Meeting a mathematical ghost

I gave up mathematics in July 1986, having completed a BA degree in it. It’s only a couple of weeks ago that I’m threw out my undergraduate notes, which probably says a lot about my hoarding tendencies. It’s 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.

It’s not just that I can’t answer this question now, it’s 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 don’t 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 don’t remember what a vector space is, I remember the girl I was then and she knew so little. She’d never been abroad or had a boyfriend when she went to university, she couldn’t look after herself, she was so shy that the only bit of the entrance interview she could shine at was the problem solving. She couldn’t 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 I’ve given above is that it’s 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 wasn’t 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 can’t siphon the knowledge back in again. I don’t 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 I’m not sure I really want to put it to the test. It’s 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.

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6 thoughts on “Meeting a mathematical ghost

  1. As a former physics major, I can grok. 🙂

    Especially the part about not being good enough to visualize stuff real mathematicians — or physicists — consider basic.

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  2. My own mathematical studies ended in 1950 but your reminiscences struck a chord with me.

    Higher Mathematics, Elementary Analysis and Additional Geometry were school subjects in the Scottish Highers system that determined my youth. Success there aided my application to study Medicine at the University of Edinburgh. That knowledge, however, played only a marginal part in my later learning and when, in my retirement, I proceeded to a degree in History of Art it seemed even less relevant. There was renaissance perspective of course, as the basis of projective geometry, a few references to the golden mean and perhaps the explanation of Velazquez’ Las Meninas by Foucault, but the connections were of tenuous relevance.

    What remains true however is that I have always believed that Mathematics was the ultimate subject of study. It is fundamental to the understanding of Natural Philosophy and equally important to Moral Philosophers. Its study demands no prior knowledge, merely a willingness to apply a logical mind.

    Perhaps I am biased. 6th.year studies in mathematics was taken by the head of the school mathematics department. He held a Doctorate and had a fierce reputation as a disciplinarian. However, there were only four of us in that class, all keen to learn, so that it had more the character of a university tutorial which both students and teacher came to enjoy. I am not talking private school here, just bog standard local authority establishment.

    Never mind the subject. I think he taught me how to enjoy studying and the topic does not matter.

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  3. I also did a math major but I realized pretty early on that it all goes out of your head pretty quickly if you let it.

    I took calculus in high school, and when I got to college they had us take a math placement test. I looked back into my notebook from my calculus class (where I had written out steps to exercises that had been assigned but no real notes) and couldn’t remember what the integral symbol meant. I couldn’t even remember what it was called. I placed out of the basic math requirement but I didn’t place out of any of the calculus sequence.

    When I took calculus in college I took very careful notes so that if I ever needed to look back I would be able to relearn the topic from my notebook, and the trend continued with all of my other math classes.

    I’m 8 years out of college, currently working toward my actuarial certification in the US, and while most of the calculus involved is pretty straight forward, I still have to pull out my notebooks and review every few months when I run across a more complicated integral. I think I’ve looked at how to take derivatives and integrals of natural logs and exponentials so many times now that it is finally sticking, but I’ve reviewed those more than I’d like to think about.

    In college I enjoyed my abstract math courses the most and didn’t do much applied stuff. I too recognize most of the terminology in your excerpt above but can’t do anything with it now. So it goes.

    But I think that, given my constant state of relearning calculus, it would be possible to get back to that level of understanding if I had the time and energy to do it. I’d bet you could too.

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    • It’s reassuring to know that it’s not just me that forgets the stuff so quickly! But that brings me to the question of why? I can remember other stuff I’ve learned for considerably longer. Even if go several months without reading German, for example, I wouldn’t have to go back and relearn all my verbs: I can get up to speed again just by looking at a few texts. And I can remember song lyrics I learned as a child.

      I wonder whether maths is much more unmemorable, perhaps because of more complicated structures/less-redundant structures? You can get by with remembering 2/3rds of your German vocabulary and inferring the rest from context, or only remembering one verse of a song, but if you don’t have every term in a mathematical formula right you’ll get the wrong answer.

      Or is that we learn maths in the wrong ways to memorise it? Undergraduate history teaching, for example, tends to focus on getting students to develop a framework of broad historical knowledge, into which specific details (the date and outcome of this particular battle) can then be fitted. In contrast, maths tends to be treated very much at the detailed level, with every theorem needing to be remembered individually. Maybe that’s inevitable with the building block approach needed to learn mathematics (you must know A, B and C before learning X), but it means that if a single block is lost, large parts of the syllabus become inaccessible. It’d be interesting to hear from some practising mathematicians whether they need to relearn stuff they don’t use frequently or whether they are able to internalise it successfully.

      You’re probably right about me being able to relearn mathematics if I really wanted to. It’s just that the thought of having to start from so far down in the foothills again, gaining almost no benefit from previous exposure (unlike with languages, for example) is particularly daunting.

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  4. Since you say it would be interesting to hear from a practising mathematician, here is my comment.

    I’m now in my first postdoc, which means I started university 10 years ago. I don’t have to relearn basic things, nor things somehow related to the field I work in. But I can easily forget a lot of things from courses I have taken say 6 years ago, even if I knew them very well at the time.

    I also find that theorems are more easily forgotten then definitions: in some way it is easier to remember what the concepts are, rather than what properties they have.

    When I needed it, I could usually study the stuff again with a small effort. This is probably due to the fact that I already studied it, and that I have a greater mathematical maturity.

    I should also add that from time to time I like to try to reconstruct these things I haven’t studied for long in my mind, for example when doing a shower. Maybe I try to reconstruct what a theorem said, or what the proof could be. This certainly helps not to forget everything.

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    • Thanks for this – it’s interesting to hear what the difference is at different levels of training: there’s clearly a big jump between first degree and doctoral level. I think that I never really mastered a lot of the undergraduate material that I studied, in the sense of it becoming part of my mental furniture. My recall was accurate, but terribly shallow, as if I’d memorised 100 phrases of a language, but never properly learnt the underlying grammar so I could generate new sentences. Perhaps the problem is that you have so much to cover so quickly in a first degree, and that you really need to do fifty examples of using a method rather than five to have it become second nature. I don’t whether if I’d exercised my mathematical muscles more in the immediate aftermath of my degree it would have helped, but I suppose it was like going to the gym – it’s easier to let it slide unless you have something very specific to achieve. It’s all too easy to get away with being remarkable mathematically immature when you still know more about maths than 95% of the rest of society.

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