In a recent post on math and spin class I touched on the question of "where do ideas (mathematical or otherwise) come from?" or more precisely, "where do my ideas come from?" and mused about connections for me between moving my body and moving my mind. As it turns out, movement is central to what is perhaps the most famous story of a mathematical lightning strike (our version of Newton and the apple): Henri Poincaré's "Bus Story". As the story goes, Poincaré (progenitor of chaos theory) has been puzzling over a deep mathematical problem. It leaves his mind for a little while as he makes plans to embark on geological expedition to some local mines. As Poincaré steps onto the bus, he strikes gold! (sorry!) The solution of the problem is revealed!
"At the moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it..."
Poincaré was one of the great science popularizers of his day and believed that as a leading scientist he had something of a responsibility to explicate the world of mathematics and science to the public. He was especially well known for his essays on special relativity and the math of space-time, perhaps so well told and well known that it may have influenced the rise of cubism in the visual arts. Poincaré takes up in general the topic of mathematical creativity in his essay "Mathematical Creation," Chapter III of his great expository work Science and Method (1908). His story of the bus is just a part of a thoughtful reflection on the role of the unconscious in the creative process.
My favorite description of mathematical epiphany is due to Winston Churchill of all people. In his memoir/auto-biography "My Early Life: 1874-1904" we find him saying, "I had a feeling once about Mathematics, that I saw it all--Depth beyond depth was revealed to me--the Byss and the Abyss. I saw, as one might see the transit of Venus--or even the Lord Mayor's Show, a quantity passing through infinity and changing its sign from plus to minus. I saw exactly how it happened and why the tergiversation was inevitable: and how the one step involved all the others. It was like politics. But it was after dinner and I let it go!" (I also like it because it brought to my attention the word "tergiversation"!)
I first learned of this quote while in graduate school, a time when I thought a lot about mathematical creativity, mainly whether or not I might ever display any or at least enough to get a degree. Churchill describes his feeling of understanding as something of a dream, indeed given all descriptions of his eating and drinking habits, probably experienced in a post-dinner haze, almost surely with whiskey in hand, presumably following a meal that included a good amount of wine. The result is an evanescent epiphany, gone almost as quickly as it was "seen" and too quickly to be grasped.
Churchill's experience resonated with some of my own in graduate school, a time at which I was so stuck into a problem that I would dream about mathematics. Most memorable were those that almost bordered on hallucination, generally taking place in that odd liminal state just before or just after sleep. I can recall quite strongly the experience of seeing a blackboard that looked full of notation (numbers, Greek letters, diagrams) that I was sure showed the way around my most recently encountered obstacle. I would will my dreamed self to approach to the board only to discover that what looked like math was actually incomprehensible, even though I was sure that at some point it really had the answer! Poincaré remarks on this particular state as well. For a while I took to going to sleep with a pad by my bed so that I might force myself into wakefulness to write down the solution that I "saw". The experiences were so tantalizing that for a time I would go to sleep I would trying to do all I could to enter into these states.
This kind of hallucinatory visual phenomenon has a name: "hypnogagia" which I learned while recently reading Hallucinations, by the recently departed and already missed Oliver Sacks. Sacks tells stories of all kinds of hallucinations. The ubiquity of geometric patterns is particularly striking, which some have suggested are linked to neural structure. I've always felt that my geometer friends were able to access a "visual" world completely different from mine. In short they saw stuff that I couldn't see. Does the neuroscience of hallucination have something to say about it?
As a mathematician who is algebraically inclined, it is fitting that my math hallucinations (now few and far between) fall into the category of "text hallucinations," a class that includes regular old text and music. In those cases too, at a distance the symbols seem full of information, but up close are unreadable or unplayable.
My hypnogagic efforts never actually succeeded in doing much other than ruining a good night's sleep, but the intention may not have been entirely off the mark. As it turns out text hallucinations are marked by hyperactivity in the word form area - "anarchical" is how Sacks describes the activity of creation of "text" ungoverned by the rules of syntax and grammar, be it in the language of math, music, or regular old common discourse; a creation of free-floating "proto-letters" that spill forth in a free-flowing stream of familiar, but ultimately uninterpretable, unplayable, or unprovable glyphs.
These are useful freedoms - although not necessary ones - when engaged in research, like when you are the early stages of putting together a puzzle, with the pieces spread out on the floor and you move them around hoping to find locks for the various keys that accrete to a finished piece of work. You may absentmindedly (OK, or perhaps doggedly) push the pieces around using the hints but also just letting connections occur as they will. This shares a lot with the early stages of problem solving (or my problem solving).
Of course you actually need to have the puzzle pieces as well as some sense of where they might go. That's the hard conscious work that goes into having an idea. There's inspiration, but generally only coupled with perspiration - and not just the kind that you make while charging away on the exercise bike.