The perfect game

[Bill Spight]

I'm ignorant about these things, but curious. If the ultimate test is to play everything out by computer, why do we need a mathematical proof? I can see that a proof might be intellectually interesting and have implications for cases that computers cannot handle yet, but it seems easiest just to wait for faster hardware, especially given the rate of progress so far.

And if there is a good reason for being able to write down a proof, would it have any value outside of go?

Here is a logical theorem. A and B are propositions.

(A is true only if B is true) if and only if (B is false only if A is false).

Most people are not used to saying "only if", but once you get over that unfamiliarity, this theorem is fairly obvious. You can verify it by the use of truth tables, showing that it is true for every truth value of A and B, which is akin to playing everything out by computer. But, OC, once this theorem is proven, we can use it in other proofs, without having to write down every truth table.

Of what value is it to know best play on the 7x7 board? Of limited value, certainly, as the corners interact in ways that the corners of larger boards do not. The best initial play on the 7x7 is surely tengen, which may not be true of the 9x9. But the 7x7 still offers lessons in tesuji and principles that apply generally, as do most small boards. The 3x3, for example, has a lesson about eye vs. no eye.

[RobertJasiek]

Computers cannot play everything out because the game space (including all legal sequences of positions) is orders of magnitude larger than the number of atoms in the universe and the operation time is orders of magnitude larger than could be executed during a duration equalling the age of existence of the human species. Mathematics is required to prune calculation space and time. Faster hardware cannot provide sufficient acceleration combined with management of more space than atoms in the universe, unless and until maybe in the far future a sort of super-quantum computer can use enough Boltzmann or quantum states for storage whilst not suffering from the uncertainty principle.

Since computers that do not apply mathematical theorems need to sample, their output can be correct or incorrect. By just looking at the output without using other interpretation means, we do not know which.

Mathematical theorems, where applicable according to their presuppositions, do tell us the truth provided the theorems are proven by mathematical proofs. Yes, they are also intellectually interesting if one appreciates effort and skill or simply the facts of having proven truths.

Proofs can have value outside go theory research if the proof techniques are new and useful. The basic proof techniques are universal to mathematics so only more sophisticated proof structures can qualify. Of course, more likely theorems can have value outside go theory research because they might be applied or modified to achieve that. I have recognised possibilities for that but it would require quite some additional research time to write down such carefully. Think in terms of years rather than days of extra time investment. It takes seconds to point out something (e.g. hypothetical-strategy) whose transcription could be useful but maybe years (in easy cases, several weeks) to work it out carefully for a different or generalised domain.