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Physics may be an empirical science, but it is a lot like math in that there is a lot of theoretical reasoning. It's like math where you can test things using experiments. Arnold said math is the part of physics where the experiments are cheap. I also am interested in the motivation for the axioms and definition. It's a lot easier to remember everything when it's not introduced by arbitrary decree. But the axioms and definitions are only the starting point. The theorems are interesting, too.

The math that people are working on today, by and large, is not foundational to other sciences. Sounds like what Gauss said, what you quoted. Math was a very different subject when Gauss was around. People were still developing the basics that are quite useful in physics and engineering. It no longer plays that foundational role for other sciences, for the most part. I hope to write a book one day about how useful recent research is and speculates on how useful current research will be. Most of the math that is really foundational to the rest of science has already been done, much of it over 100 years ago. There are some interesting applications of more recent subjects being developed, but it takes a lot of expertise to understand whether they are genuine applications or just the result of people trying to force-fit some math into things to sell the math. I used to be interested in fundamental physics, but I now find condensed matter much more compelling. I'm not sure a theory of everything with all the math it requires is that admirable or realistic of a goal. Condensed matter is a bit more practical.

My favorite areas of math are topology, complex analysis, and graph theory, for their visual appeal. Graph theory also appears to have very interesting applications.