For some remarks related to the title, see Black + Algebra + Metaphor.
There is apparently no relationship between Scholze's metaphor
and my own use of the word "diamond" in finite geometry.
For some remarks related to the title, see Black + Algebra + Metaphor.
There is apparently no relationship between Scholze's metaphor
and my own use of the word "diamond" in finite geometry.
See Eightfold 1984 in this journal.
Related material —
"… the object sets up a kind of
frame or space or field
within which there can be epiphany."
"… Instead of an epiphany of being,
we have something like
an epiphany of interspaces."
— Charles Taylor, "Epiphanies of Modernism,"
Chapter 24 of Sources of the Self ,
Cambridge University Press, 1989
"Perhaps every science must start with metaphor
and end with algebra; and perhaps without the metaphor
there would never have been any algebra."
— Max Black, Models and Metaphors ,
Cornell University Press, Ithaca, NY, 1962
Click to enlarge:
(A Prequel to Dirac and Geometry)
"So Einstein went back to the blackboard.
And on Nov. 25, 1915, he set down
the equation that rules the universe.
As compact and mysterious as a Viking rune,
it describes spacetime as a kind of sagging mattress…."
— Dennis Overbye in The New York Times online,
November 24, 2015
Some pure mathematics I prefer to the sagging Viking mattress —
Readings closely related to the above passage —
Thomas Hawkins, "From General Relativity to Group Representations:
the Background to Weyl's Papers of 192526," in Matériaux pour
l'histoire des mathématiques au XXe siècle: Actes du colloque
à la mémoire de Jean Dieudonné, Nice, 1996 (Soc. Math.
de France, Paris, 1998), pp. 69100.
The 19thcentury algebraic theory of invariants is discussed
as what Weitzenböck called a guide "through the thicket
of formulas of general relativity."
Wallace Givens, "Tensor Coordinates of Linear Spaces," in
Annals of Mathematics Second Series, Vol. 38, No. 2, April 1937,
pp. 355385.
Tensors (also used by Einstein in 1915) are related to
the theory of line complexes in threedimensional
projective space and to the matrices used by Dirac
in his 1928 work on quantum mechanics.
For those who prefer metaphors to mathematics —
Rota fails to cite the source of his metaphor.

Background— August 30, 2006—
In the 2006 post, the above seventh symbol 110000 was
interpreted as the I Ching hexagram with topmost and
nexttotop lines solid, not broken— Hexagram 20, View .
In a different interpretation, 110000 is the binary for the decimal
number 48— representing the I Ching's Hexagram 48, The Well .
“… Max Black, the Cornell philosopher, and
others have pointed out how ‘perhaps every science
must start with metaphor and end with algebra, and
perhaps without the metaphor there would never
have been any algebra’ ….”
– Max Black, Models and Metaphors,
Cornell U. Press, 1962, page 242, as quoted
in Dramas, Fields, and Metaphors,
by Victor Witter Turner, Cornell U. Press,
paperback, 1975, page 25
The algebra is certainly clearer than either I Ching
metaphor, but is in some respects less interesting.
For a post that combines both the above I Ching
metaphors, View and Well , see Dec. 14, 2007.
In memory of scholar Elinor Ostrom,
who died today—
Tina Fey to Steve Martin
at the Oscars:
"Oh, Steve, no one wants
to hear about our religion
… that we made up."
From Wallace Stevens: A World of Transforming Shapes, by Alan D. Perlis, Bucknell University Press, 1976, p. 117:
… in 'The Pediment of Appearance,' a slight narrative poem in Transport to Summer… A group of young men enter some woods 'Hunting for the great ornament, The pediment of appearance.' Though moving through the natural world, the young men seek the artificial, or pure form, believing that in discovering this pediment, this distillation of the real, they will also discover the 'savage transparence,' the rude source of human life. In Stevens's world, such a search is futile, since it is only through observing nature that one reaches beyond it to pure form. As if to demonstrate the degree to which the young men's search is misaligned, Stevens says of them that 'they go crying/The world is myself, life is myself,' believing that what surrounds them is immaterial. Such a proclamation is a cardinal violation of Stevens's principles of the imagination. 
Superficially the young men's philosophy seems to resemble what Wikipedia calls "pantheistic solipsism"– noting, however, that "This article has multiple issues."
As, indeed, does pantheistic solipsism– a philosophy (properly called "eschatological pantheistic multipleego solipsism") devised, with tongue in cheek, by sciencefiction writer Robert A. Heinlein.
Despite their preoccupation with solipsism, Heinlein and Stevens point, each in his own poetic way, to a highly nonsolipsistic topic from pure mathematics that is, unlike the religion of Martin and Fey, not made up– namely, the properties of space.
"Sharpie, we have condensed six dimensions into four, then we either work by analogy into six, or we have to use math that apparently nobody but Jake and my cousin Ed understands. Unless you can think of some way to project six dimensions into three– you seem to be smart at such projections."
I closed my eyes and thought hard. "Zebbie, I don't think it can be done. Maybe Escher could have done it."
A discussion of Stevens's late poem "The Rock" (1954) in Wallace Stevens: A World of Transforming Shapes, by Alan D. Perlis, Bucknell University Press, 1976, p. 120:
For Stevens, the poem "makes meanings of the rock." In the mind, "its barrenness becomes a thousand things/And so exists no more." In fact, in a peculiar irony that only a poet with Stevens's particular notion of the imagination's function could develop, the rock becomes the mind itself, shattered into such diamondfaceted brilliance that it encompasses all possibilities for human thought: The rock is the gray particular of man's life,
The stone from which he rises, up—and—ho,
The step to the bleaker depths of his descents ...
The rock is the stern particular of the air,
The mirror of the planets, one by one,
But through man's eye, their silent rhapsodist,
Turquoise the rock, at odious evening bright
With redness that sticks fast to evil dreams;
The difficult rightness of halfrisen day.
The rock is the habitation of the whole,
Its strength and measure, that which is near,
point A
In a perspective that begins again
At B: the origin of the mango's rind.
(Collected Poems, 528)

Stevens's rock is associated with empty space, a concept that suggests "nothingness" to one literary critic:
B. J. Leggett, "Stevens's Late Poetry" in The Cambridge Companion to Wallace Stevens— On the poem "The Rock":"… the barren rock of the title is Stevens's symbol for the nothingness that underlies all existence, 'That in which space itself is contained'…. Its subject is its speaker's sense of nothingness and his need to be cured of it."
More positively…
Space is, of course, also a topic
in pure mathematics…
For instance, the 6dimensional
affine space (or the corresponding
5dimensional projective space)
over the twoelement Galois field
can be viewed as an illustration of
Stevens's metaphor in "The Rock."
Cara:
Here the 6dimensional affine
space contains the 63 points
of PG(5, 2), plus the origin, and
the 3dimensional affine
space contains as its 8 points
Conwell's eight "heptads," as in
Generating the Octad Generator.
A Multicultural Farewell
to a winner of the
Nobel Prize for Literature,
the Egyptian author of
The Seventh Heaven:
Supernatural Stories —
"Jackson has identified
the seventh symbol."
— Stargate
Other versions of
the seventh symbol —
"… Max Black, the Cornell philosopher, and others have pointed out how 'perhaps every science must start with metaphor and end with algebra, and perhaps without the metaphor there would never have been any algebra' …."
— Max Black, Models and Metaphors, Cornell U. Press, 1962, page 242, as quoted in Dramas, Fields, and Metaphors, by Victor Witter Turner, Cornell U. Press, paperback, 1975, page 25
Philosopher Max Black,
who died on this date
in 1988
“… Max Black, the Cornell philosopher, and others have pointed out how ‘perhaps every science must start with metaphor and end with algebra, and perhaps without the metaphor there would never have been any algebra’ ….”
— Max Black, Models and Metaphors, Cornell U. Press, 1962, page 242, as quoted in Dramas, Fields, and Metaphors, by Victor Witter Turner, Cornell U. Press, paperback, 1975, page 25
A Circle of Quiet
From the Harvard Math Table page:
“No Math table this week. We will reconvene next week on March 14 for a special Pi Day talk by Paul Bamberg.”
Transcript of the movie “Proof”–
Some friends of mine are in this band. They’re playing in a bar on Diversey, way down the bill, around… I said I’d be there. Great. Imaginary number? It’s a math joke. 
From the April 2006 Notices of the American Mathematical Society, a footnote in a review by Juliette Kennedy (pdf) of Rebecca Goldstein’s Incompleteness:
^{4} There is a growing literature in the area of postmodern commentaries of [sic] Gödel’s theorems. For example, Régis Debray has used Gödel’s theorems to demonstrate the logical inconsistency of selfgovernment. For a critical view of this and related developments, see Bricmont and Sokal’s Fashionable Nonsense [13]. For a more positive view see Michael Harris’s review of the latter, “I know what you mean!” [9]….
[9] MICHAEL HARRIS, “I know what you mean!,” http://www.math.jussieu.fr/~harris/Iknow.pdf.
[13] ALAN SOKAL and JEAN BRICMONT, Fashionable Nonsense, Picador, 1999.
Following the trail marked by Ms. Kennedy, we find the following in Harris’s paper:
“Their [Sokal’s and Bricmont’s] philosophy of mathematics, for instance, is summarized in the sentence ‘A mathematical constant like doesn’t change, even if the idea one has about it may change.’ ( p. 263). This claim, referring to a ‘crescendo of absurdity’ in Sokal’s original hoax in Social Text, is criticized by anthropologist Joan Fujimura, in an article translated for IS*. Most of Fujimura’s article consists of an astonishingly bland account of the history of noneuclidean geometry, in which she points out that the ratio of the circumference to the diameter depends on the metric. Sokal and Bricmont know this, and Fujimura’s remarks are about as helpful as FN’s** referral of Quine’s readers to Hume (p. 70). Anyway, Sokal explicitly referred to “Euclid’s pi”, presumably to avoid trivial objections like Fujimura’s — wasted effort on both sides.^{32} If one insists on making trivial objections, one might recall that the theorem
that p is transcendental can be stated as follows: the homomorphism Q[X] –> R taking X to is injective. In other words, can be identified algebraically with X, the variable par excellence.^{33}
More interestingly, one can ask what kind of object was before the formal definition of real numbers. To assume the real numbers were there all along, waiting to be defined, is to adhere to a form of Platonism.^{34} Dedekind wouldn’t have agreed.^{35} In a debate marked by the accusation that postmodern writers deny the reality of the external world, it is a peculiar move, to say the least, to make mathematical Platonism a litmus test for rationality.^{36} Not that it makes any more sense simply to declare Platonism out of bounds, like LévyLeblond, who calls Stephen Weinberg’s gloss on Sokal’s comment ‘une absurdité, tant il est clair que la signification d’un concept quelconque est évidemment affectée par sa mise en oeuvre dans un contexte nouveau!’^{37} Now I find it hard to defend Platonism with a straight face, and I prefer to regard the formula
as a creation rather than a discovery. But Platonism does correspond to the familiar experience that there is something about mathematics, and not just about other mathematicians, that precisely doesn’t let us get away with saying ‘évidemment’!^{38}
^{32} There are many circles in Euclid, but no pi, so I can’t think of any other reason for Sokal to have written ‘Euclid’s pi,’ unless this anachronism was an intentional part of the hoax. Sokal’s full quotation was ‘the of Euclid and the G of Newton, formerly thought to be constant and universal, are now perceived in their ineluctable historicity.’ But there is no need to invoke nonEuclidean geometry to perceive the historicity of the circle, or of pi: see Catherine Goldstein’s ‘L’un est l’autre: pour une histoire du cercle,’ in M. Serres, Elements d’histoire des sciences, Bordas, 1989, pp. 129149.
^{33} This is not mere sophistry: the construction of models over number fields actually uses arguments of this kind. A careless construction of the equations defining modular curves may make it appear that pi is included in their field of scalars.
^{34} Unless you claim, like the present French Minister of Education [at the time of writing, i.e. 1999], that real numbers exist in nature, while imaginary numbers were invented by mathematicians. Thus would be a physical constant, like the mass of the electron, that can be determined experimentally with increasing accuracy, say by measuring physical circles with ever more sensitive rulers. This sort of position has not been welcomed by most French mathematicians.
^{35 } Cf. M. Kline, Mathematics The Loss of Certainty, p. 324.
^{36} Compare Morris Hirsch’s remarks in BAMS April 94.
^{37} IS*, p. 38, footnote 26. Weinberg’s remarks are contained in his article “Sokal’s Hoax,” in the New York Review of Books, August 8, 1996.
^{38} Metaphors from virtual reality may help here.”
* Earlier defined by Harris as “Impostures Scientifiques (IS), a collection of articles compiled or commissioned by Baudouin Jurdant and published simultaneously as an issue of the journal Alliage and as a book by La Découverte press.”
** Earlier defined by Harris as “Fashionable Nonsense (FN), the North American translation of Impostures Intellectuelles.”
What is the moral of all this French noise?
Perhaps that, in spite of the contemptible nonsense at last summer’s Mykonos conference on mathematics and narrative, stories do have an important role to play in mathematics — specifically, in the history of mathematics.
Despite his disdain for Platonism, exemplified in his remarks on the noteworthy connection of pi with the zeta function in the formula given above, Harris has performed a valuable service to mathematics by pointing out the excellent historical work of Catherine Goldstein. Ms. Goldstein has demonstrated that even a French nominalist can be a firstrate scholar. Her essay on circles that Harris cites in a French version is also available in English, and will repay the study of those who, like Barry Mazur and other Harvard savants, are much too careless with the facts of history. They should consult her “Stories of the Circle,” pp. 160190 in A History of Scientific Thought, edited by Michel Serres, Blackwell Publishers (December 1995).
For the historicallychallenged mathematicians of Harvard, this essay would provide a valuable supplement to the upcoming “Pi Day” talk by Bamberg.
For those who insist on limiting their attention to mathematics proper, and ignoring its history, a suitable Pi Day observance might include becoming familiar with various proofs of the formula, pictured above, that connects pi with the zeta function of 2. For a survey, see Robin Chapman, Evaluating Zeta(2) (pdf). Zeta functions in a much wider context will be discussed at next May’s politically correct “Women in Mathematics” program at Princeton, “Zeta Functions All the Way” (pdf).
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
quiltpattern 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 79TA37,
Notices of the
American Mathematical Society,
February 1979, pages A193, 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.
Old School Tie
"We are introduced to John Nash, fuddling flatfooted about the Princeton courtyard, uninterested in his classmates' yammering about their various accolades. One chap has a rather unfortunate sense of style, but rather than tritely insult him, Nash holds a patterned glass to the sun, [director Ron] Howard shows us refracted patterns of light that take shape in a punch bowl, which Nash then displaces onto the neckwear, replying, 'There must be a formula for how ugly your tie is.' "
— Draft of
Computing with Modal Logics
(pdf), by Carlos Areces
and Maarten de Rijke
… diamonds and boxes are upper and lower adjoints of Galois connections…."
Evariste Galois
"Perhaps every science must
start with metaphor
and end with algebra;
and perhaps without metaphor
there would never have been
any algebra."
— attributed, in varying forms
(1, 2, 3), to Max Black,
Models and Metaphors, 1962
For metaphor and
algebra combined, see
"Symmetry invariance
in a diamond ring,"
A.M.S. abstract 79TA37,
Notices of the Amer. Math. Soc.,
February 1979, pages A193, 194 —
the original version of the 4×4 case
of the diamond theorem.
Parable
“A comparison or analogy. The word is simply a transliteration of the Greek word: parabolé (literally: ‘what is thrown beside’ or ‘juxtaposed’), a term used to designate the geometric application we call a ‘parabola.’…. The basic parables are extended similes or metaphors.”
— http://religion.rutgers.edu/nt/
primer/parable.html
“If one style of thought stands out as the most potent explanation of genius, it is the ability to make juxtapositions that elude mere mortals. Call it a facility with metaphor, the ability to connect the unconnected, to see relationships to which others are blind.”
— Sharon Begley, “The Puzzle of Genius,” Newsweek magazine, June 28, 1993, p. 50
“The poet sets one metaphor against another and hopes that the sparks set off by the juxtaposition will ignite something in the mind as well. Hopkins’ poem ‘Pied Beauty’ has to do with ‘creation.’ “
— Speaking in Parables, Ch. 2, by Sallie McFague
“The Act of Creation is, I believe, a more truly creative work than any of Koestler’s novels…. According to him, the creative faculty in whatever form is owing to a circumstance which he calls ‘bisociation.’ And we recognize this intuitively whenever we laugh at a joke, are dazzled by a fine metaphor, are astonished and excited by a unification of styles, or ‘see,’ for the first time, the possibility of a significant theoretical breakthrough in a scientific inquiry. In short, one touch of genius—or bisociation—makes the whole world kin. Or so Koestler believes.”
— Henry David Aiken, The Metaphysics of Arthur Koestler, New York Review of Books, Dec. 17, 1964
For further details, see
Speaking in Parables:
A Study in Metaphor and Theology
by Sallie McFague
Fortress Press, Philadelphia, 1975
Introduction
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
“Perhaps every science must start with metaphor and end with algebra; and perhaps without metaphor there would never have been any algebra.”
— attributed, in varying forms (1, 2, 3), to Max Black, Models and Metaphors, 1962
For metaphor and algebra combined, see
“Symmetry invariance in a diamond ring,” A.M.S. abstract 79TA37, Notices of the Amer. Math. Soc., February 1979, pages A193, 194 — the original version of the 4×4 case of the diamond theorem.
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