>> I concur with Darren.
>
>> Science is the systematic investigation of the real world. Math is the
>> systematic investigation of abstract systems of axioms of an arbitrary
>> nature.
>
>    Aren't you confusing "science" with "natural sciences"? Natural sciences
> study the natural world, but "science" in general can encompass more than
> that.

Well, I suppose if you wanted to be really pedantic about it, science is 
the study of that which can be experimentally verified (or falsified). 
Mathematics is /usually/ about what can be logically proven, which isn't 
exactly the same, but... there's perhaps some overlap there.

>    As for math, would you say that, for example, the branch of mathematics
> called geometry studies how the real world works or not?

Which geometry? Euclidean geometry? Elliptic geometry? Hyperbolic 
geometry? Some sort of non-homogeneous geometry?

Pure mathematics studies these geometries purely for their own sake. One 
or other of them /may/ correspond to the real world.

(For example, Euclid presumably proposed Euclidean geometry because he 
thought it corresponded to real figures drawn on real flat surfaces. But 
of course today we know that the universe actually has negative 
curvature, so hyperbolic geometry is probably a better match.)

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