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I got a bit obsessed with this stuff (CT, HoTT, haskell, etc) and ended up studying math in college because of it. My advice is not to bother with CT. Its out o

I got a bit obsessed with this stuff (CT, HoTT, haskell, etc) and ended up studying math in college because of it. My advice is not to bother with CT. Its out of context, very abstract math, and you wont find any compelling examples or applications at any accessible level. I dont want to discourage you from learning math, but I think this isnt a good place spend too much thought at first.

youssefabdelm 5y ago
My interest in category theory is for a completely different reason. I too have had to struggle to understand it, but I don't plan on using it in an applied or programming sense. I've heard a bunch of people say that category theory is almost like a 'bird's eye view of mathematics'. That many of its concepts unify many mathematical concepts under one umbrella, and reveal the similarities and connections between mathematical subdisciplines. The main reason this interests me is that I'd like to learn math much differently than its taught. To me, to learn the same concept under different subdisciplines with different names, is O(N) learning. However, if someone put together a category theory book where they list a concept, and then they do this: "Now, any mathematician can (easily) see that every major area of mathematics is a category. - In Set Theory the arrows are functions - in Topology they are continuous functions - in Group Theory homomorphisms - in Linear Algebra they are linear transformations - in Differentiable Geometry they’re smooth maps, and so on… But what’s important is that we shifted from focusing on the objects to focusing on the functions, the ways in which we transform the objects. This is basically the category-theoretical perspective: it’s the functions that matter." (Taken from this blogpost: https://catsinthejungle.wordpress.com/2008/11/15/category-theory-for-non-mathematicians/ https://catsinthejungle.wordpress.com/2008/11/15/category-th... ) Do I understand any of that? No. But what attracts me to this format is it (MAYBE...) approximates O(1) learning in math, so long as it can also point out, for every concept in every field, how they differ/relate to every other concept. So what this does is it helps teach me the 'unifiers' of mathematics, and helps me see connections between fields which can be useful for creativity. E.g. I realize I'm doing X algorithm here from linear algebra and that is O(N), whereas if I just shift fields, and use Y algorithm from Group theory I might make it O(log n) because it differs in this way or whatever it is. Besides that I see it as a potentially efficient way of learning a lot of math.
- chobytes 5y ago
  Math is at its heart about intuition; formalism is a just tool. One uses the formalism but one has to see past it. Roughly its the difference between knowing that a group is a set with a certain binary function, and knowing that a group is a particular formalization of symmetry. While a shortcut to understanding sounds nice, I don't think you'll find it by focusing on abstract formalism.
  - youssefabdelm 5y ago
    To me this would be another way of developing intuition, via analogy across disciplines. E.g. Volk's Metapatterns draws patterns (but perhaps less rigorous and more loose) across discplines giving one intuition about many things. In this case, seeing the connections across the disciplines isn't necessarily something I see or am pursuing because I'm searching for a shortcut (although I did mention O(1)), I just see that as a side benefit. What appeals to me more is developing the intuition you mention via the analogies. I feel like if the idea is restated in different ways (not necessarily just terms, but perhaps visualizations/re-conceptualizations) a more holistic/intuitive/fuzzy idea emerges rather than anything rigorous necessarily, but you know a lot more math than me so maybe I'm saying nonsense.
    - chobytes 5y ago
      To be honest if you're sufficiently motivated and you find it interesting then go with it. Better to be motivated about an idiosyncratic method than ambivalent about the whole thing.

youssefabdelm 5y ago

My interest in category theory is for a completely different reason. I too have had to struggle to understand it, but I don't plan on using it in an applied or programming sense. I've heard a bunch of people say that category theory is almost like a 'bird's eye view of mathematics'. That many of its concepts unify many mathematical concepts under one umbrella, and reveal the similarities and connections between mathematical subdisciplines. The main reason this interests me is that I'd like to learn math much differently than its taught. To me, to learn the same concept under different subdisciplines with different names, is O(N) learning. However, if someone put together a category theory book where they list a concept, and then they do this: "Now, any mathematician can (easily) see that every major area of mathematics is a category. - In Set Theory the arrows are functions - in Topology they are continuous functions - in Group Theory homomorphisms - in Linear Algebra they are linear transformations - in Differentiable Geometry they’re smooth maps, and so on… But what’s important is that we shifted from focusing on the objects to focusing on the functions, the ways in which we transform the objects. This is basically the category-theoretical perspective: it’s the functions that matter." (Taken from this blogpost: https://catsinthejungle.wordpress.com/2008/11/15/category-theory-for-non-mathematicians/ https://catsinthejungle.wordpress.com/2008/11/15/category-th... ) Do I understand any of that? No. But what attracts me to this format is it (MAYBE...) approximates O(1) learning in math, so long as it can also point out, for every concept in every field, how they differ/relate to every other concept. So what this does is it helps teach me the 'unifiers' of mathematics, and helps me see connections between fields which can be useful for creativity. E.g. I realize I'm doing X algorithm here from linear algebra and that is O(N), whereas if I just shift fields, and use Y algorithm from Group theory I might make it O(log n) because it differs in this way or whatever it is. Besides that I see it as a potentially efficient way of learning a lot of math.

chobytes 5y ago
Math is at its heart about intuition; formalism is a just tool. One uses the formalism but one has to see past it. Roughly its the difference between knowing that a group is a set with a certain binary function, and knowing that a group is a particular formalization of symmetry. While a shortcut to understanding sounds nice, I don't think you'll find it by focusing on abstract formalism.
- youssefabdelm 5y ago
  To me this would be another way of developing intuition, via analogy across disciplines. E.g. Volk's Metapatterns draws patterns (but perhaps less rigorous and more loose) across discplines giving one intuition about many things. In this case, seeing the connections across the disciplines isn't necessarily something I see or am pursuing because I'm searching for a shortcut (although I did mention O(1)), I just see that as a side benefit. What appeals to me more is developing the intuition you mention via the analogies. I feel like if the idea is restated in different ways (not necessarily just terms, but perhaps visualizations/re-conceptualizations) a more holistic/intuitive/fuzzy idea emerges rather than anything rigorous necessarily, but you know a lot more math than me so maybe I'm saying nonsense.
  - chobytes 5y ago
    To be honest if you're sufficiently motivated and you find it interesting then go with it. Better to be motivated about an idiosyncratic method than ambivalent about the whole thing.