The Shape of Mathematical Ideas

Peter and I were talking this morning about the shape of ideas, particularly the shape of mathematical ideas. We share a despair that mathematical subjects -- even geometry and calculus -- are taught as if they are little more than a collection of formulas, as if mathematics is essentially mechanical, a functional device rather than an idea. This problem is compounded by a pervasive tendency in science to treat mathematicians as plumbers, mere technicians of thought. And this is a terribly pity not only because it cheapens mathematics but because it repels creative thinkers who would otherwise make excellent scientists.

Prior to university I was a good but unexceptional math student . Memorizing formulas did not appeal to me: I was preoccupied with questions of meaning and being. Sometimes in a math or science class I would have insights into the vast universe of ideas beyond the formulas and nomological categories we were asked to ingest and regurgitate. But my efforts to articulate those visions met with disapprobation or incomprehension. Whatever I was talking about, it wasn't considered math. And because I found make-work memory exercises tedious and repetitive, I completed them indifferently and received indifferent marks. After school, in private, I would walk through the woods and think about what I had just seen of the cosmos and the beautiful, frightening horizons I imagined stretching in all directions and longed to explore.

When I first encountered atomic physics, I was struck not by the images of protons and electrons we were asked to reproduce in primary school, but by the idea that matter was mostly hollow and held together by invisible forces. I began to think of my desk as an illusion and wondered if the solar system was really just part of part of a giant table leg. I became an existentialist at the age of eight.

When I first encountered Euclidean geometry, I thought of it as a wonderful forensic exercise. By that time I knew never to discuss my secret belief that the purpose of trigonometry was to bring one angle to justice for some dark transgression against another.

I was perplexed by differential calculus until I realised, after a few weeks, that it was analogous to a kind of mapping and was therefore inherently cartographic. Whenever I calculated differentials I thought of the clumsy necessity of fitting the curved shape of the earth to the flat text of the page, or imagined the shuttle's need, upon returning from orbit, to breach the atmosphere at precisely the right trajectory and speed. I tried to explain this to my high school calculus teacher, a very kind man who wrote little notes on my tests suggesting I apply more efforts to learning to the formulas. I absented myself shortly thereafter to take a course on surrealist sculpture offered to gifted high school students by the nearby university, and returned only to write the final exam, thinking that my weeks with the surrealists had been as instructive as more time in class would have been.

In university, encountering statistics in a rigorous way, I kept asking my tutor what standard deviation and variance really meant. Perplexed, he pointed to the formula and repeated a mantra about central tendency. But I could not help thinking of an average as a large bird and variance a measure of the spread of its wings, and I was most interested in the vast horizon beyond them. For me, statistics have always been interesting more for what they cannot capture or predict than for what they can, a view I have also written about here.

And so I became a phenomenologist rather than a scientist, although -- as I have written here and here -- I have never thought of these two domains as being especially distinct. And I married a philosopher who sees mathematics, like I do, as most useful when it is understood as a kind of metaphor for thought or an nomic utterance of a principle. And it is hardly surprising that neither of us aligns with any of the major schools of thought in the philosophy of mathematics, although I have always had a soft spot for Paul Erdos and Peter one for Bertrand Russell's idealism for eliminating idealism and Godel's realism that it is absurd to try.

And while I continue to hold empirical science in very high regard, I am still most interested in the things it cannot account for. The shift in the arc, the area outside the curve, the ineffable beauty of things that, like wind or light, can be glimpsed or felt in passing but never captured.

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Today's thought is brought to you by the number 35, which when divided by five does not necessarily produce seven, although you will get seven pieces whose average size multiplies into thirty-five.

[Book image by Sean Gwizdak.]