In Philosophical Investigations §133, Wittgenstein writes that his aim is to make philosophical problems “completely disappear.” Although mathematics is not explicitly mentioned in this remark, he very likely had in mind the philosophical problems about it too. Under this assumption,Footnote ¹ my primary goal here is to probe the extent to which this aim is attainable. More precisely, my central task is to mount a qualified defense of his contention that it is possible to dissolve, and thereby eliminate, the traditional philosophical problems about mathematics.

This book differs from other reconstructions of Wittgenstein’s views on mathematics by having a distinct agenda. While other available accounts typically aim to be comprehensive,Footnote ² this work doesn’t. It focuses on his later period and especially on the line of thought that substantiates a radical thesis: that philosophy of mathematics after Wittgenstein applied his methods to it appears devoid of its time-honored problems.

Obviously, such a claim may sound alarming to many contemporary philosophers of mathematics. It makes Wittgenstein look like a threat, especially since he is genuinely committed to this kind of ‘destructive’ or ‘eliminativist’ project. (The theme of dissolution is not a sporadic occurrence in his thinking, but a central motif: it resurfaces in the post-tractarian period,Footnote ³ after playing a prominent role earlier in the Tractatus.)Footnote ⁴ In fact, a professional philosopher of mathematics may even hope that such a project will not succeed. Who would be willing to entertain the thought, let alone accept it, that they are “under a spell” (Ms-158, 37 r) when trying to solve these problems? That these conundrums, on whose discussion one builds one’s entire career, are illusory?Footnote ⁵

Yet note that his very contention, admittedly extreme and apparently ‘antiphilosophical,’ is, after all, a philosophical one. As such, it is bound to be open to further philosophical examination – something that Wittgenstein himself actually envisaged.Footnote ⁶ Hence, even if one realizes the magnitude of the threat and becomes worried, one can rest assured that the game is far from over, so to speak. Many challenging questions can and should be asked about his ambitious ‘dissolution solution’ of these problems. Moreover, mathematics is a dynamic discipline, and new philosophical puzzles (about its concepts and proofs) will always appear on the horizon.Footnote ⁷

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As I noted, several reconstructions of Wittgenstein’s thinking on mathematics have been carried out in the past. But I should stress that although the present attempt has benefited from these enlightening works, it does not pursue the same objectives. Most importantly, I relinquished any ambition to provide a comprehensive account of his (early and late) thinking on mathematics. Virtually every analysis in the book is in the service of my primary goal, to support the eliminativist thesis stated at the outset. The aim was simply to understand why the central problems in this field vanish: that is, to show the readers how – rather than merely tell them that – these puzzles (should) dissolve. (This is of course a nod to the famous advice ‘Show, don’t tell’ meant to describe a writing technique, often attributed to Chekhov and Hemingway. It also registers my hope that the secondary literature will begin to take Wittgenstein’s eliminativist idea more seriously – in general, not only about mathematics.)Footnote ⁸

However, demonstrating that this eliminativism is a viable philosophical option depends on first being able to extract it, and articulate it, from his often-opaque remarks. So, the question to ask right at the beginning is whether there is enough substance in his later works to fuel this project. In this regard, reservations have been expressed in the past. For instance, the central work we’ll discuss here, the posthumously published RFM collection, has been described by the eminent logician Georg Kreisel as a “surprisingly insignificant product of a sparkling mind” (Reference Kreisel1958, 158).

I must admit that such harsh judgments are understandable at first glance. Even expert interpreters of Wittgenstein’s other works often wander astray in the labyrinth of his reflections on mathematics. Quite frequently we are not told what to believe, only what we should not believe. Attempts to identify clearly delineated arguments running through the remarks are generally inconclusive. Out of the many questions the reader is presented with, only a few are answered. Just as in the PI, there are sometimes distinctive characters (‘voices’) speaking, and it is a difficult task to tell which one to listen to (if any). In addition, another possible issue is the different status of the works he left behind – some are writings (either in raw manuscript form, or as typescripts), others are notes taken by people in his audiences. Wittgenstein’s literary executors – G. E. M. Anscombe, R. Rhees, and G. von Wright – tell us that only two parts out of the seven that they put together as RFM (i.e., parts I and VI) are “complete reproductions of texts of Wittgenstein’s” (RFM, “Editors’ Preface to the Revised Edition,” p. 32),Footnote ⁹ and consequently should be regarded as more authoritative than others.

The main worry, however, is about the very substance of what we inherited, regardless of how trustworthy we take the source to be. Such a concern is, I suspect, shared by many. It has been sharply summarized by Michael Potter:

These are real difficulties – and my show, don’t tell approach is meant to tackle them head-on. I believe that we should not give up yet; we can be optimistic about answering Potter’s question. As I hope to demonstrate, Wittgenstein’s later works on mathematics, when read charitably, do contain an important philosophical fixed point, that is, a reasonably coherent eliminativist vision.

My confidence is also bolstered by the fact that Potter himself lists two elements to be incorporated into such a reading – “that numbers are not objects, and that arithmetical equations are not tautologies” (Reference Potter, Kuusela and McGinn2011, 136) – ideas that Wittgenstein retained from his early and middle periods. These views (which, indeed, are stated in the ‘negative’ mode) will find a place in my reconstruction. But the reader should be warned, once again, that his contribution will not appear in the form of a comprehensive theory, as is typical in the Western philosophical tradition.

So, what was Wittgenstein for? A full answer will hopefully emerge in due course; a condensed one is as suggested: he designed, though not always explicitly pursued, a series of strategies to dissolve, and thus eliminate, the traditional problems in the philosophy of mathematics.Footnote ¹⁰ He aimed to expose them as pseudoproblems, as ill-posed questions, which appear as genuine difficulties only because the traditional schools uncritically accepted certain presuppositions regarding the working of language. The examination of these deeply entrenched assumptions consists of doing work of a descriptive, and often humdrum, nature. This enterprise, which he calls a ‘grammatical’ investigation (PI §90), amounts to ‘looking’ (at language) and ‘seeing’ (how it works) (PI §66). This is a process through which we are reminded of all-too-familiar linguistic facts and conceptual distinctions.

When it comes to applying this kind of investigation to mathematics, the result is a unique perspective on the relation between these venerable intellectual fields: philosophy no longer appears as a provider of explanatory theories aimed at solving philosophical problems raised by mathematics, but is a ‘grammatical’ endeavor aimed at achieving “complete clarity” (PI §133) about what generates these problems in the first place. And, once this clarity is obtained, the result is the dissolution of these problems. We are liberated from them upon the realization that they are somehow unreal: “You are under the misapprehension that the philosophical problem is difficult, whereas it’s hopeless” (Ms-158, 37 r; in English in original). The ‘therapeutical’ intention of this exercise is obvious, and I shall emphasize it throughout the book. Just as the goal of the doctor is not to be needed, for Wittgenstein the goal of philosophizing is reached when these problems are eliminated: “Thoughts that are at peace. That’s what someone who philosophizes yearns for” (CV 43^e (1944)).Footnote ¹¹

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Let me acknowledge five methodological commitments that guided the work on this book. First of all, what follows is an attempt to articulate Wittgenstein’s position on mathematics post-1936,Footnote ¹² and to do so by relying on substantial textual support. Yet I will sometimes go beyond the letter of the text. I will not suppress my opinions about what it seems to me that Wittgenstein would have said, or, more contentiously, about what he should have said. These elaborations are not motivated by the (laughably arrogant) intention to somehow ‘improve’ his work; I will only assume that some of his terse, often cryptic, remarks become clearer when rephrased. Hence, I will labor under the presumption that he could have said what he (seems to me to have) said in a more transparent – though admittedly less memorable – manner. In addition, I believe that his insights can be better conveyed when reshaped into a more linear train of thought, and when illustrated by more examples from the history and practice of mathematics.

Nevertheless, staying close to the texts was very important, and I will often expound on individual remarks, or parts of them. However, what follows is not primarily an exegetical exercise, nor a detailed, line-by-line, commentary of any portion of his work (on mathematics).Footnote ¹³ Even if I had tried to do this, I still could not offer any guarantee that Wittgenstein really held the views I attribute to him here – so, in this book ‘Wittgenstein’ always stands for ‘Wittgenstein as I understand him.’ But I do entertain the hope that my comprehension of the texts is correctFootnote ¹⁴ and, as far as I can make sense of his key ideas, they seem to me essentially right. Moreover, I believe that these insights are highly original and constitute a major achievement in the history of thought.Footnote ¹⁵ Yet, once again, I take my praising of his work to be compatible with occasionally voicing frustration with the “cryptic, fragmentary, and unfinished” (Anderson Reference Anderson1958, 446) presentation of his ideas. So, they need to be reconstructed properly – and, once this is done, I submit that they pose a formidable challenge to the conceptions of mathematics advanced by the traditional philosophical schools. This is a challenge that, I’m afraid, has barely begun to be recognized now, seventy-five years after his death. With very few (but notable) exceptions, contemporary philosophers of mathematics proceed as if Wittgenstein never addressed the subject.

Second, I will be very discerning about which remarks I include in my discussion. This is so not only because the argument of the book is meant to hit a well-circumscribed target, but also because I generally agree with the commentators who worry that the materials Wittgenstein left behind are of variable quality. In fact, he himself noted that not all remarks he jotted down are equally important – in a suggestive simile: “Of the sentences that I write down here, only the occasional one represents a step forward; the others are like the snip of the barber’s scissors, which must be kept in motion so as to make a cut with them at the right moment” (CV 66^e (1948)).

Indeed, it is quite clear that many of the entries in the manuscripts are such ‘snips.’ Hence, my strategy was to identify the ‘cuts’ and reconstruct his eliminativism from them.Footnote ¹⁶ Thus, not everything one finds in his later works will be, or even can be, accommodated by the present account. I must confess that I sometimes encountered lines whose role in the eliminativist project escaped me. More generally, in some remarks, to put it bluntly, I had no idea what he was talking about; so, I left them out of discussion altogether.

A third feature of my approach is that I read Wittgenstein’s oeuvre with the mindset of a contemporary philosopher of mathematics, that is, someone still struggling with these problems, and willing to understand how he addressed them (if at all). This is in direct contrast with other approaches, for example Fogelin’s, who admits that “one thing I have not tried to show is that the insights found in Wittgenstein’s later philosophy can be integrated with some of the standard forms of philosophizing that flourish on the contemporary scene” (Reference Fogelin2009, 167).

A fourth methodological maxim is that I will avoid tackling questions about the trajectory of Wittgenstein’s thinking over time – with one exception. Understanding how his early, middle, and late ‘phases’ relate to each other is an important exegetical task which, fortunately, is already well accomplished.Footnote ¹⁷ I won’t have much to say on this here, the only exception being Wittgenstein’s attitude toward Cantorian transfinite set theory. Based mostly on my perusal of the Nachlass, I shall suggest that his attitude changed from the PR and PG of the early 1930s to the end of that decade, when he delivers LFM in Cambridge in 1939, change that continued well into the 1940s. As we’ll see in Chapters 6 and 7, he begins as an unabashed revisionist, that is, as someone who would like to ‘correct,’ or revise mathematics, but ends up as a nonrevisionist (of sorts). In the early 1930s he makes harsh pronouncements against set theory; he claims, for instance, that it “builds on … nonsense” (PR §174), and that the textbook proposition “‘between the everywhere dense rational points, there is still room for the irrationals’” is “balderdash” (PG §40, p. 460; see also §41–43). Yet, as time passes, it seems to me that he gradually moves away from this combative stance toward a more nuanced position, reaching a point where he becomes what I shall call a reluctant nonrevisionist. He is a nonrevisionist insofar as he explicitly bars himself from “interfer[ing] with the mathematicians” (LFM 13) and from attempting to “prove that calculations are wrong” (RFM II-62). Nevertheless, on closer inspection, it turns out that his new stance still leaves room for some interference to happen (as RFM II, written in 1937–1938, reveals), albeit in an indirect way. In my reading, he ultimately embraces nonrevisionism, but half-heartedly.Footnote ¹⁸

Finally, the fifth methodological constraint here is related to the previous point and to the first constraint. While, as I said, my discussion relies for the most part on works that Wittgenstein produced after 1936, this investigation of his (non)revisionism will require considering remarks composed from 1929 to circa 1935, or in what has become known as his ‘middle’ or ‘transitional’ period (such as the remarks collected in abovementioned PG and PR). However, among the remarks from this period I plan to invoke only those relevant to his post-1936 conceptions. When exegetical disputes on this relevance arise, I will take sides in them – but only if I judge that this helps me reach the goals of the book: to understand why Wittgenstein believed that he could dissolve the traditional problems in the philosophy of mathematics, and also to show that there are good reasons to think that he was by and large right to believe so.

***

To paraphrase PG §141 (“Philosophy is philosophical problems”), philosophy of mathematics is philosophical problems about mathematics. It is the existence of a battery of such venerable puzzles that led to the emergence and development of various philosophies of mathematics: platonism,Footnote ¹⁹ (anti)realism, logicism, empiricism, conventionalism, intuitionism, formalism-finitism, structuralism, and so on. Consequently, an important issue to address in the book is the relation between these traditional doctrines and Wittgenstein’s own conception. The unmistakable overall impression is that he was dissatisfied with how the subject was treated in the past (and by his contemporaries). For the moment, a simplified and pictorial way of conveying this attitude would be the following: these isms populate a two-dimensional horizontal conceptual plane, unlike his own eliminativist ‘normativist’ position (as I’ll call it, for reasons to be soon clarified), which does not belong to this plane. His view is developed into a third, metadimension, so to speak; for him, to elaborate a philosophy of mathematics is to do something markedly different from what these doctrines have done – namely, to provide explanatory theories, meant to solve the philosophical puzzles raised by mathematics. He did not intend to pursue yet another explanatory project, but carried out a (meta-)eliminative-therapeutic project instead. His overall approach is exactly the opposite of the traditional approaches, in that he “do[es] away with all explanation” (PI §109), by showing that there is no need for it; as PI §126 boldly contends, “there is nothing to explain.”

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The book is divided into ten chapters, including the present Introduction and the Conclusion; each chapter contains several sections. The chapters are not meant to be read in isolation, or in a random order; no chapter (let alone section) exhausts an issue and the motivation for taking up that issue is often found in previous chapters. As I said, I did my best to present a linear narrative, but sometimes returning to the same difficulty in later chapters is unavoidable with such a topic (and such an author). Here is a rough guide to how the material is distributed in chapters and sections.

Chapter 1 is mainly devoted to articulating the eliminativist normativist view that can, and should, be attributed to Wittgenstein. The second part of the label originates in the extremely important remark that “the sentences of mathematics … are normative sentences. And this characterizes their use” (Ms-123, 49 v; emphasis in the original).Footnote ²⁰ This conveys what I take to be Wittgenstein’s fundamental insight: that true declarative mathematical sentences primarily serve (function) to formulate norms, or rules. Understanding what this means is crucial in what follows.

The first chapter signals that the eliminativist aspect of normativism has therapeutical consequences: by eliminating problems, one achieves ‘peace of mind.’ This is a reading that, I submit, puts us on the right path when it comes to understanding his later approach to mathematics. Nonetheless, as far as I can tell, this reading has never been pursued systematically, although it renders Wittgenstein’s take on mathematics a natural complement of his ideas espoused in the Philosophical Investigations – a work whose main aim is, in my view, eliminative-therapeutical too. Indeed, here I intend to present an account of Wittgenstein’s normativist eliminativism in harmony with PI.Footnote ²¹ After all, PI is especially authoritative since it contains a large chunk of text (part I) that he came very close to publishing – with Cambridge University Press. In particular, my attribution of nonrevisionism to him in the later chapters has been decisively influenced by what PI tells us – that philosophy should leave mathematics “as it is” (§124), a stance that I took to be definitive, and to supersede any revisionist-sounding pronouncements, especially those made in RFM part II.

The second chapter further articulates the normativist view, but its central task is to demonstrate its effectiveness. It illustrates the eliminativist aspect of normativism, by showing how it dissolves three central problems in the philosophy of mathematics. Section 2.1 deals with the problem of the specialness of mathematical propositions like ‘2 + 2 = 4.’ Traditionally, they are said to be necessarily true (so more than merely true) and a priori (no empirical information is needed to assess their truth-value). In a word, they are unfalsifiable, incorrigible, so significantly unlike other propositions. I first formulate this problem – roughly, as the task of explaining why such propositions are (or seem) special, while other propositions (e.g., ‘Kampala is the capital of Uganda’) are not so, but are only contingently true and a posteriori. I will explain how this problem arises and then I will show how it can be disposed of. In Section 2.3, I will be working out another dissolution, of the so-called Benacerraf problem. Yet before that (in Section 2.2) I take a detour and clarify the notion of a pseudoproblem, a central component of the eliminativist project. In Sections 2.4–2.6, I deal with a third problem, the traditional ontological question ‘do mathematical objects exist?’, and I explain how Wittgenstein’s normativism eliminates it. There I tackle three realisms: traditional platonism, Fregean logicist realism, and Quinean realism, the latter based on considerations about the indispensable role of mathematics in science and everyday life.

The next two chapters form in some respects a unit. Chapter 3 imagines, on Wittgenstein’s behalf, a possible process by which the basic arithmetical rules emerge; Chapter 4 looks into the status of such rules (are they unique? Can there be alternative arithmetics?). Thus, in Chapter 3 I return to a series of issues raised at the end of Chapter 1. The central one is about the grounds (if any) for the introduction of the basic rules (e.g., 2 × 2 = 4) we currently have in our mathematical books. Note, however, that although the answer to this question sounds very much like providing an explanation, this is not a metaphysical explanation, in the tradition of the philosophical schools (a type of endeavor that Wittgenstein rejects), but an anthropological account, as I will call it. I take Wittgenstein’s pronouncements to suggest, albeit quite obliquely, a kind of genealogy of mathematical norms, that is, a story (ahistorical, but possible) about the formation and development of the rules we currently employ. The discussion in this chapter is confined to the most primitive of these rules, of the kind we find in the multiplication table, as well as other simple multiplications based on them (e.g., 21 × 36 = 756). The more advanced rules, the mathematical theorems established by proofs, will be the subject of Chapter 5.

The discussions in the third chapter will segue naturally into Chapter 4, where I shall be confronting the possible challenges to the rules by ‘skeptics’ – reminding us of the wayward pupil of PI §185, and the much-debated issue of rule-following. In addition to taking up the issue of alternative arithmetics, I will mainly be preoccupied to show how to dispose of the problem of mathematical skepticism (‘what if we have always been wrong that, e.g., 25 x 25 = 625?’ [LFM 102]). At the end, I will explain why the connection between normativism and social constructivism (a doctrine naturally coming up in this context) is rather weak.

In Chapter 5, I shall present and defend some aspects of Wittgenstein’s view on mathematical proof. I see this view as integrated within his normativist eliminativism and most of my efforts will be devoted to elucidating his idea that “the proof creates a new concept” (RFM III-41; my emphasis). I take this to be the gist of the more general pronouncements such as “propositions of mathematics determine concepts” (RFM VII-42; my emphasis), and “mathematics … forms concepts,” where this is “essential” to it (RFM VII-33; first emphasis mine).

Understanding his ‘interactionist’ conception of proof, as I shall call it, is, I believe, the key to grasping what he means by saying that proofs “change the grammar of our language” (RFM III-31). I shall highlight this ‘grammatical’ connection between mathematical proof and mathematical necessity in order to explain how adopting this view of proof is aligned with his therapeutic-eliminativist project. Thus, the main outcome of this chapter will be an explication as to why the puzzle generated by the seemingly absolute power of a mathematical proof – that a proof creates an unbreakable link between the (concepts appearing in the) premises and the (concepts in the) conclusion of a mathematical argument – vanishes.

In terms of examples and illustrations, this chapter and the next three appeal to more advanced mathematics and deal with ‘pure’ mathematics as well. (This is a deliberate attempt to show that Wittgenstein’s views are, or can be shown to be, relevant to more complex mathematical topics, contrary to what some of his early critics claimed; see, e.g., Bernays Reference Bernays1959, 14.) An important concern here will be to draw attention to Wittgenstein’s point that proofs ensure the objectivity of (higher) mathematics: for him, they justify changes in the network of mathematical norms. (New) rules are often introduced on the basis of proofs, which, although not guaranteed to be eternal, ensure that managing and maintaining the network is a nonarbitrary process.Footnote ²²

In Chapters 6 and 7, I first complete the sketches of some of the arguments appearing in the previous chapters. More specifically, in regard to the “urge” to dismiss the complexity of language (PI §109), I will elaborate on the hypothesis I advance in Chapter 1 – namely, that what drives our inclination to assimilate the different functions of sentences and words is, ultimately, the natural and overwhelming desire to experience something I shall call ‘metaphysical awe.’ Echoing what I say in Chapters 1–5, I will take one’s succumbing to such temptations to reveal a serious flaw in one’s ‘style of thinking.’

However, the main aim of Chapters 6 and 7 is to take this hypothesis as a background for my reading of Wittgenstein’s quite controversial views on Cantor’s famous proof – that there are more real numbers than natural numbers. I argue that Wittgenstein’s main worry in RFM II is that this kind of mathematics embodies and encourages assimilationist inclinations;Footnote ²³ hence that it may serve as a source of such awe, allegedly experienced when exploring “unknown depths” (LFM 254). I take the following LFM passage to embody the spirit of RFM II. (One has only to replace ‘calculus’ by ‘transfinite set theory’ in the right places.)

These two chapters tackle more technical topics, but of central relevance to Wittgenstein: one is his sympathy for intensionalism about the real (irrational) numbers, as opposed to Cantor’s extensionalism. (This distinction will be subject to a detailed discussion in Chapter 7.) A suggestive expression of his intensionalism is the claim that, unlike their customary extensionalist understanding, “the irrational numbers are – so to speak – individual cases” (RFM V-37; I follow the Floyd and Mühlhölzer Reference Floyd and Mühlhölzer2020, 98 translation). More explicitly, the distinction corresponds to the difference in specifying the reals/irrationals: by giving them as individuating rules (of generation), or by giving them ‘extensionally,’ as infinite lists of digits.Footnote ²⁴

In Chapter 7, I will use the reconstruction of Wittgenstein’s views on proof provided in Chapter 5 and apply it to Cantor’s proof. I will argue that a good grasp of his key thought there – that proofs ‘create’ concepts – help us realize that the often-dismissive remarks about this proof in RFM II are actually less irreverent than many have (understandably) taken them to be. Thus, I shall contend that there is a way to read these remarks other than as simply denigrating Cantor’s achievements. Their sarcastic tone notwithstanding, Wittgenstein’s pronouncements are not ad hominem and do not amount to a refutation of the theorem. Instead, their intention is to make us see it in a different light – not as a result we must accept (i.e., not as having the same status as 2 + 2 = 4), but as somehow optional. This change of perspective can be conveyed by paraphrasing one of his well-known lines: “Cantor is an inventor, not a discoverer.”Footnote ²⁵ The consequence of this shift is, I submit, therapeutical: extensionalism (and, with it, Cantor’s result) – a platonist conception, almost universally, implicitly, and uncritically embraced by mainstream mathematics – will no longer appear as forced upon us as the only way to think of irrationals; intensionalism, he believes, offers an alternative.

Nevertheless, a question arises (and I will comment on it) as to whether this is a viable alternative, a concern never taken up by Wittgenstein himself. In this regard, I will note that it is far from clear whether the intensionalist mathematics is as rich as the current extensionalist one – and also that it is an open question whether this richness is a good thing or not.

The remarks in RFM II (especially sections 1–22) are very controversial indeed, and it is perhaps wise to ask how seriously one should take them. I don’t ask this in the chapter itself, where I assume, like other interpreters, that Wittgenstein fully endorsed them. Consequently, I try to elucidate them and show how they can be integrated into his post-1936 eliminativist normativism. And yet the question is worth asking, for this series of remarks has a somewhat problematic place in his body of work on mathematics. In the Nachlass they appear in Ms-117, 97–110 under the heading Ansätze. This word can be translated in a variety of ways, as ‘Attempts,’ ‘Beginnings,’ ‘Preliminaries,’ ‘First thoughts,’ and so on.Footnote ²⁶ Moreover, as Schulte (Reference Schulte2017) points out in his genetical note, this manuscript is rather isolated; it “seems to stand in no direct relation with any other manuscript.” All this indicates the very tentative status of these remarks. In fact, I suspect that Wittgenstein believed that they needed extensive work before considering them for publication.

Another consequence of these remarks being rather exploratory and provisional is that they do not clearly delineate the philosophical problem addressed in them. Although it is quite evident that he follows his usual strategy of demystifying some dramatic claims about Cantor’s proof and the infinite, it is difficult to discern what question, exactly, piqued his interest. Therefore, I take the liberty to construe the general worry in the background as ‘What is the significance of Cantor’s discovery?’Footnote ²⁷ – that there is a hierarchy of infinities. As we saw, this is a question whose virtually unanimous answer was, and is, that it marks a major conceptual breakthrough, that it transcends mathematics and reaches up to theological heights, that it is much more than a technical result by having deep metaphysical implications, and so on.

If this is the question, we can see right away why Wittgenstein thinks that it is ill posed. If we recall his insistence that Cantor did not discover anything, the question can be disposed of insofar as it relies on an invalid presupposition. To ask such a question is plain nonsense, as much as asking about the significance of Michelangelo’s discovery of David. For Wittgenstein, mathematicians, including Cantor, are not discoverers, but creators, inventors (RFM I-168).Footnote ²⁸ But, while calling someone an ‘inventor’ (‘creator’) is no less a word of praise than calling one a ‘discoverer,’ the difference is not negligible: talking in terms of ‘inventing X’ largely removes the suggestions of inevitability and naturalness about X, implications which talking about ‘discovering X’ conveys. And, moreover, it also adds a nuance of contingency, even of arbitrariness – and this is, as we’ll see, exactly Wittgenstein’s intention.

In Chapter 8, I turn to a discussion of some aspects of Wittgenstein’s view on logicism, the philosophical project aiming to show that mathematics is (reducible to) logic. I’ll examine logicism not because (later) Wittgenstein had anything illuminating to say about the technical reasons behind its failure, but because this doctrine was, and still is, the most developed attempt to tackle the age-old philosophical question ‘what is a number?’ He will take issue with the logicist solution to this problem because he believes that the pivotal notion of one-to-one correlation, used in this definition, is far from clear: as we’ll see, the innocent enough question ‘Can sets A and B be one-to-one correlated?’ is, despite appearances, ill posed (since it is incomplete).

The logicist philosophy of mathematics, as elaborated by his former mentors Frege and Russell, deserves special treatment here, since it is the doctrine for which Wittgenstein showed the most respect. Although he never thought that it might be right, he believed that logicism made several important clarificatory contributions. (For instance, in LFM 168 he praises Frege.) Yet, he is critical of the Frege–Russell definition of number and, ultimately, even of the very need for such a definition (see, e.g., LFM 24). Thus, my task in this chapter will be to explain why he believed this – and the key to understanding this admittedly radical stance will be to explicate his (anthropologically driven) analysis of the notion of one-to-one correlation (which, he implies, is uncritically taken on board both by the logicists and by Cantor).

Finally, the concluding Chapter 9 summarizes the argument of the book, adds some clarifications, and ties up loose ends.