Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.2 9/5/84; site cvl.UUCP Path: utzoo!watmath!clyde!bonnie!akgua!whuxlm!harpo!decvax!genrad!panda!talcott!harvard!seismo!umcp-cs!cvl!westling From: westling@cvl.UUCP (Mark Westling) Newsgroups: net.philosophy Subject: Re: Metaphysics Message-ID: <761@cvl.UUCP> Date: Thu, 22-Aug-85 15:10:27 EDT Article-I.D.: cvl.761 Posted: Thu Aug 22 15:10:27 1985 Date-Received: Sun, 25-Aug-85 01:31:02 EDT References: <969@sphinx.UChicago.UUCP> <608@mmintl.UUCP> Reply-To: westling@cvl.UUCP (Mark Westling) Organization: Computer Vision Lab, U. of Maryland, College Park Lines: 56 Summary: Are numbers real? Keywords: In article <608@mmintl.UUCP> franka@mmintl.UUCP (Frank Adams) writes: > Mathematical philosophers basically fall into three groups. One is the > formalists, who insist they are just manipulating rules, and that any > applicability to anything else is strictly incidental. > The intuitionists believe that mathematical objects (numbers, for short), > are solely constructs of the mind. ... > The third group, which is probably the largest, thinks numbers are real, > but doesn't really know what they are. A good answer to that question > would be most appreciated. From my limited knowledge of mathematical philosophy, it looks like the biggest group wasn't mentioned. The idea that numbers exist on their own was put forth last (I think) by Kant. Kant believed that mathematical reasoning could not be derived from logic, but rather from intuitive, a priori notions of time and space: arithmetic arises from time, and geometry arises from space. A major point was his claim that the figure is essential to all geometric proofs. He also believed in the necessity of Euclid's axioms, which are naturally based on intuition. The school of thought which was not mentioned in the original posting is the one founded by Frege and expanded by Russell. Two major results provided the incentive for dismissing Kant's ideas. Riemann and Lobachevsky showed that, in pure mathematics, non-Euclidean geometry also works, so there is no a priori need for Euclid's axioms. What happens in the real world is irrelevant; we're talking strictly about mathematical truth. Peano strengthened symbolic logic and set theory. Frege took the next step and reduced mathematics to logic. Russell continued along these lines, adding that the set theory used in his constructions was also reducible to logic. Recently, however, Quine and others have argued that the essential part is the set theory itself, not the logic. Formalists dislike this notion because they believe that logic is generally less certain than mathematics; intuitionists dispute it because it allows the existence of propositions which are unprovable (e.g. with Goedel's incompleteness theorem, or even quantification with infinite or just unmanageably big classes: how can you claim "all X are Y" if you can't test every X empirically?). Both provide alternatives to a logical or set theoretical foundation of mathematics, but neither support a special, "Platonic" existence of numbers. (At least, that's how I understand it.) It seems to me that there is more room for ontological investigation in the philosophy of logic (especially quantification) than in the philosophy of mathematics. Any thoughts? -- -- Mark Westling ARPA: westling@cvl CSNET: westling@cvl UUCP: ...!{seismo,allegra}!umcp-cs!cvl!westling