[FOM] understanding Putnam on understanding mathematics

Karin Verelst kverelst at vub.ac.be
Tue Jul 24 14:22:00 EDT 2007

In the post by Gabriel Stolzenberg occurs a quotation:

"The laws of logic,
the very laws that Aristotle taught us, do not hold beyond the domain
of the finite."

Where exactly did Hilbert make that remark?

>    In "Wittgenstein's '770' in pi" (July 15 2007) Harold Teichman
>>  In his paper "Was Wittgenstein Really an Anti-realist about
>>  Mathematics?" Putnam says:
>>     "What Wittgenstein should have said is that the mathematicians do
>>     understand the question whether 770 ever occurs in the decimal
>>     expansion of pi, and that they have learned to understand such
>>     questions by learning to do number theory..."
>>  No such statements (about decimal expansions) figure prominently in
>>  the texts on number theory that I have seen.  Could someone refer
>>  me to some actual literature in number theory (or any other part of
>>  mathematics) that has a bearing on this question?
>    Ask Putnam.
>    This has nothing to do with number theory.  You need to know the
>numerals, 0 and 7, the definition of 'sequence' and the meaning of
>'three in a row.'  For Putnam, you must also accept that the law of
>excluded middle holds for mathematics.  If you don't, then you don't
>    Even during his more than twenty years as an anti-realist (or, as
>he prefers to put it, an "internal" realist), Putnam was passionately
>committed to the law of excluded middle for mathematics, in seeming
>contrast, say, to Hilbert, who said something like, "The laws of logic,
>the very laws that Aristotle taught us, do not hold beyond the domain
>of the finite."
>    On my reading, both in Putnam's remark quoted at the beginning of
>this message and the text from which it is taken, the use of the word
>"understanding" is so vague that, despite everything we know about
>problems of meaning in a mathematics with the law of excluded middle,
>Putnam is able to make it seem that, contra Hilbert, an "understanding"
>of such a mathematics is acquired when we learn to practice it.
>    So, if Putnam is right about this, Hermann Weyl was mistaken in
>saying that Hilbert "saved" classical mathematics by shifting to a
>formal system.  His point (and Hilbert's) was that a formal system
>cannot be deficient in meaning because there is no meaning for it to
>be deficient in.
>    Gabriel Stolzenberg
>    P.S.  In at least one place, Wittgenstein uses "7777."  This is
>reminiscent of Brouwer's intuitionistic counterexamples, which use
>instead "seven 7's in a row".
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