Math on the Brain

Many people know that before I succumbed to the unremunerative way of all things literary, I was a young mathematician. I like to say that I have an affinity for math, literature, and philosophy because they're all nonempirical, pure works of the collective (and sometimes private) human imagination. Here are a couple of math-and-brain-related entries worth noting:

Jim Holt, "The Numbers Guy":

By “pure luck,” Dehaene recalls, Mehler happened to be doing research on how numbers are understood. This led to Dehaene’s first encounter with what he came to characterize as “the number sense.” Dehaene’s work centered on an apparently simple question: How do we know whether numbers are bigger or smaller than one another? If you are asked to choose which of a pair of Arabic numerals—4 and 7, say—stands for the bigger number, you respond “seven” in a split second, and one might think that any two digits could be compared in the same very brief period of time. Yet in Dehaene’s experiments, while subjects answered quickly and accurately when the digits were far apart, like 2 and 9, they slowed down when the digits were closer together, like 5 and 6. Performance also got worse as the digits grew larger: 2 and 3 were much easier to compare than 7 and 8. When Dehaene tested some of the best mathematics students at the École Normale, the students were amazed to find themselves slowing down and making errors when asked whether 8 or 9 was the larger number.

Dehaene conjectured that, when we see numerals or hear number words, our brains automatically map them onto a number line that grows increasingly fuzzy above 3 or 4. He found that no amount of training can change this. “It is a basic structural property of how our brains represent number, not just a lack of facility,” he told me.

The Reactionary Epicurean, "There are numbers and then there are Numbers...", on the stages of mathematical development:

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.