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.

Maths and Logic workshop: Emily Carson

Emily Carson gave a paper entitled ‘Arithmetic and the Possibility of Experience’. In it she sought to work out Kant’s position on how arithmetic is grounded in the transcendental structures of experience. This questions asks, very roughly, how we can explain that every ‘discursive’ thinker - as Kant puts it - possesses some arithmetical knowledge, and how arithmetic applies absolutely generally to every empirical object (every such object is quantifiable). In the case of geometry Kant’s position is fairly clear: geometry is the pure science of space and our pure intuition of space is the source of our a priori geometrical knowledge. But there is no comparable science of number for Kant, the source of which would be our pure intuition of time.

Emily argued that the source of arithmetical knowledge are the categories of quantity, categories that are involved in every synthesis of any manifold of intuition. She explained this by arguing that the threefold synthesis cited in the A-edition Deduction, a synthesis apparently necessary for any determinate thought content whatsoever, is a synthesis that minimally involves the categories of quantity. Thus the formal element of possible experience that arithmetic is grounded in is this synthesis, and this explains Kant’s cryptic statement that the formal element in experience that grounds arithmetical knowledge is‘the universal in the synthesis of one and the same thing in time and space, and the magnitude of an intuition in general (number) that arises from that.’

I for one found this a very convincing account of Kant’s position. But one question that arose was that if all that grounds arithmetic is the categorical synthesis of quantity, doesn’t that give us an unKantian answer to how arithmetic and the possibility of experience relate? That is, arithmetical capacities now start to look wholly conceptual. It came up in discussion that it seems necessary for there to be some intuitive structure to index the categories to, so that arithmetical abilities didn’t depend on contingencies like the amount of objects in the world (if there were only four objects, the sentence 7+5+12 couldn’t be formulated unless there were an infinite iterative structure like time that has enough places to index the constituent numbers of the sentence to). What was then asked was, why should this indexing structure be specifically time or space? To my mind, against Kant, this question is wrongheaded: he is not trying to show how any discursive thinker would be capable of arithmetic, but how we are.

Transcendental Arguments Workshop: Grundmann

In his paper, "Transcendental Arguments in Defence of Philosophical Intuitions", Thomas Grundmann argued that sceptical arguments against the evidential status of philosophical intuitions are epistemically self-defeating. This he did via a form of transcendental argument. A representative sceptical argument was his 'argument from sceptical possibilities':

1. It is possible that our intuitions permanently deceive us unnoticed.
2. As long as the possibility of such a permanent deception is not ruled out, intuition is not a legitimate source of evidence.
3. We cannot rule out the possibility of a permanent deception of our intuition.
So,
4. Intuition is not a legitimate source of evidence.

Grundmann argued that such arguments against the evidential status of intuitions invariably contain at least one premise (in this case, 1) that must be based on philosophical intuition (he had earlier distinguished philosophical from folk intuition). As such, one must already believe that intuitions are a good source of evidence if one is to endorse the premise. So, the argument involves what he called an 'epistemic inconsistency', and is self-defeating.

One issue concerns the extent to which the sceptic can simply reformulate his arguments to evade this charge. An argument form that does not require one to be justified in believing (or even to believe) one's premises is reductio ad absurdum. Can't then the sceptic restate the argument as a reductio of the claim that philosophical intuition is a good form of evidence. Considered in this way, it is this claim (rather than the sceptic's position) that is self-defeating.

Another potential problem concerns the relation between judgements of intuition and conceptual capacities. Some of the claims that Grundmann suggested were based on philosophical intuition look like they might plausibly said to be based merely on an understanding of concepts. If so, they might be thought to be analytic and, if that's right, the question of whether they are based on good evidence misses the point.