Many years ago I happened to visit Ulan Bator, the capital of Mongolia. The center of the city was dominated by monolithic towers, soulless Stalinist edifices meant to intimidate the civilians scurrying about in vast inhospitable plazas like figures in one of DeChirico’s paranoia-inducing paintings – nowhere to run, nowhere to hide. Some numbers are like that; they seem almost designed to put you in your place and what a small place it is, smaller than you ever imagined. The numbers are impossible (unless you’re a mathematician or cosmologist anyway) to wrap your mind around. They are so staggeringly large (indeed, many of them are larger than large insofar as they have no limit) that in the face of them we feel diminished, reduced in importance, that it’s almost impossible to reconcile our everyday world with the worlds symbolically represented by such numbers (whether they exist on a macro or microscopic scale). Not long ago, at a neighborhood restaurant, I met a lawyer from Rochester, NY who admitted that his true passion is mathematics. He pointed out that there are several ‘infinities,’ which in itself is a troubling notion since most of us wouldn’t be able to figure out what one infinity would be like, let alone a multitude of them. But there are practical aspects to this infinity business, said the mathematician-turned-lawyer (presumably for financial reasons – another type of mathematical consideration.)
Let me see if I got this right: The set of prime numbers, for instance, is infinite because even a supercomputer, crunching numbers now into…well, infinity, would never reach the end of possible prime numbers because no matter which prime number (those are the numbers that can be divided by themselves and 1) you can come up with there will always be another prime number lurking in the wings waiting to show that number up. On the other hand, there’s another type of infinity – think of the number of divisions that are possible between zero and one. You can’t. Because no matter how minute the fraction you can divide that number into you can still divide that result again…ad infinitum. (You might recall Zeno’s paradox about the race between the hare and the tortoise, the idea being that if you go half the distance and halve it again and halve that distance again you would never reach the finish line, but of course, runners – and hares and tortoises, too, if they can be steered in the right direction – do in fact get to the finish line. Calculus was designed to resolve this paradox (among other things), which it did by essentially saying that at some point the distance between you and your goal is so infinitesimal that it doesn’t count so we’ll just forget it.) That type of infinity, the lawyer asserted, is even more ‘infinite’ than the infinity of prime numbers. Since I barely scraped by in high school algebra I am in no position to weigh in on this, but the concept of infinities outdoing one another in the ‘Infinity of Infinities’ contest is the kind of thing that could keep you up at nights or at the very least give you a bad headache.