[FOM] Wittgenstein's '770' in pi

[Grant Olney Passmore]

Dear Harold,

I have two comments that may or may not be helpful.

(1) Though I'm unfamiliar with discussions like this appearing in
mainstream number theory work (which isn't saying much, as I am
unfamiliar with almost everything), I have seen such discussions come up
rather often in writings on constructivism, especially in regards to
intuitionism and Brouwerian counter-examples.  One can use the
determination of the existence of a specified sequence of digits in the
decimal expansion of pi as the basis for a Brouwerian counter-example to
the intermediate value theorem, for instance.

(2) A quick search yields that `770' appears in the decimal expansion of
pi, as digits 1307-1309 (counting `3' as digit 1).

Best wishes,
Grant




Harold Teichman wrote:
> Hilary Putnam, in several recent essays, has discussed Wittgenstein's
> perhaps naïve puzzlement over the sense of assertions like "the sequence of
> digits '770' must appear somewhere in the decimal expansion of pi" [see e.g.
> Remarks on the Foundations of Mathematics, 2nd Ed., p. 266].
>
> In his paper "Was Wittgenstein Really an Anti-realist about Mathematics?"
> [see McCarthy and Stidd, Wittgenstein in America, Oxford (2001), p. 179]
> 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?
>
> Much obliged.
>
>
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