ABSTRACT
Leaving aside conventionalism as I have described it - 1t 1s certainly quite alien to Kant's thinking on the matter - let us consider once more the two ways permitted by the positivist view of looking at the axioms and theorems of a geometry. We can, on this view, see them either as uninterpreted formulae in a pure calculus which express no propositions at all about anything; or, given a physical interpretation, as propositions about what may in a broad sense be called physical objects in space. If we ask in which of these two ways Kant looked at the propositions of Euclidean geometry, the answer is quite clearly that neither is adequate to his view of the matter. He certainly never considered seeing such propositions as uninterpreted formulae which had nothing in particular to do with space. From the start he saw them as having spatial significance. But equally clearly he did not think that the only way in which they could have spatial significance was by having a physical interpretation, i.e. by the meaning of the fundamental expressions being explained in terms of physical objects of empirical intuition, such observable or determinable objects as a taut string or the path of a light ray or, for that matter, a line drawn on paper with a ruler and a sharp pencil. He thought indeed that the propositions of Euclidean geometry were true of physical objects of empirical intuition. But he was quite firm in the belief that there was no need to have recourse to, or even to consider, such physical objects of empirical intuition, in order to ascertain the truth of the nevertheless spatially significant propositions of Euclidean geometry. Certainly we might, with great advantage, draw lines on paper (with or without a ruler) in the course of a geometrical demonstration. But the objects of empirical intuition thus provided were not the essential objects of this activity; they were there simply to provide assistance to the essential activity of pure intuition of which the objects were not physical objects at all. What then are the spatial, but not physical (nor physically determinable) objects of pure outer intuition?
