Jeff, list:
I think the problem is that your argument proceeds to prove the major (C) to belong to the middle (B) by means of the third (A) Whereas in the order of nature, one ought to prove the major term (C) to belong to the third term (A) by means of the middle (B). It has little to do with exactitude, for I suspect that the philosophical method is clearly aware of your attempt at precision because the philosopher is expected to be a complete knower. Best, Jerry R On Tue, Mar 7, 2017 at 11:10 PM, Jeffrey Brian Downard < [email protected]> wrote: > 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 > > > ----------------------------- > PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON > PEIRCE-L to this message. PEIRCE-L posts should go to > [email protected] . To UNSUBSCRIBE, send a message not to PEIRCE-L > but to [email protected] with the line "UNSubscribe PEIRCE-L" in the > BODY of the message. More at http://www.cspeirce.com/peirce-l/peirce-l.htm > . > > > > > >
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