Computations using light I recently found out about analog computers. They really intrigue me.
I've seen a few videos on how mechanical computers work.
I also know that there are electric analog computers too (I have no clue how they do multiplication though).
So, that got me thinking - can light be used to perform addition, multiplication, division and subtraction? Or can you take a bowl of water (or some other medium) of a certain shape and do math with it?
 A: You can do all kinds of things like that, but in the end you will find yourself returning to electrons. Using light in particular is not a very good idea because it is very hard to do non-linear operations with it. Almost all non-linear effects require a very large number of photons (I would guesstimate on the order of 1e8 or more), whereas one can comfortably do electronic calculations with a few dozen electrons (your computer is already approaching that limit). Moreover, light can't be stored easily, whereas electrons can, so a light computer has a serious (lack of) memory problem. 
As for how to do analog multiplications with electrons, that's very easy: pn-diodes have an almost exponential relationship between current and voltage: $I=I_S(e^{V_D/nV_T} -1)$ (Shockley ideal diode equation). For large enough currents one can neglect the $1$ and build circuits that produce an almost ideal logarithmic or exponential behavior over six orders of magnitude of current or voltage or more (usually in the 1nA-1mA range). Once we have logarithms and exponentials, one can use $ln(a*b)=ln(a) +ln(b)$, which leads to $a*b=\exp^{ln(a)+ln(b)}$. I have not seen anything like this being done with light. In reality, of course, no such circuit can compete even remotely with the precision and speed of digital electronics, so any form of analog computation is really just a footnote of computer history. It's still useful to do these things in electronic sound synthesizers and some rare analog control applications, but truthfully, one has to come up with rather esoteric reasons to go that way, these days there are certainly no valid engineering reasons whatsoever to be found for these tricks (I admit, though, that I have been paid to perform some of them, anyway). 
A: Any physical system that can imitate a Turing machine can be used to perform universal computations, including general math (a different issue is if it will become practical for actual problems or commercially useful). You can adapt almost any physical system to imitate a Turing machine. That includes optical, pneumatic, hydraulic, electric, gravitational, etc systems. In fact the opposite happens: it is actually difficult to find an example of a physics realm that cannot be adapted to do universal computation. 
