The Poetics of Chess and Math

I had a conversation with my friend William the other night about what makes poetry (or rather, the poetic) possible. It's had me thinking since we talked and I think he's really onto something that applies well to disciplines I know a little bit more about.

Basically he suggested that all people have an imaginative or creative sense within us that pushes outward. Our rational, realistic mind, or even perhaps reality itself, pushes back inward against the imaginative sense. The poetic arises from the pushing back and forth between the two. Both are necessary for the poetic, since imagination unchecked by reality is incoherent and unintelligible while without the imaginative sense we cannot see any possibilities which might create tension with the way the world is, or seems to be from a given angle. Without the imagination there would be nothing to say, no reason to feel any particular way about a given event or perspective.

This would probably express pretty well what it is that a human mind can do that I can't imagine a computer doing. Can a set of algorithms imagine? Is imagination reducible to parallel processing? Can a computer write a poem? I suppose if you assume that is happening in our brain is a set of rigid algorithms, then the answer has to be yes. But it seems to me to be the quintessentially human activity.

A computer program can try different possibilities given a set of parameters for evaluating them, but how can a program question the parameters that it is given? If it is told to evaluate the parameters of its evaluation, it has to be given parameters to judge its parameters by. How can a computer think "outside the box" of its own programmed goals? Yet I think humans are able to do this.

Why People Are Better At Chess

In any case, I think I can see all essentially human endeavors as fitting into the scheme my friend suggested. Chess is a great example, and the difference between the way a human plays and the way a computer plays is illuminating.

People are still better than computer programs at chess, despite the vast advantage in calculations per second. People can perform long-term strategy, imagining where they want to be a long ways down the road, and then, when there are bumps in the road, make imaginative connections between what they planned and what actually happened. Then they can still get where they wanted to go.

A computer cannot use imagination in the same way we do. It has to consider each possibility and evaluate it according to a set of rules. It is stuck with the realistic mode of thought without the imagination pushing out to new and unexpected places.

It cannot (although maybe someday it will) choose to go through a seemingly worse position if it cannot see exactly how it leads to a better position, while a human player's imaginative sense might push him to try.

The way chess develops is through such innovations. Chess is a very different game today because after a lot of people try a given variation, its possibilities are better understood. Sometimes it is "refuted" when a certain response is shown to be effective more often than not. Sometimes it is left open, but the intricacies become more and more subtle as it is explored in greater depths.

A computer could conceivably simulate all that, if it were powerful enough to simply try ever possible game, and this is where the analogy fails. Chess could conceivably be "solved" meaning computer methods could eventually figure out what the "right" move(s) are in any given position.

But if in the world (the physical world, the world of relationships, the socio-political world, etc.) the possibilities are not limited in number as they are in chess, it might be that computers will never be able to perform what the human imagination can, since the computer will always have to limit its considerations to possibilities that fit certain parameters. The human imagination will always be pushing against whatever the parameters are that we believe are there.

This means humans often try things that don't work, but it also allows for moments of genius, moments of the poetic, when the imagination pushes through and finds a way that the realistic mind would have had to rule out from the start.

What It Means To Do Math

I was just reading some career advice for mathematicians when I came across this (I cut out the more specific math stuff, but I thought this explanation was so close to what I wanted to say that I had better just quote the whole thing.):

"As an undergraduate one is often first taught mathematics in an informal, intuitive manner …but then told a little later that to do things "properly" one needs to work and think in a much more precise and formal manner … It is of course vitally important that you know how to think rigorously, as this gives you the discipline to avoid many common errors and purge many misconceptions.  Unfortunately, this has the unintended consequence that "fuzzier" or "intuitive" thinking (such as heuristic reasoning, judicious extrapolation from examples, or analogies with other contexts such as physics) gets deprecated as "non-rigorous".  All too often, one ends up discarding one's initial intuition and is only able to process mathematics at a formal level.  The point of rigor is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition.  It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems; one needs the former to correctly deal with the fine details, and the latter to correctly deal with the big picture." (from this page)

So we see that the actual work of doing mathematics (which is to say, proving new things or improving the communal understanding of numbers/logic/mathematical constructions, rather than mere problem-solving) involves the interplay between an imaginative sense and a realistic or rational imposition by the rules of logic/nature/what-have-you.

The imaginative part is what allows mathematicians to see possibilities of new concepts, form possible relationships between concepts, or envision striking possibilities for new ways of trying to answer certain types of questions. The realistic part is what pushes back against the imagination, cutting out poor use of intuition or careless guess-work.

The development of modern mathematics has involved the generalization and interrelation of concepts that arose independently in various fields. This just cannot be done without a bold imagination, that is, without a willingness and ability to envision what the world might be like in a radically open way. One has to be willing to say "maybe concept A and concept B are really just two examples of some larger concept C" or "solutions to problem type A might give us a starting point for what solutions to problem type B might be like" and then explore imaginatively how that might be so.

At that point the rational sense jumps in and pushes back, sometimes saying "nope, that just ain't the way it is" and sometimes saying "maybe but you have to try harder". Sometimes the imagination is spent, and the rational mind has to grind out a path based on what has worked before, and the imagination steps back and fills in the conceptual connections later. But at some beautiful points, both agree, and the imaginative and rational senses exult together at the beauty of what their long struggle produces.