I’m not one for much physical activity (see here, here, and here for more details), but after a rousing first week of Lego math research, my four lovely students have me thinking a bit about marathons and sprints.
Mathematical research involves skills relevant to (or perhaps even necessary for) both activities, it seems. Spotting patterns and forming conjectures require energy and excitement, while verifying these very claims demands patience and persistence.
Its not a perfect analogy, I concede, as there are plenty of instances where significant progress is made in a short amount of time or where conjectures and questions take quite a while to tease out. A wonderful example from the first category is the famed account of graduate student George Dantzig, a story that should be required reading for any and all seeking degrees in mathematics. A fairly good example of a story belonging to the second category might be the formulation of the Monstrous Moonshine Conjecture; while early versions of it took months to pin down, the fuller version proved by Borcherds was years in the making.
Still, something about this proposed correspondence feels right, even if only in a basic sense. Perhaps the distinction I’m tempted to draw isn’t so much between “asking questions” and “answering them,” though, as it is between “ideas” and “details.”
Formulating ideas can be a lot like sprinting, I suppose, as the ones we end up pursuing are often the same ones that quickly or instantly grabbed our attention in the first place. They appeal to our more impulsive sides, inspiring instant action or pursuit. An analogy more in my comfort zone might be flipping channels (do people do this anymore?): there I sit, remote in hand, eagerly searching for something good to watch. Similarly, when cooking up an idea, say, for the proof of a theorem, I find myself switching through different stations in my brain, ready to land on something (anything) remotely appealing. Sure, I tend to waste time on whatever the math equivalent of ALF reruns are, but every now and then I stumble across Breaking Bad.
A good idea session is short and satisfying, and it leaves you wanting more. Details, on the other hand, are a patient woman’s game. Upon initial glance your idea may seem solid and crystal clear, but some deeper consideration often reveals a wobblier, murky object. Attempts at proving any number of celebrated conjectures are in no short supply, for instance, and at the heart of the earnest ones are “good” ideas. Avenues for new research are quite easy to discover, too, and that’s why its such a shame so many of them lead down alleyways and streets that are well worn or blocked off by dead-end signs.
While working on my thesis I was treated to the following advice from Dr. Frank Jones: “Its good to save some grunt work for a rainy day.” Unfortunately, hurricanes made my time in graduate school rich with such opportunities, leaving me plenty of time to grunt. Pushing through details is necessary, of course (unless you’re a physicist), but I had always fancied myself “an idea person.” It was something of a rude awakening to sit down with a concept and push it until it broke; I had much better things to do, thank you very much, like dream up newer and crazier plans and designs.
After giving it some thought, though, I realize that most people would rank themselves similarly. Who wants to be the “detail guy” anyways? Like the myth of the floater, I suspect that despite claims to the contrary true “idea people” are in very short supply. Returning to my original analogy, there are some people who travel great distances at sprint-like speeds, but for most of us this simply isn’t the case. For us its marathon-running, like it or not.
That’s not to suggest that this is an “either-or” scenario. Indeed, the good news is that details and ideas inform one another. A rigorous, detailed pursuit of an idea is always helpful, shedding more light on the complexity if not veracity of a given conjecture. And this holds no matter how flimsy that initial idea turns out to be. Conversely, new ideas can help one become a better or faster “detail pusher.”
This dichotomy also speaks to the difference between believing something to be true and (mathematically) knowing it. Science abounds with examples of overturned conjectures, as well as cases where intuition fails to be verifiable. Consider, for instance, the following claim:
If you draw a loop that never crosses itself, then it will always pass through the corners of some square.
As mentioned on this website, the precise mathematical version of this statement remains an open problem. However, a weaker version was proved in 1989, one that “morally” confirms the above claim. It takes only a matter of minutes before one gains some intuition concerning this problem; a few doodles later, and, sure enough, it seems entirely plausible. Even likely.
I plan to post a future entry listing the most counterintuitive theorems mathematics has to offer, but for now we can focus on one particularly confounding phenomenon: Weierstrass functions. Like other “pathological monsters,” these critters give rise to continuous curves that are not smooth. Check out the animation below, stolen from Matthew Conroy’s website, for an illustration of this fact.
No matter how far we zoom in on one of these graphs, we never see a tangent line. There is something very fractal-like about this, as all we ever encounter are finer and finer oscillations. Given this, its not too surprising to learn that these functions are built out of infinite combinations of trig functions. They can be written down as summations of the form
where the constants and are chosen to ensure fails to exist for all . Standard convergence theorems from an introductory analysis course ensure that whatever other properties has, it is necessarily continuous (probably the easiest tool to use for this is the Weierstrass M-Test).
I can’t resist pointing out that the lack of differentiability can be intuitively understood, at least somewhat. Let’s consider a specific choice of parameters, say and (this corresponds to a horizontally shifted version of the animated graph displayed above). The terms in our summation take the form and, for fixed values of , produce a modified cosine wave. The factor vertically shrinks the wave while the large number horizontally compresses it. When the value of is high, these effects are dramatic, shrinking and compressing exponentially. The net result is a lot of oscillation in a tiny bit of space. Its plausible, then, that an infinite sum of such terms produces an infinite amount of oscillation on all scales, explaining the lack of tangent lines. Can you determine necessary conditions on the parameters and so that the Weierstrass function always has this property?
It boggles the mind that monsters like Weierstrass functions exist, if only upon initial consideration. Suffice it to say that its dangerous (and rather boring) to assume one’s beliefs are (mathematically) true. The finer points uncovered by logical exposition are often technical but also valuable. There’s only one thing better than trusting something’s true, and that’s knowing why its true.
So how do you prove every hand-drawn loop contains a square? Who knows? Like any open problem, much hard work and no shortage of ideas and details will likely be required. The answer might be surprising, too. Perhaps its not true that every embedded, hand-drawn loop passes through the vertices of a square.
I would not be surprised to learn that this is somewhat unique to Americans, what with our can-do attitudes and/or inflated self-images, but I wager that many people find ideas easy to come by or generate. Testing those ideas can be pretty rough, if only at first. The pain is eventually overcome and serves its purpose in the end: to prepare us for more challenges ahead.
In reality, the difficulty lies in knowing when to sprint and when to pace yourself. Exactly how difficult is this open problem? Exactly how far away is that destination? No one knows — at the very best, only once in a great while does someone know. So to all of you math majors stuck on a homework set, all of you graduate students struggling to find or finish a thesis problem, and to all of you mathematicians sighing over an unforgiving computation, I raise high my glass of wine in your honor.