[FOM] classical/constructive: what is the issue?

Gabriel Stolzenberg gstolzen at math.bu.edu
Sat Dec 17 01:55:50 EST 2005

   In FOM Digest, Vol 36, Issue 8, Message 3, Bill Tait remarked:

>                                                 It seems to me to
> be very analogous to the debate between constructivists and non-
> constructivists, where, also, one has to ask: On what non question-
> begging grounds could one possibly resolve the issue.

   What debate?  (I'm not trying to be clever. I just don't know
what Bill is talking about here.  For example, he and I once had
an exchange about his essay, "Against intuitionism: constructive
mathematics is part of classical mathematics."  Was it a debate?
I would say that it was not but I have no idea whether Bill would
agree.  And as for whether there was significant question-begging
in the exchange, I would say that, despite appearances, there was
not.  Something else derailed it.  But, again, I have no idea if
Bill would agree with this.)

   More important, what issue needs to be resolved?  I ask because,
although anyone who has learned to shift between the classical and
constructive mindsets is in a good position to discover important
strengths and weaknesses of each, having done so, there do not seem
to be any major issues left to be resolved---except the one that I
discuss below, which I doubt is what Bill has in mind.

   To me and, I believe, to nearly all mathematicians, the issue
that trumps all others is which, if either, of these two mindsets
is superior for ordinary mathematical practice---where the criteria
for judging are precisely the ones that, for more than a century,
have been invoked to claim that classical mathematics is obviously
far superior in every important respect.

   But, no matter how persuasive such comparisons have been, they
are of no value for correctly resolving the issue.  This is because
they are comparisons between classical and constructive mathematics,
both viewed in the classical mindset; whereas, for the title of
ordinary mathematics, the candidates are (the current title holder)
classical mathematics as viewed in the classical mindset and (the
challenger from another planet) constructive mathematics as viewed
in the constructive one.

   To me, this is the one great issue that remains to be resolved.
However, for there to be a meaningful debate about how to resolve
it or, more realistically, how to investigate it, the mathematical
community would first have to be disabused of its deep conviction
that this issue already has been resolved.  And how likely is that
to happen?

   Gabriel Stolzenberg

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