FOM: human well-being; constructivism; anti-foundation
Matthew Frank
mfrank at math.uchicago.edu
Tue Jan 2 15:40:55 EST 2001
I agree with Matt Insall's comments.
Andrej Bauer asked for "_specific_ remarks or comments by Hilbert and/or
Brouwer that would indicate more precisely which theorems or parts of
mathematics they expected to evaporate under intuitionism." One early
example is Brouwer's discussion of the (classical but not intuitonist)
theorem that every real number has a decimal expansion; one can find this
in one of the early selections in Mancosu's anthology From Brouwer to
Hilbert. Probably other people can come up with more interesting
examples.
Rejoinders to Ayan Mahalanobis's post:
I think most mathematicians would agree with Nerode in finding
constructive mathematics more confusing than recursive mathematics, since
most mathematicians find unlearning classical logic difficult. Of course
this is subjective, but I think it's also fairly universal--those of us
who do not find constructive math confusing are a definite minority.
As for Hilbert, he might have responded to Bishop just as he responded to
Brouwer: "with your methods most of the results of modern mathematics
would have to be abandoned, and to me the important thing is not to get
fewer results but to get more results." This would almost be more
appropriate that the original aim of the quote: Brouwerian intuitionist
mathematics got some theorems which were not classical theorems, while
Bishop-style constructive mathematics gets only classical theorems, and
not all of the classical theorems. Bishop's reformulations of many
classical results impress me a lot; still for someone with a different
mathematical aesthetic, taking them instead of the classical formulations
might well count as abandoning the classical results. So while I agree
with Bishop here more than I agree with Hilbert, I still think Hilbert's
objection has force.
--Matt
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