Category theory does not make predictions.
This is a black and white formulation, so there certainly are exceptions. Feel free to contradict.
UPDATE: As I’m watching Gromov on probability, symmetry, linearity, the first part:
I can’t stop noticing several things:
- he repeatedly say “we don’t compute”, “we don’t make computations”
- he rightly say that the classical mathematical notation hides the real thing behind, like for example by using numbers, sets, enumerations (of states for ex.)
- and he clearly thinks that category theory is a more evolved language than the classical.
Yes, my opinion is that indeed the category theory language is more evolved than classical. But there is an even more evolved stage: computation theory made geometrical (or more symmetric, without the need for states, enumerations, etc).
Category theory is some kind of trap for those mathematicians who want to say something is computable or something is, or should be an algorithm, but they don’t know how to say it correctly. Corectly means without the burden of external, unnatural bagagge, like enumeration, naming, evaluations, etc. So they resort to category theory language, because it allows them to abstract over sets, enumerations, etc.
There is no, yet, a fully geometrical version of computation theory.
What Gromov wants is to express himself in that ideal computation theory, but instead he only has category theory language to use.
Gromov computes and then he says this is not a computation.
Grothendieck, when he soaks the nut in the water, he lets the water compute. He just build a computer and let it run. He reports the results, that’s what classical mathematical language permits.
That’s the problem with category theory, it does not compute, properly, just reports the results of it.
As concerns the real way humans use category theory…
Mathematicians use category theory as a tool, or as a notation, or as a thought discipline, or as an explanation style. Definitely useful for the informed researcher! Or a life purpose for a few minds.
All hype for the fans of mathematics, computer science or other sciences. To them, category theory gives the false impression of understanding. Deep inside, the fan of science (who does not want/have time/understands anything of the subject) feels that all creative insights are based on a small repertoire of simple (apparently) tricks. Something that the fan can do, something which looks science-y, without the effort.
Then, there are the programmers, wonderful clever people who practice a new science and long for recognition from the classics 🙂 Category theory seems modular enough for them. A tool for abstraction, too, something they are trained in. And — why don’t you recognize? — with that eternal polish of mathematics, but without the effort.
This is exploited cynically by good public communicators with a creativity problem. The recipe is: explain. Take an older, difficult creation, wash it with category theory and present it as new.