Pure and Applied Chess

by David L. Duffy.

Personally, I have found some chess analogies, metaphors and thought experiments from philosophers a little unimpressive, speaking as someone who was once a middling club player. Some philosophers have been what I think of as serious players – at least one has been an International Master — but I don’t see their thoughts as widely cited as, for example, Dennett in the case of “Chmess”, or even Sellars’ “Texas Chess”. Serious chess players know there is a technical literature of many thousands of books, and databases containing ~15 million games played since people started recording them. [1] They know playing chess is intensely pleasurable and addictive, that winning well is a joy, but even losing is not so bad. Abuhamdeh and Csikzentmihalyi found that maximum pleasure in their study was felt when one’s opponent was 260 Elo points stronger (that is, one’s chance of winning was only 20%). [2] Csikzentmihalyi calls chess as played by humans autotelic; that is, intrinsically motivating. Chess players know that chess-thought combines a mixture of “intuitive” pattern recognition and brute force mental visualisation of the future chess positions that will follow particular sequences of moves (“calculation”). They know and enjoy a family of variant forms of chess – I used to play a lot of “Transfer” (known as “Bughouse” in the US), which involves players passing captured pieces to their teammates, who may place them on their board, a la Shogi, but all under a time limit, as well as Kriegspiel where you and your opponent are unaware of each others’ moves, but are informed about captures and checks. [3]

Chess has been of great interest to the computer science community as what was seen as a hard problem for “machines do[ing] the things that would require intelligence if done by men” (Minsky 1968). Now AlphaZero has reached the highest levels in just 24 hours of self-play, it is seen as a trivial mechanical task, much easier than walking without falling over, or working out that the truck in front of your car is not the sky. [4] When I played through the originally published AlphaZero chess games, I found them hilarious. They actually have a very human feel, as opposed to those played by conventional chess engines which search through the set of all possible moves a certain number of plies into the future of the current position. (Partly this was because neither program was using standard opening books.) [5]  AlphaZero’s open source successor lz0 (Leela) has a putative Elo rating of ~3400, about the same as Stockfish, while the best human is ~2800 (so the human might beat the program perhaps one game in a hundred). [6]  Computer programs don’t enjoy chess, and don’t have an aesthetic response to beautiful combinations of moves. But the pleasures of the chess player are “merely” the subjective experience of successfully meeting a meta-cognitive goal, one that drives all playfulness, and the related goals of the aesthetics of thought. We get a dopamine rush from our comprehension of elegance, simplicity and complexity, and of surprise (some may remember Koestler’s Ha! and Aha!).

The rest of this essay will be a few comments about how various philosophers have interacted with the game of chess and the playing of chess, but before that I will present a little thought experiment of my own. Many comparisons have been made between the acts of playing chess and carrying out mathematical calculations or solving mathematical problems. If one is some kind of Chess Platonist, one has the image of a Universe of approximately 1040 possible sequences of moves sitting out there. [7] The Dancing Pixies argument of Putnam, Searle, Bishop, and others is that, with the right interpretation, the interactions between atoms in a rock are computations and thus, for us computationalists, thought. [8]  They could also be interpreted as all those chess games. So I now have a concretized Platonic Chess Universe sitting in my garden. One merely needs to construct the apparatus to extract the games from the rock, and if one wished to use these to beat your human opponent, construct a further method of filtering out all the winning games arising from the current position. There are a couple of stories in Stanislaw Lem’s Cyberiad that are relevant, but Mrs Beeton’s recipe for Stewed Rabbit may also apply. [9][10]

(1) If I were asked to answer, in one sentence, the question “What was Wittgenstein’s biggest contribution?”, I should answer “His asking of the question ‘Can one play chess without the queen?’” (Wisdom 1952) [11] Well, people spot weaker players a queen all the time, and don’t think anything of it. After all, you can lose your queen during a game anyway, and the resulting play is a subtype of standard play (usually endgame), rather than a new game. If one is playing the Old Indian Opening, for example, the players can choose to enter a Queen-less game in 5 moves. You might refine Wittgenstein and ask whether introducing a new set of rules specifying starting from a particular position (that is, setup of pieces on the board), or playing under the standard rules to get there is the same. Like, there are 1040 different games lurking in there that people had never noticed before. In the Old Indian example, the players collaborate to reach a Queen-less board, but they may collaborate in all kinds of ways, such as repeatedly playing a particular chess opening variant they are both interested in (or in a grandmaster draw, as a strategy for winning a chess tournament).

(2) From Dummett’s explication of Frege on chess:

Proposition 1: Neither chess moves nor positions on the chessboard express anything; in particular they do not express thoughts that can be evaluated as true or false. Proposition 2: The theory of chess consists of meaningful propositions about chess moves and chess positions, capable of demonstration in the same way as mathematical theorems. [12]

These seem plausible, but the way I think of chess soon led me to:

Analogical Proposition 1: Neither bead moves nor positions on the abacus express anything; in particular they do not express thoughts that can be evaluated as true or false.

Analogical Proposition 2: The theory of abacus use consists of meaningful propositions about bead moves and positions, capable of demonstration in the same way as mathematical theorems.

I think it obvious that AP1 and AP2 are incorrect. My preferred interpretation for the abacus: Bead movements and positions concretely express the evolution and states of abstract entities in a virtual time and space, whose properties are chosen to simulate those of other concrete objects.

My preferred interpretation for chess: Chess moves and positions on the chessboard concretely express the evolution and states of abstract entities in a virtual time and space, whose properties are chosen to simulate the concrete objects, actions and decisions of soldiers.

A: chess is a war game.

B: The warfare modeled by the rules of chess is ancient Indian warfare, with peasant infantry, elephants, aristocratic cavalry and chariots.

C: Winning such a battle required the capture or death of the opposition king.

D: Winning usually requires the general to sacrifice many of his own soldiers in order to win, unless the opponent has made an error.

E: A general fights against another general, and usually will plan as if his opponent will make the best response to any given attack, although he will set up traps with the hope they will succumb.

F: Each wields a Theory of Mind that is domain specific, and must incorporate both low-level and high-level properties of the game or the organisation of fighting. For example, chess players and generals both divide up the elements of planning into strategy and tactics.

My preferred conclusion: Although the game is greatly abstracted, there are many high-level analogies to warfare.

As it happens, Deleuze and Guattari make much of these isomorphisms in A Thousand Plateaus , as in the State of chess versus the Nomads of Go/Wei Chi – “chess pieces entertain biunivocal relations with one another, and with the adversary’s pieces: their functioning is structural, while Go pieces… have extrinsic relations with nebulae or constellations”, and so on. [13]

My mental model of chess as simulation is therefore similar to a scientific model. There is the reality, the model that abstracts certain features of reality, and the theory that applies to the model. In a game, the pleasures arise in implementing and running the model, but are related to some of those of the reality (this is far more obvious in modern computer games). [14]

G: Chess is rational imaginative play.

H. The physical acts of moving the game tokens around are iconic – horses jump, pawns bump into one another so that they can’t move forward, children and adults both use their piece to knock over the opponent’s piece they are capturing, the latter without sound effects. Blitz players pick up the captured piece and use it to tap the clock.

I’ll point to a related line of thought that is not particularly unique to chess. For children, the pastime of chess is supposed to produce transferable, that is, generalized, principles of learning and thinking, that apply more widely than just to warfare. [15] One great goal of the AI community is to produce meta-reinforcement learners that can generalize (transfer) lessons from one domain to another domain.

I. Recall that Wittgenstein introduces Sprachspiel initially “as one of those games by means of which children learn their native language”.

J. By 2 years of age, children logically reason about outcomes of a counterfactual premise e.g. what should I do if this banana is a telephone, or which cup was emptied of imaginary tea by being knocked over.[16]

K. At a later stage of neurological development, children can imagine these little horses fight and jump following the rules of chess.

L. In children’s games, friction arises when someone else doesn’t want to play by the rules, as the reasoning carried out by the players is discordant (compare some interpretations of Dennett’s parable of Chmess).

M. This type of counterfactual analogical reasoning is the same as adult practical and scientific reasoning – I pretend light waves in the same way as water, but I’ll have to extend this mental model in a consistent fashion for it to be useful.

(3) Wilfrid Sellars considers how different chess sets (eg using cars as pieces in Texas Chess) make no difference to the play.[17] So “.Pawn.” is a linguistic type or kind (what he replaces Platonic Pawns with), and “Pawn” “is the common name of items which play the role played in our game of chess by pieces of wood”. But most strong club players have practiced playing without a board and pieces at all – blindfold chess, and their mental representations (I will use that loaded term) will vary according to peculiarities of brain development and learning. A good friend and strong chess player reported to me that he always saw a visual eidolon of his shopping list, from which he would read off the items. But I can assure you that my blindfold chess does not rely on such imagery. Rather, I must recall the list of moves by which we have reached the current position, which gives me a coordinate for that square (e.g. “g5”), from which I can then count off f6-e7-d8 – “Ah, I am pinning his Knight onto his Queen”. Possibly, a pawn for me actually is a linguistic type, but for many people it is extra-linguistic.

Living chess is another style of piece — John Brunner in The Squares of the City goes further, as did Edgar Rice Burroughs. [18] In this case, reality is being driven by the model, and in Brunner’s case, the pawns (peons!) are unaware of the fact. For modern players of computer games, there is a possibility of playing with different interfaces. That is, the game mechanics could be identical, but each player sees a completely different narrative on top of this, in terms of chosen skins , for example.

(4) Related to the latter is the point made by Fodor (1978), when he considers a computer that simulates the Six Day War and a computer that plays a game of chess, where “the internal career of a machine running one program would be identical, step by step, to that of a machine running the other.[19] I don’t think we should find such broad ranging isomorphisms of one abstraction across different physical phenomena surprising, as per my comment about a wavy nature. Fodor is, it seems pretty obvious, making a point regarding semantics versus syntax here. I have tried to push the reverse point, that the syntactic rules of chess were selected to match the semantics of warfare, that is as a simulation or a model. Tshitoyan and coworkers present a “syntactic” analysis of 3.3 million scientific abstracts published between 1922 and 2018 in the field of chemistry. [20] The words are scored in an unsupervised fashion for how close they are to one another (number of words apart) in a 200 dimensional space. Once this has been set up, the player can now make moves like “ferromagnetic – NiFe + IrMn = antiferromagnetic”, or notices that “CsAgGa2Se4” is close to “photovoltaic applications”. Using word2vec, the semantics that drives human chemists to mention certain concepts in the same sentence automatically produces a map that can be extracted mechanically.

(4) Wittgenstein also considered the mathematics-chess comparison:

It has been said very often that mathematics is a game, to be compared with chess. In a sense this is obviously false – it is not a game in the ordinary sense. In a sense it is obviously true – there is some similarity…It is something useful to compare mathematics to a game and sometimes misleading…In a sense the pure mathematics of chess makes no predictions. That is one of the important points. The pure mathematics of chess is like the pure mathematics of astronomy. The calculus makes no predictions, but by means of it you can make predictions. […] The theory of chess is not arbitrary – It’s not arbitrary, mathematics is not arbitrary, only in the sense, that is has an obvious applications. Whereas chess hasn’t got an obvious application in that way. That’s why it is a game… [21]

Elsewhere he remarks that the theory of a game is not arbitrary, although a game itself is.

It seems to me that there is a difference between the theory of chess, and, say, the theory of probability applied to a game of pure chance. In the latter case, the theory very quickly moves to a most general and most generalizable mathematics, so the probabilist’s thought experiments using urns and coloured balls are directly useful to geneticists and infectious disease epidemiologists and economists and ecologists and engineers. I have attempted to argue that it is the chess player’s lived experience that generalizes to some extent, at a very high level e.g. the abstract concept of long chains of deterministic cause and effect giving rise to foreseeable outcomes, or the strategy of offering poisoned pawns to a greedy opponent, trading freedom of activity for a temporary loss of material. The nuts and bolts of end game theory regarding, for example, forcing a mate with bishop and knight, does not offer much to any other domain. Though I start immediately thinking about knight’s tours on various tilings of the plane, and how “knight’s tour” is used as an analogy in everyday speech. The related point about bishop and knight is that one has to experience the awkwardness and indirection of the only route to success – some kind of a life lesson maybe?

(6) The signs of language, Saussure argued, are like the pieces on a chess board. In chess, the rules according to which pieces may be moved on the board are important, not the substance from which the pieces are made. Regardless of material, what matters (and this gets us back to the notion of opposition) is that each piece can be distinguished from every other on the board. ” [22] The central structuralist insight, that you can go an awfully long way with differences and relations (consider again how word2vec encodes the chemical materials ontology). Chess doesn’t really contribute much here – though I suppose the idea that the pieces are defined by their relations with each other, as well as with the space they move in, is laid bare in any any chess primer. In such a book the transnational standard abbreviations of the algebraic notation might well be introduced to English speakers, with D for the Queen, and S for the Knight.

(7) Dennett’s analogy between chess and philosophy, [23] where a chess variant (the general term as it happens is fairy chess ) stands in for a priori truths no-one is interested in, because nobody plays that variant:

Consider, as a paradigm of a priori truths, the truths of chess….All you need to know are the rules of the game…Chess is a deep and important human artifact, about which much of value has been written. But some philosophical research projects are more like working out the truths of chmess . Chmess is just like chess except that the king can move two squares in any direction, not one. I just invented it—though no doubt others have explored it in depth. I didn’t bother investigating these questions because although they have true answers, they just aren’t worth my time and energy to discover…Probably there is no investigation in our capacious discipline that is not believed by some school of thought to be wasted effort…One good test to make sure you’re not just exploring the higher-order truths of chmess is to see if people aside from philosophers actually play the game…

It is an amusing piece, and finishes with a chess problem (a smothered mate). Now chess problems are in some way “pure theory”, in that positions appear that could never arise within a legal over-the-board game. The position of the pieces is set up solely to allow a particular type of winning combination, which may be quite outré. The goal is purely aesthetic. Indeed, there was some controversy, back in the 19th century, about whether such things were a legitimate use of one’s time (they wouldn’t help with match preparation). Raymond Smullyan actually has a book of reverse chess problems, where you have to infer how one could have got to the position shown, which are extremely evil (in a good way, of course). [24]  One point is that the community that enjoyed such puzzles steadily increased in size, and now most newspaper chess columns include a match play problem (from a recent or historical game) and a set problem. The other is that the current rules of European chess have evolved to speed up play but not to change the character of the game (castling and the en passant capture rules are examples). Players of Chinese Chess (only slightly different), for example, have moved across because there are more players worldwide to compete against. If the pleasures of the game arise from high level analogies with warfare, and from puzzle-solving, then changes in the powers of the pieces won’t alter these too much. I would look to sociological features and limits of human cognition and recreation time to explain why chess was so common, and why it might become less common with the rise of computer games that get a lot closer to simulating warfare.

So there are theoretical articles on chess variants – for example, which pieces can force checkmate in Kriegspiel, or how does one play chess on a cylindrical board, or with a rider-hopper that can move more than one jump at a time. And yes, they are not widely read, and the games not widely played. Does this really help with deciding if an area in philosophy is worthwhile or not? The communities interested in any small part of human knowledge or praxis are often tiny in any one generation, but still can look backwards and forwards in time to similarly-minded thinkers (consider how J.S. Bach or B. Spinoza went underground for lesser or greater periods of time). It is possible that much fashion in philosophy is as sociological as that driving chess, but then we can always point to the justifications for the young playing complex games. But hopefully, philosophy is slightly more likely to be useful.

David L. Duffy is a research scientist who works on the statistical genetics and genetic epidemiology of traits ranging from cancer to personality.  As a result, he feels qualified to have an opinion on everything. He practiced medicine sometime back in the previous millennium, and has read far too much science fiction. You can see lists of publications and other stuff (even a couple of pastels) at https://genepi.qimr.edu.au/Staff/davidD/, and some of what he’s been reading (or doing) lately at http://users.tpg.com.au/davidd02/


[1] Hale B (ed) (2008, 2012). Philosophy Looks at Chess. Open Court. Where we read, for example, that chess is not a Utopian game in the sense of Bernard Suits, as the rules are not fun, even though playing it is (Deb Vossen).

[2] Abuhamdeh S, Csikszentmihalyi M (2012). The Importance of Challenge for the Enjoyment of Intrinsically Motivated, Goal-Directed Activities. Personality and Social Psychology Bulletin 38:317-330.

[3] Transfer Chess or Bughouse Chess https://en.wikipedia.org/wiki/Bughouse_chess – Transfer is a Utopian game, as the rules try to make play more hilarious – I recommend the three-board versions, where one must throw pieces to one’s teammate.
Kriegspiel https://en.wikipedia.org/wiki/Kriegspiel_%28chess%29, I used to play “Hopkins” Kreigspiel rules, which may not be that different from Berkeley rules.

[4] https://en.wikipedia.org/wiki/AlphaZero

[5] https://www.chessgames.com/perl/chessplayer?pid=160016

[6] https://en.wikipedia.org/wiki/Leela_Chess_Zero

[7] https://en.wikipedia.org/wiki/Shannon_number

[8] Reviewed for example at https://link.springer.com/content/pdf/10.1007/s11023-006-9009-3.pdf

[9] https://en.wikipedia.org/wiki/The_Cyberiad

[10] Apparently, she didn’t ever say it, but it was a common phrase: “First, catch your rabbit”

[11] Wisdom J (1952). Ludwig Wittgenstein, 1934-1937. Mind New Series 61:258-260.

[12] Dummett M (1996). Frege and Other Philosophers. The issues around theory and metatheory that spurred Frege to think about this material are pretty deep. But as a chess beginner, one may or may not have been told that forcing a mate with two knights is impossible, but this is usually found out by each person at the experiential level, hoping for one’s opponent to blunder. Certainly it usually takes practical experience to mentally codify the intermediate positions needed to force a mate with King, Knight and Bishop v. King, especially under time pressure. In that sense, the formalized theory of chess is not that different from the formalized theory of catching a cricket ball, catching a wave, or catching a cold.

[13] Deleuze and Guattari (1980). A Thousand Plateaus , see https://epicbaz.com/wp-content/uploads/2013/07/nomadology_read1.pdf

[14] Model here is used in the usual scientific sense, but in the case of chess, it may fit pretty directly into the apparatus of model theory.

[15] Sala G, Gobet F (2016). Do the benefits of chess instruction transfer to academic and cognitive skills? A meta-analysis. Educational Research Review 18:46-57.

[16] For example, Leslie AM (1994). Pretending and believing: issues in the theory of ToMM. Cognition 50 (1994) 211-238.

[17] Sellars W (1963). Abstract Entities. The Review of Metaphysics 16:627-671.

[18] Brunner J. The Squares of the City. https://en.wikipedia.org/wiki/The_Squares_of_the_City Burroughs ER The Chessmen of Mars. https://en.wikipedia.org/wiki/The_Chessmen_of_Mars.

[19] Rapaport WJ (2017). On the Relation of Computing to the World. In: Powers TM (ed). Philosophy and Computing: Essays in Epistemology, Philosophy of Mind, Logic, and Ethics Springer, Cham. Pp 29-64.

[20] Tshitoyan V, Dagdelen J, Weston L, Dunn A, Rong Z, Kononova O, Persson KA, Ceder G, Jain A (2019). Unsupervised word embeddings capture latent knowledge from materials science literature. Nature 571, 95-98.

[21] Wittgenstein L (1939). Cours sur les fondements des mathématiques Cambridge, établis par Cora Diamond (1975, 1976) http://www.normalesup.org/~sage/Reflexions/Maths/LWcoursFdts.pdf

[22] Caton SC (1987). Contributions of Roman Jakobson. Annual Review of Anthropology 16:223-260. https://www.annualreviews.org/doi/pdf/10.1146/annurev.an.16.100187.001255

[23] Dennett DC (2006). Higher-order truths about chmess. Topoi 2006: 39:41. https://ase.tufts.edu/cogstud/dennett/papers/chmess.pdf

[24] Smullyan R (1979). The Chess Mysteries of Sherlock Holmes These are retrograde puzzles reconstructing how the position came to be eg Indemonstrable mate in 2 moves or what is the missing piece? For example, pxd8(R)+, but captured piece must also be promoted N or B, pawn had to have got to c7 from f2 etc.


4 responses to “Pure and Applied Chess”

  1. […] Pure and Applied Chess (theelectricagora.com) […]

  2. s. wallerstein

    A very complete and well thought-out essay on chess and philosophy. I wish I knew enough about chess to make an intelligent comment about your essay, but I am the world’s worst chess player. Anyway, thank you for a very readable outline of this topic.

  3. David

    You touch directly or obliquely on many fascinating topics. Of particular interest to me are the relationship between linguistic and non-linguistic thinking, the nature and philosophy of mathematics (particularly intuitionism: Brouwer’s ideas influenced Wittgenstein, of course), the nature of games and their relationship with real life.

    On this last point, team sports play an important role in moral and social development. Like chess, they are (at least to some extent) modelled on wars –and so are zero sum — but they are closer to actual life, not so much because of their physicality but because cooperation between teammates and manners (involving both teammates and adversaries) are important elements.

    Obviously chess teaches more individual (and more intellectual) virtues.

    (On the physicality issue, I was struck by your comment to the effect that “AlphaZero has reached the highest levels in just 24 hours of self-play… a trivial mechanical task, much easier than walking without falling over…”)

    “I have attempted to argue that it is the chess player’s lived experience that generalizes to some extent, at a very high level e.g. the abstract concept of long chains of deterministic cause and effect giving rise to foreseeable outcomes…”

    I have reservations about this insofar as life does not work like this: the physics may be deterministic but the consequences of our actions in the human world are *not* foreseeable in this way. (Road to hell, good intentions.)

    “… or the strategy of offering poisoned pawns to a greedy opponent, trading freedom of activity for a temporary loss of material.”

    The “freedom of activity” here relates mainly to the details of how one’s pawns block one’s powerful pieces, so not really generalizable? Though you certainly can see (generalizable) Machiavellian principles at work.

    “The nuts and bolts of end game theory regarding, for example, forcing a mate with bishop and knight, does not offer much to any other domain. Though I start immediately thinking about knight’s tours on various tilings of the plane, and how “knight’s tour” is used as an analogy in everyday speech. The related point about bishop and knight is that one has to experience the awkwardness and indirection of the only route to success – some kind of a life lesson maybe?”

    Robert Graves’s crooked-flying cabbage white comes to mind.

  4. davidlduffy

    Hi Mark.
    “our actions in the human world are *not* foreseeable”: Yes, a lot of people have thought that games including an element of chance are better in this sense. Chess is still full of chance events in that even grandmasters make really obvious blunders from time to time (the joke runs that the winner is the person who made the second-last mistake).