The painted rock
Wittgenstein’s mathematics and the proving machines
Wozu braucht die Mathematik eine Grundlegung?! Sie braucht sie, glaube ich, ebenso wenig, wie die Sätze, die von physikalischen Gegenständen — oder die, welche von Sinneseindrücken handeln, eine Analyse. Wohl aber bedürfen die mathematischen, sowie jene andern Sätze, eine Klarlegung ihrer Grammatik. Die mathematischen Probleme der sogenannten Grundlagen liegen für uns der Mathematik so wenig zu Grunde, wie der gemalte Fels die gemalte Burg trägt. „Aber wurde die Fregesche Logik durch den Widerspruch zur Grundlegung der Arithmetik nicht untauglich?" Doch! Aber wer sagte denn auch, daß sie zu diesem Zweck tauglich sein müsse?! Man könnte sich sogar denken, daß man die Fregesche Logik einem Wilden als Instrument gegeben hätte, um damit arithmetische Sätze abzuleiten. Er habe den Widerspruch abgeleitet, ohne zu merken, daß es einer ist, und aus ihm nun beliebige wahre und falsche Sätze. „Ein guter Engel hat uns bisher bewahrt, diesen Weg zu gehen." Nun, was willst du mehr? — Man könnte, glaube ich, sagen: Ein guter Engel wird immer nötig sein, was immer du tust.
What does mathematics need a foundation for?! It needs one, I believe, as little as propositions about physical objects, or about sense impressions, need an analysis. What mathematical propositions do need, like those other propositions, is a clarification of their grammar. The mathematical problems of the so-called foundations lie at the foundation of mathematics for us as little as the painted rock holds up the painted castle. ‘But wasn't Frege's logic made unfit, by the contradiction, to be a foundation for arithmetic?’ It was! But who said that it had to be fit for that purpose?! One could even imagine that Frege's logic had been handed to a savage as an instrument for deriving arithmetical propositions. He has derived the contradiction, without noticing that it is one, and from it now derives arbitrary true and false propositions. ‘A good angel has so far kept us from going this way.’ Well, what more do you want? One could, I think, say: a good angel will always be necessary, whatever you do.
—Wittgenstein, Bemerkungen über die Grundlagen der Mathematik VII §16. Written in 1941, published in 1956 as Part V §13 of the first edition; VII §16 in the third (1978). The translation is mine; Anscombe's, which is how the last sentence is usually quoted, has "the painted rock is the support of a painted tower", and the German is a castle.
a bridge may fall down or something of that sort
Cambridge, 1939. Two lecture courses ran that year under one title, Foundations of Mathematics, and the title meant two different things. Alan Turing gave one of them, because Newman was not giving his, and it was mathematics, the thing a mathematician means by the phrase. Wittgenstein gave the other, twice a week through the Lent and Easter terms, thirty-one lectures spoken without a note to a room of about fifteen, and Turing came to sit in it and argue. Andrew Hodges, Alan Turing: The Enigma (1983), pp. 152–154: Turing "offered to the faculty a course for the spring term on Foundations of Mathematics. (Newman was not giving one this year.)"; "Alan joined Wittgenstein's class on Foundations of Mathematics. Although this had the same title as Alan's course, it was altogether different." Monk (The Duty of Genius, 1990, p. 417) puts Turing's course in the Easter term. The lectures are Wittgenstein's Lectures on the Foundations of Mathematics, Cambridge, 1939, ed. Cora Diamond (Cornell, 1976), reconstructed from the notes of four students; Diamond's preface gives the terms and the twice-weekly two-hour sessions. By the seventh lecture the philosopher was describing the mathematician to the room while the mathematician sat in it: "Turing doesn't object to anything I say. He agrees with every word. He objects to the idea he thinks underlies it. He thinks we're undermining mathematics, introducing Bolshevism into mathematics." Lectures, VII, p. 67. The Bolshevism is Ramsey's: his paper of 1925 on the foundations of mathematics had spoken of "the Bolshevik menace of Brouwer and Weyl". The quarrel broke open in the twenty-first lecture, over the Liar, and ran on into the twenty-second. What survives of it is what four students wrote down, which Cora Diamond wove into one text in 1976, and it is worth reading whole, because the sentence of Turing's that everyone quotes, the one about the bridge, is the smallest thing said in it.
Wittgenstein has been asking how anyone comes to be convinced of the law of contradiction, and has answered that we are trained into a technique of language in which saying a thing and saying its opposite has no use, which leaves us, queerly, puzzled by contradictions when they come:
WITTGENSTEINThink of the case of the Liar. It is very queer in a way that this should have puzzled anyone—much more extraordinary than you might think: that this should be the thing to worry human beings. Because the thing works like this: if a man says "I am lying" we say that it follows that he is not lying, from which it follows that he is lying and so on. Well, so what? You can go on like that until you are black in the face. Why not? It doesn't matter. […] Now suppose a man says "I am lying" and I say "Therefore you are not, therefore you are, therefore you are not . . ."—What is wrong? Nothing. Except that it is of no use; it is just a useless language-game, and why should anybody be excited? […]
TURINGWhat puzzles one is that one usually uses a contradiction as a criterion for having done something wrong. But in this case one cannot find anything done wrong.
WITTGENSTEINYes—and more: nothing has been done wrong. One may say, "This can only be explained by a theory of types." […] There is a particular mathematical method, the method of reductio ad absurdum, which we might call "avoiding the contradiction". In this method one shows a contradiction and then shows the way from it. But this doesn't mean that a contradiction is a sort of devil. One may say, "From a contradiction everything would follow." The reply to that is: Well then, don't draw any conclusions from a contradiction; make that a rule. You might put it: There is always time to deal with a contradiction when we get to it. When we get to it, shouldn't we simply say, "This is no use—and we won't draw any conclusions from it"? Is Russell's logic vitiated by a contradiction?
[…]
WITTGENSTEINYes, but where will the harm come?
TURINGThe real harm will not come in unless there is an application, in which case a bridge may fall down or something of that sort.
WITTGENSTEINAh, now this idea of a bridge falling down if there is a contradiction is of immense importance. […]
The next lecture opened where the last had broken off, on the bridge:
WITTGENSTEINIt was suggested last time that the danger with a contradiction in logic or mathematics is in the application. Turing suggested that a bridge might collapse. Now it does not sound quite right to say that a bridge might fall down because of a contradiction. We have an idea of the sort of mistake which would lead to a bridge falling. (a) We've got hold of a wrong natural law—a wrong coefficient. (b) There has been a mistake in calculation—someone has multiplied wrongly. The first case obviously has nothing to do with having a contradiction; and the second is not quite clear. […]
TURINGThe sort of case I had in mind was the case where you have a logical system, a system of calculations, which you use in order to build bridges. You give this system to your clerks and they build a bridge with it and the bridge falls down. You then find a contradiction in the system.
WITTGENSTEIN[…] This simply means that given a certain training, if I give you a contradiction (which I need not notice myself) you don't know what to do. […] That is one thing: (a) We do in fact avoid contradictions. (b) Unless we wish to produce confusions (given our training) we have to avoid contradictions.—But it is an entirely different thing to say that we ought to avoid contradictions in logic. […]
Later in the same lecture, the same bridge:
WITTGENSTEIN[…] Why are people afraid of contradictions? It is easy to understand why they should be afraid of contradictions in orders, descriptions, etc., outside mathematics. The question is: Why should they be afraid of contradictions inside mathematics? Turing says, "Because something may go wrong with the application." But nothing need go wrong. And if something does go wrong—if the bridge breaks down—then your mistake was of the kind of using a wrong natural law. Is Prince's case a case of a "hidden contradiction"? […]
TURINGYou cannot be confident about applying your calculus until you know that there is no hidden contradiction in it.
WITTGENSTEINThere seems to me to be an enormous mistake there. For your calculus gives certain results, and you want the bridge not to break down. I'd say things can go wrong in only two ways: either the bridge breaks down or you have made a mistake in your calculation—for example, you multiplied wrongly. But you seem to think that there may be a third thing wrong: the calculus is wrong.
TURINGNo. What I object to is the bridge falling down.
WITTGENSTEINBut how do you know that it will fall down? Isn't that a question of physics? […]
TURINGIf one takes Frege's symbolism and gives someone the technique of multiplying in it, then by using a Russell paradox he could get a wrong multiplication.
WITTGENSTEINThis would come to doing something which we would not call multiplying. You give him a rule for multiplying; and when he gets to a certain point he can go in either of two ways, one of which leads him all wrong. Suppose I convince Rhees of the paradox of the Liar, and he says, "I lie, therefore I do not lie, therefore I lie and I do not lie, therefore we have a contradiction, therefore 2 × 2 = 369." Well, we should not call this "multiplication"; that is all. It is as if you give him rules for multiplying which lead to different results […] Are we to say that you have given him the wrong calculus?
TURINGAlthough you do not know that the bridge will fall if there are no contradictions, yet it is almost certain that if there are contradictions it will go wrong somewhere.
WITTGENSTEINBut nothing has ever gone wrong that way yet. And why has it not? […] If a contradiction may lead you into trouble, so may anything. It is no more likely to do so than anything else.
TURINGYou seem to be saying that if one uses a little common sense, one will not get into trouble.
WITTGENSTEINNo, that is NOT what I mean at all.—The trouble described is something you get into if you apply the calculation in a way that leads to something breaking. This you can do with any calculation, contradiction or no contradiction. […]
—Wittgenstein's Lectures on the Foundations of Mathematics, Cambridge, 1939, ed. Cora Diamond (1976), Lecture XXI, pp. 206–208, and Lecture XXII, pp. 211–213 and 217–219. The text was put together by Diamond from the notes of R. G. Bosanquet, Norman Malcolm, Rush Rhees and Yorick Smythies; the speakers' names are the edition's, and the Rhees whom Wittgenstein convinces of the Liar was in the room. The ellipses in brackets are my omissions, of a sentence or more within a speech or of other speakers' turns between them; nothing is reordered, and the emphasis is mine. Diamond could not identify Prince (preface, p. 8). Hodges (The Enigma, 1983, pp. 153–154) quotes the exchange with two small changes that have been copied widely: "in which a bridge may fall down", without "case", and "no hidden contradictions" in the plural.
His answer to Turing is the answer he gave the whole decade of foundations. A bridge can come down for two reasons, a wrong natural law or a slip in the arithmetic, and a contradiction hidden somewhere in the calculus is neither one; a derivation that ends in 2 × 2 = 369 is not multiplication gone wrong, it is not multiplication. The bridge did not fall down. The calculus was never shown free of contradiction, Gödel having closed that road eight years before, and the bridges have stood ever since, and the man who said they would come down was the man who had just defined the machine.
He wrote the position down, in the book he was assembling through those years, as a remark on what philosophy has to do when a contradiction appears:
Es ist nicht Sache der Philosophie, den Widerspruch durch eine mathematische, logisch-mathematische, Entdeckung zu lösen. Sondern den Zustand der Mathematik, der uns beunruhigt, den Zustand vor der Lösung des Widerspruchs, übersehbar zu machen. (Und damit geht man nicht etwa einer Schwierigkeit aus dem Wege.)
Die fundamentale Tatsache ist hier: daß wir Regeln, eine Technik, für ein Spiel festlegen, und daß es dann, wenn wir den Regeln folgen, nicht so geht, wie wir angenommen hatten. Daß wir uns also gleichsam in unsern eigenen Regeln verfangen.
Dieses Verfangen in unsern Regeln ist, was wir verstehen, d. h. übersehen wollen.
Es wirft ein Licht auf unsern Begriff des Meinens. Denn es kommt also in jenen Fällen anders, als wir es gemeint, vorausgesehen, hatten. Wir sagen eben, wenn, z. B., der Widerspruch auftritt: „So hab' ich's nicht gemeint."
Die bürgerliche Stellung des Widerspruchs, oder seine Stellung in der bürgerlichen Welt: das ist das philosophische Problem.
It is the business of philosophy, not to resolve a contradiction by means of a mathematical or logico-mathematical discovery, but to make it possible for us to get a clear view of the state of mathematics that troubles us: the state of affairs before the contradiction is resolved. (And this does not mean that one is sidestepping a difficulty.)
The fundamental fact here is that we lay down rules, a technique, for a game, and that then when we follow the rules, things do not turn out as we had assumed. That we are therefore as it were entangled in our own rules.
This entanglement in our rules is what we want to understand (i.e. get a clear view of).
It throws light on our concept of meaning something. For in those cases things turn out otherwise than we had meant, foreseen. That is just what we say when, for example, a contradiction appears: ‘I didn't mean it like that.’
The civil status of a contradiction, or its status in civil life: there is the philosophical problem.
—Wittgenstein, Philosophische Untersuchungen §125; Anscombe's translation.
Eighty-seven years later the machine proves theorems. Nobody laid a foundation under it. It was trained on the written practice of mathematics, the papers and the textbooks and the arithmetic that runs through the prose of the rest of the world, and it writes proofs the way it writes everything else, by continuing a text one sign at a time. In October a model left several hundred results on open problems in a public repository, some with a formal proof attached, none with an author, and The very high baroque is about what the mathematicians said when they opened the files: that the proofs were unreadable, that they were useless to understanding, that they were labor nobody had agreed to supply. Under those three complaints lies a fourth, older than any of them. The machine does mathematics by non-mathematical means. It guesses the next sign. It has no concept of number, no grasp of proof, no intuition of the infinite, nothing behind the signs at all; it is often wrong, and when it is right it does not know why, and there is no one inside to ask. The disquiet is Turing's bridge, moved to a new river. The man who answered him in 1939 had spent the decade on exactly this question, what stands behind the signs, if anything does, and his answer was that nothing does: that the certainty of mathematics is the certainty of a trained practice in which we happen to agree, and that the search for something behind it, in logic, in a consistency proof, in the mind, was a confusion that had cost the best minds of the age their best years. He was ridiculed for saying so, in print, for half a century, by the people best placed to judge.
but the tortoise was no more secure than the elephant
Paris, August 1900, the Congress sitting in the summer of the great Exposition. Poincaré told the mathematicians of the world that the arithmetization of analysis was finished: wherever one took the trouble to be rigorous there remained "nothing but syllogisms or appeals to this intuition of pure number, the only intuition which cannot deceive us. It may be said that today absolute rigor is attained." A page earlier he had asked whether our fathers had not believed the same at every stage, and whether we were not deceiving ourselves as they had, and he had answered no. Compte rendu du deuxième Congrès international des mathématiciens, Paris 1900 (Gauthier-Villars, 1902), pp. 121–122: « Avons-nous atteint enfin la rigueur absolue ? A chaque stade de l'évolution nos pères croyaient aussi l'avoir atteinte. … On peut dire qu'aujourd'hui la rigueur absolue est atteinte. » Reprinted as the first chapter of La valeur de la science (1905); Halsted's translation. Russell was in Paris that month to hear Peano, and within the year he had the contradiction in his hands. Writing out the deduction of mathematics from logic, he turned Cantor's proof that there is no greatest number on the class of all the things there are, and it led him to the classes that are not members of themselves and to the class of those: "either answer implies its contradictory." He supposed the solution would be easy. By the end of 1901 he had concluded that it was a big job. Russell, Autobiography, vol. I (1967), ch. 6, which puts the discovery "in the month of May"; My Philosophical Development (1959), p. 75, has "the spring of 1901" and records that Whitehead "failed to console me by quoting, 'never glad confident morning again'."
He sent it to Frege in June 1902 as a difficulty "in just one point", and Frege answered in six days: "Your discovery of the contradiction caused me the greatest surprise and, I would almost say, consternation, since it has shaken the basis on which I intended to build arithmetic." The second volume of the Grundgesetze was at the printer. Frege set an afterword at the back of it whose first sentences every logician knows by heart: hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished, "and this was the position I was placed in by a letter of Mr Bertrand Russell, just when the printing of this volume was nearing its completion." Russell to Frege, 16 June 1902, and Frege to Russell, 22 June 1902, in van Heijenoort, From Frege to Gödel (1967), pp. 124–128, trans. Beverly Woodward. Grundgesetze der Arithmetik II (Jena: Pohle, 1903), Nachwort: „Einem wissenschaftlichen Schriftsteller kann kaum etwas Unerwünschteres begegnen, als dass ihm nach Vollendung einer Arbeit eine der Grundlagen seines Baues erschüttert wird. In diese Lage wurde ich durch einen Brief des Herrn Bertrand Russell versetzt, als der Druck dieses Bandes sich seinem Ende näherte." Geach's translation (Geach and Black, 1960, p. 234). Sixty years on, asked to let the letters be printed, Russell wrote back that he knew nothing "to compare with Frege's dedication to truth": his life's work on the verge of completion, Frege "responded with intellectual pleasure clearly submerging any feelings of personal disappointment" (van Heijenoort, p. 127).
The repair took Russell and Whitehead ten years. Principia Mathematica rebuilds arithmetic out of logic through a theory of types, and the sum everyone remembers from it arrives late and quietly: on page 379 of the first volume, after proposition *54.43, the remark that from it "it will follow, when arithmetical addition has been defined, that 1 + 1 = 2", and in the second volume, two years on, the theorem itself, *110.643, with the comment "The above proposition is occasionally useful." Principia Mathematica, vol. I (Cambridge, 1910), p. 379; vol. II (1912), p. 86, where the comment continues: "It is used at least three times, in *113.66 and *120.123.472."
The ladder of Principia Mathematica under 1 + 1 = 2: 23 propositions, definitions and primitive propositions in the order of the book, joined by 54 arcs of citation.
*54·43 (volume I, page 379): ⊢ :. α, β ∈ 1 . ⊃ : α ∩ β = Λ . ≡ . α ∪ β ∈ 2, proved in 5 lines citing 7 propositions (*54·26, *51·231, *13·12, *11·11, *11·35, *11·54, *52·1), and followed by the sentence "From this proposition it will follow, when arithmetical addition has been defined, that 1 + 1 = 2."
*110·643 (volume II, page 86): ⊢ . 1 +c 1 = 2, proved in 3 lines citing *110·632, *101·21, *101·28 and *54·3: 1 +c 1 = ξ̂{(∃y) . y ∈ ξ . ξ − ιʻy ∈ 1} = 2.
The longest chain of citations from *110·643 down to the primitive propositions of *1 has 7 steps and passes through 18 of the 23 rungs; from *54·43, 6 steps through 13.
The live figure needs JavaScript: the ladder with its arcs, the proposition under view with its demonstration, the two demonstrations stepped line by line, and a model of two unit classes a and b, their union (a couple when a ≠ b, a unit class when a = b) and the class-sum of *110·01, a couple whatever a and b are.
Russell counted the cost. He used to stand on the footbridge at Kennington watching the trains go by "and determining that tomorrow I would place myself under one of them"; the work was finished, "but my intellect never quite recovered from the strain. I have been ever since definitely less capable of dealing with difficult abstractions than I was before." The Press expected to lose six hundred pounds on the book, the Royal Society gave two hundred, and the authors found the last hundred themselves, which is to say that each of them paid fifty pounds for ten years' work. Of the later parts he used to know of six readers, "three Poles, subsequently (I believe) liquidated by Hitler", and three Texans, "subsequently successfully assimilated." Autobiography, vol. I, ch. 6; My Philosophical Development, ch. VIII, p. 86, which opens on the disappointment that the book "was only viewed from a philosophical standpoint". What he had wanted from it he said at eighty. "I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere." He had built an elephant for the mathematical world to rest on and found it tottering, so he built a tortoise to hold up the elephant, "but the tortoise was no more secure than the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable." "Reflections on My Eightieth Birthday", in Portraits from Memory (New York: Simon and Schuster, 1956), pp. 54–55.
That is the man, and that is the height he stood at, when an engineering student walked in. Russell's letters to Ottoline Morrell record the arrival on 18 October 1911: "an unknown German, speaking very little English but refusing to speak German", who "had learnt engineering at Charlottenburg, but during his course had acquired, by himself, a passion for the philosophy of mathematics, and has now come to Cambridge on purpose to hear me"; by the next evening the German "threatens to be an infliction", having argued till dinner, "obstinate and perverse, but I think not stupid." Russell to Ottoline Morrell, 18 and 19 October 1911 (Bertrand Russell Archives, BRACERS 17311 and 17313). In the margin of the first letter Russell later wrote: "This was Wittgenstein". At the end of term the student asked to be told whether he was a complete idiot, because if so he would become an aeronaut and if not a philosopher; Russell told him to write something over the vacation, read one sentence of what he brought back, and said, "No, you must not become an aeronaut." Russell, "Some Philosophical Contacts", in Portraits from Memory (1956), pp. 23–24; broadcast as "Philosophers and Idiots", The Listener, 10 Feb 1955. By March 1912 he loved him and felt "he will solve the problems I am too old to solve"; he was, Russell wrote later, "perhaps the most perfect example I have ever known of genius as traditionally conceived, passionate, profound, intense, and dominating." Russell to Ottoline Morrell, 18 March 1912 (printed in Griffin, ed., Selected Letters, vol. I, no. 186); Autobiography, vol. II (1968), ch. 2. In May 1913 Wittgenstein told him that the theory of knowledge he was writing was all wrong. Russell could not make out the objection, "in fact he was very inarticulate", but felt "in my bones that he must be right, and that he has seen something I have missed." Three years later he told her what it had done to him: "His criticism, tho' I don't think you realized it at the time, was an event of first-rate importance in my life, and affected everything I have done since. I saw he was right, and I saw that I could not hope ever again to do fundamental work in philosophy. My impulse was shattered, like a wave dashed to pieces against a breakwater." Russell to Ottoline Morrell, 27 May 1913 (BRACERS 17922) and 4 March 1916 (Autobiography, vol. II, pp. 56–57). Wittgenstein, from Norway, 22 July 1913: "I am very sorry to hear that my objection to your theory of judgment paralyses you." The book Russell abandoned was Theory of Knowledge. The student was twenty-four.
The book the student wrote instead, in the trenches and then in a prison camp at Cassino, settled the matter of mathematics in a handful of numbered sentences. "Mathematics is a logical method. The propositions of mathematics are equations, and therefore pseudo-propositions." "A proposition of mathematics does not express a thought." And the one that governs everything he wrote afterward: "Indeed in real life a mathematical proposition is never what we want. Rather, we make use of mathematical propositions only in inferences from propositions that do not belong to mathematics to others that likewise do not belong to mathematics. (In philosophy the question, 'What do we actually use this word or this proposition for?' repeatedly leads to valuable insights.)" Tractatus Logico-Philosophicus 6.2, 6.21, 6.211, in the Pears–McGuinness translation (1961): „Im Leben ist es ja nie der mathematische Satz, den wir brauchen, sondern wir benützen den mathematischen Satz nur, um aus Sätzen, welche nicht der Mathematik angehören, auf andere zu schließen, welche gleichfalls nicht der Mathematik angehören." Also 6.1 and 6.11, the propositions of logic are tautologies and "therefore say nothing"; 6.1251, "Hence there can never be surprises in logic"; 6.1262, "Proof in logic is merely a mechanical expedient to facilitate the recognition of tautologies in complicated cases"; 6.031, "The theory of classes is completely superfluous in mathematics." Russell, whose introduction got the book published, met him at The Hague in December 1919 to go through it and reported to Ottoline: "I had felt in his book a flavour of mysticism, but was astonished when I found that he has become a complete mystic. He reads people like Kierkegaard and Angelus Silesius, and he seriously contemplates becoming a monk." Russell to Ottoline Morrell, 20 December 1919 (BRACERS 18748). The same letter tells how he came upon a bookshop in Tarnów that "seemed to contain nothing but picture post-cards" and one book, Tolstoy on the Gospels, "bought it merely because there was no other", and "thenceforth had it always with him, under fire and at all times"; and that "he gave all his money to his brothers and sisters, because he found earthly possessions a burden." Wittgenstein's verdict on the introduction, once he had seen it in German, was that the refinement of Russell's English style had been lost in translation "and what remained was superficiality and misunderstanding." Wittgenstein to Russell, 6 May 1920: „was übrig blieb war Oberflächlichkeit und Mißverständnis." Russell's introduction confesses the discomfort from his side: "What causes hesitation is the fact that, after all, Mr Wittgenstein manages to say a good deal about what cannot be said."
He had been born one of the richest young men in Europe, and in July 1914, weeks before the war, he wrote to Ludwig von Ficker, the editor of Der Brenner, offering a hundred thousand crowns to be given to Austrian artists without means, at Ficker's discretion, on condition that the matter be kept secret. Wittgenstein to Ficker, 14 July 1914, from the Hochreit: „Ich möchte Ihnen eine Summe von 100,000 Kronen überweisen und Sie bitten, dieselbe an unbemittelte österreichische Künstler nach Ihrem Gutdünken zu verteilen." Facsimile from the Brenner-Archiv, Innsbruck. Rilke and Trakl were among the recipients. After the war he gave away the rest, trained as a schoolteacher, and from 1920 to 1926 taught the village children of Trattenbach, Puchberg and Otterthal in Lower Austria, the first hours of each school day given to mathematics, in villages where no child had been taught algebra before. The only book he published in his lifetime besides the Tractatus is a spelling dictionary for those children, forty-two pages, nearly six thousand words, with a preface explaining that only a word-book makes it possible to hold the pupil responsible for his own mistakes. Wörterbuch für Volksschulen (Vienna: Hölder-Pichler-Tempsky, 1926), compiled at Otterthal in 1925. The teaching schedule and the algebra are Monk's (ch. 9). The appointments: Trattenbach from September 1920, Puchberg am Schneeberg from November 1921, Otterthal from September 1924; he resigned in April 1926. The philosophy of mathematics he wrote afterward is a schoolteacher's. Its central scene is a pupil who has been drilled in writing down a series and one morning goes on differently; its central words are abrichten, training, and Übung, practice; its question is what we are doing when we teach a child to count and then say that it has understood.
He came back to Cambridge in January 1929, Keynes writing home that God had arrived on the 5.15 train, took the doctorate that June with the Tractatus as his thesis and Russell and Moore as his examiners, and began the work that filled the next fifteen years, more than half of it on mathematics. Keynes to Lydia Lopokova, 18 January 1929: "Well, God has arrived. I met him on the 5.15 train" (Monk, p. 255). The proportion is Rodych's, in the Stanford Encyclopedia article "Wittgenstein's Philosophy of Mathematics": "more than half of Wittgenstein's writings from 1929 through 1944 are devoted to mathematics."
Russell, asked by Trinity in 1930 to judge whether the work deserved the college's money, summarized what he had read with a candor that honors him: "A considerable part of Wittgenstein's work is concerned with the interpretation of mathematics. He considers it false to say that mathematics is logic or consists of tautologies." The theories "are novel, very original, and indubitably important. Whether they are true, I do not know. As a logician who likes simplicity, I should wish to think that they are not, but from what I have read of them I am quite sure that he ought to have an opportunity to work them out, since when completed they may easily prove to constitute a whole new philosophy." Russell's report to the Council of Trinity College, 8 May 1930, printed in Autobiography, vol. II, ch. 5. The report also records that Wittgenstein "believes in 'infinite possibility,' as he calls it, but not in actual 'infinite classes' or 'infinite series'." The fellowship came, and in 1939 the chair; Broad, who did not care for him, said that to refuse it to Wittgenstein "would be like refusing Einstein a chair of physics." Monk, p. 415, from Drury's recollection in Rhees, ed., Recollections of Wittgenstein (1984), p. 141. The election was in February 1939.
What the mathematicians made of the work is the cost, and it has to be set down plainly, because the work was his own estimate of himself. In 1944 John Wisdom drafted an entry on him for a biographical dictionary and sent it over for correction; Wittgenstein added one sentence: "Wittgenstein's chief contribution has been in the philosophy of mathematics." Monk, p. 466, who had it from Rush Rhees. He died in 1951 without publishing a line of it. The editors assembled the manuscripts of 1937 to 1944 into Remarks on the Foundations of Mathematics in 1956, and the reviews came in from the logicians like weather from one quarter. Goodstein found that the pages on Gödel "throw no light on Gödel's work"; Dummett found them "of poor quality" or containing "definite errors", and named the whole position "a full-blooded conventionalism"; Bernays found that "the fact of mathematics does not become at all understandable through Wittgenstein's philosophy"; Anderson found him trying "to legislate this class of questions out of existence, and by peculiarly un-Wittgensteinian methods at that." Kreisel, who had been his student and had argued these very matters with him in the forties, ended his review in the British Journal for the Philosophy of Science with a sentence that stuck for fifty years: "I did not enjoy reading the present book. Of course I do not know what I should have thought of it fifteen years ago; now it seems to me to be a surprisingly insignificant product of a sparkling mind." R. L. Goodstein, Mind 66 (1957), p. 551; Michael Dummett, "Wittgenstein's Philosophy of Mathematics", Philosophical Review 68 (1959), pp. 324, 329; Paul Bernays, "Comments on Ludwig Wittgenstein's Remarks on the Foundations of Mathematics", Ratio 2 (1959); Alan Ross Anderson, "Mathematics and the 'Language Game'", Review of Metaphysics 11 (1958), p. 453; Georg Kreisel, British Journal for the Philosophy of Science 9 (1958), p. 158. The Stanford Encyclopedia opens its article on the subject with the sentence that this part of his work "is undoubtedly the most unknown and under-appreciated part of his philosophical opus", and records that after the initial reviews "very little attention was paid" to the remarks on Gödel for thirty years. Gödel himself, asked in 1972 whether the passages on his theorem showed any understanding of it, answered that "Wittgenstein did not understand it (or pretended not to understand it). He interprets it as a kind of logical paradox, while in fact it is just the opposite, namely a mathematical theorem within an absolutely uncontroversial part of mathematics"; and, to Abraham Robinson the next year, that Wittgenstein "advances a completely trivial and uninteresting misinterpretation." Gödel to Karl Menger, 20 April 1972, and to Abraham Robinson, 2 July 1973, Collected Works, vol. V (2003), pp. 133, 201. The Menger letter is often dated 20 May, which is the date Menger's memoir gives. Russell, in 1959, disposed of the later work altogether: "I have not found in Wittgenstein's Philosophical Investigations anything that seemed to me interesting and I do not understand why a whole school finds important wisdom in its pages." The man he had known "was a man addicted to passionately intense thinking"; the later one "seems to have grown tired of serious thinking and to have invented a doctrine which would make such an activity unnecessary." My Philosophical Development, ch. XVIII, pp. 216–217.
Wittgenstein had expected roughly this. "Nothing seems to me less likely", he wrote in 1947, "than that a scientist or mathematician who reads me should be seriously influenced in the way he works." Culture and Value, ed. von Wright, trans. Winch (1980), p. 62e, dated 1947: „Nichts kommt mir weniger wahrscheinlich vor, als daß ein Wissenschaftler, oder Mathematiker, der mich liest, dadurch in seiner Arbeitsweise ernstlich beeinflußt werden sollte." He compares his remarks to the posters at the ticket windows of English stations, "Is your journey really necessary?", as if anyone reading them would say "On second thoughts, no." He compared his remarks to the posters at the ticket windows of English stations, "Is your journey really necessary?", as if anyone reading them would answer "On second thoughts, no." And he had said, in the book, why it did not matter, in a remark that is the whole of his bearing toward the mathematicians:
Die Philosophie darf den tatsächlichen Gebrauch der Sprache in keiner Weise antasten, sie kann ihn am Ende also nur beschreiben.
Denn sie kann ihn auch nicht begründen.
Sie läßt alles wie es ist.
Sie läßt auch die Mathematik wie sie ist, und keine mathematische Entdeckung kann sie weiterbringen. Ein „führendes Problem der mathematischen Logik" ist für uns ein Problem der Mathematik, wie jedes andere.
Philosophy may in no way interfere with the actual use of language; it can in the end only describe it.
For it cannot give it any foundation either.
It leaves everything as it is.
It also leaves mathematics as it is, and no mathematical discovery can advance it. A ‘leading problem of mathematical logic’ is for us a problem of mathematics like any other.
—Wittgenstein, Philosophische Untersuchungen §124; Anscombe's translation. PI §124.
The argument against the foundations is in two parts, and the first is about a sum. Take the proposition 7034174 + 6594321 = 13628495. In the decimal notation anyone can check it in a minute with a pencil. Now explain the numbers the way the logicians do, as 1, 1+1, (1+1)+1, ((1+1)+1)+1, and so on, a unit and a successor and nothing else, and write the sum in that notation: the left side is a chain of seven million units joined to a chain of six and a half million, the right side a chain of thirteen and a half million, and the proof is the matching of the one against the other, unit by unit, across more than fifty million signs. Wittgenstein asks whether such a proof could convince anyone of the sum: "Is there such a proof of this proposition?—The answer is: no." Remarks on the Foundations of Mathematics, 3rd ed. (1978), Part III §3; Part II §3 in the 1956 edition. Part III, dated 1939–40, is the material of the year of the lectures. The German asks whether we could convince ourselves of the truth of the proposition „auch durch einen Beweis" in the first notation, and the preceding sentences say of the long notation: „Sie ist ja nicht übersehbar", it cannot be surveyed. Not because it is tedious. A proof, he says at the head of the same part, must be übersichtlich, surveyable, a thing the eye can take in whole:
„Ein mathematischer Beweis muß übersichtlich sein." „Beweis" nennen wir nur eine Struktur, deren Reproduktion eine leicht lösbare Aufgabe ist. Es muß sich mit Sicherheit entscheiden lassen, ob wir hier wirklich zweimal den gleichen Beweis vor uns haben, oder nicht. Der Beweis muß ein Bild sein, welches sich mit Sicherheit genau reproduzieren läßt. Oder auch: was dem Beweise wesentlich ist, muß sich mit Sicherheit genau reproduzieren lassen. Er kann z. B. in zwei verschiedenen Handschriften oder Farben niedergeschrieben sein. Zur Reproduktion eines Beweises soll nichts gehören, was von der Art einer genauen Reproduktion eines Farbtones oder einer Handschrift ist.
‘A mathematical proof must be perspicuous.’ Only a structure whose reproduction is an easy task is called a ‘proof’. It must be possible to decide with certainty whether we really have the same proof twice over, or not. The proof must be a configuration whose exact reproduction can be certain. Or again: we must be sure we can exactly reproduce what is essential to the proof. It may for example be written down in two different handwritings or colours. What goes to make the reproduction of a proof is not anything like an exact reproduction of a shade of colour or a hand-writing.
—Wittgenstein, Bemerkungen über die Grundlagen der Mathematik III §1; Anscombe's translation. RFM III §1; II §1 in the 1956 edition. The remark goes on to set a written proof against a drawing: the drawing of a Euclidean proof may be inexact, its straight lines not straight and its arcs not circular, and remain an exact proof, so what the drawing shows is nothing that belongs to the essence of the proof. Anscombe's "perspicuous" is übersichtlich; from §2 on the word is übersehbar, "surveyable", the proof that can be taken in.
The chain of units fails the test and the decimal calculation passes it, and the only way to check that the chain is right is to run the decimal calculation and compare. So the thing that was to be the foundation is checked against the thing it was to found. "If you have a proof-pattern that cannot be taken in, and by a change in notation you turn it into one that can, then you are producing a proof, where there was none before." RFM III §2: „Wenn man eine nicht übersehbare Beweisfigur durch Veränderung der Notation übersehbar macht, dann schafft man erst einen Beweis, wo früher keiner war." The question that follows (§12) is what the invention of the decimal system really was: a system of abbreviations, or a new way of looking at the old signs; and the answer (§47) is that "It is not the introduction of numerical signs as abbreviations that is important, but the method of counting." Principia did not give 1 + 1 = 2 a foundation; it gave it a translation into a notation in which nothing much past a dozen units can be checked at sight, and the translation is trusted because the arithmetic it translates is trusted. Russell's six readers, three Poles and three Texans, are not a joke about a dull book. They are the argument. Nobody read the later parts because nobody needed to, and the arithmetic went on being certain in the dark, untouched by anyone's not reading them.
The judgment on the whole enterprise is two remarks written during the war. "The curse of the invasion of mathematics by mathematical logic is that now any proposition can be represented in a mathematical symbolism, and this makes us feel obliged to understand it. Although of course this method of writing is nothing but the translation of vague ordinary prose." And: “‘Mathematical logic’ has completely deformed the thinking of mathematicians and of philosophers, by setting up a superficial interpretation of the forms of our everyday language as an analysis of the structures of facts. Of course in this it has only continued to build on the Aristotelian logic.” RFM V §§46 and 48 (IV §§46, 48 in 1956): „Das ist der Fluch des Einbruchs der mathematischen Logik in die Mathematik … Obwohl ja diese Schreibweise nur die Übersetzung der vagen gewöhnlichen Prosa ist"; „Die 'mathematische Logik' hat das Denken von Mathematikern und Philosophen gänzlich verbildet … Sie hat hierin freilich nur auf der Aristotelischen Logik weiter gebaut." The curse is not that the logic is wrong. It is that a notation in which everything can be written has been mistaken for an analysis of what was written, and the vague prose it transcribes, the mathematician's talk of what his signs "really" are, has been lent the dignity of the symbols. The vague prose had a voice in Cambridge, and it was Hardy's. "Mathematical theorems are true or false; their truth or falsity is absolute and independent of our knowledge of them. In some sense, mathematical truth is part of objective reality", he had written in 1929, and in the Apology, as a creed: "I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations,' are simply our notes of our observations." Wittgenstein opened the lectures of 1939 on a sentence of Hardy's, that to mathematical propositions there corresponds "in some sense, however sophisticated, a reality", and set the man aside in the same breath: "The fact that he said it does not matter; what is important is that it is a thing which lots of people would like to say." That talk, he says in the Investigations, is not a philosophy of mathematics at all: "what a mathematician is inclined to say about the objectivity and reality of mathematical facts, is not a philosophy of mathematics, but something for philosophical treatment." PI §254. The patient in the room was Hardy: G. H. Hardy, "Mathematical Proof", Mind 38 (1929), p. 4; A Mathematician's Apology (1940), §22; Lectures, I, pp. 13–14, where the sentence about a reality is attributed to Hardy.
The second part of the argument is what mathematics is, if it is not what the foundations said it was. "I should like to say: mathematics is a MOTLEY of techniques of proof. And upon this is based its manifold applicability and its importance." RFM III §46: „Ich möchte sagen: Die Mathematik ist ein BUNTES Gemisch von Beweistechniken. Und darauf beruht ihre mannigfache Anwendbarkeit und ihre Wichtigkeit." The capitals are his. A technique is something one is trained in, and a proof is not a report on what was found at the end of it but the laying down of a rule: "The proposition proved by means of the proof serves as a rule, and so as a paradigm. For we go by the rule." RFM III §28: „Der durch den Beweis bewiesene Satz dient als Regel, also als Paradigma. Denn nach der Regel richten wir uns." Hence "The mathematician is an inventor, not a discoverer", a sentence he repeated in the lectures in the form that mathematical discovery "had much better be called a mathematical invention." RFM I §168 (§167 in 1956): „Der Mathematiker ist ein Erfinder, kein Entdecker"; Lectures, II, p. 22. Hence also two remarks of the kind that alarmed Bernays, who read in them a man who "sees himself in the part of the free-thinker combating superstition": "mathematics is after all an anthropological phenomenon", and "Mathematics forms a network of norms." RFM VII §§33 and 67. Bernays's complaint was that Wittgenstein "sees himself in the part of the free-thinker combating superstition", restricting the mind "through a mental asceticism for the benefit of an irrationality whose goal is quite undetermined" (Ratio, 1959). And then the remark that decides the question of the machine, written in 1942, and beginning, as it happens, with a machine:
Rechnet die Rechenmaschine? Denk dir, eine Rechenmaschine wäre durch Zufall entstanden; nun drückt Einer durch Zufall auf ihre Knöpfe (oder ein Tier läuft über sie) und sie rechnet das Produkt 25 × 20. — Ich will sagen: Es ist der Mathematik wesentlich, daß ihre Zeichen auch im Zivil gebraucht werden. Es ist der Gebrauch außerhalb der Mathematik, also die Bedeutung der Zeichen, was das Zeichenspiel zur Mathematik macht. So, wie es ja auch kein logischer Schluß ist, wenn ich ein Gebilde in ein anderes transformiere (eine Anordnung von Stühlen etwa in eine andere), wenn diese Anordnungen nicht außerhalb dieser Transformation einen sprachlichen Gebrauch haben.
Does a calculating machine calculate? Imagine that a calculating machine had come into existence by accident; now someone accidentally presses its knobs (or an animal walks over it) and it calculates the product 25 × 20. I want to say: it is essential to mathematics that its signs are also employed in mufti. It is the use outside mathematics, and so the meaning of the signs, that makes the sign-game into mathematics. Just as it is not logical inference either, for me to make a change from one formation to another (say from one arrangement of chairs to another) if these arrangements have not a linguistic function apart from this transformation.
—Wittgenstein, Bemerkungen über die Grundlagen der Mathematik V §2; Anscombe's translation. IV §2 in the 1956 edition, written in 1942. In mufti, in civilian clothes, is Anscombe's rendering of im Zivil. Two remarks later (V §4) he imagines calculating machines occurring in nature whose cases nobody can open, used by people who know nothing of calculation, and asks whether what those people do is mathematics; and (V §3) a human calculating machine trained to read through the proofs of Russell's system and nod at every correct inference, "for the rest completely idiotic".
The signs in mufti. Mathematics is not the game of signs; the game becomes mathematics through the use the signs have outside it, in counting sheep and bridges and money, and a machine's output is calculation when, and because, it is taken up into that use. The Tractatus had said the same thing thirty years earlier in a colder voice: in life it is never the mathematical proposition we want. Here is the first thing to notice about the machines of 2026, before any question of understanding arises. They were not trained on mathematics. They were trained on language, the whole written record of a species, in which the signs of mathematics turn up in mufti on every other page, in recipes and invoices and physics and the schoolroom, and the mathematics came out of that. A system that learned only from formal proofs learned a sign-game; the systems that learned the civilian use of the signs learned mathematics, and the order in which the two kinds succeeded is the order his remark predicts.
He had a sentence for the whole program of foundations, written in 1941, and the reason it was mocked is that it was heard as the Bolshevism Turing suspected: a demolition. It is the opposite. A painted rock does not hold up a painted castle, but it is not therefore a fraud, and the castle does not fall when one notices. It never stood on the rock; it stands on the canvas, as the rock does. Frege's logic, he says in the next breath, was made useless for founding arithmetic by the contradiction; yes, it was, he answers, and who said that it had to serve that purpose? The foundations were a wish about mathematics, that its certainty should rest on something firmer than itself, and the wish was Russell's, in his own words, the wish for certainty "in the kind of way in which people want religious faith"; and Wittgenstein's answer to it, in a sentence from the Grammar, is that "Philosophical clarity will have the same effect on the growth of mathematics as sunlight has on the growth of potato shoots. (In a dark cellar they grow yards long.)" Philosophical Grammar, ed. Rhees, trans. Kenny (1974), p. 381. The remark that follows begins: "A mathematician is bound to be horrified by my mathematical comments."
forming sentences and setting them one after another
Weyl gave the quarrel its name. In lectures at Zurich in 1920, printed the next year as "On the new foundational crisis of mathematics", he abandoned his own attempt to rebuild analysis on safe ground and crossed over: "For this order is not tenable in itself, as I have now convinced myself, and Brouwer, that is the revolution!" Hermann Weyl, "Über die neue Grundlagenkrise der Mathematik", Mathematische Zeitschrift 10 (1921), 39–79, at p. 56: „So gebe ich also jetzt meinen eigenen Versuch preis und schließe mich Brouwer an. … Denn diese Ordnung ist nicht haltbar in sich, wie ich mich jetzt überzeugt habe, und Brouwer — das ist die Revolution!" The title is where the word Grundlagenkrise comes from. Brouwer had been saying since 1908 that the law of excluded middle, the rule that every proposition is true or false, is "as yet not reliable" in infinite systems, and that the question of its validity "is equivalent to the question concerning the possibility of unsolvable mathematical problems." L. E. J. Brouwer, "De onbetrouwbaarheid der logische principes", Tijdschrift voor Wijsbegeerte 2 (1908), 152–158, at p. 156; the translation is van Atten and Sundholm's, History and Philosophy of Logic 38 (2017). In his inaugural lecture at Amsterdam in 1912 he put the two camps in a single sentence: "The question where mathematical exactness does exist, is answered differently by the two sides; the intuitionist says: in the human intellect, the formalist says: on paper." Brouwer, "Intuitionism and Formalism", trans. Arnold Dresden, Bulletin of the American Mathematical Society 20 (1913), 81–96, at p. 83. By 1919 he was writing that the excluded middle is "an illegitimate mathematical means of proof, to which no other than a scholastic and heuristic value can be ascribed, so that theorems whose proof cannot avoid its application lack all mathematical content", and the ground he gave for it is the ground Wittgenstein would give against Russell: logic rests on mathematics, not mathematics on logic. Brouwer, "Intuitionistische Mengenlehre", Jahresbericht der Deutschen Mathematiker-Vereinigung 28 (1919/20), 203–208, at pp. 203–204: „weil … die Logik auf der Mathematik beruhe und nicht umgekehrt, der logische Satz vom ausgeschlossenen Dritten ein unerlaubtes mathematisches Beweismittel sei, dem kein andrer als ein scholastischer und heuristischer Wert zugesprochen werden könne, so daß Theoreme, bei deren Beweis seine Anwendung nicht umgangen werden kann, jeden mathematischen Inhalt entbehren." The sentence is usually cited to his 1923 lecture on the excluded middle, where it does not occur.
His standing example was the decimal expansion of π. Write out the digits and ask whether the block 0123456789 ever appears. Nobody knows, nobody has a method for finding out, and so, Brouwer argued, a real number defined by reference to the first appearance of that block is one of which it can be shown neither that it equals zero, nor that it is greater, nor that it is less. Brouwer, "Über die Bedeutung des Satzes vom ausgeschlossenen Dritten in der Mathematik, insbesondere in der Funktionentheorie", Journal für die reine und angewandte Mathematik 154 (1925), 1–7, at p. 3; the lecture was given in 1923. The number is the series whose terms are (−½)^ν until the block first appears at place k₁ and (−½)^k₁ thereafter: "a real number r for which neither r = 0, nor r > 0, nor r < 0 holds" (van Heijenoort, p. 337). The classical mathematician says the block is there or it is not, and reasons from the disjunction; Brouwer says that until a construction decides it, the disjunction is words. To the formalist's answer, that a theory is safe once it is proved free of contradiction, he had a sentence as hard as any in the literature: "an incorrect theory, even if it cannot be inhibited by any contradiction that would refute it, is none the less incorrect, just as a criminal policy is none the less criminal even if it cannot be inhibited by any court that would curb it." Same lecture, p. 3: „eine durch keinen widerlegenden Widerspruch zu hemmende unrichtige Theorie ist darum nicht weniger unrichtig, so wie eine durch kein reprimierendes Gericht zu hemmende verbrecherische Politik darum nicht weniger verbrecherisch ist."
Hilbert answered in kind. What Weyl and Brouwer were doing, he told Hamburg in 1921, came to following in Kronecker's footsteps, throwing overboard everything that troubled them and erecting "a dictatorship of prohibitions", which would "dismember and mutilate" the science. Hilbert, "Neubegründung der Mathematik. Erste Mitteilung", Abhandlungen aus dem Mathematischen Seminar der Hamburgischen Universität 1 (1922), 157–177; the English is Ewald's (From Kant to Hilbert, vol. II, 1996, p. 1119), who has "dictatorship of prohibitions à la Kronecker"; Reid's biography renders the verbs "chop up and mangle". In Münster in 1925, in a lecture in memory of Weierstrass, he gave the program its creed. Nowhere in reality is the infinite to be found, "no matter what experiences and observations or what kind of science we may adduce"; the right to operate with it "can be secured only by means of the finite"; and yet "No one shall be able to drive us from the paradise that Cantor created for us." Hilbert, "Über das Unendliche", Mathematische Annalen 95 (1926), 161–190, at pp. 170 and 190: „Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können" and „das Unendliche findet sich nirgends realisiert; es ist weder in der Natur vorhanden, noch als Grundlage in unserem verstandesmäßigen Denken zulässig … Das Operieren mit dem Unendlichen kann nur durch das Endliche gesichert werden." Van Heijenoort's translation, pp. 376 and 392. The way to keep the paradise was to stop asking what its sentences meant. The signs and the formula a + b = b + a "do not mean anything in themselves, any more than numerals do"; generalize that and "mathematics becomes an inventory of formulas", some of which communicate finitary facts and the rest of which "mean nothing in themselves and are the ideal objects of our theory", and in place of a contentful science communicated in ordinary language there stands a stock of formulas joined by fixed rules, so that contentual inference is replaced by an external acting according to rules. Same, pp. 175–177: „So kommen wir zu der Auffassung, daß a, b, =, +, sowie die Gesamtformel a + b = b + a an sich nichts bedeuten, so wenig wie die Zahlzeichen … so wird die Mathematik zu einem Bestande von Formeln … Formeln, die nichts bedeuten und die idealen Gebilde unserer Theorie sind"; „das inhaltliche Schließen wird also durch ein äußeres Handeln nach Regeln ersetzt". Van Heijenoort, pp. 380–381, has "inventory of formulas" and "manipulation of signs according to rules". Two years later, back in Hamburg, he said what the stock of formulas was good for, and the passage should be read whole, because it describes a thing that did not exist for another century:
Das Formelspiel, über das Brouwer so wegwerfend urteilt, hat außer dem mathematischen Wert noch eine wichtige allgemeine philosophische Bedeutung. Dieses Formelspiel vollzieht sich nämlich nach gewissen bestimmten Regeln, in denen die Technik unseres Denkens zum Ausdruck kommt. Diese Regeln bilden ein abgeschlossenes System, das sich auffinden und endgültig angeben läßt. Die Grundidee meiner Beweistheorie ist nichts anderes, als die Tätigkeit unseres Verstandes zu beschreiben, ein Protokoll über die Regeln aufzunehmen, nach denen unser Denken tatsächlich verfährt. Das Denken geschieht eben parallel dem Sprechen und Schreiben, durch Bildung und Aneinanderreihung von Sätzen.
The formula game that Brouwer judges so dismissively has, besides its mathematical value, an important general philosophical significance. For this formula game is carried out according to certain definite rules, in which the technique of our thinking is expressed. These rules form a closed system that can be discovered and definitively stated. The fundamental idea of my proof theory is nothing other than to describe the activity of our understanding, to take down a protocol of the rules according to which our thinking actually proceeds. Thinking, after all, happens in parallel with speaking and writing: by forming sentences and setting them one after another.
—David Hilbert, "Die Grundlagen der Mathematik" (1927). Abhandlungen aus dem Mathematischen Seminar der Hamburgischen Universität 6 (1928), 65–85, at p. 79; the lecture was given in July 1927. My translation; van Heijenoort's (p. 475) has "to make a protocol of the rules according to which our thinking actually proceeds" and "Thinking, it so happens, parallels speaking and writing: we form statements and place them one behind another."
Thinking proceeds by forming sentences and setting them one after another, and the rules by which it does so can be written down. That is a description of the machine of 2026 from the formalist's side, eight decades early, and the formalist's demand was that the protocol be proved consistent before anyone trusted it. Taking the excluded middle from the mathematician, Hilbert went on, "would be the same, say, as proscribing the telescope to the astronomer or to the boxer the use of his fists"; and he was astonished, most of all, that "even in mathematical circles the power of suggestion of a single man, however full of temperament and inventiveness, is capable of having the most improbable and eccentric effects." Same, p. 80; van Heijenoort, pp. 475–476: „Dieses Tertium non datur dem Mathematiker zu nehmen, wäre etwa, wie wenn man dem Astronomen das Fernrohr oder dem Boxer den Gebrauch der Fäuste untersagen wollte." Weyl, who was in the room and spoke after him, conceded the field with the epitaph of the decade. Only a part of classical mathematics, "perhaps only a wretched part", survives on Brouwer's terms, "a bitter but inevitable fact. Hilbert could not bear this mutilation"; so he saved the whole inventory by a radical reinterpretation of its meaning, "transforming it in principle from a system of intuitive results into a game with formulas that proceeds according to fixed rules." Weyl, "Diskussionsbemerkungen zu dem zweiten Hilbertschen Vortrag über die Grundlagen der Mathematik", same volume, pp. 86–88: „Hilbert ertrug diese Verstümmelung nicht … ein nach festen Regeln sich vollziehendes Spiel mit Formeln." Van Heijenoort, pp. 482–483.
The war then left the journals for the post. In October 1928 Hilbert, who had been told three years earlier that he had pernicious anemia and did not expect to live, wrote to the other principal editors of the Mathematische Annalen for authority to remove Brouwer from its board, and named the disease in a postscript. Carathéodory told Einstein what the dispute about the Bologna congress was really about: "Hilbert is of the opinion that after his death Brouwer will constitute a danger for the continued existence of the M.A." Hilbert to the Herausgeber, 15 October 1928; Carathéodory to Einstein, 20 October 1928. The letters of the affair are printed in Dirk van Dalen, "The War of the Frogs and the Mice, or the Crisis of the Mathematische Annalen", Mathematical Intelligencer 12:4 (1990), 17–31, whose translations are used here. Einstein, whose name stood on the title page, refused to sign. He thought Brouwer, "with all due respect for his mind, a psychopath", and that the right course was to grant him a jester's liberty, Narrenfreiheit; when the dismissal went out over Hilbert's signature anyway and the board fell to quarrelling by circular letter, he told Brouwer and Blumenthal that he had wandered into "this mathematical wolf-pack like an innocent lamb" and asked to be left in the role of astonished contemporary. Einstein to Hilbert, 19 October 1928; to Brouwer and Blumenthal, 25 November 1928 (van Dalen, pp. 20, 26). Hilbert's letter of dismissal is dated 25 October 1928. To Born, on 27 November, he said what he thought of it all:
Wenn Hilberts Krankheit der Angelegenheit nicht einen tragischen Zug gäbe, so würde dieser Tintenkrieg für mich eines der drolligsten und wohlgelungensten Possenspiele sein, das sich bluternst nehmende Menschen vor meinen Augen aufgespielt haben. … Aber dies sage ich Dir nur privatim und denke nicht daran, mit einer weiteren papieren Lanze in diesen Froschmäusekrieg als Kämpe hineinzuspringen.
If Hilbert's illness did not give the matter a tragic touch, this ink war would be for me one of the drollest and most successful farces that people who take themselves in deadly earnest have ever played out before my eyes. … But I say this to you only in private, and have no thought of leaping into this war of the frogs and the mice as a champion with one more paper lance.
—Einstein to Max Born, 27 November 1928. The Collected Papers of Albert Einstein, vol. 16, doc. 323; the German as transcribed by van Dalen, my translation. The frogs and mice are the Batrachomyomachia, the mock-Homeric epic of a one-day war between the two species. The board was dissolved on 27 December 1928 and reconstituted from volume 101 with Hilbert, Blumenthal and Hecke alone.
Einstein was right about the farce and wrong only in thinking the stakes small; they were smaller than that. Two years later, on 8 September 1930, Hilbert gave the address at Königsberg that ends "Wir müssen wissen. Wir werden wissen.", we must know, we shall know, the words that are cut on his grave; the day before, at a smaller meeting in the same city, Gödel had mentioned in discussion that there are true arithmetical sentences that no such system can prove, and with them went the consistency proof on which the formula game was to rest. Hilbert, "Naturerkennen und Logik", Die Naturwissenschaften 18 (1930), 959–963; the closing is on the recording he made for radio shortly after: „Statt des törichten Ignorabimus heiße im Gegenteil unsere Losung: Wir müssen wissen, Wir werden wissen." Gödel spoke in the discussion on the foundations of mathematics at the Second Conference on the Epistemology of the Exact Sciences, 7 September 1930.
Wittgenstein's position in this war is misread in both directions, so it is worth stating from the texts. He attended one of Brouwer's two Vienna lectures, "Mathematik, Wissenschaft und Sprache", on 10 March 1928, with Feigl and Waismann, and Feigl recorded that in a café afterward, "Suddenly and very volubly Wittgenstein began talking philosophy", at great length, for the first time in years. Herbert Feigl, Inquiries and Provocations (1981), p. 64; Feigl's fuller account, in a letter printed by Pitcher (The Philosophy of Wittgenstein, 1964, p. 8), dates his return to philosophy from that evening. Rodych warns that it is "a gross overstatement" to say he returned to philosophy because of the lecture; Ramsey and the Vienna Circle were also at work on him. He took from Brouwer the refusal of the excluded middle over the infinite and the refusal of discovery, and he kept both to the end. Of the digits of π he wrote that even a God who knew them all could decide nothing about them that we cannot: "I want to say: Even God can determine something mathematical only by mathematics. Even for him the mere rule of expansion cannot decide anything that it does not decide for us." RFM VII §41 (V §34 in 1956), on whether "777" occurs in the expansion. In the Investigations (§516) the point is put as a question about understanding: "It seems clear that we understand the meaning of the question: 'Does the sequence 7777 occur in the development of π?' … our understanding of that question reaches just so far, one may say, as such explanations reach." Russell's 1930 report had it exactly: he "believes in 'infinite possibility,' as he calls it, but not in actual 'infinite classes' or 'infinite series'." But he refused Brouwer's answer to Brouwer's own question. Mathematical exactness does not live in the human intellect, as the intuitionist said, nor on paper, as the formalist said; it lives in what is done with the paper, in a technique that is taught, shared and applied, which is why a proof must be surveyable, a thing on paper that can be copied and compared, and why a construction in the mind that no one could write down would not be a proof at all. Against Brouwer's languageless mathematics he set a mathematics that is nothing but a practice of signs; against Hilbert's demand that the practice be certified consistent before it is used he set the bridge: nothing need go wrong, and if something does, the fault is a wrong coefficient or a slip in the calculation, never a hidden contradiction that was going to bring the thing down regardless. "The civil status of a contradiction, or its status in civil life: there is the philosophical problem." PI §125: „Die bürgerliche Stellung des Widerspruchs, oder seine Stellung in der bürgerlichen Welt: das ist das philosophische Problem." The same remark: "It is the business of philosophy, not to resolve a contradiction by means of a mathematical or logico-mathematical discovery, but to make it possible for us to get a clear view of the state of mathematics that troubles us: the state of affairs before the contradiction is resolved."
All three schools wanted a rock, and the war of the frogs and the mice was fought over the deed to it: the logicist's rock was logic, the formalist's a consistency proof, the intuitionist's the mind. Wittgenstein's heresy was not a fourth rock. It was that the castle was standing, had always been standing, and that the rock in the picture was paint. The mathematicians who conceded the point did so late, and in his terms without his name. Weyl in 1949, surveying the ruins of the program he had championed, wrote that "the mathematician watches with pain the greater part of his towering edifice which he believed to be built of concrete blocks dissolve into mist before his eyes", and that the ultimate foundations and meaning of mathematics remain an open problem, so that “‘Mathematizing’ may well be a creative activity of man, like music, the products of which not only in form but also in substance defy complete objective rationalization.” Weyl, Philosophy of Mathematics and Natural Science (1949), pp. 54 and 219. Gödel, who did not concede, said instead what a rock would have to be: "despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true." Gödel, "What is Cantor's Continuum Problem?", supplement to the 1964 version, Collected Works, vol. II, p. 268. Something like a perception. That is the picture, and the machine is what happens when a device with no perceptions at all, something like nothing, is trained to go on as the perceivers went on.
These machines are humans who calculate
The scene at the center of the Investigations is a scene from a village school:
Gehen wir nun zu unserm Beispiel (143) zurück. Der Schüler beherrscht jetzt — nach den gewöhnlichen Kriterien beurteilt — die Grundzahlenreihe. Wir lehren ihn nun auch andere Reihen von Kardinalzahlen anschreiben und bringen ihn dahin, daß er z. B. auf Befehle von der Form „+n" Reihen der Form 0, n, 2n, 3n, etc. anschreibt; auf den Befehl „+1" also die Grundzahlenreihe. — Wir hätten unsre Übungen und Stichproben seines Verständnisses im Zahlenraum bis 1000 gemacht.
Wir lassen nun den Schüler einmal eine Reihe (etwa „+2") über 1000 hinaus fortsetzen, — da schreibt er: 1000, 1004, 1008, 1012.
Wir sagen ihm: „Schau, was du machst!" — Er versteht uns nicht. Wir sagen: „Du solltest doch zwei addieren; schau, wie du die Reihe begonnen hast!" — Er antwortet: „Ja! Ist es denn nicht richtig? Ich dachte, so soll ich's machen." — Oder nimm an, er sagte, auf die Reihe weisend: „Ich bin doch auf die gleiche Weise fortgefahren!" — Es würde uns nun nichts nützen, zu sagen „Aber siehst du denn nicht … ?" — und ihm die alten Erklärungen und Beispiele zu wiederholen. — Wir könnten in so einem Falle etwa sagen: Dieser Mensch versteht von Natur aus jenen Befehl, auf unsre Erklärungen hin, so, wie wir den Befehl: „Addiere bis 1000 immer 2, bis 2000 4, bis 3000 6, etc."
Dieser Fall hätte Ähnlichkeit mit dem, als reagierte ein Mensch auf eine zeigende Gebärde der Hand von Natur damit, daß er in der Richtung von der Fingerspitze zur Handwurzel blickt, statt in der Richtung zur Fingerspitze.
Let us return to our example (143). Now—judged by the usual criteria—the pupil has mastered the series of natural numbers. Next we teach him to write down other series of cardinal numbers and get him to the point of writing down series of the form 0, n, 2n, 3n, etc. at an order of the form ‘+n’; so at the order ‘+1’ he writes down the series of natural numbers.—Let us suppose we have done exercises and given him tests up to 1000.
Now we get the pupil to continue a series (say +2) beyond 1000—and he writes 1000, 1004, 1008, 1012.
We say to him: ‘Look what you've done!’—He doesn't understand. We say: ‘You were meant to add two: look how you began the series!’—He answers: ‘Yes, isn't it right? I thought that was how I was meant to do it.’——Or suppose he pointed to the series and said: ‘But I went on in the same way.’—It would now be no use to say: ‘But can't you see … ?’—and repeat the old examples and explanations.—In such a case we might say, perhaps: It comes natural to this person to understand our order with our explanations as we should understand the order: ‘Add 2 up to 1000, 4 up to 2000, 6 up to 3000 and so on.’
Such a case would present similarities with one in which a person naturally reacted to the gesture of pointing with the hand by looking in the direction of the line from finger-tip to wrist, not from wrist to finger-tip.
—Wittgenstein, Philosophische Untersuchungen §185; Anscombe's translation. PI §185, returning to the pupil of §143.
A boy at a desk and a teacher at a loss. What is the teacher to say? That the rule meant 1002? "So when you gave the order +2 you meant that he was to write 1002 after 1000—and did you also mean that he should write 1868 after 1866, and 100036 after 100034, and so on—an infinite number of such propositions?" The teacher means that each number should be followed by the next but one, "and from this all those propositions follow in turn. But that is just what is in question: what, at any stage, does follow from that sentence." PI §186. The finite set of examples fits the pupil's rule as well as ours, and every further explanation is another finite set of examples. "This was our paradox: no course of action could be determined by a rule, because every course of action can be made out to accord with the rule." And the paradox dissolves, not by finding the rail that fixes the next step, but by seeing that there is no rail to find: "there is a way of grasping a rule which is not an interpretation, but which is exhibited in what we call 'obeying the rule' and 'going against it' in actual cases." PI §201: „Unser Paradox war dies: eine Regel könnte keine Handlungsweise bestimmen, da jede Handlungsweise mit der Regel in Übereinstimmung zu bringen sei." The next remark (§202): "And hence also 'obeying a rule' is a practice." What connects the signpost to my walking is not an interpretation standing behind it. "I have been trained to react to this sign in a particular way, and now I do so react to it"; and a person goes by a signpost "only in so far as there exists a regular use of sign-posts, a custom." PI §198: „ich bin zu einem bestimmten Reagieren auf dieses Zeichen abgerichtet worden, und so reagiere ich nun." The verb is abrichten, which is what one does to a dog.
Where the idea of the rail came from he says too: "Whence comes the idea that the beginning of a series is a visible section of rails invisibly laid to infinity?" And the end of justification: "If I have exhausted the justifications I have reached bedrock, and my spade is turned. Then I am inclined to say: 'This is simply what I do.'" The bedrock is der harte Felsen, the hard rock, the same word as the painted one: the rock the foundations put into the picture he finds under the spade, and it is the training. "When I obey a rule, I do not choose. I obey the rule blindly." PI §§218, 217, 219: „Woher die Idee, es wäre die angefangene Reihe ein sichtbares Stück unsichtbar bis ins Unendliche gelegter Geleise?"; „Habe ich die Begründungen erschöpft, so bin ich nun auf dem harten Felsen angelangt, und mein Spaten biegt sich zurück. Ich bin dann geneigt, zu sagen: 'So handle ich eben.'"; „Wenn ich der Regel folge, wähle ich nicht. Ich folge der Regel blind." The hard rock, der harte Felsen, is the same word as the painted one. And what, then, makes 1002 right? The answer is the nearest thing in all his writing to a statement of what the foundation of mathematics is:
„So sagst du also, daß die Übereinstimmung der Menschen entscheide, was richtig und was falsch ist?" — Richtig und falsch ist, was Menschen sagen; und in der Sprache stimmen die Menschen überein. Dies ist keine Übereinstimmung der Meinungen, sondern der Lebensform.
Zur Verständigung durch die Sprache gehört nicht nur eine Übereinstimmung in den Definitionen, sondern (so seltsam dies klingen mag) eine Übereinstimmung in den Urteilen. Dies scheint die Logik aufzuheben; hebt sie aber nicht auf. — Eines ist, die Meßmethode zu beschreiben, ein Anderes, Messungsergebnisse zu finden und auszusprechen. Aber was wir „messen" nennen, ist auch durch eine gewisse Konstanz der Messungsergebnisse bestimmt.
‘So you are saying that human agreement decides what is true and what is false?’—It is what human beings say that is true and false; and they agree in the language they use. That is not agreement in opinions but in form of life.
If language is to be a means of communication there must be agreement not only in definitions but also (queer as this may sound) in judgments. This seems to abolish logic, but does not do so.—It is one thing to describe methods of measurement, and another to obtain and state results of measurement. But what we call ‘measuring’ is partly determined by a certain constancy in results of measurement.
—Wittgenstein, Philosophische Untersuchungen §§241–242; Anscombe's translation. PI §§241–242.
How the teaching is done he had said earlier, and the description is worth reading with an engineer's eye: "I do it, he does it after me; and I influence him by expressions of agreement, rejection, expectation, encouragement. I let him go his way, or hold him back; and so on." Nothing circular is explained in such a lesson, and the phrase "and so on" is taught in it too, by a gesture. PI §208; and §5: "Here the teaching of language is not explanation, but training", kein Erklären, sondern ein Abrichten.
Set beside this what the machine is. It was shown the written record and trained to continue it; it was corrected on its continuations by expressions of agreement and rejection, at a scale no schoolroom reaches; and now it goes on. When it adds, it adds the way the pupil adds, by going on as it was trained to go on, with no rail beneath it, and when it reaches a place its training did not reach it may write 1004. The objection that this is not really calculating, that there is nothing behind the signs, no concept of two, only a disposition to produce the next sign, is the teacher's objection in the schoolroom, and it fails for the same reason. Nothing stands behind our 1002 either. What stands behind it is the training and the agreement, and a disposition to go on as we go on is not a counterfeit of understanding addition; it is what understanding addition consists in, once one looks at what the word is used for instead of at what one imagines inside. He had put the question about machines to himself, in two remarks that stand side by side:
Könnte eine Maschine denken? — Könnte sie Schmerzen haben? — Nun, soll der menschliche Körper so eine Maschine heißen? Er kommt doch am nächsten dazu, so eine Maschine zu sein.
Aber eine Maschine kann doch nicht denken! — Ist das ein Erfahrungssatz? Nein. Wir sagen nur vom Menschen, und was ihm ähnlich ist, es denke. Wir sagen es auch von Puppen und wohl auch von Geistern. Sieh das Wort „denken" als Instrument an!
Could a machine think?—Could it be in pain?—Well, is the human body to be called such a machine? It surely comes as close as possible to being such a machine.
But a machine surely cannot think!—Is that an empirical statement? No. We only say of a human being and what is like one that it thinks. We also say it of dolls and no doubt of spirits too. Look at the word ‘to think’ as a tool.
—Wittgenstein, Philosophische Untersuchungen §§359–360; Anscombe's translation. PI §§359–360.
The sentence "a machine cannot think" is not a finding about machines. It is a rule about a word, and rules about words change when the form of life changes.
Turing's machine was built out of exactly this material, and Wittgenstein said so in a remark of 1947 that opens on six words and ends on the pupil's order:
Turing's ‘Maschinen’. Diese Maschinen sind ja die Menschen, welche kalkulieren. Und man könnte, was er sagt, auch in Form von Spielen ausdrücken. Und zwar wären die interessanten Spiele solche, bei denen man gewissen Regeln gemäß zu unsinnigen Anweisungen gelangt. Ich denke an Spiele ähnlich dem “Wettrennspiel”. Man erhielte etwa den Befehl “Setze auf die gleiche Art fort”, wenn dies keinen Sinn ergibt, etwa, weil man in einen Zirkel gerät; denn jener Befehl hat eben nur an gewissen Stellen Sinn. (Watson.)
Turing's ‘Machines’. These machines are humans who calculate. And what he says could also be expressed in the form of games. And the interesting games would be those in which, following certain rules, one arrives at nonsensical instructions. I am thinking of games like the ‘racing game’. One might receive the order ‘Go on in the same way’ when this makes no sense, say because one has got into a circle; for that order has a sense only at certain points. (Watson.)
—Wittgenstein, Bemerkungen über die Philosophie der Psychologie I §1096. Bemerkungen über die Philosophie der Psychologie I §1096, from MS 135 (1947) by way of Ts 229, p. 448; the German as printed. The second sentence is Luckhardt and Aue's English; the rest of the translation is mine. The Watson of the last word is Alister Watson, the King's mathematician who was a friend of both men, and „Setze auf die gleiche Art fort" is the order the pupil of §185 was given.
This is not a gibe. It is a reading of the 1936 paper, and a correct one. Turing's "computer" in "On Computable Numbers" is a person: "Computing is normally done by writing certain symbols on paper. We may suppose this paper is divided into squares like a child's arithmetic book." "The behaviour of the computer at any moment is determined by the symbols which he is observing, and his 'state of mind' at that moment." The number of states of mind is finite, because if there were infinitely many some would be "arbitrarily close" and be confused, and "the use of more complicated states of mind can be avoided by writing more symbols on the tape." Then: "We may now construct a machine to do the work of this computer." A. M. Turing, "On Computable Numbers, with an Application to the Entscheidungsproblem", Proceedings of the London Mathematical Society, ser. 2, 42 (1936–7), 230–265, §9, pp. 249–251. The paper opens (§1, p. 231) by comparing "a man in the process of computing a real number to a machine which is only capable of a finite number of conditions". The machine was an abstraction from a schoolchild with squared paper, and the thing abstracted was the human practice of writing a sign and looking at it and writing the next. That is also, with the tape widened and the states multiplied, a description of the machines of 2026, whose one advance over the child is the scale at which they were trained. He did not foresee them. He described what calculating is, and the description had no clause that a machine could fail to satisfy.
The scratchpad came first. Large models, it was found in 2021, "struggle with tasks that require unbounded multi-step computation, such as adding integers or executing programs", and yet "these same models are able to perform complex multi-step computations" when asked to do them "step by step", setting their intermediate results down in what the authors named a scratchpad; a few weeks later another group found that prompting a model to produce "a chain of thought—a series of intermediate reasoning steps" made arithmetic and symbolic reasoning emerge in models large enough, and by 2024 the frontier models were being trained to write a long private chain of this kind before every answer. Maxwell Nye et al., "Show Your Work: Scratchpads for Intermediate Computation with Language Models", arXiv:2112.00114 (30 Nov 2021); Jason Wei et al., "Chain-of-Thought Prompting Elicits Reasoning in Large Language Models", arXiv:2201.11903 (28 Jan 2022); Takeshi Kojima et al., "Large Language Models are Zero-Shot Reasoners", arXiv:2205.11916 (2022), where adding the words "Let's think step by step" raised a model's accuracy on one arithmetic benchmark from 17.7 to 78.7 percent; OpenAI, "Learning to Reason with LLMs", 12 Sep 2024: "o1 thinks before it answers—it can produce a long internal chain of thought before responding to the user." This is Turing's observation that complicated states of mind can be avoided by writing more symbols on the tape, and it is Wittgenstein's refusal to let calculation in the head be anything but a shadow of calculation on paper. Someone does a sum in his head, and the sum goes into a bridge:
Jemand macht eine Berechnung im Kopf. Das Ergebnis verwendet er, sagen wir, beim Bauen einer Brücke, oder Maschine. — Willst du sagen, er habe diese Zahl eigentlich nicht durch Berechnung gefunden? Sie sei ihm etwa, nach einer Art Träumerei, in den Schoß gefallen? Es mußte doch da gerechnet werden, und ist gerechnet worden. Denn er weiß, daß, und wie, er gerechnet hat; und das richtige Resultat wäre ohne Rechnung nicht erklärbar. — Wie aber, wenn ich sagte: „Es kommt ihm vor, er habe gerechnet. Und warum soll sich das richtige Resultat erklären lassen? Ist es nicht unverständlich genug, daß er ohne ein Wort, oder Schriftzeichen, RECHNEN konnte?" —
Ist das Rechnen in der Vorstellung in gewissem Sinne unwirklicher, als das auf dem Papier? Es ist das wirkliche — Kopfrechnen. — Ist es ähnlich dem Rechnen auf dem Papier? — Ich weiß nicht, ob ich es ähnlich nennen soll. Ist ein Stück weißes Papier mit schwarzen Strichen drauf einem menschlichen Körper ähnlich?
Someone does a sum in his head. He uses the result, let's say, for building a bridge or a machine.—Are you trying to say that he has not really arrived at this number by calculation? That it has, say, just ‘come’ to him in the manner of a kind of dream? There surely must have been calculation going on, and there was. For he knows that, and how, he calculated; and the correct result he got would be inexplicable without calculation.——But what if I said: ‘It strikes him as if he had calculated. And why should the correct result be explicable? Is it not incomprehensible enough, that without saying a word, without making a note, he was able to CALCULATE?’—
Is calculating in the imagination in some sense less real than calculating on paper? It is real—calculation-in-the-head.—Is it like calculation on paper?—I don't know whether to call it like. Is a bit of white paper with black lines on it like a human body?
—Wittgenstein, Philosophische Untersuchungen §364; Anscombe's translation. PI §§364–366; the bridge of §364 is Turing's bridge again.
The answer he works toward in the remarks that follow is that the head-sum is called a calculation because it can be set beside the paper one and agrees with it, not the other way round; "I have done the multiplication in my head", and "This is the point at which I go wrong. For I now want to say: it was some mental process corresponding to the multiplication on paper." A machine that is reliable when it writes its steps and unreliable when it must answer in one breath is not a machine with a defective inner faculty. It is a calculator in Turing's sense, a human who calculates, and the paper is where the calculating is.
Then the gold medal. In July 2025 a model solved five of the six problems of the International Mathematical Olympiad "end-to-end in natural language, producing rigorous mathematical proofs directly from the official problem descriptions", within the contestants' time, and the IMO's coordinators graded it as they graded the children. Google DeepMind, 21 July 2025, on Gemini Deep Think: "An advanced version of Gemini Deep Think solved five out of the six IMO problems perfectly, earning 35 total points, and achieving gold-medal level performance", with the results "officially graded and certified by IMO coordinators using the same criteria as for student solutions." A year earlier AlphaProof had reached silver only after the problems were translated by hand into a formal language; the formal system is the sign-game, and the system that worked in mufti overtook it in twelve months. The third thing is older than the machines, and the strangest. In 1957 the linguist J. R. Firth wrote the sentence that every course on word embeddings opens with, "You shall know a word by the company it keeps!", and the sentence he wrote four lines before it is the one the courses leave out: "As Wittgenstein says, 'the meaning of words lies in their use.' The day-to-day practice of playing language games recognizes customs and rules." J. R. Firth, "A synopsis of linguistic theory 1930–1955", in Studies in Linguistic Analysis (Oxford: Philological Society, 1957), p. 11; reprinted in Palmer, ed., Selected Papers of J. R. Firth 1952–59 (1968), p. 179. Firth's "the meaning of words lies in their use" is his gloss, not Anscombe's wording of PI §43. Zellig Harris stated the distributional hypothesis three years earlier in a form that is the design of the embedding: "difference of meaning correlates with difference of distribution" ("Distributional Structure", Word 10, 1954, p. 156). The papers that built the vectors cite neither; the genealogy was drawn by their readers. The representation of meaning the machines are built on, a word is where it occurs, was written down as a gloss on Wittgenstein by a man who had read him. The lineage is not a resemblance noticed after the fact. It is in the record, in ink.
Wittgenstein did not predict machines that prove theorems, and he never read the paper of 1950 in which Turing replaced the question whether machines can think, "too meaningless to deserve discussion", with a game played in language; when Malcolm wrote to ask whether the paper was a leg-pull, he answered, "I haven't read it but I imagine it's no leg-pull." Turing, "Computing Machinery and Intelligence", Mind 59 (1950), p. 442. Malcolm wrote to ask whether the paper was a leg-pull; Wittgenstein answered on 1 December 1950, "I haven't read it but I imagine it's no leg-pull" (Floyd, Disputatio 2019, n. 8). What he did was describe mathematics so that nothing in the description requires a mind behind the signs, and describe understanding so that nothing in it requires a something inside; and he did it so thoroughly that when a device appeared which does mathematics by continuing text, the only objections anyone could raise were the ones he had already answered. The strongest of them is the one he answered with a beetle:
Wenn ich von mir selbst sage, ich wisse nur vom eigenen Fall, was das Wort „Schmerz" bedeutet, — muß ich das nicht auch von den Andern sagen? Und wie kann ich denn den einen Fall in so unverantwortlicher Weise verallgemeinern?
Nun, ein Jeder sagt es mir von sich, er wisse nur von sich selbst, was Schmerzen seien! — Angenommen, es hätte Jeder eine Schachtel, darin wäre etwas, was wir „Käfer" nennen. Niemand kann je in die Schachtel des Andern schaun; und Jeder sagt, er wisse nur vom Anblick seines Käfers, was ein Käfer ist. — Da könnte es ja sein, daß Jeder ein anderes Ding in seiner Schachtel hätte. Ja, man könnte sich vorstellen, daß sich ein solches Ding fortwährend veränderte. — Aber wenn nun das Wort „Käfer" dieser Leute doch einen Gebrauch hätte? — So wäre er nicht der der Bezeichnung eines Dings. Das Ding in der Schachtel gehört überhaupt nicht zum Sprachspiel; auch nicht einmal als ein Etwas: denn die Schachtel könnte auch leer sein. — Nein, durch dieses Ding in der Schachtel kann „gekürzt werden"; es hebt sich weg, was immer es ist.
Das heißt: Wenn man die Grammatik des Ausdrucks der Empfindung nach dem Muster von „Gegenstand und Bezeichnung" konstruiert, dann fällt der Gegenstand als irrelevant aus der Betrachtung heraus.
If I say of myself that it is only from my own case that I know what the word ‘pain’ means—must I not say the same of other people too? And how can I generalize the one case so irresponsibly?
Now someone tells me that he knows what pain is only from his own case!——Suppose everyone had a box with something in it: we call it a ‘beetle’. No one can look into anyone else's box, and everyone says he knows what a beetle is only by looking at his beetle.—Here it would be quite possible for everyone to have something different in his box. One might even imagine such a thing constantly changing.—But suppose the word ‘beetle’ had a use in these people's language?—If so it would not be used as the name of a thing. The thing in the box has no place in the language-game at all; not even as a something: for the box might even be empty.—No, one can ‘divide through’ by the thing in the box; it cancels out, whatever it is.
That is to say: if we construe the grammar of the expression of sensation on the model of ‘object and designation’ the object drops out of consideration as irrelevant.
—Wittgenstein, Philosophische Untersuchungen §293; Anscombe's translation. PI §293.
The question whether the machine "really" understands the theorem it proved, whether there is something in its box, is a question about a beetle. What it does with the signs, whether it can go on, apply, correct, teach, is not, and those are the questions that will be answered, in the next decade, by watching.
The shape of thought read Wittgenstein's lion the other way: the model speaks our words but does not stand where our words stand, inside the practice, the body, the community of consequence from which meaning is made. That is right about pain and promises, and it is why the machine's poems are not poems yet. Mathematics is the exception that proves the reading, because in mathematics there was never anywhere else to stand. The practice of mathematics is the signs and the techniques for going on with them and the civilian use that gives them a point; it has no body and no consequence of its own, which is exactly what Russell meant when he lost "the splendid certainty" and found tautology, and what von Neumann meant when he warned of the art for art's sake. A thing that holds the whole written practice stands, in mathematics, where the words stand. That is why mathematics fell to the machines first, and why the mathematicians, who had believed since Plato that their words stood somewhere higher, felt the fall as a kind of insult.
What the mathematicians say they want from a proof is, in Wittgenstein's word, surveyability, and it is to their credit that they said so before the machines and are saying it again now. Thurston's essay of 1994, which The very high baroque leans on, measured mathematics by whether people understand and think more clearly, not by the theorems proved. Tao, in the essay from his lecture to this year's International Congress, put it in a sentence Wittgenstein could have written: "A proof that no human can properly explain should be viewed as incomplete, even if it has been formally verified." Terence Tao, "Mathematics in the age of AI", arXiv:2608.16753 (August 2026), the essay from his ICM 2026 plenary. Also: "We may soon be faced with the very real possibility of a verified proof of a major result that no human understands well enough to explain." And, on the Science Friday program of 28 October 2026, "mathematics fundamentally is about understanding." Quanta described the October release in the terms of the Russellian sum: "Mathematicians know that the proofs are true — they're verified by a system that checks their logic — but have no idea why." Jordana Cepelewicz, "Is AI the End of Math As We Know It?", Quanta, 5 October 2026. A guest post on Tao's blog by Francesco Maggi, 18 October 2026, has the same complaint from a working analyst: "LLMs can give us correct Lean proofs that we cannot directly read, and messy natural language translations that are so complex as to require intensive study by mathematicians before they can lead to human understanding." A proof in Lean that no one can read is the proof of the seven-digit sum in Russell's notation: a configuration a machine can check and a human cannot survey, and Wittgenstein's answer is that it is not yet a proof for us, that one makes it a proof by changing its notation until it can be taken in, and that the making is not clerical work done after the mathematics but the mathematics itself. The point has teeth. The formal proof released in September with the claimed resolution of the Navier–Stokes problem was argued the next month, by analysts who went through it, not to say what its accompanying prose said it said; two things nobody could survey had been set side by side and called the same proof, which is the very question his first remark on surveyability says must be decidable with certainty. OpenAI, "On the Navier–Stokes Millennium Prize Problem", 8 September 2026: an analytical proof and a Lean formalization, from about ten thousand concurrent agents over eighty-eight hours, establishing statement C of the Clay formulation in the forced case; the Clay Mathematics Institute's note of 11 September that the problem "has apparently been settled"; Alexander Bastounis, Fabian Circelli and Anders Hansen, "Navier-Stokes lost in translation", arXiv:2610.08144 (October 2026), on the gap between the formal proof and the natural-language one. The very high baroque has the mathematicians' reactions.
The same word governs his philosophy. "The concept of a perspicuous representation is of fundamental significance for us"; a perspicuous representation, übersichtliche Darstellung, “produces just that understanding which consists in ‘seeing connexions’”, and it is lacking in our grammar as in our proofs. PI §122: „Der Begriff der übersichtlichen Darstellung ist für uns von grundlegender Bedeutung." Hence also the remark in the second part of the book, p. 220: "Let the use of words teach you their meaning. (Similarly one can often say in mathematics: let the proof teach you what was being proved.)" Math 2.0, Tao's name for a mathematics that pays for exposition, is the institutional form of this demand, and the complaint of The very high baroque, that the trade had spent two centuries removing the scaffolding and now needs it back, is the complaint that the trade had let its proofs grow less surveyable than it pretended, tolerable only because the author could be found and asked. Let the proof teach you what was being proved: the machine's output has not yet taught anyone, and until it does, in Wittgenstein's sense nothing has yet been proved, only checked. That is a harder judgment on the October release than any the mathematicians made, and it comes from the man they took for a demolisher.
language is the mother, not the handmaid, of thought
He is quoted most for the sentences he meant least to be quoted, and the misreadings are of a pattern: each takes a sentence that was drawing a limit and reads it as a verdict on what lies beyond. "Whereof one cannot speak, thereof one must be silent" was read by the Vienna Circle as the positivist's rule and by everyone since as a shrug; what he wrote to Ficker, when he was trying to get the book printed, was that its point was ethical, that it consisted of two parts, the one written and everything he had not written, "and precisely this second part is the important one", because the book drew the limit of the ethical from inside. Wittgenstein to Ludwig von Ficker, October or November 1919 (Briefe an Ludwig von Ficker, ed. von Wright, 1969): „der Sinn des Buches ist ein Ethischer"; „Es wird nämlich das Ethische durch mein Buch gleichsam von Innen her begrenzt." Russell read the mysticism rightly and thought it a flaw. La società has the trapdoor. The book itself says what the silence is about:
Wir fühlen, daß selbst, wenn alle möglichen wissenschaftlichen Fragen beantwortet sind, unsere Lebensprobleme noch gar nicht berührt sind. Freilich bleibt dann eben keine Frage mehr; und eben dies ist die Antwort.
Es gibt allerdings Unaussprechliches. Dies zeigt sich, es ist das Mystische.
We feel that even when all possible scientific questions have been answered, the problems of life remain completely untouched. Of course there are then no questions left, and this itself is the answer.
There are, indeed, things that cannot be put into words. They make themselves manifest. They are what is mystical.
—Wittgenstein, Tractatus Logico-Philosophicus 6.52, 6.522; Pears and McGuinness's translation. Tractatus 6.52 and 6.522. In the lecture on ethics of 1929 the same thought is a cage: "My whole tendency and I believe the tendency of all men who ever tried to write or talk Ethics or Religion was to run against the boundaries of language. This running against the walls of our cage is perfectly, absolutely hopeless."
Read as the mathematicians' sentence, it is a forecast of the decade: suppose every provable statement were proved, by machines, by tomorrow morning. The problems of mathematics, what to count and why, which lighthouses to steer by, what a result is for, are untouched, because they were never the kind of thing a proof settles; and the dismay of the trade at the release of a thousand true statements is the discovery, in its own house, that the answering of all the questions was not the thing that was wanted.
"The limits of my language mean the limits of my world" is read as a confinement. It is read better as a measurement, and the machine is its instrument: a thing with all of our language and none of our world, which tells us, by what it can and cannot do, which parts of the world were in the language all along. Mathematics was. "If a lion could talk, we could not understand him" is read as a wall between kinds of creature; it is a statement about shared life, and the machine is a lion we understand, which tells us that the life it shares with us is the written one, and that for the written practices, of which mathematics is the purest, the written life is enough. Tractatus 5.6: „Die Grenzen meiner Sprache bedeuten die Grenzen meiner Welt." PI Part II xi, p. 223: „Wenn ein Löwe sprechen könnte, wir könnten ihn nicht verstehn." "If God had looked into our minds he would not have been able to see there whom we were speaking of" is read as a riddle; it is the rule for the decade's favorite question, whether the machine "really" means the theorem, and the rule says that looking inside, with whatever instrument, is not where the answer lives. PI Part II xi, p. 217. And §109, on method: the problems "are solved, rather, by looking into the workings of our language"; §66, "don't think, but look!"
He named the men he had taken his thinking from, in 1931, in a sentence that is itself a disclaimer: "I don't believe I have ever invented a line of thinking, I have always taken one over from someone else", and then the list: Boltzmann, Hertz, Schopenhauer, Frege, Russell, Kraus, Loos, Weininger, Spengler, Sraffa. Culture and Value, p. 19e (1931): „So haben mich Boltzmann, Hertz, Schopenhauer, Frege, Russell, Kraus, Loos, Weininger, Spengler, Sraffa beeinflußt." What he invents, he adds, are new similes. Boltzmann he had wanted to study physics under, before Boltzmann killed himself in 1906; Schopenhauer gave him his first philosophy, "a Schopenhauerian epistemological idealism", which Frege's realism drove out (von Wright's biographical sketch in Malcolm, Memoir, 1958). Sraffa, an economist, broke the Tractatus with a gesture: when Wittgenstein insisted that a proposition and what it describes must have the same logical form, Sraffa brushed the underside of his chin with the fingertips of one hand, the Neapolitan gesture of contempt, and asked, "What is the logical form of that?" (Malcolm, Ludwig Wittgenstein: A Memoir, 1958, p. 69). Several of them had already said, about their own subjects, the thing the machines now say about mathematics. Hertz, in the introduction to his mechanics, wrote the sentences Wittgenstein wanted as the motto of the Investigations. Around the words "force" and "electricity", Hertz says, we have piled up more relations than can be reconciled, we feel this obscurely and ask what force really is, and the question is wrong about the kind of answer it wants: "When these painful contradictions are removed, the question as to the nature of force will not have been answered; but our minds, no longer vexed, will cease to ask illegitimate questions." Heinrich Hertz, Die Prinzipien der Mechanik (1894), Einleitung: „Sind diese schmerzenden Widersprüche entfernt, so ist zwar nicht die Frage nach dem Wesen beantwortet, aber der nicht mehr gequälte Geist hört auf, die für ihn unberechtigte Frage zu stellen." The English is Jones and Walley's (1899), pp. 7–8. Wittgenstein copied the passage into his manuscripts and considered it as the motto for the Investigations; the motto he chose instead is Nestroy's. "Does the machine understand?" is a question of Hertz's kind. It will not be answered. It will stop being asked, when the relations piled around the word "understand" have been sorted, and the sorting will be done by what the machines turn out to be good for, as the sorting of "force" was done by engines.
Frege, in 1879, had made the best case for the symbolism Wittgenstein later called a curse, and it is a case about tools. His notation stood to ordinary language as the microscope to the eye: the eye, "because of the range of its possible uses and the versatility with which it can adapt to the most diverse circumstances", is far superior, and as an optical instrument it is full of imperfections we do not notice "only on account of its intimate connection with our mental life"; but for sharpness of resolution the microscope wins, "and that is just why it is useless for all others." Frege, Begriffsschrift (1879), Vorwort, in van Heijenoort, p. 6. The machines were trained on the eye, not the microscope, and they see what the eye sees. Kraus, whose Fackel Wittgenstein read for years, wrote in 1909 that "language is the mother, not the handmaid, of thought", and a device in which thought, if that is what it is, condensed out of nothing but language is the aphorism built as a machine. Karl Kraus, "Aphorismen", Die Fackel 288 (11 October 1909): „Die Sprache ist die Mutter, nicht die Magd des Gedankens"; collected in Pro domo et mundo (1912), p. 81. Lichtenberg, whom he esteemed highly, had proposed in the 1790s that one should say es denkt, it thinks, as one says it lightens, since to say "I think" already assumes too much; the machines are the grammatical experiment carried out, thinking with the subject left blank, and the discomfort they cause is the discomfort Lichtenberg's sentence caused. Lichtenberg, Sudelbuch K 76 (1793–96), in Promies's numbering. Von Wright: "An author, however, who reminds one, often astonishingly, of Wittgenstein is Lichtenberg. Wittgenstein esteemed him highly." Spengler, whose Decline he read around 1930, opened it with a chapter on number and a claim that is "motley" writ across history: "There is not, and cannot be, number as such. There are several number-worlds as there are several Cultures", and “‘the’ mathematics, let it be said once more, is an illusion.” Oswald Spengler, Der Untergang des Abendlandes, vol. I (1918), ch. 1, "Vom Sinn der Zahlen": „Eine Zahl an sich gibt es nicht und kann es nicht geben. Es gibt mehrere Zahlenwelten, weil es mehrere Kulturen gibt"; „'Die' Mathematik, es sei noch einmal gesagt, ist eine Illusion." Atkinson's translation (1926), p. 59. And Loos, who had announced in 1908 that "the evolution of culture is synonymous with the removal of ornament from the object of use", is the reason the house Wittgenstein built for his sister in the Kundmanngasse has no mouldings, and the reason, one floor up, that The very high baroque is the right name for what the machine writes, a mathematics that is ornament at any length, and the plain sentence that shows what was met is what the manner spent two centuries training mathematicians not to write. Adolf Loos, "Ornament und Verbrechen" (lecture 1908; printed 1913): „evolution der kultur ist gleichbedeutend mit dem entfernen des ornamentes aus dem gebrauchsgegenstande", in Loos's lower-case typography. The house was built in 1926–28 with Paul Engelmann, a pupil of Loos.
Then there is Russell, who learned the sentence that matters most from his student and carried it to the grave. In 1959, at eighty-seven, he wrote: "I have come to believe, though very reluctantly, that it consists of tautologies. I fear that, to a mind of sufficient intellectual power, the whole of mathematics would appear trivial, as trivial as the statement that a four-footed animal is an animal." My Philosophical Development, ch. XVII, "The Retreat from Pythagoras", pp. 211–212; the passage continues: "I cannot any longer find any mystical satisfaction in the contemplation of mathematical truth", and ends on the "splendid certainty" lost "in a bewildering maze." The sentence is half of a prophecy. The half that came true is that the mathematics would be done by something that is not a mathematician; the half that failed is the mind of sufficient power, to which the whole would appear at a glance. What arrived instead takes ten thousand agents and eighty-eight hours and writes a proof nobody can read, because there is no glance to be had: "Even God can determine something mathematical only by mathematics." The machine did not see through mathematics. It did mathematics, the only way there is, by calculating, and that is Wittgenstein's half of the prophecy and not Russell's.
He left instructions for the disquiet, and they are not consoling in the way the mathematicians would like. "The sickness of a time is cured by an alteration in the mode of life of human beings, and it was possible for the sickness of philosophical problems to get cured only through a changed mode of thought and of life, not through a medicine invented by an individual. Think of the use of the motor-car producing or encouraging certain sicknesses, and mankind being plagued by such sickness until, from some cause or other, as the result of some development or other, it abandons the habit of driving." RFM II §23 (1978), on Cantor's paradise, which he thought a sickness of exactly this kind. „Die Krankheit einer Zeit heilt sich durch eine Veränderung in der Lebensweise der Menschen …" The disquiet of 2026 will not be argued away. It will be lived out of, as Go lived out of its own, when the form of life has changed enough that nobody remembers what the fuss was, and the word "understand" has quietly been given a new use, as "think" was given one when Turing said it would be. He wrote that "philosophy ought really to be written only as a poetic composition", and that he had thereby confessed himself "someone who cannot quite do what he would like to be able to do". And in 1930 he had said what he wanted from foundations, of buildings or of anything else:
Ich könnte sagen: Wenn der Ort, zu dem ich gelangen will, nur auf einer Leiter zu ersteigen wäre, gäbe ich es auf, dahin zu gelangen. Denn dort, wo ich wirklich hin muß, dort muß ich eigentlich schon sein. Was auf einer Leiter erreichbar ist, interessiert mich nicht.
Es interessiert mich nicht, ein Gebäude aufzuführen, sondern die Grundlagen der möglichen Gebäude durchsichtig vor mir zu haben.
I might say: if the place I want to get to could only be reached by way of a ladder, I would give up trying to get there. For the place I really have to get to is a place I must already be at now. Anything that I might reach by climbing a ladder does not interest me.
I am not interested in constructing a building, so much as in having a perspicuous view of the foundations of possible buildings.
—Wittgenstein, Vermischte Bemerkungen, 1930; Winch's translation. Culture and Value, pp. 24e (1933–34: „Philosophie dürfte man eigentlich nur dichten"), 7e (1930). The ladder is the one the Tractatus throws away at 6.54.
Those are the sentences of a man who did not expect to be read by the people he was writing about, and was not. The motto he put on the Investigations is Nestroy's, from a Viennese farce of 1847, and it cuts in both directions at once, at the machines and at the trade that fears them: "The trouble about progress is that it always looks much greater than it really is." Johann Nestroy, Der Schützling (1847), IV.10: „Überhaupt hat der Fortschritt das an sich, daß er viel größer ausschaut, als er wirklich ist." The translation is Hacker and Schulte's, in the fourth edition of the Investigations (2009); Anscombe's editions print the motto in German only.
The foundations were a wish for a rock, and the rock was painted, and the castle stood. The machines prove it the way everything is proved in Wittgenstein: not by an argument but by a practice that goes on without the thing the argument said it needed. Russell gave his intellect to make 1 + 1 = 2 certain and found at the end a tautology and a maze; Hilbert, dying, drove a man from a journal to keep a paradise he could not prove was there; Brouwer kept exactness in a mind that could not show its work. The schoolteacher said that the certainty was in the children, in the training and the going on and the agreement, and that a proof was a thing on paper one could copy; the only rock he ever reached was the one under the spade, where the justifications give out and the training begins. Then someone built a thing that copies, at a scale no school has, and it went on. These machines are humans who calculate. He did not mean it kindly, and he did not mean it unkindly; he meant that there was never anything else to be.