Calculation and Conceptualization

This week I recorded a ChessLecture that should come out in a month or so, called “How to Tell When the Moment is Right.” The question I was looking at in the lecture is, how do you tell when it’s time to calculate detailed variations, and how do you tell when you shouldn’t calculate variations and you should just go on general principles?

Of course, I don’t want to give away the content of the lecture. But I will say this: I’m a calculator. On almost every move of every tournament game, I’m trying to calculate variations, even in positions where there is nothing to calculate. I think that this is one of my flaws as a chess player, and it’s the reason that I get into time trouble so much.

As I thought about it some more after the lecture, I realized that this dichotomy goes way beyond chess. There are two complementary approaches to almost any problem that requires conscious reasoning. We could call them the “top-down” approach and the “bottom-up” approach or, as in the title to this post, conceptualizing and calculating.

The top-down approach is to start out with the big picture and then modify it with specific information. This is the viewpoint of, say, a CEO, who has to set the direction of his company but cannot be bothered (except in exceptional circumstances) with the details of what individual people do.

The bottom-up approach is to start out with the nitty-gritty details and assemble them into a big picture. This is the viewpoint of a farmer. Sow lots of plants, water them day after day, and eventually you feed the world.

The dichotomy is very strong in mathematics, which is something I was also thinking about a great deal this week. In mathematics, too, my style was always to calculate, calculate, calculate. On a few rare occasions — about three or four in my career — the calculations came together into a glorious big picture. One of those times was just a couple years ago, when I was working on the problem that recently turned into this paper (my first mathematical research paper in 18 years). Roughly a third of the way through the paper there is something called the Symmetry Lemma, which started out as a mass of calculations but then got abstracted into a result that barely involves any calculation at all.

But calculation doesn’t always lead to a big picture. Usually, it doesn’t. One virtue of calculation is that at least you get a lot of small pictures out of it. You aren’t left empty-handed. Notice that the paper I just referred to is 56 pages long. Of that, maybe six pages are devoted to the big-picture result that I just mentioned, the Symmetry Lemma. The rest is just massive calculations. These have value, too; they solve the problem I was trying to solve. On the other hand, they don’t point to any general, overarching principles beyond this particular problem. To invoke the chess analogy, the paper is like a single chess game that I won. Imbedded within it, from moves 20 to 26, is this general principle called the Symmetry Lemma that I could use to reason about other, similar positions (and maybe win them, too).

There is another approach to mathematics, the grand systematizing approach, which in my view is represented best by the French school and especially Alexandre Grothendieck. If you haven’t heard of him, you should definitely read this amazing two-part series by Allyn Jackson that appeared in the AMS Notices in 2004. There’s an anecdote in the second part that illustrates what I’m talking about. Grothendieck always reasoned in terms of grand structures and never talked about specific examples, which were beneath him. Once, Jackson writes,

In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. “You mean an actual number?” Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested, “All right, take 57.”

Of course, the punch line is that 57 isn’t prime. (It’s 3 x 19.) People started calling it a “Grothendieck prime.”

If you’re a calculator, the joke is on Grothendieck. How can this world-famous mathematician not know his basic arithmetic? But if you’re a conceptualizer, the joke is on the person who was talking with Grothendieck. Of course, the CEO of a corporation can’t be bothered with how many light bulbs they’re using or who is installing them. Similarly, Grothendieck can’t be bothered with which numbers are actually prime.

As an aside, I think that number theory (a field very close to Grothendieck’s work) is a prime example of a part of mathematics where the conceptualizers have won. The great theorems of number theory are proved by creating vast abstract concepts and then specializing. The subject is replete with terms like sheaves and schemes and idèles and adèles and étale cohomology that mean almost nothing to me. Look at how Andrew Wiles proved Fermat’s Last Theorem, the greatest number theory result of our lifetime. He proved that “all semisimple elliptic curves are modular,” and deduced the numerical result known as Fermat’s Last Theorem as a very incidental example. You don’t need to understand what “semisimple,” “elliptic curve,” and “modular” mean to get my point. These are very broad, conceptual terms.

I wrote at the outset that calculating and conceptualizing are complementary. The beauty of mathematics is that there is plenty of room for both of them. And chess, too, is a game of sufficient richness that both approaches are feasible and reasonable. We all learn some general chess concepts: develop your pieces, control the center, move your king to safety, look for your worst-placed piece and try to make it better, bishops are better than knights in an open position, etc.

I think that there are “conceptual” players who have an even broader repertoire of concepts, although they might not be able to formulate them precisely and they may be mistaken at times (a little bit like Grothendieck thinking 57 was prime). For example, I think of Jesse Kraai as a great conceptualizer. That was one of the things that made his ChessLectures so wonderful and so special: He was always looking for simple rules, expressed in common English (like the “angry bishop”), that governed the game. But I also think that he did a lot more calculating than he realized. He would use his general principles to get to a critical position, and then there would be this little tactic here and that little tactic there, which he would almost brush off in his lectures as inconsequential. Perhaps they were inconsequential compared to the general themes he was elaborating. But if you don’t spot those tactics and calculate them accurately, all of your “general principles” and “simple chess” won’t do you any good.

Anyway, as said before, I’m in the opposite situation. I need to work more on trusting and using general principles. That will be my goal for my next tournament, the Larry Evans Memorial in Reno from April 18 to 20.

For the rest of you, it might be good to pause and ask yourself for a moment: What kind of player am I? Which facet of the game do I need to work more on? If you’re frustrated by the fact that you seem to know more chess than your opponents but you keep making tactical errors or missing opportunities, then y0u’re probably a top-down person and you need more work on calculating. If you’re quick with the combinations and can solve tactics puzzles like an expert, yet somehow you never get the positions where you can use those abilities, then you’re probably a bottom-up person and need more work on conceptualizing.