[FOM] re "understanding Putnam on understanding mathematics

Vladimir Sazonov V.Sazonov at csc.liv.ac.uk
Sat Jul 28 15:21:11 EDT 2007

Quoting Gabriel Stolzenberg <gstolzen at math.bu.edu> Fri, 27 Jul 2007:

>   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.)

I do not agree with "removing the meaning" by formalization. Which so 
valuable meaning was removed? Nothing was lost at all. Formalizing any 
intuition only improves it, even corrects, makes it more reliable, 
conscious, refined and powerful. I consider such blames against 
formalization and the formalist view on mathematics as having no 
grounds at all.

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
> paragraph.)

Of course insufficient. Without intuition formalization is not 
interesting. Without formalizability intuition is too poor and vague to 
be considered as mathematical.

>   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

Which reinterpretation? I do not understand. Everything essential was 
preserved and even improved.

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.

"Game with formulas" assumes "meaningless". But that is wrong. 
Interplay of intuition with formalism is not a meaningless game. Who 
said that formalization excludes or removes the intuition? The blamers 
themselves. This is the essence of mathematics which always pretended 
to be rigorous. What is rigour if not formalizability? Mathematics 
always was sufficiently formal. (Rules for the integrals, algebra, 
etc.) Hilbert and others made just the final (if it is really final) 
step in this direction.

>   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.

Thanks to these circumstances! Anyway, this step was logically 
inevitable by the very nature of mathematics. Can we imagine now that 
the mankind will reject all these formalisms, say, in the form of 
computer software?

I can understand the objections of Weyl only as a claim that there is 
something better. But can we imagine even Intuitionism (if it is really 
better than ZFC) without being eventually formalized? For mathematics 
there is no other way. And what then these objections against 
formalization and formalist view to mathematics mean at all?

Vladimir Sazonov

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