I had a strange idea for a new type of rpg. It's a very weird idea, and I'm not sure if it's a very good idea. In fact, I'm almost sure it's a very bad idea for actually playing in (certainly for any long period of time). However, it could be good if you wanted to put it within your roleplay as, for example, a game that a totally alien culture (such as demigods or some other quasi-immortals) plays in its spare time. You thus wouldn't have to totally flesh out the rules, but perhaps your PCs, if they were in the realm of the quasi-immortals for whatever reason and talking to them, could be invited to sit down and join them in a game of roleplay - you could then introduce them to this (just plunge them in) briefly, before they made their excuses and left.
Anyway, here goes:
Instead of roleplaying a person (human, elf, etc.) you roleplay a mathematical function such as x^2, sin(x), etc. Actually, the function that you would roleplay would have to be much more complex than this with at least a dozen terms, most of them being compound terms. For example, a player might roleplay f(x), where f(x) = 3x^2exp(x) + sin(x)/x! + 1/2exp(cot(2x) + 5/3x^4P(3)(x) + cos(x) + ......
N.B. You should probably use complex numbers really, but I will not here for simplicity.
Anyway, each term would represent something. For example, sin(x) could represent fighting ability (simplifying enormously). Constants greater than 1 would represent being good at it (e.g. sin(3x) ), constants worse than 1 being bad at it. Everything about a character (i.e. attributes, personality traits, etc.) would be represented. Let us suppose that x^2 represented being confident. You can build complex traits out of the simpler ones by making more complicated terms. For example, a character whose function included the term 0.2x^2sin(4x) would have very low self-esteem about their fighting abilities, even though they were could at them.
You could maybe have functions of other variables, which would be like races. So, f(x,y) might be something like a "half-elf".
However, this is much more than a pointlessly complicated way of representing stats. Your functions will all wander about function space and have their adventures and quests, but whilst there they will interact with the other denizens and obstacles of this space (which would mainly be operators). For example, lets suppose you were wandering around and bumped in to a differential operator. You would be differentiated, and this would mean that your entire character altered, both your traits and your abilities. Because they alter predictably, it could sometimes be useful to meet an operator, providing the operator was what you thought it would be. An operator such as d/dy would be devastating to f(x), so maybe is companion g(y) would have to keep it occupied whilst f(x) slipped by. It would be very hard to die, but your character would be in constant flux, which you would have to constantly roleplay. There are all sorts of operators you would find wandering function space: differential operators, Hamiltonians, etc.
I'm presuming of course that these functions, operators et al are intelligent and have their own goals, desires and means of interacting with each other. I'm not sure what these goals and desires are, but they'd probably be much too abstract for the likes of us to understand. The quests of roleplayers et by their DM are probably equally weird and ethereal. Constants are probably like inanimate objects - you can interact with them and just pick them up if you want and take them somewhere else. I'm not sure what function space looks like either, though I know it's infinitely dimensioned.
Of course, function space isn't the only plane in this reality. Maybe our functions which we're roleplaying go planar wandering, and decide to enter Fourier space. It's not as simple as entering the astral plane however; your entire personality and traits will be reversed to be the opposite of what they were before, using the maxim "wide in normal space goes to narrow in Fourier space" and vice-versa. Maybe they like it better there, although there will no doubt be operators there, too, to change them. There are other "planes" as well, eg. Hilbert space which I have heard of but don't know anything about.
A function can also self modify itself - their equivalent of us undergoing training. For example, sin(2x) = 2sin(x)cos(x), two different terms which would represent different traits. It would probably take a little bit of time to do this, but most functions could probably do it. Some functions, however, those which were good at self modification, could change almost all of their terms in to new terms by means of Fourier analysis and the like. They could represent their terms as sin and cos functions, or as Legendre Polynomials, etc. This would take a bit of time for a function to do as well, though the better ones would be quicker. Of course, the thing that would determine whether a function would be good at self-modification would be some of its terms (probably complicated ones like x^3cot(exp(x))sin(4x)ln(cos(x)). A very good function might even be able to modify the terms of itself that related to self-modification.
That's about all I've thought of. I did say it was very weird - but as a device to introduce on the periphery of your world it might just work. I think I could probably create some conversations of some quasi-immortal roleplayers playing a game using this system, even though there's no way I could (or would want to!) play a game in it myself.