Mathematics, AI and Chess

Shannon's number is an estimate of the possible number of games of chess, arrived at by the mathematician and engineer Claude Shannon (1916-2001). Shannon postulated an average of 1000 possible moves for one move by White followed by one move for Black. Then he postulated a typical length of a chess game of 40 moves, and came up with his very famous number, his very famous estimate: there are at least 10 to the 120th power possible different chess games. 

I think Shannon's number is complete garbage. I think it tells us little beyond the fact that Shannon and other mathematicians didn't know much about chess, and that few chess players know much about math. Otherwise, Shannon's number would never have become famous to begin with, and, chastened by so much derisive laughter, he would've headed back to the drawing board to try again. 

In some positions on the chess board, there are many possible moves. For White's first move, there are 20 possible moves: 16 by the Pawns and 4 by the Knights. 20 choices also for Black's first move. There are other position in which a player would have far move than 20 possible moves. For example, if a player had 3 Queens, 2 Bishops, 2 pawns and a lot of space.  

In other positions, a player has only 1 possible move: if his King is under attack and there is one 1 possible way to defend it. Or if there is only 1 possible move which would not expose his King to attack. And there are positions where only 2 moves could defend the King against attack. Or 3, or 4, and so forth.

Or, instead of threats to the King, the number of moves could be limited by his pieces being blocked by his opponent's pieces, or by his own pieces. 

How would one get an average number of moves out of all of these different kinds of positions? How many different positions are there with just 1 possible move? How many positions yield 50 or more possible moves? I have no idea. Not the faintest idea. Furthermore, I have yet to see anyone even asking this very basic kind of question when trying to determine the number of possible chess games. I'm not saying I'm the only person who's asked these questions. I'm saying that not enough people have been asking them insistently enough for the evidence of their existence to have come to my attention.

Okay, now, the number of moves in a game. Average it out at 40, like Shannon? That's ridiculous. Checkmate can happen after 2 moves, or 4 moves. It's not unheard of for checkmate to happen after 10 or 15 moves, or 20 or 30. Conversely, some games have gone on for hundreds of moves. Has anyone even attempted to calculate the number of ways in which a game could last for over 100 moves? Or the number of different ways in which a game could go on, limited by the rule that one player can claim a draw if 50 moves go by during which neither player captures and piece or moves a Pawn? Did you notice that I said that a player CAN claim a draw under those conditions? We may have to make such draws mandatory and automatic if we wish to make the number of different games finite -- or perhaps not, I'm not competent to say.

These are just a few examples of the different numbers which would need to be calculated before one could attempt to combine them all and come up with any sort of reasonable estimate of the number of possible chess games.

I don't believe that AI is here. I haven't seen a product designed by AI which wasn't hideously ugly, haven't read a poem written by AI which wasn't ridiculous, haven't interacted with a search engine or automated call center which wasn't infuriatingly stupid. 

And I haven't seen an impressive attempt yet to estimate the number of possible games of chess, let alone solve the game by coming up with the moves which will always win, or always draw against perfect moves by the opponent, the way that checkers has just recently been solved. And when those things finally do happen, which they will if we don't kill ourselves off first, it being ultimately just a matter of crunching very, very big numbers, actual human-like communication and creativity will still be far off, or, perhaps, ultimately inaccessible to mathematics.

Buy books about AI and chess on Amazon: https://amzn.to/4jaiCKx

The Science-Humanities Split

Perhaps you've heard: STEM -- Science, Technology, Engineering and Mathematics -- and the humanities -- art, literature, history, music, etc -- have split apart from one another.

Perhaps you've just read the previous sentence, and asked: Whaddya talkin' about, Steve? Was there some time when science and art actually got along?

Oh yes. The time was up until the eighteenth century, and can perhaps be seen most dramatically in Western civilization -- I really don't have much of a clue about non-Western civilizations, but I'm trying to catch up -- in the example of philosophy, and of individual philosophers. Up until a few centuries ago, the leading philosophers were also the leading mathematicians and scientists, and people generally took for granted that this was so. Descartes, Spinoza, Leibniz were the leading philosophers and the leading mathematicians of their time. Newton was a leading scientist and mathematician, but he left scarcely a mark in what today is generally considered to be philosophy. The split seems to be beginning already in Newton's time. Kant, Schopenahuer, Marx, Nietzsche and the other most prominent 19th-century philosophers are not, to my knowledge, enthusiastically read today by most scientists. Bertrand Russell and Alfred North Whitehead were prominent 20th-century mathematicians and philosophers, but they were very unusual in being both at the same time. In the 21st century, Stephen Hawking and Neil deGrasse and Lawrence Krauss Tyson have said that philosophy is worthless, without causing much of an uproar among their fellow scientists, which shows you how little you can know about philosophy today and still be a brilliant scientist.

My brother, an engineer, goes probably even farther than Hawking and deGrasse Tyson and Krauss in his ignorant dismissal of philosophy -- and that's all this is: ignorance. If Hawking or deGrasse Tyson or Krauss or my brother knew very much at all about philosophy, they wouldn't say such things.

This unfortunate split, this destructive antagonism between two vital types of human endeavor is not, of course, all the fault of the scientists. Those who have objected to the dismissal philosophy by prominent scientists have included other prominent scientists. And it's certainly not as if all philosopher, artists, musicians, poets etc, have a decent appreciation of STEM. There is plenty of fault on both sides of the split.

I've tried to bring the sciences and the humanities back together, but I could've done much more. I stopped studying math in school just as soon as I was allowed to stop studying it, after completing 10th-grade geometry. I usually had the best math grades in my class -- the only exception I can remember was in 9th-grade algebra. The teacher posted a constantly-updated list of the members of the class by our current grade. I don't remember whether A was 90% and up, or 94% and up, or what exactly. I do remember that it was possible to score above 100% with extra-credit work, and that the 2 of us at the top of the list were over 100%, and that I wasn't on top. That felt very strange, not being the best math student in sight.

That 9th-grade algebra teacher, and some other math teachers I had, talked to me very excitedly about how far I would be able to go in math. They didn't realize that I didn't enjoy math at all. It was my undiagnosed autism which allowed me to make those grades without trying and without being interested.

The 9th grade was 45 years ago. Since then I've made a few feeble attempts to make more progress in math, which, it seems to me, would amount to developing an interest in and enjoyment of math. I was talking to a math teacher the other day, and he said, You have to enjoy math to go far in it.

My brother was valedictorian in high school and got 2 degrees from MIT. He enjoys math. During one of those periods when I was trying to develop an enjoyment, my brother gave me his copy of the 5th edition of Calculus and Analytic Geometry by Thomas and Finney, one of his former MIT textbooks. He's a good brother, even though he is an appalling philistine when it comes to the arts.

Pages 355 through 362 of this book are missing. Did my brother remove these pages before giving me the book? Are there things on pages 355 to 362 which, my brother decided, must remain hidden from librul artistic types such as me?

I still haven't made that big breakthrough, to where I enjoy math. Although, in the past year or so, chess, mildly interesting to me already for decades, has become much more interesting, and a large part of chess, or maybe all of it, is math. (Well, no, not all of it. There's also psychology in sizing up one's opponent.)

And when people like Melvin Schwarz -- co-recipient of the 1988 Nobel Prize in Physics -- are writing about things like vectors, I actually understand part of it. So, hey, lookit that, I actually have learned some calculus! Schwarz also writes things like: "Electromagnetic theory is beautiful!" And I believe him even thought I still don't understand it.


And I still want to understand. So that I can enjoy math at last, and for many other reasons.

Who knows: maybe, if I understand things like advanced physics, I'll become much better at helping people like Neil deGrasse Tyson appreciate things like existentialism.

Computers and Language

So-called artificial intelligence programs are still a long way from tackling language. And by language I mean languages which are spoken by humans, like English or Japanese. "Computer languages," sets of instructions followed by computers, are not the same as human languages. It has not been demonstrated yet that the languages which we speak can be reduced to such sets of instructions. If they can, we're still a long way away from doing it. If they can't, then, in my opinion, that would be one of the reasons not to worry about the machines eventually rising up and killing us all. If a computer was capable of having a conversation with me which was indistinguishable from a conversation with a human, then I'd be startled. And possibly a little spooked as well. I would include written conversations like those in chat rooms.

Math is exact and language is not. Often times letters, the symbols used to record some languages, have been used as mathematical symbols. Roman numerals are one well-known example of that. But while the symbols used in language and math may be the same, what they refer to is not. X + IV = XIV means exactly the same as 10 + 4 = 14, and any number of different systems of notation can be used to express exactly the same thing as 10 + 4 = 14. However, many times the simplest sentences are untranslatable from one language to another. And very many, perhaps most sentences cannot be exactly translated. Furthermore, in many cases, perhaps most, the best translation is a matter of opinion. Highly-qualified experts in linguistics and literature routinely disagree about whether this translation of a poem or novel is better than that one.

As long as we're talking about translation made by humans, that is. With all of the astounding advances made in computing, the best computer translation programs still routinely produce results which are comically bad and far inferior to any work done by any competent professional human translator. The same goes for original written compositions by computers compared with those written by ordinary lit students.

Computers are far beyond humans now when it comes to playing chess. (And if recent headlines have not misled me, computers are about to pass us as Go players as well.) But computers play chess differently than humans, by crunching enormous amounts of data. How do we humans do it? Well, we don't know yet.

Perhaps human intelligence would be less mysterious if the possibility were more often considered that it involves things which aren't reducible to math. Perhaps researchers sometimes resist considering that, because one of the things it would mean is that we're not, in fact, on the brink of developing artificial intelligence. Well, actually, many people think that we're already well past the brink, and that artificial intelligence has already existed for some time.

No doubt, information technology has produced many amazing things, and continues to do so with no end in sight. Maybe there's no reason for me to object to calling some of those things artificial intelligence. I don't go around angrily telling IT people to stop using the term "computer language."

But as I've said, a computer language is a fundamentally different thing than a language spoken by humans. And artificial intelligence, if we want to use that term for things which already exist, and why not -- I mean, people are using that term for things which exist, whether we approve or not -- is still fundamentally different from what human brains do. Playing chess, computers win. Writing poems, computers still have not given us any competition. Perhaps the things needed in order to write great poems are quantifiable. But perhaps they're not.

And perhaps the latter possibility is too often overlooked.

Nerdly Arrogance

"Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral ∫ a b f ( x ) d x {\displaystyle \int _{a}^{b}\!f(x)\,dx} \int _{a}^{b}\!f(x)\,dx is defined informally as the signed area of the region in the xy-plane that is bounded by the graph of f, the x-axis and the vertical lines x = a and x = b. The area above the x-axis adds to the total and that below the x-axis subtracts from the total."

Oh, is that all!

Welcome to more of me failing to learn advanced math. Well, okay, it's not 100% accurate to say that I'm failing, but I'm being thwarted and blocked just a bit by unnecessary obtuseness such as that just quoted. I suspect that there may be a definition of definite integrals out there somewhere which is somewhat more comprehensible to people who don't already know what definite integrals are. I also suspect that communication with the general public is not a strong point among STEM (Science, Technology, Engineering and Math) nerds, and I suspect that it may not be a strong point because, generally speaking, they despise us. They don't particularly want to help. Maybe I'm completely wrong about that because what the Hell do I know about math anyway because for the last 40 years, ever since I finished 10th grade and all of the math I was ever required to study, I've been running away from math. Or maybe it's not the general public at all which nerds tend to despise, but me in particularly, because I in particular tend to offend nerds in some way.

Then again, maybe I'm right. I'm not the first to suggest such a thing. For example, some people have noticed how computers tend to be made by nerds for nerds, and not for the general public; that is to say, the general public has difficulties with computers not because these difficulties are inherent but because the nerds who made the computers don't care much, generally speaking, about the general public and its difficulties. Which is somewhat shocking when you consider that it is the general public which is directly responsible for the nerds making all of those gazillions of dollars, euros, yen and so forth. But they don't have to care because the general public has not yet caught up with the nerds enough to be able to choose the more user-friendly ones from among them, to borrow a nerdly phrase. And don't come at me with Apple, saying that Apple is that user-friendly brand of nerd right before my eyes which I refuse to see. Apple is a rip-off, and ripping people off ain't friendly.

There are not yet enough computer nerds that they have to compete with each other for the approval of the general public. 100 years ago, auto mechanics were just as smug and unbearable as computer nerds are now. A meager supply of mechanics and a huge demand for their services gave them elite status, and they abused the situation and were assholes about it, preferring to be moody geniuses rather than to be helpful and nice and have lots of friends. Then more people learned how to make and fix automobiles, and all of a sudden it wasn't an elite profession any more, and those who had so recently thought of themselves as geniuses suddenly had trouble finding work, and they had no friends to help them.

Those who do not learn from history are condemned to repeat it, STEM nerds.

Update On My Struggle With Advanced Math

I got a book on Fourier series because I liked the way the cover looked.

Yesterday I started reading David Bohm's Quantum Theory and was encouraged by his assertion in the Preface that his approach de-emphasized a dependency on advanced math. But then on page 1 of Chapter 1 he mentions that, despite this approach, some familiarity with Fourier series cannot be avoided in order to understand the book. Then before he got to the Fourier series there was what seemed to me to be an awful lot of complex math for someone who was de-emphasizing math. For all I know it might have been a great de-emphasis indeed for a textbook on quantum theory.

My point is that I still hate math and that that is still hampering my study about things like electromagnetic fields and waves and quantum theory. Perhaps it's in part that 55 is a very advanced age to pick up mathematical studies which one broke off in the 10th grade. I think I know now what that capital sigma means in mathematical... equations. I don't know whether "equations" is the right term for all that weird stuff which Good Will Hunting and famous physicians scribble all over their blackboards.

I've been thinking lately about various math teachers of mine. I got along well with all of them. Maybe that had more to do with my talent than my personality. All of them were disappointed when they saw that I wasn't planning on an extensive career in mathematics. I feel bad about disappointing them. But such a career was never a serious possibility. There's only so far you can go, hating what you do. I imagine that successful mathematicians and physicists, when they see one of those blackboards covered with all of those squiggles, or a difficult paper or book on Fourier series or quantum theory or what have you, feel something somewhat like what I feel looking at an ancient text or a commentary on that text: intense interest, a strong desire to immerse myself in the subject at hand. I can't imagine that a person could get very far on ability alone, unaccompanied by a love for the subject.

Am I Finally Developing An Interest In Math?

I've always been freakishly good at doing arithmetic in my head. Not quite as good as Rain Man, but definitely unusual. However, I've never found mathematics to be interesting. I wonder whether that's an unusual combination of aptitude and disinterest. I stopped taking math classes in high school as soon as I was allowed to, at the end of 10th grade, when the algebra and geometry courses I had completed met the minimum requirements for graduation.

My younger brother took more advanced math courses. Much more advanced. My brother is literally a rocket scientist. He's got a Bachelor's and Master's of Science from MIT. As an undergrad he had a summer internship working for Martin Marietta and NASA on the Space Shuttle. Then between the Bachelor's and the Master's he took two years off from school and worked for a private company which has sent all sorts of things into orbit. A genuine rocket scientist. We're very proud of the little genius.

Every couple of years, I get an urge to study some more advanced math, and engineering and physics. The urge usually passes very quickly, but then again, it keeps coming back. About 30 years ago I had the urge, and my brother gave me this:

It's the 5th edition of Calculus and Analytic Geometry by George B Thomas and Ross L Finney, both professors at MIT when the 5th edition was published in 1979. It was a worn-out copy, my brother was done with it. I don't know whether he had studied this book in high school in preparation for MIT, or if it was the textbook for a freshman class he took at MIT, or maybe both.

I still have that old worn copy of the Thomas/Finney that my brother gave me. But I still haven't looked at it much. I'm currently having another one of those urges to make myself interested in math. But that's just the problem: math remains excruciatingly boring to me. But now I've been looking at that textbook, paging through it. And also looking at other books such as Blatt and Weisskopf's Theoretical Nuclear Physics, Rojensky's Electromagnetic Fields and Waves and Tolstov's Fourier Series. Looking for something which I can honestly say that I find interesting.

I may have found something. Thomas and Finney may have been rather sly when it came to education. There are a lot of word problems for the students to solve in their textbook, problems demonstrating some applications of calculus and analytic geometry, and one of those problems has actually caught my attention. That's right: something in a math book has begun to interest me.

I can't find that problem right now. I think it's somewhere in the first 50 pages or so of this textbook which runs to well over 900 pages. And it's a collage freshman textbook. Freshmen at MIT, which is certainly not the same thing as freshmen everywhere, but still. Early on in a freshman math textbook, there was a problem which I don't know how to solve.

Yes, it was arrogant of me, but I had wondered whether, in addition to boring me, this textbook would also teach me anything, or not. Arrogant, yes. But, for example, I was factoring 3-digit numbers in my head as a small child, years before a math teacher introduced me to the term "factor." Without finding it interesting. Just because it was there in my head.

But somewhere toward the front of Thomas/Finney 5th ed is a problem which, reconstructed from memory, goes something like this: a person of height X is walking at speed Y directly toward a streetlamp of height Z. Determine the rate at which the length of X's shadow decreases.

I can't do that. But apparently the first chapter or two of this textbook will show me how to do it. (Assuming I'm smart enough to understand what the book says.)

And that is interesting. That is definitely an example of something this textbook could teach me. And, apparently, that's just the beginning of introductory calculus. Just scratching the surface.

That's pretty cool.

So, you realize what this means, right? That's right: I'm going to be the first person to win a Nobel Prize in Literature and another one in Physics, plus a Fields Medal.

Newton, Leibniz, Wolff, Mathematics, Leibniz' Reputation and Epistemology

I often think about epistemological subjects: What do we know? What can we know? Why do we think we know what we think we know? In particular, I wonder why some people seem so sure that they know the thoughts, feelings and motivations of others, without believing in telepathy, the notion of which I also reject, pending much stronger evidence than anything I've seen so far. I think about this when I hear about jury verdicts being overturned by things like DNA evidence. I think about it when I hear scientists talking about Newtonhaving invented calculus, and rarely mentioning Leibniz,who claimed that he had invented calculus independently of Newton. During his lifetime and since, this claim of Leibniz' has often been called a lie.

In this earlier Wrong Monkey post, as I waited for this volume of letters between Leibniz and Wolffto arrive from Amazon, I speculated on Christian von Wolff'spossible role in the decline of Latin as an academic vernacular. When the book arrived and I read its introduction by C.I. Gerhardt, it became plain that Gerhardt blamed Wolff for damaging Leibniz' reputation. Indeed, it seems Gerhardt may have gathered these particular letters and published this book for no other reason than to expose Wolff's bad behavior and rehabilitate Leibniz' reputation -- his unjustly tarnished reputation, in Gerhardt's opinion. It is Gerhardt's thesis that Wolff, early in his academic career, was weak in mathematics, too weak to justify the academic positions in mathematics and philosophy which he occupied, and that he basically used Leibniz during this period as an unpaid math tutor, and that after Leibniz' death he claimed many of Leibniz' mathematical achievements as his own and downplayed the help he had received from Leibniz. Gerhardt maintains that this misrepresentation of the facts not only helped Wolff acquire and hold academic posts for which he was gravely underqualified, but that it also gave ammunition to those who maintained that Newton alone had invented calculus and that Leibniz had been lying when he claimed otherwise. Gerhardt maintains that the letters between Wolff and Leibniz which he presents on this volume clearly demonstrate all of this.

Do they? I don't know, in large part because my knowledge of math is not extensive enough to allow me to follow all of the math contained in the letters written in Latin bewteen Wolff and Leibniz and collected in Gerhardt's book. My knowledge of math would've been nowhere near cutting-edge 300 years ago when those letters were new, much less is it cutting-edge now, when all these world-class mathematicians and physicists seem quite dismissive of any notion that anyone but Newton had any part in inventing calculus. Then again, those physicists and mathematicians have almost all been American or British. I haven't heard any present-day German experts weigh in on the Newton-Leibniz controversy. And Gerhardt, who published his volume in 1860 with a thesis of Leibniz having been wronged, by Wolff and also by those who praised Newton at his expense, was German. National sentiments were and are widespread, pervasive and often subtle, much more widespread than the obvious hatreds of extremists fringes. And Newton seems to me to have been the sort apt to fight a bitter feud with or without significant cause, like the one he fought against Leibniz until Leibniz died in 1716, and Leibniz seems like the sort who would not feud without cause, who would be reluctant to fight even with cause, and who would cheerfully admit it when and if some laurels had been bestowed upon him which he had not earned.

But how on Earth do I think I know so much about Newton's and Leibniz' personalities and motivations and about their respective characters? Could it not well be that I am predisposed to like Leibniz and dislike Newton because of some other things each of them wrote which have nothing to do with calculus, so that in this quarrel I am judging Newton too harshly and Leibniz too well? Could it not well be that I too am much too hasty to think that I know this or that? that for instance I am completely unjustified in claiming that national sentiment may have tipped the scales in favor of Newton in the judgment of all those expert mathematicians and physicists?

It could be. Of course I still think I'm right and that I am unusually free of prejudice and unusually attuned to the prejudices of others. But I know I haven't proven anything of the sort. I don't think this essay will change many minds about Newton or Leibniz, or Wolff, or Gerhardt, or math in general. But perhaps it will persuade some readers to ponder more often the nature of things like knowledge and certainty. I think that would be a good thing, although I don't think I can prove that either.