief and excitement specified as W=T=UV, W=T=Uv, W=T=uV and W=T=uv but also in order to negate or eliminate the antecedent displeasures of fear and anxiousness seen in Eqs135138 as W=T= −(−UV), W=T= −(−Uv), W=T= −(−uV) and W=T=−(−uv).
This has us revise and expand our generalization of a few paragraphs back now to people spending their money and time and being motivated not just on the pursuit of the pleasures of excitement and relief but also by the avoidance of the displeasures of anxiousness and fear. Understanding behavior to be motivated by the pursuit of pleasure and the avoidance of displeasure is the essence of hedonism. It should be made clear that this sense of hedonism is not an encouragement for people to seek pleasure and avoid displeasure, but rather a conclusion drawn from the foregoing mathematical analysis that people just do behave so as to achieve pleasure and avoid displeasure as the essence of human neurobiology. To generalize hedonism you need, of course, to also take into consideration the visceral emotions we have that motivate our behavior like hunger, the pleasure of eating, feeling cold and the pleasure of warmth, pain and relief from it, sex and love and the displeasure of their frustrations and failures. That last topic will be the subject of our next section.
6. Mixed Emotions and Behavioral Selection
Our expectations determine our behavioral selections, what we decide to do. To show how, let’s expand the lucky number game in two ways. In the first, we will have the roll of a lucky number of 4, 7 or 10 give a prize of V=$240 and the failure to roll a lucky number a penalty of v=$60. The expected value in this case is a combination of the E=ZV hopeful expectation of Eq4 and the E= −Uv fearful expectation of Eq73.
140.) E = ZV – Uv
For the above game, the amount of money one can expect to win on average, the expected value of the game, E, is
141.) E_{1} = Z_{1}V – U_{1}v = (1/3)$240 – (2/3)$60 =$80 − $40 = $40
We use the subscript, 1, with the variables Z, U and E because this is Game #1 used to illustrate behavioral selection. One can also play a Game #2 whose lucky numbers are 3, 6, 11 and 12. Their probabilities of being roiled are 2/36, 5/36, 2/36 and 1/36, which sum to the probability of rolling a lucky number in Game #2 of
142.) Z_{2} = 2/36 + 5/36 + 2/36 + 1/36 = 9/36 = 1/4
And the improbability or uncertainty of rolling one of the 3, 6, 11 and 12 lucky numbers in Game #2 is
143.) U_{2} = 1 – Z_{2} = 3/4
With a prize of V=$240 for a successful roll and a penalty for failure of v=$60, the expected value or average payoff for Game #2 is
144.) E_{2} = Z_{2}V – U_{2}v = (1/4)$240 – (3/4)$60 =$60 − $45 = $15
Now to best develop an understanding of behavioral selection let us consider competitive play between two people. Player #1 is given Game #1 to play with lucky numbers, 4, 7 and 10, and Player #2 is made to play Game #2 with lucky numbers, 3, 6, 11 and 12. Each player is paid the V=$240 prize by the other when he rolls a lucky number. And each player pays the penalty of v=$60 when he fails to roll a lucky number to the other player. The two take turns rolling the dice with every pair of rolls producing on average a win for Player #1 that is the difference in the expected values of Eq141 and Eq144 of
145.) E_{1} – E_{2} = (Z_{1}V – U_{1}v) – (Z_{2}V – U_{2}v) = $40 − $15 = $25
And the outcome for Player #2 after every pair of rolls is
146.) E_{2} – E_{1} = (Z_{2}V – U_{2}v) – (Z_{1}V – U_{1}v) = $15−$40 = −$25
That
is, Player #2 loses $25 in every pair of rolls in this competition. Now we
assigned these two games to the two players. What if each had a choice of games
to play? To emphasize which game selection would be optimal for either player,
let’s go back to the games as they were assigned and give Player #1 a bankroll
of $100 to play with and Player #2 a bankroll of $900 with an added rule to
this game that the first player to wipe out the other’s bankroll wins the
competition. If the average wins and losses hold true Player #1 player in
consideration of winning the expected value of $25 in every pair of rolls will
be the winner of this competition as made clear in the graph below of average
outcomes.
Figure 147. Two Player Lucky Number Competition
After 36 pairs of rolls with average outcomes Player #2 in blue goes broke and
Player #1 in red wins the competition. This might be called competitive
selection in the sense that the parameters of the game select Player #1 as
the winner on the basis of the positive sign of the E_{1} –E_{2}=$25
expected value difference in Eq145.
And another kind of selection is also illustrated here, behavioral selection or intelligent selection. We mandated Player #2 playing Game #2 in order to show which game was superior. But if Player #2 had a choice between the two games, he would obviously choose to play Game #1 because of its greater expected value or average return. This behavioral selection can also be thought to arise because of the greater pleasure it gives in expectation and/or as is functionally and emotionally related because playing Game #2 resulted in a loss, on average in every pair of rolls and ultimately in Player #2 having his bankroll wiped out, and in feelings of displeasure from loss that we showed earlier.
The emotional or hedonistic basis of behavioral selection is obvious. We do what is pleasant and avoid doing what is unpleasant, generally selecting a more pleasant option over a less pleasant as made clear above and select a less unpleasant penalty game over a more unpleasant penalty when we must choose one of them, as we will make clear in with the following example of such a Hobson’s choice illustrated with the two Lucky Numbers games of Game #1 and Game #2 but with the v in both for failure raised to v=$100. This example considers only one individual who must play either of these two v=$100 penalty modified Lucky Number games. The expectation aroused for Game #1 where the lucky numbers are 4, 7 and 10 with probability of Z=1/3 and the penalty is raised to v=$150 is a variation of Eq141 as
148.) E_{1} = Z_{1}V – U_{1}v = (1/3)$240 – (2/3)$180 =$80 − $120 = −$40
And in similar fashion, that for the penalty modified Game #2 is as an altered Eq144
149.) E_{2} = Z_{2}V – U_{2}v = (1/4)$240 – (3/4)$180 =$60 − $135 = ‑$75
Now the individual will choose Game #1 on the basis of its lower cost and hedonistically or emotionally for a number of interrelated reasons. One the one hand it is because there is less displeasure in the fearful expectation of U_{1}v=−$120 of Game #1 in Eq148 than in the U_{2}v=−$135 of Game #2 in Eq149. There is specifically a difference of
150.) U_{1}v−U_{2}v =−$120 – (−$135)= +$15
This is the origin of the positive emotion for choosing the lesser of two evils, why that selection operation feels positive or pleasant, which is the neurobiological instrument for getting one to select to do something that is itself a penalty or loss for you. We can also see that there is more positive expectation in Game #1
151.) Z_{1}V – Z_{2}V = $80 − $60 =+$20
The combination of the two is just E_{1} of Eq148 minus E_{2} of Eq149, this E_{1}−E_{2} difference, though, differing from its arrangement in Eq145 by showing the E_{1} ‒ E_{2} expectation to be a compound function of combined relative pleasure and displeasure as
152.) E_{1} – E_{2} = (Z_{1}V – U_{1}v) – (Z_{2}V – U_{2}v) = (Z_{1}V – Z_{2}V) – (U_{1}v – U_{2}v)
The first E_{1} ‒ E_{2} expression was developed in Eq145 as the difference in expected value, quite impossible to argue with, and the second entirely equivalent expression represents what people actually feel in comparing the relative pleasure and displeasure content of competing possible behaviors for selection to be executed. The equivalence of the two expressions shows the mind to work in a remarkably ordered way in three basic modes: choose greater pleasure over less; choose less displeasure over more; and always choose pleasure over displeasure. This is in effect a mathematical definition of hedonism, the basic mechanism that determines the world’s behavior. To understand war, the danger of impending nuclear war and how and why the elimination of all weapons from the earth for people is the only remedy for it, that basic understanding of what makes us tick  the carrot and the stick  must be understood and accepted, mathematically and intuitively.
There are two broad variations of the competition between relative pleasures and displeasures that should also be mentioned. One of them is sacrifice where displeasure the displeasure of some form of penalty, the general nature of which we’ll consider thoroughly in the next section, is suffered in the present for the pleasure of ultimate success and its reward expected in the future. And the other is coercive restriction where pleasure is forgone in the present because of fearful expectation of a consequent penalty be it a punch, a stay in prison or deprivation of survival needs like housing via lack of income from being fired from one’s job. The equations that explain this are grade school simple and readily extracted from Eqs140‒152.
To keep the introduction of this section’s material simple, we kept the values of the V prizes and v penalties of games compared equal. The more general case is when any game or situation has rewards specific to it and also with any associated penalties. This generalizes Eq152 to
153.) E_{1} – E_{2} = (Z_{1}V_{1} – U_{1}v_{1}) – (Z_{2}V_{2} – U_{2}v_{2}) = (Z_{1}V_{1} – Z_{2}V_{2}) – (U_{1}v_{1} – U_{2}v_{2})
This leads readily to understanding why people fail in many cases. And that’s because they select doing something on the basis of E_{1} ‒ E_{2} being positive from their suppositions about the value of the variables in Eq152 they may be in error from inexperience or from obtaining bad information from others, intentionally disingenuous or unintentionally in error. But that gets way ahead of the time and we must wait to discuss the darker aspects of man’s emotional nature until we well lay out well its broadest general principles.
7. Survival Emotions
Many of our most basic emotions are associated with surviving or staying alive. We derive the pleasant and unpleasant emotions that drive survival behavior from the primitive Law of Supply and Demand, specifically of the form of it in Eq138.
154.) W=T = −(−Uv) =Uv
We do that by applying it not to avoiding the loss of v dollars but to avoiding the loss of one’s v*=1 life. That is, the penalty for failing at a survival behavior like getting food to eat or air to breathe is the loss of your own v*=1 life respectively by starvation or suffocation. The other terms in Eq138 are also asterisked to show that they are associated with avoiding the loss of one’s V*=1 life rather than the loss of v dollars.
155.) W*=T*= U*v*= −(−U*v*)
Rather than describing the asterisked terms in the most general way immediately, it is easier to introduce them with specific survival behaviors and save the generalizations of what these variables until we have paid out some primary kinds of survival behaviors. Let’s start with breathing air. Consider Eq155 with the U* in it as the immediate uncertainty or lack of probability of getting air or the scarcity of air. First we’ll take up then U* is very great as when a person is underwater drowning or having a critical asthmatic attack or has a pillow placed forcibly over his or her face. In this case of suffocation the U* scarcity of air as great can be represented as U*=.999 understood intuitively as the probability of losing one’s v*=1 life under such circumstances.
The T*=U*v* transition emotion in Eq155 is experienced when a behavior is done to obtain air under this U*=.999 circumstance. In parallel to T=Uv relief of Eq87, T*=U*v* of Eq155 is the pleasurable feeling of great relief in getting air to breathe when one is suffocating. While not all have had this experience of suffocation in one form or another followed by escape many people have had it and all of them will attest to the great intensity of the pleasant relief felt. One measure of this great relief is with Eq155 evaluated for the v*=1 life saved by the relieving behavior and its prior U*=.999 scarcity of air or uncertainty in getting it.
156.) T*=U*v*=.999.
We can also put a cash value on this relief by putting a cash value or price on your own v*=1 life that you don’t want to lose, let’s say a high value of v*=$100,000. That calculates a cash value for the T*=U*v* relief from suffocation that parallels the cash value of the T=Uv=$1166.67 relief from avoiding the loss of v=$1200 in Eq90 of
157.) T*= –(–U*v*)=U*v*=(.999)$100,000=$99,900
Of course the value put on one’s own v*=1 life of $100,000 is arbitrary as is the T*=$99,900 value of the intensity of the relief felt from alleviation of suffocation. But they convey the central aspect of such relief being as great as you would feel if your life were spared some threatening disaster, which is what suffocating to death unarguably is.
The –U*v* term in Eqs155&157 negated or resolved by the behavior of escaping suffocation to T*= –(–U*v*)=U*v* relief is a measure of the unpleasant panic fear instinctively felt in suffocation that parallels the −Uv fear in Eq73 of losing money in the Lucky Numbers penalty game.
Eqs155&157 also specify from W*=T* of Eq155 the W*price one would pay to save one’s life, namely W*=T*=$99,900. If the value of what one would pay to save his or her life was v*=$100,000, then the price one would pay for escape from suffocation and death as W*=T*=$99,900 is entirely reasonable assuming the person suffocating does not have the lives of a love and/or family to concern himself and his bankroll with.
The W*=T* equivalence of Eq155 also makes clear that the W*=T*=U*v function that governs the emotional dynamic is an expression of the Law of Supply and Demand with the demand for some commodity, good or service, that provides escape from suffocation and the loss of one’s v*=1 life being measured as the instinctively great value a person places on his or her life and the supply of what is needed to preserve that life as measured inversely by the scarcity in this case of air to breathe or the uncertainty in getting it of U*.
The W*_{ }= T*_{}=U*v*= −(− U*v*) Supply and Demand Law of Eq155 also holds when there is no scarcity of air, no uncertainty in the body’s cells getting oxygen, no probability of losing one’s v*=1 life from lack of oxygen, U*=0, which is the situation for most people most of the time. Inserting U*=0 into Eq88 obtains
158.) W*=T*=U*v*= −(−U*v*)=0
This formula quite perfectly fits normal breathing having no unpleasant fearful feeling, −U*v*=0, no noticeable relief in breathing air, T*=U*v*=0, W*=$0 a person is willing to pay for air to breathe under the normal circumstances of plenty of air available to breathe, W*=U*v*=0. The mathematically derived conclusions for U*=.999 suffocation and U*=0 normal breathing universally fit observable human experience.
And so does the intermediate situation with air in short supply but not critically scarce as say U*=.2 as might be found in COPD or Chronic Obstructive Pulmonary Disease. In this case the –U*v* displeasure is felt as pulmonary distress but less horribly unpleasant than the panic fear of U*=.999 suffocation. Also significant is the T*= −(−U*v*)=U*v* relief felt when oxygen is supplied to a COPD sufferer with bottled oxygen. And we also see in this common ailment for older people that they are willing to pay a W*=U*v* price for relief, a lot if necessary though not every last penny a person has as a person would if their life was critically threatened with U*=.999 level suffocation. Again we may reasonably understand the U*=.2 as an approximate probability of death for untreated COPD.
The U*=.999 case of suffocation provides a clear example of how emotion facilitates survival through negative feedback control. In parallel to the E= −Uv fearful expectation of Eq87 of losing v dollars in the Lucky Numbers penalty game, the –U*v* panic fear from suffocation provides an expectation of losing one’s v*=1 life. Each situation generates the unpleasant sense of fear that motivates avoidance of the fearful situation, the E= ‒Uv fear, avoidance of playing the penalty game if at all possible and the E*=‒U*v* fear, avoidance of the suffocating situation.
An important difference between the two fearful expectations is the origin of the uncertainty variable in them. In E= ‒Uv, the U uncertainty or probability of losing v dollars from failing to roll a lucky number is a mental calculation that derives from elementary probability theory or from an observed failure rate of repeated tosses. In E*= ‒U*v*, the U* uncertainty of getting oxygen to the body’s cells or probability of losing one’s v*=1 life derives from a physiological measure of the amount of oxygen in the blood stream. To understand how this U* measure motivates behavior that obtains the oxygen needed as a feedback control system let’s write the E* expectation U* is the primary variable of as,
159.) E*= –U*v*= 0 – U*v*
Expressing the instinctive panic fear felt from getting insufficient air to breathe appreciates the 0–U*v* term in the above as the “error function” in classic feedback control theory where the error is what one wants to get rid of or “zero out” in the parlance of negative feedback control theory, that accomplished by reducing the –U*v* fear term to zero with appropriate behavior that returns one to normal breathing, thus returning the E* expectation of losing one’s v*=1 life by suffocation to zero, E*0.
This confluence of biological control theory with the Law of Emotion derived Law of Supply and Demand understanding of breathing under difficult circumstances is another strong validation of Lucky Numbers. This analysis of emotional control as negative feedback control extends also to all emotion functions derived for obtaining money and avoiding its loss and also to the visceral emotional systems we’ll consider next. This and the overall understanding in Lucky Numbers of behavior as emotion mediated phenomena makes it clear that man has been designed to operate in an automated machine like fashion, whether by the passive design of Darwinian evolution or the intelligent design of an unseen creator God with an excellent sense of systems engineering, which of the two is more improbable, a heated debate between professional biologists and the Duck Dynasty crowd these days.
Temperature regulation as avoidance of extremes of heat and cold is like breathing centrally important for avoiding the loss of one’s v*=1 life. Temperature in the range of 68^{o }−82^{o}_{ }Fahrenheit is optimal for man. If the temperature falls below 68^{o}, the heat needed by the body to function well is in short supply or scarce with the uncertainty of the body getting the heat needed specifiable as U*_{ }>0 in Eq155, whatever numerical estimate we may wish to insert being an increasing function of how cold and life threatening the temperature is. That is, the colder the skin temperature is, the greater the U* scarcity of heat and from W*=T*=U*v*= −(−U*v*) of Eq155, the greater is the –U*v* unpleasant feeling of cold.
The –U*v* unpleasant sensation of cold is not quite a feeling of patent fear as was the –Uv fear of losing money in Eq73 but it has the same effect as fear in making one want to avoid the cold as though you did fear it. The range of the discomfort of cold extends to extreme cold represented as a U*=.99 scarcity of heat as a feeling approaching pain that makes one moan little differently than if one were being hit with a club as those who have experienced such cold will attest to.
Negating the –U*v* displeasure of cold by getting warm provides via Eq155 the T*= −(–U*v*)=U*v* pleasure of the feeling of the relief of warmth, a pleasure greater in intensity as U*v* the greater the displeasure of the −U*v* antecedent cold, as fits universal experience. As further validates this understanding of temperature regulation, a person is quite willing to pay a W*=U*v* price from Eq75 to alleviate the −U*v* displeasure of cold and obtain the U*v* pleasure of warmth, the amount of money willing to be paid being proportional to the U* scarcity of heat in –U*v* felt as antecedent cold.
And by understanding the W* money spent to get warm when one is cold to be directly proportional to the time spent to make that money, Eq155 also tells us as fits universal experience that a person is willing to spend time to get warm directly as by cutting wood to burn in a fireplace and/or by getting or making clothes to put on to stay warm.
It is also universal experience that the pleasant feeling of warmth is not felt when a person is in the optimal 68^{o}−82^{o}F temperature to begin with. Being continuously in the optimal temperature range represents a situation of no scarcity of heat, U*=0, which dictates no unpleasant sense of cold via –U*v*=0 nor any pleasant feeling of warmth T*= U*v*=0.
Temperature regulation also requires that the temperature be less than the high end of the 68^{o}82^{o}F optimal range. Above that there is a scarcity of coolness required by the body to operate optimally, U*>0, with −U*v*>0 displeasure from Eq155 manifest as feeling hot and with the pleasurable alleviation or negation of such unpleasant overheating as −(−U*v*)= U*v*, that by appropriate cooling felt as pleasantly cool relief, T*= −(−U*v*)=U*v*>0. And it is clear from Eq155 as fits experience that a person is willing to pay for air conditioning to stay cool, W*=U*v*>0. Temperature regulation as feedback control has been understood as such for decades and is also derivable from Eq155 in the same way that we did for regulation of breathing.
Obtaining food for the body to keep an individual from losing his or her v*=1 life from critical lack of it also follows Eq155, but not in as simple and direct manner as with breathing and temperature regulation because of the complicating factor of the intermediate storage of the food in various organs and tissues of the body, short term in the stomach and long term in fat and the liver. We dodge that problem by minimizing the effect of storage on the emotions involved as follows.
When one hasn’t eaten for some time, the glucose or blood sugar in the blood vessels of the body becomes in short supply or scarce for the body’s cells, U*>0. Then the emotion of feeling hungry arises as −U*v*>0 of Eq155. This −U*v* feeling of being hungry, quite unpleasant as hunger in high intensity, is negated or relived to the T*=−(−U*v*)=U*v* pleasure of eating that includes both the deliciousness of food taste and the relief of the filling of the stomach.
The T*= −(−U*v*)= U*v* equivalence in Eq155 tells us that the intensity of the pleasure of eating, T*=U*v*, is greater, the greater the antecedent −U*v* hunger. This is readily validated by those who have had genuine hunger and experienced marked pleasure in eating to relieve the hunger even with just a piece of stale bread or cracker, which tastes very delicious under that circumstance. And almost all of us have experienced the fact that feeling hungry before eating makes the food taste better. Eq155 also tells us that people are willing to spend W* dollars to obtain food and also to spend time to that end whether to get the money needed to purchase food or, as our primitive huntergatherer ancestors did by gathering plants and hunting animals to spend time directly to get food. .
When blood sugar levels are high and the stomach full, U*=0, that is, there is no scarcity of food chemicals for the body’s cells and under normal circumstances, hence, no feeling to eat, −U*v*=0. Under these circumstances, eating food pretty much lacks the T*=U*v*>0 pleasure produced when one does have a −U*v*>0 appetite. In such a state, absent the abnormal, constantly present hunger that is pathologically responsible for modern man’s epidemic obesity, there is neither a pleasure nor displeasure motivation to eat.
Lastly as a survival behavior we want to consider physical trauma like a fracture that causes pain. Pain is signified as –U*v* with U*>0 the uncertainty or scarcity of a normal healthy condition mechanically that threatens incurring the penalty of losing one’s v*=1 life. In this way pain is in obvious parallel to the scarcity of air, warmth or food, all unhealthy circumstances that threaten the loss of one’s v*=1 life,. Behavior that eliminates or negates the −U*v* pain as with not putting mechanical pressure on the fracture as −(−U*v*)=U*v*>0 produces U*v*>0 relief from the pain, which is felt as pleasant in proportion to the antecedent pain that pampering the fracture relieves.
Now let us make it clear that the unpleasant emotions of suffocation, hunger, cold and physical trauma and the pleasant emotions of their alleviation, both sets of which derive from W*=T*= U*v*= −(−U*v*) of Eq155 are different from the emotions of the behaviors that obtain the commodities needed for these basic survival activities. When one is hungry, for example, eating proceeds in a direct and immediate fashion when food is readily available. But one must have possession of food first before one can eat. Explaining the functional relationship between the emotions for getting food and the emotions for eating food is best done with a concrete example of a food procurement behavior. We shall use a behavior we are familiar with in playing a dice game that gives food as a reward for rolling a lucky number of 4, 7 or 10.
Eating this food gotten as a prize alleviates a hunger of –U*v* to produce the eating pleasure of T*= −(−U*v*)=U*v*. This behavior to get the food has a Z=1/3 probability of success and an improbability or uncertainty of U=(1−Z)=2/3. One’s expectations in this game are not via E=ZV an expectation of getting V dollars but rather of getting the W*=T*=U*v* pleasure of eating the food. Because this T* pleasant emotion gotten has an implicit dollar value from W*=T*=W*v* of Eq155 we can substitute W* for the V dollar value in the E=ZV expectation expression to obtain our hopes of pleasure as
160.) E=ZV=ZW*=ZT*=ZU*v*=(1−U)U*v*=U*v*−UU*v*
This E* expectation r hopes of obtaining T*=U*v* food pleasure to probability Z stands in comparison to E=ZV=V−UV of Eq69 as the hopes of getting V dollars. In the latter, the desire is for V dollars while in the former of Eq160 the desire is for T*=W*=U*v* food pleasure as the negation of one’s −U*v* hunger. Then much as the pleasant desire for V dollars is reduced by the –UV meaningful uncertainty about getting the money to one’s probability tempered hopes of E=ZV, so is the U*v* pleasant desire of eating the food reduced by −UU*v* meaningful uncertainty in getting the food to the probability tempered hopes of ZU*v*.
This latter term is the intensity of pleasure felt in one’s hopes of satisfying one’s hunger by a particular behavior of getting food, here by playing the dice game to get food to eat. And this is exactly how the mind works in seeking pleasure by a particular behavior characterized by some Z probability of success in achieving that pleasure. We can also develop a T transition emotion felt when one rolls a lucky number and gets the food. From the Eq75, Law of Emotion, T=R−E, what is realized following a successful throw of the dice is eating the food and the R=U*v* pleasure of eating it. But also because there is U uncertainty in getting the food, there is a thrill or excitement in obtaining the food to eat of
161.) T=R−E= U*v*− (U*v*−UU*v*) =UU*v*
When there is no uncertainty in getting the food as in reaching into the refrigerator to pull out a ham sandwich, there is no excitement involved in the act of getting the food to eat. Contrast this to a hunt for food for people who have no immediate food store or a search to gather berries to eat under the same circumstances of otherwise having nothing to eat. Then upon making the kill for meat or the finding of berry bush, there is great excitement.
In that sense UU* in UU*v in the above is a compound improbability, the U* improbability of your body’s cells getting what they need in food chemicals because your blood stream is low on blood sugar and the U uncertainty or improbability of your getting food to eat in order to replenish your blood stream with the blood sugar it needs supply the body’s cellular needs.
The T=UU*v* excitement in getting the food, hence, is a function of the U=2/3 uncertainty in getting the food and of the T*=U*v* pleasure in eating the food, itself a function of the –U*v* antecedent hunger via T*=U*v*= −(−U*v*) of Eq155. One gets both the R=U*V* pleasure of eating the food and the T=UU*v* thrill of obtaining it under uncertainty, which is what our hunter gatherer ancestors surely felt when searching for vegetative food or hunting for animal food with uncertainty, U. One can picture such a group having an exciting meal or feast following a successful hunt or search. In contrast if there is no U uncertainty in getting food, from T=UU*v*=0, there is no excitement or thrill in getting the food despite the R=U*v* pleasure in eating it, much as when one needs but to open the door of one’s refrigerator to grab an apple to eat if one is hungry.
Note also in the food prize dice game the disappointment that is felt, assuming the game can only be played once, when the lucky number is not rolled and food is not obtained under a condition of –U*v* hunger. In that case with E*=ZU*v* and R=0 for no prize realized, from the Law of Emotion, T=R−E,
162.) T=R−E*= 0−ZU*v*= −ZU*v*
This tells us that beyond the factor of your Z confidence in getting the food, the more –U*v* hungry you are and the more U*v* pleasure anticipated, the greater is the T= –ZU*v* disappointment in failing to get the food. And it also is the case that if there is no hunger for the food, there is no disappointment in not getting it, assuming the food can’t be stored for later consum7ption.
In a coming section we will consider the various behaviors used to alleviate –UU*v* survival anxieties or needs. These include not only direct and obvious behaviors like hunting or searching for food when one is hungry, but also complex, many part, behaviors analogous to the sequence of rolling three lucky numbers to obtain a V prize, social behavior that obtains food by getting help from another and aggressive behavior that gets food via aggressing on another and stealing his food. And in a later section we will also consider reproductive behavior in terms of the E=ZV function but as E*=Z*V* where V* represents obtaining V*=1 life as that of one’s child.
TO BE CONTINUED