Comment by chas | Hacker News Reader

Comment by chas | Hacker News Reader

Parent Full thread

Type theory is a framework for doing logic that predates computers as we know them. If you have used first-order predicate logic before (e.g. ∀x∀y(P(f(x))→¬(P(x)→ Q(f(y),x,z)))), it performs a similar role to type theory and is the same sort of mathematical thing, it just has different properties. Logical frameworks are interesting to programmers and computer scientists because logical systems and programming languages/computation are related through the Curry-Howard correspondence. http://math.stackexchange.com/questions/166430/curry-howard-...

This[1] isn't a perfect description of type theory in general because it is aimed at a particular branch of type theory called homotopy type theory, but I feel like it does a pretty good job of explaining the differences between type theory and set theory and what motivated those differences. [1]http://planetmath.org/11typetheoryversussettheory

Category theory is a particularly abstract part of abstract algebra primarily concerned with extremely general mathematical structures. It concerns itself with identifying and understanding the core structures common to a large number of mathematical objects and operations such as the one shared by multiplication, the cartesian product, least common multiple, logical conjunction (&&), and structs (or record types) in programming. This structure is usually referred to as the categorical product.

Category theory is often brought up when discussing type theory because there is a close relationship between these sorts of abstract structures like the one linking structs and conjunction and the structures that are described by type theory. In general, there is a close relationship between type theories and certain types of categories so you can learn interesting things about type theory from studying category theory and vice versa, but category theory contains many things which are not primarily useful for or associated with type theory.

> It concerns itself with identifying and understanding the core structures common to a large number of mathematical objects and operations such as the one shared by multiplication, the cartesian product, least common multiple, logical conjunction (&&), and structs (or record types) in programming. This structure is usually referred to as the categorical product.

This sounds more like universal algebra (http://www.encyclopediaofmath.org/index.php/Universal_algebr...) than category theory, which, almost by definition, is interested in studying structure-preserving morphisms, without too much attention to exactly what structure is being preserved.

Category theory is about categories,. One notion which makes sense in the context of a category is the categorical product (http://en.wikipedia.org/wiki/Product_%28category_theory%29). All the examples given are examples of categorical products within suitable categories [e.g., Cartesian product within the category of sets and functions between them (amounting to multiplication of cardinals, if one just cares about the action on objects), least common multiple within the category of positive integers ordered by divisibility (a partial ordering being just a special kind of category), logical conjunction within the category of truth values (which can be thought of as sets with at most one element), and structs or record types in the category whose objects are the types of your favorite programming language and morphisms are the programs between them].

Certainly, I didn't mean to imply that you (EDIT: I mean chas) were wrong, just that it didn't feel like the examples that you brought up were in the 'spirit' of category theory. As I mentioned in a sister comment (https://news.ycombinator.com/item?id=8780829), any sufficiently powerful formalisation can encode any other (proof: definition of 'sufficiently powerful'), so that there is no mathematical structure of which it can be said definitively that it is not an instance of category theory—but that doesn't mean that every structure should be so viewed!

For example, to pick on the lcm example (just because it's the one that caught my attention): as you mention, posets are automatically categories, but I don't think that the theory of posets is best viewed as a part of category theory; and, similarly, the product is just a special case of the limit over a discrete category, but I don't think that describing the least common multiple, say, as a limit will be educational to anyone! By contrast, viewing the lcm as part of an unusual algebraic structure on the natural numbers I think can be instructive.

I absolutely think it's instructive to view the LCM as a specific case of meets in a semilattice, and in turn to view these as specialization of a general theory of categorical products. (Why wouldn't it be instructive? If you like, the "unusual algebraic structure" on the natural numbers which might be worth considering is that of a category with products). And I also think that recognition of such analogies and general patterns ubiquitous across mathematics is a large part of the spirit of category theory.

Reply on news.ycombinator.com