Friday, February 11, 2011

Kant's Philosophy of Mathematics

On the way of completing my first chapter of the Ph.D thesis by mid April, I started to read Posy's collection of essays on Kant's Philosophy of Mathematics. It's the best compilation in my area including papers from Hintikka, Parsons, M. Friedman, Melnick, Posy and by many others. I do not know if there is a *right* interpretation of Kant in terms of his philosophy of mathematics. His scholars could not even agree what Kant means by "Intuition" in analyzing mathematical objects and methods. There is a big debate started in the mid 60s between Parsons and Hintikka and more work followed by other scholars based on this debate. Hintikka, in his paper "Kant on the Mathematical Method" refers Kantian intuitions as particulars and claims that there is nothing intuitive about intuitions (p.23, Posy). He uses Kant's definition in Logic: "Every particular idea as distinguished from general concepts is an intuition". (ibid) He refers to other sources which carry the similar meaning: Kant's Dissertation 1770 Section 2 Paragraph 10, Critique A320/ B376-7 and Prolegomena  Paragraph 8

He claims that if we read Kant's Transcendental Aestetic in the Critique, the intuitions are referred as "mental images or an image before our minds eye" (p.26, Posy) but then he argues that in connection with Kant's mathematical views the definition of intuition which should be taken into account is the one at the end of the Critique, in the Transcendental Doctrine of Method. Then we will have no problems with Kant basing Arithmetic and Algebra on intuition, he claims. Otherwise, if the intuitions are taken as mental images, there is no way to interpret Arithmetic and Algebra being based on intuitions (ibid). Hence according to Hintikka, singularity of intuitions are essential and immediacy is a by product of singularity. It follows that intuitions, then can be given logical formulations with instantiations and existential quantifier eliminations . 

Parsons disagrees and argues that immediacy of intuitions are essential and in interpreting Kantian intuitions for his philosophy of mathematics, one needs more than logic, a method similar to perception. 

I am halfway with the Hinttikka's essay. So, let's dig in and see how  he expresses these logical formulations of intuitions. 

image from here

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