A Critique of the Kantian View of Geometry
David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
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 disciplinesÂ—logic, arithmetic and geometryÂ—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 logicÂ—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..
|Keywords||No keywords specified (fix it)|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library||
References found in this work BETA
No references found.
Citations of this work BETA
No citations found.
Similar books and articles
Kristina Engelhard & Peter Mittelstaedt (2008). Kant's Theory of Arithmetic: A Constructive Approach? [REVIEW] Journal for General Philosophy of Science 39 (2):245 - 271.
Amit Hagar (2008). Kant and Non-Euclidean Geometry. Kant-Studien 99 (1):80-98.
Yuval Steinitz (1994). Russell's Reductionism Revisited. Grazer Philosophische Studien 48:117-122.
Jeremy Heis (2011). Ernst Cassirer's Neo-Kantian Philosophy of Geometry. British Journal for the History of Philosophy 19 (4):759 - 794.
Carol A. Van Kirk (1986). Synthesis, Sensibility, and Kant's Philosophy of Mathematics. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1986:135 - 144.
John L. Bell, Two Approaches to Modelling the Universe: Synthetic Differential Geometry and Frame-Valued Sets.
J. E. Wiredu (1970). Kant's Synthetic a Priori in Geometry and the Rise of Non-Euclidean Geometries. Kant-Studien 61 (1-4):5-27.
Lisa Shabel (2004). Kant's "Argument From Geometry". Journal of the History of Philosophy 42 (2):195-215.
M. Giaquinto (2007). Visual Thinking in Mathematics: An Epistemological Study. Oxford University Press.
Joongol Kim (2006). Concepts and Intuitions in Kant's Philosophy of Geometry. Kant-Studien 97 (2):138-162.
Seung-Kee Lee (2009). The Synthetic a Priori in Kant and German Idealism. Archiv für Geschichte der Philosophie 91 (3):288-328.
Sun-joo Shin (1997). Kant's Syntheticity Revisited by Peirce. Synthese 113 (1):1-41.
Added to index2010-12-22
Total downloads9 ( #163,591 of 1,100,004 )
Recent downloads (6 months)0
How can I increase my downloads?