First, I’m a graph theorist. I’ve always been interested in graph struture theory, mainly along the lines of the theorems of Robertson and Seymour (Neil Robertson was my advisor). I’m interested in what category theory has to say about the structure of graphs, but I’ve not found anything of note that it has to say.

I purchased Mac Lane’s “Categories for the Working Mathematician” a long time ago, but my copy is in near mint condition. Mostly, I’ve been too busy with other things to read it, but part of my failure to study the work has been an inability to find any truth to its title.

I first encountered categories at the beginning of an algebraic topology course. My instructor briefly discussed some of the basics, stated that category theory was useful, and then dropped it. At that time, it appeared to me that category theory was way to0 general to be able to use in the course. Plus, I didn’t understand it anyway.

Once, Mac Lane came to give a talk. During the talk, in front of a packed audience, he stated that matroid theory wasn’t good or important mathematics, pissing off several faculty who worked in matroid theory. I found this comment to be very bizarre. Here was an advocate of a vast generalization of dubious importance dismissing a generalization of vector spaces that has tremendous importance. Was Mac Lane jealous or just cutting down people he didn’t like?

Now, I’m not putting down category theory, I just don’t see how it is useful in my work. I find set theory to be immensely useful, so maybe, as others have noted here, my mindset is too set theoretical. But category theory seems to me to say little about everything. But maybe I just lack knowledge of category theory. Perhaps, one day, someone will show me that category theory says something important about the structure of graphs. How wonderful that day will be.

Scott Randby

At some point Joachim Lambek realized that the lambda calculus was
basically just another way of talking about cartesian closed categories.
More precisely, if you ask in what sort of category you can do lambda
calculus, the answer is: a cartesian closed category.

He told me that he presented this result at a conference on lambda
calculus, naively hoping that henceforth the subject would be recognized
as a small branch of category theory and basically go away. As an older,
wiser man, he now realizes that not everyone doing lambda calculus *wants* to see it subsumed by category theory!

(I don’t know how else to interpret Lawvere’s plan to make category theory the foundations of mathematics).

I think a better way to phrase Lawvere’s plan, as I understand it, is to make CT a foundation for mathematics; different approaches to mathematical formulation bring out different insights more readily.

Any intellectual field that becomes “trendy” can sometimes generate more heat than light in the stampede to get it into theses ;-), but I agree with other posters who think that one of the fundamental insights of CT, the discovery that mapping the same mathematical idea into a different formulation can bring out a better understanding of the idea, is a profound mathematical insight.

Having multiple equivalent “foundations” for mathematics is a natural extension of this idea.

See “From the Erlangen Program to Category Theory,” a weblog entry from Dec. 3, 2002, at
http://www.xanga.com/m759/7315944/item.html.

This weblog entry at
http://www.xanga.com/m759/7315944/item.html
from Dec. 3, 2002, seems relevant.

From the Erlangen Program
to Category Theory

See the following, apparently all by Jean-Pierre Marquis, Département de Philosophie, Université de Montréal:

* Introduction:
http://www.math.mcgill.ca/rags/seminar/JPMarquis_Introduction.htm
* Chapter 1: Klein’s Erlangen program:
http://www.math.mcgill.ca/rags/seminar/JPM-Chapitre1.rtf
* Chapter 2: Eilenberg & Mac Lane’s methodological extension:
http://www.math.mcgill.ca/rags/seminar/JPM-Chapitre2.rtf
* Chapter 3: Basic principles of category theory:
http://www.math.mcgill.ca/rags/seminar/JPM-Chapitre3.rtf

See also the following by Marquis:

* Category Theory (article in the Stanford Encyclopedia of Philosophy):
http://plato.stanford.edu/entries/category-theory/
and
* Categories, Sets, and the Nature of Mathematical Entities (abstract):
http://www.univ-nancy2.fr/ACERHP/colloques/symp02/abstracts/marquis.pdf

“What are the properties of this collection of objects?”

considerable value could be got by answering instead this question:

“What transformations of the objects in the collection leave them unchanged? “

The second question, unlike the first, is essentially a categorical one.