October 24, 2006
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Two-part invention:
Another illustration
of the previous entry‘s concept of
a “critical mass” of weblog entries,
a concept reflected in
the saying
“You can’t win the lottery
if you don’t buy a ticket.”Mathematics and Narrative:
A Two-Part Invention
Here are today’s
numbers from the
Keystone State:
Here is an interpretation
of those numbers:8/21 — Mathematics:“The
geometrization conjecture, also known as Thurston’s geometrization
conjecture, concerns the geometric structure of compact 3-manifolds. The geometrization conjecture can be considered an analogue for 3-manifolds of the uniformization theorem for surfaces. It
was proposed by William Thurston in the late 1970s. It ‘includes’ other
conjectures, such as the Poincaré conjecture and the Thurston
elliptization conjecture.”The second sentence, in bold
type, was added on 8/21 by yours truly. No deep learning or
original thought was required to make this important improvement in the
article; the sentence was simply copied from the then-current version
of the article on Grigori Perelman (who has, it seems, proved the geometrization conjecture).This
may serve as an example of the “mathematics” part of the above phrase
“Mathematics and Narrative” — a phrase which served, with associated
links, as the Log24 entry for 8/21.7/23 — Narrative:
“Each
step in the story is a work of art, and the story as a whole is a
sequence of episodes of rare beauty, a drama built out of nothing but
numbers and imagination.” –Freeman DysonThis quotation appeared in the Log24 entry for 7/23,
“Dance of the Numbers.” What Dyson calls a “story” or “drama” is
in fact mathematics. (Dyson calls the “steps” in the story “works of
art,” so it is clear that Dyson (a former student of G. H. Hardy)
is discussing mathematical steps, not paragraphs in someone’s
account– perhaps a work of art, perhaps not– of mathematical
history.) I personally regard the rhetorical trick of calling the
steps leading to a mathematical result a “story” as contemptible
vulgarization, but Dyson, as someone whose work (pdf) led to the particular result he is discussing, is entitled to dramatize it as he pleases.For related material on mathematics, narrative, and vulgarization, click here.
The
art of interpretation (applied above to a lottery) is relevant to
narrative and perhaps also, in some sense, to the arts of mathematical
research and exposition (if not to mathematics itself). This art
is called hermeneutics.
For more on the subject, see the Stanford Encyclopedia of Philosophy article on Hans-Georg Gadamer, “the decisive figure in the development of twentieth-century hermeneutics.”“Foreword” in Gian-Carlo Rota,
Indiscrete Thoughts,
Boston: Birkhäuser Verlag,1996, xiii-xvii, and
“Gadamer’s Theory of Hermeneutics” in
The Philosophy of Hans-Georg Gadamer,
edited by Lewis E. Hahn,
The Library of Living Philosophers, Vol. 24,
Chicago: Open Court Publishers,
1997, 223-34.
