I had this interesting discussion today with one of the instructors from my summer course in physics. She was a math major, then became a physics major (I’m not sure if she finished one before starting the other). And she claimed very strongly that the things that you do with math in physics simply are not math.
Now I know that she was deliberately making a very broad statement simply to exclaim the difference, but I couldn’t let it go.
At the high school algebra-based level, there is no difference. Especially as some schools move towards more reality-based math construction, the difference between physics and math is growing smaller.
But even the approximations and wacky things physicists do with derivatives and integrals make sense on a physical level. It doesn’t fit the “rules” of math as they were taught to me by mathematicians, but it fits with the purpose and meaning of the math. Isn’t that what makes math work?