Abstract
In the scholarship on Wittgenstein’s later philosophy of mathematics, the dominant interpretation is a theoretical one that ascribes to Wittgenstein some type of ‘ism’ such as radical verificationism or anti-realism. Essentially, he is supposed to provide a positive account of our mathematical practice based on some basic assertions. However, I claim that he should not be read in terms of any ‘ism’ but instead should be read as examining philosophical pictures in the sense of unclear conceptions. The contrast here is that basic assertions that frame philosophical ‘isms’ are propositional such that they are subject to normal argumentative evaluation, while pictures in Wittgenstein’s sense are non-propositional—they lack a clear truth condition. They, therefore, need clarification rather than argumentation. In this paper, I provide a detailed analysis of Wittgenstein’s treatment of philosophical pictures with special focus on his argument on contradiction. I begin by explaining the problem with this trend of theoretical interpretation, taking Steve Gerrard’s otherwise excellent interpretation as a representative example and pointing out why it is problematic. Next, I will argue that those problems do not arise if we take Wittgenstein’s task as the clarification of philosophical pictures. I do this, first, by explaining Wittgenstein’s method using his argument concerning the Augustinian Picture in Philosophical Investigations and then pointing out that the same method can be identified in the crucial arguments in his philosophy of mathematics. Finally, in order to connect my interpretation with the current scholarship, I will explain the relation of my interpretation with those of New Wittgensteinian scholars.
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Notes
In providing the notion of philosophical picture, rather than that of grammatical rules, a prominent role, my reading falls somewhere along the newer reading. It is true that traditional readers sometimes mention the notion of picture as a source of philosophical confusion. However, they do not provide it the central role in Wittgenstein’s methodology, contrary to my reading. See, for example, Baker and Hacker (2009): pp. 277–299.
See Monk (2007) for a similar diagnosis.
This is not to say that these basic assertions are assumed without philosophical justification. For example, Dummett’s famous ‘acquisition argument’ and ‘manifestation argument’ make a case for the basic assertion of anti-realism. (Both arguments can be found in Dummett 1973; see also Wright 1986: pp. 13–23, for a helpful discussion.)
See Shapiro (2000) for a useful survey of contemporary ‘isms’ in the philosophy of mathematics.
See LFM: pp. 91–92, 138–139, 239–240. Cf. AWL: p. 222, 225.
Perhaps, Gerrard means that a picture is unsophisticated compared with an ism (Gerrard 1991, p. 128). However, this contrast does not seem to be substantive. For, if ‘unsophisticated’ only means that a picture is simplistic, then a picture is just a poor ism.
Gerrard characterises middle Wittgenstein’s ‘calculus conception’ by four principles, and then later explicates Wittgenstein’s ‘language-game conception’ as their modifications and discards (Gerrard 1991: pp. 133–136). Again, Gerrard shows no implication that those principles are anything different from basic assertions.
Wittgenstein’s works are indicated by their usual abbreviations, which are listed in References.
See also LFM: p. 103, where he denies that he is offering an opinion.
See also RFM: pp. 373–4.
For the sake of simplicity, I focus here on the assertive uses of a sentence.
In PI 38, Wittgenstein, criticizing Russell’s theory, notes that the word ‘name’ has many different uses.
Fischer (2006), pp. 472–473 also notes that one is unwittingly led by a philosophical picture and draws a conclusion which is not warranted.
My explanation above might be backed up by the stereotype theory in psychology (Rosch 1975; Rosch and Mervis 1975). Garfield (2000) mentions the possibility of connecting Wittgenstein’s philosophy of language with the stereotype theory. Fischer et al. (2015), Fischer and Engelhardt (2016) explore a kind of (Wittgensteinian) philosophical therapy in the philosophy of perception based on the stereotype theory.
Wittgenstein explicitly denies that he is concerned with introducing a new expression to solve philosophical problems. (PI 132)
There is a gap between the understanding of the linguistic meaning of a sentence and understanding what is said by using it on a particular occasion; therefore, a complete sentence may not express a proposition. See Conant (2002), Dobler (2013), Ohtani (2016), Travis (2006), Whiting (2010) for the discussions of Wittgenstein’s view on this gap.
Note that the notion of a picture should be distinguished from that of ‘prose’ (WVC: p. 149) or ‘gas’ (LFM: pp. 13–14), which is another key concept in Wittgenstein’s philosophy of mathematics. A picture in the philosophy of mathematics involves philosophical remarks such as ‘to a mathematical proposition corresponds a reality’, while prose or a gas in Wittgenstein’s sense is an interpretation of a calculus or a proof and typically appears in the last line of it. Thus, prose is presented as a part of mathematics, and some in fact constitute mathematics. One prose example that Wittgenstein discusses in several places (from his middle to later writings) is the interpretation of the conclusions of mathematical inductions: In the simplest cases, from F (1) and F (k) \(\rightarrow \) F (k + 1), we conclude, ‘for all n, F (n)’, which, according to Wittgenstein can be misleading because it suggests that we can use shortcuts for all of the, possibly infinite, steps of mathematical induction (BT: pp. 443–481, LFM: pp. 287–290, PG: pp. 395–450, RFM: pp. 186–187, WVC: pp. 134–136). The discussion on prose specifically concerns the interpretation of calculi and proofs, and thus we should distinguish it from the discussion of philosophical pictures in the philosophy of mathematics in general. See Dawson (2016), Floyd (2001), and Shanker (1987, ch. 5) for illuminating discussions on the notion of prose.
Wittgenstein says ‘wrong picture’ or ‘false picture’ here as well as in other places (PI 604, OC 249, Z 20, 111). This may raise a doubt about my interpretation which sees the problem with a picture as its lack of clarity. I admit that my wording does not correspond with that of Wittgenstein’s everywhere. As for the passage quoted here, he refers to what I call ‘model’ by the word ‘picture’. However, I think my wording is justified because (i) Wittgenstein at least sometimes refers to an unclear conception by the word ‘picture’ as we have seen in his discussion about the Augustinian Picture, and (ii) Wittgenstein does not use the word ‘picture (Bild)’ as a technical term, using it as an ordinary word, so that ‘wrong picture’ means roughly ‘misleading way of seeing things’. (He also says ‘bad picture’ in PI 136.) My aim in this paper is to highlight Wittgenstein’s method in the philosophy of mathematics by distinguishing picture and model, rather than to record his uses of the word ‘picture’.
Dawson (2014) argues that Wittgenstein’s remarks on the application of mathematics should not be taken as a general claim by citing the notion of an ‘object of comparison’. (PI 130–131) My interpretation has some similarity with Dawson’s view, but he does not distinguish between a picture and what I call a ‘model’, a distinction that, in my view, is important for understanding what Wittgenstein is doing in his philosophy of mathematics. See Kuusela (2008) for a good discussion of the notion of an object of comparison.
Juliet Floyd offers a fuller interpretation of Wittgenstein’s view on the impossibility proof of the trisection of an angle from a similar perspective as mine (Floyd 1995).
Note that empirical invention, such as the invention of the car engine, is different from mathematical invention in several respects. Particularly, the way in which the result of invention is used is different. What is invented in the case of the car engine is not a norm but a design that is used to produce a physical object, whereas in the mathematical invention, we have new norms that guide our actions: once a proof is made and accepted, it functions as a norm for our action. The case of the invention of the car engine is a model and only features the salient characteristics of our mathematical activity. We should not think that everything in the model is to be found in our actual mathematical activities.
Wittgenstein says, comparing proofs to riddles, that ‘I shall know a good solution if I see it’ (LFM: p. 84). See Diamond (1977) for more on this aspect of mathematical proof.
See Floyd (1995) for the details of historical facts that support my reading here.
Wittgenstein contrasts the search for a white lion with the mathematical search for the construction of the heptagon in his lecture. (LFM: pp. 64–68)
We can distinguish two kinds of situations here: (i) We may want a contradiction in our calculus. For example, finding a contradiction in a system may perhaps have an aesthetic point (cf. RFM: p. 214), the purpose of the practice. In that case, the possibility of a hidden contradiction is a condition for a system to be interesting. (ii) We can ignore a contradiction. For example, suppose ZFC set theory contains a hidden contradiction. Even after finding a contradiction in ZFC, we do not conclude that all of our previously determined theoretical reasoning, which are believed to be done in ZFC, are useless or destroyed. We may find surrogates for (most of) them in a new system designed to avoid the new contradiction. See Maddy (1997), ch. 2, on the idea that the purpose of a foundational system is not to justify our mathematics but to offer a theory in which surrogates of various forms of mathematical reasoning are expressed.
It might be useful here to mention Felix Mühlhölzer, who offers a series of highly enlightening interpretations of Wittgenstein’s view of mathematics (Mühlhölzer 2005, 2015). Although Mühlhölzer explicitly rejects the ascription to Wittgenstein of taking a stance in the realism / anti-realism debate (Mühlhölzer 2015: p. 206), it seems to me that he ends up attributing a kind of conventionalism to Wittgenstein, according to which mathematical theorems are linguistic rules based on our stipulations (see Mühlhölzer 2015.) It is true that, as Mühlhölzer points out, Wittgenstein remarks that mathematical theorems function as rules or paradigms for our linguistic practices (Mühlhölzer 2005: pp. 71–73, 2015: p. 191); but I disagree with Mühlhölzer in that, unlike him, I take Wittgenstein’s remarks as being presented as little more than useful pictures that begin to clarify some (not all) aspects of mathematics. As such, in one lecture, Wittgenstein explicitly states, “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...’, (LFM: p. 55)” thus indicating that his comments should be taken as material to be investigated rather than as a definite assertion.
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Acknowledgements
This work was supported by the Japan Society for the Promotion of Science (JSPS KAKENHI Grant Number 25370029). I am indebted to Musashino University for the sabbatical leave, which has enabled me to write this paper. I am also indebted to the University of East Anglia for accepting me as an academic visitor during my sabbatical year. I am grateful to Ryan Dawson, Tamara Dobler, Eugen Fischer and Oskari Kuusela for helpful comments on the earlier versions of this paper.
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Ohtani, H. Philosophical pictures about mathematics: Wittgenstein and contradiction. Synthese 195, 2039–2063 (2018). https://doi.org/10.1007/s11229-017-1317-z
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DOI: https://doi.org/10.1007/s11229-017-1317-z