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 rules of chess, go, and bridge are also sufficiently precise that they can be stated in FOL. And the expert players of those games have proved many theorems about them. There is no definition of pure mathematics that would exclude those four games (and many others). JBD
What is more, the playing of those games does not need to a science that deduces theorems from hypotheses. They can be played on the basis of hunches, where the goal is simply to win and not to prove anything of a more general sort.
Of course. That point is also true of arithmetic, geometry, and other versions of mathematics. Many people play games with numbers and figures. Do you remember the column on mathematical games in the _Scientific American_? For many years, that was a very popular feature by Martin Gardner. In fact, I discovered Peirce's existential graphs from reading his column in 1978. Games are an excellent way to teach and learn math. JBD
Lacking an explicit statement of the hypotheses, we can only rely on unstated assumptions as unanalyzed common notions. Those will often suffice for practical purposes, but they won't suffice for developing mathematics as a pure science.
I suspect that your view of mathematics was perverted by studying the Bourbaki (or something similar). As remedial reading, I recommend George Polya's books. See the references and quotations in http://www.jfsowa.com/talks/ppe.pdf See below for an excerpt from slide 2 of ppe.pdf John ______________________________________________________________________ “Mathematics — this may surprise or shock some — is never deductive in its creation. The mathematician at work makes vague guesses, visualizes broad generalizations, and jumps to unwarranted conclusions. He arranges and rearranges his ideas, and becomes convinced of their truth long before he can write down a logical proof... the deductive stage, writing the results down, and writing its rigorous proof are relatively trivial once the real insight arrives; it is more the draftsman’s work not the architect’s.” Paul Halmos (1968) Mathematics as a creative art, _American Scientist_, vol. 56, pp. 375-389. http://www-history.mcs.st-andrews.ac.uk/Extras/Creative_art.html
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