the difference between mathematics and physics

Think about it.

What mathematics tells you is objective truth. Not all mathematical theorems may be applicable in all situations (for instance, if you're trying to navigate on the surface of the Earth, clearly spherical trigonometry and spherical geometry are relevant, while hyperbolic geometry is not). But all theorems are true whenever applicable. Mathematics deals with reality.

What physics tells you is (usually) empirical truth. When you get down to the bottom of matters, physics only generalises measurements performed on the real world. And these measurements could always be wrong, or worse, be measurements of the wrong thing. For instance, verification of Newtonian mechanics didn't deal with high velocities or very small scales, because of an incorrect homogeneity assumption. And relativistic verification always seems to assume anisotropy (of space), for very good reasons. Physics deals with perceived reality.

Of course, a side-effect of all this is that physics can say a lot more about the world than mathematics. It goes out on a limb, and this is the payoff.

NOTE

Actually, the difference between math and physics is the same as the difference between the English language and a shelf of books. The difference is that you can actually pick up physics and touch it.

Physics is the science of the physical universe, a collection of theories, laws, theorems and axioms that explain how things move, how energy actually works, why the universe is expanding, what everything is made of, and so forth. Today, it's as much theoretical as empirical. But physics, like every other science to a certain degree, is described using the language of mathematics.

Math, you see, isn't about objective truth. Most people think math is somehow intristic to the world around us, when it's actually an invention, something man created in order to quantify the world around us. Mathematics does not deal with reality; it deals entirely with unreal quantities and ideas, with abstract numbers and shapes and forces and formulas.

Mathematics is not tied to any actual things, but it's the language of logic on which all the sciences are built -- and physics most of all.

A famous quote (from Feynman, I think) is that physics is to maths as sex is to masturbation.

By this is meant that the processes are similar, but physics produces tangible results whereas maths is just for fun (for the analogy I think we have to ignore the tangible results of masturbation). It sounds good, but it's not a brilliant analogy.

The real difference is that physics is a science, and maths is not. Physics therefore works on the principle of formulating a hypothesis, testing it, and discarding it if disproven. Maths works on the principle of playing around, sometimes with an aim (perhaps to prove something you suspect), sometimes without.

The confusing part is that the two are so closely linked. A scientific hypothesis can usually be best framed mathematically, and once that's done you can apply mathematical principles to it, which will usually work with reality as well. For example, if you hypothesise about something circular you can use geometry (in maths) to turn that into a statement about π and angles. This isn't always the case, because sometimes mathematics makes assumptions different to what holds in the real world. If I draw a triangle around the sun, the angles will add up to more than 180° because gravity curves space. Mathematics works only within the assumptions you put into it.

The beauty is that the real world often does correspond to something elegant mathematically. However, it is important to remember that the processes are quite different.