Category Theory and Structuralism in Mathematics: Syntactical Considerations

Abstract

Thus, to be is to be related and the “essence” of an “entity” is given by its relations to its “environment”. This claim is striking: it seems to describe perfectly well the way objects of a category are characterized and studied. Consider, for instance, the fundamental notion of product in a category C: a product for two objects A and B of C is an object C with two morphisms p ₁: C → A and p ₂: C → B such that for any other pair of morphisms f: D → A and g:D→B, there is a unique morphism h:D→C such that f = p ₁h and g = p ₂h. What is crucial in this specification is the pair of morphisms <p ₁,p ₂> and the universal property expressed by the condition, for it is those which are used in proofs involving products. Thus to be a product is, in an informal sense, to be a position in a category. It is to be related in a certain manner to the other objects or positions in the category. Moreover, a product for two objects is defined up to isomorphism and it does not make sense to ask what is the product of two objects. It simply does not matter as far as mathematical properties are concerned. Now, if mathematics can be developed within category theory and if we can show that all the crucial concepts are given by universal properties, or, equivalently, come from adjoint situations, then we would have substantiated the above claim considerably.

“In mathematics, I claim, we do not have objects with an ‘internal’ composition arranged in structures, we have only structures. The objects of mathematics, that is, the entities which our mathematical constants and quantifiers denote, are structureless points or positions in structures. As positions in structures, they have no identity of features outside of a structure”² (Resnik, 1981, 530).

The author would like to thank the SSHRC and FCAR for their financial support while this work was done.