Since Euclid's axiomatization of space, we have developed a sophisticated mathematical model of space. Given a category of structures (measures), local space is modeled the spectrum of measurements that these structures can possibly make. Global space is construed as patches connected via transport, which identifies measurements across patches.


I'm troubled that I have not come across any axiomatization of time. Assuming that mathematics is *a priori* science, the great varieties of theories of space in physics can be attributed to our sophisticated mathematical model of space. There is relativity, string theory, quantum theory and M theory.


Perhaps the reader may object that these theories are theories of space-time, rather than theories of space. However, I wish to note that in these theories, time is essentially treated in the same manner as space. In classical physics, time is but another dimension of space. In relativity, time is distinguished from time by the (3,1) signature, but this is just a metric. Riemannian geometry is still considered a theory of space rather than a theory of time.


I'm wondering, then, whether you have encountered is a mathematical axiomatization of time, that treats time in a way not that is not inherently spatial? Assuming once more that mathematics is *a priori* science, perhaps such an axiomatization can lead to breakthroughs in physics and finance.


Finally, there is a physical theory that I think comes close to a model of time. Namely, entropy. Just as space is dual to measures, we can think of time as dual to entropy. Given as entropy can be defined using combinatorics and probability, this could be viewed as a mathematical theory.


EDIT: Steve mentioned that perhaps one can view entropy as a theory of time via the Thermal Time Hypothesis. Other than entropy, are there any other axiomatizations of time?


ANOTHER EDIT: In the answers given below, most of the models of time are archimedean. I'm wondering, this models can be tweaked to allow a cyclic conceptualization of time. Many ancient cultures, eg from India, consider time to be cyclic rather than archimedean. Should I ask this as a separate question? 


I think of this cyclic/archimdean dichotomy as something like Euclidean/non-Euclidean geometry. 



Is there a mathematical axiomatization of time (other than, perhaps, entropy)?
In probability, time is usually handled as a nested sequence of $\sigma$-algebras (say $B\_t$, with $B\_t \subset B\_s$ if $t\leq s$), and to find the reality (call the reality $f$, and it includes the state at all times past and future) at time $t$, one takes the conditional expectation $f\_t := E[f | B\_t ]$. The sequence $(f\_t)$ is then a martingale (a uniformly integrable martingale, more precisely), and this construction is the essence of what the big deal is about martingales.


Brownian motion is a martingale that you've probably heard of, but this also handles simpler situations. For example, consider the experiment: toss a coin repeatedly, and keep track of how many heads you've thrown, minus how many tails. We can capture this experiment in the following way: For $0\leq x <1$, let $f\_n(x)$ be the number of 1's minus the number of 0's among the first $n$ digits of the binary expansion of $x$, and let $B\_n$ be the $\sigma$-algebra (in this case, a boolean algebra) generated by the intervals $[i/2^n,(i+1)/2^n)$. Then $f\_n$ is $B\_n$-measurable, and $E[f\_t | B\_s]=f\_s$ for any natural numbers $s < t$, and the sequence $(f\_n)\_{n=1}^\infty$ is a martingale (albeit different from the type mentioned above). If you want to play any fair game on the "coin tosses" as they come up (allowing use of knowledge of all previous-in-time tosses), then your fortune at time $t$ is still a martingale.


In other words, the passage of time is captured as un-conditional-expectating a function.


For a practical introduction to martingales, I recommend Williams' "Probability with martingales." It is a marvel of writing, and in my humble opinion should be taken as a model for how to write a monograph.


