Wittgenstein on pure and applied mathematics

Wittgenstein on pure and applied mathematics

Ryan Dawson

Received: 3 March 2014 / Accepted: 1 July 2014 / Published online: 29 July 2014
© Springer Science+Business Media Dordrecht 2014

Abstract

Some interpreters have ascribed to Wittgenstein the view that mathematical statements must have an application to extra-mathematical reality in order to have use and so any statements lacking extra-mathematical applicability are not meaningful (and hence not bona fide mathematical statements). Pure mathematics is then a mere signgame of questionable objectivity, undeserving of the name mathematics. These readings bring to light that, on Wittgenstein’s offered picture of mathematical statements as rules of description, it can be difficult to see the role of mathematical statements which relate to concepts that are not employed in empirical propositions e.g. set-theoretic concepts. I will argue that Wittgenstein’s picture is more flexible than might at first be thought and that Wittgenstein sees such statements as serving purposes not directly related to empirical description. Whilst this might make such systems fringe cases of mathematics, it does not bring their legitimacy as mathematical systems into question.

Keywords Wittgenstein $\cdot$ Mathematics $\cdot$ Pure $\cdot$ Applied $\cdot$ Maddy $\cdot$ Foundations

1 Introduction

My goal in this paper is to defend Wittgenstein from a particular revisionary doctrine that has been attributed to him. This is a doctrine which says that only applied mathematics is really mathematics and pure mathematics is a mere sign-game of questionable objectivity, detached from reality and undeserving of the name mathematics (Maddy 1993, p. 67; Rodych 1997, p. 217; Steiner 2009, p. 23). I do not mean to question the view that Wittgenstein saw it as of immense importance that mathematics is

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tied to empirical applications. What I wish to argue against is the claim that Wittgenstein laid down a requirement to the effect that mathematical statements must have an application to extra-mathematical reality in order to have use and so any statements lacking extra-mathematical applicability are not meaningful (and hence not bona fide mathematical statements).

I will concentrate on Maddy’s reading since she gives most direct focus to the question of how pure and applied mathematics are to be distinguished and their respective roles for Wittgenstein. Wittgenstein’s view, as I see it, does make a case that mathematics with direct applications should be seen as more central to our concept of mathematics, and in this sense more important, than mathematics which lacks external applications (for example, foundational systems). But the legitimacy of pure mathematics (including foundational systems) is not called into question. In order to bring to light a reading of Wittgenstein as paying full respect to pure mathematics, it is first important to consider why he emphasises the connection between mathematics and empirical applications in the way that he does.

2 Mathematical statements and empirical applications

Wittgenstein famously claimed that philosophy, at least as he means to practise it, “leaves mathematics as it is” (PI §124) and that philosophy “cannot give it any foundation either” (PI §124). Anti-scepticism is an important theme of Wittgenstein’s philosophical project in regard to mathematics (WVC, p. 149), although it is not always an easy one to understand. I would like to suggest that the key element of this antiscepticism is a resistance to dogmatic philosophical accounts. When we are inclined to apply a particular picture dogmatically, then anything that does not fit that picture particularly well comes to look strange and in need of philosophical justification (for instance, justification by a neater integration into the picture). Thus the commitment to a single model across a range of cases (of which only some cases are a good fit for the model) can be seen to give rise to scepticism and the appearance that a philosophical project is needed in order to give a foundation for our using expressions. ${ }^{1}$ I take this to be Cavell’s point when he says that “scepticism for Wittgenstein is the intellectual twin of metaphysics” (2005, p. 195).

Wittgenstein’s way of working can be seen to address sceptical problems by resisting the grip of a particular picture, thereby addressing both the scepticism and the appearance of a need for a foundation (since both are products of the inappropriate application of the picture). Wittgenstein thinks that we are inclined to view mathematics as having a subject-matter like physics does and this determination to see mathematics as like physics is a key theme of the pictures that tend to be applied dogmatically (with this attraction to a dogmatic reliance upon a particular picture becoming especially prominent even among mathematicians during the foundations crisis $^{2}$ ). Wittgenstein aims in various ways in his work on mathematics to undermine

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the attraction of this picture by highlighting the differences between the ways that empirical descriptions are used and the ways that mathematical statements are used.

One of the most illustrative discussions is in $L F M$ when Wittgenstein responds to what he takes to be a suggestion by Godfrey Hardy that “a reality corresponds to” mathematical theorems:

We have here a thing which constantly happens. The words in our language have all sorts of uses; some very ordinary uses which come into one’s mind immediately, and then again they have uses which are more and more remote… A word has one or more nuclei of uses which come into everyone’s mind first.

So if you forget where the expression “a reality corresponds to” is really at home-
What is “reality”? We think of “reality” as something we can point to. It is to this, that.

Professor Hardy is comparing mathematical propositions to propositions of physics. This is extremely misleading. (LFM, pp. 239-240)

The expression “a reality corresponds to” is one which we associate most clearly with cases of being able to point to objects in physical reality in comparison with our empirical descriptions-cases like using an expression “there is a green sofa” and pointing to the green sofa in order to justify the assertion. Wittgenstein accepts that it is possible to talk of a reality corresponding to mathematical propositions but thinks that it misleads us about mathematical propositions to think of mathematical propositions as made true or false by mathematical facts. He takes this up in various ways but what he seems to be especially concerned about is that when we regard mathematical propositions as justified by a mathematical reality then it is natural to start to ask questions about this alleged ‘reality’-questions such as where it is, how it is it that we are able to talk about it and ‘see’ it and why statements about it (i.e. mathematical statements) should be so useful in science and in everyday life. The mathematical reality starts to look 'fishy’3 (LFM, p. 145) and this ‘fishiness’ gets transposed onto mathematics itself, making it appear that we need to determine the true nature of mathematical entities in order to secure confidence in mathematics. Wittgenstein saw this as a key motivation that drove, for example, Frege and Russell to undertake their foundation projects (LFM, p.273; AWL, pp.150-152) with the idea being that the projects “would at long last give us the right to do arithmetic as we do” (PG, p.296). Wittgenstein wants to show that these problems arise because of our attachment to the picture of mathematical statements as descriptions and are thus problems with our picture rather than with mathematics (Kuusela 2008, p. 198). ${ }^{4}$

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A central theme of Wittgenstein’s criticism of understanding mathematical statements as descriptions is that mathematical entities then look like they are part of a kind of ‘shadow reality’. This shadow quality is in part because the picture undermines the variety and creativity that we see in mathematics (as Wittgenstein suggests at ( $L F M$, p. 145)). Another contribution to the fishiness is that we say that true mathematical propositions are not just true but ‘necessarily’ true, whereas we don’t say this about empirical descriptions. ${ }^{5}$ If we construe mathematical propositions as descriptions then it looks as though this ‘necessity’ were a feature of mathematical objects, as though mathematical objects were special objects that were “not subject to wind and weather” (RFM, p. 74). Wittgenstein sees this as a distortion that misrepresents a difference in the use of certain kinds of expressions as a difference in types of entity. The relevant difference is better characterised (though not explained or justified), he proposes, by saying that mathematical statements are such that we do not allow any empirical fact to count for or against them (RFM, pp. 47-50). ${ }^{6}$

One of the key differences that Wittgenstein points to between expressions like ‘there is a green sofa’ and ’ $2+2=4$ ’ is that there is no empirical observation which we would take to confirm or disconfirm ’ $2+2=4$ ’ (RFM, pp. 96-97). Whilst we often explain arithmetical expressions like ’ $2+2=4$ ’ by way of illustrations like putting two apples together with another two apples, if one day we were to do this and find that we had three apples then we would say that we had lost an apple somehow ( $R F M, \mathrm{p}$. 97). We would appeal to some physical explanation first and if we found that it kept happening then we would simply say that apples were not good things to count with (as we do say with regard to objects that easily fall apart or mix into one another like blobs of jelly). Whereas if I say that there is a green sofa in the next room and then we go into the next room and see no sofa or a red sofa then I would have to concede that I was mistaken. Empirical descriptions are called true or false in the light of empirical observations, whereas all that can count for or against a mathematical proposition is a proof (RFM, pp. 98-99).

When the immunity of mathematical propositions from empirical revision is presented as arising from a feature of mathematical entities, then both mathematical entities and mathematical propositions start to look fishy. As noted, mathematical entities look fishy because properties are ascribed to them that we do not have any clear model of, since it is not clear what it means for an object to exist necessarily (RFM, p. 64). But mathematical expressions can start to look fishy too, since then it seems mysterious that we should be able to describe or refer to these entities in a ‘shadowy’ realm.

In order to release the grip of the picture of mathematical propositions as descriptions, Wittgenstein offers an alternative picture (only as an alternative picture, rather than a replacement of one dogmatic account by another ${ }^{7}$ ) which is intended to avoid

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the ontological difficulties and present mathematical language in a more down to earth fashion. ${ }^{8}$ Wittgenstein’s suggestion is that we see mathematical propositions on the model of statements that he calls ‘preparations for description’ or ‘rules of description’, of which he gives examples such as ‘red is’ (RFM, p. 64) and ‘there is no reddishgreen’ (LFM, p. 245).

Whilst ‘there is no reddishgreen’ might initially look like an empirical description, there is no single empirical fact that clearly corresponds to it:

The correspondence is between this rule and such facts as that we do not normally make a black by mixing a red and a green; that if you mix red and green you get a colour which is “dirty” and dirty colours are difficult to remember. All sorts of facts, psychological and otherwise. (LFM, p. 245)

When Wittgenstein says that the expression might be said to correspond to the fact that ‘if you mix red and green you get a colour which is “dirty”’, he does not mean that this is what the proposition says in the way that ‘there is a sofa in the next room’ says that there is a sofa in the next room. He is suggesting rather that the expression shows us something concerning how we use the terms ‘red’ and ‘green’ and other colour terms. We tend to give names to clearly identifiable colours and use those colours as points of reference. Whilst mixing samples of red and green will naturally result in something that might be called a colour, it is not a colour that is useful to us as a point of reference and so we don’t give it any name. The statement ‘there is no reddishgreen’ might be said to be part of our system of colour terms, which we use for describing objects as having colours. ${ }^{9}$

Carrying this idea over to mathematics, we might say that mathematical propositions like ’ $2+2=4$ ’ set up a system which shows how ’ 2 ’ and ’ 4 ’ are to be used and we apply this system when we describe situations in the world in terms of numbers - statements such as ‘there are 2 apples over there’. The system of arithmetic being what it is allows us to make the transition (barring any apples going missing or any miscounting) from ‘there are two apples over there’ and ‘there are two apples here’ to ‘there are four apples altogether’ (AWL, p. 154). Arithmetic is set up such that it is applicable in this way, and if we want to talk of a reality corresponding to arithmetic then we should say that this reality is to be seen in our using arithmetic in the way that we do ( $L F M$, pp. 248-249). ${ }^{10}$ This is naturally connected with many facts about us and the world, much as our using the colour terminology that we do is connected with many facts about the world, our visual systems and our brains.

Whilst arithmetic is a favourite source of comparisons, Wittgenstein wants the idea of mathematical statements as rules of description to be seen to be illuminating with

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regard to more than just arithmetic and seems to have regarded kinematics as another key case. As Diamond puts it:

Abstract

Wittgenstein’s idea that in mathematics we are developing our means of description should be seen with his view that there are many different kinds of description, in which a variety of techniques are used. One technique of description, for example, is the formulation of some kind of “ideal case,” which enables us to describe actual cases as departures of one or another sort from the ideal. Wittgenstein thinks of kinematics, for example, as providing such a means of description… Mathematics is integrated into the body of standards for carrying out methods of arriving at descriptive propositions, for locating miscounts (for example), or mistakes or inaccuracies of measurement. (Diamond 1991, p. 234)

It is notable that Wittgenstein does not pin down mathematical propositions to particular kind of rule of description (for example, in terms of ‘ideal cases’ as with kinematics, or propositions that license one term to be substituted for another as seems to fit better with arithmetic). If Wittgenstein were advocating a definitive (i.e. dogmatically-applied) account of mathematical propositions as rules of description, then one might expect him to do so. Despite having often been read as advocating a definitive account of mathematical propositions as rules of description (for example, Maddy (1993) and Steiner (2009)), Wittgenstein seems to disavow such an aim:
…the whole point is that I must not have an opinion… I have no right to want you to say that mathematical propositions are rules of grammar. I only have the right to say to you, “Investigate whether mathematical propositions are not rules of expression, paradigms-propositions dependent on experience but made independent of it. Ask whether mathematical propositions are not made paradigms or objects of comparison in this way.” Paradigms and objects of comparison can only be called useful or useless. ( $L F M$, p. 55)

The passage is somewhat confusing but Wittgenstein’s claim that it is his “whole point” that he “must not have an opinion” is quite clear. His model of mathematical propositions is not meant to be true but only “useful or useless”. It will be useful if it manages to dislodge the picture of mathematical propositions as descriptions and show us a way out of our sceptical problem. If it itself becomes a picture that we advocate as a dogmatic philosophical account then it risks giving rise to sceptical problems like the ones that we were supposed to be avoiding. I think the way that Maddy (1993) reads Wittgenstein does give rise to this kind of sceptical problem. I want to press that Wittgenstein’s picture can be taken in a more flexible way which avoids the revisionary doctrine that she ascribes to him (namely the doctrine that any alleged mathematical propositions which are not applied empirically cannot be truly mathematical). The key is that Wittgenstein’s picture is applied in a way that means it does not need to fit

all of mathematics neatly. ${ }^{11}$ Instead it just needs to bring out key characteristics (such empirical applicability and immunity from empirical revision) in a way which enables us to see the dogmatic character of claims such as that mathematical propositions are (in all respects) descriptions.

3 Pure and applied mathematics

Understanding something of Wittgenstein’s motivations for presenting a picture of mathematical propositions as rules of description will now help us to understand the shortcomings in Maddy’s reading and, through an exploration of Maddy’s reading, to better understand Wittgenstein’s attitude towards pure mathematics.

Maddy gives focus to what she sees as a stipulation that the applications of mathematical statements (whether conceived as rules or not) have to be extra-mathematical in order for the statements to count as legitimately mathematical - a claim that “the parts of mathematics without application are just empty games with meaningless signs” (1993, p. 67). Her primary motivation (1993, pp. 67-68) for reading Wittgenstein as requiring an extra-mathematical use for mathematical statements is that she cannot see room for any other kind of use. She cites (1993, p. 68) Wittgenstein’s criticisms of Platonism ( $L F M$, pp. 239-240) and takes it that if there is no mathematical reality for mathematical statements to apply to (i.e. if Platonism is rejected) then the only way in which they can have use is by applying indirectly to empirical reality - mathematical propositions need to have demonstrable “effectiveness in science” (Maddy 1993, p. 68). For this reason she takes Wittgenstein’s remarks criticising Platonism’s conception of mathematics as the study of mathematical objects not just as directed at Platonism but also as a rejection of pure mathematics itself.

Wittgenstein says that we must distinguish the work that a mathematician does from what a mathematician is inclined to say about the objectivity of mathematics (PI, §254), the latter being a subject for philosophical treatment:

What we ‘are tempted to say’ in such a case is, of course, not philosophy; but it is its raw material. Thus, for example, 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.

Maddy takes the treatment to reveal that applied mathematics is mathematics because it “can be made clear” and shown to have a “real use”, whereas pure mathematics only has a prose-constructed illusion of use and so philosophical treatment will reveal that it can be “pruned away” (Maddy 1993, p. 67). She ascribes to Wittgenstein the view that pure mathematical statements, with their associated prose shrouding, are aimed to be ‘about’ mathematical objects (their illusion of justification is an imagined

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“application in a purely mathematical realm”) and since these objects don’t exist we must take the statements to be meaningless (Maddy 1993, p. 68).

Maddy’s reading of (PI §254) runs contrary to the anti-sceptical way that Wittgenstein speaks of the prose-mathematics distinction (WVC, p. 149):

It is a strange mistake of some mathematicians to believe that something inside mathematics might be dropped because of a critique of the foundations. Some mathematicians have the right instinct: once we have calculated something it cannot drop out and disappear! And in fact, what is caused to disappear by such a critique are names and allusions that occur in the calculus, hence what I wish to call prose. It is very important to distinguish as strictly as possible between the calculus and this kind of prose. Once people have become clear about this distinction, all these questions, such as those about consistency, independence, etc., will be removed. (WVC, p. 149)

Prose is a subject for philosophical examination because what a mathematician is inclined to say about the objectivity of mathematics is typically linked to some form of metaphysical thesis (commonly some form of Platonism). Prose is not part of mathematics itself and can be done away with without detriment to the mathematics ( $P G$, p. 422). Maddy would have Wittgenstein saying that pure mathematics is dependent upon its associated prose for our acceptance of it, this acceptance being based upon a deception by prose. But it is surely against the spirit of Wittgenstein’s philosophy, and his conception of ‘prose’ in particular, to say that any mathematics should depend upon prose for its acceptance. If the prose-mathematics distinction is to be used to call the validity of certain mathematics into question, then what ‘drops out and disappears’ would be mathematics and not philosophical scepticism. Maddy reads Wittgenstein as using the distinction to subject pure mathematics to a sceptical challenge, whereas Wittgenstein’s comment indicates that the distinction is a part of Wittgenstein’s efforts to show that sceptical problems can be dissolved and should be of no concern to mathematicians. ${ }^{12}$

Aside from misreading the significance of the prose-mathematics distinction, Maddy also misreads the significance of Wittgenstein’s criticism of the Platonist picture of mathematical entities as analogous to physical entities. Contrary to how Maddy would have it, when Wittgenstein criticises the Platonist view of the existence of mathematical entities he is also criticising the view that mathematical statements aim to be descriptive of a mathematical reality. Maddy takes Wittgenstein to criticise the Platonist ontology but leave the Platonist model of pure mathematical statements (as descriptions) untouched in the case of pure mathematics (1993, p. 68). But it is exactly this image of the functioning of mathematical statements that is Wittgenstein’s focus. Wittgenstein’s rejection of a Platonist mathematical reality does not have to be a rejection of mathematical statements as meaningless, since he emphasises the idea that mathematical statements (and I see no reason to exclude statements of pure mathemat-

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ics) function not as descriptions of a mathematical reality but as rules. Wittgenstein’s aim is to move away from seeing mathematics as dependent upon a special mathematical realm, to make us see that this is a philosophical misconception. Maddy takes him to be saying that pure mathematics really is so-dependent. ${ }^{13}$

It could be argued that this response to Maddy does not get to the heart of the matter. Maddy might say that even if we follow the conception of mathematical statements as rules in the case of applied mathematics, it is less obvious where pure mathematics fits into matters. Maddy could suggest that we have no reason to speak of a statement as functioning as a rule for forming descriptions unless we form descriptive statements using that rule. Many of Wittgenstein’s examples appear to be drawn from arithmetic and geometry so it is easy to form descriptive statements which show the mathematical statements being applied in descriptions. How then do we make sense of statements in higher mathematics? Perhaps it is because Maddy cannot see how such statements can have application that she takes Wittgenstein to read such statements as meaningless attempts to form descriptive statements about some fanciful Platonistic reality. She appears to be influenced in this view by some of Wittgenstein’s remarks related to higher-order set theory (1993, p. 69), this being a favourite example of a branch of mathematics limited on applications outside of mathematics. ${ }^{14}$

I don’t want to venture too far into Wittgenstein’s remarks on set theory, since this is a significant topic in itself, but Maddy’s use of certain remarks needs to be addressed. Whilst these remarks have widely been read as highly revisionary, Wittgenstein says that set theory is “evidently mathematics” ( $R F M$, p. 264). His question is not whether set theory is mathematical since its applications are not all clear, his question is how best to understand that it is mathematical even though its applications are not all clear:

And why is it evidently mathematics?-Because it is a game with signs according to rules? But isn’t it evident that there are concepts formed here-even if we are not clear about their application?

But how is it possible to have a concept and not be clear about its application? (RFM, p. 265)

We may not be clear about the application, or better “intended application” (RFM, p. 259), of certain concepts in set theory but this does not disqualify set theory from being mathematics. Even if this unclarity in intended applications were to disqualify set theory from being mathematics (and I don’t think it does), being unclear about the intended application of some of the concepts formed in set theory does not mean that we are unclear about the intended application (or the application) of the concepts in all of pure mathematics (this is a point which I will return to)-to assume this would be to make an interpretative leap.

Another remark to which Maddy (1993, p. 67) gives prominence (as does Rodych (1997, p. 217)) is the following:

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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 $25 \times 20$.

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. (RFM, p. 257)

On a first reading this statement (particularly the ‘it is essentially to mathematics’) appears to have an air of finality to it, appearing to proclaim that all mathematical signs must be employed in descriptive statements about the world if the signs are to count as mathematical. But we can put pressure on this idea by noting the ‘I want to say’. Morris (1994) has argued that Wittgenstein often uses expressions like this when he is drawing our attention to one of the senses of a family-resemblance term (stricly she says to one possible picture but presumably a picture is possible because it captures one strand of a family-resemblance expression). And indeed this seems to be the case here-Wittgenstein is suggesting that we can take it as a characteristic of mathematics that its signs are employed in mufti, not because this holds of all mathematics but because this holds of certain key exemplars of the family-resemblance term ‘mathematics’. ${ }^{15}$

Of all the exemplars of the term ‘mathematics’, arithmetic is what comes most immediately to mind. And we can argue that it is arithmetic which Wittgenstein has in mind by noticing that the context of the cited remark relates only to arithmetic. The calculating machine (the one which has ‘come into existence by accident’) is not said to be a general theorem-proving machine, just a machine capable of producing the symbols corresponding to arithmetical operations. That this is the scope of Wittgenstein’s remark becomes even more likely when we see that Wittgenstein is using the point to make a criticism of Russell and Whitehead’s attempted reduction of arithmetic to logic in Principia Mathematica:

But is it not true that someone with no idea of the meaning of Russell’s symbols could work over Russell’s proofs? And so could in an important sense test whether they were right or wrong?

A human calculating machine might be trained so that when the rules of inference were shewn it and perhaps exemplified, it read through the proofs of a mathematical system (say that of Russell), and nodded its head after every correctly drawn conclusion, but shook its head at a mistake and stopped calculating. One could imagine this creature as otherwise perfectly imbecile.

We call a proof something that can be worked over, but can also be copied. (RFM, p. 258)

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Whilst the point here is not to discuss Wittgenstein’s views on the Principia project of reducing arithmetic to logic at any length ${ }^{16}$, it is worth giving some explanation (in order to follow through on the point that it is primarily arithmetic in play in the previously-cited remark). The attempted reduction of arithmetic to logic in Principia relied on being able to construct ‘proofs’ which would relate statements of logic to corresponding statements of arithmetic. Whilst these proofs would be impossibly long, Russell imagines that the process of constructing these proofs is mechanical one and so it does not matter that we cannot actually carry it out. Wittgenstein’s objection to Russell is that we need to be able to see the significance of these proofs if they are to count as a reduction of arithmetic to logic, which means that they would need to be ‘surveyable’ in Wittgenstein’s sense. Wittgenstein’s point with the calculating machine is that proof is not a matter of just constructing certain symbols. The symbols have to be used in a way which gives them significance, which is why surveyability is necessary in proofs. The calculating machine is an extension of the mechanistic view of proof that Wittgenstein finds in Russell’s thought.

So the reason why Wittgenstein says that it “is the use outside mathematics, and so the meaning of the signs, that makes the sign-game into mathematics” is because this is a plausible thing to say about arithmetic (as well as geometry, kinematics and other parts of mathematics) and it is arithmetic that is at issue in this context. Russell’s logical equivalents of arithmetical statements would not be usable as arithmetical statements (since they would be far too long) and hence they cannot be taken to capture the meaning (use) of arithmetical statements.

Wittgenstein furnishes us with an argument that it is application which makes systems like arithmetic into mathematics. In a discussion with the Vienna Circle (WVC, p. 170), Wittgenstein suggests that we might imagine wars being fought using chess. The suggestion is that if that were to happen then chess would no longer be just a game. Mathematics is not just a sign game (as Sudoku is) because we do not treat it as just a game. It does not matter what the intentions were behind chess as regards whether it is a game or not, it matters how we actually use it in our lives. To put the idea another way (not Wittgenstein’s own way), we might say that a religious text is marked out as a religious text by the way that it is consulted and used rather than by its content. Wittgenstein’s suggestion is that mathematical systems are not just games because we do not use them as just games. ${ }^{17}$

My interpretation of Wittgenstein’s ‘in mufti’ remark (RFM, p. 257) might seem to have the downside that Wittgenstein said ‘mathematics’ when he should have said ‘arithmetic’ or ‘certain core parts of mathematics such as arithmetic’. But it is important to stress Morris’s point (1994) that Wittgenstein tends to use expressions like 'I want

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to say’ when he’s pointing to one instance of a family. There is nothing unclear or confused about Wittgenstein’s expression if we see it as in line with his typical way of using expressions like ‘I want to say’. Even if there were something awkward about reading Wittgenstein as expressing his point in this way (and I don’t think there is), then I would see such an awkwardness as preferable to reading the remark in a way which attributes to Wittgenstein a rejection of pure mathematics-this strikes me as uncharitable in itself and also deeply out of step with Wittgenstein’s methodological promises to do philosophy of mathematics in a non-revisionary way (PI §124).

There is a second statement in $R F M$ which can appear problematic for my reading of the ‘in mufti’ remark ( $R F M$, p. 257) as saying that it is only characteristic of certain core members of the family ‘mathematics’ for which it is essential to their nature that they are applied. The potentially problematic remark reads:

Now how about this-ought I to say that the same sense can only have one proof? Or that when a proof is found the sense alters?

Of course some people would oppose this and say: “Then the proof of a proposition cannot ever be found, for, if it has been found, it is no longer the proof of this proposition”. But to say this is so far to say nothing at all.- It all depends what settles the sense of a proposition, what we choose to say settles its sense. The use of the signs must settle it; but what do we count as the use?-

That these proofs prove the same proposition means, e.g.: both demonstrate it as a suitable instrument for the same purpose.

And the purpose is an allusion to something outside mathematics. (RFM, pp. $366-367)$

It is the last sentence of the remark which is of importance but we have to understand the rest of the remark first. Here Wittgenstein’s notion that a proof demonstrates how to use a statement ( $R F M$, pp. 305-307) is in play. He notes that this conception can lead us to ask how it is that we can have multiple proofs of the same statement, since each proof must bring out some different use for the statement. His answer (elaborated more after the quoted passage) is that it all depends upon the other connections that the statement has, the various roles that are ascribed to it. One proof may be geometrical and another algebraic but this need not prevent us from taking them as proofs of the same statement, depending upon the setting (i.e. depending upon the rest of the mathematical system in which the statement/s play a part ${ }^{18}$ ).

So why does Wittgenstein say that the purpose revealed by a proof is an allusion to something outside mathematics? I think that Wittgenstein is suggesting that mathematical statements play roles in systems and that the proofs of those statements establish their roles. When a mathematical system is looked at in a more general way then the system itself will have some kind of purposes behind it since mathematicians don’t invent mathematical systems arbitrarily. It may be tempting to take Wittgenstein as saying that mathematical systems have to have direct applications to empirical

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descriptions or else they are arbitrary but the following remark suggests that is not what Wittgenstein meant by ‘outside mathematics’:

But then why doesn’t it need a sanction for this? Can it extend the network arbitrarily? Well, I could say: a mathematician is always inventing new forms of description. Some, stimulated by practical needs, others, from aesthetic needs, -and yet others in a variety of ways. And here imagine a landscape gardener designing paths for the layout of a garden; it may well be that he draws them on a drawing-board merely as ornamental strips without the slightest thought of someone’s sometime walking on them. (RFM, p. 99)

Aesthetic purposes are not the sorts of purposes that come to mind when Wittgenstein says ‘outside mathematics’ and yet the above remark reveals that aesthetic purposes are part of his picture. ${ }^{19}$ Wittgenstein seems to have thought that some systems might be (aesthetic) ends in themselves. Perhaps Wittgenstein even had in mind ‘aesthetic’ ends like the parsimony that can be achieved by setting up concepts to link previouslyunrelated mathematical systems. ${ }^{20}$

That Wittgenstein was not dismissive of pure mathematics is suggested by his ways of talking about ‘pure mathematics’ in $R F M$. At one point he suggests that pure mathematics can be characterised as the practice of deriving mathematical rules from other mathematical rules:

Any proof in applied mathematics may be conceived as a proof in pure mathematics which proves that this proposition follows from these propositions, or can be got from them by means of such and such operations; etc. (RFM, p. 436)

Rather than regarding pure mathematics as incidental to the core of mathematics, this conception seems to be an immensely wide one, under which most branches of mathematics would be pure mathematics. The idea that statements of applied mathematics can be transformed into statements of pure mathematics is not an obvious one. It suggests that Wittgenstein means something very different by ‘applied mathematics’ than what we might naturally assume him to mean.

Light is shed on Wittgenstein’s distinction between pure and applied mathematics when he considers whether we could imagine a people who have only an applied mathematics and not a pure mathematics ( $R F M$, p. 232). He imagines this people using a co-ordinate system to express physical predictions but he takes it that the theorems we associate with co-ordinate geometry would not be part of their mathematics, implying that they are not part of applied mathematics. Wittgenstein asks whether they would arrive at the commutative law of multiplication and suggests that they would not

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formulate it as a rule since it is “not a rule of their notation” nor “a proposition of their physics” and would hence have to be pure mathematics. ${ }^{21}$

What Wittgenstein seems to be driving at is the question of whether mathematical expressions need to be such that we see the point in them, as opposed to following a practice blindly (albeit a practice that yields desirable results). Wittgenstein seems to be considering whether our being able to see the point of certain propositions and derive others from them is something crucial to what we call mathematics. Regarding the commutative law, he says that the imagined people “do not need to obtain any such proposition-even if they allow the shift of factors” ( $R F M$, p. 232). He is trying to imagine that their “mathematics were done entirely in the form of orders” ( $R F M$, p. 232). These orders are used to make predictions but he says that it “does not matter at all how these people have arrived at this method of prediction” ( $R F M$, p. 232). The idea of expressions in the form of orders comes up again later and an illustration is then given-Wittgenstein wonders whether we could do mathematics just by formulating instructions like "let $10 \times 10$ be 100 " ( $R F M$, p. 276). Wittgenstein says that the “centre of gravity of their mathematics lies for these people entirely in doing” ( $R F M$, p. 232). The ‘doing’ here is presumably to be contrasted with activities like formulating, deriving, abstracting and seeing-as Wittgenstein says, “these people are not supposed to arrive at the conception of making mathematical discoveries” (RFM, p. 233).

So the transformation from applied to pure mathematics is being understood as a transformation from application of instructions to rules formulated so as to allow derivations (i.e. propositions). The people who lack pure mathematics do mathematics by moving from empirical statement to empirical statement without ever formulating the rules by which they make the transitions as propositions (formulating them instead only as instructions). Wittgenstein suggests that some techniques employed by physicists and engineers are like those employed by the tribe (in that the rule is never formulated as a proposition), citing specifically the determination of resultant force on an object by means of drawing a polygon to represent the individual forces:

Take the construction of the polygon of forces: isn’t that a bit of applied mathematics?

And where is the proposition of pure mathematics which is invoked in connexion with this graphical calculation? Is this case not like that of the tribe which has a technique of calculating in order to make certain predictions, but no propositions of pure mathematics? (RFM, p. 265)

The tribe might lack co-ordinate geometry and the commutative law but presumably they might have the technique of drawing force polygons.

Given that the theorems of co-ordinate geometry and the commutative law are considered to be pure mathematics by Wittgenstein, we have to wonder whether the sorts of examples that Wittgenstein uses in his illustrations of his idea of mathematical

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statements functioning as rules really are, as Maddy seems to assume, ${ }^{22}$ seen by him as statements of applied mathematics. Perhaps surprisingly, it appears not:
…the question “are there a hundred times as many marbles here as there?” is surely not a mathematical question. And the answer to it is not a mathematical proposition. A mathematical question would be: “are 170 marbles a hundred times as many as three marbles?” (And this is a question of pure, not of applied mathematics.)
Now ought I to say that whoever teaches us to count etc. gives us new concepts; and also whoever uses such concepts to teach us pure mathematics? (RFM, p. 412)

Clearly Wittgenstein regards at least some of pure mathematics as quite easy to relate to applications in empirical propositions.

It is interesting that Wittgenstein does not come to a firm view about whether we can imagine a people who lack a pure mathematics. He seems to be inclined towards the idea that it is essential to what we call mathematics that we formulate at least some of it as propositions. He says that it “is clear that mathematics as a technique for transforming signs for the purpose of prediction has nothing to do with grammar” (RFM, p. 234). If we are simply working with instructions and predictions then there is no element of necessity:

What is the transition that I make from “It will be like this” to “it must be like this”? I form a different concept. One involving something that was not there before. When I say: “If these derivations are the same, then it must be that…”, I am making something into a criterion of identity. (RFM, p. 237)

It seems that if we lacked a pure mathematics, at least as Wittgenstein uses the term in $R F M$, then what we would have would only be mathematics in a lesser sense, if it were mathematics at all.

The interpretation of Wittgenstein’s attitude towards pure mathematics being offered here is a long way from Maddy’s suggestion (1993, p. 68) that Wittgenstein regards pure mathematics as devoted to ‘fanciful applications’. Maddy might well object that how we use the term ‘pure mathematics’ is not important. What is important, she might suggest, is that Wittgenstein regards some of mathematics to be devoted to ‘fanciful applications’ and that Wittgenstein is scornful of this. But this is a misidentification of Wittgenstein’s target - Wittgenstein may be mindful to point out cases where we attribute ‘fanciful applications’ to a piece of mathematics and remind us that they are no more than fanciful (being as they are cases where our prose is a poor translation of the mathematics) but such misguided prose need not affect the validity of the mathematics. Even if the mathematical system in question lacks extra-

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mathematical application, it is enough that the mathematical system has connections ${ }^{23}$ to other mathematical systems (and that those systems have applications):

I have asked myself: if mathematics has a purely fanciful application, isn’t it still mathematics?-But the question arises: don’t we call it ‘mathematics’ only because e.g. there are transitions, bridges from the fanciful to non-fanciful applications? That is to say: should we say that people possessed a mathematics if they used calculating, operating with signs, merely for occult purposes?

But in that case isn’t it incorrect to say: the essential thing about mathematics is that it forms concepts?-For mathematics is after all an anthropological phenomenon. Thus we can recognize it as the essential thing about a great part of mathematics (of what is called ‘mathematics’) and yet say that it plays no part in other regions. This insight by itself will of course have some influence on people once they learn to see mathematics in this way. Mathematics is, then, a family; but that is not to say that we shall not mind what is incorporated into it. (RFM, p. 399)

It is not clear that this last remark (that mathematicians of the future might be more careful about what new mathematics is developed) has to be read as a prediction that mathematicians of the future will not count set theory (Wittgenstein refers to the axiom of choice shortly after this passage) as mathematics. As we have seen, Wittgenstein himself counts set theory as mathematics so there is no reason for him to make such a prediction. But Wittgenstein does clearly hope that mathematicians of the future will see mathematics with direct extra-mathematical applications to be core to mathematics and he hopes that this will lead them to direct mathematical enquiry differently. Wittgenstein is most likely suggesting that seeing mathematics as a family (specifically a family that has mathematics with extra-mathematical applications at the core) would lead to less significance being attributed to foundational pursuits, which were receiving a great deal of (philosophically-motivated) attention in Wittgenstein’s day, and more significance being attributed to mathematics with important extramathematical applications. ${ }^{24}$

Why might mathematicians want to direct mathematical enquiry differently if they saw systems with extra-mathematical applications to be core to mathematics? In part because mathematicians are likely to want to do work which is most central in the sense of being most characteristic of mathematics. But also in part because if mathematicians did see it as characteristic of mathematics to be linked to extra-mathematical applications then this would put pressure on the dogmatic application of certain pictures which Wittgenstein takes to be at the heart of the foundations crisis. Mathematicians would become less interested in building foundational systems, since they would have

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available a picture which can lead them to reminders of the variety of mathematics and so enable them to resist the scepticism that goes along with the dogmatic use of the pictures. Wittgenstein is suggesting that the motivation to unify all of mathematics under foundational systems (such as has been attempted with set theory) would be reduced if it could be seen that the quest for unity that goes along with such projects is itself a source of the scepticism that motivates the projects (by making certain parts of mathematics look problematic because they do not fit a particular picture). Understanding mathematics as a family would mean accepting the variety of mathematics, which would take away much of the motivation for attempting to reduce mathematics to a particular foundational picture.

Wittgenstein takes set theory as a fringe case of mathematics because it lacks empirical applications and as such it is only an awkward fit for the model of mathematical propositions as rules of description. Wittgenstein is giving the model of mathematical propositions as rules of description preferential treatment over the others in that he takes this model to bring out features that are most characteristic of mathematics. This preferential treatment is not ungrounded-as we have seen, Wittgenstein argues that mathematics would be a game if it were not applied, much as chess would no longer be a game if we used it to fight battles. Moreover, it is perfectly consistent with Wittgenstein’s methodology for him to give preferential treatment to a particular picture-Wittgenstein needs to emphasise the picture/s that he takes to be most illuminating, the picture/s which can do most work in resolving the problem/s at hand. The central problems of Wittgenstein’s philosophy of mathematics relate to scepticism arising from inappropriate models of mathematical propositions (such as the model of mathematical propositions as descriptions). The model of mathematical propositions as rules of description has a key role to play in showing that mathematics is a family more various and subtle than the description model makes it seem. Wittgenstein’s use of the rules of description model also shows set theory to be on the fringes of mathematics and not at its centre.

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