> On Mar 7, 2017, at 9:10 PM, John F Sowa <s...@bestweb.net> wrote: > > On 3/7/2017 3:19 PM, Jeffrey Brian Downard wrote: >> pure mathematics starts from a set of hypotheses of a particular sort, >> and it does not seem obvious to me that these games are grounded >> on such hypotheses. > > More precisely, pure mathematics starts with axioms and definitions. > A hypothesis is a starting point for a proof that also uses those > axioms and definitions. > > JBD >> Peirce... uses tic-tac-toe in the Elements of Mathematics as >> an example of how to take a kid's game, and then to examine it >> in a mathematical spirit. Does this make the game a part of >> mathematics? > > It certainly does. The axioms and definitions of tic-tac-toe > can be stated in FOL. From those axioms, you can prove various > theorems. For example, "From the usual starting position, if > both players make the best moves at each turn, the game ends > in a draw."
The problem with the game theoretical view of mathematics is the question of realism. This is why Godel made his argument about things not provable since he assumed they were true. While of course Wittgenstein’s model of language isn’t opposed to realism within mathematics there’s a difference between how we use the language of mathematics and what the objects of mathematics are. That is what are the relationship between the game and reality. Where this comes up is in semi-empirical methods such as Putnam suggested we apply to mathematics. As a practical matter there are unproven (and for all we know unprovable) mathematical theorems that are used as premises for other mathematical proofs. Perhaps this is still limited but I suspect it will accelerate in the future. Again returning to language games of course while the notion can be abused a robust notion of language games is compatible with realism. But I think we have to think through carefully what sort of game we are playing if we’re going to use that as our metaphor.
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