# Time in Mathematics

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?

• I don't know what the word "believe" means here, but plenty of mathematicians either use objects which can easily be thought to be evolving in time (e.g. solutions to PDEs) or otherwise use dynamic intuition, so I don't really see the first point you're trying to make. – Qiaochu Yuan Nov 16 '10 at 12:25
• "Intuition" is one thing, "thorough definition" another. And aren't solutions to PDEs just sets-with-a-structure or functions-as-sets (see example 2)? – Hans-Peter Stricker Nov 16 '10 at 12:45
• Anyway: I'm going to change the "teaser". – Hans-Peter Stricker Nov 16 '10 at 12:47
• en.wikipedia.org/wiki/Dynamical_system – Abdulh Khazzak Gustav ElFakiri Oct 20 '13 at 13:31

## 6 Answers

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.

• "Time 'feeling'" is a good term! (There you meet Qiaochu's "dynamic intuition".) So my question should read: "Where do you have a time feeling or dynamic intuition?" – Hans-Peter Stricker Nov 16 '10 at 14:01
• Of interest is the recent development of directed homotopy theory. This looks at non-reversible maps. It also interacts with Theoretical COmputer Science (TCS) see other answers. – Tim Porter Apr 30 '11 at 11:21
• Whoa! Are there accessible references for this answer? This has to be one of the most interesting things I've ever read! – bright-star Nov 6 '13 at 8:11
• Homotopy theory is really all over the place, I probably couldn't give a better reference than Hatcher's Algebraic Topology. For Lie theory, I would suggest John Stillwell's Naive Lie Theory. It only requires linear algebra and calculus and focuses on examples. – Vhailor Nov 16 '13 at 4:39

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.)

• That's an interesting comment by Atiyah. But how can one objectively say that algebra is about one and geometry is about the other? Spacetime (eg Minkowski manifold) has an algebraic aspect and a geometric aspect. – alancalvitti Jan 8 '13 at 1:44

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.

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.

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:

Projections of $n$-dimensional objects in a $(n-k)$-dimensional Euclidean space also have in my humble opinion this kind of "dynamic intuition".

For instance when we observe a tesseract ($4$-dimensional "cube") through its $3D$ projection, if we want to "observe" the whole object, it is required to apply a complete rotation around the object to have an intuition of its structure (an example of the methodology applied is here). This can be applied in general to higher dimensions projected into lower dimensions. It is required to rotate and project. I believe that the required rotation implies the concept of "dynamic intuition" you mention.

For instance, below there is a projection of a tesseract and a point inside the tesseract. They are $4$-dimensional objects and for this reason we rotate the object in $4D$ and then we project it to $3D$ (indeed what we can see below is a $2D$ simulation of the $3D$ projection).

In essence, we are aware of the remaining fourth dimension (or in general, the remaining dimensions) that we can not see in the $3D$ projection through the dynamic rotation (and the rotation implies time as well!). Another example of a $4D$ projection that can be a good example of the concept of "dynamic intuition" here.