Wittgenstein and Stenlund on Mathematical Symbolism


  • Martin Gullvåg Sætre University of Bergen




Wittgenstein, symbolic mathematics, ontological mathematics, symbolism, mathematical symbolism, mathematical progress


In recent work, Sören Stenlund (2015) contextualizes Wittgenstein’s philosophy of mathematics as aligned with the tradition of symbolic mathematics. In the early modern era, mathematicians began using purely formal methods disconnected from any obvious empirical applications, transforming their subject into a symbolic discipline. With this, Stenlund argues, they were freeing themselves of ancient ontological presuppositions and discovering the ultimately autonomous nature of mathematical symbolism, which eventually formed the basis for Wittgenstein’s thinking. A crucial premise of Wittgenstein’s philosophy of mathematics, on this view, is that the development of mathematical concepts is independent of any ontological implications and occurs in principle without normative connections to empirical applicability. This paper critically examines this narrative and arrives at the conclusion that Stenlund’s view of mathematical progress is in stark contrast to the later Wittgenstein’s writing, which emphasizes links between symbolisms and their applications.

Author Biography

Martin Gullvåg Sætre, University of Bergen

Martin Gullvåg Sætre is a PhD candidate at the University of Bergen, where he is a member of the project Mathematics with a Human Face: Set Theory within a Naturalized Wittgensteinian Framework. His dissertation, Wittgenstein on the Anthropological Grounds of Mathematics, delves into the later Wittgenstein’s writings and lectures on mathematics and situates them within Wittgenstein’s anthropologically oriented philosophy.


Abdulaziz, A. A., 2010. “The Plimpton 322 tablet and the Babylonian method of generating Pythagorean triples”. arXiv:1004.0025.

Cajori, F., 1993/1929. A History of Mathematical Notations. Mineola, NY: Dover Publications.

Carraher, D. W., Martinez, M. V., Schliemann, A. D., 2008. “Early algebra and mathematical generalization”. ZDM - Mathematics Education 40, 3–22.

Conant, J., 2019. “Reply to Gustafsson”. In: S. Miguens, ed., The Logical Alien: Conant and His Critics. Cambridge, MA: Harvard University Press, 863–947.

Dantzig, T., 1930/2005. Number: The Language of Science. J. Mazur ed., New York, NY: Pi Press.

Dieudonné, J., 1998. Mathematics – The Music of Reason. H.G. Dales and J.C. Dales trans., Berlin: Springer.

Gustafsson, M., 2019. “Wittgenstein on Using Language and Playing Chess: The Breakdown of an Analogy and Its Consequences”. In: S. Miguens, ed., The Logical Alien: Conant and His Critics. Cambridge, MA: Harvard University Press, 202–221.

Heeffer, A., 2009. “On the nature and origin of algebraic symbolism”. In: B. V., Kerkhove, ed., New Perspectives on Mathematical Practices: Essays in Philosophy and History of Mathematics. Singapore: World Scientific Publishing, 1–27.

Katz, V. J., 2007. “Stages in the history of algebra with implications for teaching”. Educational Studies in Mathematics 66, 185–201.

Klein, J., 1936/1968. Greek Mathematical Thought and the Origin of Algebra. Cambridge, MA: MIT Press.

Nakano, A. L., 2020. “Wittgenstein, formalism, and symbolic mathematics”. Kriterion: Journal of Philosophy 61, 31–53.

Nesselmann, G. H. F., 1969/1842. Die Algebra der Griechen. Berlin: Minerva.

Stenlund, S., 2014. “The Origin of Symbolic Mathematics and the End of the Science of Quantity”. Uppsala Philosophical Studies 59.

Stenlund, S., 2015. “On the Origin of Symbolic Mathematics and Its Significance for Wittgenstein’s Thought”. Nordic Wittgenstein Review 4, 7–92.

Unguru, S., 1975. “On the need to rewrite the history of Greek mathematics”. Archive for History of Exact Sciences 15, 67–114.

Unguru, S., 1991. “Greek Mathematics and Mathematical Induction”. Physis 28, 273–289.

Unguru, S., 1994. “Fowling after Induction, Reply to D. Fowler’s comments”. Physis 31, 267– 272.

Wittgenstein, L., 1921/2021. Tractatus Logico-Philosophicus [TLP]. Side-by-side ed., K. C. Klement ed., C. K. Ogden, D. F. Pears and B. F. McGuinness trans. [online] Available at: <https://people.umass.edu/~klement/tlp/tlp.pdf> [February 2023].

Wittgenstein, L., 1939/1976. Wittgenstein’s Lectures on the Foundations of Mathematics, Cambridge, 1939: From the Notes of R.G. Bosanquet, Norman Malcolm, Rush Rhees, Yorick Smithies [LFM]. C. Diamond ed., Ithaca, NY: Cornell University Press.

Wittgenstein, L., 1953/2009. Philosophical Investigations [PI]. 4th ed., G. E. M. Anscombe, P. M. S. Hacker and J. Schulte trans., P. M. S. Hacker and J. Schulte ed., Oxford: Wiley- Blackwell.

Wittgenstein, L., 1956/1978. Remarks on the Foundations of Mathematics [RFM]. 3rd ed., G. H. von Wright, R. Rhees and G. E. M. Anscombe eds., G. E. M. Anscombe trans., Oxford: Basil Blackwell.

Wittgenstein, L., 1974/1980. Philosophical Grammar [PG]. R. Rhees ed., A. Kenny trans., Malden, MA: Blackwell Publishing.

Wittgenstein, L., 1979. Wittgenstein and the Vienna Circle: Conversations Recorded by Friedrich Waismann [WWK]. B. McGuinness ed., J. Schulte and B. McGuinness trans., Oxford: Basil Blackwell.