As part of my on-going attempt to convince myself that my chess reading is not a complete waste of time (even for my chess-playing ability!), I offer the following thoughts on the important relationship between chess strategy, computers, and spiritual knowledge.

Of late, I have been reading a carefully annotated account of Garry Kasparovâ€™s last encounter with Deep Blue, the IBM super computer that ultimately bested Kasparov, who is widely regarded as the best chess player of all time. To understand why this was such a big deal, you have to understand something about chess strategy and chess computers.

Despite some failed attempts to create artificial-intelligence approaches to chess, virtually all chess computers play through a combination of brute memorization and calculation. Here is how it works. The opening of a chess game is extremely important. It is also fairly manageable in the sense that the universe of â€œsoundâ€? moves is relatively small. For example, opening a chess game by moving your queenâ€™s rookâ€™s pawn forward two spaces (a2-a4) is just dumb. As a result there are various opening systems (the English Opening, the Queenâ€™s Gambit, the Ruy Lopez, the Sicilian Defense, etc.) that have been studied in great detail and consist of a long lines of pre-determined moves. A good chess computer has databases filled with thousands of these variations, which it can recall with perfect accuracy.

Once a computer is out of the opening, it plays by brute calculation. It looks at the board and examines every possible move. The position resulting from each move is judged by a combination of material advantage, tempo, king safety, and space. This process is then repeated for all of the available moves in each resulting position and so on. Some lines are discarded as worthless, but most are examined. As you might imagine, the set of positions that the computer is analyzing rapidly becomes astronomically large. However, with enough micro-processors and memory chips, the computer can â€œseeâ€? several moves into the future, far more moves than can a human being engaged in the same calculation.

Traditionally, this brute-calculating aspect of chess-computers has led to a simple rule of thumb: Computers play spectacularly well in open positions, but are often befuddled by closed ones. An open position is one where many or most of the pieces (especially pawns) have been removed from the board and play is active and fluid. This is the kind of game where attacking combinations are most effective because it is easier to get at an opponentâ€™s pieces (especially her king). As a result, the ability to precisely plot many moves into the future becomes decisive. A closed position is one in which many pieces remain on the board, and the center is clogged by lines of stationary pawns that make maneuvering difficult. In a closed position, deep strategic thinking triumphs over precise tactical calculations. The essence of such strategic thinking is an ability to spot and exploit tiny structural changes in the position of the pieces that will pay off with important advantages many, many moves later. What is required is not calculation but intuition.

Traditionally, computers have been prone to aimless play in closed positions, and grandmasters such as Kasparov who have an intuitive grasp of positional play have been able to keep the game closed until the computer stumbles into a losing position. Or at least this was the tradition until Deep Blue. During its games against Kasparov (or at least some of them), Deep Blue didnâ€™t play like a computer. It didnâ€™t make meandering moves in the closed positions. Rather, it deftly and subtly maneuvered its pieces as if motivated by an intuitive understanding of the position, making moves whose advantage could be articulated only at the level of airy strategic abstraction, e.g. â€œpawn to b5 strengthens the queen side defensesâ€? etc. The wonder of this lies in the fact that Deep Blue did not have any â€œdeep strategyâ€? algorithm. Its programmers did not design it to think strategically. It simply performed a move by move calculation in exactly the same manner as does any $5.99 chess program. At some point, millions of calculations per minute transformed itself into intuition.

The divide between the tactical calculation of open position and the strategic intuition of a closed one arguably mirrors an ancient distinction in philosophy between two sorts of knowledge: episteme and techne. Episteme refers to concrete propositional knowledge. For example, something like â€œThere are three members of the godhead: Father, Son, and Holy Ghostâ€? is an example of episteme. In contrast, techne refers to knowledge that takes the form of some skill that cannot be reduced to propositions or rules. Knowledge of how to live with the Spirit or live a righteous life are examples of techne. These are activities that cannot be stated in terms of simple procedures.

Mormon thinkers eager to lionize the seeming absence of a well-developed LDS theological tradition have frequently latched on to something like this distinction. Important religious knowledge, they insist, does not consist in the abstractions of theology (or what many LDS call â€œdoctrineâ€?), but rather is best seen in spiritual skills and practices that cannot be easily reduced to â€“ or deduced from â€“ such abstractions. Thus, the largely atheoretical nature of our religious thinking is tagged as an intellectual virtue that accommodates a laudable emphasis on practice and techne over abstraction and episteme.

Deep Blue, however, seems to present a challenge to the neat dichotomy upon which this apologetic rests. It suggests that in the end intuition may be achieved through calculation, techne may emerge from the mere aggregation of episteme. Of course chess, despite its delightful complexity is really a rather simple phenomena. (As I recall a mathematician in the 1920s performed a proof that shows that there are a finite number of possible chess games, although no one knows how many there are are, let alone how they were played.) Hence, one might dismiss this â€œchallengeâ€? as irrelevant. Position judgement in chess is simple compared to spiritual skill. Still, if Deep Blue is a valid example of intuition emerging from calculation, then there may be a deeper affinity between the practical and the theoretical â€“ between episteme and techne â€“ than we have been led to assume. And there may be more room for abstract theology in our practical religion than we have often allowed ourselves to believe.

Wow, that’s quite the interesting proposition. I don’t know what I can really say in terms of furthering or adding insight to the lines of reasoning you spell out because I’m not sure anyone really can at this point. It’s worth looking into, however. And, the explanation of Kasparov and Deep Blue was worth the read in and of itself. Thanks!

The maximum bound of possible chess games, I believe, is 64!/32! * 200 which is about 10^55

64! is pronounced 64 factorial and is 64*63*62…*3*2*1, 64 being the number of possible positions (i.e., squares).

32! is 32*31*…3*2*1, 32 being the number of pieces.

200 is the upper bound of the number of moves allowed (100 each I think) in a modern chess game.

PS. Which online chess services do you use? (playchess.com?, yahoo.com?)

Daylan,

Is the 10^55 number the upper bound of the possible number of chess games or the number of actual chess games? I freely confess that the math here is far too complicated for my poor brain. Law school, as they say, is the ultimate refuge for the math-phobic.

Godel’s theorem merits some mention here, a simplisitc explanation of it being that for any formal system, you’re going to end up with either unprovable truths or provable falsehoods. Thus any mechanistic system will have either inconsistency or will be incomplete. This also points to the idea that you can’t trust mere calculation for understanding, and in fact, Godel derailed an entire movement of mathematicians going for just that very goal when he made his case.

Interestingly enough, this leads right back to an argument in discussions about artificial intelligence and consciousness: Roger Penrose builds an argument off of Godel’s theorem which says that human beings do something when thinking other than lots and lots of computation.

Here’s an article about the differences between chess and Go and the difficulties of programming a computer to play a game with human-like intuition:

http://www.chessbase.com/newsdetail.asp?newsid=450

Here’s the New York Times article that the previous link I provided references. This article might in fact be more useful:

http://tech2.nytimes.com/mem/technology/techreview.html?res=9D01EEDC1F38F932A3575BC0A9649C8B63

Here is an interesting statement of the kind of mathematical problems involve in the brute calculating strategy:

“The main problem of chess programming is the very large number of continuations involved. In an average position there are about 40 legal moves. If you consider every reply to each move you have 40 x 40 = 1600 positions. This means that after two ply (half-moves), which is considered a single move in chess 1600 different positions can arise. After two moves it is 2.5 million positions, after three moves 4.1 billion. The average game lasts 40 moves. The number of potential positions is in the order of 10^128 (10 to the power of 128), which is vastly larger that the number of atoms in the known universe (a pitiful 10^80).”

Nate, I think your hypothesis about the apologists’ dichotomy (eventually) breaking down is correct.

The fact/value distinction is probably due to our epistemology, and not due to an ontological distinction between facts and values. Were we to have God’s infinite knowledge, we would probably accept God’s values

necessarily. You might say that all truth can be circumscribed in one whole,by someone with an IQ of about 10^128. Hence the foolishness of men and our dependence on revelation.Godel’s theorem, though, would apparently say that 10^128 isn’t even enough (though it might be for all practical limits of human endeavor in any of our lifetimes). If you want true completeness and consistency out of a rule-based system — well, to transcend the problem, you have to have an

infiniteset of axioms.Or you have to have something else going on other than computation…

Weston: I think that you may be pushing the Impossibility Theorem farther than it will actually go. All that it demonstrates is that there exists no set of logical axioms from which one can derive all true mathematical statements. I know that there are frequent attempts to draw more impressive philosophical conclusions from this result, but I am not sure that they are supportable. Why can’t I simply say that the impossilibility theorem is a statement about mathematics, but not a statement about chess or spiritual knowledge?

Still, if Deep Blue is a valid example of intuition emerging from calculation…The key word is “emerging.” What we’re talking about here is

emergence, the idea that entities with very simple properties can combine to form an entity with entirely different, and sometimes apparently intelligent, properties. (If the system is dynamic, then you could have anadaptivesystem that becomes progressively more intelligent. Reproduction with random mutations and natural selection can provide these dynamic properties. I don’t think that chess programs are typically adaptive.)A typical chess program is emergent, but not completely. It’s true that it involves millions of dumb calculations that combine to form intelligent strategies, but it also utilizes prefabricated heuristics and has an overseer that directs the process. Thus, some of the intelligence is emergent, but the rest of it is injected into the system by the human programmer.

Proponents of “intelligent design” argue that something as complex as a human being cannot be the product of pure emergence, but rather requires an intelligent overseer. There are many emergent system apps that one can run on a home computer that, in my option, refute this notion.

As a proponent of emergence, I agree with Nate’s speculation that vague spiritual strategies can emerge from simple, concrete doctrines.

(P.S. I assume that when chess games are enumerated, those that involve both players repeatedly going through the same cycle are not counted. Otherwise the number of possible chess games would be infinite. And I don’t believe that Godel’s Incompleteness Theorem applies to chess, since the rules of chess specify legality but make no truth claims.)

As Will pointed out, Gödel’s Theorem really doesn’t apply. The whole point of Gödel was that there are notions we have about the entities of mathematics that can’t be proven one way or the other. However this is because the formal systems and our

ideasabout the formal systems are different. i.e. our idea of a circle and the way circles are expressed formally in geometry are different. For example we tend to think of geometry in terms of real circles, real lines, and typically think about them graphically. Yet when doing geometry proofs, such things don’t enter in. The whole reason Gödel brought this up was because he fell mathematical entities like π or circles wererealand not just formal definitions. Of course most mathematicians disagreed with him. But that was what he was getting at – that there was somethingmorethat our definitions could never fully capture.In chess we are fully

withinthe formal system. i.e. all we care about is whatcanbe arrived at via a series of moves. The situation in a fashion analogous to Gödel would be those statements in arithmetic whichcanbe proven true or false. I’m sure that would could find a way to apply Gödel’s theorem. However I’m not quite sure what the equivalent entities would be in chess. I’m open to any meta-moves if someone can think of them. But generally we don’t concern ourselves with truth in such games, merely possible legal moves. (Which, I should add, many philosophers of mathematics feel is all we should concern ourselves with in mathematics as well)Clark: It seems that there are a host of statements about chess that one can meaningfully make that are both in a sense internal to the system and not reducible merely to legal moves. Consider the statement, “The King’s Gambit (1. e4 e5 2. f4 exf4) is an inherently unsound opening and unless he commits an error, Black should be able to win.” This is a controversial statement about chess strategy, but it is one to which any number of expert players would assent to. (Bobby Fischer, for example, argued that the King’s Gambit was a dead-end.) I would submit that this statement can be either true or false, but that it is difficult if not impossible to reduce it to a simple statement about the rules of chess. The reason for this is that the concept of “an error” is not defined in terms of the rules of chess, but in terms of chess strategy. I am not sure that this means that Godel has any application to chess (I am skeptical), but I am not convinced that the play of chess is merely a matter of legal moves. It seems that there are other concepts in play.

If one could explain that view in terms of probability (i.e. the aggregate of possible moves including this move) and suggesting the likelihood of a given output, then it seems it is describable in terms of the rules of the system. If one discusses likelihood in terms of something external to the system (i.e. practical probability in terms of

humanrather than rational strategy) then I suspect Gödel’ would apply, simply because something that appears meaningful in terms of the rules really is only meaningful in terms of the rules and something else.Of course you are right, I’m not sure Gödel’ is the right way to express this. (If only because Gödel’ is so misapplied that invoking his name is almost like talking about perpetual motion machines)

It occurs to me that one of the ways in which statements like “The King’s Gambit is unsound” are purportedly tested is by looking at the outcomes in a very large number of master-level games in which the opening is played.

I think Clark’s point is that chess is indisputably nothing more than a set of formal definitions that claims no relationship to non-chess reality. Since all possible chess games can, theoretically, be enumerated, that set of games tells us everything there is to know about the chess universe. We can look at that set and answer any possible question that can be posed about chess, including the question of whether the King’s Gambit is a sound opening. In that sense, Incompleteness doesn’t apply.

Not precisely, Will, because many games in that set will open with a King’s Gambit but still proceed on to White victory. That is to say, lots and lots of possible chess games are pretty silly because no one would actually play them that way.

This reminds me of a demonstration I once attended that talked about Agents (not the Secret kind, but the mathematical kind).

While I’m sure there are other variations, this presenter talked about a very simple algorhythm to model organic growth. They use a simple grid made up of cells, or squares. Given about three rules (if there are 0 or 3 ‘neighbors’, then the ‘cell’ dies; If there is 1 neighbor, the cell doubles, if there are 2, it stays the same) the model produces groupings of cells (organisms) that, over iterations of applying the rules, appear to move in apparently very complex ways.

OK, thus far, not very exciting, I admit.

The really cool thing is that, in a certain configuration that, ultimately wasn’t so complex (just more complex than I could have come up with. . .) these 3 simple rules produced a ‘factory’ of organisms that pumped out scores of new organisms that took off on their own. No change of the rules, just a certain, specific beginning configuration.

At the time I couldn’t get Matthew 11: 30 javascript:selectVerse(30,74592); out of my head. The actual rules were so simple. . .

I guess I understand html about as well as I understood the math. . .

The truth claims aren’t inherent in the description of the chess game, certainly. I suppose what I’m seeing is a question about the algorithms for chess games compared to a question about spiritual understanding. In both cases, the question I’m thinking about is “Does there exist some finite set of axioms by which I can (a) play a winning game of chess and (b) play (live) a winning (redeemable) life?” And the questions are in fact pretty different, as there’s no apparent formalism for the latter… but hey, the possibility of analogy from one to the other was the impetus for this thread, which I’m enjoying, so I don’t feel bad about the comparison. :) And the truth/falsehood of a statement claim comes into chess when you look at what you’re asking a computer to do (or rather, asking a computer programmer to formalize as code): create a set of axioms, embodied in an algorithm, which are so designed so that the computer can evaluate (by computation) the truth or falsehood of the proposition “move M will lead to a winning game.”

Yet I’d have to agree that since it’s true that sheer enumeration would in fact seem to at least theoretically present a finite, complete, consistent means for evaluating a series of moves, the problem of creating such a formalism isn’t bound by Godel.

Still…. I can’t see victory by massive enumeration as an encouraging sign for logic and abstraction as tied to episteme. We *do* take in and process massive amounts of data on the earth in the course of our experience, and that informs us and shapes us, but we don’t do anything remotely close to formally enumerating possibilities presented by our choices well, and if we had to depend on that for spiritual progression, we’d need considerably longer mortal lives and probably different brains.*

But conversely, (and fortunately) I don’t think that the value of abstraction is at all divorced from techne, and so I agree with Nate’s premise that there’s more room for it… though I still don’t see how we arrive at that conclusion via the accomplishments of Deep Blue. : )

* unless, as I’ve sometimes wondered from a quote attributed to Joseph Smith after receiving a revelation — “I was so full of light I could see out the ends of my fingertips” — the enlarged faculty brought by sheer quick computation manifests itself at the conscious level in the form of instant perception of the array of possibilities…..

Regarding the statement, “The Kingâ€™s Gambit (1. e4 e5 2. f4 exf4) is an inherently unsound opening and unless he commits an error, Black should be able to win”: In zero-sum, deterministic, two-player games like chess, there is a well-defined sense in which a board position can be said to be a win for Black or White, namely, given a board position, if Black can always find a sequence of moves such that no matter what White does, Black can win, then the position is said to be a win for Black. To prove that a position is a win or a draw requires in general that all possible move sequences be enumerated.

For example, the game of Tic-Tac-Toe at the beginning of the game is known to be a draw for both sides, meaning that either side can force a draw (that is, not lose) if it plays optimally. The opening position of the game Nim is known to be a win for the first player. This does not mean that the first player will always win, just that the first player can always win if it plays optimally. The first player can make a non-optimal first move, in which case the next move is a win for the second player.

Tic-tac-toe and nim are “solved”, meaning that the optimal strategy for playing these games is known. I believe that checkers is solved or is close to being solved (I’m not sure if it is a win for the first player or not). Chess is nowhere near being solved.

A little research reveals that the Chinook project claims to be close to solving checkers. Following the link, you will see a discussion of an opening in checkers that casts some light on the discussion of the King’s Gambit.

Wow Bryce. Thanks for the link. I enjoyed reading about that Chinook project.

Of course the grand game of Go is even further away from being solved than Chess… It’s also much, much harder to program a reasonable Go player as compared to a reasonable Chess player.

Wolfram’s “A New Kind of Science” comes to mind. I only know the book through John Derbyshire’s unfavorable review: http://www.olimu.com/Journalism/Texts/Reviews/NewKindOfScience.htm

An excerpt from Derbyshire shows why it comes to mind:

“Well, consider the universe at large. At some point in time, it â€” all its untold bazillions of particles â€” is in a certain configuration. At some later instant, it is in a different configuration. How did it get from the one state to the other? The traditional answer is: by obeying certain very complicated physical laws, which you need a ton of calculus and a truckload of algebra to understand. No, no, says Wolfram: Itâ€™s a cellular automaton! ”

Or consider biological evolution. At a certain point in its history, the typical member of a certain species has such-and-such a genetic code. A million years later, it has a different one. How did it evolve from the first to the second? By natural selection working on random mutations, says traditional biology. Wolfram is shaking his head again: cellular automaton! ”

Derbyshire’s closing line is “The universe a vast cellular automaton? As Diderot remarked in a similar case: ‘Nothing is that simple; certainly not everything!’â€?

Back to the gospel application. The article’s final lines are “Still, if Deep Blue is a valid example of intuition emerging from calculation, then there may be a deeper affinity between the practical and the theoretical â€“ between episteme and techne â€“ than we have been led to assume. And there may be more room for abstract theology in our practical religion than we have often allowed ourselves to believe.”

This idea may be supported by President Packer who has stated that the study of the doctrine would more quickly improve behavior than a study of behavior would improve behavior, or something like that. See Ensign, May 1997, 9; Ensign, May 2004, 77; Ensign, Nov. 1986, 17.

Similiarly, Elder Oaks says “Well-taught doctrines and principles have a more powerful influence on behavior than rules.” Ensign, Nov. 1999, 78.

Finally, President Ezra Taft Benson: â€œThe Lord works from the inside out. The world works from the outside in. â€¦ Christ changes men, who then change their environment. The world would shape human behavior, but Christ can change human nature.â€? Ensign, November 1985, 6.

That’s an interesting quote by Pres. Packer I’d missed. Now I don’t feel so bad about studying theology. I think that Elder Oaks comment contextualizes it somewhat. Principles vs. rules. I’ve long thought that was why one of the big things restored with the gospel through the early years of the church was a basic notion of the plan of salvation. Something I suspect earlier generations in Israel or among the Nephites had in a much more fragmentary way.

I just found a pretty good site that describes the game of Go. I thought I’d throw the link in here (since I mentioned it earlier in the comments):

http://www.usgo.org/resources/whatisgo.asp