Contradictions in games?
in [Math]
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From: Counterclockwise on 1 Aug 2010 06:45 > what is the difference [between math and chess]? Since you've at least appeared to be respectful, I'll answer this. Chess is only a game. It is a single abstraction of two nations at war with a sea between them, (The bishops were originally ships), to practice the strategy behind troop placement. Math is somewhat more complicated, and this is how I think about it, (others may have differing opinions): Mathematics is about abstractions. And abstractions of abstractions. And so on, until we've abstracted out as far as we can, or as far as we need to, whichever one comes first. That there are numbers in a lot of math isn't important; numbers are a method we can use in math to ensure correct analysis and objectivity, they aren't even present in some math, and in higher levels of math, we tend to abstract them away anyway, so that we deal only with the ideas of numbers, (And sometimes abstract those ideas away again). As long as you can create a sensible mapping from a system to numbers, you can use mathematical rules to analyze and predict the system. The purpose behind math is as . . . a toolbox is a good word to use, so I'll continue that. Math, in itself, does nothing to the world. And math applied in the world is always impure and incorrect. Which isn't to say it is useless, but that we can always get a more precise and better answer through a better application of math to the world. The purpose of applied math is to provide better applications of math for this toolbox, while pure math focuses on expanding the toolbox. While areas of pure math may not have any use yet, they probably will sometime in the future, no matter how abstract they are. The scope of math is such that every other study uses math. Whether it be music, or chemistry, or psychology, or poetry, they all use mathematical concepts, though they might not use numbers. As a mathematician and a poet, I like to describe the analysis of mathematics and poetry as using the same algorithm, but different databases of keywords and meanings. Mathematics and poetry both focus on complex relationships between non-equal entities, (for example: consider metaphors, analogies, similes, allegories, etc., which all try to define two different things as being alike,(equal), in some properties), but while math traditionally deals with numbers, poetry deals with emotions, with dreams & thoughts, with ideas so hard to define we can best represent them using symbols, and tries to explain them universally through the poem to other people, as the author feels them to be. (I'm trying to outline both the vast differences in subject matter between poetry and math there, but also the essential sameness in the way they both deal with the subject matter, by making equalities, comparing it to known relationships or constants, breaking it down to smaller independent problems, etc.) Sorry for the long-winded answer, I guess I haven't thought too much before about I would define math. In any case, have I given you a clearer understanding of math? TL;DR: Math is a series of abstractions following from a small amount of logical rules that we use to understand the entire world and everything in it. Chess, on the other hand, is a single abstraction that helps us understand, through practice, the importance of troop movements.
From: David Bernier on 1 Aug 2010 12:40 Robert Kaufman wrote: > Hi > > I guess what all these replies have brought out is that there is > a vast difference between mathematics and chess,but exactly > what is the difference? [...] In mathematics, there is the idea of a proposition, and its negation. P: "1999 is a prime number". ~P: "1999 isn't a primes number". A contradiction arises when, starting from premises and applying rules of logic, axioms, the premises (either correctly or incorrectly), one arrives as the two consequences (1) P (2) ~P or not(P). From (1), (2) we get: "P And not(P)". [a contradiction]. In chess, if in a game some interesting position is reached, what could the position's negation mean? If there's a nice answer, I don't know it. David Bernier
From: Robert Kaufman on 1 Aug 2010 10:39 Hi Thanks for your input. I've been thinking about this problem,and I think first of all you have to establish whether all all configurations of the chess pieces are ultimately possible. If not all configurations are possible and assuming we can identify a configuration as a "theorem" in chess then maybe a contradiction would be as follows: Let C be a given configuration. Then C is possible and C is impossible is a contradiction. Of course, in this case chess would be incomplete. If all configurations are possible, then chess would be complete and no contradictions are possible. As far as applying mathematical reasoning in some form to chess in order to establish that C is possible or C is impossible I don't think that this is possible right now, since applying mathematics to chess would have to involve multivalued functions for which we have a choice as to which one of the values of the function we wish to choose subject to certain restrictions. I'm not familiar with any such multivalued functions within mathematics. Has anyone else heard of such mathematics? Respectfully, Robert Kaufman
From: quasi on 1 Aug 2010 17:00 On Sun, 01 Aug 2010 17:17:37 +0300, Aatu Koskensilta <aatu.koskensilta(a)uta.fi> wrote: >Robert Kaufman <Yearachmeel(a)verizon.net> writes: > >> I guess what all these replies have brought out is that there is a >> vast difference between mathematics and chess,but exactly what is the >> difference? > >What's the difference between mathematics and water polo? Perhaps a more >pertinent question is: just how is mathematics at all like chess? If a position in chess is claimed to be a winning position (for the player whose move it is), a "proof" is required. In math, if a statement in a given formal system is claimed as true, a proof is required. However such a proof may or may not be obtainable if the axioms and allowable rules of inference are not sufficiently powerful -- that is, if the formal system is incomplete. Since chess has only finitely many positions, a proof or disproof is always possible -- hence chess is "complete". quasi
From: James Waldby on 1 Aug 2010 18:52
On Sun, 01 Aug 2010 14:39:20 -0400, Robert Kaufman wrote: > I've been thinking about this problem,and I think first of all you have > to establish whether all all configurations of the chess pieces are > ultimately possible. If not all configurations are possible and assuming > we can identify a configuration as a "theorem" in chess then maybe a > contradiction would be as follows: Let C be a given configuration. Then > C is possible and C is impossible is a contradiction. Of course, in this > case chess would be incomplete. If all configurations are possible, then > chess would be complete and no contradictions are possible. It is easy to find configurations that cannot occur in any legitimate chess game. For example, any configuration with both kings in check, or configurations with all pawns in their initial squares and some pieces other than knights not so. > As far as applying mathematical reasoning in some form to chess in order > to establish that C is possible or C is impossible I don't think that > this is possible right now, since applying mathematics to chess would > have to involve multivalued functions for which we have a choice as to > which one of the values of the function we wish to choose subject to > certain restrictions. I'm not familiar with any such multivalued > functions within mathematics. Has anyone else heard of such mathematics? I don't understand your "applying mathematics to chess would have to involve multivalued functions" claim or "not familiar with any such multivalued functions within mathematics" notion, so won't comment on that paragraph. -- jiw |