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.