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> about bringing together concepts Well, yes, obviously. My analogy is just that: it's like bug collection, bringing together a bunch of samples from many fiel

• synetic 3y ago
  I get the complaint. In the old days people would sometimes say “abstract nonsense” when talking about category theory related topics. It’s increasingly an important part of a basic mathematical education at the graduate level. Eventually there may be a spillover into other areas like computer science where so called practical uses occur. But maybe not. It’s still too early to say. There’s a famous book called Categories for the Working Mathematician. Maybe someday there will be a book called Categories for the Working Programmer. I like that you call it the string theory of math. It’s a more modern way of saying “abstract nonsense”.
  • consilient 3y ago
    > I like that you call it the string theory of math. It’s a more modern way of saying “abstract nonsense”. The sociological analogy is a decent one - they're both ambitious sub-fields of little interest to the average practitioner, with a handful of committed opponents and a wildly overgrown internet hatedom - but their positions in their respective disciplines aren't remotely similar. String theory is a physical theory like any other: pick a spacetime, fill it with dynamical objects, impose some symmetries, and away you go. Category theory, at its most ambitious, wants to be a setting for doing ordinary mathematics in: it belongs in the same general grouping as set theory and higher order logic, not calculus or linear algebra.
  • dkjaudyeqooe 3y ago
    There's this, but the programmer doesn't have to be working: https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/ https://bartoszmilewski.com/2014/10/28/category-theory-for-p... https://github.com/hmemcpy/milewski-ctfp-pdf https://github.com/hmemcpy/milewski-ctfp-pdf
• catgary 3y ago
  I mean, the modern language of algebraic geometry and homotopy theory is based on advanced category theory. David Spivak, the author of this book, wrote a great thesis on cobordisms and used categorical techniques to sidestep hard problems in intersection theory. You can also look at some of Ezra Getzler’s work to see how setting up clean categorical abstractions can make it easier to attack hard problems in geometry. In CS, abstractions like monads and effect handlers come from category theory. Plotkin and Power did a lot of super technical work on syntax/semantics adjunctions while sorting out the foundations of effect handlers.
• aleph_minus_one 3y ago
  > Category theory is something I've always wanted to succeed. In mathematics, category theory is a success story; just have a look at the revolutionary reformulation of algebraic geometry be Alexander Grothendieck and his descendents. Whether category theory is useful to you in computer science depends a lot on the things that you work on and your "taste in programming style": If you want to develop new even-higher-level abstractions that unify handling of very different concepts in a unified way, it is in my opinion a great idea to have category theory at your disposal - at least as a source of inspiration, and plausibly for more. Similarly, category theory might be helpful if you want to generalize some cool solution technique to a much larger (and seemingly unrelated!) class of problems. On the other hand, I can easily find problems in computer science where at least I cannot see how category theory might help me with them (at least with my limited intelligence; I am no Grothendieck ;-) ).
• consilient 3y ago
  > My criticism, I think, is still valid. Just because you've renamed things in a bunch of fields with a consistently obtuse verbiage doesn't mean you've understood anything It's unification, not renaming. As a somewhat tortured analogy, consider two concepts: "games", and "mass". Particular concrete games are prior to "games" in general: we learn nothing about chess or basketball by noting that we happen to have a label broad enough to include both, and the worldview of an alien visitor accustomed to thinking only in terms of board-games or ball-games but never simply games is simply different, not incomplete. But if someone has concepts for mass-of-people and mass-of-planets but doesn't realize that they're all instances of a more general notion of mass-of-things, the situation is different. People-mass and planet-mass are both "mass" not only by convention, as with games, or by brute coincidence, but by physical law. And someone who doesn't know that has a hole in their understanding of physics. Category-theoretic concepts should be understood in the latter sense. Expecting knowing what a monoid is to deepen your understanding of string concatenation, for instance, is getting things backwards. (This is really an example from abstract algebra, not category theory proper, but the same principle applies). The point is to use your knowledge of string concatenation to get knowledge about {string concatenation | linked lists | regular expressions | MapReduce | max/min | addition | multiplication | convex hulls | ... }, all at once and (once you've paid the upfront cost) all for free.
• ndriscoll 3y ago
  > The thing is, AD predated CT substantially, and was well-understood for a long time. The original theoretical description of AD is something I understood, and the CT description of it was impenetrable, and not useful. I was the one who linked that. I wasn't familiar with AD before his talk, but I find it perfectly understandable (modulo Haskell's sometimes foreign-to-me syntax). I think the relevant background here is that I was a math major and had a course in abstract linear algebra (i.e. general vector spaces over fields) and two semesters of abstract algebra (groups, rings, fields) plus dipped my toes in some CT on my own. He basically notes that D had a slightly nicer chain rule if you write D'f(a)= (f(a), Df(a)) and use D' instead. Then his definition of AD is just the chain rule and the product rule, plus the fact that Df = f for all linear f, applied recursively. So forward mode AD is just writing down facts you learned in calc 1 as a recursive definition (with, I guess, some base cases like sin/cos hard coded). His description of reverse mode AD is more complicated, but IMO still accessible to someone who's seen dual spaces, product/coproduct vector spaces and maybe enough CT to understand that the map from a vector space to its dual (and matrices to their adjoint) is a contravariant functor. The basic observation is that he can write the chain rule and product rule as a compositions of linear maps with product/coproduct spaces (which are the same thing for vector spaces). Then he says a biproduct category is one that has (equal) product/coproduct, and notes that the dual map preserves that structure. The point is that biproducts are what he needs to write chain/product rules as compositions of linear maps. Since he wrote D' recursively in terms of linear maps, and dualizing/taking the adjoint still gives him the structure he needs to do that, he's able to write reverse mode AD as the adjoint representation of D'. He just needs to write the adjoints of the pieces he used, and again compute recursively. I know at least at my school, the engineering program I was in had a first semester graduate class that dealt some with dual representations (using "test functions" etc. when solving differential equations), so I think it could be distilled down to something appropriate for masters level engineering while retaining the character of what he presented. Like I said, the Haskell syntax was probably the most obscure thing for me. It's all a matter of perspective and background, but IMO it's sort of like how if you're familiar with linear algebra (including function spaces), then the Fourier transform is just writing a function as a sum of projections onto an orthonormal basis. No need to remember the formulas for it; they're obvious and have simple geometric meaning.
• pnin 3y ago
  Since you provide a zoological metaphor, let me offer an alternative. Category theory is much more like Goethe's work on the /Urpflanze/ (primeval plant). Linné had developed a way of systematically naming species of plants, but Goethe was not satisfied by this approach. Goethe wanted to find an underlying pattern common to all plants which would explain how plants grow and develop. Something like a "universal grammar" (to draw an anachronistic parallel to Chomsky) of plants. Goethe called this his "morphological" method and wrote about it in "On the metamorphosis of plants". One caveat: this paints a slightly too esoteric picture of category theory. Goethe was very idiosyncratic as a scientist, whereas category theory, far from being esoteric, is a common language for all of modern mathematics.