Is there any connection between Structuralism and Category Theory?

Question

Having only the a very cursory knowledge of Structuralism, there does appear to be some points of coincidence:

Structuralism:

1. Individual elements of culture must be placed within a System/Structure.
2. The individual elements of culture must be understood by their inter-relationships within this System, and not by their individual identity, that is their identity is supressed.

and compare this with Category Theory:

1. Individual objects of mathematical interest must be placed with a Category.
2. Though these individual objects have their own character, this knowledge is supressed, and only their relationships (called morphisms) have import.

There seems to me a clear correspondance here. Of course, it could mean that both paradigms evolved independently from some prior philosophy. I'm thinking of Leibniz rather than Kant, from whom Saunders MacLane, one of the two cofounders of Category Theory, purloined the word 'Category' for his own uses. He also studied at Gottingen, which from my limited knowledge of German philosophical history, was a centre of philosophy, presumably due to Kant.

Some more evidence from Structuralism, by Sturrock:

'What is a structure, then, for Husserl, and 'in general'? The broadest definition is that a structure is an abstract model of organisation including a set of elements and the law of their composition...What stands out in a structure is that the relationships between the elements are more important than the intrinsic qualities of each element'.

and the definition of a category can be further elaborated as:

3.Morphisms between objects (i.e. the relationship) follow a law of composition.

Answer 1

Consult this paper by Andrei Rodin for an interpretation of category theory without structures.

His central claim (from the abstract) is that while structuralism in the philosophy of mathematics studies "invariant form" (for instance, the sentences of a categorical theory are invariant across isomorphic models), categorical mathematics studies covariant and contravariant transformations, which in general have no such invariants.

If his reading is plausible, it provides reason to think Category Theory needn't be wed to structuralism. If his stronger claim is correct, then Category Theory shouldn't be wed to structuralism.

I'm not sure that connecting category theory to structuralism in the humanities is of much interest. The similarities, though they are present, are at a vague and broad enough level to be virtually meaningless.

Answer 2

A letter by Jean-Michel Kantor, Bourbakis Structure and Structuralism at springerlink.com/content/x60030547jl61071/that throws a great deal of light on this:

he says,'When I asked Claude Levi-Strauss about the origin of the word ‘‘structure’’ in his work, he answered (letter to the author, Nov. 16, 1990): ‘‘Ne croyez pas un instant que Bourbaki m’ait emprunte´ le terme ‘‘ structure’’ ou le contraire, il me vient de la linguistique et plus pre´cise´ment de l’Ecole de Prague.’’ (Do not believe for one minute that Bourbaki borrowed the word ‘‘structure’’ from me, or the contrary; it came to me from linguistics, more precisely, from the School of Prague.'' '

Bourbaki was a society of mathematicians that were intent on putting the whole of mathematics on a rigourous foundation. The idea of Structure pervaded their thinking, in particular the idea of homomorphism=structure preserving map.

It looks like that the idea of Structure becoming pervasive and prominent in two unrelated fields at roughly the same time can be put down to the workings of the Zeitgeist.

Answer 3

I think there is a "philosophy of abstract algebra," or "structuralist philosophy," if you prefer, that guides us in asking the right questions. It says, "It's not about what a real number is, its about how real numbers relate to one another; it's about plus, times, less-than, etc."

I think category theory takes this philosophy to its logical conclusion. Let me illustrate.

If we have a metric space, we can ask, "What is an open set?" A good answer would be, "The open sets are precisely those whose every point has enough space for a little ball around it." From this definition, we can prove some facts about open sets, such as: their union is an open set, finite intersections are open sets, and many other things, too!

After a while of proving things, you realize that many of the interesting statements about open sets follow from just two observations, namely that their union is an open set, and that finite intersections are open sets.

So you have an epiphany. You realize: It's not about what an open set is. Its about how they relate. So you define a topological space. You tell the reader what a topological space is, rather than telling them what an open set is. This is the philosophy of algebra in action.

After a while, you start thinking about continuous maps between topological spaces, homeomorphisms, Cartesian products, etc. And at some point, you have another epiphany. You realize: it's not about what a topological space is. It's about how they relate. So you define the category of topological spaces.

In conclusion, I think category theory takes the philosophy of abstract algebra (or "structuralism", if you prefer) to its logical conclusion.

Answer 4

In my knowledge of mathematics it is rather difficult to find something that is not a Category. It will not surprise me that Structuralism is one. In other words: to say of Structuralism that it is a Category is to say very little.