FOM: the "normal" mathematician (reply to Friedman)
Martin Davis
martin at eipye.com
Fri May 12 14:54:29 EDT 2000
Harvey invokes the figure of the "normal" mathematician as opposed to the
set theorist and makes much of his views regarding such things as V=L as an
axiom and how arbitrary mathematical objects should be regarded.
I regard the concept of the normal mathematician as quite like the
proverbial man-in-the-street. It makes about as much sense to make serious
mathematical judgements based on a poll (real or imagined) of typical
mathematicians on such questions as it would to base scientific judgements
in biology on a similar poll of ordinary Americans on the validity of
Darwinian evolution. In both cases one is dealing with people, who however
clever and well-informed about other matters, are utterly ignorant
concerning the matter at hand. V=L? Most mathematicians to whom I have
spoken are utterly astonished to learn that everything they are doing can
be regarded as internal to a countable structure. They assume that
Goedel-Cohen have settled CH and are truly surprised to learn that Cohen's
forcing models are typically countable.
It isn't exactly news that mathematicians tend to be mostly interested in
well-behaved objects. The move to arbitrary sets, functions etc. developed
historically out of the need for various kinds of completeness. When I used
to be on committees examining graduate students, my favorite question in
real analysis was:
Why do analysts prefer to use the Lebesgue integral rather than the
Riemann integral?
The typical student reply: well, there are functions that are Lebesgue
integrable but not Riemann integrable. I would ask the student to name one.
The reply would almost invariably mention the characteristic function of
the irrationals. Then, I'd gleefully close the trap asking: Do you really
think that analysts care about that function? A good student would quickly
recover and point out that the Lebesgue integral has these terrific limit
theorems. The full system of real numbers (of course set-theoretically
equivalent to the power set of omega) is needed not because the "normal"
mathematician has any interest in undefinable reals, but because the full
continuum is needed for completeness.
Martin
Martin Davis
Visiting Scholar UC Berkeley
Professor Emeritus, NYU
martin at eipye.com
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http://www.eipye.com
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