FOM: human well-being; constructivism; anti-foundation
Matthew Frank
mfrank at math.uchicago.edu
Tue Jan 2 00:37:49 EST 2001
Some responses to Ayan Mahalanobis's recent post:
Mahalanobis asks: "What are the objections of formalists and recursive
function theorists to Bish[op-style math]?"
Some recursive function theorists (Nerode, for instance) think that
Bishop-style math is about computability, and that recursive mathematics
is a more powerful, more precise, and less confusing way of discussing it.
Many formalists believe in the principle of the excluded middle and
therefore find constructive math pointless. Hilbert responded to Brouwer
(according to Reid's biography, p. 184) with a slight variation of this:
"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."
Simpson had said: "Bridges/Richman do not comment on *why* one might
choose one system [of constructive math] over another, except to say that
Bish[op-style math] may be better, because it assumes less."
I would argue that working on Bishop-style mathematics brings more
conceptual benefits to classical mathematics than working on other
varieties of constructive math.
Mahalanobis says: "Is your point...that subjectivism is interesting and
merits a study[?] If so then I can't agree more, but I will also point
that it probably has very little or nothing to do with modern day
constructivism or Intuitionistic logic as it stands today."
These casual claims that something "merits a study" (especially when
coupled with claims of irrelevance!) annoy me. I hear them far too often
on this list and elsewhere; I have wanted to respond to them for a while.
We should be serious with claims about how to spend our energy and
resources. Can we collectively make a New Year's resolution to do this?
--Matt
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