My answer to a question on Quora: Is it natural for a mathematician to forget specifics of some parts of math?
I think it is natural. I love to surprise my students by saying that I hardly remember any trigonometric formula beyond \(\sin^2x+\cos^2x=1\)– but I can deduce most standard trigonometric formulae on the spot. On a number of occasions I offered my students a game: give me a wrong formula, say, for \(\sin(x+y)\) — and I will instantly explain you why it is wrong. I have always won. I am not special; I believe that the vast majority of my mathematician colleagues can recover statements of L’Hopital theorems even if they do not remember them exactly.
And there is one more aspect of mathematical memory. The great mathematician Andrew Wiles said in a recent interview Andrew Wiles: what does it feel like to do maths?:
I really think it’s bad to have too good a memory if you want to be a mathematician. You need a slightly bad memory because you need to forget the way you approached [a problem] the previous time because it’s a bit like evolution, DNA. You need to make a little mistake in the way you did it before so that you do something slightly different and then that’s what actually enables you to get round [the problem].
So if you remembered all the failed attempts before, you wouldn’t try them again. But because I have a slightly bad memory I’ll probably try essentially the same thing again and then I realise I was just missing this one little thing I needed to do.