Wednesday, April 27, 2022

Discrete distributions, probability mass and so-called reality, etc.

I was reading about the probability mass function (pmf) in a statistics textbook yesterday and suddenly had a flash of insight regarding the entities of experience, what you might call objects of reality. I had been thinking (while stumbling about my house seeking more coffee at various times during the day) about the philosophic tradition of considering material objects to be more or less defined by their extension, the apparently plain fact that they occupy space and resent attempts to force sharing that space (a concept not politically correct these days). 

I knew from my study of quantum mechanics and high-energy physics, in particular field theory, that the idea of solidity of matter breaks down at the level of particles, which of course are the constituents of matter. If you were to divide a chair into smaller and smaller pieces, until you arrived at atoms, and then at the neutron, proton and electron of which the atoms are in turn comprised, and then the neutron and proton into quark triples, bound together by gluon slinkies, you would be led to consider these objects more statistical phenomena. That is, instead of resting your hand on a chair, you would metaphorically graph how often your hand passed through the chair without resistance, noting the number of times this was easily done and the number of times it required some effort.

Luis Walter Alvarez, Nobel Prize Physics 1968, helped develop the present-day partly statistical art that is high-energy physics. There is now less promise in high energy physics, we as a race (to the extent that generalization can be made these days) having run out of energy as we examine smaller and smaller regions looking with some desperation for hints as to what we are missing in our quest to understand why the Universe is beginning to inflate again, as if God decided to blow into the balloon again at this late date (13.7 billion years roughly) and accelerate our pre-existing antisocial relations with other galaxies, absent the occasional collision in the eternal night, the banging cars of Drive by Truckers and 21st Century USA, as it were (was, and not necessarily ever shall be). 

Alvarez had been skeptical of the histograms experimental physicists used to plot possible masses from the number of events (and character of trajectories visible in bubble chambers suggesting momenta). He asked Gerry Lynch for a computer program (c. 1950's) to allow him to simulate such histograms (histograms plot vertical bars whose heights represents the number of events and the x-axis energy or other independent variable) with Monte Carlo simulations (those having been developed by an acquaintance and colleague of his, Johnny von Neumann, along with Stanislaw Ulam, in their work on the atomic bomb project during WWII). Alvarez found the faked bumpy plots (physicists interpreted the bumps as possible particles, objects if you will) showed as much structure as those drawn from experimental data. Alvarez pushed for experimenters to have their experimental plots randomly placed in a pile of printouts of faked plots based on similar parameters and attempt to pick out the "real" plots, thereby reducing many of the erroneous results published at the time (and beginning, in my opinion, the art of statistical high energy physics experimental practice). Here is one of the early plots:

That figure (Figure 15 from Alvarez' December 11, 1968 Nobel Lecture) shows distribution of energy  in experiments leading to discovery of the K particle. Notice the vertical axis is number of events observed and the x axis horizontal the derived mass of the particles observed, in million electron volts (remember that mass and energy are equivalent so in high energy physics masses are always given in electron volts).

In digital signal analysis, you can plot histograms also:

The relation to the probability mass function in statistics is

In any case, what caught my attention was that the statistics text described the function that generates the probability mass function (right hand figure above) is discontinuous at points described as mass. That is, for a discrete distribution there are points in the sample (the horizontal axis) where the probability of occurrence (vertical axis above) is zero and points where there is a probability. You can distinguish that idea from that of the probability distribution function, which describes a similarly shaped plot (a so-called normal curve of Gaussian) that has no discontinuities, but is rather composed of real numbers, continuous. 

Those so-called real numbers were a, well, real problem for the ancient Greeks, who had been quite content with measuring and calculating ratios of numbers that described the lengths of sticks (ironically one might call them real objects) compared to one another. This problem manifested in their failed attempt to find some sticks that could identify the length of the diagonal in a square with sides of length 1. You can read about the mathematics involved in this problem in a paper I wrote describing the use of the automated theorem prover, Isabelle at Bentley paper on Isabelle software . The link should open the pdf in your browser, but if it doesn't you can click the "download" button and just get a copy locally. I would recommend downloading it instead anyway and reading it in your pdf viewer of choice, because the paper is hosted at Github, which strips the internal links of the document---I hate to read anything that does not have links internally with an index, table of contents, etc, because I don't have time to waste at my age and with the world in slow motion disaster. The length of the diagonal in this case turns out to be irrational, a word that might have already occurred to you if you have read this far in my post, grin.

The insight I spoke of to begin with was that the Universe, the stage upon which our little parts as conscious beings (some of us anyway) are played, really has no solid objects, but only local energy field peaks which interact with other fields. For example, we can sit on a chair without passing through to the floor immediately because the Pauli exclusion principle makes electrons (the peaks in "position" of the electron fields) vehemently refuse to be crowded, i.e., the electron shells of the atoms of the chair resist the electron shells of the atoms in our buttocks. The gravitational field of the Earth, meanwhile, really would prefer that we continue on down to the floor, but a détente is reached, for the moment. 

Speaking of détente, I look back with some nostalgia on those days of my youth when we merely faced the Russians with mutual assured destruction our treaty of coexistence (wrote and recorded a rock song about those days some time back Dalton Bentley & Chas Thomas play Last Martini at Amazon). Now we have that murderous coward Putin slaughtering women and children in Ukraine, while blustering that we better not interfere or "there is a risk of nuclear war." Really? I spit every time I hear Putin or his puppets make such threats. I say, "why don't you take a shot at an American or NATO soldier and see what happens to you when you are not using missiles on women and children." I just hope Biden has made it clear in some way that the US will use any and all force required in the event of the use of nuclear weapons by Russia. I know Biden is being advised by many to speak softly, but I can see it in his face (and that of his wife) that he is as disgusted and angry as I am at what is happening in Ukraine (and also fed up with the blustering threats of Russia).

When I am unable to work (chronic fatigue syndrome among other curses of age), I have been watching the Lord of the Ring series of 6 DVD movies in recent weeks. It is eerie to see the similarity in the actions of evil forces in those movies with that of the evil Putin. Aside from that, Tolkien (and the producers of the movie versions) captured the mystic beauty and power of the Virgin and the goddess archetype in the two women featured as more or less goddess-like benefactors. It seems like lifetime ago that I lay in a bed in an Army hospital in 1968, broken hip bolted back together after a motorcycle accident (a Chevy El Camino field displaced my immediate field), reading a parody of the series (Bored of the Rings by Henry Beard and Douglas Kenney, who later founded National Lampoon). Movies help me get through the increasing periods of incapacity lately, but I don't want to make a hobbit of it (grin). I do feel compelled to do as much useful work as possible on the way out, which at this point is really only possible through research and writing. The fact that my work is all complete futility in the event of the collapse of civilization is depressing, but has not deterred me. 

I am more and more disconnected, out of place, out of time. Cannot stand the stupidity of television for the most part, every time I see another new company ending in a cutsy "y" (EverlyWell, Newsy, etcetera-y) I want to gag. Cannot watch PBS anymore, the constant distortion of facts for their pc/multicultural, antiwhite anti male antiwestern civilization evil agenda drives me immediately to apoplexy. The Story Channel has no agenda, but is so inept that it is equally objectionable (their fractured attempts at history feature hemming and hawing gesticulating morons never quite arriving at a factual and precise statement). Is it always this way for the old? I can imagine a world where I watched the steady advance of man and felt some comfort in knowing there would be beauty, nobility, knowledge, love and the full measure of life into the future. Possibly that is simply yet another curse, i.e., the ability to see what could be, what should be, yet be planted in a perverse parody of the idea. 

Hearing their songs outside the other day, I wondered if the birds are really the lyric poets of the dinosaurs and have been singing the same lament for the Day of Destruction 65 million years ago (hope to see Attenborough's BBC special, The Final Day, in May on PBS, Attenborough is a man uncontaminated by the general PBS agenda, like myself, his days are numbered). Turned on my Sandisk mp3 player FM last week in the early am hours during a period of waking (secondary insomnia) and was uneasy after Peter Van De Graaff, the host of the classical music show, played March to the Scaffold by Hector Berlioz, an all percussion version. Is reality trying to tell me something, grin?

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