One of the most striking features of the two theories which revolutionised physics at the beginning of this century (relativity theory and quantum mechanics) is the extent to which they are mathematical theories. For this reason, no discussion of the philosophy of contemporary science can ignore the role of mathematics in science. Any such account will have to say something about the way in which mathematics is applied in physics and as such will be based on, or will tacitly presuppose some view of the nature of, the (pure) mathematics being applied.

Bachelard sees the role of mathematics in contemporary scientific thought as extending beyond the organisation and expression of particular theories to the provision of frameworks for rational thought which reach outside those theories. This is to accord mathematics a central epistemological role in science, a role in theory construction and development. Mathematics, Bachelard says, provides the space within which scientists dream: ‘… the poetic art of Physics is done with numbers, with groups, with spins’ (PN p.39). He further holds that these mathematical, ‘anagogical’ reveries must be sharply distinguished from ordinary reveries, those which form the subject-matter of depth psychology. Grasping this distinction is, he says, crucial to understanding the psychology of the scientific mind. But just why is this so crucial?

It is because, as we have just seen (pp.62–5), at the growth point of scientific theorising the active role of the subject has to be reconciled with the demands of objectivity.