A Critique of the Kantian View of Geometry


A survey of Kant's views on space, time, geometry and the synthetic nature of mathematics. I concentrate mostly on geometry, but comment briefly on the syntheticity of logic and arithmetic as well. I believe the view of many that Kant's system denied the possibility of non-Euclidean geometries is clearly mistaken, as Kant himself used a non-Euclidean geometry (spherical geometry, used in his day for navigational purposes) in order to explain his idea, which amounts to an anticipation of the later discovery of the general concept of non- Euclidean geometries. Kant's view of geometry and arithmetic as synthetic was, I believe, essentially correct, in that geometry and arithmetic are both synthetic a priori if considered as branches of mathematics independent of the rest of mathematics. However, the view that somehow logic is analytic, while mathematics is synthetic for Kantian reasons, is mistaken. All three disciplines—logic, arithmetic and geometry—are synthetic as disciplines independent from one another. However, they have a common basis, recursion theory, which I prefer to identify with mathematics as a whole. As a result, I do not say, as is often considered to be the Kantian view, that mathematics is synthetic while logic is analytic. Rather, I prefer to say that mathematics is analytic, while logic is synthetic. This is perfectly consistent with Kant's system, since it was arithmetic and geometry individually that he argued were synthetic. What Kant called the analytic is recursion theory, which could be considered as a basic formulation of mathematics or logic—or better, both mathematics and logic could be recognized as essentially the same discipline. However, if "logic" is taken to mean "predicate logic", as is often the case in modern times, then it is mathematics that is closer to Kant's analytic, not logic. Such ambiguities, of course, can be avoided by simply associating Kant's analytic with recursion theory, and avoiding the controversies as to what counts as mathematics or logic..



    Upload a copy of this work     Papers currently archived: 92,227

External links

  • This entry has no external links. Add one.
Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

  • Only published works are available at libraries.

Similar books and articles

Kant’s Theory of Arithmetic: A Constructive Approach? [REVIEW]Kristina Engelhard & Peter Mittelstaedt - 2008 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 39 (2):245 - 271.
Kant and non-euclidean geometry.Amit Hagar - 2008 - Kant Studien 99 (1):80-98.
Russell's Reductionism Revisited.Yuval Steinitz - 1994 - Grazer Philosophische Studien 48 (1):117-122.
Ernst Cassirer's Neo-Kantian Philosophy of Geometry.Jeremy Heis - 2011 - British Journal for the History of Philosophy 19 (4):759 - 794.
Essays on the foundations of mathematics.Moritz Pasch - 2010 - New York: Springer. Edited by Stephen Pollard.
Kant's "argument from geometry".Lisa Shabel - 2004 - Journal of the History of Philosophy 42 (2):195-215.
The synthetic a priori in Kant and German idealism.Seung-Kee Lee - 2009 - Archiv für Geschichte der Philosophie 91 (3):288-328.


Added to PP

9 (#1,258,729)

6 months

Historical graph of downloads
How can I increase my downloads?

Author's Profile

Allan Randall
York University

Citations of this work

No citations found.

Add more citations

References found in this work

No references found.

Add more references