Game of Theorems

This article is part of a series on The Structure of Theorems. See also:
1. Theorems’ Almanack; 2. The Greatest Theorem; 3. Game of Theorems

Few (currently) practicing mathematicians – in my experience – deign to concern themselves with issues surrounding set theory and the foundations of mathematics. In these areas reside the very definitions upon which the rest of our discipline rests; in any case, our discipline proceeds nonetheless, despite its practitioners’ regrettable ignorance. Mathematicians are pragmatic people. In 1949, Bourbaki – perhaps apprehending a subtle need to defend itself – titled a paper Foundations of Mathematics for the Working Mathematician [1].

This state of affairs, unfortunate as it is, explains the surprise and intrigue I often feel when I take time to explore foundational issues. The definition of the so-called Axiom of Determinacy particularly struck me. This set-theoretic axiom is formulated in terms of a certain type of two-player game – an infinite sequential game, in fact, in which two players take turns playing integers, leading ultimately to a sequence of integers of infinite length. The axiom (indeed, it’s something we might choose to suppose) states that every such game – that is, every choice of a victory set, a distinguished collection of possible infinite sequences whose members define the winning outcomes for the first player – is determined, in the sense that one player or the other in the game has a dominant strategy.

I’ll explain these terms below. The important thing, here, is that this mathematical property is defined by the existence of dominant strategies for a certain class of two-player games. This intrusion of an apparently economic, or game-theoretic, notion – that of the two-player game – into mathematics surprised me.

This intrusion, in retrospect, should have been less than surprising. Continue reading

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Conway’s Game of Coinage

It was a long time before I talked to Professor Emeritus John Boardman. I’d seen him walking in the halls before, sure. Wisps of white hair ran across the top of his high forehead, and he walked with a stoop. Sometimes his eyes appeared pried too far open, as if he were struggling to look ahead despite the downward incline of his head. He had a habit of incessantly clearing his throat, even as he walked, and even as he sat in his office; his office was adjacent to the math help room and I often looked up from an undergrad’s work only to notice the familiar sound again. I’d never seen anyone else talk to him, either.

This changed one evening as we all sat down for dinner in a sparsely patronzied restaurant after an invited speaker’s seminar talk. There were about six or seven of us at the table, and the others quickly became distracted in conversation. Professor Boardman, alone at the end of the table, sat to my left.

The first thing I noticed was his sharp, almost Russell-esque British accent. Continue reading

Buried Treasure

A week ago, I took the USMLE step 1, an 8-hour Goliath of a test. As daunting of a prospect as that is, the study process was much more extensive. I studied for twelve hours a day for six weeks.

I had intended to spend those six weeks memorizing a whole lot of facts. But I eventually, I found that I wasn’t just learning facts; I was learning structures. This took a lot of the drudgery away, since the latter are quite a bit more fun to study.

Now that boards are over, I find myself stepping away from medicine and looking towards other fields. Do other fields, like medicine, produce elaborate structures from the underlying facts and principles? Must they? Are some resultant structures better than others? In medicine, there often is a right answer (especially on boards). Is the same true of other fields? Where, if at all, does the rubber meet the road? Continue reading

The World-Builders

This article is part of a series entitled The Unlimited Mind. See also:
1. On Memory; 2. The Genius Within; 3. The World-Builders

I’ve been fascinated with expertise since childhood. And it started over the chessboard. My dad would beat me—swiftly, crushingly, and above all, effortlessly—time and time again. He understood lines and positions in a way that I just couldn’t, and, as it seemed to me, would never be able to. My first question at the end of most games was: “where did I go wrong?”

globe-chess

Chess has served as a popular topic of study for those seeking to understand expertise.

Almost more unnerving than my dad’s ability was the fact that there were people out there who could, just as easily, beat him. “In college in Russia, I played a classmate of mine, who was a master,” he told me once. “I would think all night about my move, and then the next day in class, he’d move right away. Still, he beat me easily.”

Thus my interest in expertise was born. It seemed that some just had some sort of divine gift, which beckoned them onto a higher plane of understanding. For me to attempt to reach those heights would be futile. I could only watch in awe from below.

As I grew older, my skills improved. My games with my dad grew stricter and cleaner, until, one day, I beat him. In time, whether I won or lost, I was always able to give him a fair fight. I came to appreciate chess as an incredibly rich and rewarding game.

But my view of expertise—now that I had a taste of it—had lost a bit of its sparkle. Continue reading

The Research Player

This article is part of a series entitled 20th Century Chess Greats. See also:

  1. Mikhail Tal: The Deep Dark Forest
  2. Bobby Fischer: The American
  3. Tigran Petrosian: The Iron Fortress
  4. Mikhail Botvinnik: The Research Player

In a previous article, I addressed the classic nature vs. nurture dichotomy, in which skill is attributed to both genetics and environment. I noted that, while talk of nature often concerns only genetic inclination towards talent, we might also consider genetic tendency towards drive, which prompts the skill-seeker to alter her environment such that she might increase her skill beyond that which her combination of environment and natural talent would otherwise allow.

For example, Fischer became a great player only because the three stars aligned. His environment led him towards chess; his talent, presumably, brought strong results early-on; and finally, his incessant drive allowed him to keep studying long after most would-be champions would have put the board away.

Today, I advance an even stronger argument: perhaps, natural talent doesn’t even exist; perhaps we’re dealing only with environment and drive. Natural talent is simply an apparition, a phantom, often confused with natural drive, but not even existing in its own right. Or, more profoundly, perhaps the existence of natural talent is not scientifically-supportable, and, whether or not it exists, we need not believe in it. Continue reading

The Iron Fortress

This article is part of a series entitled 20th Century Chess Greats. See also:

  1. Mikhail Tal: The Deep Dark Forest
  2. Bobby Fischer: The American
  3. Tigran Petrosian: The Iron Fortress
  4. Mikhail Botvinnik: The Research Player
Petrosian

Petrosian as a youth in Moscow

1963 world chess champion Tigran Petrosian was nearly deaf, a trait that produced several interesting tournament tales. In one tournament game, Petrosian offered Serbian grandmaster Svetozar Gligorić a draw. Gligorić was taken aback and declined, but seconds later he reevaluated and accepted the draw offer. Petrosian, however, had already entered “concentration mode” and had shut off his hearing aid. He didn’t hear Gligorić’s re-offer, kept playing, and went on to win the game.

At a candidate’s match in Seville, Petrosian played Robert Hübner, who was at the time a frail 22-year-old college student. Hübner was driven to distraction by the noise produced by the crowds and by the streets outside. Ultimately, he overlooked a winning move, burst into tears, and forfeited the tournament. Petrosian, meanwhile, hadn’t heard a sound [1].

Petrosian’s performance on the chessboard matched closely his external demeanor. No matter how destructive and chaotic was his opponent’s attack, Petrosian’s iron fortress would remain standing. Continue reading

Rise of the Silicon Dynasty

blood vomit

The final position of the famous “Blood-Vomiting Game” of 1835 *

Chess computer programs have been beating our grandmasters since 1997. One game, though, still eludes artificial intelligence. This is the ancient Chinese game of Go. Despite its incredibly simple rules, master Go players still destroy our best computers.

Is man’s dominance temporary? Or will Go always be our territory?

The exploration of why humans still win at Go, and whether or not they always will, promotes discussion about game theory, computation and human psychology.

Continue reading