Andy's Math/CS page

Sunday, April 08, 2007

Math for Little People

I'm 22. There's a part of me that's asking: "Why haven't you proved any great theorems yet?" But I can't beat myself up too much over this--with Republicans in the White House, and with so much HBO magic happening lately, the odds are stacked against me.

But there's a different kind of math anxiety creeping up: one of these days, I might have kids. Those kids are going to be curious about the world around them, and they're going to look to me for answers--at least when their mother isn't around.

For the most part, I'll be happy to fend them off with a mix of idle speculation and propaganda. But... what if they ask why honeycombs have six sides?

What if they want to know why scooting towards the middle of the see-saw makes you go up? What if they ask why the rubber chain-links on the swing don't ever come apart, if they're not 'connected'? Could I in good conscience give sloppy answers to questions I know involve beautiful math?

I know the Feds wouldn't come and take them away; I know the kids won't be too disappointed; I know their appetite for rigorous proof will come slowly, if at all. But could I live with myself? No, I've got to get a better grasp of the 'basic facts' of life in our spatial world before I help bring someone new along, if only for my own sake.

But what are these? It's up for grabs, but here's the category I've had in mind mostly: qualitative features of structures, arrangements, and movement, observable by 'medium-sized' agents like children, and generated by/consistent with a naive 'block-world' understanding of matter and physical law (not referring to the AI environment of that name, but in a similar spirit). This is the conceptual space in which I've lived most of my life (and thought about discrete math and computer science), and if it's good enough for me it's good enough for my kids. Molecules and lightning bolts I won't sweat so much, but Block World I want to get right.

So then... how to pick out, think about, and explain its most important facts?

Some of them are, I think, topological, as I've alluded with the swing example. But kids won't swallow homotopy theory any more than they'll eat lima beans, even if I can still remember it by then. Anyways, there's a potential save here: Block World is basically discrete, so the space of 'topological' deformations of objects is much smaller and more well-behaved (I've already posted on discretization in topology).

A good place to start would be the Jordan Curve Theorem, famed for its difficulty in spite of its surface obviousness: what could be clearer than the fact that a non-self-intersecting loop in the plane divides the plane into two components, 'inside' and 'outside'?

As one discrete formulation, consider the loop as a path on the grid, where each step in any direction is of even length. Then it appears that the inside is path-connected, also by grid paths (without the even-length restriction), and separated from the outside. For example:

I think this problem is excellent fodder for thought, and I would encourage anyone who's been scared away by the Theorem in the past to work on it in this friendlier setting.

No hints, though--would your kids respect you?

(Clarification for concerned parties: there are no children on the horizon; I do believe in responsible parenting; and I don't get HBO, although I do swear by Curb Your Enthusiasm.)

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  • I'm 22. There's a part of me that's asking: "Why haven't you proved any great theorems yet?"

    I'm 25. I enjoy math, but I only minored in it. Now I'm thinking it would be nice to go back to school and become a mathematician. Part of me says this will work out well, whether or not I ever do any "important" work. Another part of me is telling me not to waste my time because I'm too old.

    Would I be making a horrible mistake? I don't know anyone who can advise me on this. Obviously it's a pretty major decision.

    By Anonymous Anonymous, at 1:44 PM  

  • Hi anon,
    going back to school is a major decision. But it's not necessarily a binary all-or-nothing decision, nor does going back to school require such a decision. You could potentially take a year of additional school as

    -an investigation into the field, your interest in it, and your aptitude;
    -an investment in your overall intellectual growth/capital;
    -a hedonistic act.

    By any of these lights it would seem that trying grad school is unlikely to be a 'horrible mistake'.

    I have no scientific data on age & achievement in math, but I seriously doubt being 25 is a fundamental barrier to entry, especially if you enjoy math and minored in it.

    It's hard to give you more advice--I would need to know more about you. But the most basic advice I have is, expose yourself to math and give it a chance to grow in your life. If it grows, follow up with school etc.

    In my view, the most fertile and authentic mathematical experiences are active ones. Regularly solving problems (even very modest ones) empowers, while reading lay treatments of the most advanced theories and profiles of famous mathematicians can engender feelings of envy and inadequacy. That's not to say one shouldn't try to get a sense of the larger field and its successes; but personal involvement is in my view key both to success and to insight into one's own abilities and tastes. This is why I post puzzles on my page.

    Hope this helps, and email me if you'd like more specific advice.

    By Blogger Andy D, at 4:09 PM  

  • Hi,
    thanks for your comments. I will follow your advice. I was relieved that you didn't say 25 is too old, which I was half-expecting you to say. I think you're quite right about lay treatments being discouraging (aside from "Men of Mathematics", which is how I got turned on to math in the first place). Maybe they are the reason I'd come to believe that I was too old. Also, when I posted my comment, I'd just read a very discouraging essay about mathematicians by alfred adler. Choice quotes:

    "The mathematical life of a mathematician is short. Work rarely improves after the age of twenty-five or thirty. If little has been accomplished by then, little will ever be accomplished."

    "Each generation has its few great mathematicians, and mathematics would not even notice the absence of the others. They are useful as teachers, and their research harms no one, but it is of no importance at all. A mathematician is great or he is nothing."

    Discouraging stuff!
    I will make an effort to ignore it.

    By Anonymous Anonymous, at 2:23 AM  

  • By the way, I've also read Adler's essay and can report that he has no idea what he's talking about (in my experience as a professional mathematician). He's done an unusually large amount of damage to the mathematical community by spreading discouraging rumors.

    He's right that few mathematicians do such incredibly innovative work that if they hadn't lived, nobody else would have thought of it for a long time, but so what? Just because somebody else would have made the same discovery sooner or later, it doesn't mean your work isn't valuable. There are thousands of mathematicians making fundamental, important research contributions, even though only a handful stand out as geniuses.

    It's also true that if you spend ten years trying to do research but end up with only minor accomplishments, then you are not likely to have great accomplishments in the future (although there are exceptions). So if you work on research from ages 25 to 35 without much success, it's probably not your calling. On the other hand, if you reach 35 without having had an opportunity to do much research, this doesn't really limit your future success. Eventually old age will catch up with you - few 80-year-olds can compete with their former 20-year-old selves intellectually - but 35 is far from old. 25 isn't old at all, so you shouldn't consider age a barrier.

    The only serious disadvantage to getting an older start is the tenure system. Let's say you start graduate school at 26, and spend six years there (the average is probably around five, but let's add a year since you may need to fill in some background). At age 32 you will either get a tenure-track job or a two to three-year postdoc (the latter if you are aiming for a job at a top research university, the former otherwise). Your tenure case will be decided about six years after you start a tenure-track job.

    So the net effect is that you may well not have a job until age 32 or a permanent job until age 38-41. In the meantime, you may be getting married, starting a family, etc. For some people, this is no big deal; for others, it's a real issue. My impression is that this is the main thing to keep in mind regarding the effect of age on starting a Ph.D. program.

    By the way, one temptation may be to start a masters program and then transfer to a Ph.D. program if it goes well. At some schools this is a common option, but it is often better to start a Ph.D. program and then leave with a masters degree if you don't like it (nobody will ever know from your CV that your plans changed). Math faculty often treat Ph.D. students better, and it is easier for a Ph.D. student to get a fellowship to help pay for graduate school.

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