Philosophy of Mathematics 1: Geometry Kant’s View of Geometry

Recap Rationalism Empiricism Descartes Knowledge – especially maths – primarily comes from logical deduction. Explains why maths is universally true, and why its conclusions seem impossible to doubt; but Fails to explain why maths is useful – that is, why it should have any application to reality. Empiricism Locke Knowledge – even maths – primarily comes from induction from experience. Explains why maths is useful. Fails to explain what makes it true, and why our knowledge of it seems so certain.

Recap Kant Critical philosophy aims to draw a line separating what can count as knowledge from what can’t (e.g. faith or preference). It does this by seeking the conditions of possibility for knowledge: what must logically be in place for knowledge to happen. Transcendental Idealism locates those conditions of possibility in the knower. It is an idealism because these are properties of the mind, broadly conceived, not of the physical world. It is transcendental because it goes beyond individual minds – any “knower” (including animals, aliens and AIs) would logically have to share it.

Overview We begin today by defining some important jargon that is still in regular use throughout philosophy. We will then look at Kant’s own description of the role of space (and geometry) in the conditions of possibility for knowledge.

Jargon: Analytic vs Synthetic These are ways a statement can be true. Many statements can be thought of as associating a predicate (something that can be true or false) to a subject (a thing). “All bachelors are unmarried” “All swans are white” Analytic The predicate is part of the definition of the subject. “All unicorns have horns” – can be true by definition, whether unicorns exist or not. The Synthetic The predicate lies outside or beyond the definition of the subject. Technically for Kant we should say the concept of the subject rather than definition, since this has nothing to do with language. If it’s true, something else makes it so besides just semantics.

Jargon: a posteriori vs a priori These are ways a statement can be known. A posteriori Can only be known based on empirical evidence or observation. Empiricists generally think all real knowledge is a posteriori. “All swans are white” requires us to check the colours of actual swans. A priori Can be known without any empirical evidence. Rationalists generally think all real knowledge is a priori. “All bachelors are unmarried” can be deduced simply from the meanings of the words; no need to interview bachelors!

Synthetic a priori? Analytic (Trivial; semantics) Synthetic (Interesting) A priori (Known through reason) Tautologies Conditions of possibility for knowledge. A posteriori (Known through experience) Nothing falls into this category. Empirical science. Kant thinks the conditions of possibility for knowledge can’t be analytic, because in a sense they’re non-trivial; they aren’t just embedded in the definition of knowledge itself. So they must be synthetic. He also thinks they must be knowable purely by rational thought, so they must be a priori. (Otherwise we would have an infinite regress, attempting to use scientific methods to determine the foundations of science itself.)

Transcendental Aesthetic This is where Kant tries to work out what fundamental structure must be present in order for us to have intuitions. By “intuition” he means experiences that we take to be representations of the external world – that is, phenomena. If we had no intuitions, we would have no a posteriori knowledge at all – and it’s not clear we would “have anything to think about”. So the structure of intuition is part of the conditions of possibility for knowledge.

p.81 “Space” in this context will be a very general prerequisite for having any kind of “outer” experience at all – that is, for having a representation in my mind of something that is not me. This must be “prior to all actual perceptions” because it makes perceptions possible!

p.77 “Space” here is a very general notion: the ability to separate objects from each other and from ourselves. This ability can’t be something we learned as knowledge, because to have knowledge at all seems to require it. (That doesn’t rule out the possibility that children develop this faculty after birth; as far as I know Kant doesn’t consider this question.)

p.78-9 Geometric truths are deduced from space as the structure of experience, not from experiences themselves (a posteriori), nor from their definitions (analytic).

Transcendental Doctrine of Method This passage describes various attempts to deduce simple geometric properties of the triangle such as the sum of its angles. (See p.4) The philosopher struggles because the triangle’s properties cannot be arrived at simply by analysing the definition of “triangle” (or other relevant words). This is not analytic knowledge. The scientist struggles because empirical measurements of actual triangles yields only a posteriori knowledge, which is imprecise and uncertain. Geometry is, rather, the study of the form of outer intuition – space – itself. This is shared by all – it is universal and identical for all knowers. This is usually taken to imply that Euclidean geometry is universally true for all knowers.

Summing Up For Kant geometry has a special status. It is the study of the form of outer intuition, which is a condition of possibility for knowledge as such. This form – “space” – is shared by anyone or anything capable of knowledge. So it is universal in a strong sense. Geometry involves synthetic a priori knowledge. It doesn’t work like empirical science, generalizing from observations of our experience. This is because it studies, not experiences, but the very structure that makes experiences possible. But it also doesn’t work only at the level of definitions. It is about something real, not just a semantic game.