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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
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[]
	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
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