Your favorite part of it?
I was never very good as a
practitioner of the mathematics, but the history and philosophy of mathematics intrigues me to no end. How mathematical concepts get discovered (e.g. how people tried to prove Euclid's fifth and instead accidentally invented hyperbolic geometry), how they get argued, questions like how
real mathematical entities are (i.e. ontology)- these strike me as fascinating metaphysical questions.
One thing I really want to look into is the problem of applicability of mathematics to the real world. Consider this. Pythagoreans thought reality must, in a fundamental way, mirror the natural numbers. In their zeal to see patterns and find connections everywhere, they posited the existence a tenth planet- a counter-earth- just because they thought the celestial bodies were perfect and must correspond to a neat, round number like 10. You may scoff at their eccentricity if not ignorance, but fast forward to the twentieth century. When Paul Dirac was solving for the charge of an electron, he found that the equation yields two results- one was the expected value, the other was the same value except positive. Any lesser mathematician or scientist would discount this as an anomaly or a mistake. Not Dirac. He went ahead and postulated the existence of positron- a particle with equal amount of charge as the electron, except positive. And sure enough, a few years later scientists ended up confirming the existence of positron based on experimental evidence.
I know, the two cases aren't exactly same, or even similar. There's clearly a method to the madness in Dirac's case. But it's still intriguing to see how precisely mathematical concepts map to how things really are, no matter how ethereal the connections may be. I've been reading Roger Penrose's
The Road to Reality: A Complete Guide to the Laws of the Universe as of recent, and the same theme kept coming up. Mathematical solutions that are chosen with considerations like simplicity or elegance in mind often end up reflecting how reality actually is.
This is where, in my opinion, mathematics is at its most beautiful.