Category theory represented a huge change in the way the community thought about mathematics, leaving its the set theoretic nature behind and bringing up the importance of arrows between the objects rather than the objects themselves.

I believe this radical change must have originated in philosophy and/or art and was eventually absorbed by mathematics (or even science?). So, my questions are:

Is this true? if it is, which philosophical schools are behind the revolutionary ideas of category theory? How did this schools evolve? Did philosophy behind math evolve accordingly?

Were Eilenberg and Mac Lane part of any non-exclusively-mathematical community which might have influenced their work?

How did category theory influence philosophy?

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    $\begingroup$ You can see : Jean-Pierre Marquis, From a Geometrical Point of View : A Study of the History and Philosophy of Category Theory (2009) but it seems to me that (in spite of the title) there is no references to a "philosophical influence" outside of phil of math... $\endgroup$ – Mauro ALLEGRANZA Dec 15 '14 at 9:19
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    $\begingroup$ I once looked in detail into this question and I came to the conclusion that the belief that "this radical change must have originated in philosophy and/or art and was eventually absorbed by mathematics" is in fact incorrect. I found no evidence on the influence of philosophy or social science on the origin of category theory. Of course, such a negative result is hard to substantiate. $\endgroup$ – Olivier Dec 15 '14 at 9:35
  • $\begingroup$ @MauroALLEGRANZA Thank you for the reference, I'll give it a look. $\endgroup$ – hjhjhj57 Dec 15 '14 at 9:39
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    $\begingroup$ There is a PhD thesis turned 400 page book, which tracks the development of the conception of algebra from Galois to Grothendieck: Corry - Modern Algebra Rise Mathematical Structures. $\endgroup$ – Nikolaj-K Dec 25 '14 at 2:17
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    $\begingroup$ @Javier: Yes. The title of Corrys book is actually a pun! Modern algebra was an influential textbook and one relevant points in time where people a full axiomatic perspective on abstract concepts such as groups and fields really became mainstream. The book (somewhat tediously at parts, im my opinion), details the content of several textbooks leading up to this. Using the book, a broad answer to your question would probably be that the philosophical change that happend was the perception of what an "axiom" is. "self-evident" vs. "theory defining" $\endgroup$ – Nikolaj-K Dec 25 '14 at 2:46

There are some great references in the comments above, by Nikolajk and Mauro. I recommend them too.

But as mentioned, mostly inspiration for category theory is philosophy of mathematics, in the sense of logic, at least so far as MacLane talks about it in his autobiography.

I speculate also that the philosophy of Ernst Mach, as evident in his first quasi-popular book, seems to be the only EXTRA-mathematical philosophy (other than the coherence theory of truth, due to Joachim and Blanshard, which however probably had little influence on the mathematicians, unlike Mach) that inspired category theory. (Basically, it's relational realism more or less, itself related to structural realism.) It's true, I agree, that category theory mostly is inspired by philosophy of mathematics, not the extra-mathematical philosophy, which mostly centers on theory of truth, ideas, ethics, and aesthetics (and so has no bearing on the subject, if the theory of truth was not relational fundamentally). Mach was and is very influential on thinking of mathematicians, especially those think about physics, so I think his is the main (extra-mathematical) philosophical influence. Leibnitz seemed to have in turn been an indirect influence on Mach...

To some degree this is by elimination of philosophies directly opposed to the conceptual approach of category theory. The radical empiricism of the instrumentalists and logical atomists (Wittgenstein, Russell, etc.) is generally opposed to the key information being in the arrows, in a static world of related events, so very few popular philosophies really had any influence on category theory, while the classical (Aristotelian realism, abstract Platonism) and neo-classical (Kant, Hegel, Schopenhauer) were not especially interested in putting any emphasis on relations of events instead of the events themselves (the Ens). Others like Ortega-y-Gasset stressed a return to metaphorical Platonism.

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