Math is immutable truth, and it’s everywhere in physics, chemistry, statistics, and engineering, subjects that I love.  Once a theorem is proven, it does not change.  The laws of physics seem to change a bit every 100 years.  The recommendations of medicine change all the time.  Math theorems are permanent and very well-defined.  The language itself is very precise and can be made perfectly precise with tools like Coq or Lean.

Math teaches you how to write proofs–irrefutable arguments that prove your theorems.  Once you write the theorem and spell out all the little steps in a proof, you can be 95% or more sure that your conjecture is correct.  If you want more certainty, you can run your proof by another mathematician or resort to Coq or Lean.  The search for proofs makes you really understand what is going on.  Once you write your conjectures and prove them, you have a strong feeling of certainty about the subject and a deep understanding.  I have seen this happen in finance, business, physics, engineering, and games.

I love formulas–so much compressed knowledge crunched down into just a few symbols.  F = ma, $c^2 = a^2 + b^2$, f(x) = f(x_0) + (x – x_0)f'(x_0) + …, Gauss’s Law, the chain rule, Cramer’s Rule, $E = mv^2/2, F = kx, (f g)’ = f’g + g’f$, Planck’s Law $B_v(T) = (2h v^3/c^2) / (\exp(hv/(kT)) – 1)$, Shannon Capacity $C = B \log_2(1 + S/N)$, and thousands more.  Researchers and mathematicians spent years looking for many of these formulas and then found them.  You can combine them together and manipulate them with plain old algebra to solve a huge number of problems.  Add in calculus, and you can solve even more questions and derive hundreds of additional formulas.

When you are studying Category Theory, Logic, Linear Algebra, Topology, Manifolds, or Functional Analysis, you feel like you are studying God–the ultimate, irrefutable knowledge–the most important concepts known to humanity that would be useful to any alien in this universe, or any other universe.  After reaching these heights of thought, you can sometimes astound employers, colleagues, and friends with the things you discover.  Sometimes you uncover new theorems about a subject that you thought you knew, casting it in a new light.  Other times you give people new insights into their own areas of study and passion.

Lastly, I also love how so many things in math can be explored and tested on a computer and how beautiful it looks when written in TeX.