[FOM] Brief Reply to Comments on my Opinion 57

[Torkel Franzen]

  Doron Zeilberger says:

 >The existing body of mathematics is every bit as contingent as
 >the rules of chess, with the difference that human math is less deep
 >since humans pick problems that they can do.

I see that you're a staunch adherent of Hilbert's point of view!

     Take any definite unsolved problem, such as the question as to the
     irrationality of the Euler-Mascheroni constant C, or the existence of
     an infinite number of prime numbers of the form 2^n+1. However
     unapproachable these problems may seem to us, and however helpless we
     stand before them, we have, nevertheless, the firm conviction that
     their solution must follow by a finite number of purely logical
     processes. ...This conviction of the solvability of every mathematical
     problem is a powerful incentive to the worker. We hear within us the
     perpetual call: There is the problem. Seek its solution. You can find 
     it by pure reason, for in mathematics there is no ignorabimus.

A less optimistic view would be that we may well be utterly incapable of
solving many of the problems that occur to us and that we find fascinating,
in mathematics and elsewhere.

About foundations and problems in finite mathematics, there is
indeed a tradition of regarding such problems as in some philosophical
(or mathematical but highly ethereal) sense trivial. This tradition is
most pronounced in constructivism. Thus Brouwer simply stated that "in
so far as only finite discrete systems are introduced, the
investigation whether an embedding is possible or not can always be
carried out and admits a definite result", and Dummett remarks in
passing, quite absurdly, that we can "if we choose" or "at will"
determine the truth value of (in principle) decidable statements.
But today perhaps most people with an interest in foundations are well
aware of the difficult and interesting foundational problems associated
with finite mathematics.

Then of course there are also those who like to say that mathematical
problems that are not reducible to questions about the properties of
some suitable initial segment of the natural numbers are
"meaningless", thereby eliminating difficult and interesting
foundational problems concerning finite mathematics through a tactic
that neatly complements the view of those who would dismiss all
problems in finite mathematics as trivial.