Wittgenstein/Foundations of Mathematics

Quotes taken from Wittgenstein's Lectures on the Foundations of Mathematics, Cambridge, 1939

Cornell University Press

Lecture 1

What kind of misunderstandings am I talking about? They arise from a tendency to assimilate to each other expressions which have very different functions in the language. We use the word "number" in all sorts of different cases, guided by a certain analogy. We try to talk of very different things by means of the same schema. This is partly a matter of economy; and, like primitive peoples, we are much more inclined to say, "All these things, though looking different, are really the same" than we are to say, "All these things, though looking the same, are really different." Hence, I will have to stress the differences between things, where ordinarily the similarities are stressed, though this, too, can lead to misunderstandings.

- p. 15

Suppose Smith tells the municipal authorities, "I have provided all Cambridge with telephones - but some are invisible." He uses the phrase "Turing has an invisible telephone" instead of "Turing has no phone."

There is a difference of degree. In each case he has done something but not the whole. As he does less and less, in the end what he has done is to cahnge his phraseology and nothing else at all.

To think this difference is irrelevant because it is a difference of degree is stupid.

Lecture 2

Should you... say, "I believe that I intend to play chess, but I don't know. Let's just see" - ? Just as Russel once suggested that we don't know what we wish, don't know whether we want an apple or not.

Suppose we said, "What he said was just a description of his state of mind." But why should we call the state of mind he is in at present "intending to play chess"? For playing chess is an activity...

One might say, "Intending to play chess is a state of mind which experience has shown generally to precede playing chess." But this will not do at all. Do you have a peculiar feeling and say, "This is the queer feeling I have before playing chess. I wonder whether I'm going to play"? - this queer feeling which precedes playing chess one would never call "intending to play chess."

...I have been considering the word "intend" because it throws light on the words "understand" and "mean". The grammar of the three words is very similar; for in all three cases, the words seem to apply both to what happens at one moment and to what happens in the future.

Suppose I teach Lewy to square numbers by giving him a rule and working out examples. And suppose these examples are taken from the series of numbers from 1 to 1,000,000. We are then tempted to say, "We can never really know that he will not differ from us when squaring numbers over, say, 1,000,000,000. And that shows that you never know for sure that another person understands."

But the real difficulty is, how do you know that you yourself understand a symbol? Can you really know that you know how to square numbers? Can you prophesy how you'll square tomorrow? - I know about myself just what i know about him, namely, that I have certain rules, that I have worked certain examples, that I have certain mental images, etc etc. But if so, can I ever know if I have understood? Can I ever really know what I mean by the square of a number? Because I don't know what I'll do tomorrow.

- p. 27

Does the formula $ y = x^2 $ determine what is to happen at the 100th step?

It might mean, "Is there any rule about it?"

If it means, "Do most people after being taught to square numbers up to 100, do so-and-so when they get to 100?", it is a completely different question. The former is about the operations of mathematics but the latter is about people's behavior.

-p. 29

We have all been taught a technique of counting in Arabic numerals. We have al of us learned to count - we have learned to construct one numeral after another. Now how many numerals have you learned to write down?

Turing: Well, if I were not here, I should say $ \mathfrak{N}_0 $

...I did not ask, "How many numerals are there" This is immensely important. I asked a question about a human being, namely, "How many numerals did you learn to write down?"

Lecture 3

{{{1}}}

Watson: ...for instance, suppose one said, "There are in this room as many people as the number of moves which are needed for Black to checkmate White from such and such a position," would that be called an application of chess?

Wittgenstein: Why, certainly it would be. Indeed it might be the case that we discovered a fixed correlation between the number of moves needed to checkmate from certain positions and the number of, say, atoms in certain molecules. Then, in order to discover the number of atoms in such and such a molecule, it might be easiest to set the chessmen in such-and-such a position and play chess. That would certainly be an application of chess.

- p. 34

We call these things proofs because of certain applications; and if we couldn't use them for predicting, couldn't apply them, etc, we wouldn't call them proofs. The word "proof" is taken from ordinary everyday language, and it is only used because the thing proves something in the ordinary sense.

- p. 38