Terence Tao

The hardest part of solving a problem is often figuring out what the right question is.

In harmonic analysis, understanding how a function oscillates is often more important than its size.

A good analogy is like a bridge: it allows you to cross from the familiar to the unfamiliar.

The twin prime conjecture is one of those problems that is incredibly simple to state, but seems to require a completely new idea to solve.

Persistence is more important than talent.

The interplay between structure and randomness is a central theme in many areas of mathematics.

When stuck, try to solve a simpler version of the problem first.

A mathematical model is only as good as its assumptions.

The beauty of mathematics is that you can sometimes prove that something exists without being able to explicitly construct it.

I view mathematics as a giant interconnected web of ideas, not as a linear sequence of topics.

Compression of ideas is a key part of mathematical progress.

The Navier-Stokes existence and smoothness problem is a monument to our ignorance about fluid dynamics.

A proof is a story where the characters are mathematical objects and the plot is logical deduction.

In research, it’s often more useful to have a good heuristic than a messy rigorous proof, at least in the early stages.

The goal of mathematics is to find patterns and make them precise.

Sometimes, the key to a difficult problem is to forget about it for a while and let your subconscious work on it.

A counterexample is often more valuable than a proof, because it tells you where the boundaries of a theory lie.

The most exciting breakthroughs are the ones that connect two areas of mathematics that were previously thought to be unrelated.

Rigor is to the mathematician what hygiene is to a surgeon. It is a necessary condition for doing good work, but not the goal itself.