What's the relationship between Chess Theory and the Mathematics of Chess?

Hi all,

A bit of background: I'm doing a PhD in the philosophy of maths, I'm interested in and have a degree of education in Mathematical Logic and Set Theory. I'm casually trying to teach myself to be better at chess. When I started trying to teach myself chess I went and bought some chess books (I'm working through Nimzowitch's "My System" at present). I was really surprised by how unlike a maths book it was. I was expecting formally stated conjectures, lemmas and theorems, and that's just not how Chess Theorists work. This surprised me and so I started thinking about the relationship between the Mathematics of Chess and Chess Theory. I've included my thoughts. Disclaimer: I'm coming at this as someone who knows more about maths and what makes for interesting/good maths than about Chess. I'm a distinctly average Chess player. If you think I'm misrepresenting Chess Theory, please tell me I'd be interested to hear the way in which you think I'm misrepresenting it.

Important difference 1: Mathematics is (usually) non-Constructive, Chess Theory is highly Constructive.

What are constructive vs non-constructive theories? Constructive Mathematics restricts proofs to only positive proofs, e.g. no proofs by contradiction. Mathematicians, who tend to work non-constructively, are satisfied simply by knowing that something exists, even if they can't tell you what it is. For instance, a proof that there is a winning strategy for White (say) could look like one of two things. It could literally gives you an algorithm that a White player could follow and always win, or it could prove White does have a winning strategy (e.g. because the contrary leads to a contradiction) but not tell you what that strategy is. The first is a constructive proof, I take it that would be of interest to Chess Theory (given one reservation outlined below). The second is a non-constructive proof. That might be minimally interesting to Chess Theory, but would be unlikely to have any real impact. (It might have some. If, for instance, the proof had a constructive element. So, say, there's a constructive part of the proof that says you can force a win from being a minor piece up, but the lemma that you can force being a minor piece up is non-constructive; I take it the constructive part of the proof would potentially be interesting).

Chess Theory is generally constructive. Mathematics is usually non-constructive.

Caveat on this: Chess Theory might allow the Axiom of Choice, which amounts to saying things like "pick an arbitrary possible board position". AoC is often seen as non-constructive, but that's contestable.

Important difference 2: Chess Theory is highly interested in limiting the Algorithmic Complexity of solutions. (A.K.A Maths can work with ideal agents in a way Chess Theory can't)

Suppose I had a constructive proof that White has a winning strategy, i.e. I had a winning algorithm for white. The problem is that, even for the best computer in the world, let alone a human, the algorithm has a run time of a few days, which far out runs the time I'm allowed for the game. A Chess Player using a flawed but quick to calculate algorithm will beat a player using a perfect but slow algorithm, as the clock will run out.

Important difference 3: Chess Theory wants to be able to accommodate mistakes in the algorithm, Mathematics doesn't need to do that.

Suppose I have a quick algorithm that is perfect (i.e. if applied correctly will always win). However, it's complicated to calculate in my head and I make a mistake 1 move in 10. This algorithm could have one of two properties. It could be "Safe", in the sense that blundering 1 move in 10, but making optimal moves the rest of the time, still gives me a high chance of winning. It could be "unsafe" in the sense that even small mistakes generate a significant drop in win chance. If you think about this in terms of possibility space, there's an island of guaranteed win that the algorithm is trying to take you to, but all the nearby possible games involve you loosing. If you think about this in terms of usual algebraic notation, there might be a move that looks !? that, contingent upon some exact sequence of moves, is !!, but, if that's the point at which you blunder makes the whole sequence a blunder.

Chess Theory is interested in Algorithms with a high degree of Safety. Both the Mathematics of Chess and the Computer Science of Chess can tolerate a much lower degree of Safety.

Important Difference 4: Mathematics of Chess is interested in generalisations in a way Chess Theory isn't

If a mathematician had a proof, constructive or otherwise, for, say, White winning, the next thing they'd look to do would be to generalise this, in some way. For instance, a proof of which "chess like" games that involve checkmating a king, but have alternative movement rules, board sizes and piece numbers, yield a winning strategy for white. I take it such a proof would be very interesting to Mathematicians but not very interesting to Chess Theory (though probably very interesting to Chess Theorists, as people!).

Characterisation:

So, if we were to characterise Chess Theory as a branch of Mathematics / Computer Science, it would be as a problem in a type of fault resistant, low-complexity constructive decision theory. I add the Computer Science bit, because this kind of problem does exist in CompSci, though maybe not with this much fault resistance or low-complexity, unless you're building something to operate on the bottom of the sea, in outer space or in nuclear winters!

What could Maths offer Chess Theory?
Not knowing that branch of mathematics well, I couldn't point to a theorem or area and say "Chess Theorists should learn from this". Maybe it's just the verificationist in me, but it maybe would be nice to state certain concepts in Chess more formally (e.g. what counts as "development", exactly and without any vagueness?). If we did, we could then statistically test if the best chess computers (that play better than the best GMs, reliably) are actually following these "rules of thumb". Without a precise definition, there's no way to do that (well, there is, you can teach a neural network to do it, but that's then not helpful for moving forwards). Maybe there are formal definitions of these things and I just don't know about them. This would be a limited methodology, I could expand upon that in more detail but won't.

I'm spitballing in this section. Tell me to shut up if this is obviously wrong for some reason I don't know, or has already been done!

How can you have a blitz rating of 752 if you are brilliant( in maths)

Hey  If you read through the post, you can see what I think are some of the big differences between Maths and Chess. If you're wondering how someone could cognitively be able to do one well and not both well, that should give you some incite! See also my disclaimer  

Many great chess players have been mathematicians. The President of my local chess club is a Professor of Mathematics. But mathematics is not just abstract proofs, construction & existence. When I studied math (over half-a-century ago) I was fascinated by how mathematics applied to the real world.

At a lunch table a few years ago I mentioned that most math has some application in the real world. A man at the table asked about prime numbers and how could they have any use in the real world.  He had never heard of Rivest, Shamir or Adleman .

As Richard Reti showed us, the geometry of the chess board yields surprises. Almost a hundred years later discussions of geometry are still fuzzy in chess books & blogs. See my post in this forum General Chess Discussions of June 13, 2018  Distance in a metric space... . 

The book you are looking for is:

The Grand Tactics of Chess: An Exposition of the Laws and Principles of Chess Strategetics, the Practical Application of These Laws and Principles to the Movement of Forces: Mobilization, Development, Maneuver and Operation.

By Franklin Knowles Young.

https://www.amazon.ca/Grand-Tactics-Chess-Strategetics-Application/dp/1146152469

It is not taken seriously by mainstream chess theoreticians, though.

@garethpearce Chess theory and practice are mostly not about "exist" or "does not exist", but the probabilities of something to happen when the final goal is to develop an unstoppable initiative that ends in checkmate.

The logic is similar to the one we use every day. We don't have all the information nor have the resources to develop algorithms supporting every decision. Yet we make decisions towards a goal based on the probabilities of success at each step.

… so it's more like Quantum Theory, then?

Not really. If you apply mathematics reasoning to chess, once you get material advantage (to give an example) you may say: "I have a win, I just may not know where it is". Which may be true in a K+Q vs. K ending, but not necessarily in every position where a side is a queen up.

Point being the number of variables in chess is high enough as to elude –at this moment– an algorithm and certainty when making a decision. So we simplify the matter by taking a few elements in a position to make a forecast of how the situation should (or may) evolve.

It's not a perfect method, but at least works in most cases. Now, how much we don't know can be seen every time a GM, making comments on a live game, watches an engine's line and says something like "where did that come from?" or "that's engine's move". Another hint can be seen in the opening's preparation: No matter the tens of thousands of engines constantly analyzing the main lines, we still get new moves and new ideas.