German Idealism and the Origins of Pure Mathematics: Riemann, Dedekind, Cantor

Document Type : Research Paper

Author

PhD in Philosophy, Allameh Tabataba'i University, Tehran, Iran.

Abstract

When it comes to the relation of modern mathematics and philosophy, most people tend to think of the three major schools of thought—i.e. logicism, formalism, and intuitionism—that emerged as profound researches on the foundations and nature of mathematics in the beginning of the 20th century and have shaped the dominant discourse of an autonomous discipline of analytic philosophy, generally known under the rubric of “philosophy of mathematics” since then. What has been completely disregarded by these philosophical attitudes, these foundational researches which seek to provide pure mathematics with a philosophically plausible justification by founding it on firm logico-philosophical bases, is that the genuine self-foundation of pure mathematics had been done before, namely during the 19th century, when it was developing into an entirely new and independent discipline as a concomitant of the continuous dissociation of mathematics from the physical world. This self-foundation of the 19th-century pure mathematics, however, was more akin to the German-idealist interpretations of Kant’s transcendental philosophy, than the post-factum, retrospective 20th-century researches on the foundations of mathematics. This article aims to demonstrate this neglected historical fact via delving into the philosophical inclinations of the three major founders of the 19th-century pure mathematics, Riemann, Dedekind and Cantor. Consequently, pure mathematics, with respect to its idealist origins, proves to be a formalization and idealization of certain activities specific to a self-conscious transcendental subjectivity.

Keywords

Main Subjects

References

Bolzano, B. (1851). Paradoxien des Unendlichen, Leipzig: C. H. Reclam.
-    Cantor, G. (1932). Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. Germany: Springer.
-    Dedekind, R. (1932). Gesammelte mathematische Werke. Bd. 3. Herausgegeben von R. Fricke, E. Noether, and Ö. Ore. Braunschweig: Vieweg. Dugac, Pierre.
-    Dieck, T. T. (2013). “Die Göttinger Mathematiker Gauß, Riemann, Klein und Hilbert”. In Die Geschichte der Akademie derWissenschaften zu Göttingen. Teil 1, Herausgegeben von Christian Starck und Kurt Schönhammer, Berlin: De Gruyter, pp. 135-160.
-    Ewing, A. C. (1938). A Short Commentary on Kant's Critique of Pure Reason. Chicago: University of Chicago Press.
-    Heidegger, M. (1967). Sein und Zeit. Tübingen: Max Niemeyer Verlag.
-    Henrich, D. (2008). Between Kant and Hegel: Lectures on German Idealism. United States: Harvard University Press.
-    Kant, I. (1956). Kritik der reinen Vernunft. Herausgegeben von Jens Timmermann. Hamburg: Felix Meiner.
-    Kant, I. (1998). Critique of Pure Reason. Edited and translated by Paul Guyer, Allen W. Wood. United States: United Kingdom: Cambridge University Press.
-    Kline, M. (1990). Mathematical Thought From Ancient to Modern Times. Vol. 3. United Kingdom: Oxford University Press.
-    Kuhn, T. S. (1996). The Structure of Scientific Revolutions. Chicago: University of Chicago Press.
-    McCarty, D. C. (1995) “Mysteries of Dedekind”, in From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics. Edited by Jaakko Hintikka. Netherlands: Springer Netherlands.
-    Paugh, C. C. (2002). Real Mathematical Analysis. New York: Springer-Verlag.
-    Pesic, P. (2007). Beyond Geometry: Classic Papers from Riemann to Einstein. United States: Dover Publications.
-    Pippin, R. (2014). “The Significance of Self-Consciousness in Idealist Theories of Logic”. Proceedings of the Aristotelian Society, 114, new series, 145-166.
-    Riemann, B. (2007), “On the Hypotheses That Lie at the Foundations of Geometry” in Beyond Geometry: Classic Papers from Riemann to Einstein. Edited by Peter Pesic. New York: Dover Publications.
-    Shapiro, S. (2000), Thinking about Mathematics: The Philosophy of Mathematics, Oxford University Press.
-    Spivak,  M. (1979). A Comprehensive Introduction to Differential Geometry. Vol 2.  Berkeley: Publish or Perish.
-    Steiner, H. G. (1980). Mengenlehre. In Historisches Wörterbuch der Philosophie (Band 5). Hg. Joachim Ritter, Karlfried Gründer. Basel/ Stuttgart: Schwabe Verlag.
Send comment about this article
CAPTCHA Image
Journal of Philosophical Investigations
Volume 15, Issue 36 - Serial Number 36
Special English Issue: Interaction Between Sciences and Philosophy
2021
Pages 171-188

Files

Share

How to cite

Statistics