FOM: the "normal" mathematician (reply to Friedman)

[Martin Davis]

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