4 ms·

I'm obliged, I suppose, to list off a few big wins. From the 50s and 60s, we have classic theorems which use abstract nonsense to generalize big statements abou

I'm obliged, I suppose, to list off a few big wins. From the 50s and 60s, we have classic theorems which use abstract nonsense to generalize big statements about entire classes of objects, like Freyd's adjoint functor theorem [0], Yoneda's lemma [1], and Lawvere's fixed-point theorem [2]. Yoneda's lemma is the slogan that "an object is equivalent to the arrows coming/leaving it", but formal. Lawvere's theorem is a deep and permanent generalization of Gödel, Tarski, Turing, et al. on incompleteness and undecideability. Starting in the 60s and continuing to the present, there's been a theme of exploring category theory as a prime foundation for maths. There's been complete foundations like the Elementary Theory of the Category of Categories (ETCC) [3], and also much smaller but pointed presentations that focus on just simplifying set theory. I like [4] in particular. On a deeper level, in terms of structure and philosophy, the entire existence of homotopy type theory (HoTT) relies on categorical presentations and memes. HoTT is, more than any other type theory, an investigation into what equivalence means, and it dovetails wonderfully with 2-category theory and the understanding that algebraic laws can be transformed into transformations. [0] https://en.wikipedia.org/wiki/Formal_criteria_for_adjoint_functors https://en.wikipedia.org/wiki/Formal_criteria_for_adjoint_fu... [1] https://en.wikipedia.org/wiki/Yoneda_lemma https://en.wikipedia.org/wiki/Yoneda_lemma [2] http://tac.mta.ca/tac/reprints/articles/15/tr15.pdf http://tac.mta.ca/tac/reprints/articles/15/tr15.pdf [3] https://ncatlab.org/nlab/show/ETCC https://ncatlab.org/nlab/show/ETCC [4] https://arxiv.org/abs/1212.6543 https://arxiv.org/abs/1212.6543

spekcular 6y ago
Those aren't really "big wins" though. To most mathematicians, generalizations and abstraction are only interesting insofar as they are useful, and I don't see any results there someone working in mainstream mathematics would or should care about. (Obviously the Yoneda lemma is used in algebraic topology and geometry, but that's the odd one out on your list, and you don't really need categorical language to state the interesting examples.) Also, EETC/HoTT are basically junk, as explained in detail by Harvey Friedman in various postings on the foundations of math mailing list. We already have a perfectly good foundation for mathematics and EETC/HoTT don't improve on it in any way (and in fact are worse in many respects).
- Kednicma 6y ago
  Friedman and Simpson are, unfortunately, dinosaurs who don't grok categorical concepts. This became clear on the FOM list when Pratt and other abstract algebraists who know category theory but do not depend on it were able to pry apart the problem. Friedman and Simpson deny that certain mathematical objects exist and are equivalent to each other, and while I won't begrudge them their nearsight due to spending so much time with weak/reverse maths, I won't excuse the mistakes. I had to go find a blow-by-blow of the drama because it's good. Simpson cannot imagine topoi which don't implement standard set theory [3]. Friedman demonstrates a total misunderstanding of sets vs. categories [4]. McLarty and Feferman claim that categorical FOM make sense once one is used to categories [1]. Finally, the truth is laid bare: Friedman and Simpson simply don't agree with us on whether categorical logic is philosophically valid [2]. For more on this perspective, check out Pratt's take on Yoneda's lemma [0]. (An interesting aside: Simpson is an Objectivist who hates postmodernism! I wonder if this is part of what causes them to reject topos theory, where there are many different logics and collections, with a single barren plain Boolean set theory?) [0] http://boole.stanford.edu/pub/yon.pdf http://boole.stanford.edu/pub/yon.pdf [1] https://cs.nyu.edu/pipermail/fom/1998-March/001309.html https://cs.nyu.edu/pipermail/fom/1998-March/001309.html [2] https://cs.nyu.edu/pipermail/fom/1998-March/001467.html https://cs.nyu.edu/pipermail/fom/1998-March/001467.html [3] https://cs.nyu.edu/pipermail/fom/1998-January/000782.html https://cs.nyu.edu/pipermail/fom/1998-January/000782.html [4] https://cs.nyu.edu/pipermail/fom/1998-January/000835.html https://cs.nyu.edu/pipermail/fom/1998-January/000835.html
  - spekcular 6y ago
    I read the links you provided (except [0], which I do not have time for at the moment). I do not think Awodey and McLarty come across as well as you think they do, and I do not think Friedman misunderstands anything. Moreover, you omit some of Friedman's best postings on this topic. The challenge is to show what advantages this proposed theory has over the standard foundations, or to demonstrate some interesting problem or conceptual issue that it resolves. I did not see any serious replies to this. (Incidentally, the same challenge is my reply to the other comment to my previous post, about the ZX-calculus.) Here's an analogy. Why do mathematicians consider groups interesting, but general Moufang loops not interesting? I propose the answer is that the additional generality provided by Moufang loops doesn't aid in addressing any interesting questions external to their theory. On the other hand, the utility of the group concept is obvious to anyone who studies a subject with connections to algebra. This reasoning is roughly why the vast majority of mathematicians don't give two hoots about category theory, beyond picking up the few tricks like Yoneda's lemma that are actually useful. You can – as certain members of the category theory community have shown – take any mathematical concept, consider some generalization, and play various definition pushing games. But that's pointless without a good motivation (e.g. a concrete problem). The previous two paragraphs concern "actual" mathematics, but similar remarks apply to foundations of mathematics. Friedman is pointing out that we already have a perfectly good foundational theory of mathematics and asking what gain we get by introducing categorical concepts.

spekcular 6y ago

Those aren't really "big wins" though. To most mathematicians, generalizations and abstraction are only interesting insofar as they are useful, and I don't see any results there someone working in mainstream mathematics would or should care about. (Obviously the Yoneda lemma is used in algebraic topology and geometry, but that's the odd one out on your list, and you don't really need categorical language to state the interesting examples.) Also, EETC/HoTT are basically junk, as explained in detail by Harvey Friedman in various postings on the foundations of math mailing list. We already have a perfectly good foundation for mathematics and EETC/HoTT don't improve on it in any way (and in fact are worse in many respects).

Kednicma 6y ago
Friedman and Simpson are, unfortunately, dinosaurs who don't grok categorical concepts. This became clear on the FOM list when Pratt and other abstract algebraists who know category theory but do not depend on it were able to pry apart the problem. Friedman and Simpson deny that certain mathematical objects exist and are equivalent to each other, and while I won't begrudge them their nearsight due to spending so much time with weak/reverse maths, I won't excuse the mistakes. I had to go find a blow-by-blow of the drama because it's good. Simpson cannot imagine topoi which don't implement standard set theory [3]. Friedman demonstrates a total misunderstanding of sets vs. categories [4]. McLarty and Feferman claim that categorical FOM make sense once one is used to categories [1]. Finally, the truth is laid bare: Friedman and Simpson simply don't agree with us on whether categorical logic is philosophically valid [2]. For more on this perspective, check out Pratt's take on Yoneda's lemma [0]. (An interesting aside: Simpson is an Objectivist who hates postmodernism! I wonder if this is part of what causes them to reject topos theory, where there are many different logics and collections, with a single barren plain Boolean set theory?) [0] http://boole.stanford.edu/pub/yon.pdf http://boole.stanford.edu/pub/yon.pdf [1] https://cs.nyu.edu/pipermail/fom/1998-March/001309.html https://cs.nyu.edu/pipermail/fom/1998-March/001309.html [2] https://cs.nyu.edu/pipermail/fom/1998-March/001467.html https://cs.nyu.edu/pipermail/fom/1998-March/001467.html [3] https://cs.nyu.edu/pipermail/fom/1998-January/000782.html https://cs.nyu.edu/pipermail/fom/1998-January/000782.html [4] https://cs.nyu.edu/pipermail/fom/1998-January/000835.html https://cs.nyu.edu/pipermail/fom/1998-January/000835.html
- spekcular 6y ago
  I read the links you provided (except [0], which I do not have time for at the moment). I do not think Awodey and McLarty come across as well as you think they do, and I do not think Friedman misunderstands anything. Moreover, you omit some of Friedman's best postings on this topic. The challenge is to show what advantages this proposed theory has over the standard foundations, or to demonstrate some interesting problem or conceptual issue that it resolves. I did not see any serious replies to this. (Incidentally, the same challenge is my reply to the other comment to my previous post, about the ZX-calculus.) Here's an analogy. Why do mathematicians consider groups interesting, but general Moufang loops not interesting? I propose the answer is that the additional generality provided by Moufang loops doesn't aid in addressing any interesting questions external to their theory. On the other hand, the utility of the group concept is obvious to anyone who studies a subject with connections to algebra. This reasoning is roughly why the vast majority of mathematicians don't give two hoots about category theory, beyond picking up the few tricks like Yoneda's lemma that are actually useful. You can – as certain members of the category theory community have shown – take any mathematical concept, consider some generalization, and play various definition pushing games. But that's pointless without a good motivation (e.g. a concrete problem). The previous two paragraphs concern "actual" mathematics, but similar remarks apply to foundations of mathematics. Friedman is pointing out that we already have a perfectly good foundational theory of mathematics and asking what gain we get by introducing categorical concepts.