# What is there to be studied?

The prominent British mathematician G. H. Hardy (1877-1947) said he had never done anything useful in his life. But don’t judge him. He did many wonderful things but is somehow, unfortunately, most famous for his discovery of Srinivasa Ramanujan (1887–1920). Hardy was a number theorist, and number theory is considered by many as the queen of mathematics. Carl Friedrich Gauss (1777–1855) once said: "Mathematics is the queen of the sciences and number theory is the queen of mathematics."

Well, for a long time, number theory was pure mathematics. But times change. No mathematician in the 1940s could imagine that in a couple of decades some highly advanced and abstract mathematics discovered in one of the French military prisons would be used by pretty much everyone on the planet. And yet it is so. André Weil (1906–1998) refused to serve in the army during World War II and was put in prison in 1940. This is where he made some of his most important discoveries. The work was about pairings, and today pairings are used heavily in computer security, more precisely – cryptography. By the way, Weil famously said that every mathematician should spend a couple of years in prison.

And that’s actually what fascinates me the most – mathematics that used to be considered pure and looked like it could never be anything but pure, subsequently turned out to have unexpected applications.

I had ups and downs in my relationship with mathematics. I was a lousy student for most of my years at college and I didn’t touch mathematics for quite some time after I graduated. However, the desire to learn now seems to be getting stronger every year.

But what is it that I would like to learn? Around three years ago, my wish list was something like:

- Understand the mathematics behind Weil pairing.
- Understand the proof of Gödel incompleteness theorems.
- Learn how to use Lean to verify nontrivial proofs.
- Understand how homotopy type theory is used in formal verification in mathematics.

This list is ridiculous. I know that now. At that point, three years ago, I knew some cryptography and some abstract algebra, but most of the things that I’d learned as a student were pretty much just a blur. Somehow, at that time, I stopped learning cryptography in my free time (but continued to do it for a living) and went more toward the points above. But given the complexity of these topics and having a full-time job, the plan was just ridiculous.

So what’s the situation now?

Until 18 months ago, I’d been studying Abstract Algebra by Dummit and Foote page by page with the goal of preparing to study The arithmetic of elliptic curves by Joseph H. Silverman. Today, I still haven’t moved beyond the second quarter of Abstract Algebra. It seems like I will reach the elliptic curves sometime in the 25th century. This is depressing. Not all is lost, though; I am now busy studying the isogeny-based cryptography, and my knowledge of elliptic curves and algebra in large is slowly improving. That means I am kind of moving towards pairings from the other side – from the application point of view instead of from building the fundamentals first (but also working on the fundamentals at the same time).

As for the second point – Gödel’s proof left me speechless; it is so very different from anything else I’ve seen in mathematics. That said, I doubt I will continue studying logic; it is incredibly fascinating but also somehow disconnected from other areas of mathematics. On the other hand, in Paul Halmos’ wonderful autobiography, I’ve read about how he handled logic with algebraic structures. Sounds interesting, but I am not sure I will ever explore that.

Studying Gödel’s proofs at that time actually led me to the lambda calculus, which finally brought me to the third and fourth points of the above list. The third point is about how to verify proofs using a computer. And that is totally exciting. The math that is used to enable such verifications seems to be crazy difficult and beyond exciting. Sadly, I’m only able to use Lean for very simple proofs; anything nontrivial is beyond my reach at this point. And I haven’t touched it in months, so I might have already forgotten most of what I knew.

I am no better at the fourth point either. I have no idea about homotopy type theory. This one was actually the longest shot by far. I started studying Topology by James Munkres, Lambda-Calculus and Combinators by J. Roger Hindley and Jonathan P. Seldin, and Type Theory and Formal Proof by Rob Nederpelt and Herman Geuvers – to pave the way for me to understand homotopy type theory. But I’ve given up on this one. Too little time, too much of something I know almost nothing about.

I haven’t given up entirely, though. I left cryptography for some time, but now I am back at it. And I like it a lot. I like cryptography, the math behind it, and the applicability of it. I’m sorry, Mr. Hardy. Applicability aside, I want to properly understand the abstract algebra that is behind cryptography, particularly behind pairing-based and isogeny-based cryptography. I just wish I could process all this beautiful knowledge faster.