An expository article by the the algebraic geometer Yuri Manin always has something interesting in it, and his latest, entitled The notion of dimension in geometry and algebra is no exception.
In this article Manin discusses various ideas related to the notion of dimension, ranging over fractal geometry, non-commutative geometry and theoretical physics. He begins with a quote from Glenn Gould, which is quite amusing, but of obscure relation to the notion of dimension. Then he goes on to some history, from Euclid to Leibniz, finally veering off into a fascinating discussion of the relation of algebra and geometry, and ending with the sociological comment that visual mass media is leading to a dominance of right-brain mental faculties, and thus “projects us directly into dangerously archaic states of collective consciousness.”
The body of the article includes comments on Hausdorff dimension, dimensional regularization of path integrals, the theory of operator algebras, non-commutative geometry, a weird digression on databases, and supergeometry. He also discusses “Spec Z” (the “space” naturally associated to Z, the ring of integers) making various comments about it and giving arguments for its dimension being 1, 3 and infinity. Next there are some comments on modular forms, and finally a section on fractional dimensions in homological algebra.
Its not clear how seriously one should take all of this, but Manin’s article is definitely thought-provoking.
Mirror Symmetry of Yuri Manin is reminiscent of Occam’s razor principle.”Entities should not be multiplied unnecessarily.”
Peter, don’t be so literal!
You write that Manin “begins with a quote from Glenn Gould, which is quite amusing, but of obscure relation to the notion of dimension.”
Your own quick summary of Manin’s choice of topics shows that there’s a whole lot of dimension-discussing (and other-stuff-discussing) going on in the article.
Manin decided to begin his wandering journey by making a little self-deprecating joke.
I once observed that many interesting fractals living in 2D have Haussdorf dimension of the form D = (100 – n^2)/48, for n integer between 2 and 10. Apart from obvious D = 2 and D = 0, this series includes the percolation cluster (D = 91/48), the percolation hull (D = 7/4), linear polymers (D = 4/3), and red links, which cut the percolation cluster in two (D = 3/4). The formula comes from conformal field theory; D = 2 – 2h, where h is in the discrete series in the formal c -> 0 limit.
This was very much in the air in 1986, and I later learned that Hubert Saleur made the same observation a couple of months before me, but at least my letter was submitted when his appeared in print. And this observation did earn me a four-year postdoc, so it was important to me.
Glenn Gould was notoriously hard on himself. Beethoven would slam the piano lid down when someone expected him to play. The idea of “disgust” is built into some of Beethoven’s late work – he said about his early work “why does it make such a bad impression on me? From this day forward I mean to strike out in a new direction.” Gould may have been similar. Actually, he must surely have understood that his fans didn’t care how he played.
His first and last recordings were of the Goldberg variations. He was a hypochondriac who died of a stroke.