RE: What is the difference between physics and mathematics?
Such a fascinating/perplexing question... I'm so frustrated because I got an undergrad degree in philosophy but didn't get into these sorts of deeper metaphysical questions until after I graduated, almost 2 yrs ago now.
One of the best things I've read that I think gets to the heart of the relationship between mathematics and the physical world is Eugene Wigner's, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" whose title sort of speaks for itself!
Here's two quotes from Wigner's paper linked above that you might find relevant,
It is difficult to avoid the impression that a miracle confronts us here, quite comparable in its striking nature to the miracle that the human mind can string a thousand arguments together without getting itself into contradictions, or to the two miracles of the existence of laws of nature and of the human mind's capacity to divine them. The observation which comes closest to an explanation for the mathematical concept's cropping up in physics which I know is Einstein's statement that the only physical theories which we are willing to accept are the beautiful ones. It stands to argue that the concepts of mathematics, which invite the exercise of so much wit, have the quality of beauty. However, Einstein's observation can at best explain properties of theories which we are willing to believe and has no reference to the intrinsic accuracy of the theory.
The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.
I'm not a mathematician but I love admiring from afar, and I've been seriously considering diving back into the subject. I would guess that if I have a genuine interest then cranking out "homework" will be a bit more bearable than in high school haha
Thanks for the post! It was an enjoyable read!