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 change 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 Russell 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

This is a part of our ordinary speech; it is no more a mathematical proposition than "You have on a number of shoes which satisfies the equation $x+5=7$" is a mathematical proposition.

- p. 34

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

There is no "general proof." The word "proof" changes its meaning, just as the word "chess" changes its meaning. By the word "chess" one can mean the game which is defined by the present rules of chess or the game as it has been played for centuries past with varying rules.

We fix whether there is to be only one proof of a certain proposition, or two proofs, or many proofs. For everything depends on what we call a proof.

It is not the case that there are two facts - the physical fact that if one counts the squares one gets 756 and the mathematical fact that 21 times 36 equals 756.

- p. 39

Lecture 4

The point is that the proposition 25 x 25 equals 625 may be true in two senses. If I calculate a weight with it, I can use it in two different ways.

First, when used as a prediction of what something will weigh - in this case it may be true or false, and is an experiential proposition. I will call it wrong if the object in question is not found to weigh 625 grams when put in the balance.

In another sense, the proposition is correct if calculation shows this - if it can be proved - if multiplication of 25 by 25 gives 625 according to certain rules.

...It is of course in the second way that we ordinarily use the statement that 25 x 25 equals 625. We make its correctness or incorrectness independent of experience. In one sense it is independent of experience, in one sense not.

Independent of experience because nothing which happens will ever make us call it false or give it up.

Dependent on experience because you wouldn't use this calculation if things were different. The proof of it is only called a proof because it gives results which are useful in experience.

We often put rules in the form of definitions. But the important question is always how these expressions are used.

Suppose someone knew logic but not mathematics. Could we teach him to multiply simply by definitions? Can the decimal system be taught by definitions? If Russell can do all mathematics in Principia Mathematica, he ought to be able to work out 25 squared equals 625. But can he? How could decimal numbers be introduced into Principia Mathematica? Russel and Frege said that by introducing some more definitions into their systems they could prove such things as 25 squared equals 625. But we cannot teach anybody to multiply by definitions.

Mathematics and logic are two different techniques. The definitions are not mere abbreviations; they are transitions from one technique to another.

...It is immensely important to realize that definitions join two quite different techniques. Sometimes the difference is important and sometimes it is trivial (as when we write c instead of a times b). But the fact that the difference is trivial should not blind us to the fact that these are two different techniques.