Time in Mathematics

Question

I claim that it is commonly believed that Mathematical objects can be seen as genuinely static, with no "Platonic" time in which they do genuinely evolve.

Nevertheless time has its place in mathematics:

1. An endomorphism of a set (seen as a set of states of a system) into itself can be seen as evolution of the system in discrete time steps.

2. For a function of a totally ordered set into a set (seen as above) the ordered set can be seen as "time".

3. as the time-like component in Minkowski space

Questions (slightly modified after Qiaochu's comment and Vhailor's answer):

Which other constructs do give you a "time feeling" or give rise to "dynamic intuition" admit a comparable straight-forward interpretation as "time"?

The examples above are set-theoretical ("concrete"). Is there a more abstract modelling of "time", maybe in category theory?

Answer 1

I know of two other concepts that have a "time" feeling to them.

The definition for homotopy involves a parameter which can very intuitively be interpreted as time.

A lie group also has to me some feeling of time, since it adds on top of classical geometry the idea that isometries must be a sort of continuous motion through time, not just a "teleportation" between two states.

Answer 2

I wish I had this reference handy, but Atiyah said something to the effect that algebra is about time and geometry is about space. The "processes" in mathematics are, in the broadest sense, the things concerned with time. (This doesn't take into account the idea of "reification", the transformation of a process into an object, which is central to mathematical practice.)

Answer 3

If you consider TCS to be maths, then formal semantics, in particular small-step or big-step semantics, might be of interest. Those map input data and a series of operations to a sequence of universes/states, on per time step.

Answer 4

Hermann Weyl and LEJ Brouwer dwelled on this matter. You can search

• The Continuum (Weyl) here
• PhilPapers on Brouwer's works here

For modern perspective:

• Malament-Hogarth spacetime

Answer 5

In general, anywhere that you see a morphism there is probably a time-like interpretation.

For example, discrete-time dynamical systems could be viewed as a category with one object (the state space) and a distinguished morphism $f$ representing evolution by one time unit, and composition of morphisms giving the evolution by multiple time units.

The identity morphism gives us the concept of 'no time passing' and the associative property of morphism composition (i.e. $f\circ (g\circ h) = (f\circ g)\circ h$) ensures that 'time' behaves in a familiar way (i.e. advancing by $n$ time units and then $m$ time units is the same as advancing $m$ time units and then $m$ time units). In this framework, reversible dynamical systems are naturally seen as categories in which every morphism is invertible (i.e they are groups).

In computer science, the morphisms in a particular category are seen as abstracting the idea of sequential computation (i.e. perform this operation, then this other one). The generalization to arrows takes this further by also allowing parallel and recursive computations, and there are various interesting theorems to be proved about when a computation can be parallelised, and whent he order of two computations matters.