[FOM] re "understanding Putnam on understanding mathematics

Gabriel Stolzenberg gstolzen at math.bu.edu
Fri Jul 27 11:04:12 EDT 2007

On Fri, 27 Jul 2007, William Tait wrote:

> On Jul 25, 2007, at 6:24 PM, Gabriel Stolzenberg wrote:
> >    The exact remark is in "On the Infinite."  In "From Frege to
> > Goedel,"
> > it's in the second paragraph on p. 379.  Here is part of it.
> >
> >        In any case, those logical laws that man has always used
> >        since he began to think, the very ones that Aristotle
> >        taught, do not hold.
> The other part of it, which is essential for understanding what
> Hilbert was saying, is "In the domain of finitary propositions". He
> is speaking about the methodological position he wants to take
> (finitism) in order to prove the consistency of what he calls the
> laws of Aristotelian logic. Further down in the same paragraph: "we
> just do not want to renounce the use of the simple laws of
> Aristotelian logic..."

> Bill Tait

   I think I more or less agree.

   Sure, Hilbert doesn't want to renounce the use of any part of
classical mathematics.  ("Noone though he speak with the tongues of
angels....)  As I said earlier, Hermann Weyl talked about Hilbert
not renouncing classical mathematics but "saving" it and, in
particular, the laws of Aristotlean logic, by removing the meaning
and working in a formal system. (See below.)   And re consistency,
Weyl says that it is necessary but not sufficient.  ("Commentary on
Hilbert's second lecture," in "From Frege to Goedel," p. 484, 2nd

   The following two paragraphs are from the same lecture by Weyl.
("From Frege to Goedel," p. 483, 3rd paragraph.)

   [Hilbert] succeeded in saving classical mathematics by a
   radical reinterpretation of its meaning without reducing
   its inventory, namely, by formalizing it, thus transforming
   it in principle from a system of intuitive results into a
   game with formulas that proceeds according to fixed rules.

   Let me now by all means acknowledge the immense significance
   and scope of this step of Hilbert's, which evidently was made
   necessary by the pressure of circumstances.

   Gabriel Stolzenberg

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