A Brief Aside About Mathematics
I am skipping over so much in my effort to lay out what I see as the core ideas in the Critique that every so often I get to feeling a little queasy. After posting today's segment, I reflected that I had said nothing at all about Kant rather strange doctrine of manifolds of pure intuition and its relationship to the epistemological peculiarity of mathematical knowledge. Let me try to remedy that here. This is in the nature of an aside, and not really a part of the line of exposition I have been developing.
Kant thinks that mathematics is significantly different from physics, in the following way. We could not, he thinks, possess the concepts of mass, velocity, and the rest that turn up in the propositions of the physical sciences without first having had sense experience of physical objects. In this respect, he agrees with the empiricists in the great debate between the empiricists and the rationalists. As he says in the famous opening sentence of the portion of the Introduction to the Critique added in the Second Edition, "There can be no doubt that all our knowledge begins with experience." [B1] [He goes on, in the first sentence of the second paragraph, to add, "But though all our knowledge begins with experience, it does not follow that it all arises out of experience."] And so he is on his way to the argument that there is a mind-dependent, mind-contributed element in our knowledge, the element that makes knowledge of experienced objects possible a priori.
But Kant thinks that mathematics is knowable a priori in a stronger sense than this, because the concepts we deploy in Geometry -- circle, line, angle, triangle, and the rest -- can be constructed by the mind in intuition entirely independently of all experience. We may derive the concept of mass from experience, but we do not, in the same sense, derive the concept of a triangle from experience. [Even though, to be sure, we as living human beings do not start thinking about logic and mathematics and philosophy until we have lived for a bit and had lots of sensory input.] He thus agrees with Plato and the rationalists about this. We can construct in intuition, he believes, mathematical objects we have never encountered in sense experience. [Remember Plato's argument that we can never abstract the concept of a perfect circle or a perfectly straight line from our sense experience of plates or sticks.]
How can this be? Kant concludes that over and above space and time being the pure forms of sensible intuition, space and time are manifolds of pure intuition given to the mind by itself, and completely devoid of sensory content. What are manifolds of pure intuition? Manifolds of relations that are not relating anything to anything, as far as I can understand what he is saying. This is not a dumb view, although it is very difficult to understand. I may be totally wrong, but it is my impression that the great Dutch mathematician L. E. J. Brouwer held a similar position. Perhaps if Charles Parsons is reading this, he will weigh in and correct me, since he really does know a very great deal about Brouwer.
Just to repeat, Kant's motivation for advancing this view is, in the first instance, his desire to explicate the special epistemic status of pure mathematics, although -- waste not, want not -- the doctrine of pure intuition crops up later on and plays an important role in Kant's theory of a priori synthesis.
Well, I just thought I would mention it.