Yes. It’s at the heart of the experience of learning and doing maths.

But it’s also the great unspoken, almost like a taboo. I’ve even heard the following: “When a mathematician says he* doesn’t understand something, it’s because he thinks it’s wrong”.

We are not supposed to admit that we don’t understand. It almost seems that if you misunderstand, you don’t exist, mathematically speaking, you’re not there, not in the place where mathematics happens, that austere realm where IF YOU CAN BUT SEE, beautiful objects exist in timeless purity.

“That is a pack of lies!” Ah, where is Zorba the Greek when we most need him, to protect us against Plato the Greek?

We do misunderstand. It’s an essential stage. When we first encounter a new idea or a difficult problem or even when we do a calculation that should be routine but seems to go wrong, we always reach understanding through misunderstanding. It is by accepting that misunderstanding will happen, in the confidence that with patience and cunning you can outwit it, that you really learn what mathematics is, what it means to you, as a living, breathing, feeling human being with a human body. It is a frustrating and painful experience, but intensely rewarding.

It is also the route by which you make mathematics your own. The end point of mathematical theory and practice is the beautiful theorem, the elegant solution, which can be communicated in any language and means exactly the same thing every time. (Try that with poetry, advertisements or political speeches). That, after all, is the goal of mathematics: to think of the number 7, the theorem of Pythagoras, n-dimensional vector space and so on as things that are exactly the same for every single person on earth, now and forever and since time began.

But this agelessness makes it inhuman. More inhuman than even the weather and the stars, which after all differ from place to place and, as long as you yourself are attached to some places more than others, give you your own weather and stars. You can’t have your own theorem of Pythagoras, not in its pure mathematical meaning and truth. But can and do have your own history with that theorem, which must include moments of confusion and misunderstanding, and this leads you to your own favoured proof (or, if you’re lucky, your own preferred proofs and generalisations), and in that sense you can and will have your own theorem of Pythagoras.

*This is a verbatim quote, hence the pronoun. Besides, it fits with the aggression implicit in the statement.