Wittgenstein’s Lectures on the Foundations of Mathematics, Cambridge 1939 by Lectures on the Foundations of Mathematics

Author:Lectures on the Foundations of Mathematics [ed. Diamond] (Chicago, 1989)
Language: eng
Format: epub
Publisher: The University of Chicago Press
Published: 1976-02-15T00:00:00+00:00

And you can use this in all sorts of ways. About how it is to be used, Russell tells us nothing—except where he uses it himself. And he uses it in mathematics itself.

It is said to be a consequence of Russell’s theory that there are as many even numbers as cardinal numbers, because to every cardinal number I can correlate an even number.

But suppose I say, “Well, go on—correlate them.” Is it at once clear what I mean? Is there only one technique for correlating cardinal and even numbers?

You can interpret “correlate” in such a way that you’ll say, “Yes, there are as many . . .” But in what sense can you say you have proved this? You do a new thing and you call it “correlating them one-one”; and you call an entirely new thing “having the same number”. All right. But you have not found two classes which have the same number; you have only invented a new way of looking at the thing.

In fact, if you say you have one-one correlated the even numbers to the cardinal numbers—you have shown us an interesting extension of this idea of one-one correlation. But you haven’t even yet correlated any two things.

Supposing Lewy has learned to count. Then at a certain point we would say, “He can now count indefinitely.” And we can then say, “He has now learned ℵ0 numbers.”

I wanted to describe today the relation between the actual use of the word “counting” outside mathematics and its use inside mathematics.

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