I interpret the "attenuation" mentioned in the poem as a reference to all the kinds of idealization/abstraction that we engage in as we collectively turn real-world experiences into mathematical concepts. I'll probably come up with a better example, but consider a circle drawn in the sand (which incidentally might have been created by a combination of human actions and natural process: see http://www.maa.org/community/columns/maa-found-math/maa-found-math-gallery-2...). We first replace it in our heads by a perfect circle, as the Greeks did, sitting in an ideal Euclidean plane. Then we coordinatize that plane, as Descartes and his contemporaries did, replacing it by a set of ordered pairs (x,y) where x and y are real numbers satisfying x^2+y^2=1: an algebraic variety over the reals. Then (and here I'm on shaky ground; it's one of many things I'll need to understand better before I'm qualified to explain it) we follow the modern algebraic geometers and get rid of the numbers, or at best move them to the sidelines; instead, we have a scheme for associating varieties with number fields in general. At each stage of this process of attenuation, something is lost and something is gained. The losses are obvious; the gains are less so. (In fact, I don't really know what schemes are "good for". What insights made available by Grothendieck et al. were so useful for doing old-style algebraic geometry that they made converts among hard-headed, practical mathematicians who didn't value abstraction and generality for their own sake?) Someone should explain what those gains are, to the extent that one can do this for a general audience. I don't know if I'll tell the "circle story" or a different one. But that's the use of "thrice-attenuated shades" that I intend to make at some point in the future. Jim On Wednesday, June 17, 2015, Hilarie Orman <ho@alum.mit.edu> wrote:
"Thrice attenuated shades of things" may mean that mathematics is written as lines (one dimension), but "things" are three dimensions. "Shades" is just poetic.
Hilarie
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