It may be conceded to the mathematicians that four is twice two. But two is not twice one; two is two thousand times one.
That's a line from G. K. Chesterton's The Man Who Was Thursday, and it's a great line. Then he follows it up with "That is why, in spite of a hundred disadvantages, the world will always return to monogamy," because that's my lovely friend Gilbert for you, always bringing it back round to Catholic orthodoxy somehow.
The context? The protagonist, Gabriel Syme, who has felt himself to be alone as an undercover detective amidst a den of anarchists, realizes that there's another like him, and he's overwhelmed with a feeling of camaraderie and strength. One person, alone, needs to be a tremendous fighter indeed to face an entire mob. Two persons, back to back, just might be able to cover the whole field. Chesterton is right: two is much, much stronger than one.
It's observations such as this one which prompt me to think that there's something genuine about numerology, which as a youthful science nerd I was taught to flee with horror, the same way astronomers are taught to despise astrology and chemists are taught that alchemy is obsolete mediaeval junk. Every major science, I suppose, has its twilight counterpart with evil connotations. I'm sympathetic to astrology these days but I'm also forced to admit that it's a tool of bıgotry. The shadow companion to biology is truly frightening—Lysenko, "race science", and eugenicism live with biology's sinister twin. Numerology conjures up other horrors, like people irrationally frightened of the number 13 or who gamble away entire fortunes because they have feelings about odd integers. It's tempting to think, as the science nerds think, that it's all nonsense.
Yet I've come to think there's value in numerology, though I'm still uncertain about how best to express it. Observations like Syme's remind us that there's a subtle difference in quality and mood to numbers and how they interrelate. Two is not merely one more than one. And three is not merely one more than two. x² + y² = z² has infinitely many solutions among the integers; x³ + y³ = z³ has none, and proving this was the work of centuries. Ordinary mortals can't even understand the proof—I certainly don't. The "two-body problem" is an undergraduate textbook example; the "three-body problem" has no general analytical solution, though many special cases are soluble. Once again we sense that going from 2 to 3 is a fearfully complicated business in reality. It's not simply counting.
There's an opposing extreme to numerological complication, I daresay, and it's to be seen in the bean-counters and the computer programmers who seem to regard numbers as dead objects, meaning nothing more than quantities to be shuffled about with calculators or computers and plugged into simple formulae. How many programmers realize that the sequence 1, 2, 3 is not actually simple? It's not merely incrementing an object. In reality, altering numbers (even by a simple increment of one) produces drastic changes sometimes. The computer code, however, makes it seem quite dull. It's merely...advanced bean-counting. It's not surprising that the programming world has virtually merged with the business and financial worlds, and commits all the same sins.
I—well, the Pnictogen Wing—have daydreamed about attempting to formalize astrology and even alchemy in some way, establishing a more solid bridge between these disreputable domains and the world of empirical science. Perhaps structure can be applied to numerology as well.
~Chara of Pnictogen