Maths and Logic workshop: Brian King
Brian King gave a paper entitled ‘Making Mathematics Sensible: a Kantian manifesto’. In it Brian defended the Kantian thesis that for mathematics to apply to the world there must be a link between sensibility (our access to the world) and the axioms or fundamental concepts of mathematics. Brian’s line here was fairly classically Kantian: the form of our intuitions is an a priori structure which belongs to sensibility and the objects to which sensibility relates. This guarantees a link between mathematics and the world because mathematical concepts are given their meaning by reference to this form.
What was interesting in the paper was Brian’s manifesto for making a Kantian philosophy of mathematics viable to the contemporary philosopher of mathematics. He argued that Kant had too heavily relied on empirical intuition (the drawing of lines, the counting of beads etc.) and that this reliance seems to vitiate his philosophy of mathematics, due to the fact that much of mathematics since Kant is not possible in terms of such crude methods. However, he suggested that if we effect a split between sensible mathematics, the concepts of which gain their meaning by reference to pure intuition, and intellectual mathematics, the concepts of which gain their meaning by having the logical form common to all concepts, then we can link mathematics to the world and account for much of modern mathematical practice. He further suggested that the sensible part of mathematics could receive a propadeutic justification of its application to the world a la Frege, and the intellectual part could be understood along the lines of Hilbert’s axiomatic method of justification.
The paper was very rich. One question that came up in different forms was whether, when we consider that much of modern mathematics would fall on the intellectual side of the divide, there really is any need for the sensible side. Brian’s answer was that as philosophers of mathematics we should be concerned with showing how mathematical concepts apply to the world and that a translation of intellectual concepts into sensible ones would play a large role in providing such a grounding.
What was interesting in the paper was Brian’s manifesto for making a Kantian philosophy of mathematics viable to the contemporary philosopher of mathematics. He argued that Kant had too heavily relied on empirical intuition (the drawing of lines, the counting of beads etc.) and that this reliance seems to vitiate his philosophy of mathematics, due to the fact that much of mathematics since Kant is not possible in terms of such crude methods. However, he suggested that if we effect a split between sensible mathematics, the concepts of which gain their meaning by reference to pure intuition, and intellectual mathematics, the concepts of which gain their meaning by having the logical form common to all concepts, then we can link mathematics to the world and account for much of modern mathematical practice. He further suggested that the sensible part of mathematics could receive a propadeutic justification of its application to the world a la Frege, and the intellectual part could be understood along the lines of Hilbert’s axiomatic method of justification.
The paper was very rich. One question that came up in different forms was whether, when we consider that much of modern mathematics would fall on the intellectual side of the divide, there really is any need for the sensible side. Brian’s answer was that as philosophers of mathematics we should be concerned with showing how mathematical concepts apply to the world and that a translation of intellectual concepts into sensible ones would play a large role in providing such a grounding.
Labels: Frege, Hilbert, Intuitions, Kant, mathematics, Sensibility
posted by matt at 11:25 am
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