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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?

Rodrigo de Azevedo
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hjhjhj57
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    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... – Mauro ALLEGRANZA Dec 15 '14 at 09:19
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    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. – Olivier Dec 15 '14 at 09:35
  • @MauroALLEGRANZA Thank you for the reference, I'll give it a look. – hjhjhj57 Dec 15 '14 at 09:39
  • @Olivier that would actually be even more amazing and demonstrate the creative power of mathematics, but it would also require some kind of proof. This may not be that hard for someone with some expertise in philosophy. – hjhjhj57 Dec 15 '14 at 09:41
  • @hjhjhj57 Negative statements are by nature hard to establish, yet alone prove. My particular experience has been to look hard and long in the original texts and correspondences of the relevant authors, and being unable to find any indication that they were influenced by any extra-mathematical ideas. – Olivier Dec 15 '14 at 12:55
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    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. – Nikolaj-K Dec 25 '14 at 02:17
  • That's actually pretty cool. According to the Mathematical Genealogy Project, his thesis was The Origins of Category Theory as a Mathematical Discipline, the book must containt the most important parts of it. Have you read it? – hjhjhj57 Dec 25 '14 at 02:27
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    @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" – Nikolaj-K Dec 25 '14 at 02:46
  • Btw. I initially came across that book 2 years ago in the first Math SE answer here. – Nikolaj-K Dec 25 '14 at 02:48
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    Mathematics use to be alive. Now it is a collection of bones to be categorized. – Euclid Looked On Beauty Bare Sep 08 '21 at 13:19
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    Didn't I read somewhere that the motivation for defining categories was to get a rigorous formulation of the [already existing notion] "natural transformation"? – Gerald Edgar Feb 16 '23 at 12:38
  • @GeraldEdgar Mac Lane says it at the end of section 4 of chapter I of Categories for the Working Mathematician, attributing it to himself and Eilenberg (though not further specifying any place it had been previously written). – Robert Furber Feb 28 '23 at 18:47

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The conventional view is that categories were introduced by Samuel Eilenberg and Saunders Mac Lane in the 1940s as a tool for the study of algebraic topology. What we now call functors and natural transformations were manifesting themselves, but there was not yet any precise language for defining them formally. So Eilenberg and Mac Lane invented that language.

Category theory is now often thought of as being relevant to the foundations of mathematics more generally, and comes up a lot in philosophical discussions. But this was not true in the early days. Eilenberg and Mac Lane were initially motivated by technical questions in a particular branch of mathematics, and were not trying to come up with a new way of thinking about all of mathematics. Even as category theory developed further, with advances in homological algebra and algebraic geometry, there were always concrete mathematical problems driving the developments. The notion that category theory might "overthrow" set theory and lead to a new way to think about mathematics was a rather late development.

Of course, it's possible that if someone digs deeper into the historical raw data, this conventional narrative could be upended. But I rather doubt that it can be shown that the origins of category theory came from philosophy or art. You might have better luck trying to show that William Lawvere's view of category theory was partially influenced by extra-mathematical ideas. That is, not category theory itself, but the use of category theory for philosophical ends, might have been partially motivated by philosophical considerations coming from outside of mathematics.

EDIT: I was just reading the Introduction to Johnstone's book Stone Spaces, and he says that Stone's work in the 1930s already contained examples of what we would nowadays call functors and equivalence of categories. Johnstone writes, "All this was proved in detail by Stone, although the categorical language in which we now express it was not introduced until the following decade; but Stone's Theorem was undoubtedly one of the major influences which prepared the mathematical world for the introduction of categories by Eilenberg and Mac Lane." (But note the careful wording; later in the same Introduction, Johnstone mentions that "the categorical ideas present in Stone's 1937 paper were not directly followed up by the founders of category theory.")

So you could argue that categorical thinking predated Eilenberg and Mac Lane, and was implicit in Stone's work. But Stone's work still betrays no evidence of influence from philosophy or art.

Timothy Chow
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It's only a 'huge' change because set theory formalities, aka ZFC, was so hegemonic.

Had a wider discussion ensued as to what to take as axiomatic, the function or the set or both, then the sea change that ensued after the discovery of category theory might not be so shocking.

In the large scheme of things, one might read set theory and category theory as part of a debate and philosophy about how to collect mathematical objects into collections. I certainly think that is a healthier way to think about this rather than an unhealthy way of setting category theory as a competitor to set theory.

Mozibur Ullah
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    Yes. Although set theory seemed to be the only available "foundation" for several decades, in practice mathematics was done mostly "without foundations". Category theory, in a rudimentary sense, does seem to address more relevant aspects of mathematical objects, maps, etc. Not that it's magic, nor need be "a foundation". It's just that thinking of $0,1,2,\ldots$ as ${}, {0}, {1,2}$ doesn't seem to tell us much about numbers. :) In contrast, universal mapping properties seem very useful. :) – paul garrett Feb 17 '23 at 03:02
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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.

Gottfried William
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  • I actually think that the change between early and late Wittgenstein is analogous to the change from "static" set theory to "relational" category theory. Not saying there was a direct influence one way or the other, but this change of perspective was somehow in the air, after a hypertrophy of the "atomic" "static" view. – Torsten Schoeneberg Feb 18 '23 at 06:23