13*. Ludwig Wittgenstein

Ludwig Wittgenstein was born in 1889, son of parents of Jewish extraction but not Jewish religion. Asked how his family came by the name ‘Wittgenstein’ Ludwig said they had been court Jews to the princely family and so had taken the name when Jews were required by law to have European-style names. The father, Karl, was a Protestant, the mother a Catholic. The Jewish blood was sufficient to bring the family later on into danger under Hitler’s Nuremberg Laws. They did not think of themselves as Jews or belong to the Jewish community in Vienna. The children were brought up sort-of Catholic though so far as I know only the eldest, Hermine, towards the end of her life, took this seriously and made a profession of faith before friends and household. At 9 years of age Ludwig and Paul, a year or two older than Ludwig, talked together and decided that their religion was all nonsense. Paul became a pianist of some fame, but soon after his debut in Vienna he became a wounded prisoner on the Russian front and his arm was lopped off by a surgeon who did not know he was a pianist. Their father, Karl, who died in 1913, thought the only proper career for a son of his was that of a civil engineer. He had himself started the Austrian iron and steel industry and become immensely rich. Two of his sons committed suicide, one upon the armistice in 1918. Ludwig was sent to Berlin before the war to study engineering. He went from there to Manchester for the same purpose and while there got extremely interested in explaining the mathematics that he was studying. He also flew kites and invented a jet engine for an aeroplane. He was told that if the questions about mathematics that he seemed so concerned with were truly his interest, he should go to Cambridge and study with Bertrand Russell. This he did. As I heard the story, it was Russell who drew his attention to Frege. He read and went to see Frege. They had a discussion, of which Wittgenstein said ‘He wiped the floor with me’. He was very delighted that Frege said he should come again. Frege wrote him some letters during the War expressing a remarkable respect for him. Later he received a copy of the Logisch-Philosophische Abhandlung (pretty universally known as the Tractatus Logico-Philosophicus, a title proposed by G.E. Moore).[1] Frege said he could make nothing of it.

It is a book that stands, one might say, halfway between Frege and Russell—at least in some ways.

A principal concern of the Tractatus is the relations of sense to reality—truth and falsehood. Only propositions have sense, and if the structure of a sentence is permissible and the sentence—the proposition—doesn’t make sense, this must be because no meaning has been given to some sign or signs as they occur in it. ‘Meaning’— the translation of Frege’s Bedeutung and also a word constantly used by Russell for standing-for—belongs to names, which have meaning only in the context of propositions.

A significant proposition—sinnvoller Satz—would prove on final analysis to be a kind of arrangement of names of simple objects, or a truth-function of several such, or indeed of all possible such arrangements.

The arrangement of names in what is (simply) such an arrangement is called its structure.

The question arises, how the propositional connection comes into existence. Wittgenstein mentions this question but does not answer it.

The possibility of the structure is called the form of representation.

The form of representation is also the possibility that the simple objects named by the names—which are guaranteed to name permanent simples—are arranged as the names are in the proposition.

Thus the possibility of the structure is the possibility of the arrangement of the simple objects which are named in the significant proposition.

This is the possibility of the fact expressed by the significant proposition.

It will also be possible that the objects named in the proposition are not so arranged. Then the significant proposition will be false. If—as at present—we are speaking of arrangement, then the propositions we are considering are elementary—atomic—and the non-arrangement of the objects as their names are arranged is falsehood and will be the non-existence of elementary, or atomic, facts. (The expression ‘atomic fact’ used by Ramsey was taken from Russell.)

If you had the totality of elementary propositions you could express the sense of any one of them by putting it down as a conjunct with all the huge array of disjuncts of the remaining elementary propositions and their negations, e.g. thus: p&q v -q&r v -r&s v -s&t v -t ... and so on. This would be a significant proposition and could be offered to illustrate the thesis that all logical constants are already somehow present in the compoundedness of any proposition. (5.47) Any significant proposition thus reaches through the whole of logical space.

I once mentioned to Wittgenstein that I had come across the philosopher Proclus saying that a name is a logical image of its object. I hesitated to say ‘logical picture’ because the Tractatus doctrine that propositions are logical pictures of (possible) facts was so famous, but he at once responded ‘I have so often had that thought’—the thought, namely, that a name is a logical picture of what it names. I was slow to realise that I had been wrong in assuming that the objects, the simples, spoken of in the Tractatus, were uniform characterless atoms, whose arrangement alone produced the characters of familiar things. (These characters indeed Wittgenstein called ‘external’.) The assumption was absurd—the internal characters of objects are not of the same logical form (2.0233)—in fact, it looks as if their logical form and their internal character were the same thing. The possibility of a given fact must be ‘prejudged’ in the things that can occur in such a fact. (2.012) This at least suggests that it is not possible for every simple object to occur in just any fact. Rather, as holders of their names too, the objects can only enter into certain compositions. We cannot illustrate this with elementary propositions, as we do not know any, but we might construct analogues, using only the sorts of names we do know; we may note that ‘Mount Everest chased Napoleon to Cairo’ does not express a possible fact—unless we change the meaning of, say, ‘Mount Everest’.

That the simple objects of the Tractatus are diverse in logical form is actually quite obvious. For example, we are told ‘Its possibility of occurring in elementary facts (Sachverhalte) is the form of an object.’ (2.0141) And ‘Space, time and colour are forms of objects.’ (2.0251) These thoughts are quite near to ‘Names are (logical) pictures of objects’ if you grant the character of a name only to names of simples—even though you cannot produce an example of such a name.

This truth—that for the Tractatus there is a diversity of forms of object—allows a corresponding diversity in the characters of names, even of simple objects. Such names would be the elements of a ‘fully analysed’ proposition—a sprinkle of names on a logical network, as Wittgenstein put it later on. Remember that he calls a proposition a logical picture of a (possible) fact. This means that there isn’t a problem arising from the ‘isomorphism’ between language, thought and reality, as many—including myself—have felt there is for the Tractatus theory. The problem was constituted by the isomorphism’s being two-way. If a figure x is isomorphic with a figure y. then equally y is isomorphic with x. So how does x’s isomorphism with y show that x is a picture of y any more than y is a picture of x?

In some cases we must make the admission: which is a picture of which is not determined. If you have a simple spatial picture of another spatial arrangement, and you exhibit the correlations by lines of projection, then the second spatial arrangement is as much a projection of the first as the first is of it. Similarly with arrangements of colours. But here the forms of representation are not purely logical; they include the forms signified by the terms ‘spatial’ and ‘coloured’. If you have a tune, with a temporal order of notes, and you see this represented by a line of musical notation which is spatial, there is no form of representation other than the logical form connecting the two things—the tune and the line of notation. (These considerations help us to understand the remark ‘Space, time and colour are forms of objects’.) The pattern in the tune and in the line of notation is also, Wittgenstein says, in the grooves of the gramophone record of the tune and the sound waves. That need not concern us. The marks belonging to the line of notation signify the notes of the tune and not the other way round. You have to understand such and such a mark as the sign (name) of a note in order to know what these marks are coordinated with. And similarly, if you wanted to say: a fact declared by a proposition was just as much a picture of the proposition as the proposition was of the fact, you would have to call the objects in the fact ‘names of names’—for it is only as names that certain elements of the propositional sign are elements of the picture of the fact. But you could not make out what the elements of the picture were independently of its being a picture. No such difficulty arises about the elements of the fact which the picture presents. Thus the argument from isomorphism’s being two-way fails—except in cases where it is harmless and either structure may be used as a picture of the other.

Let us return to the relation between structure and form of representation. The structure of a picture is the way its elements combine, the way they relate to one another. The form of representation is both the possibility of the structure and the possibility that the objects in the reality being represented are related to one another in the same way as the elements of the picture.

In what way do the elements of a proposition, an elementary proposition, relate to one another? It is composed only of names of simples. They are connected together in a ‘logical’ arrangement.

If that is so, then the simples in the corresponding fact (if the proposition is true) are connected together also in a ‘logical’ arrangement—the same ‘logical’ arrangement as that of the names. We ought to be amazed.

This announcement makes the connection between thought and a thinkable reality. The possibility that the elements of the reality combine as they do equals, i.e. is identical with, the possibility of the logical picture’s elements combining as they do. The picture’s very form of representation is identical with the (logical) possibility of things combining in the way its elements do. And the possibility of things combining in the way that would constitute the actual fact, is its form. (2.033)

So much for the Tractatus on simples and pictures. Let us however think again of Proclus’ remark: The name is a logical picture of its object. As the Tractatus lays down what ‘logical picture’ is to mean, that will not have been true of its names and objects. One might, as I did, translate Proclus’ phrase by ‘logical image’—the Greek will have been icon logike. But whatever we do, and even if we follow the Tractatus, there is something about names and their objects which is not a matter of a simple relation effected arbitrarily in the manner assumed by John Locke and John Stuart Mill. Mill said that proper names have only denotation, not connotation. Wittgenstein, as I heard him in his classes, denounced this. ‘It is a great deal of information about a word that it is a proper name, and still more, what kind of thing it is a proper name of—a man, a battle, a place, etc., etc.’ In the Tractatus, names being restricted to simple objects, we can’t say what their objects are, only give propositions presenting configurations of them. ‘A proposition cannot say what a thing is, only how it is.’ In his later work, Wittgenstein certainly gave up his simple objects. But even they had logical forms, which would have come out in the propositions that could have been formed out of their names—if we could in fact have named them. And propositions were descriptions of possible elementary facts by their internal properties.

This has not simply died in the later work. Earlier, he had spoken of structure; later he spoke of grammar, and said ‘Essence is expressed in grammar’. This, we may say, was made clear in the first place by Frege in the case of the essence connected with the general notion of an arithmetical function. Of course, Frege did not produce that sentence about ‘essence’. I am inclined to say that he laid an egg, in such writings as Funktion und Begriff and Was ist eine Funktion?,[2] an egg which Wittgenstein hatched. In Funktion und Begriff, Frege pointed to the difference between, say, 2 + x4 and 2 + 34. The former is an expression of a numerical function of which the latter is an example. The first has no numerical value, the second has one:

2+ 34 = 83

The difference of meaning between the expressions of instances of a numerical function—in this case e.g. 2 + 14, 2 + 54, 2 + 104, etc—and the expression of a numerical function is not an example of equivocation like ‘John gave three rings’, when it was a door-bell he is described as ringing, and ‘John gave three rings’, when it was a present of rings for the fingers. The difference between 2 + x4 and 2 + 34 is highly significant because the point of the former is to signify the form of such expressions as the latter. This is a grammatical difference, as can be clearly seen in the joke about the teacher who says ‘Suppose there are x pounds of sugar in a box’ and the pupil who puts up his hand and says ‘But sir, suppose there aren’t?’ The pupil hasn’t yet grasped the grammar of x used as it is in expressions of a function for example—or he is making a cheeky joke. Even so, it would be a grammatical joke.

That essence is expressed in grammar was clear enough in the case: arithmetical function. But it is also clear in most cases of familiar concepts of substances and kinds of stuff. Examples: acid, wood, metal, milk, animal, plant, peacock, man, flea, banana-tree. N.B. that when we come to plants and animals the identity of an individual is of a different kind from the identity of a lump of lead, say. Here I have been mentioning ‘substantial’ terms. The notion of essence is certainly not confined to those, as the example of numerical functions shows. Let us consider another example of a mathematical essence. It occurs in Plato’s Meno: a square.

One which is twice the area of a given square is the square of its diagonal. I once undertook to demonstrate Plato’s point in the Meno with a nine year old girl who, like Meno’s slave, had never learned any geometry. I began as Socrates did, drawing a rough square and asking: how long will the side be of a square twice as big? To my astonishment and pleasure she answered just as the slave did, and we proceeded just as the dialogue did, because she always said the next thing that the slave did. I became convinced that this famous bit of the dialogue is no fiction.

What did she end up knowing? One might say: if I drew the squares etc. quite accurately, she ended up knowing that this square and this one (the first and the second guesses) weren’t twice the original square, but this last one was. But, first, I wasn’t being accurate in my drawing, and second, we could ask how she knew what we are saying she ended up knowing. Was it by the way they looked? If so, would she have any reason to suppose it would look the same another time? You might say it would have to. But suppose another time I drew them in a different colour and a different size. ‘Oh’, you might say, ‘we don’t mean “look the same” in those ways.’ What way of ‘looking the same’ do we mean? ‘The same, in that square on the diagonal was (and so at least roughly looked) twice the size of the original square.’ But how will it look twice the size? You reply ‘By being composed of triangles, each being half the size of the original square, and a quarter of the new one.’

If you don’t draw it so, or at least ask questions which the child answers so, then I am not asking about the geometrical proposition. (For this, accurate drawing doesn’t matter.)

What I am eliciting by my questions—which are not ‘leading questions’ containing the wished for answers—is an essence, part at any rate of the essence of a plane square.

Wittgenstein says in Part I of the book Remarks on the Foundations of Mathematics, remark 32, that mathematicians produce essences. We can see what he means in the examples: numerical function and plane square. Functions emerged, as a mathematical topic, I believe, in the seventeenth century. I didn’t say that Frege ‘produced’ such essences, but only that he showed what they were, and how to avoid confusing sign and thing signified. The square of Euclidean geometry was an essence produced many centuries before.

Mathematicians have ‘produced’ such essences by using a grammar; the first formulator of the geometrical notion of a square was presumably extending and adding to a grammar already in use. It is a curious thing that people can build grammar without knowing what they are doing. There is a remark something like this in the Tractatus at 4.002: ‘man possesses the capacity of building languages in which any sense can be expressed, without any idea how and what each word means.—And one speaks, without knowing how the individual sounds are produced.’

This may be verified, up to a point, in examples of mathematical concepts, and in a host of others. That language as such was a human invention seems enormously doubtful, as does the expression ‘build languages in which any sense can be expressed’. Languages don’t fail to be languages because they need to be built onto in order to express physics in its present state. There may be in this remark about expressing any sense a sign of conviction that anything that is a language can say anything sayable. The later Wittgenstein, like Descartes, rather makes a comparison with an old city, the centre full of narrow twisty streets and odd corners, while the suburbs are all straight wide streets.

However, I am more interested in the similarities than the differences. And I put it forward that ‘grammar’ hasn’t got a special new sense, it is only more extensive than the rather thin grammar children learn at our schools. And grammar, as Wittgenstein considers it, corresponds to the ‘structure’ of pictures, of which he wrote in the Tractatus. In that book, maybe we can say objects have essences, if we are allowed to say anything about objects; Wittgenstein speaks not of essences there but rather of logical forms, and there is little about them. What have essences rather are propositions and elementary facts; and this fits in well with the analogue of structure to grammar.

We might truly observe that the Tractatus has a sort of simplicity. It offers a strange and rather powerful theory of sense, truth and falsehood for what it calls ‘significant propositions’; and a rather simplistic conception of mathematics as consisting of nothing but equations. (This I do not understand, so I won’t dwell on it.) It also gives an account of propositions of logic, or logically true propositions, which makes them non-significant though not nonsensical. They are not significant, because they can’t be true or false like ‘significant propositions’. This doctrine, together with the special meaning Wittgenstein invented for ‘tautology’, has been widely embraced. According to it, p v -p is not significant, because it excludes nothing. Equally, the contradictory p&-p is not significant, because it excludes everything. Still, neither tautology nor contradiction are nonsensical, ‘unsinnig’ in German; they have a role to play in logical exposition.

Having said that much, I can characterise Wittgenstein’s latest period—not his middle period or periods—as marked by the realisation consequent upon the middle work, that ‘it’s not as simple as all that’. Much in the Tractatus remains valuable and the book fascinates people like me, who do not believe a lot of it; I believe the valuable stuff so far as I can identify it, but remain fascinated by much I cannot discern as valuable. However, noting the excessive simplicity of the central picture of sense, truth and falsehood, I also note what the book does not cover, even though it covers astonishingly much. It shows a half belief that experimental psychology is a natural science. The observation ‘Theory of knowledge is philosophy of psychology’ I think indicates this—along with the statement that psychology is no nearer philosophy than is any other natural science. When he was writing one of the short pieces which we put at the end of Part II of Philosophical Investigations (for it had no definite place given it by Wittgenstein in his MS of that work) he wrote that experimental psychology is marked by experimental methods and conceptual confusion. He called it analogous to set theory, which he said was marked by conceptual confusion and methods of proof.

However, the observation that theory of knowledge is philosophy of psychology can be taken in a more sophisticated sense than I have given it. Once in conversation with Wittgenstein I asked how he’d describe the difference he had made—I meant, and he understood me—in his later work. His answer indeed fits the Tractatus too, from which I have taken the remark about theory of knowledge. He said in answer to me, that if you looked at the titles of most of the famous works of philosophy in recent centuries, you found that they tended either to contain the word ‘principles’ or some reference to the human mind. This last is correct enough even of Hume’s title A Treatise of Human Nature, no less than of Locke’s Essay. Reflecting on the matter, I reminded myself that for the centuries since Descartes, theory of knowledge has been queen in philosophy—as metaphysics had been in earlier centuries.

Now let us consider some principal features of this phenomenon. One is that extraordinary leap by which Descartes turned so many successors into believers in the certainty only of immediate experience. In his second Meditation he asks what he attributed to the soul. His answer includes nutrition. I am always surprised that this doesn’t bring people up with a jolt. Why nutrition? The answer is simple; he had something of an Aristotelian training at La Flèche. Nutrition is one of the marks of the vegetative soul. Two pages later, Descartes is reciting what he knows, which includes ‘I see’. But he remembers his method of doubt, and corrects it to ‘I seem to see’ and then says that that’s what seeing really is.

Contrast with this Wittgenstein’s late investigation into the difference of category between two different ‘objects’ of sight: first, ‘I see this’ I assert and offer a description, ‘a sleeping cat’, say, or I draw a picture. Second, ‘I’ve suddenly seen a likeness between this face and that’. A new aspect, which the man I say this to does not see. In the Tractatus Wittgenstein had regarded the two ways of seeing a cube as seeing ‘two different facts’. Is one really just seeing, which is all one thing; or is one’s seeing partly thought? We remember ruefully what all Descartes was willing to call thoughts.

And later philosophers were worse. At least Descartes did not count knowledge and memory among his cogitationes. Early twentieth century philosophers sometimes did make just that mistake. Whether Descartes thought an act of will—simply of the will itself— was involved as an immediate precedent to a voluntary or intentional action I don’t remember; John Stuart Mill certainly did, and I could recite twentieth century philosophers who do so treat voluntariness, and even intention. It is not that they think there can be an act of resolution sometimes involved in bringing yourself to do something; it is that they think that it is essential to the performance of a voluntary act. But voluntariness doesn’t require any such thing. Wittgenstein was clearly as subject to temptation here as any of us. ‘Doing itself seems not to have any volume of experience. It seems like an extensionless point, the point of a needle. This point seems to be the real agent …’ Yet ‘ ‘I do’ seems to have a definite sense, separate from all experience’. (Philosophical Investigations, 1.620) Again he asks ‘when I raise my arm, ... are the kinaesthetic sensations my willing?’

There is a little book, probably little known, called The Practice of the Presence of God. It is mostly records of what was said to people who came to visit him by a Carmelite friar, a lay-brother of little ‘education’ who worked in the kitchen of his monastery. If that phrase was his, what did he mean by it? Reading what he said to people, it is clear that he meant often speaking to God in prayer in the course of his work. But suppose he was having to concentrate on some tricky job? Did this involve an intermission in his ‘practice’? I should judge not; that he thought the practice should be all the time. Was he all the time imagining God observing him all the time while he was busy? After some reflection I realised that I was still the victim of a Cartesian type of assumption—it must be a constant state of consciousness, and that must have been a continuous Cartesian cogitatio. I then saw that no such thing was necessary. What one is doing in one’s activity, the reasons one has for what one does, may, as Wittgenstein occasionally observed, be merely elicited from one by a question. ‘Why did you pull that cord?’—’To change the notice on the door’—’Did you notice the notice?’—’No, I always change it, round about now’. ‘But isn’t there an act “peculiar to the will” which is nothing but a turning towards doing something, an act which proceeds from an interior starting point of cognition?’ How is one supposed to learn there is such a thing? What one knows, what one mentions to explain one’s action, may not have been ‘in one’s consciousness’ at all. And suppose one’s action is inaction—in some situations one votes by doing nothing, saying nothing, making no movement of the hand, for example. One knows this, but does not have to be thinking of it so to vote. Indeed one did learn about it at some time; one was told and one would probably give it in explanation if someone asked.

To say this is not to say there is no such thing as an event of thinking of something and being ‘therefore just about to’ do what one does. The mistake is to think that there must always be some such thing.

Even when there is such an antecedent to a voluntary action, it may itself consist in different things on different occasions. It may for example be a memory or a reminder, which ‘prompts’ a behaviour; sometimes it may be a catching sight of something, sometimes a thought which struck one. The ‘interior act of will’ is invoked because of the being prompted which one refers to in the explanation of one’s movement which one truthfully gives: it is not necessarily an experience, but it resides in the ‘because’ of such explanations.

There is a similar mistake that threatens us when we seek to explain what suddenly understanding something is.

In the sense in which there are experiences characteristic of understanding, Wittgenstein remarks, understanding is not an experience. Further, it is the circumstances in which one has an experience characteristic of understanding that justify taking the experience as one of ‘understanding’.

An intelligent logician once responded to my speaking of an explanation as sometimes being ‘elicited’ after the event, by calling it then a ‘rationalisation’. That it might on occasion be that I would of course not deny; that it must be that, if it was not present in the consciousness of the person at the moment of the action that he is explaining, is a thought showing a powerful ‘Cartesian’ influence.

There are things in the early Wittgenstein which he never gave up: for example, the equivalence of p and ‘It is true that p’. And some things which lasted with him for a long time: ‘To be able to say: ‘p’ is true (or false) I must have determined in what circumstances I call ‘p’ true, and with that I determine the sense of the proposition.’ (Tractatus 4.063) Much later, when someone mentioned the ‘verification principle’ at the Moral Science Club, Wittgenstein asked who invented it, and having it attributed to himself, explained ‘Who? Me?’ in a tone of outrage. I do not know what was the formulation given. But it seems to me that he must have forgotten something he had written in the work Philosophical Remarks, which was published after his death: ‘A proposition (Satz) is a draft upon a verification’. He wrote that book around 1930 and at some time gave it to Moore, indicating that he didn’t think it was particularly good. Nevertheless, that remark does seem to be related to the ‘verification principle’. When he wrote the Philosophical Investigations, he wrote: ‘Asking about the kind and possibility of the verification of a proposition is only a special form of the question ‘How d’you mean?’ The answer is a contribution to the grammar of the proposition.’ (I, §353)

The equivalence of ‘p’ and ‘It is true that p’ must of course be understood as confined to where ‘“p” is true’ makes surface-grammatical sense. ‘Oh damn!’ would not be a substitution instance. On the other hand, cases like ‘I’ve got a pain’ where ‘there is no distinction between truth and truthfulness’ as Wittgenstein put it in Philosophical Investigations Part II, would be examples of the equivalence. At a very early stage (in Last Writings on the Philosophy of Psychology, Volume II) he mentioned the thought that belief in the existence of God is ‘an attitude’ with the comment that one who believed in God might say that if that was what someone thought, then he did not believe in God.

One thing I have observed among people who have been influenced by Wittgenstein: a certain tendency to think that if such and such a ‘language game is played’ that fact is justification of it. First, I deprecate talk of language games except where the speaker is able to describe them—that is, to describe the procedure into which words are woven in a certain fashion. But, more importantly, it should not be supposed that when a certain language game is played—i.e. when there is a certain use of words—that constitutes a title so to use the words. In Zettel 608–610 we have a passage where Wittgenstein says: ‘No supposition seems to me more natural than that there is no process in the brain correlated with associating or with thinking’, and further on asks ‘Why should there not be psychological regularity to which no physiological regularity corresponds?’ He remarks: ‘If this upsets our concepts of causality, then it is high time they were upset.’ One could not refute this comment by saying ‘But we do play these language games with “causality” and explanation’.

To come to an end: I once heard someone ask Wittgenstein what it all came to, what was, so to speak, the upshot of the philosophy he was teaching in the 1940s. He did not answer. I am disposed to think that there wasn’t an answer he could give. That, namely, he did not think out a total position as in writing his first book; that, rather, he was constantly enquiring; some things he was pretty sure of, but much was in a state of enquiry. I therefore deprecate attempts to expound Wittgenstein’s thought as a finished thing. He himself in his classes sometimes said he was as it were giving examples of ‘five-finger exercises’ in thinking. These were certainly not limited in number like the set a piano teacher might employ, and were not like automatic formulae of investigation. Predictions of ‘what Wittgenstein would say’ about some question one thought of were never correct.

 

* First published in Philosophy 70 (1995): 395–407 and reprinted by permission of the publishers, Cambridge University Press. Originally delivered as a lecture in Cambridge in 1991 as part of a series of lectures on Cambridge philosophers. Misprints in the original printed version corrected in the light of the author’s final typescript.

1 The works of Wittgenstein referred to here are as follows: Tractatus Logico-Philosophicus, tr. Frank Ramsey, ed. C.K. Ogden (London: Kegan Paul; New York: Harcourt Brace, 1922); Remarks on the Foundations of Mathematics, ed. G.H. von Wright, R. Rhees, and G.E.M. Anscombe (Oxford: Blackwell, revised edition, 1978); Philosophical Investigations, tr. G.E.M. Anscombe (Oxford: Blackwell, 1953); Philosophical Remarks, ed. R. Rhees (Oxford: Blackwell, 1964); Last Writings on Philosophy and Psychology, ed. G.H. von Wright and H. Nyman (Oxford: Blackwell, 1982); Zettel, ed. G.H. von Wright and G.E.M. Anscombe (second edition, Oxford: Blackwell, 1981).

2 Translated as ‘Function and Concept’ and ‘What is a Function?’ respectively in Translations from the Philosophical Writings of Gottlob Frege, tr. P.T. Geach and M. Black (Oxford: Blackwell, 1952).