Something Must Be Going On

The heart of OCD is a feeling of "not being right" or repeating a ritual until it "feels right." A creative mathematician experiences intuition as a feeling there is "something going on here but I don't know what it is" according to William Byers in How Mathematicians Think. You were probably taught in high school that mathematics is a rigorous and logical endeavor and that for every mathematical principle there is a proof. It was implied to you that mathematicians seek out new principles by following threads of logic from an existing proof to a new proof. You were taught a myth. Most mathematical breakthroughs began with an intuition. Only later, after the instruction was explored well enough to believe it was true, to believe it was worth proving, perhaps even after it was proved to the satisfaction of the mathematician was an "official" proof created for the record. Proof comes after the fact, not before it. An interesting relationship between obsessive compulsive disorder and mathematics.

Moreover, instruction plays a vital role in creative mathematics. Just as in other creative arts, a shift of frame is required to turn the ordinary into the novel. The author relates the story of how he along with fellow mathematician John McKay noticed something curious about a single number. If you express adding one to 196884 as an equation you get 196884 = 196884 + 1. On the surface, it hardly seems worth the interest of a mathematician. You can add one to any integer on to infinity, something obvious to even non-mathematicians. What is so fascinatingly curious about this instance? As Byers writes, "...these are not just any two numbers. They are significant mathematical constants that are found in two different areas of mathematics." The relationship of the constants could not be a coincidence, thought McKay, who began a line of inquiry leading to a series of conjectures, which went under the fanciful but telling name of "monstrous moonshine." I want to linger a moment on this point. Here we have a mathematician who sees something curious, which prompts a "gut feeling" something systematic must be going on, a suggestion there may be a relationship between two systems of mathematics, who starts inquiring into the possibility, and as he finds more support for the reality of the intuition, he begins to make conjectures about how the two systems might be connected through the curiosity he discovered. At this point, we can hardly blame a mathematician for feeling he was chasing "moonshine." But that is exactly what creative people do. They chase moonshine and rainbows. Yet, somehow they end up driving the process of scientific rational, mathematical and artistic discovery. McKay's conjectures were later proved.

Byers does relate mathematical creativity to artistic creativity, observing good mathematicians (the creative ones) are very sensitive to the feeling of something going on, and ties mathematical intuition to the poet's, quoting the poet Denise Levertov saying "You can smell a poem before you see it."

This is all a blow to anyone raised on the rhetoric of rationalism. The human mind is a reasoning machine. Human beings are rational actors seeking the most efficient path. This ought to be nonsense to any carnival barker or snake oil salesman, but for most educated people it is a conceit they sustain because they enjoy the belief they are rational. Reason has become a virtue and virtues cannot be questioned.

At the bottom of human irrationality may be rational decisions, observations, the machinery of the mind is not metaphysical, but the abstract layers above the fine grain of deterministic reasoning are irrational. The mind is connected to a body. People get "gut feelings" as their mind tries to tell itself something from its emotional, pattern recognizing centers. How else could the pattern recognizing centers of the brain communicate with this supremely rational being, other than by kicking it in the gut?

I take away from this you will not be a creative scientist, mathematician or musician unless you learn to use your intuition. Exercise your curiosity. Keep a childlike sense of astonishment about the world around you or the inner worlds you explore. Experiment. Follow instruction. Don't worry about the result, the path to a Nobel prize in mathematics is not by seeking that which is likely to win a prize, but by following up an intuition, seeing where the thread will lead, without any thought to where it will go, other than to satisfy curiosity and that feeling of something must be going on.