I gave up mathematics in July 1986, having completed a BA degree in it. Its only a couple of weeks ago that Im threw out my undergraduate notes, which probably says a lot about my hoarding tendencies. Its always odd when you look back at things from your past, but this was positively spooky. The name on the pages is my maiden name, the handwriting is recognisable, if slightly juvenile (I was twenty when I graduated). But who was the girl who had done this studying? It was someone, apparently, who could answer the following question:
Let V be a finite dimensional vector space over C, and let α and β be linear transformations in V with αβ = βα. Show that α and β have a common eigenvector. Suppose the subspace W of V is invariant under both α and β. Show that α and β induce linear transformations ᾱ and β̄ of V/W.
Its not just that I cant answer this question now, its that I can barely comprehend even what it means. The terminology bounces through my brain, stirring vague imprecise echoes: a linear transformation preserves…something or other, and the eigenvector of a transformation is a vector which remains the same when the linear transformation is performed on it (but I dont know how it is performed) and V/W is…completely meaningless.
This is probably one of the simpler questions I came across, one of the ones I can actually work out how to type. And yet the girl of the files understood a lot of this stuff, judging from the many tick marks, could reel it off from memory in exams, and not just algebra like this, but probability and logic and complex analysis. How did she do it? Because though I dont remember what a vector space is, I remember the girl I was then and she knew so little. Shed never been abroad or had a boyfriend when she went to university, she couldnt look after herself, she was so shy that the only bit of the entrance interview she could shine at was the problem solving. She couldnt cook a meal or raise a child or give a lecture or write an article the way I can.
Nor was she some kind of prodigy, even though she went to some of the same lectures as Ruth Lawrence. The really terrifying thing about the problem Ive given above is that its second year undergraduate stuff (or at least it was in the mid 1980s). To be a research mathematician, you have to be able to work at a level almost unimaginably above the common mathematical abilities. The girl thought she wasnt good enough to do that, and she was probably right. So she turned to other things, eventually found a subject she loved and where innate talent was less vital. And in the process I forgot 99% of what I learnt in my maths degree, which seems a waste. Or maybe it was just that that kind of mathematical knowledge never really belonged in my brain, but was only forced into it temporarily, and the moment the pressure to learn and remember was removed it naturally siphoned out.
Throwing away the notes is a belated recognition that I cant siphon the knowledge back in again. I dont know if somewhere deep in me there is still a mathematical core (an invariant subspace?) and the ability would come flooding back if I seriously put my mind to relearning mathematics. And Im not sure I really want to put it to the test. Its disconcerting enough to come across the ghost of your own self. It would be even more disconcerting to know for sure that part of your intellect is now forever closed off to you.