Category Theory and Time's Arrow

A fascinating discussion, from 2009, about why some kinds of thoughts (and some kinds of math) seem more "natural" than others — Stephen Harris writes in the n-Category Café:

I guess I lean to the position (really a truism) that so much of mathematics as we know it is dictated by the way humans are wired — the way in which cognitive processes are rooted in vision and in physical actions, and driven by emotional needs, that it seems much safer to me to suppose these sorts of imbalances are due to human habits or characteristics. In other words, if computers were one day to do independent mathematical research, they might explore all sorts of possibilities which would seem very strange or would never occur to us, and redress some of the apparent imbalances.

John Baez responds:

... living in a world where the future and past are so drastically different, people are bound to come up with math that has this asymmetry built in.

... Part of my reasons is that language, from which many higher mental functions seem to be evolved, can cope with resolving quite deep "stacks" of definite elements but are very poor at resolving much less deep "stacks" which involve tracking multiple different possibilities at each level. ...

I agree, but I again see the arrow of time here. We need to predict the future much more than we need to postdict the past. Why? Because we want to keep on surviving in the future, not the past. Why? It's an arrow of time thing.

So, trees of possibilities that branch forwards in the future are more worrisome than trees that branch backwards into the past. So, we find it very nice when a cause has just one possible effect: it simplifies our life. We don't care as much whether an effect has just one possible cause — unless we're doing a little piece of detective work, or history.

In short, we don't need to think about physics explicitly to develop a preference for functions over cofunctions: the physics of the world we live in almost guarantees it.

On the contrary, it's only when we start thinking about physics explicitly that we notice that the laws are time-symmetric, or nearly so, and we start developing an interest in categories that are self-dual. First the additive group of real numbers: the time line, which stretches symmetrically in both directions. Then the concept of groups, where every element has an inverse. Then the idea that time evolution is a symmetry, i.e. the action of a group. Then the development of matrix mechanics, where every operator has an adjoint. Then the realization that dagger-categories are important in physics, and the idea of 'reversible computing'.

and Todd Trimble notes:

I like John's arrow-of-time or entropy idea, where in favoring many-to-one relations (functions), we are just doing the normal human thing, putting different things into one box [which may be a physical box, like this morning when I was picking up my kids' toys, or a mental box, i.e., an abstraction], and thus disregarding or losing information or distinctions which we deem irrelevant or unimportant for the time being. It may be superfluous to give examples, but one of the first that comes to mind is putting all fractions that represent the same rational number together in a box: 2/6 and 1/3 are for all intents and purposes regarded as the same.

Decategorification is a kind of normal human act where we identify (and forget) by way of abstracting. Categorification can be seen as an act of remembering where the abstractions originally came from.

It may seem as though words like entropy, losing information, forgetting, and decategorifying are being used here as pejoratives, but of course I don't mean that: life, and science in particular, would be impossible without such basic maneuvers. There's that haunting short story by Borges, "Funes the Memorious", where a young Uruguayan savant gets thrown off a horse and is in a coma for a few days; when he comes to, he is physically paralyzed but endowed with an infallible memory. He can retrieve, with perfect fidelity, anything he has ever seen, heard, felt, thought, etc., down to the most minute detail ...

(cf Greatest Inventions (2011-06-09), Simplicity via Abstraction (2016-01-07), Cakes, Custard, and Category Theory (2016-02-14), Category Theory Concepts (2016-04-25), Bird's-Eye View (2016-07-20), Category Theory for Programmers (2017-05-12), Ultimate Abstraction (2017-08-24), Put the Vast Storehouse in Order (2017-10-04), Category Theory is like a Lighthouse (2018-12-24), Macro vs Micro (2019-02-03), Why Care about Category Theory (2019-03-03), ...) - ^z - 2020-05-01