Wittgenstein's Lectures on the Foundations of Mathematics: Cambridge, 1939 / Edition 1

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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.

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Editorial Reviews

Peter T. Geach, A.C. Jackson, and K. Shah kept meticulous notes from the last formal course that Wittgenstein taught at Cambridge in 1947. To preserve the actual words of Wittgenstein, this volume compiles all three sets of notes with no attempt to conflate or edit them beyond rendering them into lucid English. Topics covered by the notes in this volume include the private language argument, the grammar of sensation statements, certainty and experimentation in psychology. No index. No bibliography. Reprint of the Cornell U. Press 1976 edition. Using notes taken by four students, Diamond, (philosophy, U. of Virginia) has reconstructed thirty-one lectures given by Ludwig Wittgenstein in 1939. The result is a clearer exposition of Wittgenstein's philosophy of mathematics than can be found in the manuscript selections published as Remarks on the foundations of mathematics. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9780226904269
  • Publisher: University of Chicago Press
  • Publication date: 10/28/1989
  • Edition description: 1
  • Edition number: 1
  • Pages: 300
  • Sales rank: 803,796
  • Product dimensions: 5.62 (w) x 8.60 (h) x 0.65 (d)

Meet the Author

Cora Diamond is professor of philosophy at the University of Virginia.

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

From the Notes of R. G. Bosanquet, Norman Malcolm, Rush Rhees, and Yorick Smythies

By Cora Diamond

The University of Chicago Press

Copyright © 1976 Cora Diamond
All rights reserved.
ISBN: 978-0-226-90426-9


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. 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 a telephone." What if some houses have only a stand with a number on it? Would we still say, "Every house has a telephone"?

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 telephone".

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.

Suppose we said, there being only a difference of degree, "Smith has provided all Cambridge with telephones." If such a difference is allowed, couldn't one say, "How did you come by this? I don't yet know what you mean."

We learn our ordinary everyday language; certain words are taught us by showing us things, etc.—and in connexion with them we conjure up certain pictures. We can then change the use of words gradually; and the more we change it, the less appropriate the picture becomes, until finally it becomes quite ridiculous. In the earlier cases we should say Smith was exaggerating or using high-flown language; finally we should say that he was simply using sophistry to cheat us.

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

This can only be said to confuse yourself or cheat yourself. If you do say it, it is only because you like to say you have provided the whole of Cambridge with telephones.

To understand a phrase, we might say, is to understand its use.

Suppose a man says that he has flown to the North Pole and has seen tulips there; and it turns out he means he saw there certain vortices of air and cloud which looked like tulips from his airplane. He says, "You mustn't think these tulips grow. They can only be seen from above. No seed, etc."—Here he is cheating in his use of this word. We should say we hadn't understood him. And if he was in the habit of saying this sort of thing, we should have the right, when he told us something which seemed surprising, to say to him, "I do not know what you mean. Tell me exactly what you mean, or else I may be cheated."

If a man says, "I flew to the North Pole", then one immediately thinks that one knows a lot about it, for example, that he crossed the Arctic Circle, etc.—If he said, "In my way of flying this doesn't hold", and he has been in a Cambridge laboratory all the time—he has described a new scientific process in old words, and we would say we didn't understand him. The picture he makes does not lead us on.

How much do we know of what he's talking about? By the words of ordinary language we conjure up a familiar picture—but we need more than the right picture, we need to know how it is used.

Suppose I said, "This is a picture of Moore.

* * *

It's an exact picture, but in a new projection. You mustn't think ..."—If I say, "This is a picture of him", it immediately suggests a certain way of usage. For instance, I might say, "Go and meet So-and-so at the station; you will know him because this is a picture of him." Then you may take the picture and use it to find him. But you couldn't do the same with my picture of Moore. You don't understand my picture of Moore because you don't know how to use it.

Similarly, you only understand an expression when you know how to use it, although it may conjure up a picture, or perhaps you draw it.

An expression has any amount of uses. How, if I tell you a word, can you have the use in your mind in an instant? You don't. You may have in your mind a certain picture or pictures, and a piece of the application, a representative piece. The rest can come if you like.

What is a 'representative piece of the application'? Take the following example. Suppose I say to Turing, "This is the Greek letter sigma", pointing to the sign σ. Then when I say, "Show me a Greek sigma in this book", he cuts out the sign I showed him and puts it in the book.—Actually these things don't happen. These misunderstandings only immensely rarely arise—although my words might have been taken either way. This is because we have all been trained from childhood to use such phrases as "This is the letter so-and-so" in one way rather than another.

When I said to Turing, "This is the Greek sigma", did he get the wrong picture? No, he got the right picture. But he didn't understand the application.

Similarly if I say to Lewy, "What is a Greek sigma?" and Lewy writes σ in the corner of the blackboard, then we say that Lewy knows what a sigma is. But it might turn out that he thought that the sign was only a sigma when written in the corner of the blackboard—perhaps because his schoolmaster wrote it there or something of the sort. Then we should say that after all he did not understand.—Or he draws sigmas like this:

* * *

He had the right picture in his mind, namely a picture of the sign σ; but he put it to the wrong use.

I say

(1) He understands it if he always uses it right in ordinary everyday life, millions of times.

(2) If he does this [Wittgenstein drew a sigma], we take his doing this as a criterion of his having understood.

Because in innumerable cases it is enough to give a picture or a section of the use, we are justified in using this as a criterion of understanding, not making further tests, etc.

I will be concerned with cases where having a picture is no guarantee whatever for going on in the normal way.


Excerpted from Wittgenstein's Lectures on the Foundations of Mathematics, Cambridge, 1939 by Cora Diamond. Copyright © 1976 Cora Diamond. Excerpted by permission of The University of Chicago Press.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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Table of Contents

The Lectures, I-XXXI

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  • Anonymous

    Posted June 10, 2001

    Wittgenstein at his best

    This set of 31 lectures is very well compiled and illustrates Wittgenstein's post-Tractatus views brilliantly. In fact, I think this text makes understanding Wittgenstein's 'Philosophical Investigations' even easier than the Blue and Brown books. Of particular interest in this text is the dialogue between Wittgenstein and Alan Turing(of WWII Enigma code fame and considered to be the father of modern computing). Buy this now!

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