Terence Tao

In the pre-rigorous stage, mathematics is taught in an informal, intuitive manner. The rigorous stage is where one is taught the proper, formal way of doing mathematics. The post-rigorous stage is where one uses the intuitive understanding again, but now informed by the rigor.

It’s not about being smart, it’s about being stubborn and curious.

The ‘soft analysis’ way of thinking is more about estimates and qualitative understanding, while ‘hard analysis’ is about precise bounds and quantitative control.

I don’t have any magical ability. I look at a problem, and it looks something like one I’ve already done. I think maybe the idea that worked before will work here.

The power of mathematics is often in transforming an impossible-looking problem into a manageable one through a sequence of simplifications.

If you want to be a good mathematician, you have to solve problems. If you want to be a great mathematician, you have to find good problems.

Technical skill is mastered through practice, but insight and intuition come from a deeper engagement with the subject.

In analysis, we sometimes use the ‘epsilon of room’ trick: if you can prove something for all epsilon > 0, you can often prove it for epsilon = 0.

A common theme in mathematics is that of duality: two seemingly different perspectives turn out to be essentially equivalent.

The primes are not truly random, but they behave in many ways as if they were.

You don’t need to be a genius to do mathematics; you just need to have a taste for it.

Good notation can make a deep idea seem simple, and bad notation can make a simple idea seem impenetrable.

The purpose of abstraction is not to be vague, but to create a new semantic layer in which one can be absolutely precise.

In mathematics, the journey from a heuristic argument to a rigorous proof can be very long, but it is the journey that provides the real understanding.

Sometimes, the most profound breakthroughs come from asking a ‘stupid’ question that nobody thought to ask before.

A mathematical theory is not complete until you can explain it to the first person you meet on the street.

The density increment argument is a fundamental tool in additive combinatorics: if a set is not structured, you can find a subset where it is denser, and iterate.

I try to learn something from every paper I read, even if it’s just a neat trick or a way of thinking about a problem.

The Cauchy-Schwarz inequality is probably the most useful inequality in mathematics.

Mathematics is a social activity. You learn by talking to others, by reading, and by explaining.