John S, List, If my view of mathematics has been perverted, then the perversion wasn't caused by studying the works of the Bourbaki group (or something similar). Rather, I'm trying to interpret Peirce's remarks about the importance of stating the mathematical hypotheses of a system precisely for the purpose of drawing conclusions with exactitude. That, I take it, is the kind of advance that was made by Euclid and his predecessors in stating the postulates, definitions and common notions with considerable (although still far from perfect) precision.
One of the great advantages, I take it, of working with mathematical hypotheses as opposed to philosophical conceptions is that the mathematician can make the hypotheses that serve as the starting points of the inquiries as clear as is needed. The philosopher, on the other hand, must accept the vague conceptions that are part and parcel of his inquiries--warts and all. --Jeff Jeffrey Downard Associate Professor Department of Philosophy Northern Arizona University (o) 928 523-8354 ________________________________________ From: John F Sowa <[email protected]> Sent: Tuesday, March 7, 2017 9:10 PM To: [email protected] Subject: Re: [PEIRCE-L] Truth as Regulative or Real; Continuity and Boscovich points. 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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