Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.1 6/24/83; site petsd.UUCP Path: utzoo!watmath!clyde!burl!ulysses!mhuxr!ihnp4!houxm!vax135!petsd!cjh From: cjh@petsd.UUCP (Chris Henrich) Newsgroups: net.math Subject: Re: Nova's Mathematical Mystery Tour Message-ID: <460@petsd.UUCP> Date: Thu, 7-Mar-85 14:08:36 EST Article-I.D.: petsd.460 Posted: Thu Mar 7 14:08:36 1985 Date-Received: Fri, 8-Mar-85 05:18:17 EST References: <143@ihlpa.UUCP> Organization: Perkin-Elmer DSG, Tinton Falls, N.J. Lines: 76 [] Lew Mammel asks some interesting questions, which I will paraphrase before giving my $.02 worth. 1. Are most mathematicians Platonists, in the sense of believing that mathematics is about some objective reality, not just participation in a game of manipulating symbols? I think so. Existentially, the answer is "yes, surely" - mathematicians do their thing as if it did have an objective meaning. In particular, though parts of mathematics can be formalized, i.e. described in terms of rules for manipulating symbols, the rules have not remained constant over the history of mathematics, and there have been lively debates over what are the "right" rules. These debates must appeal to some standard outside the rules themselves. The experience of doing mathematics certainly feels to me as if it is connected to an outside reality. There are too many puzzles and mysteries and surprises in math to account for on the premise that it is all subjective. When you try to solve a problem, especially on the frontier of research, you are wrestling with something that is very tough and subtle. Formality and formalisms are good techniques for doing this, but they are not the thing with which you are wrestling. Then, there is the relevance of mathematics to the natural sciences, what Wigner called "The Unreasonable Effectiveness" of mathematics. If mathematics were merely a game, like chess, then the first students of subatomic physics would have been no more likely to find mathematical phenomena inside the atom than to find chess pieces there. Finally, I think mathematical logicians are existential Platonists also. They study formalisms, to be sure. But they use mathematical methods (look at Goedel for instance) and regard their conclusions are "really" true. 2. What is the standing of a statement like the continuum Hypothesis, which has been shown to be "undecidable;" is it really either true or false, only we cannot get at the truth about it? The work of Goedel and Cohen showed that you can have a consistent set theory with or without this axiom. The situation is like that in geometry, where you can have a consistent theory with or without the parallel postulate. When mathematicians got used to this discovery, they found that non-euclidean geometries were as interesting and elegant as the Euclidean kind, and actually more useful for some applications. The consistency of set theory without the Continuum Hypothesis was discovered much more recently, and it is my impression that non-Cantorian set theories still seem exotic and "pathological" (i.e. you only see them if you go looking for trouble). But this is a matter of taste. 3. How does the question of the Continuum hypothesis relate to issues such as "constructivism?" I think they are loosely coupled. The "constructivist" position, at its strongest, is that a statement, that something with property X exists, ought not to be made unless a means of "constructing" such a thing can be given. Even if the contrary statement can be shown to result in a contradiction, the affirmative statement is not really true until the "construction" is provided. This is a minority position among mathematicians. It bears on set-theoretic questions, notably the Axiom of Choice, which asserts the existence of lots of things without making the slightest effort to construct them. Regards, Chris -- Full-Name: Christopher J. Henrich UUCP: ..!(cornell | ariel | ukc | houxz)!vax135!petsd!cjh US Mail: MS 313; Perkin-Elmer; 106 Apple St; Tinton Falls, NJ 07724 Phone: (201) 870-5853