ROMNEY, OBAMA AND POLITICAL ENTROPY:
Distinguishing the lies from truth in this election using
entropy’s precise characterization of information

   `
         Boltzmann’s Entropy Inscribed Tombstone

     

By Dr. Peter V. Calabria, PhD, Biophysics
Contact: petercalabria@matrix-evolutions.com

Many things in nature we have no problem understanding in modern times were once great mysteries. The back and forth movement of the planets in the night sky confusedly thought in the Middle Ages to come from angels pushing the planets is now explained in a very precise way with Isaac Newton’s equation for gravitation. And magnetic attraction once thought to be magical is now explained in a very precise way with Maxwell’s equations for electromagnetism. Entropy, which explains why heat flows out of your body in winter and into your body in summer, also has a mathematical representation in the equation for it inscribed above Boltzmann on his tombstone, S=klogW.

To see how entropy explains information, a very important concept because all our conscious perception, thought and feeling in life is information for us, you have to resolve the error in Boltzmann’s entropy that makes entropy still seem confusing and almost mystical today despite its mathematical formulation. Once you fix it, the new entropy equation makes everything meaningful fall into place including the dirt on Romney’s wingnut influenced repeated contradictions of reality and the Obama corporate puppet’s well Vaselined promises of the moon again.

At least the new entropy equation will enlighten the men and children reading this, for females old enough to have the hormones of adolescence run through them tend to judge the validity of things more on a touchy feely basis with the majority of women having no trouble rejecting even a mathematical proof if something if it is in contradiction with their emotional feelings.  This is not a biased condemnation of them, just a hard fact, which has an occasional exception in the female population.   

Once it is refined Boltzmann’s entropy provides the path to truth by operationally distinguishing information from misinformation. Boltzmann’s entropy is in modern terminology 

1.)       S = k_BlnW

The k_B term is just a constant, the important term that is the crux of Boltzmann’s entropy being W, the number of states that a system of K energy units randomly distributed over N molecules can exist in. For the simple system of K=12 energy units distributed over N=3 molecules, one of the states has each molecule getting the same number of energy units, the (4, 4, 4) state, and another state is of the K=12 energy units being distributed over the N=3 molecules as (8, 3, 1). There are 89 other states in this system, W=91 states in all, as can be obtained from the shortcut formula,

2.)       W = (K + N −1)!/K!(N − 1)!                

It calculates the W number of states in the K=12, N=3 distribution as

3.)       W = (K + N −1)!/K!(N − 1)! = 14!/12!2! = 91

The natural logarithm of W or lnW is just

4.)       lnW = ln[(K + N −1)!/K!(N − 1)!]

This lnW term can be expressed in a simpler way using Sterling’s Approximation of the logarithm of the factorial of any number, n,

5.)       ln(n!) ≈ nln(n) − n

For example for n=150, ln(n!)=605 is rendered by nln(n)−n=602, a good approximation. For very large K and n systems, as realistic thermodynamic systems are, Sterling’s Approximation obtains the lnW term in the Boltzmann entropy, S=k_BlnW, as

6.)       lnW ≈ Nln(K/N)

The K/N term is the average number of energy units per molecule, which is generally interpreted in physical chemistry theory as the absolute temperature of the system, T. This specifies Boltzmann’s S=k_BlnW entropy as

7.)       S = k_BlnW = k_BNlnT

The k_BNlnT term is a tangible expression for macroscopic entropy found in every elementary physical chemistry textbook (though more accurately in differential form) that effectively validates Boltzmann’s microscopic entropy equation. There is no argument with it in that regard. The gripe with the Boltzmann entropy, rather, is that despite its mathematical fit to macroscopic entropy based on measurable parameters like temperature, lnW has no intuitively sensible meaning physically or mathematically. This makes it very confusing except to elderly physics professors who have become used to Boltzmann’s entropy as they might to a wife not loved very much but accepted after many years of having to live with her as better than nothing.                 

Boltzmann is to be forgiven for this shortcoming in his entropy equation because the mathematics that corrects it, the Simpson Diversity Index, was not developed by the British statistician, Hugh Simpson, until a half century after Boltzmann’s death or rediscovered by us in matrix form until a few years ago. We are talking specifically about Simpson’s Reciprocal Diversity Index, D, which takes the form of the abscissa of the quadratic Renyi entropy, which we are not dwelling on because we want rather to move quickly to the simpler function of the average of this D Simpson’s diversity derived from centuries old multinomial distribution theory that specifies the molecular energy diversity of the representative configuration of the distribution.  

8.)       D_RC= NK/(K+N−1)

This .998 correlation is emphasized when an orderly variation of D_RC with K and N is graphed to show the near direct proportionality between the two implied by the .998 correlation.  

Plot of lnW versus D_RC for a Distribution of K=40 to K=91 Energy Units over N=40 Molecules

Doubters of the high correlation between D and lnW are urged to test it themselves with any thought experiment of their choosing as will quickly develop a hands on sense for them that the correct understanding of entropy must be the average energy diversity of K energy units over N molecules, which Boltzmann effectively approximated with his lnW function because the correct diversity functions did not come into mathematical existence until long after his death. The rejection of this diversity based explanation for entropy because of this author’s outspoken sociopolitical views, should not be tolerated, for its proof, as we shall see, is as firm as the proof of the Pythagorean Theorem,.       

We begin that proof by first developing Simpson’s Diversity Indices from the distinctions and similarities you intuitively sense between the objects you see in your visual field. We develop this fresh mathematical take on all the confusing things in life from entropy to cleverly lying politicians by quantifying the distinctions and likenesses sensed between objects based on color in the tableau shown below.   

Figure 9. K=6 Variously Colored Objects

This visual tableau is described with a simple equation that is a function of the K=6 objects counted along with the y=11 pairs of objects that are distinct in color and the λ=4 pairs of objects that are alike in color.       

10.)    2y + 2λ + K = K² 

The y=11 distinct pairs arise from each of the 3 red objects being distinct in color from 3 non-red objects and from the purple object being distinct from the 2 green objects. And the λ=4 pairs that are alike in color consist of 3 red pairs and 1 green pair of objects. This verifies the equation as     

11.)    2y + 2λ + K = K² = 2(11) + 2(4) + 6 = 36

We call 2y+2λ+K=K² The Matrix Equation because a comparison matrix of the set of objects in Figure 9 represented as (a b c, d e, f) systematically generates the K, y and λ terms in the equation.        

Figure 12. Comparison Matrix of (a b c, d e, f)

The (a b c, d e, f) axes of the matrix represent the x₁=3 red, x₂=2 green and x₃=1 purple objects in the visual tableau. These K=6 objects are systematically compared to each other in the matrix with the K²=36 comparisons generated consisting of Y=22 distinct pairs of objects of different color, Λ=8 pairs of objects alike in having the same color and K=6 identities, which result from comparing an object to itself, (aa bb cc, dd ee, ff), as run down the diagonal of the matrix from left to right.

 

The intuitive distinctions we make between two objects of different color don’t depend on the order in which the objects are compared. To wit, the distinction we make between the red object, a, and the green object, d, represented by ad in the matrix is no different than the distinction we make between the green object, d, and the red object, a, as represented by da. The order of comparison doesn’t matter so the number of intuitive distinctions we make is half the Y=22 distinctions in the matrix or y=Y/2=22/2=11.

And in the same way our intuitive sense of likeness between objects is independent of the order of comparison, the likeness in color between a and b represented by ab being essentially the same as the likeness in color between b and a represented by ba with λ=Λ/2=4 being the number of intuitive likenesses in the matrix. That is how the y=11 distinctions and Λ=4 likenesses we see in the visual tableau are represented in the matrix.  

And the K=6 objects, (a b c, d e, f), we see in the visual tableau of Figure 9 are also represented in the matrix as the corresponding set of K=6 identities, (aa bb cc, dd ee, ff), an identity relationship of an object being in every operational way a specification of the object itself. The y, λ and K variables in the matrix are, hence, related to each other with the same equation that represents them in the visual field itself.   

The matrix equation, 2y+2λ+K=K², is valid for all visual fields, but takes a different form that more directly derives the Simpson Diversity Indices and the correct functions for entropy and information when the set of objects compared in the matrix are not perfectly the same and the identity relationship does not appear. We will explain and develop this variation of the matrix and its matrix equation with a game that compares a guess about what will happen with the reality of what actually happens.   

Consider K=6 objects in N=3 colors sitting in a bag, (■■, ■■, ■■), one of which is chosen at random with a blind pick. The color of the object is guessed at before the pick and then compared to the pick once made to see if the guess was right or wrong. The matrix of the guesses compared to the outcomes is

Figure 13. Comparison Matrix of (■■, ■■, ■■).

The (■■, ■■, ■■) set of objects on the horizontal axis is the average distribution of 6 picks made from the bag of (■■, ■■, ■■) and the set of objects, (■■, ■■, ■■), on the vertical axis is the average frequency or distribution of colors guessed at in 6 guesses, each color assumed equally likely to be guessed given that each has an equal probability of being picked from (■■, ■■, ■■).

Each of the K²=36 comparison pairs in the matrix, then, represents a guess followed by a pick. If the guess and the pick are the same as with ■■, red guessed, red picked, the guess is correct. And if the guess and the pick are not the same or distinct as with ■■, red guessed, green picked, the guess is incorrect.

We count Y=24 distinctions or incorrect guesses as come about on average and ε=12 samenesses or correct guesses out of the K²=36 guesses made. Hence when there are no identities in a matrix as there are not in this kind of a comparison, the Y distinctions and ε samenesses must add up to the K² total number of comparisons.    

14.)    Y + ε = K²

This is the same as the matrix equation of Eq10 with Y=2y and ε=2λ+K. A fractional measure of the ε=12 samenesses relative to the K²=36 total number of comparisons is  

15.)    Z = ε/K²

For our color guessing game, Z = ε/K² = 12/36 = 1/3 = .333. This Z=1/3 fraction of correct guesses assumed to arise from actual guessing experience becomes the probability of guessing correctly in subsequent tries, much as a baseball player’s batting average computed from actual at-bats is a measures our sense of the probability or likelihood of the batter getting a hit in future tries. And Z is also the first Diversity Index developed by Hugh Simpson back in 1949. That is, it measures the color diversity of the (■■, ■■, ■■) set of objects. Next let’s see how Z measures diversity by playing the guessing game using a bag of K=6 objects in N=2 colors, (■■■, ■■■), to pick from. Its comparison matrix is     

Figure 16. Comparison Matrix of (■■■, ■■■).

The number of correct guesses is now ε=18 with a fractional measure relative to the K²=36 total guesses of Z = 18/36 =1/2 = .5. On the one hand this is the probability of a correct guess. And on the other hand as Simpson’s Z Diversity Index, it is a measure of the color diversity in (■■■, ■■■). Now note carefully that Simpson designed Z as an inverse measure of diversity as we see in N=2 color (■■■, ■■■) with Z= .5 being less diverse color-wise than N=3 (■■, ■■, ■■) with Z= .333.

Simpson also around the same time developed his Reciprocal Diversity Index, D, which is an increasing rather than an inverse measure of diversity,

17.)    D = 1/Z          

Using D=1/Z as the diversity measure gives the more diverse set of objects, (■■, ■■, ■■), a higher diversity value, D=1/Z=3, than the less diverse set, (■■■, ■■■), which has D=1/Z=2, this in keeping with our intuitive sense that (■■, ■■, ■■) is more diverse than (■■■, ■■■). It is this diversity function, D, whose average value for a thermodynamic system as D=KN/(K+N−!) of Eq8 has a .998 correlation with Boltzmann’s entropy. The sense of diversity of these sets of colored objects fits well with the probabilities associated with guessing from them in an inverse way for the more diverse a set of possible outcomes, the less the probability of guessing which outcome will be realized.     

A fractional measure of the Y=24 distinctions or incorrect guesses from the (■■, ■■, ■■) set to K²=36 total guesses in Figure 13 is the average frequency of incorrect guesses, which projected to future guessing is the probability of guessing incorrectly or the uncertainty in guessing, to which we give the symbol, U.   

18.)    U = Y/K²

Note from Eqs14,15&18 that

19.)    U = 1 − Z

Much as the D=1/Z diversity index of Eq17, the inverse of Z diversity, is an increasing function of diversity, so also is U, the arithmetic converse of Z, which also happens to be the third diversity index that Simpson developed shortly after WWII. For the guessing game from (■■, ■■, ■■), the uncertainty is U = Y/K² = 24/36 = 2/3 = .667.

This matrix analysis applies to all kinds of attempts to achieve a goal as readily illustrated with a game that tries to obtain from the (■■, ■■, ■■) bag an object of a given color, success in such an effort being marked by the sameness between the goal sought and the outcome realized at a probability of success of Z=ε/K².

And Z=ε/K² is also the probability of a supposition about reality communicated being correct via a comparison of the supposition with realizations of it perceived. Let’s look at the matrix of (■■, ■■, ■■) redisplayed below with this in mind.  

Figure 13. Comparison Matrix of (■■, ■■, ■■).

 

Recall that to see if the guess of one color is correct, you have to match the guess, which derives from the mind’s imagination of what will be matching or being the same as the color of the object picked that is seen. It’s the same with making a supposition. It is correct only to the extent that it fits reality seen, whether in the past or in the near future, but after your death when you can’t really see anymore other than by magic that is as absurd as saying that 2+3 can be something other than 5, if God wants it. And that goes for God, too. You no more should think your supposition (belief) about God is true when nobody has ever seen God than you can think your guess of red, ■, correct without red, ■, being seen to be picked from the bag. Nobody accepts supposition without proof, at least not for long, in anything else that matters. In a court of law, it’s what is seen, eye witness up top, not what is supposed that counts as hard evidence, and beyond eye witness accounts with circumstantial evidence the acceptance of it is only when logic is firmly applied to tie it to reality.

Only God gets an exemption from validation from logic and observation in the gibble gabble about whether He’s real or just a lot of smoke and mirrors at a carnival in Atlantic City or Las Vegas where they’re also out to rob the people with a license from government exempting the robbers from punishment, the ministers being no different than the casino people or the bankers who took all that money and hope from people who bought mortgages the banks knew were going to turn sour, certainly did because they had bet in that direction, a lot of real money.    

So that’s the lie detector. Truth is what’s said being a match to reality observed. Certainly none of the God bullshit from the otherwise bullshit fundamentalist conservatives is the same as reality observed. These are all lies or errors in thinking by the right and big ones in either case. And you can judge the rest of what they say by its being wrapped one way or the other in and being as absurdly magical in the end as the God nonsense. The political entropy lie detector says that the rightwing, the Republicans, en masse these days, are liars.

And the Democrats are honest people caught up in the reality of their success relative to other labor class people so much as to accept the hierarchy of power. And they dance their way around the totem pole as Obama does in his fellatio on the military that is 180 degrees from his promises about war, expressed and implied. These people you can only kick down the stairs because however you may judge them and whatever their true motives, giving them full benefit of the doubt and a high likeability rating, they are ineffective at stopping the predators from feeding on us. For them there is no sameness between their promises and realization as the entropy matrix tells us must be the case to be the truth. The lie detector, then, says insulting words about the Obama’s not worthy to be uttered by one as analytical as myself.  

From this political analysis of entropy we see that the matrix and its distinction and sameness functions, Y, y, ε, Λ, λ and K, provide a foundation for specifying not only what we see in our visual field but also for guessing and other goal directed behaviors including speaking the truth such as microstate thermodynamics and entropy and, as we shall see, for Darwinian natural selection, for emotion driven behavioral selection or making choices and for electronic circuit behavior. This matrix based mathematical understanding of nature - physical, biological and human - in terms of the matrix functions is important for revamping the entirety of science in a unified way.

Next we shall focus on how the matrix develops simple mathematical functions for the basic human emotions of hope, disappointment, anxiety, excitement, fear, relief, security, dismay, joy and depression. This begins in the simplest way with returning to the sameness between a guess and an outcome like ■■ or ■■ or ■■ that determines success and understanding that every successful guess or attempt that has meaningful consequences feels pleasant and that every failed guess or attempt as represented by a distinction like ■■ or ■■ or ■■ feels unpleasant. These emotional properties of successful and failed guesses are universal and consistently observed empirically on TV game shows. 

It is these individual instances of pleasure in a correct guess (that is the same as the outcome) and of displeasure in an incorrect guess (that is distinct or different than the outcome) that are the bases of the pleasure in the Z=ε/K² probability or confidence we have in attaining success and of the displeasure in the U=Y/K² probability of failure or uncertainty about achieving success. We will cover this topic of human emotion next before going to an in-depth analysis of the entropy problem we began with because the basis of deciding whether an assertion is a lie or the truth is very much dependent on the emotions felt in the decision making process.

But before we begin the mathematics of the human emotions in a formal way, let me tell a bit about myself and the family who developed this new mathematics.

 

Our main fear goes beyond the corporate ruling class with their lackey politicians bleeding us dry and their police thugs beating us up to great worry about the fellow who will have his finger on the button for the next four years. This will almost certainly be Romney as the election polls shaped by mass misinformation from the money bag privileged are making clear and Romney’s talk at conservative Liberty University makes it clear that he’s either as crazy as the rest of the fundamentalist loonies in being happy to make it to Heaven prematurely if it be God’s will to end the world with nukes or more than willing to bend to what they want him to do, which will get us to the same end.

My rebel attitude came about way back in my twenties after having had research stolen by my Jew businessman laboratory boss, Dr. Aaron Posner, may he come down with pancreatic cancer on the way out. And it intensified after I was set up by my fundamentalist brother-in-law lawyer, Don Graf of Lubbock. Three years in federal prison, and some of the worst, on a first time charge at age 56 with a PhD in biophysics, and this was after writing newspaper articles like this:

But nobody could be fouler than Wendell Williams. This is the guy the FBI used to put a hit on me. Screwed us out of $70,000 in family money. We get the money back because he’s in the wrong to begin with. I got angry over being scammed. Who wouldn’t with Wendell goading me into anger once the money had been whittled down to $1500 on repeated margin calls on unauthorized trades. This is along with his douche bag wife, with whose mother the two lived and without kids, for many years, a bit weird from this Sicilian’s perspective. 

And the locals went wild over an RPI professor angrily threatening this half blind asshole (millionaire also unfortunately), licensed by the FBI with whom he had close affiliation in being your handy money laundering joint for Wendell bought jewelry and such for cash and sold gold and silver coins as part of Ferris Stamp. This is how close this pedophile was to the FBI, indeed, very close in regard to some then recent arrest of a Ten Most Wanted fugitive. For me it was a matter of the LCMS, Lutheran Church Missouri, having the power of the FBI as tapped to do so by brother-in-law, Don Graf, the humiliation of whom in a fight over my wife led to his wife, Ruby divorcing him back after 1980 and he never remarrying despite his lawyer level bank account. Sounds like a joke, but wasn’t. I hate ‘em. Show me a fundamentalist conservative and I’ll show you someone who’ll beat a three year old dumb on visitation to get back at the mother for divorcing him. No, like a rattler on the road, you run over these sixteen times, back and forth, starting the minute you see them. Fucking evil disguised with a halo.            

Here’s me attempting to preach the gospel of revolution to a crowd of 500 black people at the Trayvon Martin rally in Las Vegas where I made the point that there could only be A Political Path to Justice for Trayvon Martin, a radical political path. I could waste two paragraphs on police torture and repeated attempted rape through inmates they controlled, unsuccessful, thank the God who doesn’t exist, and the violence in me that once took blood our of the eyes in streams that ran down this thug hired by the landlord’s cheeks. This bastard parked himself outside our bathroom window, was rude to my wife, insulted me in front of her and then passed GO when he made the baby cry by frightening his mother when she held him in her arms, which got me to take him to the optometrist, so I saw him to be, with eyeglasses on, a changed man, nearly a year later. I didn’t enjoy prison, the stories so dramatic and dangerous it would sound as though I were making them up, though I wouldn’t be. All snakes in the road. Imagination couldn’t construct asshole jerks bigger than these torturers.

How close was asshole millionaire weirdo Wendell Williams to the FBI? He was the mastermind of a notorious stamp scam on eBay as the “long time upstate New York stamp dealer” referred to in the msnbc article you can link to. Also note that this scam of Williams was no more followed up and prosecuted than his swindle of us because of the protection he had as a junior subagent of the FBI. You know, like the agent in the matrix, Wendell had the same feel as that guy. I didn’t know to run, unfortunately. But it’s just a matter of time when your name’s on the list, the one that’s not published.

Worst is when the hit is on those you love.

I like Professor Boltzmann for having been so dedicated to scientific truth that he committed suicide when his equation for entropy was initially rejected by his peers at the turn of the 20^th Century. For if he hadn’t killed himself the reconsideration given the equation following his suicide that got it accepted and preserved for us a century later might not have been.

But I have no patience with today’s pompous professors. The creep’s name was Barnet, Professor Alexander Barnett, totally LGBT, the icky kind, and a craven coward who as a result of a mathematical argument I had with him up in his office at Dartmouth that certainly involved no threat to him called in the Hanover police who then did their job as Dartmouth’s lap dogs for their salaries to find a pretext to have Child Protective take my grandson away from his mom at gunpoint.

After Thomas disappeared for nearly a week and looking like we were never going to ever get him back, his mom started coming apart at the seams with horrified shrieking and I was on the verge of jumping into the Connecticut River at the thought that I had caused it all with the stupid argument I’d had with Barnett.

What saved Thomas was a sympathetic email sent me by Kenneth Bazinet, then the White House correspondent for the NY Daily News, one of those guys you see sitting in the White House press room on TV, whom I had told about the situation. Bazinet’s email and my threatening to use him to broadcast this police state hit job on us got Thomas sent home lickety split, no further contact needed, once the cops and the child protective monsters saw that I was not the littlest fish in the sea to be swallowed whole without a fight.      

I want to point out Barnett seemed outraged at the thought of a quantification of the distinctions people intuitively make and lectured me in a strident tone that mathematics wasn’t philosophy and didn’t stupid me, not a mathematics professor but a biophysicist, know that. But perhaps Barnett was seething underneath rather from a weak person’s jealousy of a confident smiling fellow up in his office that day with his pretty wife in tow. People with little self-confidence, and most LGBT I have run into are such, can be very painful to deal with when they have the power, the institutional power, to aggressively act out on their jealousies towards those who have personal and family success.    

There’s a lot more that could be said in anecdote form and if you can stand amateur music, we have the political message as Songs from a Rebels Past, but best is the next section on the mathematics of our feelings, which in the end are everything in life.

Those who wish to support our family’s disarmament, scientific and political efforts with a contribution can send a check or money order to Ruth Calabria, P.O. Box 3035, Albany, NY, 12203.    

The Mathematics of Human Emotion

We make that analysis using a slightly more complicated guessing game from the (■■, ■■, ■■) bag of objects that requires one to guess three objects picked from the bag, one at a time with replacement, in the sequence they are picked in, with a correct guess being rewarded with a prize of V=$2700 dollars. Below are the 27 sequences that can be picked, the correct guess of which gets the V=$2700 prize.  

Figure 20. The 27 Ways n=3 Colors can be Picked in Sequence

The probability of guessing the 1^st color in the sequence picked from (■■, ■■, ■■) is Z_A=1/3; of the 2^nd color in the sequence, Z_B=1/3; and of the 3^rd color in the sequence, Z_C=1/3 with the probability of guessing all three colors in the sequence and winning the prize given the symbol, Z,        

21.)    Z = Z_AZ_BZ_C = (1/3)(1/3)(1/3) = 1/27

 

Note that this guessing is from a balanced set of objects, (■■, ■■, ■■), whose N=3 colors have an equiprobable chance of being picked. For the general case of guessing from an unbalanced or non-equiprobable set of objects like (■■■■, ■■■, ■■), the probabilities are more complicated but still derive from the matrix function, Z, of Eq15 as will be considered in detail later after we develop the basics of the mind’s emotional processes from the simpler (■■, ■■, ■■) equiprobable set.

 

In the role of a detective trying to explain human nature as it is determined by the emotions, we next introduce mathematical expectation, E, the product of a V prize or payoff and the Z probability of getting it.   

 

22.)    E = ZV

For this three color guessing game with a V=$2700 prize, the mathematical expectation is

23.)    E = ZV = (1/27)($2700) = $100

The mathematical expectation, E, is objectively the average payoff per guess when one plays the guessing game repeatedly. If you play the guessing game 27 times on average you will get the V=$2700 prize once, which averages out to E=$100 per guess. There is also a subjective interpretation of E=ZV=$100 as a measure of the intensity of the pleasure of your hopes or expectations of winning the V=$2700 prize with probability, Z=1/27. This measure of an E=$100 pleasure in expectation of getting a $2700 prize with a Z=1/27 probability should be considered to be equal more or less to the pleasure of actually getting $100. 

The greater the V prize for guessing the winning triplet match, the higher your hopes and greater your pleasure in anticipation. So it’s a more pleasant expectation you would have of possibly getting a V=$27,000 payoff to probability, Z=1/27, of  

24.)    E = ZV = (1/27)($27,000) = $1000

And also from Eq22, the greater the Z probability of winning that is supposed, the higher your hopes and the greater the pleasure you feel in anticipation. If the V=$2700 prize was given for guessing just the first color picked from (■■, ■■, ■■) rather than the sequence of three colors, which has a probability of Z_A=1/3, the mathematical expectation of winning and the pleasure intensity in anticipating the win is 

25.)    E = Z_AV = (1/3)($2700)=$900

 

Next consider how you’d feel about getting the V=$2700 prize with a Z=1/27 probability and you fail to guess correctly. In that case where your E=ZV hopes are dashed, to use a colloquial expression, and what you feel is the emotion of disappointment. As should not be too surprising, that feeling of hopes dashed or disappointment is specified as a negation of the E=ZV hopes you had to begin with,    

26.)    T = −ZV         

In the guessing game to get V=$2700 with probability Z=1/27, the intensity of the displeasure of the disappointment is

27.)    T = −ZV = −(1/27)($2700) = −$100

The T symbol used for T= −ZV disappointment indicates it to be a transition emotion, a category of emotion we will explain in detail shortly. The negative sign in T= −ZV tells us that disappointment is an unpleasant emotion. The T= −ZV function for disappointment fits well how people actually feel when their hopes are dashed, for T= −ZV disappointment is greater and more unpleasant the greater the V prize hoped for and greater the Z probability the person feels of winning. If the person feels a Z_A=1/3 probability of winning as in the game of guessing just one color, the disappointment over not winning is

28.)    T = −Z_AV = −(1/3)($2700) = −$900

This is considerably greater than the T= −$100 disappointment felt in not getting the V=$2700 prize when the probability of winning is Z=1/27 as fits the universal sense of low expectation of success leading to less disappointment upon failure.

When the player does guess the triplet sequence and wins the V dollar prize, he or she feels excitement, a thrill from the success. The bigger the V payoff is, the greater the pleasure of the excitement. Also affecting the intensity of the excitement is the uncertainty the guesser has beforehand, U, as the probability of failing to guess the triplet sequence correctly,

29.)    U = 1 −Z

If you have a Z=1/27 probability of winning the prize, then you have a U=1−Z=28/29 uncertainty or probability of not winning. This uncertainty of winning is the other factor in the excitement of winning,    

30.)    T = UV           

 

The T in T=UV also indicates excitement to be transition emotion, a category of emotion that will be explained shortly. The intensity of pleasurable excitement in winning in the Z=1/27, U=1−Z=26/27, color guessing game is

 

31.)    T = UV = (26/27)($2700) = $2600

Note that T=UV is a positive function, which specifies excitement to be a pleasant feeling, much as T= –ZV disappointment’s negative sign indicates it to be an unpleasant feeling. To illustrate excitement’s dependence on the U uncertainty felt beforehand, consider feeling absolutely sure, Z=1, with no uncertainty, U=1−Z=0, of getting a $2,700 monthly salary. While you still feel pleasure in getting money when there is no uncertainty, it is not an exciting thrill like winning $2,700 in a guessing game that has significant uncertainty like U=26/27. So the intensity of the pleasure of the excitement felt in winning is a function of the size of the V prize won and of how much U uncertainty you had beforehand.   

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

The excitement in winning money under uncertainty is an emotion felt over and above the elementary pleasure of just getting money, which is pleasurable, a joy in itself, in proportion to how many V dollars are received. This pleasurable emotion of getting money that is independent of any uncertainty felt ahead of time is represented with the symbol, R, as indicates an emotion felt from getting or realizing money. 

32.)    R = V

This R=V pleasure of getting money is greater the greater the V amount of money gotten. Another realization or outcome of guessing is not getting any money when you guess wrong, which produces no emotion in itself as specified as

33.)    R = 0  

This no emotion outcome in getting no money is independent of the T= −ZV disappointment felt, as will be clarified momentarily. In brief review we see that when you have E=ZV expectation and fail to guess successfully, realizing no money, R=0, you feel T= −ZV disappointment. And that when you have E=ZV expectation and win the V prize, R=V, you feel T=UV excitement.

Now if you look carefully at the relationships between the E expectation emotion, the R realization emotion and the T transition emotion whether as T= −ZV disappointment or T=UV excitement, you note that both T transition emotions of disappointment and excitement are generated as the arithmetic difference of the R realization emotion and E expectation.

34.)    T = R − E       

This is called the Law of Emotion. With expectation as E=ZV we see that when no money is won, R=0, the Law of Emotion of Eq34 tells us that the T transition emotion produced is T= –ZV disappointment.

35.)    T = R – E = 0 – ZV = −ZV     

And when the V dollar prize are won, R=V, the Law of Emotion of Eq34 tells you via U=1−Z of Eq29 that the T transition emotion is the T=UV excitement.

36.)    T = R – E = V – ZV = (1−Z)V = UV           

The Emotions of Partial Success

Next we give further evidence of the validity of the T=R−E Law of Emotion by using it to derive the excitement a person feels when he guesses the 1^st color in the triplet sequence game even though guessing it gets no money in itself. The probability of guessing the 1^st color is Z_A=1/3. Once the first color is guessed and known, the probability of winning the V prize by guessing the next two colors in the triplet sequence correctly is

37.)    Z_BZ_C = (1/3)(1/3) = 1/9    

Now we want to ask what, if anything, is realized when the 1^st color is guessed. It is not R=V, for the V prize is not won when the 1^st color is guessed. And it’s not R=0, which is what is realized when the guesser fails to guess the triplet sequence and does not get the V prize. Rather what is realized when the 1^st color is guessed is an increased expectation of guessing the triplet sequence, one that is greater than the original expectation of E=ZV=$100 of Eq23 as

38.)    E_A = Z_BZ_CV = (1/9)($2700) = $300        

As this increased expectation, E_A, is what is realized in guessing the 1^st or A color, we can also specify E_A as a realization using the R symbol as R_A.

39.)    R_A = E_A = Z_BZ_CV = $300     

Now we can use the Law of Emotion, T= R−E, of Eq34, to generate the T_A transition emotion felt when the R realization in the Law of Emotion is the R_A=E_A=Z_BCV increased expectation and the E in the Law of Emotion is the E=ZV original expectation. With Z=Z_AZ_BZ_C from Eq21 and U_A=1−Z_A=2/3 from Eq29,        

40.)    T_A = R_A – E = Z_BZ_CV – ZV = Z_BZ_CV – Z_AZ_BZ_CV = (1 − Z_A)Z_BZ_CV = U_AZ_BZ_CV = (2/3)(1/3(1/3)($2700) = $200

What kind of emotion might T_A=U_AZ_BZ_CV be? To find out let us recall that the excitement of getting the triplet match and winning the R=V payoff by guessing the three color sequence in one fell swoop is T=UV from Eq31 written with V=R from the R=V equivalence,

41)      T = UV = UR = (26/27)($2700) = $2600         

To find out what kind of emotion T_A=U_AZ_BZ_CV is, let us recall from Eq39 that R_A = Z_BZ_CV, which allows us to express T_A=U_AZ_BZ_CV as

42.)    T_A = U_AZ_BZ_CV = U_AR_A          

The parallel of the T_A= U_AR_A partial success transition emotion to the T=UR complete success emotion of excitement in Eq41 makes clear that T_A=U_AR_A is the excitement felt in correctly guessing the 1^st or A color in the triplet sequence as a partial success even though it doesn’t pay off the V=$2700 or any part of it. Note that the intensity of the partial success excitement of T_A=$200 in Eq40 is significantly less than the T=$2600 intensity of excitement of guessing the triplet sequence that actually gets the guesser the V=$2700 prize.

This T_A=U_AR_A excitement from partial success is universal and readily observed in slot machine players noticing when the first two icons in a three icons the same payout are the same and on TV game shows like “The Price is Right.” In the latter we see partial success that provides no prize in itself eliciting excitement as when a contestant gets visibly excited about entry into the Showcase Showdown with the high spinner number even though entry itself has no prize associated with it. And there are many instances of such partial excitement in other TV game shows and in life, itself, much of what happens there being partial success with hope for some end goal never to be achieved and all the worse in the disappointment in the let down, once you’re put in your place in the hierarchy and learn to live with a collar around your neck at the end of leash with a finger up your ass. I doubt it’s just genetic.

Anyway, observation of this T_A=U_AR_A partial excitement in people and on TV game shows and in Las Vegas casinos, which is predicted by the Law of Emotion of Eq34, constitutes an empirical validation of it.

To further validate the Law of Emotion, T=R−E, we can use it to predict the excitement felt in guessing the 2^nd or B color in the triplet sequence. After getting the 1^st or A color, we saw the expectation increase in Eq39 to E_A=R_A=Z_BZ_CV. Now we want to derive the emotion felt when the 2^nd or B color is also guessed from the Law of Emotion, T=R−E. The realized outcome in that case is a further increase in expectation of

43.)    R_B = E_B = Z_CV = (1/3)($2700)=$900

The transitional emotion that comes from guessing the 2^nd or B color after having guessed the 1^st or A color is from the T=R−E Law of Emotion expressed with T as T_B, R as R_B=Z_CV and E as E_A=Z_BZ_CV   

44.)    T_B = R_B−E_A = Z_CV − Z_BZ_CV = (1−Z_B)Z_CV = U_BZ_CV = (2/3)(1/3)($2700) = $600

The rewriting of T_B = U_BZ_CV from R_B=Z_CV of Eq44 as T_B=U_BR_B and its parallel to the excitement equation, T=UR, of Eq41, makes it clear that T_B=U_BZ_CV =U_BR_B is the partial excitement felt when the 2^nd or B color is guessed. Note that the intensity of the T_B partial excitement of T_B=$600 is greater than the T_A=$200 partial excitement, which makes intuitive sense given that the T_B=$600 excitement comes about from having guessed two of the three colors needed to win.   

And we can also use the Law of Emotion, T=R−E, to predict the excitement felt in guessing the 3^rd or C color in the triplet sequence after having guessed the first two colors. The realized outcome in that case is getting the V prize or R=V and the expectation that precedes the last guess is from Eq43, E_B=Z_CV. Hence, 

45.)    T_C = R−E_B = V – Z_CV = (1−Z_C)V = U_CV = (2/3)($2700) = $1800

The rewriting of T_c = U_CV from R=V of Eq32 as T_C=U_CR and its parallel to the excitement equation, T=UR, of Eq41, makes it clear that T_c = U_CR is the excitement felt when the 3^nd or C color is guessed to complete the color sequence and win the V=$2700 prize. Note that the intensity of the T_C=$1800 partial excitement is much greater than that of T_B=$600 and T_A=$200 partial excitements. This makes sense intuitively in that the T_C=$1800 partial excitement comes about from completing the triplet strategy and winning the V=$2700 prize. And it is also empirically observed in “The Price is Right” game show where winning the Showcase Showdown prize has the winner jumping up and down and running around on the stage screaming excitedly. 

The validity of the Law of Emotion, T=R−E and this analysis based on it is made further clear in the three partial excitements adding up to the T=UV=$2600 excitement of guessing the triplet sequence in one fell swoop of Eq41.

46.)    T_A + T_B + T_C = $200 + $600 + $1800 = $2600 = T             

 

Note, though, what happens if you guess the first two colors correctly but get the 3^rd color wrong. Let’s go back to Eq45, but have R=0 in the applicable Law of Emotion for R to generate the transition emotion of counting chickens before they hatch disappointment as      

 

47.)    T_C = R−E_B = 0 – Z_CV = – Z_CV = −(1/3)($2700)= −$900

This let down, so to speak, is considerably more disappointment than the T = −$100 of Eq27, the difference between the two being the take back of the T_A=$200 and T_B=$600 excitements of Eqs40&44, what happens to the mass of suckers in the slave colony, generally in the large and frequently in the small. These pictures of the emotions drawn in mathematical language are stunning in their fidelity to reality as a validation of the correctness of the Law of Emotion and the analysis that flows from it.    

The Law of Supply and Demand

The validity of the Law of Emotion of Eq34 is also reinforced from its deriving the Law of Supply and Demand for commodity pricing in market economies. The commodity priced is the information of what the 1^st color is. Assume some agent blindly picks the object from (■■, ■■, ■■) before the guess is made and then after the color guess is made tells the guesser whether he is right or wrong. Assume also that the agent can sell the information on the 1^st color is to the guesser for some price. To determine the fair price that will be charged, note that knowing the 1^st color produces a mathematical expectation from Eq39 of E_A=$300 in contrast to the expectation when the 1^st color isn’t known from Eq23 of E=$100.

Understanding mathematical expectation as the average payoff per guess computes the fair price for the 1^st or A color, W_A to be what keeps the new average payoff of E_A=$300 minus the W_A price the same as the initial average payoff of E=$100. 

48.)    E_A – W_A = E

And solving this for W_A gives the fair price via Eq40 as  

49.)    W_A = E_A−E = Z_BZ_CV – ZV = T_A = U_AE_A = $200

Of course the guesser would want to pay less than this fair price of W_A=$200 for the 1^st color, as little as possible, and the agent as seller would want to charge more than W_A=$200, as much as possible. But this W_A=$200 price is the fair price in its keeping the average payoff per guessing the same at E=$100.

The W_A=U_AE_A fair price formula is the most primitive form of the Law of Supply and Demand. In Economics 101 the Law of Supply and Demand specifies the price of a commodity as an increasing function of the demand for it and a decreasing function of the supply. Or put another way, the Law of Supply and Demand specifies price to be an increasing function of demand and scarcity, with scarcity just an inverse measure of the supply of the commodity.

The buyer’s emotional sense of the scarcity of a commodity is the uncertainty he feels in obtaining it, the scarcer the commodity is, the more uncertain the buyer is about being able to obtain. In our case of the commodity being the information on the 1^st color, the uncertainty of getting it as a measure of its scarcity or difficulty in getting it is U_A=1−Z_A=2/3, which is one of the variables in W_A=U_AE_A of Eq49.

And the buyer’s demand for the commodity can be measured by its value or usefulness or utility once it is obtained. The value of the information on the 1^st color is the E_A=Z_BZ_CV=$300 expectation of Eq38 of getting the V=$2700 payoff getting the 1^st color provides, which is the other term in the W_A=U_AE_A fair price of Eq49.

Hence W_A=U_AE_A specifies the fair price of a commodity from The Law of Supply and Demand in terms of U_A as the scarcity or inverse of supply and E_A as the demand. This primitive derivation of the Law of Supply and Demand, a firmly empirical law that everybody agrees with, in terms of the human feelings of uncertainty and expectation, is a powerful validation of the Law of Emotion of Eq34.

Now note the equivalence in Eq40 of the W_A=$200 fair price for 1^st color and the T_A=W_A=$200 excitement in guessing the 1^st color, which indicates excitement felt in obtaining the 1^st color by buying it. This suggests that the fair price or value of a commodity is a measure of the amount of pleasurable excitement gotten from obtaining it by purchase. Or put another way, the greater the pleasure of excitement provided by a commodity, the greater the price a person is willing to pay for it. This very much fits economic reality as, for example, in TV ads that incessantly tout the exciting nature of the product they are trying to sell.

This primitive Law of Supply and Demand, W_A=U_AE_A has broad application and extends not only to W_A as a measure of the money people spend to get something of value but also of the time they spend to get something of value via the equivalence of money spent to time spent implied in income being generally earned as dollars per hour wages or dollars per month salary. This suggests from the W_A=U_AE_A Law of Supply and Demand that people spend time on things that are exciting, which nobody sane and honest would deny.

The above verifications of the Law of Emotion of Eq34 that derives this primitive Law of Supply and Demand understand it to be a totally basic equation for science on the level of the Law of Gravitation and Maxwell’s Equations in giving us with great mathematical preciseness and clarity the elements of the emotional machinery of the mind. Specifically it enlightens us in a way we never understood before that the human mind has three basic categories of emotion: R realized emotion that derives from outcomes, E expectation emotion from anticipation of a possible outcome and T transition emotion as the T=R−E arithmetic difference between realization and prior expectation.

No understanding of emotion driven behavior can be valid that is not based on the Law of Emotion, a conclusion drawn from the above precise analysis that calls into question the validity of the foundation precepts of all disciplines of the human sciences as they currently stand in their not being based on the Law of Emotion. 

An Alternative Expectation Function

Expressing the E=ZV expectation of Eq22 in terms of the U=1−Z uncertainty of Eq29 provides an alternative sense of how we anticipate getting something of value like a V dollar prize from a guessing game.    

50.)    E = ZV = (1−Z)V = V−UV

This alternative expression of a person’s E=ZV hopes as E=V–UV splits one’s expectation of success into two component emotions. One component is the V in E=V−UV that represents one’s desire for the V prize, the pleasure of which in daydreaming is a function of the size of the V prize desired. This V of desire in E=V–UV is accompanied by and diminished in its pleasure by the –UV term, which represents the anxiety a person feels over the possibility of not getting the desired V payoff. What we are calling anxiety is colloquially referred to with a number of other words like worry, apprehension, unease and fear. It is good, hence, when appropriate to use a technical term for –UV, meaningful uncertainty, that is, uncertainty, U, made meaningful by its association with V dollars in −UV, money being a meaningful item generally.

The negative sign of −UV anxiety specifies it as an unpleasant emotion with the intensity of the displeasure greater the greater the V payoff that one is anxious about getting and the greater the U uncertainty there is about getting it. Note that expressing E=ZV expectation as a function of V desire and –UV anxiety in E=V−UV derives the T=UV excitement of Eq36 that results from winning the R=V prize in a slightly different way from the Law of Emotion, T=R−E, as

51.)    T = R−E = V−(V−UV) = UV           

This derivation of T=UV excitement tells us that it comes about as the elimination or negation of antecedent –UV anxiety meaningful uncertainty. Adventure movies generate their excitement for the audience in this way by being front loaded with anxiety or dramatic tension over the hero’s meaningfully uncertain situation, which when the uncertainty is resolved by success at the end of the movie brings about excitement and thrills for the audience.

Meaningful Uncertainty & Meaningful Information

Consider the sequential changes in expectation from knowing the 1^st color, E_A = Z_BZ_CV = (1/9)($2700) = $300 of Eq39, to knowing the 1^st and the 2^nd colors, E_B = Z_CV = (1/3)($2700)=$900 of Eq43. The increase in expectation is via Eq44

52.)    ΔE = E_B − E_A = Z_CV − Z_BZ_CV = U_BZ_CV = T_B

We can also express this ΔE increase in expectation with the simplest of algebraic manipulation as a decrease in the function’s U=1−Z uncertainties.     

53.)    ΔE = Z_CV − Z_BZ_CV = (1− Z_BZ_C)V − (1− Z_C)V = U_BZ_CV = T_B

In the above (1− Z_BZ_C) is the uncertainty of getting the sequential triplet once you already know the 1^st color with (1− Z_BZ_C)V being the meaningful uncertainty or anxiety felt about getting the sequential triplet once you know the 1^st color. And similarly (1− Z_C) is the uncertainty of getting the sequential triplet once you know the 1^st and 2^nd colors with (1− Z_C)V the meaningful uncertainty or anxiety felt about getting the sequential triplet once you know the 1^st and 2^nd colors. Clearly we see in Eq53 that the ΔE increase in expectation is equal to the reduction in meaningful uncertainty.        

In classical information theory, information is said to come about as the resolution or reduction in uncertainty. This is intuitively sensible. If you have no uncertainty about the date Valentine’s Day falls on, Feb. 14, are sure of it, and somebody tells you that Valentine’s Day is on Feb. 14, that message is not information for you because you already knew it, had no uncertainty about it. On the other hand if you did have uncertainty about the date Valentine’s Day falls on, weren’t sure, and somebody tells you it falls on Feb. 14, it is information for you.

So what’s told you that eliminates uncertainty is information for you. Let’s now generalize a bit more to say that what’s told you that reduces uncertainty is information for you. This would be saying that finding out the 2^nd color by guessing it correctly or paying for it is information for you because you had a U_A=1−Z_A uncertainty about it before you knew it.

Now let’s go one step further to say that what reduces meaningful uncertainty is meaningful information for you. But what reduces meaningful uncertainty increases expectation, so what increases expectation, ΔE>0, as in finding out the 2^nd color, is meaningful information for you, which is hard to argue about given that knowing the 2^nd color is certainly information for you and meaningful information in its enabling you to get the V=$2700 prize.

Next note that the ΔE>0 meaningful information in Eq53 of the 2^nd color is equal to the T_B excitement in finding out the second color, which defines meaningful information in terms of the emotion we feel in association with it. And also note that meaningful information is specified in terms of Z probability and U uncertainty parameters that are defined most primitively as Z=ε/K² and U=Y/K² matrix functions. Then recalling that Z=1/D from Eq17 and that the average energy diversity, D_RC=KN/(K+N−1), of a thermodynamic system has a correlation of .998 with Boltzmann’s entropy shows the connection between information and entropy that we will flesh out in greater detail as we proceed.   

TO BE CONTINUED 

Copyright, May 18, 2012, Calabria