From pictures to employments: later Wittgenstein on ‘the infinite’

ABSTRACT

With respect to the metaphysics of infinity, the tendency of standard debates is to either endorse or to deny the reality of ‘the infinite’. But how should we understand the notion of ‘reality’ employed in stating these options? Wittgenstein’s critical strategy shows that the notion is grounded in a confusion: talk of infinity naturally takes hold of one’s imagination due to the sway of verbal pictures and analogies suggested by our words. This is the source of various philosophical pictures that in turn give rise to the standard metaphysical debates: that the mathematics of infinity corresponds to a special realm of infinite objects, that the infinite is profoundly huge or vast, or that the ability to think about infinity reveals mysterious powers in human beings. First, I explain Wittgenstein’s general strategy for undermining philosophical pictures of ‘the infinite’ – as he describes it in Zettel; and then show how that critical strategy is applied to Cantor’s diagonalization proof in Remarks on the Foundations of Mathematics II.

KEYWORDS:

• Wittgenstein
• the infinite
• philosophy of mathematics
• history of analytic philosophy

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Compare Hilbert’s (1926) claim that Cantor’s work presents a kind of mathematical ‘paradise’, Cantor’s thought that the mathematics of infinity reveals to us the infinite nature of God (Moore 1990, 127), or Nagel’s suggestion that it is mysterious (and possibly inexplicable) how we so much as have an idea of the infinite (Nagel 1997, 76). Compare also Leopold Kronecker’s early critique of Cantor’s work on infinity as a kind of theology rather than genuine mathematics, famously remarking that ‘God made the integers, all the rest is the work of man’ (see Moore 1990, 120). Boolos, Burgess, and Jeffrey (2007, 19) recognize the potential of Cantor’s proof to play with our imagination, and even (cautiously) indulge in such extravagances (involving the powers of divine beings), albeit for strictly pedagogical reasons.

2 Compare Dummett (1978, 168): ‘Certainly in his discussion of Cantor he displays no timidity about “interfering with the mathematicians”.’

3 See fn. 1. My reading thus bears similarities to Shanker’s (1987) discussion of Wittgenstein on (Cantorian) infinity, especially in his emphasis on Wittgenstein’s suspicions about the suggestive ‘prose’ used to describe a mathematical result rather than the mathematical ‘proof’ itself. However, I have avoided that distinction (between ‘prose’ and ‘proof’) to emphasize that later Wittgenstein’s concern (see fn. 5 below) is more specifically about the confused pictures that such ‘prose’ tends to suggest – in which case even standard ‘prose’ can be tolerated if it is not misunderstood. For example, via misleading analogies across distinctive uses of the relevant vocabulary. (Compare PI 79, an epigraph to this paper.)

4 See Bold (2022) for a comprehensive reading of later Wittgenstein's philosophy of mathematics grounded in his distinctive therapeutic methodology. See Bold (2023) and (2024) for applications of that method to traditional questions regarding mathematical necessity and mathematical reality, respectively.

5 For all that I will say, Moore’s objections might be perfectly valid regarding Wittgenstein’s early or transitional-period remarks on ‘the infinite’. By Wittgenstein’s ‘early’ remarks, I have in mind Notebooks 1914–16 and Tractatus Logico-Philosophicus. By his ‘transitional-period’, I have in mind his Philosophical Remarks. What about writings such as Philosophical Grammar and The Big Typescript? It is difficult to draw a sharp line between the true ‘later’ Wittgenstein and the works of his so-called ‘transitional’ period. But it is uncontroversial that Wittgenstein saw his Philosophical Investigations as a profound shift in his philosophical thought that contrasts significantly with his ‘older way of thinking’ (PI, Preface). For my purposes, it is sufficient to consider the author of the Philosophical Investigations and writings thereafter (such as most of those collected in Zettel and Manuscript 117) as the ‘later Wittgenstein’. The reading that follows thus differs in general orientation from that offered by Marion (1998, xii) in that it takes seriously (and as its sole focus) Wittgenstein’s later philosophical methodology – including his explicit aspirations to avoid any ‘theory’ (PI 109) – and attempts to make sense of particular (later) remarks about infinity by showing how they are expressions of that methodology at work. As with Moore, for all I say about the later Wittgenstein, Marion may be perfectly correct in his readings of Wittgenstein’s early or transitional-period remarks on ‘the infinite’.

6 Compare Monk’s (2007, 285) reading on which Wittgenstein aims to ‘redescribe’ mathematics so as to dislodge misbegotten pictures of it.

7 See especially Frascolla (1994, 142) for a useful explanation of ‘Finitism’ as it is typically understood and how it contrasts with Wittgenstein’s later philosophy of mathematics.

8 As noted by Moore (2011, 108) and Wheeler (2022, 324).

9 My reading thus contrasts with those on which Wittgenstein endorses ‘Finitism’, e.g., Ambrose (1935), Dummett (1997), Kielkopf (1970), Kreisel (1958), Wang (1958), and Wright (1980). My reading agrees with those that emphasize Wittgenstein’s explicit disavowal of ‘Finitism’ as a confused doctrine, e.g., Frascolla (1994, 143), Klenk (1976, 18–24, 92–123), Kripke (1982, 107) and Monk (2007, 276).

10 Compare especially Frascolla (1994, 143–144): ‘The denial of the existence of infinite sets is a mistaken way to draw a grammatical distinction which, though it may be opportune, should be done differently: by showing that the grammar of the word “infinite” cannot in the slightest be clarified by taking into account only the picture of something huge, a picture which usually accompanies the use of the word.’

11 Compare also Papineau (2012): “The example of the reals shows that infinite sets come in different sizes. There is the size shared by all the denumerable sets. But the real numbers are bigger again” (27, emphasis added); “There is the infinite number that characterizes the denumerable sets, and the distinct and bigger infinite number that characterizes all the sets whose members can be paired up with the real numbers” (32, emphasis added).

12 See Moore (2016, 326): ‘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’; and Boolos, Burgess, and Jeffrey (2007, 19): ‘[T]here is no need to refer to [a hypothetically infinite] list, or to a superhuman enumerator: anything we need to say about enumerability can be said in terms of the functions themselves; for example, to say that the set P* is not enumerable is simply to deny the existence of any function of positive integers which has P* as its range. Vivid talk of lists and superhuman enumerators may still aid the imagination, but in such terms the theory of enumerability and diagonalization appears as a chapter in mathematical theology.’

13 This addition strikes me as the simplest way to make sense of the passage and does nothing to interrupt the major philosophical point.

14 Anscombe’s translation. But Mühlhölzer (2023, 571–572) strongly prefers the more literal translation: ‘mathematics is a COLOURFUL mix [ein BUNTES Gemisch] of techniques of proof’, which suggests ‘the permanent remixing and restructuring of what we do in mathematics’ rather than a ‘motley’ (ibid, fn. 2). Both translations, however, suggest a variety of techniques (‘COLOURFUL’ rather than, say, colourless or monochromatic).

15 This helps to address Moore’s (2016) objection that if mathematicians are engaged in the ‘modification of [pre-existing] concepts’ (e.g., ‘big’), then ‘the use of the relevant vocabulary will after all be essential to what mathematicians are doing’ (Moore 2016, 327). The objection is irrelevant if all Wittgenstein hopes to do is to highlight the differences between the uses of certain vocabulary to upset the temptation to think they share a singular rule, definition, or essence.

16 My reading thus disagrees with Schroeder’s (2021, 154): ‘[Later] Wittgenstein’s view is that the diagonal method does not yield a new irrational number’. The point, as I understand it, is that the diagonal method yields a ‘number’ by stipulation – not by sharing some strict, underlying essence with other ‘numbers’ (in short: ‘number’ is a family-resemblance concept; there is no hard-and-fast rule that prevents the concept ‘number’ from being extended in this manner). However, Schroeder seems to acknowledge this lesson soon after his initial claim above: ‘Note that here Wittgenstein does not object to that usage […] [since] regarding the diagonal decimal expansion as a number would also strike us as a natural use of the word ‘number’. However, we should realise that it involves a conceptual extension’ (2021, 155).

17 As Gerrard (1991) notes, Wittgenstein’s later philosophy of mathematics centrally aims to undermine the ‘Hardyian picture’ of an independent and preexisting mathematical reality. See also Bold (2022), (2023), and (2024).

18 See fn. 13.

19 As noted by Floyd (1995, 388–389): ‘Archimedes [e.g.] gives us a perfectly good geometrical construction; he does trisect the angle. Only not in what we now call “the relevant sense”.’ Compare Schroeder (2021, 155).

20 Compare Schroeder (2021, 144ff). My reading of these passages contrasts with Mühlhölzer’s (2020, 134), who thinks that the task stated ‘Name a number that [dis]agrees with $\sqrt{2}$ at every second decimal place’ is obviously not satisfied by the suggested answers.

21 Compare Schroeder (2021, 62): ‘It is of course true that elsewhere Wittgenstein calls grammar “arbitrary” […]. But what he means by that is that it is essentially a human artefact that cannot be assessed as true or false to nature. That, however, does not mean that anybody will be allowed to introduce any kind of grammatical rule as the whim takes him.’

22 This point is made most forcefully by Maddy (2014, Ch. 5-6). See also Floyd (2021, 60).

23 See fn. 14.

24 Of course, if by ‘Conventional-ism’ someone means the watered-down view that ‘to some extent (often overlooked by Realists or Platonists in mathematics) our use of mathematical terms depends on human convention’, then Wittgenstein can be called a ‘Conventional-ist’. But to avoid confusion, I think it is best to avoid this attribution entirely. Especially seeing as someone could just as easily attribute the following ‘Realism’ to Wittgenstein: ‘to some extent (often overlooked by Conventional-ists) our use of mathematical terms depends on facts of nature’. Neither is the typical understanding of the relevant ‘ism’. That said: I take no issue with Schroeder’s (2021, 125) suggestion that Wittgenstein be considered a ‘moderate conventionalist’ – to the extent that doing so does not conflict with the considerations above. (Though, again, one should consider how much is gained by this label if we can, via similar conditions and caveats, call him a ‘moderate realist’). See Diamond (1986) and Anscombe (1981) for further discussion. See also Ben-Menahem (1998) on ‘convention’ more generally in Wittgenstein’s later philosophy.

25 C.f., RFM p. 358: ‘How can you say that “ … 625 … ” and “ … 25 × 25 … ” say the same thing? – Only through our arithmetic do they become one.’

26 This contrasts with readings on which Wittgenstein simply identifies the meaning of a mathematical proposition (or its ‘truth’) with its proof (or with some other specific aspect of its use). See especially Dummett (1978), Moore (2016), and Schroeder (2021, 57–58).

27 See Bold (2023) and (2024) for Wittgenstein's critique of mathematical reality via the notions of mathematical ‘necessity’ and ‘truth’ respectively.

28 This point about ‘correlation’ or ‘correspondence’ is made frequently in the LFM. See especially LFM VII, XVI, XVII, XXVII, and XXX. The point is essentially that ‘one-to-one correspondence’ (as much as ‘number’, ‘denumerability’, etc.) is a family resemblance concept: the instances of which do not adhere to a strict, unifying essence, but are connected by evolving and overlapping similarities (PI 66-7).

29 “[Wittgenstein:] Suppose you had correlated cardinal numbers, and someone said, ‘Now correlate all the cardinals to all the squares.’ Would you know what to do? Has it already been decided what we must call a one-one correlation of the cardinal numbers to another class? Or is it a matter of saying, ‘This technique we might call correlating the cardinals to the even numbers’?

Turing: The order points in a certain direction, but leaves you a certain margin.

Wittgenstein: Yes, but is it a mathematical margin or a psychological and practical margin? That is, would one say, ‘Oh no, no one would call this a one-one correlation’?

Turing: The latter.

Wittgenstein: Yes. – It is not a mathematical margin” (LFM, p. 168).

30 See fn. 26.

31 As is suggested by Moore (2011, 119).

32 Mühlhölzer (2020, 162) is right that Wittgenstein’s examples do not exhaust the applications of diagonalization. The major lesson of the passage still stands: our excitement about Cantor’s proof should go no further than our excitement about its applications. (I leave the reader to decide their own level of excitement about its ‘central place in set theory […] [and] recursion theory’ (Mühlhölzer (2020, 162) – by contrast with the excitement that might be inspired by its alleged ability to reveal secrets of the various and hidden realms of infinity – or of God (see fn. 34)).

33 C.f., the thought experiment of RFM V 7: ‘Imagine set theory’s having been invented by a satirist as a kind of parody on mathematics. – Later a reasonable meaning was seen in it and it was incorporated into mathematics. (For if one person can see it as a paradise of mathematicians, why should not another see it as a joke?) […] But isn’t it evident that there are concepts formed here – even if we are not clear about their application?’.

34 See for instance Moore (1990, 127): ‘And this Absolute that had revealed itself in his own formal work, in a way that was so reminiscent of more traditional views of the infinite, was embraced by Cantor as a vital part of his conception of God.’

35 Compare again Boolos, Burgess, and Jeffrey’s (2007, 19) pedagogical use of theological imagery to illustrate the proof’s significance.

36 Compare Fogelin (1987, 220): ‘But surely nothing forces us to extend our concepts in these ways, and thus the idea that Cantor has proved the existence of a hierarchy of transfinite cardinals is simply an exaggeration.’ On my reading, this is an exaggeration only if it suggests the wrong picture to one’s imagination – a picture that is significantly dampened by reminders about the various uses of relevant vocabulary and the role of stipulation in proof.

37 Compare Stern’s (2004, 169) characterization of the so-called ‘quietist’ position: ‘Wittgenstein’s invocation of forms of life is not the beginning of a positive theory of practice […] but rather is meant to help his readers get over their addiction to theorizing about mind and world, language and reality.’ For a similar conclusion arrived at via the concepts of mathematical necessity and truth, respectively, see Bold (2023) and (2024).

38 Thanks to Chris Dorst, Tamara Fakhoury, Ram Neta, and Mario Von Der Ruhr for helpful comments. Thanks also to three anonymous reviewers for suggestions that significantly improved the paper. Special thanks are owed to Alan Nelson for many fruitful conversations that led to the major ideas of this paper as well as comments on multiple drafts.