# After Non-Euclidean Geometry: Intuition, Truth and the Autonomy of Mathematics

### Abstract

The mathematical developments of the 19th century seemed to undermine Kant’s philosophy. Non-Euclidean geometries challenged Kant’s view that there is a spatial intuition rich enough to yield the truth of Euclidean geometry. Similarly, advancements in algebra challenged the view that temporal intuition provides a foundation for both it and arithmetic. Mathematics seemed increasingly detached from experience as well as its form; moreover, with advances in symbolic logic, mathematical inference also seemed independent of intuition. This paper considers various philosophical responses to these changes, focusing on the idea of modifying Kant’s conception of intuition in order to accommodate the increasing abstractness of mathematics. It is argued that far from clinging to an outdated paradigm, programs based on new conceptions of intuition should be seen as motivated by important philosophical desiderata, such as the truth, apriority, distinctiveness and autonomy of mathematics.### References

Benacerraf, Paul and Hilary Putnam, eds., 1983. Philosophy of Mathematics: Selected Readings. Cambridge: Cambridge University Press.

Bitbol, Michel, Pierre Kerszberg, and Jean Petitot, eds., 2009. Constituting Objectivity. Dordrecht: Springer.

Brouwer, L. E. J., 1913. “Intuitionism and Formalism.” Reprinted in Benacerraf and Putnam (1983), pp. 77–89.

Carson, Emily, 1999. “Kant on the Method of Mathematics.” Journal of the History of Philosophy 37: 629–52.

De Paz, Maria and Robert DiSalle, eds., 2014. Poincaré, Philosopher of Science: Problems and Perspectives. New York: Springer.

Ewald, William, ed., 1996. From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols. Oxford: Clarendon Press.

Folina, Janet, 2006. “Poincaré’s Circularity Arguments for Mathematical Intuition.” In Friedman and Nordman (2006), pp. 275–93.

———, 2012. “Newton and Hamilton: In Defense of Truth in Algebra.” The Southern Journal of Philosophy 50: 504–27.

———, 2014. "Poincaré and the Invention of Convention." In de Paz and DiSalle (2014), pp. 25–45.

Frege, Gottlob, 1884. The Foundations of Arithmetic, translated by J. L. Austin. Evanston, IL: Northwestern University Press, 1968.

Friedman, Michael, 1985. “Kant’s Theory of Geometry.” The Philosophical Review 94: 455–506.

———, 2001. The Dynamics of Reason. Stanford, CA: CSLI Publications.

———, 2002. “Kant, Kuhn, and the Rationality of Science.” Philosophy of Science 69: 171–90.

———, 2009. “Einstein, Kant and the Relativized A Priori.” In Bitbol, Kerszberg and Petitot (2009), pp. 253–67.

———, 2016. “Ernst Cassirer.” In The Stanford Encyclopedia of Philosophy, edited by E. N. Zalta, https://plato.stanford.edu/archives/fall2016/entries/cassirer/, accessed 19 December 2017.

Friedman, Michael and Alfred Nordman, eds., 2006. The Kantian Legacy in Nineteenth Century Science. Cambridge, MA: MIT Press.

Guyer, Paul, ed., 2006. The Cambridge Companion to Kant and Modern Philosophy. New York: Cambridge University Press.

Hamilton, William Rowan, 1837. “Theory of Conjugate Functions.” Reprinted in The Mathematical Papers of Sir William Rowan Hamilton, vol. 3, edited by H. Halberstam and R. E. Ingram, pp. 3–96. Cambridge: Cambridge University Press, 1967.

Heis, Jeremy, 2011. “Ernst Cassirer’s Neo-Kantian Philosophy of Geometry.” British Journal for the History of Philosophy 19: 759–94.

Helmholtz, Hermann von, 1876/1878. “The Origin and Meaning of Geometrical Axioms.” Reprinted in Ewald (1996), pp. 665–89.

Kant, Immanuel, 1781/1787. Critique of Pure Reason, translated by Norman Kemp Smith. Basingstoke: Palgrave Macmillan, 2007.

———, 1783. Prolegomena to any Future Metaphysics, translated by P. Carus and J. Ellington. Indianapolis: Hackett, 1977.

Patton, Lydia, 2016. “Hermann von Helmholtz.” In The Stanford Encyclopedia of Philosophy, edited by E. N. Zalta, https://plato.stanford.edu/archives/win2016/entries/hermann-helmholtz/, accessed 19 December 2017.

Poincaré, Henri, 1894. “On the Nature of Mathematical Reasoning,” translated by G. Halsted. In Ewald (1996), pp. 972–81.

———, 1898. “On the Foundations of Geometry,” translated by T. J. McCormack. Reprinted in Ewald (1996), pp. 982–1011.

———, 1900. “Intuition and Logic in Mathematics,” translated by G. Halsted. In Ewald (1996), pp. 1012–1020.

———, 1902. Science and Hypothesis, translated by W. J. G. New York: Dover.

———, 1905–1906. “Mathematics and Logic,” 3 parts, translated by G. Halsted and W. Ewald. In Ewald (1996), pp. 1021–1071.

———, 1913. Mathematics and Science: Last Essays, translated by J. Bolduc. New York: Dover, 1963.

Russell, Bertrand, 1897. An Essay on the Foundations of Geometry. Cambridge: Cambridge University Press.

Shabel, Lisa, 1998. “Kant on the ‘Symbolic Construction’ of Mathematical Concepts.” Studies in the History and Philosophy of Science 29: 589–621.

———, 2006. “Kant’s Philosophy of Mathematics.” In Guyer (2006), pp. 94–128.

Weyl, Hermann, 1918. The Continuum, translated by S. Pollard and T. Bole. Kirksville, MO: Thomas Jefferson University Press.

Bitbol, Michel, Pierre Kerszberg, and Jean Petitot, eds., 2009. Constituting Objectivity. Dordrecht: Springer.

Brouwer, L. E. J., 1913. “Intuitionism and Formalism.” Reprinted in Benacerraf and Putnam (1983), pp. 77–89.

Carson, Emily, 1999. “Kant on the Method of Mathematics.” Journal of the History of Philosophy 37: 629–52.

De Paz, Maria and Robert DiSalle, eds., 2014. Poincaré, Philosopher of Science: Problems and Perspectives. New York: Springer.

Ewald, William, ed., 1996. From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols. Oxford: Clarendon Press.

Folina, Janet, 2006. “Poincaré’s Circularity Arguments for Mathematical Intuition.” In Friedman and Nordman (2006), pp. 275–93.

———, 2012. “Newton and Hamilton: In Defense of Truth in Algebra.” The Southern Journal of Philosophy 50: 504–27.

———, 2014. "Poincaré and the Invention of Convention." In de Paz and DiSalle (2014), pp. 25–45.

Frege, Gottlob, 1884. The Foundations of Arithmetic, translated by J. L. Austin. Evanston, IL: Northwestern University Press, 1968.

Friedman, Michael, 1985. “Kant’s Theory of Geometry.” The Philosophical Review 94: 455–506.

———, 2001. The Dynamics of Reason. Stanford, CA: CSLI Publications.

———, 2002. “Kant, Kuhn, and the Rationality of Science.” Philosophy of Science 69: 171–90.

———, 2009. “Einstein, Kant and the Relativized A Priori.” In Bitbol, Kerszberg and Petitot (2009), pp. 253–67.

———, 2016. “Ernst Cassirer.” In The Stanford Encyclopedia of Philosophy, edited by E. N. Zalta, https://plato.stanford.edu/archives/fall2016/entries/cassirer/, accessed 19 December 2017.

Friedman, Michael and Alfred Nordman, eds., 2006. The Kantian Legacy in Nineteenth Century Science. Cambridge, MA: MIT Press.

Guyer, Paul, ed., 2006. The Cambridge Companion to Kant and Modern Philosophy. New York: Cambridge University Press.

Hamilton, William Rowan, 1837. “Theory of Conjugate Functions.” Reprinted in The Mathematical Papers of Sir William Rowan Hamilton, vol. 3, edited by H. Halberstam and R. E. Ingram, pp. 3–96. Cambridge: Cambridge University Press, 1967.

Heis, Jeremy, 2011. “Ernst Cassirer’s Neo-Kantian Philosophy of Geometry.” British Journal for the History of Philosophy 19: 759–94.

Helmholtz, Hermann von, 1876/1878. “The Origin and Meaning of Geometrical Axioms.” Reprinted in Ewald (1996), pp. 665–89.

Kant, Immanuel, 1781/1787. Critique of Pure Reason, translated by Norman Kemp Smith. Basingstoke: Palgrave Macmillan, 2007.

———, 1783. Prolegomena to any Future Metaphysics, translated by P. Carus and J. Ellington. Indianapolis: Hackett, 1977.

Patton, Lydia, 2016. “Hermann von Helmholtz.” In The Stanford Encyclopedia of Philosophy, edited by E. N. Zalta, https://plato.stanford.edu/archives/win2016/entries/hermann-helmholtz/, accessed 19 December 2017.

Poincaré, Henri, 1894. “On the Nature of Mathematical Reasoning,” translated by G. Halsted. In Ewald (1996), pp. 972–81.

———, 1898. “On the Foundations of Geometry,” translated by T. J. McCormack. Reprinted in Ewald (1996), pp. 982–1011.

———, 1900. “Intuition and Logic in Mathematics,” translated by G. Halsted. In Ewald (1996), pp. 1012–1020.

———, 1902. Science and Hypothesis, translated by W. J. G. New York: Dover.

———, 1905–1906. “Mathematics and Logic,” 3 parts, translated by G. Halsted and W. Ewald. In Ewald (1996), pp. 1021–1071.

———, 1913. Mathematics and Science: Last Essays, translated by J. Bolduc. New York: Dover, 1963.

Russell, Bertrand, 1897. An Essay on the Foundations of Geometry. Cambridge: Cambridge University Press.

Shabel, Lisa, 1998. “Kant on the ‘Symbolic Construction’ of Mathematical Concepts.” Studies in the History and Philosophy of Science 29: 589–621.

———, 2006. “Kant’s Philosophy of Mathematics.” In Guyer (2006), pp. 94–128.

Weyl, Hermann, 1918. The Continuum, translated by S. Pollard and T. Bole. Kirksville, MO: Thomas Jefferson University Press.

Published

2018-02-05

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