# Philosophical pictures about mathematics: Wittgenstein and contradiction

## 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.

## Keywords

Wittgenstein Contradiction Philosophical picture Philosophical methodology Theory in the philosophy of mathematics The Hardyian Picture## Notes

### 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.

## Compliance with ethical standards

## Conflict of interest

The author declares that there are no conflicts of interest.

## Ethical standards

I assert that I am the sole author of this work and that it is my original work. I assert that the article has not received prior publication and is not under consideration for publication elsewhere. This research has not been submitted for publication nor has it been published in whole or in part elsewhere.

## References

- Baker, G. P. (1991). Philosophical investigations §122: Neglected aspects. In R. L. Arrington & H. J. Glock (Eds.),
*Wittgenstein’s ‘Philosophical Investigations’: Text and Context*. London: Routledge. Reprinted in G. P. Baker (2004). K. Morris (Eds.),*Wittgenstein’s method: Neglected aspects*, (pp. 22–51). Oxford: Blackwell.Google Scholar - Baker, G. P. (2001). Wittgenstein: Concepts or conceptions?
*Harvard Review of Philosophy*,*9*. Reprinted in G. P. Baker (2004). K. Morris (Eds.),*Wittgenstein’s method: Neglected aspects*(pp. 260–278). Oxford: Blackwell.Google Scholar - Baker, G. P., & Hacker, P. M. S. (2009).
*Wittgenstein: Understanding and meaning, part 1: Essays*(2nd ed.). Oxford: Wiley-Blackwell.Google Scholar - Chihara, C. (1977). Wittgenstein’s analysis of paradoxes in his lectures on the foundations of mathematics.
*The Philosophical Review*,*72*(3), 365–381.CrossRefGoogle Scholar - Conant, J. (1997). On Wittgenstein’s philosophy of mathematics.
*Proceedings of the Aristotelian Society*,*97*, 195–222.CrossRefGoogle Scholar - Conant, J. (2002). The method of the tractatus. In E. H. Reck (Ed.),
*From Frege to Wittgenstein: Perspectives on early analytic philosophy*(pp. 374–462). Oxford: Oxford University Press.CrossRefGoogle Scholar - Dobler, T. (2013). What is wrong with Hacker’s Wittgenstein? On grammar, context and sense-determination.
*Philosophical Investigations*,*36*(3), 231–250.CrossRefGoogle Scholar - Dawson, R. (2014). Wittgenstein on pure and applied mathematics.
*Synthese*,*191*(17), 4131–4148.CrossRefGoogle Scholar - Dawson, R. (2016). Wittgenstein on set theory and the enormously big.
*Philosophical Investigations*,*39*(4), 313–334.CrossRefGoogle Scholar - Diamond, C. (1977). Riddles and Anselm’s Riddle.
*Proceedings of the Aristotelian Society*,*51*. Reprinted in C. Diamond (1991).*The realistic spirit*(pp. 267–289). Cambridge: The MIT Press.Google Scholar - Diamond, C. (1981). What nonsense might be. Philosophy, 56. Reprinted in C. Diamond,. (1991).
*The realistic spirit*(pp. 95–114). Cambridge: The MIT Press.Google Scholar - Dummett, M. (1959). Wittgenstein’s philosophy of mathematics. The Philosophical Review, 68(3). Reprinted in Dummett, M., (1978).
*Truth and other enigmas*(pp. 166–185). Cambridge: Harvard University Press.Google Scholar - Dummett, M. (1973). Philosophical basis of intuitionistic logic. In H. E. Rose & J. C. Shepherdson (Eds.),
*Proceedings of the Logic Colloquium, Bristol*. Reprinted in Dummett, M. (1978).*Truth and other enigmas*(pp. 215–247). Cambridge: Harvard University Press.Google Scholar - Egan, D. (2011). Pictures in Wittgenstein’s later philosophy.
*Philosophical Investigations*,*34*(1), 55–76.CrossRefGoogle Scholar - Fischer, E. (2008). Wittgenstein’s ‘Non-cognitivism’–Explained and vindicated.
*Synthese*,*162*(1), 53–84.CrossRefGoogle Scholar - Fischer, E., & Engelhardt, P. E. (2016). Intuition’s linguistic sources: Stereotypes, intuitions and illusions.
*Mind and Language*,*31*(1), 67–103.CrossRefGoogle Scholar - Fischer, E., Engelhardt, P. E., & Herbelot, A. (2015). Intuitions and illusions. From explanation and experiment to assessment. In E. Fischer & J. Collins (Eds.),
*Experimental philosophy, rationalism and naturalism*(pp. 259–292). London: Routledge.Google Scholar - Floyd, J. (1991). Wittgenstein 2, 2, 2.: The opening of remarks on the foundations of mathematics.
*Synthese*,*87*(1), 143–180.CrossRefGoogle Scholar - Floyd, J. (1995). On saying what you really want to say: Wittgenstein, Gödel and the trisection of the angle. In J. Hintikka (Ed.),
*From Dedekind to Gödel: Essays on the development of the foundations of mathematics*(pp. 373–423). Dordrecht: Kluwer.CrossRefGoogle Scholar - Floyd, J. (2001). Proof versus prose: Wittgenstein on Gödel, tarski and truth.
*Philosophia Mathematica*,*9*(3), 280–307.CrossRefGoogle Scholar - Floyd, J. (2005). Wittgenstein on philosophy of logic and mathematics. In S. Shapiro (Ed.),
*Oxford handbook of philosophy of mathematics and logic*(pp. 75–128). Oxford: Oxford University Press.CrossRefGoogle Scholar - Frascolla, P. (1994).
*Wittgenstein’s philosophy of mathematics*. London: Routledge.Google Scholar - Frascolla, P. (2001). Philosophy of mathematics. In H.-J. Glock (Ed.),
*Wittgenstein: A critical reader*(pp. 268–288). Oxford: Blackwell.Google Scholar - Garfield, J. (2000). Particularity and principles: The structure of moral knowledge. In B. Hooker & M. O. Little (Eds.),
*Moral particularism*(pp. 178–204). Oxford: Clarendon Press.Google Scholar - Gerrard, S. (1991). Wittgenstein’s philosophies of mathematics.
*Synthese*,*87*(1), 125–142.CrossRefGoogle Scholar - Glock, H.-J. (1991). Philosophical investigations section 128: ‘Theses in philosophy’ and undogmatic procedure. In R. L. Arrington & H.-J. Glock (Eds.),
*Wittgenstein’s philosophical investigations: Text and context*(pp. 69–88). London: Routledge.Google Scholar - Hacker, P. M. S. (2012). Wittgenstein on grammar, theses and dogmatism.
*Philosophical Investigations*,*35*(1), 1–17.CrossRefGoogle Scholar - Hutchinson, P. (2007). What’s the point of elucidation?
*Metaphilosophy*,*38*(5), 691–713.CrossRefGoogle Scholar - Kuusela, O. (2008).
*The struggle against dogmatism: Wittgenstein and the concept of philosophy*. Cambridge: Harvard University Press.Google Scholar - Maddy, P. (1997).
*Naturalism in mathematics*. Oxford: Oxford University Press.Google Scholar - Marion, M. (1998).
*Wittgenstein, finitism, and foundations of mathematics*. Oxford: Oxford University Press.Google Scholar - Marion, M. (2009). Radical anti-realism, Wittgenstein and the length of proofs.
*Synthese*,*171*(3), 419–432.CrossRefGoogle Scholar - McDowell, J. (2009).
*Wittgensteinian quietism. Common Knowledge*,*15*(3), 365–372.CrossRefGoogle Scholar - Monk, R. (2007). Bourgeois, bolshevist or anarchist? The reception of Wittgenstein’s philosophy of mathematics. In G. Kahane, E. Kanterian, & O. Kuusela (Eds.),
*Wittgenstein and his interpreters*(pp. 269–294). Oxford: Blackwell.CrossRefGoogle Scholar - Moyal-Sharrock, D. (2013). Beyond Hacker’s Wittgenstein.
*Philosophical Investigations*,*36*(4), 355–380.CrossRefGoogle Scholar - Mühlhölzer, F. (2005). “A mathematical proof must be surveyable” What Wittgenstein meant by this and what it implies.
*Grazer Philosophische Studien*,*71*, 57–86.Google Scholar - Mühlhölzer, F. (2015). Putnam, Wittgenstein, and the objectivity of mathematics. In R. E. Auxier, D. R. Anderson, & L. E. Hahn (Eds.),
*The philosophy of Hilary Putnam*(pp. 181–211). Chicago: Open Court.Google Scholar - Ohtani, H. (2016). Wittgenstein on context and philosophical pictures.
*Synthese*,*193*(6), 1795–1816.CrossRefGoogle Scholar - Panjvani, C. (2006). Wittgenstein and strong mathematical verificationism.
*Philosophical Quarterly*,*56*(224), 406–425.CrossRefGoogle Scholar - Potter, M. (2011). Wittgenstein on mathematics. In O. Kuusela & M. McGinn (Eds.),
*Oxford handbook of Wittgenstein*(pp. 122–137). Oxford: Oxford University Press.Google Scholar - Putnam, H. (2001). Was Wittgenstein really an anti-realist about mathematics? In T. McCarthy & S. C. Stidd (Eds.),
*Wittgenstein in America*(pp. 140–194). Oxford: Clarendon Press.Google Scholar - Rosch, E. (1975). Cognitive reference points.
*Cognitive Psychology*,*7*, 532–547.CrossRefGoogle Scholar - Rosch, E., & Mervis, C. B. (1975). Family resemblances: Studies in the internal structure of categories.
*Cognitive Psychology*,*7*, 573–605.CrossRefGoogle Scholar - Russell, B. (1918). The philosophy of logical atomism. In J. Slater (1986) (Ed.).
*The collected papers of Bertrand Russell, Vol. 8: The philosophy of logical atomism and other essays 1914–19*(pp. 157–244). London: Routledge.Google Scholar - Schönbaumsfeld, G. (2010). A “Resolute” later Wittgenstein?
*Metaphilosophy*,*41*(5), 649–668.CrossRefGoogle Scholar - Shanker, S. (1987).
*Wittgenstein and turning-point in the philosophy of mathematics*. Albany: State University of New York Press.Google Scholar - Shapiro, S. (2000).
*Thinking about mathematics: The philosophy of mathematics*. Oxford: Oxford University Press.Google Scholar - Travis, C. (2006).
*Thought’s footing: A theme in Wittgenstein’s philosophical investigations*. Oxford: Oxford University Press.CrossRefGoogle Scholar - Whiting, D. (2010). Particular and general: Wittgenstein, linguistic rules, and context. In D. Whiting (Ed.),
*The later Wittgenstein on language*(pp. 114–132). Hampshire: Palgrave Macmillan.Google Scholar - Witherspoon, E. (2000). Conceptions of nonsense in Carnap and Wittgenstein. In A. Crary & R. Read (Eds.),
*The new Wittgenstein*(pp. 315–349). London: Routledge.Google Scholar - Wittgenstein, L. (1969).
*The blue and brown books*(2nd ed.). Oxford: Blackwell [BB].Google Scholar - Wittgenstein, L. (1974).
*Philosophical grammar*. In R. Rhees (Ed.), A. Kenny (Trans.), Oxford: Blackwell [PG].Google Scholar - Wittgenstein, L. (1975).
*Wittgenstein’s lectures on the foundations of mathematics: Cambridge 1939*. In C. Diamond (Ed.). Chicago: The University of Chicago Press [LFM].Google Scholar - Wittgenstein, L. (1978).
*Remarks on the foundations of mathematics*(3rd ed.), G. H. von Wright, R. Rhees, & G. E. M. Anscombe (Eds.), G. E. M. Anscombe (Trans.). Oxford: Blackwell [RFM].Google Scholar - Wittgenstein, L. (1979a).
*Ludwig Wittgenstein and the Vienna Circle*. In B. McGuiness (Ed.), J. Shulte & B. McGuiness (Trans.). Oxford: Basil Blackwell [WVC].Google Scholar - Wittgenstein, L. (1979b).
*Wittgenstein’s Lectures: Cambridge 1932–1935. From the Notes of Alice Ambrose and Margaret Macdonald*. In: A. Ambrose (Ed.). Chicago: University of Chicago Press [AWL].Google Scholar - Wittgenstein, L. (1981).
*Zettel*(2nd ed.) G. E. M. Anscombe (Trans.). Oxford: Blackwell [Z].Google Scholar - Wittgenstein, L. (2005).
*The Big Typescript: TS 213*. In C. G. Luckhardt & M. A. E. Aue (Eds.), Oxford: Wiley-Blackwell [BT].Google Scholar - Wittgenstein, L. (2009).
*Philosophical investigations*(4th ed.) G. E. M. Anscombe, P. M. S. Hacker & J. Schulte (Trans.). Oxford: Wiley-Blackwell [PI].Google Scholar - Wright, C. (1980).
*Wittgenstein on the foundations of mathematics*. London: Duckworth.Google Scholar - Wright, C. (1986). Introduction. In C. Wright (Ed.),
*Realism, meaning and truth*(pp. 1–43). Oxford: Blackwell.Google Scholar