Alain Connes - How a Mathematician works

A typical exercise, for example, is... to understand any subject, above all, a mathematician SHOULD NOT pick up a book and read it. It is the worst error!

No, a mathematician needs to look in a book, and to read it backwards. Then he sees the statement of a theorem. And, well, he goes for a walk. And, above all, he does not look at the book. He says, "How the hell could I prove this?" He goes for his walk, he takes two hours... He comes back and he has thought about how he would have proved it. He looks at the book. The proof is 10 pages long. 99% of the proof, pff, doesn't matter. Tak!, here's the idea!

But this idea, on paper, looks the same as everything else that is written. But there is a place where this little thing is written that will immediately translate in his brain through a complete change of mental image that will make the proof.

So, this is how we operate. Well, at least some of us. Math is not learned in a book, it cannot be read from a book. There is something active about it, tremendously active.

If you try to understand a proof in a passive manner, you'll have a guided tour through the proof but you will not understand. You'll follow line by line without understanding anything. You'll be able to follow the logic of a proof, similar to how a computer could check this logic. In a crude form, you can see it. But you will never be able to understand.

The real way to understand is to look at a statement that gives you a certain mental image, you think about it over some time, and then come back, you look at the book, and then you have a chance to understand.

But the essential role is this TRANSITION between something that is written, which seems incomprehensible, and the mental images we create.

And this is already at an artistic level, let's say. There is a creation, there's a great creative act, and that creates in the brain something, a mental image that represents this thing that is written.

And while you do not have it, you can not think. While you have not created in the brain the elements of understanding that will then fit together, you can not understand.

And the only way you have of creating these elements of understanding in your brain is to be STUCK on a problem!

What does that mean? It means you're here, I ask you a question, and you can not answer it... "How can this be solved?", etc.

Then you say, "I don't understand, so I waste my time." Most people think that because they do not understand, they are losing time. This is completely false!

It is while you're trying to understand and you can't do it, that you drill the hole. It's as if you tried to make a hole in a wall with a drill. Until you reach the end, you have not yet understood. But you are moving forward!

And if you back away because you say "I don't understand", you will never understand! You'll never understand anything!

You need to have the patience, the perseverance to reach the end. And if sometimes you do not reach the end, look it up in a book. But you will read 100 times faster than reading the book from the beginning. And you'll understand at a speed 100 times greater.

So this is the mathematician's work. It is a work almost inside the brain—of creation of mental images, of elaboration of things that are created, etc.

And all that is already at an artistic level, of course. But it is not transferable from one person to another. It's a personal, individual work.