On the Origin of Symbolic Mathematics and Its Significance for Wittgenstein’s Thought


  • Sören Stenlund Department of Philosophy Uppsala University




calculation, symbolism, formalism, blind thought, infinity, symbolic mathematics, Wittgenstein Ludwig, Hertz Heinrich, Weyl Hermann


The main topic of this essay is symbolic mathematics or the method of symbolic construction, which I trace to the end of the sixteenth century when Franciscus Vieta invented the algebraic symbolism and started to use the word ‘symbolic’ in the relevant, non-ontological sense. This approach has played an important role for many of the great inventions in modern mathematics such as the introduction of the decimal place-value system of numeration, Descartes’ analytic geometry, and Leibniz’s infinitesimal calculus. It was also central for the rigorization movement in mathematics in the late nineteenth century, as well as for the mathematics of modern physics in the 20th century.

However, the nature of symbolic mathematics has been concealed and confused due to the strong influence of the heritage from the Euclidean and Aristotelian traditions. This essay sheds some light on what has been concealed by approaching some of the crucial issues from a historical perspective. Furthermore, I argue that the conception of modern mathematics as symbolic mathematics was essential to Wittgenstein’s approach to the foundations and nature of mathematics. This connection between Wittgenstein’s thought and symbolic mathematics provides the resources for countering the still prevalent view that he defended an uttrely idiosyncratic conception, disconnected from the progress of serious science. Instead, his project can be seen as clarifying ideas that have been crucial to the development of mathematics since early modernity.

Author Biography

Sören Stenlund, Department of Philosophy Uppsala University

Sören Stenlund was appointed professor of theoretical philosophy at Uppsala University (Sweden) in 2000. He has acted as head of the Philosophy Department in Uppsala and was the Director of Graduate Studies in the Department of Philosophy and the coordinator of the seminar for theoretical philosophy until his retirement in 2008. Since his dissertation in 1972, Stenlund has published books and articles on various themes in the philosophies of language, logic and mathematics, such as Language and Philosophical Problems (Routledge 1991). Apart from these themes, Stenlund has discussed a vast array of other philosophical issues concerning the nature and history of philosophy and the philosophy of culture.


Aristotle, 1984, The Complete Works of Aristotle, Vols. 1 and 2. Ed. Jonathan Barnes. Princeton: Princeton University Press.

Bell, E.T., 1937. Men of Mathematics. London: Victor Gollancz LTD.

Bottazzini, U., 2003. “Complex Function Theory, 1780-1900”. In Jahnke, ed. 2003, pp. 213-259.

Boyer, C.B., 1968. A History of Mathematics. New York: Wiley, cop.

Brauer, R. and Weyl, H., 1935. “Spinors in n Dimensions”. American Journal of Math. 57, pp. 425-449.

Bussey, W. H. 1917. “The Origin of Mathematical Induction”. American Mathematical Monthly 24, pp. 199–207.

Calinger, R., 1999. A Contextual History of Mathematics to Euler. Upper Saddle River: Prentice Hall, cop.

Carnap, R., 1959. The Logical syntax of Language. English ed. first published in 1937, transl. by A. Smeaton. London: Routledge & Kegan Paul.

Cauchy, A. 1882-1919. Oeuvres complètes d’Augustin Cauchy. Paris: Gauthier-Villars.

Cassirer, E. 1957. The Philosophy of Symbolic Forms, Vol. 3: The Phenomenology of Knowledge. New Haven and London: Yale University Press.

Cajori, F. 1918. “Origin of the Name Mathematical Induction”. American Mathematical Monthly 25, pp. 197–201.

Corry, L., 2006. “Axiomatics, Empiricism, and Anschauung in Hilbert’s Conception of Geometry: Between Arithmetic and General Relativity”. In Ferreirós and Gray (2006), pp. 133-156.

De Morgan, A., 1849. Trigonometry and Double Algebra. London: Taylor, Walton, & Maberly.

Diamond, C. (ed.), 1976. Wittgenstein’s Lectures on the Foundations of Mathematics, Cambridge 1939. Chicago and London: The University of Chicago Press.

Eddington, A.S. 1931. The Nature of the Physical World. Cambridge: Cambridge University Press.

Engberg-Pedersen, T., 1979. “More on Aristotelian Epagoge”, Phronesis, Vol. 24 (3), pp. 301-319.

Epple, M., 2003. “The End of the Science of Quantity: Foundations of Analysis 1860-1910”. In Jahnke (ed.) 2003, pp. 291-323.

Ernst, P., 1982. “Mathematical Induction. A recurring theme.” Mathematical Gazzette Vol. 66 (436), 120-125.

Esquisabel, O.M., 2012. “Representing and Abstracting. An Analysis of Leibniz’s Concept of Symbolic Mathematics”. In Casanave (ed.) 2012, pp. 1-49.

Euclid. Elementa. Post I. L. Heiberg edit E. S. Stamatis. 5 vol. Leipzig-Stuttgart: B. G.Teubner, 1969–1977.

Ferreiros, J. & Gray, J.J., 2006. The Architecture of Modern Mathematics. Oxford: Oxford University Press.

Fowler, D., 1994. “Could the Greeks Have Used Mathematical Induction? Did They Use It?” Physis XXXI, pp. 253–265.

Frege, G., 1960. Translations from the Philosophical Writings of Gottlob Frege, Geach, P. and Black, M. (eds.). Oxford: Basil Blackwell.

Frege, G., 1979. Posthumous Writings, H. Hermes (ed.), transl. by P. Long. Oxford: Basil Blackwell.

Frege, G., 1980. Philosophical and Mathematical Correspondence, G. Gabriel, et al. (eds.). Oxford: Blackwell publishers.

Freudenthal, H. 1953. “Zur Geschichte der vollständigen Induktion”. Archives Internationales d’Histoire des Sciences 6, pp. 17–37.

Freudenthal, H. 1962. “The main trends in the foundations of geometry in the 19th century”. In Ernst Nagel et al., (eds.) Logic, methodology and philosophy of science. Stanford: Stanford Univ. Press, pp. 613-621.

Goodstein, R.L., 1956. “The Arabic Numerals, Numbers and the definition of Counting”, The Mathematical Gazette 40 (332), pp. 114-129.

Grassmann, H., 1844. Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik dargestellt. Leibzig Otto Wiegand. <https://archive.org/details/dielinealeausde00grasgoog> (Accessed July 17, 2015).

Grattan-Guinness, I., 2004, “The mathematics of the past: distinguishing its history from our heritage”. Historia Mathematica, vol. 31, pp. 163-185.

Guicciardini, N., 2003. “Conceptualism and contextualism in the recent historiography of Newton's Principia” Historia Mathematica 30 (4), pp. 407-431. http://dx.doi.org/10.1016/S0315-0860(03)00051-X

Hertz, H., 1956. The Principles of Mechanics Presented in a New Form. New York, Dover (English translation of Die Prinzipen der Mechanik in neuem Zusammanhange dargestellt, Leipzig, 1894.)

Hestenes, D. and Sobczyk, G., 1984. Clifford Algebra to Geometric Calculus. A Unified Language for Mathematics and Physics. Dordrecht: D. Reidel Publishing Company.

Hilbert, D., 1903. Grundlagen der Geometrie, Leibzig: B. G. Taubner. <https://archive.org/details/grunddergeovon00hilbrich> (Accessed July 17, 2015).

Hilbert, D., 1922. “The New Grounding of Mathematics. First Report.” In Ewald, W., 1996, From Kant to Hilbert. A Source Book in the Foundations of Mathematics, Volume II. Oxford: Oxford University Press, pp. 1115-1134.

Hilbert, D., 1923. “The Logical Foundations of Mathematics”. In Ewald, W., 1996. From Kant to Hilbert. A Source Book in the Foundations of Mathematics, Volume II. Oxford: Oxford University Press, 1996. Pp. 1134-1148.

Hilbert, D., 1926. “On the infinite”, in van Heijenoort, Jean (ed.), 1967, From Frege to Gödel. A Source Book in Mathematical Logic, 1897–1931. Cambridge, Mass.: Harvard University Press.

Hilbert, D., 1927. “The Foundations of Mathematics”. In van Heijenoort, 1967, From Frege to Gödel. Cambridge (Mass.): Harvard University Press, pp. 464-479.

Hopkins, B.C., 2011. The Origin of the Logic of Symbolic Mathematics: Edmund Husserl and Jacob Klein. Bloomington, Indiana: Indiana University Press.

Ishiguro, H. 1990. Leibniz’s Philosophy of Logic and Language. Cambridge, New York: Cambridge University Press.

Jahnke, H.N. (ed.), 2003. A History of Analysis. History of Mathematics Volume 24. Providence Rhode Island: Amer. Mathematical Society, London Math. Society.

Jahnke, H. N., 2003. “Algebraic Analysis in the 18th Century”. In Jahnke (ed.) 2003, pp. 105-136.

Kastanis, N. & Thomaidis, Y.: 1991, ‘The Term “Geometrical Algebra”, Target of a Contemporary Epistemological Debate’, in N. Kastanis (ed.), On the Historiography of Ancient Greek Mathematics (pp. 1–26), Group for the history of mathematics of Greek society for the history of science and technology. <http://users.auth.gr/~nioka/Files/GEOMALGE.pdf> (Accessed July 6, 2015).

Klein, J., 1968. Greek Mathematical Thought and the Origin of Algebra. Cambridge (Mass.), MIT Press.

Kreisel, G., 1958. “Hilbert’s Programme”, Dialectica 12 (3-4), pp. 346-372.

Krämer, S., 1996. “Mind, Symbolism, Formalism: Is Leibniz a Precursor of Artificial Intelligence?”. Knowl. Org. 23 (2), pp.83-87.

Leibniz, G. W. 1962. Matematische Schriften, Hildesheim: Olm.

Loemker, L. E. (ed.), 1969. Leibniz. Philosophical Papers and Letters. Dordrecht/Boston/London, D. Reidel Publishing Company.

Lützen, J., 2003. “The Foundation of Analysis in the 19th Century”. In Jahnke 2003, pp. 155-195.

Mancosu, P. 1996. Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century. New York: Oxford University Press.

Mancosu, P. 1998. From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s. Oxford and New York: Oxford University Press.

Mancosu, P. & Ryckman, T., 2002. “Mathematics and Phenomenology. The correspondence between Oskar Becker and Hermann Weyl,” Philosophia Mathematica 10, pp. 130-202.

Müller, A. (1923). “Über Zahlen als Zeichen”. Mathematische Annalen 90, pp. 150-159.

Mueller, I., 1981. Philosophy of Mathematics and Deductive Structure in Euclid’s Elements. Cambridge (Mass.), MIT Press.

Netz, R. 2004. The Transformation of Mathematics in the Early Mediterranean World. Cambridge: Cambridge UP.

Ortega y Gasset, J. 1971. The Idea of Principle in Leibnitz and the Evolution of Deductive Theory. New York: W.W. Norton & Company, Inc. (Transl. Mildred Adams.)

Putnam, H., 2012. Philosophy in an Age of Science, Cambridge MA: Harvard University Press.

Rabinovitch, N. L., 1970. “Rabbi Levi Ben Gerson and the Origins of Mathematical Induction”. Archive for History of Exact Sciences 6, pp. 237–248.

Scholz, E. 2006. “Practice-related symbolic realism in H. Weyl’s mature view of mathematical knowledge”. In: Ferreirós and Gray 2006, pp. 291-309.

Stenlund, S., 2012. “The “middle Wittgenstein” and Modern Mathematics”. In: Dybjer et al. (eds.) Epistemology versus Ontology, Essays on the Philosophy and Foundations of Mathematics in Honour of Per Martin-Löf. Dordrecht: Springer Science + Business Media, pp.139-159.

Stenlund, S., 2012b. “Different senses of finitude: An inquiry into Hilbert’s finitism”. Synthese 185, pp. 335-363.

Stenlund, S., 2014. The Origin of Symbolic Mathematics and the End of the Science of Quantity, Uppsala Department of Philosophy 2014: <http://urn.kb.se/resolve?urn=urn%3Anbn%3Ase%3Auu%3Adiva-221570>

Thomae, J., 1898. Elementare Theorie der analytischen Functionen einer complexen Veränderlichen, 2nd Ed. Halle a.S.: Nebert.

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

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

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

Vieta, F. (Viète, François), 1968. Introduction to the Analytic Art, English translation in appendix to Klein 1968.

Van der Waerden, B.L., 1975. “Defence of a ‘Shocking’ point of view,” Archive for History of Exact Sciences 15, pp. 199-210.

Van der Waerden, B.L., 1985. A History of Algebra. From Al-Khwarizmi to Emmy Noether. Berlin: Springer-Verlag.

Weir, Alan, 2011. “Formalism in the Philosophy of Mathematics”. In: Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy (Fall 2011 Ed.),


Weyl, H., 1918. Das Kontinuum. Veit & Co., Leipzig, Reprinted 1987. 2 edn, de Gryter & Co., Berlin, 1932. English translation: The Continuum: A Critical Examination of the Foundation of Analysis, translated by Stephen Pollard and Thomas Bole, Thomas Jefferson University Press: 1987. (Re-publication, Dover 1994.)

Weyl, H., 1927. “The Current Epistemological Situation in Mathematics”. In: Mancosu 1998: 123-142.

Weyl, H., 1949. Philosophy of Mathematics and Natural Science (Revised and Augmented English Edition). Princeton: Princeton Univ. Press.

Weyl, H., 1949. ”Wissenschaft als symbolische Konstruktion des Menschen”. In: Weyl, H. 1968, Gesammelte Abhandlungen, Band IV. Berlin: Springer-Verlag, pp. 289-345. (First published in Eranos-Jahrbuch 1948, pp. 375-431 (1949).)

Weyl, H., 1953.” Über den Symbolismus der Mathematik und mathematischen Physik”. Studium Generale 6, pp. 219-228.

Weyl, H., 2009. Mind and Nature: Selected Writings on Philosophy, Mathematics, and Physics, Edited with an introduction by Peter Pesic. Princeton and Oxford: Princeton Univ. Press.

Wittgenstein, L., 1969 [TLP]. Tractatus Logico-Philosophicus, trans. D.F. Pears and B.F. McGuinnes. London: Routledge & Kegan Paul.

Wittgenstein, L., 1975 [PR]. Philosophical Remarks, 2nd ed., R. Rhees (ed.), trans. R. Hargreaves and R. White. Oxfod: Blackwell, Oxford.

Wittgenstein, L., 1974 [PG]. Philosophical grammar, R. Rhees (ed.), trans. A.J.P. Kenny. Oxford: Blackwell.

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

Wittgenstein, L., [NB], 1998. Notebooks 1914-1916, 2nd ed., G.H. von Wright et al. (Eds.). Oxford: Basil Blackwell.

Wittgenstein, L., [BT] 2012. The Big Typescript: TS 213, German English Scholar’s Edition, Trans. C. Grant Luchart and M.E. Aue. Oxford: Wiley-Blackwell.