The difference between mathematicians and other people is not that they are specially clever, it is that they have learned to suffer the journey from misunderstanding to understanding. When encountering new mathematics, mathematicians accept that there will be a time that they are mystified, uncomprehending, in a foreign country where a language is spoken they barely understand. They also know that with persistence, understanding will come.

Of course they are better at this journey than non-mathematicians. How could it be otherwise? But this is not a matter of gift (though gift helps), it is that they have learned how to learn mathematics, how not to get stuck in misunderstanding, how to recognise misunderstanding. A few simple tricks are part of this, like writing out your own examples and trying to explain the new ideas to a friend. Other methods are elaborate, such as making complicated diagrams that relate old ideas to new, or that relate unsolved problems to each other in detail.…

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.…