Pragmatism and category theory
In the introduction, I said that the way mathematicians work with categories reveals interesting insights into their implicit philosophy (how they interpret mathematical objects, methods, and the fact that these methods work). On the grounds of the evidence presented, we can now observe that the history of CT shows a switch in this interpretation: at first, objects of categories were always interpreted as sets (as in the case of the representations of Eilenberg and Mac Lane; see section 188.8.131.52); the purely formal character of categorial concepts was acknowledged but not consequently stressed. What was stressed positively is that concerning the categories themselves, the “all” is to be taken seriously <#20 p.237>. One was not aware of the fact that the difference between set theory and formal CT allowed for an interpretation of CT beyond sets (as far as the objects are concerned). This changed with Grothendieck on the one hand and Buchsbaum on the other. Grothendieck was interested in infinitistic argumentation and tried to extend the scope of the (formal) concept of set. Buchsbaum was interested in formal purity. The result of this development is a new technical intuition. This paradigm change took a different shape in the American and the French community, respectively.
KeywordsMathematical Object Structural Category Category Theory Categorial Concept Methodological Principle
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