Axiomatic Method and Category Theory

group and develop the corresponding general theory, which unifies a large spectrum of significant mathematical results from different areas of mathematics.

As we see Bourbaki points here on a general phenomenon, which is not specific to the Axiomatic Method (and moreover to Hilbert’s Formal Axiomatic Method), namely to the mathematical concept-building and its unifying role. Such basic mathematical concepts as number, figure and the like can be similarly seen as abstractions generalizing upon more specific examples like sets of dots or sets of strokes (for number) and circles, polygons, etc. (for figures). Although the generalization upon and abstraction from specific features of previously known examples is not the only way in which emerge new mathematical concepts this way of emergence is a major one.

And in Bourbaki’s example it works through the Axiomatic Method. Let us see how it works more attentively.

First of all we need to distinguish between two ways in which an axiomatic theory unifies its content. When a set of contentual propositions is logically deduced from certain propositions belonging to the same set and chosen as axioms this unifies all these propositions into a single (contentual) theory. As we have seen Bourbaki recognizes this fact but in the last quote he clearly points to a different way of unification, which is equally made possible by the Axiomatic Method.

This different way of unification is made possible by the formal character of Axiomatic Method, where “formal” is to be understood in the sense of Hilbert’s Foundations of 1899 rather than in the sense of his Foundations of 1927 [100] (which is referred to in the above quote in the expression “logical formalism”). This second axiomatic unification amounts to the following: a formal (as opposed to contentual) axiomatic theory unifies its interpretations (models) by identifying certain common features of these interpretations and abstracting from all other specific features of those interpretations. The usual talk of interpretation of a given formal theory takes it for granted that the formal theory is given first and interpreted next. Now we reverse the perspective and consider the (would-be) interpretations as given (as contentual mathematical theories and fragments of such theories) and then think how to make up a formal theory, which captures common features of these things and thus unifies them. The example of GT is used by Bourbaki to illustrate the latter but not the former way of unification.

Thus Bourbaki shows - in my view quite correctly - that the Formal Axiomatic Method has a unifying capacity, which is absent from contentual versions of the Axiomatic Method. However Bourbaki’s version of Axiomatic Method is not identical to Hilbert’s! Let me now describe the difference.

As an example of a theorem of GT Bourbaki mentions this proposition P: 82