"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
See as well, in this journal, Deathly Hallows, Relativity Problem, and Space Cross.
A related quote: "This is not mathematics; this is theology."
A remark on coordinatization linked to by John Baez today —
This suggests a more historical perspective:
See as well a search for Interpenetration in this journal.
|
From some 1949 remarks of Weyl— "The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time." — Hermann Weyl, "Relativity Theory as a Stimulus in Mathematical Research," Proceedings of the American Philosophical Society , Vol. 93, No. 7, Theory of Relativity in Contemporary Science: Papers Read at the Celebration of the Seventieth Birthday of Professor Albert Einstein in Princeton, March 19, 1949 (Dec. 30, 1949), pp. 535-541 Weyl in 1946—: "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 |
For some context, see Relativity Problem in this journal.
In the case of PG(3,2), there is a choice of geometric models
to be coordinatized: two such models are the traditional
tetrahedral model long promoted by Burkard Polster, and
the square model of Steven H. Cullinane.
The above Wikipedia section tacitly (and unfairly) assumes that
the model being coordinatized is the tetrahedral model. For
coordinatization of the square model, see (for instance) the webpage
Finite Relativity.
For comparison of the two models, see a figure posted here on
May 21, 2014 —
Labeling the Tetrahedral Model (Click to enlarge) —
"Citation needed" —
The anonymous characters who often update the PG(3,2) Wikipedia article
probably would not consider my post of 2014, titled "The Tetrahedral
Model of PG(3,2)," a "reliable source."
“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
See also Relativity Problem and Diamonds and Whirls.
From the former date above —
|
Saturday, September 17, 2016 |
From the latter date above —
|
Tuesday, October 18, 2016
Parametrization
|
From March 2018 —
"But unlike many who left the Communist Party, I turned left
rather than right, and returned—or rather turned for the first time—
to a critical examination of Marx's work. I found—and still find—
that his analysis of capitalism, which for me is the heart of his work,
provides the best starting point, the best critical tools, with which—
suitably developed—to understand contemporary capitalism.
I remind you that this year is also the sesquicentennial of the
Communist Manifesto , a document that still haunts the capitalist world."
— From "Autobiographical Reflections," a talk given on June 5, 1998, by
John Stachel at the Max Planck Institute for the History of Science in Berlin
on the occasion of a workshop honoring his 70th birthday,
"Space-Time, Quantum Entanglement and Critical Epistemology."
From a passage by Stachel quoted in the previous post —
From the source for Stachel's remarks on Weyl and coordinatization —
Note that Stachel distorted Weyl's text by replacing Weyl's word
"symbols" with the word "quantities." —
This replacement makes no sense if the coordinates in question
are drawn from a Galois field — a field not of quantities , but rather
of algebraic symbols .
"You've got to pick up every stitch… Must be the season of the witch."
— Donovan song at the end of Nicole Kidman's "To Die For"
Or: Coordinatization for Physicists
This post was suggested by the link on the word "coordinatized"
in the previous post.
I regret that Weyl's term "coordinatization" perhaps has
too many syllables for the readers of recreational mathematics —
for example, of an article on 4×4 magic squares by Conway, Norton,
and Ryba to be published today by Princeton University Press.
Insight into the deeper properties of such squares unfortunately
requires both the ability to learn what a "Galois field" is and the
ability to comprehend seven-syllable words.
The term "parametrization," as discussed in Wikipedia,
seems useful for describing labelings that are not, at least
at first glance, of a vector-space nature.
Examples: The labelings of a 4×4 array by a blank space
plus the 15 two-subsets of a six-set (Hudson, 1905) or by a
blank plus the 5 elements and the 10 two-subsets of a five-set
(derived in 2014 from a 1906 page by Whitehead), or by
a blank plus the 15 line diagrams of the diamond theorem.
Thus "parametrization" is apparently more general than
the word "coodinatization" used by Hermann Weyl —
“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
Note, however, that Weyl's definition of "coordinatization"
is not limited to vector-space coordinates. He describes it
as simply a mapping to a set of reproducible symbols .
(But Weyl does imply that these symbols should, like vector-space
coordinates, admit a group of transformations among themselves
that can be used to describe transformations of the point-space
being coordinatized.)
Despite the blocking of Doodles on my Google Search
screen, some messages get through.
Today, for instance —
"Your idea just might change the world.
Enter Google Science Fair 2014"
Clicking the link yields a page with the following image—
Clearly there is a problem here analogous to
the square-triangle coordinatization problem,
but with the 4×6 rectangle of the R. T. Curtis
Miracle Octad Generator playing the role of
the square.
I once studied this 24-triangle-hexagon
coordinatization problem, but was unable to
obtain any results of interest. Perhaps
someone else will have better luck.
* For a rather different use of this word,
see Hermann Weyl in the Stanford
Encyclopedia of Philosophy.
A sequel to last night's post The 4×4 Relativity Problem —
In other words, how should the triangle corresponding to
the above square be coordinatized ?
See also a post of July 8, 2012 — "Not Quite Obvious."
Context — "Triangles Are Square," a webpage stemming
from an American Mathematical Monthly item published
in 1984.
The sixteen-dot square array in yesterday’s noon post suggests
the following remarks.
“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
The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—
The 1977 matrix Q is echoed in the following from 2002—
A different representation of Cullinane’s 1977 square model of the
16-point affine geometry over the two-element Galois field GF(2)
is supplied by Conway and Sloane in Sphere Packings, Lattices and Groups
(first published in 1988) :
Here a, b, c, d are basis vectors in the vector 4-space over GF(2).
(For a 1979 version of this vector space, see AMS Abstract 79T-A37.)
See also a 2011 publication of the Mathematical Association of America —
Today's previous post on coordinate systems
suggests a look at the phrase "Galois coordinates."
A search shows that the phrase, though natural,
has apparently not been used before 2011* for solutions
to what Hermann Weyl called "the relativity problem."
A thorough historical essay on Galois coordinatization
in this sense would require more academic resources
than I have available. It would likely describe a number
of applications of Galois-field coordinates to square
(and perhaps to cubical) arrays that were studied before
1976, the date of my Diamond Theory monograph.
But such a survey might not find any such pre-1976
coordinatization of a 4×4 array by the 16 elements
of the vector 4-space over the Galois field with two
elements, GF(2).
Such coordinatizations are important because of their
close relationship to the Mathieu group M 24 .
See a preprint by Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of Kummer
surfaces in the Mathieu group M 24 ," with its remark
denying knowledge of any such coordinatization
prior to a 1989 paper by R. T. Curtis.
Related material:
Some images related to Galois coordinates, excerpted
from a Google search today (click to enlarge)—
* A rather abstract 2011 paper that uses the phrase
"Galois coordinates" may have some implications
for the naive form of the relativity problem
related to square and cubical arrays.
(Continued from 1986)
|
S. H. Cullinane This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.
— H. Weyl, The Classical Groups , In finite geometry "points" are often defined as ordered n-tuples of a finite (i.e., Galois) field GF(q). What geometric structures ("frames of reference," in Weyl's terms) are coordinatized by such n-tuples? Weyl's use of "objectively" seems to mean that such structures should have certain objective— i.e., purely geometric— properties invariant under each S. This note suggests such a frame of reference for the affine 4-space over GF(2), and a class of 322,560 equivalent coordinatizations of the frame. The frame: A 4×4 array. The invariant structure: The following set of 15 partitions of the frame into two 8-sets.
A representative coordinatization:
0000 0001 0010 0011
The group: The group AGL(4,2) of 322,560 regular affine transformations of the ordered 4-tuples over GF(2). |
S. H. Cullinane This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.
— H. Weyl, The Classical Groups , In finite geometry "points" are often defined as ordered n-tuples of a finite (i.e., Galois) field GF(q). What geometric structures ("frames of reference," in Weyl's terms) are coordinatized by such n-tuples? Weyl's use of "objectively" seems to mean that such structures should have certain objective— i.e., purely geometric— properties invariant under each S. This note suggests such a frame of reference for the affine 4-space over GF(2), and a class of 322,560 equivalent coordinatizations of the frame. The frame: An array of 16 congruent equilateral subtriangles that make up a larger equilateral triangle. The invariant structure: The following set of 15 partitions of the frame into two 8-sets.
The group: The group AGL(4,2) of 322,560 regular affine transformations of the ordered 4-tuples over GF(2). |
For some background on the triangular version,
see the Square-Triangle Theorem,
noting particularly the linked-to coordinatization picture.
In memory of William S. Knowles, chiral chemist, who died last Wednesday (June 13, 2012)—
Detail from the Harvard Divinity School 1910 bookplate in yesterday morning's post—
"ANDOVER–HARVARD THEOLOGICAL LIBRARY"
Detail from Knowles's obituary in this morning's New York Times—
William Standish Knowles was born in Taunton, Mass., on June 1, 1917. He graduated a year early from the Berkshire School, a boarding school in western Massachusetts, and was admitted to Harvard. But after being strongly advised that he was not socially mature enough for college, he did a second senior year of high school at another boarding school, Phillips Academy in Andover, N.H.
Dr. Knowles graduated from Harvard with a bachelor’s degree in chemistry in 1939….
"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
From Pilate Goes to Kindergarten—
The six congruent quaternion actions illustrated above are based on the following coordinatization of the eightfold cube—
Problem: Is there a different coordinatization
that yields greater symmetry in the pictures of
quaternion group actions?
A paper written in a somewhat similar spirit—
"Chiral Tetrahedrons as Unitary Quaternions"—
ABSTRACT: Chiral tetrahedral molecules can be dealt [with] under the standard of quaternionic algebra. Specifically, non-commutativity of quaternions is a feature directly related to the chirality of molecules….
In the Beginning…
"As is well known, the Aleph is the first letter of the Hebrew alphabet."
– Borges, "The Aleph" (1945)
From some 1949 remarks of Weyl—
"The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time."
— Hermann Weyl, "Relativity Theory as a Stimulus in Mathematical Research," Proceedings of the American Philosophical Society , Vol. 93, No. 7, Theory of Relativity in Contemporary Science: Papers Read at the Celebration of the Seventieth Birthday of Professor Albert Einstein in Princeton, March 19, 1949 (Dec. 30, 1949), pp. 535-541
Weyl in 1946—:
"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
Coxeter in 1950 described the elements of the Galois field GF(9) as powers of a primitive root and as ordered pairs of the field of residue-classes modulo 3—
"… the successive powers of the primitive root λ or 10 are
λ = 10, λ2 = 21, λ3 = 22, λ4 = 02,
λ5 = 20, λ6 = 12, λ7 = 11, λ8 = 01.
These are the proper coordinate symbols….
(See Fig. 10, where the points are represented in the Euclidean plane as if the coordinate residue 2 were the ordinary number -1. This representation naturally obscures the collinearity of such points as λ4, λ5, λ7.)"
Coxeter's Figure 10 yields...
The Aleph
The details:
Coxeter's phrase "in the Euclidean plane" obscures the noncontinuous nature of the transformations that are automorphisms of the above linear 2-space over GF(3).
A footnote was added to Finite Relativity—
Background:
Weyl on what he calls the relativity problem—
“The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time.”
– Hermann Weyl, 1949, “Relativity Theory as a Stimulus in Mathematical Research“
“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, 1946, The Classical Groups , Princeton University Press, p. 16
…. A note of Feb. 20, 1986, supplied an example of such coordinatizations in finite geometry. In that note, the group of mediating transformations acted directly on coordinates within a 4×4 array. When the 4×4 array is embedded in a 4×6 array, a larger and more interesting group, M 24 (containing the original group), acts on the larger array. There is no obvious solution to Weyl’s relativity problem for M 24. That is, there is no obvious way* to apply exactly 24 distinct transformable coordinate-sets (or symbol-strings ) to the 24 array elements in such a way that the natural group of mediating transformations of the 24 symbol-strings is M 24. ….
Footnote of Sept. 20, 2011:
* R.T. Curtis has, it seems, a non-obvious way that involves strings of seven symbols. His abstract for a 1990 paper says that in his construction “The generators of M 24 are defined… as permutations of twenty-four 7-cycles in the action of PSL2(7) on seven letters….”
See “Geometric Interpretations of the ‘Natural’ Generators of the Mathieu groups,” by R.T. Curtis, Mathematical Proceedings of the Cambridge Philosophical Society (1990), Vol. 107, Issue 01, pp. 19-26. (Rec. Jan. 3, 1989, revised Feb. 3, 1989.) This paper was published online on Oct. 24, 2008.
Some related articles by Curtis:
R.T. Curtis, “Natural Constructions of the Mathieu groups,” Math. Proc. Cambridge Philos. Soc. (1989), Vol. 106, pp. 423-429
R.T. Curtis. “Symmetric Presentations I: Introduction, with Particular Reference to the Mathieu groups M 12 and M 24” In Proceedings of 1990 LMS Durham Conference ‘Groups, Combinatorics and Geometry’ (eds. M. W. Liebeck and J. Saxl), London Math. Soc. Lecture Note Series 165, Cambridge University Press, 1992, pp. 380–396
R.T. Curtis, “A Survey of Symmetric Generation of Sporadic Simple Groups,” in The Atlas of Finite Groups: Ten Years On , (eds. R.T. Curtis and R.A. Wilson), London Math. Soc. Lecture Note Series 249, Cambridge University Press, 1998, pp. 39–57
Weyl on what he calls the relativity problem—
"The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time."
— Hermann Weyl, 1949, "Relativity Theory as a Stimulus in Mathematical Research"
"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, 1946, The Classical Groups, Princeton University Press, p. 16
Twenty-four years ago a note of Feb. 20, 1986, supplied an example of such coordinatizations in finite geometry. In that note, the group of mediating transformations acted directly on coordinates within a 4×4 array. When the 4×4 array is embedded in a 4×6 array, a larger and more interesting group, M24 (containing the original group), acts on the larger array. There is no obvious solution to Weyl's relativity problem for M24. That is, there is no obvious way to apply exactly 24 distinct transformable coordinates (or symbol-strings) to the 24 array elements in such a way that the natural group of mediating transformations of the 24 symbol-strings is M24.
There is, however, an assignment of symbol-strings that yields a family of sets with automorphism group M24.
R.D. Carmichael in 1931 on his construction of the Steiner system S(5,8,24)–
"The linear fractional group modulo 23 of order 24•23•11 is often represented as a doubly transitive group of degree 24 on the symbols ∞, 0, 1, 2,…, 22. This transitive group contains a subgroup of order 8 each element of which transforms into itself the set ∞, 0, 1, 3, 12, 15, 21, 22 of eight elements, while the whole group transforms this set into 3•23•11 sets of eight each. This configuration of octuples has the remarkable property that any given set of five of the 24 symbols occurs in one and just one of these octuples. The largest permutation group Γ on the 24 symbols, each element of which leaves this configuration invariant, is a five-fold transitive group of degree 24 and order 24•23•22•21•20•48. This is the Mathieu group of degree 24."
— R. D. Carmichael, 1931, "Tactical Configurations of Rank Two," in American Journal of Mathematics, Vol. 53, No. 1 (Jan., 1931), pp. 217-240
"As noted previously, in Figure 2 viewed as a lattice the 16 digital labels 0000, 0001, etc., may be interpreted as naming the 16 subsets of a 4-set; in this case the partial ordering in the lattice is the structure preserved by the lattice's group of 24 automorphisms– the same automorphism group as that of the 16 Boolean connectives. If, however, these 16 digital labels are interpreted as naming the 16 functions from a 4-set to a 2-set (of two truth values, of two colors, of two finite-field elements, and so forth), it is not obvious that the notion of partial order is relevant. For such a set of 16 functions, the relevant group of automorphisms may be the affine group of A mentioned above. One might argue that each Venn diagram in Fig. 3 constitutes such a function– specifically, a mapping of four nonoverlapping regions within a rectangle to a set of two colors– and that the diagrams, considered simply as a set of two-color mappings, have an automorphism group of order larger than 24… in fact, of order 322,560. Whether such a group can be regarded as forming part of a 'geometry of logic' is open to debate."
The epigraph to "Finite Relativity" is:
"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
The added paragraph seems to fit this description.
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 space-time 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 group-theoretical 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 space-time].‘[22]
22. The significance of the notion of invariance and its group-theoretic 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, 139-149. Reprinted in E. Castellani (ed.), Interpreting Bodies: Classical and Quantum Objects in Modern Physics, Princeton, NJ: Princeton University Press, 1998, pp. 155-167.
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. 410-421.
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. 45-86–
“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 Mathematiker-Vereinigung 19: 281-300. [Reprinted: Physikalische Zeitschrift 12 (1911): 17-27].
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.
Earendil_Silmarils:
Les Anamorphoses:
“Pour construire un dessin en perspective,
le peintre trace sur sa toile des repères:
la ligne d’horizon (1),
le point de fuite principal (2)
où se rencontre les lignes de fuite (3)
et le point de fuite des diagonales (4).”
_______________________________
Serge Mehl,
Perspective &
Géométrie Projective:
“… la géométrie projective était souvent
synonyme de géométrie supérieure.
Elle s’opposait à la géométrie
euclidienne: élémentaire…
La géométrie projective, certes supérieure
car assez ardue, permet d’établir
de façon élégante des résultats de
la géométrie élémentaire.”
Similarly…
Finite projective geometry
(in particular, Galois geometry)
is certainly superior to
the elementary geometry of
quilt-pattern symmetry
and allows us to establish
de façon élégante
some results of that
elementary geometry.
Other Related Material…
from algebra rather than
geometry, and from a German
rather than from the French:
“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 U. Press, 1946
Evariste Galois
Weyl also says that the profound branch
of mathematics known as Galois theory
“Perhaps every science must
start with metaphor
and end with algebra;
and perhaps without metaphor
there would never have been
any algebra.”
For metaphor and
algebra combined, see
A.M.S. abstract 79T-A37,
Notices of the
American Mathematical Society,
February 1979, pages A-193, 194 —
the original version of the 4×4 case
of the diamond theorem.
“When approaching unfamiliar territory, we often, as observed earlier, try to describe or frame the novel situation using metaphors based on relations perceived in a familiar domain, and by using our powers of association, and our ability to exploit the structural similarity, we go on to conjecture new features for consideration, often not noticed at the outset. The metaphor works, according to Max Black, by transferring the associated ideas and implications of the secondary to the primary system, and by selecting, emphasising and suppressing features of the primary in such a way that new slants on it are illuminated.”
— Paul Thompson, University College, Oxford,
The Nature and Role of Intuition
in Mathematical Epistemology
That intuition, metaphor (i.e., analogy), and association may lead us astray is well known. The examples of French perspective above show what might happen if someone ignorant of finite geometry were to associate the phrase “4×4 square” with the phrase “projective geometry.” The results are ridiculously inappropriate, but at least the second example does, literally, illuminate “new slants”– i.e., diagonals– within the perspective drawing of the 4×4 square.
Similarly, analogy led the ancient Greeks to believe that the diagonal of a square is commensurate with the side… until someone gave them a new slant on the subject.
Finite Relativity
Today is the 18th birthday of my note
“The Relativity Problem in Finite Geometry.”
That note begins with a quotation from Weyl:
“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
Here is another quotation from Weyl, on the profound branch of mathematics known as Galois theory, which he says
“… 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 space-time dealt with by ordinary relativity theory.”
— Weyl, Symmetry, Princeton University Press, 1952, p. 138
This second quotation applies equally well to the much less profound, but more accessible, part of mathematics described in Diamond Theory and in my note of Feb. 20, 1986.
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