A Critique of the Kantian View of Geometry
David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
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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..
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