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| Mathematics and Narrative, continued:Knight Moves:
The Relativity Theory
of Kindergarten Blocks
(Continued from January 16, 2008)
Something:
From Friedrich Froebel,
who invented kindergarten:
Click on image for details.
An Unusually
Complicated Theory:
From Christmas 2005:
Click on image for details.
For the eightfold cube
as it relates to Klein's
simple group, see
"A Reflection Group
of Order 168."
For an even more
complicated theory of
Klein's simple group, see
Click on image for details.
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| Annals of Mathematics:
Bertram Kostant, Professor Emeritus of Mathematics at MIT, on an object discussed in this week's New Yorker:
" A word about E(8). In my opinion, and shared by others, E(8) is the most magnificent 'object' in all of mathematics. It is like a diamond with thousands of facets. Each facet offering a different view of its unbelievably intricate internal structure."
Hermann Weyl on the hard core of objectivity:
"Perhaps the philosophically most
relevant feature of modern science is the emergence of abstract
symbolic structures as the hard core of objectivity behind-- as
Eddington puts it-- the colorful tale of the subjective storyteller
mind." (Philosophy of Mathematics and Natural Science, Princeton, 1949, p. 237)
Steven H. Cullinane on the symmetries of a 4x4 array of points:
A Structure-Endowed Entity
"A guiding principle in modern mathematics is this lesson: Whenever you
have to do with a structure-endowed entity S, try to determine its group of
automorphisms,
the group of those element-wise transformations which leave all
structural relations undisturbed. You can expect to gain a deep
insight into the constitution of S in this way."
-- Hermann Weyl in Symmetry
Let us apply Weyl's lesson to the following "structure-endowed entity."
What is the order of the resulting group of automorphisms?
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The above group of
automorphisms plays
a role in what Weyl,
following Eddington,
called a "colorful tale"--
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| Mathematics and Narrative, continued:Hard CoreDavid Corfield quotes Weyl in a weblog entry, " Hierarchy and Emergence," at the n-Category Cafe this morning: "Perhaps the philosophically most relevant feature of modern science is the emergence of abstract symbolic structures as the hard core of objectivity behind-- as Eddington puts it-- the colorful tale of the subjective storyteller mind." (Philosophy of Mathematics and Natural Science [Princeton, 1949], p. 237)
For the same quotation in a combinatorial context, see the foreword by A. W. Tucker, "Combinatorial Problems," to a special issue of the IBM Journal of Research and Development, November 1960 ( 1-page pdf). See also yesterday's Log24 entry. | | |
| Annals of Philosophy:
CHANGE
FEW CAN BELIEVE IN |
Continued from June 18. Jungian Symbols
of the Self -- Compare and contrast: Jung's four-diamond figure from Aion
-- a symbol of the self --
Jung's Map of
the Soul, by Murray Stein: "... Jung thinks of the self as
undergoing continual transformation during the course of a lifetime.... At the
end of his late work Aion, Jung presents a diagram to illustrate the
dynamic movements of the self...." | | |
| Elliptic comment:My comment on a discussion of elliptic curves and modular forms at Secret Blogging Seminar, about 10 PM tonight:
How does this affect popularized discussions of the Taniyama-Shimura
conjecture-- for instance, Ivars Peterson's, in "Curving Beyond Fermat,"
November 1999-- which claim, for instance, that "Elliptic curves and
modular forms are mathematically so different that mathematicians
initially [in the 1950's, the early days of the conjecture] couldn't
believe that the two are related."?
Update of about 10:45 PM tonight:
A reply by the author of the discussion, Scott Carnahan:
I don’t think anyone doubted that there is a connection between
elliptic curves and modular forms on the level I described above.
However, the Taniyama-Shimura conjecture refers to a more advanced idea
about a deeper connection. Carnahan then gives a one-paragraph summary, definitely not popularized, of the deeper connection.
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