Archive for September, 2011

Wilderness Mathematics

September 23, 2011 Leave a comment

A few years ago, while I was reading Boole, Frege, Russell, and Hilbert, I would have cited perhaps the most common reason mathematicians cornered by interviewers cite for their initial interest in mathematics: its problems are simpler, its foundations more clear and secure, and its solutions–if difficult to achieve–are easier than in the mess of the “real world” and our human concerns. I still enjoy that feeling of clarity and simplicity when it comes, but lately I’ve strayed into areas where math is seldom used and I find myself wanting to be one of the mathematician colonists.

Most straightforwardly, my use of “math in the wild” alludes to one of my primary focuses, using math, physics and statistics to study evolution and life on the organismic level and above. But I am also concerned with how math is actually done “in the wild”: the neuro/psychological processes we use in generating and learning mathematics and perceiving the objects we study, and the historical evidence that sheds light on how the concepts we find cleaned up in textbooks came into being–usually after much heated debate and often accompanied by Kuhnian paradigm shifts. I strongly believe that bringing a deeper understanding of the historical contingency and cognitive basis of our mathematics and science knowledge into the teaching of mathematics enriches the experience for teachers and students and makes the learning process more effective. Besides that, though, I’m thrilled when I’m looking at a primary source and realize that the received story of how mathematics has gone ignores the messier, more revealing story: for example, Descartes, for all that he is credited for the Cartesian coordinate system, still had a geometric idea of how to locate points in the plane that didn’t at all involve ordered pairs of numbers. This history is not only a curiosity: it can reveal patterns of thought and how they change, thereby giving us insight into the mind at work.

Lastly, I want to figure out how to do math in the wilderness of fields that haven’t traditionally been looked at mathematically, and how those fields can re-teach us how we do our science. Vygotsky, the developmental psychologist, is famous for his “naturalistic experiments” where the setup was open-ended and the goal was to elicit detailed descriptions of individual thought patterns. It’s much harder to get appropriately “scientific” results when you’re working outside the laboratory and haven’t controlled for everything. But very little of what we experience in life can be faithfully recreated in laboratories: if we are really going to study the neuroscience of storytelling, patterns in literature, our use of metaphor, or what happens in the brain as we listen to music, we need to be able to deal with messy, qualitative data sets, to find a mathematics of the psychiatric case study and the naturalist’s description. Conversely, works of fiction and the wealth of treatises on aesthetics, epistemology, and meaning in various contexts can be, with a lot of work, sources of unconsidered hypotheses as well as rich data sets: imaginative works of all stripes teach us how to consider alternative possibilities and bring some rigor to open-ended studies.

I hope this blog will eventually include pieces involving the history and philosophy of science, as well as expositions of some of my favorite problems from other fields that people have treated with mathematics, however simple. There will be a lot of biology and psychology, and thus a lot of clarification: I find that many mathematicians are bothered by lack of rigor in definitions when they encounter other fields of science, and in music psychology and evolutionary theory there are still heated debates over issues that would seem basic. There will also be book synopses–a personal memory aid–and, on separate pages, writings about specific works of literature and music, my inspiration for studying these fields rigorously. I might digress into beautiful bits of mathematics that are part of the pure mathematics canon, but I’ll try to keep any extensive notes in that vein as papers on other pages of this site. Mainly, I hope this will provide glimpses of scientists as they go from the earliest stages of speculation to definitions and testable hypotheses in their fields, in the spirit of the great naturalist E.O. Wilson:

“The discipline of the dark envelope summoned fresh images from the forest of how real organisms look and act. I needed to concentrate for only a second and they came alive as eidetic images, behind closed eyelids, mving across fallen leaves and decaying humus. I sorted the memories this way and that in hope of stumbling on some pattern not obedient to abstract theory of textbooks. I would have been happy with any pattern. The best of science doesn’t consist of mathematical models and experiments, as textbooks make it seem. Those come later. It springs fresh from a more primitive mode of thought, wherein the hunter’s mind weaves ideas from old facts and fresh metaphors and the scrambled crazy patterns of thought, which in turn dictate the design of the models and experiments.” (The Diversity of Life, p.5)

There’s no time like the present to begin.

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