There are numbers and then there are Numbers...

This post by James Poulos and the ongoing tragicomedy of my Galois Theory class have reminded me that different people interact with numbers in very different ways. Without further ado (and with apologies to Sigmund Freud), I present the Stages of Mathematical Development:

Stage 1: Algorithmic Understanding and Rote Memorization

As the article Poulos links to makes clear, multiplication is an unnatural operation for human beings. The same goes for long division (which I never learned -- yes, you can be a math major and not know long division!), factorization, and other common operations.

In the first stage of development, students learn to apply memorized algorithms to transform symbols into "solutions" with no knowledge of the structure of the underlying operations. The failure of New Math was its inability to understand the fact that the apparently mindless repitition of algorithmic process led to a familiarization of the brain with the underlying structure of all algorithms. Human beings excel at analogies, pattern matching, and the like. It's astonishing to watch the speed with which 4-5 yr. old children abstract their knowledge away from the algorithmic level to the meta-algorithmic level -- but this cannot happen without a lot of sweat.

I lump most high-school and university-level calculus courses under this stage, as few of them make the crucial leap beyond formalistic approaches to limits, differentiation, etc. Also, many simple theorem-proving techniques rely on an approach similar to the "exhaust all possibilities" strategy used by Chess-playing computers. They fall under this stage as well.

Stage 2: Geometric Intuition

We all recognized the kids in grade school who were "good at math" -- they grasped the concepts with a facility which we all envied, and performed the meta-algorithmic leap early and often. I would contend that these kids possessed the rudiments of geometric intuition. A classic example of this kind of child was Benoit Mandelbrot:

"Benoit was never taught the alphabet and never learned multiplication tables past fives. Even today he claims not to know the alphabet, so that it is difficult for him to use a telephone book.... He had a visual mind, a geometric mind, in a school setting where this was discouraged. He solved problems with great leaps of geometric intuition, rather than the "proper" established techniques of strict logical analysis. For instance, in the crucial entrance exams he could not do algebra very well, but still managed to receive the highest grade by, as he puts it, translating the questions mentally into pictures." ~ Fractal Wisdom

Mandelbrot's story is an extreme case of a fairly common phenomenon -- that of the student who sees past the formalisms into the underlying mathematical structures of an operation. However the intuition is weak, as it remains irrevocably tied to a geometric conception of space and number. Thus, only mathematical structures amenable to geometric interpretation (in the extremely limited sense that our brains can perform such interpretation -- intuitive understandings of higher-dimensional spaces don't count as geometric) can be grokked intuitively, and for the rest the student regresses to Stage 1. This is a painful experience, and the visually-inclined student of mathematics is often extremely frustrated when he can only understand a proof in a formalistic or technical sense. The underlying structures, once familiar friends, now seem like mere shadows of something much greater and more profound.

Stage 3: Abstract Intuition

At some point, every visually-inclined student of mathematics hits a wall. It happened to me at the end of my freshman year in college (I had to integrate a function over a 5-dimensional hypertorus in my Vector Analysis final, and nearly passed out from the visual strain), and it's almost an impossible feeling to describe if you haven't experienced it yourself.

The bottom of your stomach drops out as you realize that mathematics is deeper and more sublime than the constrained, geometric mockery of mathematics you have been doing up to this point. There is a moment of existential horror; an understanding that the "gift" of visualization you possessed was actually a curse, that it kept your understanding of conceptual structures chained down and crippled for so long. At this point there is a choice...

The only way to develop abstract intuition that encompasses and goes beyond what you have is to tear down the geometric intuition that you worked so hard for throughout your educational life. The devoted student must return to the formalistic methods of Stage 1 and apply them over and over until true conceptual insight breaks through and floods the brain like dawn breaking over the arctic after 6 months of darkness. The process is something akin to meditation, and beyond the first epiphany there are countless higher epiphanies. Each layer of abstraction and conceptual depth must be absorbed by the brain in its entirety, and the absorption requires a return to Stage 1 and an abandonment of the hard-won inuition that has served the student to this point.

There are a very few individuals who possess a natural mathematical intuition that surpasses the visual and rests on a deep understanding of mathematical structures. These lucky few are geniuses and savants -- but they are rare on the ground. I have been privileged to know 2 such individuals, and both stunned me by their humility and respect for the subject. As our intuition develops, greater and greater vistas of uncharted land are revealed, even while the land which seemed commonplace before now appears awe-inspiring and sublime.

Stage 4: Motivation and Artistic Sense

As G.H. Hardy reminds us, mathematics is at its core a creative art. The full weight of that statement becomes clear when students are asked to state, motivate, and then prove their own theorems (this is the second great drop-out point in math education -- the first being the transition from Stage 2 to Stage 3). The student with abstract intuition excels at solving problems and proving theorems when asked to prove them, but balks when asked to come up with a suggestive hypothesis himself. Knowing what is worth proving and why it is worth proving (the answer is aesthetic) is the crux of Stage 4, and success here requires a deep familiarity with the artistic nature of mathematics qua mathematics.

Conclusion: Mathematics is the union of art and theology.

Conclusion 2: Every biography ever written of a mathematician is a tragedy.

EDIT: Some people have asked me where meta-mathematics goes in this framework. The answer is that it is not a separate stage, but can be approached at every stage. This is unsurprising since Godel proved that meta-mathematics and mathematics are the same thing.