Charles Matthews's question summarizing the Erlangen Program,
Current Validity for Erlangen…? (March 28, 2011)
has been removed from mathoverflow.net.
A cached copy is available at Log24.com. Enjoy.
Charles Matthews's question summarizing the Erlangen Program,
Current Validity for Erlangen…? (March 28, 2011)
has been removed from mathoverflow.net.
A cached copy is available at Log24.com. Enjoy.
The previous post suggests an example of
extreme aesthetic distance.
The word "mosaic" in Max Black —
The same word in a very different author —
Related historical remarks, for the Church of Synchronology —
The above death reportedly occurred on Sunday, Sept. 13, 2015.
This journal at 11 AM on that date —
Some background —
Item from the New Orleans TimesPicayune
in January 2013:
Chinese new year celebrated Welcoming the Chinese New Year, 4710, the Academy of Chinese Studies will hold a celebration Feb. 6 from 2 to 4 p.m. at Dixon Hall, at Tulane University. A student talent show, and lucky "Red Envelope" will be featured. 
See also this journal on the reported* date
of the above celebration, Feb. 6, 2013:
Wednesday, February 6, 2013
A Bus Named Desire

* The reported celebration date was later changed to Feb. 3 , 2013.
For a NewOrleansrelated Log24 post from that date, plus backstory,
see posts now tagged Trophy.
Mathematics:
A review of posts from earlier this month —
Wednesday, September 4, 2013

Narrative:
Aooo.
Happy birthday to Stephen King.
For Kiefer Sutherland, Hasty Pudding Man of the Year, 2013
Act I:
Current Validity for Erlangen…?
… MathOverflow question dated March 28, 2011
Act II:
… Starring Elke Sommer, former Erlangen student
Act III:
… See also Bus 318 and 3/18 in 2012.
Act IV:
… Log24 post dated March 28, 2011
From the previous entry:
“If it’s a seamless whole you want,
pray to Apollo, who sets the limits
within which such a work can exist.”
— Margaret Atwood,
author of Cat’s Eye
Happy birthday
to the late
Eugene Wigner
… and a belated
Merry Christmas
to Paul Newman:
— Eugene Wigner, Nobel Prize Lecture, December 12, 1963
“The secret to life, and
to love, is getting started,
keeping going, and then
getting started again.”
Nobel Laureate
Seamus Heaney
at Sanders Theatre,
Harvard College,
September 30, 2008
On Elke Sommer:
“…Young Elke… studied
in the prestigious
Gymnasium School
in Erlangen….”
Erlangen Prize Lecture:
Variations on a Theme of
Plato, Goethe, and Klein
(Background:
Christmas Knot, Sept. 26,
and Hard Core, July 1718.)
Today is the 21st birthday of my note “The Relativity Problem in Finite Geometry.”
Some relevant quotations:
“This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.”
— Hermann Weyl, The Classical Groups, Princeton University Press, 1946, p. 16
Describing the branch of mathematics known as Galois theory, Weyl says that it
“… is nothing else but the relativity theory for the set Sigma, a set which, by its discrete and finite character, is conceptually so much simpler than the infinite set of points in space or spacetime dealt with by ordinary relativity theory.”
— Weyl, Symmetry, Princeton University Press, 1952, p. 138
Weyl’s set Sigma is a finite set of complex numbers. Some other sets with “discrete and finite character” are those of 4, 8, 16, or 64 points, arranged in squares and cubes. For illustrations, see Finite Geometry of the Square and Cube. What Weyl calls “the relativity problem” for these sets involves fixing “objectively” a class of equivalent coordinatizations. For what Weyl’s “objectively” means, see the article “Symmetry and Symmetry Breaking,” by Katherine Brading and Elena Castellani, in the Stanford Encyclopedia of Philosophy:
“The old and natural idea that what is objective should not depend upon the particular perspective under which it is taken into consideration is thus reformulated in the following grouptheoretical terms: what is objective is what is invariant with respect to the transformation group of reference frames, or, quoting Hermann Weyl (1952, p. 132), ‘objectivity means invariance with respect to the group of automorphisms [of spacetime].^{‘[22]}
22. The significance of the notion of invariance and its grouptheoretic treatment for the issue of objectivity is explored in Born (1953), for example. For more recent discussions see Kosso (2003) and Earman (2002, Sections 6 and 7).
References:
Born, M., 1953, “Physical Reality,” Philosophical Quarterly, 3, 139149. Reprinted in E. Castellani (ed.), Interpreting Bodies: Classical and Quantum Objects in Modern Physics, Princeton, NJ: Princeton University Press, 1998, pp. 155167.
Earman, J., 2002, “Laws, Symmetry, and Symmetry Breaking; Invariance, Conservation Principles, and Objectivity,’ PSA 2002, Proceedings of the Biennial Meeting of the Philosophy of Science Association 2002, forthcoming [Abstract/Preprint available online]
Kosso, P., 2003, “Symmetry, objectivity, and design,” in K. Brading and E. Castellani (eds.), Symmetries in Physics: Philosophical Reflections, Cambridge: Cambridge University Press, pp. 410421.
Weyl, H., 1952, Symmetry, Princeton, NJ: Princeton University Press.
See also
Archives Henri Poincaré (research unit UMR 7117, at Université Nancy 2, of the CNRS)–
“Minkowski, Mathematicians, and the Mathematical Theory of Relativity,” by Scott Walter, in The Expanding Worlds of General Relativity (Einstein Studies, volume 7), H. Goenner, J. Renn, J. Ritter and T. Sauer, editors, Boston/Basel: Birkhäuser, 1999, pp. 4586–
“Developing his ideas before Göttingen mathematicians in April 1909, Klein pointed out that the new theory based on the Lorentz group (which he preferred to call ‘Invariantentheorie’) could have come from pure mathematics (1910: 19). He felt that the new theory was anticipated by the ideas on geometry and groups that he had introduced in 1872, otherwise known as the Erlangen program (see Gray 1989: 229).”
References:
Gray, Jeremy J. (1989). Ideas of Space. 2d ed. Oxford: Oxford University Press.
Klein, Felix. (1910). “Über die geometrischen Grundlagen der Lorentzgruppe.” Jahresbericht der deutschen MathematikerVereinigung 19: 281300. [Reprinted: Physikalische Zeitschrift 12 (1911): 1727].
Related material: A pathetically garbled version of the above concepts was published in 2001 by Harvard University Press. See Invariances: The Structure of the Objective World, by Robert Nozick.
From the Erlangen Program
to Category Theory
See the following, apparently all by JeanPierre Marquis, Département de Philosophie, Université de Montréal:
See also the following by Marquis:
Symmetry, Invariance, and Objectivity
The book Invariances: The Structure of the Objective World, by Harvard philosopher Robert Nozick, was reviewed in the New York Review of Books issue dated June 27, 2002.
On page 76 of this book, published by Harvard University Press in 2001, Nozick writes:
"An objective fact is invariant under various transformations. It is this invariance that constitutes something as an objective truth…."
Compare this with Hermann Weyl's definition in his classic Symmetry (Princeton University Press, 1952, page 132):
"Objectivity means invariance with respect to the group of automorphisms."
It has finally been pointed out in the Review, by a professor at Göttingen, that Nozick's book should have included Weyl's definition.
I pointed this out on June 10, 2002.
For a survey of material on this topic, see this Google search on "nozick invariances weyl" (without the quotes).
Nozick's omitting Weyl's definition amounts to blatant plagiarism of an idea.
Of course, including Weyl's definition would have required Nozick to discuss seriously the concept of groups of automorphisms. Such a discussion would not have been compatible with the current level of philosophical discussion at Harvard, which apparently seldom rises above the level of cocktailparty chatter.
A similarly low level of discourse is found in the essay "Geometrical Creatures," by Jim Holt, also in the issue of the New York Review of Books dated December 19, 2002. Holt at least writes well, and includes (if only in parentheses) a remark that is highly relevant to the Nozickvs.Weyl discussion of invariance elsewhere in the Review:
"All the geometries ever imagined turn out to be variations on a single theme: how certain properties of a space remain unchanged when its points get rearranged." (p. 69)
This is perhaps suitable for intelligent but ignorant adolescents; even they, however, should be given some historical background. Holt is talking here about the Erlangen program of Felix Christian Klein, and should say so. For a more sophisticated and nuanced discussion, see this web page on Klein's Erlangen Program, apparently by JeanPierre Marquis, Département de Philosophie, Université de Montréal. For more by Marquis, see my later entry for today, "From the Erlangen Program to Category Theory."
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