A.W. MOORE
The philosophy of mathematics was of colossal importance to Wittgenstein. Its problems had a peculiarly strong hold on him; and he seems, at times, to have thought that it was in addressing these problems that he produced his greatest work. Thus Rush Rhees recounts that, in the mid‐1940s, when John Wisdom had written a short paragraph on Wittgenstein for inclusion in a biographical dictionary, he (Wisdom) sent the paragraph to Wittgenstein for comments, whereupon Wittgenstein recommended just one change, namely to add at the end: “Wittgenstein’s chief contribution has been in the philosophy of mathematics” (see Monk, 1990/91, p.466; and 2007, p.273 and n.2).
Yet Wittgenstein’s writings in the philosophy of mathematics stand in a curious relation to this self‐assessment. By 1938 he had written an early version of his masterwork Philosophical Investigations, the second half of which was on the philosophy of mathematics (see also Chapter 1, WITTGENSTEIN’S PHILOSOPHICAL DEVELOPMENT). This material did not however survive into the version of Philosophical Investigations that was eventually published after his death. Instead it appeared, modified in various ways, along with notes that he wrote during World War II, as Remarks on the Foundations of Mathematics, another posthumous publication, assembled by his literary executors. Apart from this there are scattered remarks in other material that he had produced earlier, while his ideas were beginning to take shape, and there are notes taken by some of those who attended his lectures on the philosophy of mathematics (see especially PR, PG, and LFM). None of this was submitted for publication by Wittgenstein himself. And, be his own relation to this body of work as it may, its early reception, when it did appear (starting with RFM in 1956), was largely dismissive, if not positively contemptuous. Michael Dummett, in a passage that was not at all unrepresentative, wrote:
Many of the thoughts [expressed in Remarks on the Foundations of Mathematics] are expressed in a manner which the author recognized as inaccurate or obscure; some passages contradict others; some are quite inconclusive; […] other passages again, particularly those on consistency and on Gödel’s theorem, are of poor quality or contain definite errors.
(Dummett, 1959, p.166; see also Monk, 2007, sec.IV)
My own view is that Wittgenstein’s reflections on the philosophy of mathematics, for all the disarray with which they have been passed on to us, can indeed be seen as incorporating some of his greatest insights; and that the opposition that they provoked when they first appeared, and that they have continued to provoke since, is due largely to the combined difficulty and radicalness of these insights. My chief concern in this chapter is not however to substantiate that view. Instead I want to do something more oblique. I want to look at some questions of Wittgensteinian exegesis on which his philosophy of mathematics has a unique and critical bearing. These are questions in the first instance about his philosophy of philosophy (see also Chapter 13, PHILOSOPHY AND PHILOSOPHICAL METHOD).
Wittgenstein famously says that “philosophy leaves everything as it is” (PI §124). Immediately after saying this he makes the same point specifically in connection with mathematics: “It also leaves mathematics as it is” (§124). He is expressing his well‐known conviction that the proper role of philosophy is to save us from the confusions into which we fall when we misconstrue the functioning of our own language. Philosophy should not try to modify our language, still less to take issue with anything that is said in the proper exercise of it. It should just guide us to a clear view of it. In particular, philosophy has no business challenging any of the developments that mathematics has undergone. Its business is to challenge the extra‐mathematical deliverances of those who, when they reflect on the nature of these developments, or on the nature of any other part of mathematics, misperceive the workings of the concepts being exercised and then mangle them in their struggle to provide a coherent account of what is going on there. (See PI §§89–133, and PG 369.)
So far, so familiar. So far, one might think, so reasonable. But here is the rub. Wittgenstein himself, in his own reflections on the nature of mathematics, makes claim after claim to outrage the working mathematician. Sometimes the mathematician’s complaint would be that Wittgenstein misunderstands what it is to practice mathematics. Sometimes the complaint would be that he misunderstands the mathematics itself. Thus Gödel, in a letter to Abraham Robinson, dismisses Wittgenstein’s remarks on his (Gödel’s) famous incompleteness theorem on the grounds that they arise from “a completely trivial and uninteresting misinterpretation” (see Dawson, 1989, p.89). I do not myself believe that Wittgenstein misunderstood Gödel’s theorem. (For a helpful corrective see Floyd, 2001; and Kienzler and Grève, 2016. See also Moore, 1998, sec.5, for reflections of my own on this matter.) But even if I am right about that and Gödel is wrong, what about all the rest of what Wittgenstein says to give mathematicians umbrage? Time after time he seems, either through incompetence or by design, to violate his own philosophical precept that philosophy should leave mathematics as it is. But how credible is it that he should really have done so – so often, and so flagrantly? I hope I am not exhibiting undue deference to the master by registering my skepticism on this score. It is a question of how plausible it is that someone should be as steeped in such a distinctive conception of philosophy as this and then not be sensitive to ways in which his own philosophical work, including what may even be some of his greatest philosophical work, flies in the face of it. If Wittgenstein makes claim after claim to outrage the working mathematician, then the explanation had surely better not be either that he is simply oblivious to the fact or that, despite his own philosophical scruples, he is bent on reform of mathematical practice.
In fact, of course, another explanation is available, and one that looks entirely consonant both with his conception of philosophy and with his practice of it. There is mathematics; and there is what people are inclined to say about mathematics. Wittgenstein’s target consists of confusions that beset the latter. And it would not be the least surprising if the people most prone to these confusions were mathematicians themselves. They are the people most likely to have opinions about the nature of mathematics, and there is no reason whatsoever why, in arriving at these opinions, they should be any less susceptible to the kinds of confusions that Wittgenstein is concerned to combat than the rest of us. Just the opposite in fact. “A mathematician is bound to be horrified by my mathematical comments,” Wittgenstein writes, “since he has always been trained to avoid indulging thoughts and doubts of the kind I develop” (PG, 381–2). Again: “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, emphasis in original).
That last comment has a quite specific target. In an article in Mind the celebrated mathematician G.H. Hardy wrote that “[the] truth or falsity [of mathematical theorems] is absolute and independent of our knowledge of them,” adding that “in some sense, mathematical truth is part of objective reality” (Hardy, 1929, p.4, emphasis in original). The same view has been stoutly defended more recently by another celebrated mathematician, Roger Penrose, who describes the way in which, in mathematics, “human thought [seems to be] […] guided towards some eternal external truth – a truth which has a reality of its own, and which is revealed only partially to any one of us,” and who, in the old debate about whether mathematics is invention or discovery, accordingly places himself, with only minor qualifications, in the latter camp (Penrose, 1989, pp.95–6). This view is an anathema to Wittgenstein. “The mathematician is an inventor,” Wittgenstein insists, “not a discoverer” (RFM I §168; cf. RFM II §38). Not that Wittgenstein’s stance on this issue puts him at odds with all mathematicians. There are distinguished mathematicians who have been as keen as he is to reject the picture of mathematics as discovery (see e.g., Cohen, 1967, and Davies, 2003). When mathematicians reflect on the nature of their own discipline, some of them incline one way in this debate, some of them the other. What can plausibly be said to put Wittgenstein at odds with all mathematicians, or at least with all but the most atypical of mathematicians, is not his stance on this issue, but the way in which he maintains it.
Eschewing the picture of mathematics as discovery, he denies that the propositions of mathematics have a subject matter in anything like the way in which the propositions of physics or geography have a subject matter. Rather, in establishing the truth of a mathematical proposition, we are forming new concepts, establishing new ways of making sense of things, contributing to “a network of norms” (RFM VII §67). And, in asserting a mathematical proposition, we are not saying how things are, still less saying how things are independently of us; we are enunciating one of our rules of representation (see e.g., M 72). It follows, for Wittgenstein, that we need to look to the proof of a mathematical proposition to find out what was being proved, and hence that, in one sense of “understand,” we cannot really be said to understand a mathematical proposition unless we are in command of a proof of it (e.g., RFM V §§42–6, and PG 369–76; cf. PI §578). That is certainly at odds with what the typical mathematician thinks. Very few people are in command of the proof of Fermat’s last theorem, for instance, but most mathematicians would feel no compunction about crediting anyone who has mastered basic high‐school mathematics with an understanding of what the theorem states, in any but an absurdly and unhelpfully demanding sense of “understanding.” Or consider Goldbach’s conjecture. At the time of my writing this, the conjecture has been neither proved nor disproved. Yet most mathematicians would not think twice about saying that the conjecture is already fully intelligible, again in any but an absurdly and unhelpfully demanding sense of “intelligible.” How else, the mathematician is liable to ask, can anyone be engaged in trying to settle it, something that plenty in the profession are, indeed, engaged in trying to do? (Not that Wittgenstein is unaware of this objection, incidentally: see e.g., RFM VI §13; and cf. PI §578. He acknowledges the less demanding senses of “understanding” and has characteristically helpful comments to make about these. Nevertheless, he continues to insist on the significance of the more demanding sense.)
It is plain, then, that there is a crucial role to be played in Wittgenstein’s philosophy of mathematics by the distinction between what the mathematician says when strictly engaged in mathematical practice and what the mathematician says, however instinctively, and with however little sense of departure from such practice, when not so engaged. Wittgenstein notes that mathematicians themselves are alive to this sort of distinction. There is a reference in one of his lectures to the way in which mathematicians look upon “interpretations of mathematical symbols [as] […] some kind of gas which surrounds the real process, the essential mathematical kernel” (LFM 1). Here he is once again echoing Hardy, who, in the article from which I have already quoted, defines “gas” as “rhetorical flourishes designed to affect psychology, pictures on the board in the lecture, devices to stimulate the imagination of pupils” (Hardy, 1929, p.18). Elsewhere, in a similar vein, Wittgenstein distinguishes between what he calls the “calculus” and what he calls the surrounding “prose” (WVC 149). “Prose” is perhaps a more suitable term than “gas,” because it is not pejorative (or at any rate, not relevantly pejorative). Wittgenstein never suggests that there is anything wrong with such prose in itself. Nor should he. As the quotation from Hardy testifies, the prose that surrounds the calculus may play an indispensable heuristic role. The point, however, is that it is the prose that will harbor any confusions of the kind that Wittgenstein is concerned to combat. It is the calculus, and the calculus alone, that can be regarded as sacrosanct.
The question that will primarily concern me is how robust this distinction is. It is clear that Wittgenstein would count, say, a proof of the irrationality of √2 as part of the calculus and a claim to the effect that there is therefore at least one gap in “the everywhere dense rational points” as part of the prose (e.g., PG 460 and, more generally, PG 460–74). But why? By what criteria?
We had better not say, what the word “calculus” might encourage us to say, that authentic mathematics comprises all and only the formal proofs that belong to some formal system or other. That is both too broad and too narrow. It is too broad because some such proofs, indeed all but an infinitesimal minority of such proofs, while they may be of mathematical interest in their own right and thus apt objects of mathematical study, are too complex to have a place in real mathematical practice. In fact Wittgenstein even balks at dignifying them with the label “proofs” (see RFM III §§1–62 passim). Here we see that Wittgenstein is not thinking of authentic mathematics as an idealization of whatever engages the working mathematician; it is itself what engages the working mathematician. It has no features that the working mathematician cannot in practice recognize it as having (cf. RFM III §1, and PI §126). But this also helps to explain why authentic mathematics comprises not only less, but more, than is indicated in the characterization above. It is, as Wittgenstein puts it, “a MOTLEY of techniques of proof” (RFM III §46, capitalization in original). It comprises the many varied procedures whereby mathematicians actually establish their results; and what survives in the formal proofs of any given formal system is liable to abstract from differences between these procedures.
But this last point merely serves to reinforce the concerns we might have about how robust the distinction between the calculus and the surrounding prose is. For the differences in question, between these various proof procedures, might well be thought to include the flourishes, pictures, and other devices to which Hardy alludes. Or if not, why not?
At one point Wittgenstein says something that may appear to settle the matter in a very simple and neat way:
Mathematics consists entirely of calculations.
In mathematics everything is algorithm and nothing is meaning; even when it doesn’t look like that because we seem to be using words to talk about mathematical things. Even these words are used to construct an algorithm.
(PG 468, emphasis in original; cf. TLP 6.2ff, and PR §157)
Here the suggestion is that anything that a mathematician says that is not a contribution to the establishment or implementation of an algorithmic procedure for manipulating symbols is a part of the accompanying prose. Moreover, anything that a mathematician says to intimate that there is more to authentic mathematics than that – in particular, that these symbols are related to an independent reality in the way in which the words and phrases in an empirical proposition such as “she walked to the station” are related to an independent reality – is not just a part of the accompanying prose; it is a breeding‐ground for all the confusions that beset the philosophy of mathematics and it can legitimately be challenged by the philosopher. This applies to the case considered earlier. Suppose a mathematician says, “We know exactly how things would have to be in mathematical reality for Goldbach’s conjecture to be true or false. What we do not know is which of the two it is.” This is paradigmatic prose. Wittgenstein is well within his rights, on his own principles, to take issue with it.
This all appears relatively straightforward. But now consider: must even talk of truth and falsity themselves count as part of the prose? Wittgenstein seems to think so. In his discussion of Gödel’s theorem, in Appendix III to Part I of Remarks on the Foundations of Mathematics, he urges that our ascription of truth or falsity to mathematical propositions rests on nothing more than their superficial grammatical similarity to other propositions, a similarity that we can readily imagine away: it is not an integral part of the mathematics itself (cf. RFM IV §§15–16, and RFM V §13). And he further insists that, if we are going to “play the game of truth functions” with mathematical propositions (cf. RFM I III §2, and PI §136), then we had better understand all ascriptions of truth or falsity to them as relative to some formal system. For a mathematical proposition can count as true or false only insofar as something can count as asserting it, and the only thing that can count as asserting a mathematical proposition is producing it as the result of a proof in such a system (RFM I III–§6; cf. PG 366–8). Again the message seems clear: the concepts of truth and falsity have no purchase in mathematics beyond certain analogies that strike us when we compare mathematical practices with practices of other kinds; all that sustains application of the concepts within mathematical practice is the obtaining of certain proof‐relations between mathematical propositions and formal systems.
Yet what if we shift our attention from Gödel’s theorem to Tarski’s theorem – that arithmetical truth resists being defined in a certain way? The notion of truth involved here goes beyond provability in any given formal system. So is this not a case in which a fully fledged conception of truth has to be seen as part of the calculus itself, not just as part of the prose? (For related discussion see Steiner, 2001, and Floyd, 2001, sec.III.)
Or consider the law of the excluded middle. We naturally acquiesce in this law when we consider mathematical propositions. To take the stock example: we naturally assume that any given sequence of digits either occurs somewhere in the decimal expansion of π or does not. Wittgenstein expresses reservations about this, which make clear that he sees this and other such assumptions as a contribution, not to any calculus, but to the accompanying prose (e.g., RFM V §§9–28). He again reminds us that asserting a mathematical proposition is not a way of saying how things are; it is a way of stating a rule. This makes the claim that any given sequence must either occur somewhere in the decimal expansion of π or not do so akin to the totally unwarranted claim that either “The opening move shall be a pawn move” or “The opening move shall not be a pawn move” must be a rule of chess. (Of course, one could insist that the law of the excluded middle had application only to propositions whose assertions were a way of saying how things are. This would leave one free to say that Wittgenstein’s critique leaves the law completely unchallenged. There are times when Wittgenstein himself suggests that we should adopt this stance: see e.g., RFM V §17; and cf. PR §173. But it would be little more than a terminological stance. It would not gainsay the fact that his critique does present a challenge to the law if the law is construed as having application wherever “the game of truth functions is played” – which is how I am construing it.) The problem for Wittgenstein is that, although some mathematicians have themselves had reservations about the law of the excluded middle, most notably Brouwer, it is hard to see why such reservations do not count as reservations about standard mathematical practice. Mathematicians standardly adopt classical logic, including the law of the excluded middle, when they are establishing and implementing their algorithmic procedures. How is this a fact about extra‐mathematical prose and not a fact about – precisely – their establishment and implementation of algorithmic procedures?
We can turn to the infinite for a third example. Wittgenstein is very uncomfortable with the way in which set theorists claim to have shown that some infinite sets are bigger than others, as though they were astrophysicists claiming to have shown that some distant galaxies are bigger than others. He writes: “The dangerous, deceptive thing about [such an idea] […] is that it makes the determination of a concept – concept formation – look like a fact of nature” (RFM II §19; cf. PG 287). Again he would say that he is casting doubt on the prose surrounding the calculus. Again the concern is that he is casting doubt on the calculus itself. That some infinite sets are bigger than others would be accepted by any orthodox set theorist as an unassailable result of set theory.
There is a fourth example, which might appear as compelling as any. In fact I think that Wittgenstein has ways of addressing it – at least in his own terms – that cannot be extended to the other three examples, though it is worth a digression to see why. The example concerns consistency. Like truth, consistency appears to have a substantive role to play in mathematics, a role whereby its ascriptions are answerable to the investigator‐independent layout of mathematical reality. If a mathematician asserts that Zermelo–Frænkel set theory is consistent, for instance, say as a prelude to proving that the continuum hypothesis is independent of it, then his or her assertion seems to be at the mercy of whether Zermelo–Frænkel set theory is consistent; of whether there is in fact, quite independently of what he or she or any of the rest of the mathematical community might be disposed to say about the matter, a set‐theoretical proposition that admits of both a proof and a refutation within the theory. For that matter, the very idea that mathematics consists of algorithmic procedures seems to entail that there is an issue for mathematicians, if not about the truth or falsity of their propositions, at least about the consistency or inconsistency of their procedures, where the consistency or inconsistency of a procedure is a mathematically investigable feature of it that is quite independent of mathematicians themselves. There seems, then, to be a notion at work within mathematics – within mathematics, not just within the surrounding prose – which embodies the very picture of mathematics, as answerable to an independent reality, that Wittgenstein is concerned to repudiate.
As I said, I think Wittgenstein has ways of addressing this fourth example. What are they? They are largely a matter of his biting various bullets. Among these are bullets that he notoriously does bite and bullets that I think he would be happy to bite. To begin with the latter: I think he would simply accede to the idea that, when consistency features within mathematics, it is no more answerable to an independent mathematical reality than any other mathematical notion. Thus we are at just as much liberty to declare Zermelo–Frænkel set theory to be consistent as we are to declare the successor function to be one:one. That declaration can serve as a piece of legislation, a contribution to an algorithmic procedure or to a family of algorithmic procedures. Of course it is natural to protest, “But what if Zermelo–Frænkel set theory is not consistent?” Here, however, Wittgenstein can precisely appeal to his distinction between calculus and prose. For there are two corresponding ways of taking this question. If it is taken as a question within mathematics, then there is plenty to be said in response to it, for instance that if Zermelo–Frænkel set theory is not consistent, then it is finitely axiomatizable. This poses no threat whatsoever to Wittgenstein. Taken in this way, the question is just an invitation to do more mathematics, mathematics that can sit alongside whatever mathematics we might do on the strength of our declaration that Zermelo–Frænkel set theory is consistent. If the question is taken as a contribution to the prose, on the other hand, then it adverts to the possibility that we shall one day acknowledge both a proof within Zermelo–Frænkel set theory and a refutation within Zermelo–Frænkel set theory of one and the same proposition. And this is where we find the bullets that Wittgenstein notoriously does bite. He is prepared to meet what he calls “the superstitious dread and veneration by mathematicians in the face of contradiction” (RFM I III §17) with a studied nonchalance. His stance, roughly, is that, as long as we do not find any such conflict in our procedures, we do not need to worry about the possibility, and, if ever we do find such a conflict in our procedures, then we can decide how to proceed (RFM VII §§12ff, and PG 303–5). He is even prepared to countenance our proceeding by simply circumventing the conflict. In one of his lectures he says: “If you can draw any conclusion you like from [a contradiction], […] I would say, ‘Well then, just don’t draw any conclusions from a contradiction’” (LFM 220). This may seem literally laughable: it is reminiscent of the Tommy Cooper joke in which a patient tells his doctor that his arm hurts whenever he raises it and the doctor replies, “Well then, don’t raise it.” But actually, Wittgenstein’s nonchalance does not seem untoward once we rid ourselves of the idea that mathematical propositions are related to an independent reality in the way in which empirical propositions are. As long as we think of mathematicians as establishing and implementing algorithmic procedures, then Wittgenstein’s nonchalance can simply be seen as his way of sanctioning mathematicians’ continued use and periodic revision of any given procedure until such time as it no longer serves their purposes. And lest it seem utterly fanciful to suppose that mathematicians should work with inconsistent procedures even while fully aware of the inconsistencies, worse still that they should do so by simply negotiating the inconsistencies as they see fit, let us not forget that this is precisely what they did in the seventeenth and eighteenth centuries when the notion of an infinitesimal difference, as both equal to zero and not equal to zero, still informed work on the differential calculus. (See Moore, 1990/2001, ch.4, secs.1–2.)
The fourth example seems to me not telling, then. But the other three remain – as no doubt do variants on them.
There is one obvious way for Wittgenstein to rise to this collective challenge. However robust the distinction between the calculus and the surrounding prose, the prose may infect the calculus; or, more strictly, the prose may infect how we couch the calculus. Thus in all three of the troublesome examples considered in the previous section Wittgenstein can say that the trouble lies, not in the calculus itself, but in our choice of certain vocabulary to express it: “true,” “false,” “either… or…,” “bigger.” This vocabulary has a use in non‐mathematical contexts that resonates loudly from there. And it harbors a certain view of the calculus that is strictly inessential to it. So the very fact that we say that some infinite sets are “bigger” than others, to take that example, is fair game for Wittgenstein’s animadversions on what goes beyond the calculus, however securely lodged within the calculus the result itself may be. There is even an issue about whether we should use the word “infinite” in a strictly mathematical context. “Ought the word ‘infinite’ to be avoided in mathematics?,” Wittgenstein’s interlocutor asks at one point. “Yes,” Wittgenstein replies, “where it appears to confer a meaning upon the calculus; instead of getting one from it” (RFM II §58).
Moreover – this is a separate point – it is Wittgenstein’s firm conviction that, if only we were to recast much of the mathematics that most captivates us, by removing the offending vocabulary in favor of some purpose‐specific mathematical jargon (cf. PG 468–9), then interest in it would wane. It would lose what Wittgenstein calls its “schoolboy charm” (LFM 16). We feel a certain heady pleasure when we are told that some infinite sets are bigger than others. We feel considerably less pleasure when we are told that certain one:one correlations yield elements that are not in their ranges. Though Wittgenstein’s principal concern is to combat philosophical confusions attending mathematics, he does also see it as part of his mission as it were to cut mathematics down to size. (The two things are related. This is for reasons that we have just seen. As Wittgenstein nicely puts it at one point: “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.)” (PG 381).)
One might wonder how even this consists with Wittgenstein’s non‐revisionary insistence that philosophy should leave mathematics as it is. But there is one marvelous remark in which he makes clear how they consist. The remark is proffered in response to Hilbert, who famously said in connection with the work by Cantor in which transfinite set theory was founded, “No one shall be able to drive us from the paradise that Cantor has created for us” (Hilbert, [1926] 1967, p.376). Wittgenstein replies:
I wouldn’t dream of trying to drive anyone from this paradise… I would do something quite different: I would try to show you that it is not a paradise – so that you’ll leave of your own accord. I would say, ‘You’re welcome to this; just look about you.’
(LFM 103)
Elsewhere he puts the point by saying: “What I am doing is, not to show that calculations are wrong, but to subject the interest of calculations to test” (RFM II §62, emphasis in original; cf. LFM 141).
Wittgenstein may appear vindicated then. Although he is keen to warn mathematicians about the dangers of transferring vocabulary from one context to another, and although he knows, indeed intends, that heeding his warning will make them reconsider the value of some of their work, he does not himself want to issue a direct challenge to any of that work.
There is still a problem for Wittgenstein, though. Mathematical use of the vernacular is never just a matter of transferring vocabulary from one context to another – or, if it is, that is not Wittgenstein’s concern. (No harm accrues from the fact that the word “exponent” has a quite different use in mathematical contexts from the use it has in non‐mathematical contexts.) When set theorists describe some infinite sets as “bigger” than others, they are not just choosing an arbitrary label which happens to have a use elsewhere. They take themselves to be appropriating a concept with which we are already familiar and extending its application. In fact that is precisely what gives Wittgenstein pause (PG 464). But why does it give him pause? Wittgenstein himself urges that mathematics involves the formation of concepts (RFM VII §67). Why should this formation of concepts not include the modification of concepts as well as their creation? And if it does, then the use of the relevant vocabulary will after all be essential to what the mathematicians are doing. To claim that that vocabulary can be peeled off from the underlying calculus is to issue a direct challenge to their work.
For a clear example of what I have in mind, consider Wittgenstein’s reluctance, which we noted earlier, to dignify all the formal proofs of any given formal system with the label “proofs”. That cannot but be heard as a challenge, not only to each of the systems, but also to proof theory, the branch of mathematics in which our informal notion of a proof is at once idealized and codified.
Or consider this:
Does the relation m = 2n correlate the set of all numbers with one of its subsets? No. It correlates any arbitrary number with another, and in that way we arrive at infinitely many pairs of sets, of which one is correlated with the other, but which are never related as set and subset.
(PR §141, emphasis in original, “class” replaced by “set”)
Here Wittgenstein is balking at the standard way of couching the result that each natural number can be paired with its double. The standard way of couching this result makes reference to a one:one correlation between the complete set of natural numbers and one of its proper subsets, that which contains only the even natural numbers. But Wittgenstein refuses to sanction this use of the word “set,” if it is understood as involving application of a single concept to both the finite case and the infinite case: if we do talk of both “finite sets” and “infinite sets,” then these two uses of “sets” must be understood as having fundamentally different grammars from each other (PG 463–5). This is certainly a bold stance. But the issue for us is not whether it is bold or not; nor whether it is justified or not; nor even whether it is a stance to which most set theorists would take exception. The issue, for us, is whether it is a direct assault on set theory. And surely it is.
Wittgenstein could of course beg all the relevant questions and insist that the very challengeability of what he is challenging, in his capacity as a philosopher, ensures that it is not an essential part of any authentic mathematics. That would be uninteresting – save in so far as it highlights what may in any case be a circle that afflicts his philosophy. Wittgenstein believes that, qua philosopher, he is entitled to take issue with that which perverts or is in danger of perverting either mathematical thinking or our thinking about mathematical thinking, but that he is not entitled to take issue with mathematical thinking itself. He can take issue with the prose, but not with the calculus. The apparent circularity is this: there is no way, in practice, of respecting this distinction without having a grasp of the calculus; and there is no way of acquiring a grasp of the calculus without being suitably sensitive to authentic mathematical practice; and there is no way of being suitably sensitive to authentic mathematical practice without knowing how to screen those parts of mathematical practice that do not constitute proper exercise of the calculus; and there is no way of knowing which parts to screen without already being able to respect the original distinction.
I do not claim that this apparent circularity is vicious. I do not even claim that it is real. Each step in the sequence can be disputed. For example, Wittgenstein might say that we can tell which parts of mathematical practice to screen because there is a distinctive discomfort that eventually manifests itself when the prose gets out of control (cf. PI §§54 and 123). Perhaps there is – though even then, of course, “distinctive” is the operative word, with its own threat of circularity. (Mathematicians can display plenty of discomfort when they are wrestling with bona fide mathematical problems.) The point, however, is that whether the circularity is real or not, the distinction between calculus and prose is not just a piece of theory for Wittgenstein. It is a tool that he needs to be able to implement in practice, in his attempt to rid the philosophy of mathematics of the confusions that beset it. And the mere threat of such circularity is surely enough to disturb the confidence that proper handling of this tool requires. It is surely enough to call into question the very project of trying to approach the philosophy of mathematics with that self‐conscious detachment which his philosophy of philosophy demands. In sum: there is a real practical issue for Wittgenstein about the effectiveness of this distinction that is so crucial to his philosophy of mathematics, the distinction between calculus and prose.
Is part of the problem excessive censoriousness on Wittgenstein’s part? How would it be if his attitude were much more one of laissez‐faire, so that, instead of regarding whatever could be seen as incidental to any given algorithmic procedure as part of the prose, he regarded whatever could be seen as a feature of some algorithmic procedure, however incidental, as part of the calculus? This would mean that set theorists could just be left to get on with their business, be the interest of the exercise as it may. The only point at which philosophers would need to get involved would be the point at which someone reflecting on the exercise began to mishandle the conceptual apparatus involved in it and got into a muddle as a result. The threat of circularity just considered would remain, but it would be mitigated by the fact that the distinction between calculus and prose would need to be drawn much less frequently: only when there was a troublesome uncertainty about how to proceed and the issue was whether it was an uncertainty calling for mathematical insight or an uncertainty calling for philosophical clarification. No doubt this would leave Wittgenstein himself feeling uneasy, but would it be contrary to the strict letter of anything in his philosophy of mathematics?
Well, yes, it would. There is an issue about the application of mathematics that I have not mentioned at all so far. When Wittgenstein says that mathematics involves algorithms, he means to specify a necessary condition for something to count as part of mathematics, not a sufficient condition. He would not reckon the mere algorithmic manipulation of symbols a part of mathematics unless those symbols also had a use in non‐mathematical contexts. Wittgenstein puts the point as follows:
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‐games 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.
(RFM V §2, emphasis in original; cf. RFM V §25)
Let us not pause to consider what tension there might or might not be between this demand that mathematical vocabulary have a use outside mathematics and the worries that we saw Wittgenstein express earlier about the mathematical use of any vocabulary that has a use outside mathematics. Of more immediate concern is the fact that what we have here is essentially an assault on the very idea of pure mathematics. And it immediately furnishes a new complaint for Wittgenstein to level against transfinite set theory. For, to date, there is no serious use of any of set theory’s heavy‐duty measuring apparatus in non‐mathematical contexts.
This assault on the idea of pure mathematics, which is strictly independent of anything that we have considered hitherto, seems to me problematical for a number of reasons. Here are two. First, even if the algorithmic manipulation of symbols needs to have application to count as a proper part of mathematics and not just a game, it is not clear why it needs to have application outside mathematics as opposed to elsewhere within it. Second, a branch of mathematics often remains unapplied until well after its development, so that even those who think that the application of mathematics is what gives it its point should acknowledge the importance of allowing unapplied mathematics to have free rein. Both of these points, whatever general force they might have, have specific force in the case of transfinite set theory, which is both vigorous in its application to other branches of mathematics and relatively young. There is far more to be said about both points, obviously. But there is also a third point that needs to be made in this context, of even greater significance, namely that the assault on the idea of pure mathematics is an assault on mathematicians’ very self‐image. Hardly any mathematician would agree with Wittgenstein that “it is essential to mathematics that its signs are also employed in mufti.” This of course brings us back to square one. For yet again Wittgenstein has made a claim in his philosophy of mathematics to which the typical working mathematician would take exception. This in itself need not worry him. What the typical working mathematician is prepared to count as mathematics was always going to be a clear candidate for classification as prose rather than calculus. But it is another stark reminder of how subversive Wittgenstein is prepared to be in his critique of what mathematicians themselves actually think and say; and of how hard he therefore makes it for himself to draw the distinctions that he needs to draw in order to maintain his precept that philosophy leaves mathematics as it is. There is much that lies deep in the territory within and around mathematics that Wittgenstein’s philosophy does not leave as it is. It has been one of the main burdens of this chapter to show this.
It remains for me to make one very brief but very significant final point. My aim has been to highlight a tension that I claim to have discerned between Wittgenstein’s philosophy of mathematics and his philosophy of philosophy. But even if I have succeeded in this aim, it is a further question where the fault lies. Despite the various reservations that I have voiced about Wittgenstein’s philosophy of mathematics, there is much in it that seems to me to embody insights of the most profound kind. I think that what we have been witnessing are, in large part, problems with his philosophy of philosophy. In particular, I think that we have been seeing manifestations of a continual struggle that his philosophy of philosophy has with its own highly distinctive brand of conservatism. But that, as they so often say, is a topic for another occasion.