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

Ludwig Wittgenstein, Cora Diamond, Cora Diamond

• Publisher: University of Chicago Press
• Year: 2015
• ISBN:
  9780226308609
• Topic: Philosophy
• Subtopic: Philosophy History & Theory

For several terms at Cambridge in 1939, Ludwig Wittgenstein lectured on the philosophical foundations of mathematics. A lecture class taught by Wittgenstein, however, hardly resembled a lecture.He sat on a chair in the middle of the room, with some of the class sitting in chairs, some on the floor. He never used notes. He paused frequently, sometimes for several minutes, while he puzzled out a problem. He often asked his listeners questions and reacted to their replies. Many meetings were largely conversation.These lectures were attended by, among others, D. A. T. Gasking, J. N. Findlay, Stephen Toulmin, Alan Turing, G. H. von Wright, R. G. Bosanquet, Norman Malcolm, Rush Rhees, and Yorick Smythies. Notes taken by these last four are the basis for the thirty-one lectures in this book.The lectures covered such topics as the nature of mathematics, the distinctions between mathematical and everyday languages, the truth of mathematical propositions, consistency and contradiction in formal systems, the logicism of Frege and Russell, Platonism, identity, negation, and necessary truth. The mathematical examples used are nearly always elementary.

I

I am proposing to talk about the foundations of mathematics. An important problem arises from the subject itself: How can I—or anyone who is not a mathematician—talk about this? What right has a philosopher to talk about mathematics?

One might say: From what I have learned at school—my knowledge of elementary mathematics—I know something about what can be done in the higher branches of the subject. I can as a philosopher know that Professor Hardy can never get such-and-such a result or must get such-and-such a result. I can foresee something he must arrive at.—In fact, people who have talked about the foundations of mathematics have constantly been tempted to make prophecies—going ahead of what has already been done. As if they had a telescope with which they can’t possibly reach the moon, but can see what is ahead of the mathematician who is flying there.

That is not what I am going to do at all. In fact, I am going to avoid it at all costs; it will be most important not to interfere with the mathematicians. I must not make a calculation and say, “That’s the result; not what Turing says it is.” Suppose it ever did happen—it would have nothing to do with the foundations of mathematics.

Again, one might think that I am going to give you, not new calculations but a new interpretation of these calculations. But I am not going to do that either. I am going to talk about the interpretation of mathematical symbols, but I will not give a new interpretation.

Mathematicians tend to think that interpretations of mathematical symbols are a lot of jaw—some kind of gas which surrounds the real process, the essential mathematical kernel.¹ A philosopher provides gas, or decoration—like squiggles on the wall of a room.

I may occasionally produce new interpretations, not in order to suggest they are right, but in order to show that the old interpretation and the new are equally arbitrary. I will only invent a new interpretation to put side by side with an old one and say, “Here, choose, take your pick.” I will only make gas to expel old gas.

I can as a philosopher talk about mathematics because I will only deal with puzzles which arise from the words of our ordinary everyday language, such as “proof”, “number”, “series”, “order”, etc.

Knowing our everyday language—this is one reason why I can talk about them. Another reason is that all the puzzles I will discuss can be exemplified by the most elementary mathematics—in calculations which we learn from ages six to fifteen, or in what we easily might have learned, for example, Cantor’s proof.

Another idea might be that I was going to lecture on a particular branch of mathematics called “the foundations of mathematics”. There is such a branch, dealt with in Principia Mathematica, etc. I am not going to lecture on this. I know nothing about it—I practically know only the first volume of Principia Mathematica.

But I will talk about the word “foundation” in the phrase “the foundations of mathematics”. This is a most important word and will be one of the chief words we will deal with. This does not lead to an infinite hierarchy. Compare the fact that when we learn spelling we learn the spelling of the word “spelling” but we do not call that “spelling of the second order”.

I said “words of ordinary everyday language”. Puzzles may arise out of words not ordinary and everyday—technical mathematical terms. These misunderstandings don’t concern me. They don’t have the characteristic we are particularly interested in. They are not so tenacious, or difficult to get rid of.

Now you might think there is an easy way out—that misunderstandings about words could be got rid of by substituting new words for the old ones which were misunderstood. But it is not so simple as this. Though misunderstandings may sometimes be cleared up in this way.

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.

There is one kind of misunderstanding which is comparatively harmless. For instance, many intelligent people were shocked when the expression “imaginary numbers” was introduced. They said that clearly there could not be such things as numbers which are imaginary; and when it was explained to them that “imaginary” was not being used in its ordinary sense, but that the phrase “imaginary numbers” was used in order to join up this new calculus with the old calculus of numbers, then the misunderstanding was removed and they were contented.

It is a harmless misunderstanding because the interest of mathematicians or physicists has nothing to do with the ‘imaginary’ character of the numbers. What they are chiefly interested in is a particular technique or calculus. The interest of this calculus lies in many different things. One of the chief of these is the practical application of it—the application to physics.

Take the case of the construction of the regular pentagon. Part of the interest in the mathematical proof was that if I draw a circle and construct a pentagon inside it in the way prescribed, a regular pentagon as measured is the result under normal circumstances.—And of course the same mathematical statement may have a number of different applications.

Another interest of the calculus is aesthetic; some mathematicians get an aesthetic pleasure from their work. People like to make certain transformations.

You smoke cigarettes every now and then and work. But if you said your work was smoking cigarettes, the whole picture would be different.

There is a kind of misunderstanding which has a kind of charm:

“The line cuts the circle but in imaginary points.” This has a certain charm, now only for schoolboys and not for those whose whole work is mathematical.

“Cut” has the ordinary meaning:

. But we prove that a line always cuts a circle—even when it doesn’t. Here we use the word “cut” in a way it was not used before. We call both “cutting”—and add a certain clause: “cutting in imaginary points, as well as real points”. Such a clause stresses a likeness.—This is an example of the assimilation to each other of two expressions.

The kind of misunderstanding arising from this assimilation is not important. The proof has a certain charm if you like that kind of thing; but that is irrelevant. The fact that it has this charm is a very minor point and is not the reason why those calculations were made.—That is colossally important. The calculations here have their use not in charm but in their practical consequences.

It is quite different if the main or sole interest is this charm—if the whole interest is showing that a line does cut when it doesn’t, which sets the whole mind in a whirl, and gives the pleasant feeling of paradox. If you can show there are numbers bigger than the infinite, your head whirls. This may be the chief reason this was invented.

The misunderstandings we are going to deal with are misunderstandings without which the calculus would never have been invented, being of no other use, where the interest is centered entirely on the words which accompany the piece of mathematics you make.—This is not the case with the proof that a line always cuts a circle. The calculation becomes of no less interest if you don’t use the word “cut” or “intersect”, or not essentially.

Suppose Professor Hardy came to me and said, “Wittgenstein, I’ve made a great discovery. I’ve found that . . .” I would say, “I am not a mathematician, and therefore I won’t be surprised at what you say. For I cannot know what you mean until I know how you’ve found it.” We have no right to be surprised at what he tells us. For although he speaks English, yet the meaning of what he says depends upon the calculations he has made.

Similarly, suppose that a physicist says, “I have at last discovered how to see what people look like in the dark—which no one had ever before known.”—Suppose Lewy says he is very surprised. I would say, “Lewy, don’t be surprised”, which would be to say, “Don’t talk bosh.”

Suppose he goes on to explain that he has discovered how to photograph by infra-red rays. Then you have a right to be surprised if you feel like it, but about something entirely different. It is a different kind of surprise. Before, you felt a kind of mental whirl, like the case of the line cutting the circle—which whirl is a sign you haven’t understood something. You shouldn’t just gape at him; you should say, “I don’t know what you’re talking about.”

He may say, “Don’t you understand English? Don’t you understand ‘look like’, ‘in the dark’, etc.?” Suppose he shows you some infra-red photographs and says, “This is what you look like in the dark.” This way of expressing what he has discovered is sensational, and therefore fishy. It makes it look like a different kind of discovery.

Suppose one physicist discovered infra-red photography and another discovered how to say, “This is a portrait of someone in the dark.” Discoveries like this have been made.

I wish to say that there is no sharp line at all between the cases where you would say, “I don’t know at all what you’re talking about” and cases where you would say, “Oh, really?” If I’m told that Mr. Smith flew to the North Pole and found tulips all around, no one would say I didn’t know what this meant. Whereas in the case of Hardy I had to know how.—In the case of the dark he only got an impression of something very surprising and baffling.

There is a difference in degree.—There is an investigation where you find whether an expression is nearer to “Oh, really?” or “I don’t yet . . .”

Some of you are connected on the telephone and some are not.—Suppose that every house in Cambridge has a receiver but in some the wires are not connected with the power station. We might say, “Every house has a telephone, but some are dead and some are alive.”—Suppose every house has a telephone case, but some cases are empty. We say with more and more hesitation, “Every house has...