[FOM] understanding Putnam on understanding mathematics

Gabriel Stolzenberg gstolzen at math.bu.edu
Sun Jul 22 18:15:31 EDT 2007

   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".

More information about the FOM mailing list