Background reading — Math’s Big Lies and, more generally, Mazur.
Related news for fans of Language Games —
Background reading — Math’s Big Lies and, more generally, Mazur.
Related news for fans of Language Games —
RED
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GRAY
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Arya on Rothko
Recent posts now tagged Black Fire suggest some context . . .
Meditations on the Portals of the Imagination
by Grace Dane Mazur, A K Peters/CRC Press;
first edition November 8, 2010
Interviewer's questions to the author (Feb. 4, 2011) —
"This book fuses together literature, art, science, history,
certainly the underworld–so many different points of obsession
for you, and you move so swiftly among them. It feels like a
magnum opus in that way. Where do you go from here?
After the hinges of hell, what comes next?"
The reviews?
(Continued from a post of Pi Day 2009, "Flowers for Barry,"
and from a post of July 5, 2019, "Darkly Enchanting") —
From this journal on 5 juillet 2019 —
Related material —
Grace Dane ("Gretchen") Mazur on Black Fire —
"Here, modernism is defined as an autonomous body
of ideas, having little or no outward reference, placing
considerable emphasis on formal aspects of the work
and maintaining a complicated—indeed, anxious—
rather than a naïve relationship with the daytoday
world, which is the de facto view of a coherent group
of people, such as a professional or disciplinebased
group that has a high sense of the seriousness and
value of what it is trying to achieve. This brisk definition…."
— Jeremy Gray, Plato's Ghost: The Modernist
Transformation of Mathematics , Princeton, 2008
"Even as the dominant modernist narrative was being written,
there were art historians who recognized that it was inaccurate.
The narrative was too focused on France . . . . Nor was it
correct to build the narrative so exclusively around formalism;
modernism was far messier, far more multifaceted than that."
— Jane Kallir, https://www.tabletmag.com/
jewishartsandculture/visualartanddesign/
269564/theendofmiddleclassart,
quoted here on the above date — Sept. 11, 2018.
From some related Log24 posts —
Update of 10:18 AM the same day —
See also Logicomix in this journal and, at Harvard,
http://www.math.harvard.edu/~mazur/ —
Update of 10:48 AM the same day —
See Log24 on the date of Tieszen's death.
" The origin of new ways of doing things may often be
a disciplinary crisis. The definition of such a crisis
provided by Barry Mazur in Mykonos (2005) applies
equally well to literary creation. '[A crisis occurs] when
some established overarching framework, theoretical
vocabulary or procedure of thought is perceived as
inadequate in an essential way, or not meaning
what we think it means.' "
— Circles Disturbed :
The Interplay of Mathematics and Narrative
Edited by Apostolos Doxiadis & Barry Mazur
Princeton University Press, 2012. See
Chapter 14, Section 5.1, by Uri Margolin.
See also "overarching" in this journal.
Peter Galison, a Harvard professor, is a defender of
the Vienna Circle and the religion of Scientism.
From Galison's “Structure of Crystal, Bucket of Dust,” in
Circles Disturbed: The Interplay of Mathematics and Narrative ,
edited by Apostolos Doxiadis and Barry Mazur, pp. 5278
(Princeton: Princeton U. Press, 2012) …
Galison's final paragraph —
"Perhaps, then, it should not surprise us too much if,
as Wheeler approaches the beginningend of all things,
there is a bucket of Borelian dust. Out of this filth,
through the proposition machine of quantum mechanics
comes pregeometry; pregeometry makes geometry;
geometry gives rise to matter and the physical laws
and constants of the universe. At once close to and far
from the crystalline story that Bourbaki invoked,
Wheeler’s genesis puts one in mind of Genesis 3:19:
'In the sweat of thy face shalt thou eat bread, till thou
return unto the ground; for out of it wast thou taken:
for dust thou art, and unto dust shalt thou return.' "
For fans of Scientism who prefer more colorful narratives —
From a piece in The Harvard Crimson today:
"The refrain of one of my favorite songs goes,
'Tension is to be loved when it is like a passing note
to a beautiful, beautiful chord.' I don’t want to seek
the allusion [sic ] of finding balance anymore:
It can’t be found."
Allusion —
"No angel born in Hell
Could break that Satan's spell"
— "American Pie"
A passing note —
"The name diabolus in musica ("the Devil in music")
has been applied to the interval from at least the
early 18th century…."
— Wikipedia on the tritone
See also Music & Noise at a physics site and the square root
of two on page 56 of Barry Mazur's Imagining Numbers
(Picador imprint, Farrar, Straus and Giroux, 2003) —
— The New Yorker , May 19, 1997 issue, page 52
See also Hollander in this journal.
(This post was suggested by a search for
"Barry Mazur" + "TwoFaced.")
"Celebrate National Library Week 2015 (April 1218, 2015)
with the theme "Unlimited possibilities @ your library®."
See also Library of Hell.
A page from Princeton University Press on March 18, 2012:
… "mathematics and narrative…." (top of page xvii).
I prefer the interplay of Euclidean and Galois mathematics.
Continued from yesterday.
The passage on Claude Chevalley quoted here
yesterday in the post Dead Reckoning was, it turns out,
also quoted by Peter Galison in his essay "Structure of Crystal,
Bucket of Dust" in Circles Disturbed: The Interplay of
Mathematics and Narrative (Princeton University Press, 2012,
ed. by Apostolos Doxiadis and Barry Mazur).
Galison gives a reference to his source:
"From 'Claude Chevalley Described by His Daughter (1988),'
in Michèle Chouchan, Nicolas Bourbaki: Faits et légendes
(Paris: Éditions du Choix, 1995), 36–40, translated and cited
in Marjorie Senechal, 'The Continuing Silence of Bourbaki:
An Interview with Pierre Cartier, June 18, 1997,'
Mathematical Intelligencer 1 (1998): 22–28."
Galison's essay compares Chevalley with the physicist
John Archibald Wheeler. His final paragraph —
"Perhaps, then, it should not surprise us too much if,
as Wheeler approaches the beginningend of all things,
there is a bucket of Borelian dust. Out of this filth,
through the proposition machine of quantum mechanics
comes pregeometry; pregeometry makes geometry;
geometry gives rise to matter and the physical laws
and constants of the universe. At once close to and far
from the crystalline story that Bourbaki invoked,
Wheeler’s genesis puts one in mind of Genesis 3:19:
'In the sweat of thy face shalt thou eat bread, till thou
return unto the ground; for out of it wast thou taken:
for dust thou art, and unto dust shalt thou return.'"
See also posts tagged Wheeler.
The late Colin Wilson appears at the head
of this afternoon's New York Times obituaries —
Margalit Fox's description this afternoon of
Wilson's first book, from 1956—
"The Outsider had an aim no less ambitious
than its scope: to delineate the meaning of
human existence."
This suggests a review of Log24 posts on "The Zero Theorem"
that yields—
See also Log24 on the date of Wilson's death.
Related material: Devil's Night, 2011.
A recent addition to Barry Mazur's home page—
"December 1, 2013: Here are rough notes for
a short talk entitled The Faces of Evidence
(in Mathematics) ([PDF]) to be given at the
Cambridge Scientific Club, Dec. 5 2013."
The PDF link does not work, but some earlier remarks by
Mazur on this topic have been published elsewhere:
Related material:
The Proof and the Lie (St. Andrew's Day, 2003), and
a recent repetition of the lie in Wikipedia:
"Around 1955, Japanese mathematicians Goro Shimura
and Yutaka Taniyama observed a possible link between
two apparently completely distinct, branches of mathematics,
elliptic curves and modular forms."
This statement, from the article on Algebraic number theory,
was added on Oct. 22, 2013 by one "Brirush," apparently a
Temple University postdoctoral researcher, in what he rightly
called a "terrible history summary."
From a book by Harvard mathematician Barry Mazur —
"Part of the self leaves the body when we sleep…"
See also the Saturday evening post "Fingo."
Continued from August 29th
"The general mood was summed up by fan
Debra Kay, who tweeted simply: 'R.I.P. Frederik Pohl.
Thanks for the stories.'" — The Guardian today
Pohl reportedly died Monday.
A synchronicity check: Quoted here Monday—
See also this journal on August 5th, 2008—
From The New York Times Sunday Book Review of Sept. 1, 2013—
THE GAMAL Reviewed by Katharine Weber Ten years ago, when Mark Haddon’s “Curious Incident of the Dog in the NightTime” turned up on the bestseller list and won a number of literary awards, the novel’s autistic narrator beguiled readers with his unconventional point of view. Today, even as controversy surrounds the revised classification of autism in the latest version of the American Psychiatric Association’s Diagnostic and Statistical Manual of Mental Disorders, the quirky yet remarkably perceptive points of view of autistic narrators have become increasingly familiar in every category of fiction, from young adult to science fiction to popular and literary fiction. Like Haddon’s Christopher Boone, the narrator of Ciaran Collins’s remarkable first novel, “The Gamal,” has been encouraged by a mental health professional to write his story for therapeutic purposes. Charlie McCarthy, 25, is known in the West Cork village of Ballyronan as “the gamal,” short for “gamalog,” a term for a fool or simpleton rarely heard beyond the Gaeltacht regions of Ireland. He is in fact a savant, a sensitive oddball whose cheeky, strange, defiant and witty monologue is as disturbing as it is dazzling. … 
The Gamal features a considerable variety of music. See details at a music weblog.
This, together with the narrator's encouragement "by a mental health professional
to write his story for therapeutic purposes" might interest Baz Luhrmann.
See Luhrmann's recent film "The Great Gatsby," with its portrait of
F. Scott Fitzgerald's narrator, and thus Fitzgerald himself, as a sensitive looney.
The CarrawayDaisyGatsby trio has a parallel in The Gamal . (Again, see
the music weblog's description.)
The Times reviewer's concluding remarks on truth, lies, and unreliable autistic
narrators may interest some mathematicians. From an Aug. 29 post—
A different gamalog , a website in Mexico, is not entirely unrelated to
issues of lies and truth—
(Mathematics and Narrative, continued from May 9, 2013)
See also Scriba's The Concept of Number and,
from the date of his death, The Zero Theorem.
"Why history?
Well, the essence of history is story ,
and a good story is an end in itself."
— Barry Mazur, "History of Mathematics as a tool,"
February 17, 2013
This journal on February 17, 2013:FROM Christoph Waltz"Currently in postproduction": The Zero Theorem. For Christoph WaltzRaiders of the Lost Tesseract continues… SOCRATES: Is he not better off in knowing his ignorance? 
See also today's previous post.
National…
International…
Click medal for some background. The medal may be regarded
as illustrating the 16point Galois space. (See previous post.)
Related material: Jews in Hyperspace.
Sarah Tomlin in a Nature article on the July 1215 2005 Mykonos meeting on Mathematics and Narrative—
"Today, Mazur says he has woken up to the power of narrative, and in Mykonos gave an example of a 20year unsolved puzzle in number theory which he described as
Michel Chaouli in "How Interactive Can Fiction Be?" (Critical Inquiry 31, Spring 2005), pages 613614—
"…a simple thought experiment….*
… If the cliffhanger is done well, it will not simply introduce a wholly unprepared turn into the narrative (a random death, a new character, an entirely unanticipated obstacle) but rather tighten the configuration of known elements to such a degree that the next step appears both inevitable and impossible. We feel the pressure rising to a breaking point, but we simply cannot foresee where the complex narrative structure will give way. This interplay of necessity and contingency produces our anxious— and highly pleasurable— speculation about the future path of the story. But if we could determine that path even slightly, we would narrow the range of possible outcomes and thus the uncertainty in the play of necessity and contingency. The world of the fiction would feel, not open, but rigged."
* The idea of the thought experiment emerged in a conversation with Barry Mazur.
Barry Mazur in the preface to his 2003 book Imagining Numbers—
"But the telltale adjective real suggests two things: that these numbers are somehow real to us and that, in contrast, there are unreal numbers in the offing. These are the imaginary numbers .
The imaginary numbers are well named, for there is some imaginative work to do to make them as much a part of us as the real numbers we use all the time to measure for bookshelves.
This book began as a letter to my friend Michel Chaouli. The two of us had been musing about whether or not one could 'feel' the workings of the imagination in its various labors. Michel had also mentioned that he wanted to 'imagine imaginary numbers.' That very (rainy) evening, I tried to work up an explanation of the idea of these numbers, still in the mood of our conversation."
See also The Galois Quaternion and 2/19.
New York Lottery last evening
Part 3 of 5 (See also Part 1 and Part 2) begins as follows…
"Incommensurable. It is a strange word. I wondered, why did Kuhn choose it? What was the attraction?
Here’s one clue. At the very end of 'The Road Since Structure,' a compendium of essays on Kuhn’s work, there is an interview with three Greek philosophers of science, Aristides Baltas, Kostas Gavroglu and Vassiliki Kindi. Kuhn provides a brief account of the historical origins of his idea. Here is the relevant segment of the interview.
T. KUHN: Look, 'incommensurability' is easy.
V. KINDI: You mean in mathematics?
T. KUHN: …When I was a bright high school mathematician and beginning to learn Calculus, somebody gave me—or maybe I asked for it because I’d heard about it—there was sort of a big twovolume Calculus book by, I can’t remember whom. And then I never really read it. I read the early parts of it. And early on it gives the proof of the irrationality of the square root of 2. And I thought it was beautiful. That was terribly exciting, and I learned what incommensurability was then and there. So, it was all ready for me, I mean, it was a metaphor but it got at nicely what I was after. So, that’s where I got it.
'It was all ready for me.' I thought, 'Wow.' The language was suggestive. I imagined √2 provocatively dressed, its lips rouged. But there was an unexpected surprise. The idea didn’t come from the physical sciences or philosophy or linguistics, but from mathematics ."
A footnote from Morris (no. 29)—
"Those who are familiar with the proof [of irrationality] certainly don’t want me to explain it here; likewise, those who are unfamiliar with it don’t want me to explain it here, either. There are many simple proofs in many histories of mathematics — E.T. Bell, Sir Thomas Heath, Morris Kline, etc., etc. Barry Mazur offers a proof in his book, 'Imagining Numbers (particularly the square root of minus fifteen),' New York, NY: Farrar, Straus and Giroux. 2003, 26ff. And there are two proofs in his essay, 'How Did Theaetetus Prove His Theorem?', available on Mazur’s Harvard Web site."
There may, actually, be a few who do want the proof. They may consult the sources Morris gives, or the excellent description by G.H. Hardy in A Mathematician's Apology , or, perhaps best of all for present purposes, the proof as described in a "sort of a big twovolume Calculus book" (perhaps the one Kuhn mentioned)… See page 6 and page 7 of Volume One of Richard Courant's classic Differential and Integral Calculus (second edition, 1937, reprinted many times through 1970, and again in a Wiley Classics Library Edition in 1988).
The Magic Lyre
(Click image for context.)
See also Saturday's post—
as well as Solemn Dance
and Mazur at Delphi.
(This last is apparently based on
a talk given by Barry Mazur at Delphi
in 2007 and may or may not appear in
a book, Mathematics and Narrative,
to be published in 2010.)
Suggested tune for the lyre–
"Send me the pillow
that you dream on,"
in memory of Hank Locklin,
who died on this date last year.
[Note: Janus is Roman, not Greek, and
the photo is from one “Fubar Obfusco”]
Click on image for details.
Religion and Narrative, continued:

Context:
Notes on Mathematics and Narrative
(entries in chronological order,
March 13 through 19)
(Cf. Sinatra’s birthday, 2004)
One for his baby:
Ron Silver as
Alan Dershowitz in
“Reversal of Fortune”
suggests the epigraph of
The Particulars of Rapture:
Reflections on Exodus —
two stanzas from attorney
Wallace Stevens
quoted here yesterday afternoon.
One more for the road:
A link that appeared in a
different form in Saturday’s
“Flowers for Barry“–
Speed the Plow.
This leads to
A Hanukkah Tale
containing the following:
On Time
(in Mathematics and Literature) “… I want to spend these twenty minutes savoring, and working up, the real complexity of the metaphorical relationship of time and distance– to defamiliarize it for us. And then I will give a few examples of how imaginative literature makes use of the inherent strangeness in this relationship: Time ↔ Distance.
And finally I will offer my opinion (which I think must be everyone’s opinion) about why we derive significant– but not total– comfort from this equation.” — Barry Mazur, March 8, 2009, draft (pdf) of talk for conference on comparative literature* 
Another version of
Mazur’s metaphor
Time ↔ Distance:
— Steven H. Cullinane,
October 8, 2003
For some context in
comparative literature,
see Time Fold
(Oct. 10, 2003)
and A Hanukkah Tale
(Dec. 22, 2008).
Related material:
Rat Psychology
yesterday.
Prominent literary intellectuals often like to make familiar reference to the technical terminology of mathematical logic or philosophy of language. A friend of mine overheard the following conversation in Cambridge during l’affaire Derrida, when the proposal to grant an honorary degree to that gentleman met serious academic opposition in the university. A journalist covering the fracas asked a Prominent Literary Intellectual what he took to be Derrida’s importance in the scheme of things. ‘Well,’ the PLI confided graciously, unblushingly, ‘Gödel showed that every theory is inconsistent unless it is supported from outside. Derrida showed that there is no outside.’Now, there are at least three remarkable things about this. First, the thing that Gödel was supposed to show could not possibly be shown, since there are many demonstrably consistent theories. Second, therefore, Gödel indeed did not show it, and neither did he purport to do so. Third, it makes no sense to say that an inconsistent theory could become consistent by being ‘supported from outside’, whatever that might mean (inconsistency sticks; you cannot get rid of it by addition, only by subtraction). So what Derrida is said to have done is just as impossible as what Gödel was said to have done.
These mistakes should fail you in an undergraduate logic or math or philosophy course. But they are minor considerations in the world of the PLI. The point is that the mere mention of Gödel (like the common invocation of ‘hierarchies’ and ‘metalanguages’) gives a specious impression of something thrillingly deep and thrillingly mathematical and scientific (theory! dazzling! Einstein!) And, not coincidentally, it gives the PLI a flattering image of being something of a hand at these things, an impresario of the thrills. I expect the journalist swooned.
“The word explicit is from the Latin explicitus related to the verb explicare meaning to ‘unfold, unravel, explain, explicate’ (plicare means ‘to fold’; think of the English noun ‘ply’).”
Related material: Mark Taylor’s Derridean use of “le pli” (The Picture in Question, pp. 5860, esp. note 13, p. 60). See also the discussion of Taylor in this journal posted on Dec. 19.
A Penny for My Thoughts? by Maureen Dowd “If an online newspaper in Pasadena, Calif., can outsource coverage to India, I wonder how long can it be before some guy in Bangalore is writing my column….” — New York Times teaser for a column of Sunday, November 30, 2008 (St. Andrew’s Day) 
DH News Service, Bangalore, Tuesday, Dec. 2, 2008:
“Monday evening had a pleasant surprise in store for skywatchers as the night sky sported a smiley, in the form of a crescent moon flanked by two bright planets Jupiter and Venus…” 
Meanwhile, at National Geographic:
Jupiter, Venus, Moon Make “Frown”
A Midrash for Maureen:
Related material on Pasadena:
Happy birthday, R. P. Dilworth.
Related material on India:
The Shining of May 29 (2002) and
A WellKnown Theorem (2005).
— Barry Mazur in 2000 as quoted today at the University of St. Andrews
“It is said that the students of medieval Paris came to blows in the streets over the question of universals. The stakes are high, for at issue is our whole conception of our ability to describe the world truly or falsely, and the objectivity of any opinions we frame to ourselves. It is arguable that this is always the deepest, most profound problem of philosophy.”
— Simon Blackburn, Think (Oxford, 1999)
Michael Harris, mathematician at the University of Paris:
“… three ‘parts’ of tragedy identified by Aristotle that transpose to fiction of all types– plot (mythos), character (ethos), and ‘thought’ (dianoia)….”
— paper (pdf) to appear in Mathematics and Narrative, A. Doxiadis and B. Mazur, eds.
Mythos —
A visitor from France this morning viewed the entry of Jan. 23, 2006: “In Defense of Hilbert (On His Birthday).” That entry concerns a remark of Michael Harris.
A check of Harris’s website reveals a new article:
“Do Androids Prove Theorems in Their Sleep?” (slighly longer version of article to appear in Mathematics and Narrative, A. Doxiadis and B. Mazur, eds.) (pdf).
From that article:
“The word ‘key’ functions here to structure the reading of the article, to draw the reader’s attention initially to the element of the proof the author considers most important. Compare E.M. Forster in Aspects of the Novel:
[plot is] something which is measured not be minutes or hours, but by intensity, so that when we look at our past it does not stretch back evenly but piles up into a few notable pinnacles.”
Ethos —
“Forster took pains to widen and deepen the enigmatic character of his novel, to make it a puzzle insoluble within its own terms, or without. Early drafts of A Passage to India reveal a number of false starts. Forster repeatedly revised drafts of chapters thirteen through sixteen, which comprise the crux of the novel, the visit to the Marabar Caves. When he began writing the novel, his intention was to make the cave scene central and significant, but he did not yet know how:
When I began a A Passage to India, I knew something important happened in the Malabar (sic) Caves, and that it would have a central place in the novel– but I didn’t know what it would be… The Malabar Caves represented an area in which concentration can take place. They were to engender an event like an egg.”
— E. M. Forster: A Passage to India, by Betty Jay
Dianoia —
“Despite the flagrant triviality of the proof… this result is the key point in the paper.”
— Michael Harris, op. cit., quoting a mathematical paper
Online Etymology Dictionary:
flagrant c.1500, “resplendent,” from L. flagrantem (nom. flagrans) “burning,” prp. of flagrare “to burn,” from L. root *flag, corresponding to PIE *bhleg– (cf. Gk. phlegein “to burn, scorch,” O.E. blæc “black”). Sense of “glaringly offensive” first recorded 1706, probably from common legalese phrase in flagrante delicto “redhanded,” lit. “with the crime still blazing.”
A related use of “resplendent”– applied to a Trinity, not a triviality– appears in the Liturgy of Malabar:
— The Liturgies of SS. Mark, James, Clement, Chrysostom, and Basil, and the Church of Malabar, by the Rev. J.M. Neale and the Rev. R.F. Littledale, reprinted by Gorgias Press, 2002
Judy Davis in the Marabar Caves
In mathematics
(as opposed to narrative),
somewhere between
a flagrant triviality and
a resplendent Trinity we
have what might be called
“a resplendent triviality.”
For further details, see
“A FourColor Theorem.”
"Mazur introduced the topic of prime numbers with a story from Don Quixote in which Quixote asked a poet to write a poem with 17 lines. Because 17 is prime, the poet couldn't find a length for the poem's stanzas and was thus stymied."
— Undated American Mathematical Society news item about a Nov. 1, 2007, event
Desconvencida,
Jueves, Enero 17, 2008
Horses of a Dream
(Log24, Sept. 12, 2003)
Knight Moves
(Log24 yesterday–
anniversary of the
Jan. 16 publication
of Don Quixote)
Windmill and Diamond
(St. Cecilia's Day 2006)
“Mazur introduced the topic of prime numbers with a story from Don Quixote in which Quixote asked a poet to write a poem with 17 lines. Because 17 is prime, the poet couldn’t find a length for the poem’s stanzas and was thus stymied.”
— Undated American Mathematical Society news item about a Nov. 1, 2007, event
“The Wu Li Masters know that physicists are doing more than ‘discovering the endless diversity of nature.’ They are dancing with Kali [or Durga], the Divine Mother of Hindu mythology.” –Gary Zukav, Harvard ’64 
Veronica 
“I think transformation becomes the main word in my life, transformation.
Because you don’t want to just put a mirror in front of people and say, here, look at yourself. What do you see?
You want to have a skewed mirror. You want a mirror that says, you didn’t know you could see the back of your head. You didn’t know that you could… almost cubistic, see all aspects at the same time.
And what that does for human beings is it allows them to step out of their lives and to revisit it and maybe find something different about it.” —Julie Taymor
Related material:
The previous two entries and
readings for the Feast of
the Triumph of the Cross
in 2006 and in 2003.
Solemn Dance
Virgil on the Elysian Fields:
Some wrestle on the sands, and some in play And games heroic pass the hours away. Those raise the song divine, and these advance In measur'd steps to form the solemn dance.
(See also the previous two entries.)
"The cover of this issue of the Bulletin is the frontispiece to a volume of Samuel de Fermat’s 1670 edition of Bachet’s Latin translation of Diophantus’s Arithmetica. This edition includes the marginalia of the editor’s father, Pierre de Fermat. Among these notes one finds the elder Fermat’s extraordinary comment [c. 1637] in connection with the Pythagorean equation
— Barry Mazur, Gade University Professor at Harvard
Mazur's concluding remarks are as follows:
Mazur has admitted, at his website, that this conclusion was an error:
"I erroneously identified the figure on the cover as Erato, muse of erotic poetry, but it seems, rather, to be Orpheus."
"Seems"?
The inscription on the frontispiece, "Obloquitur numeris septem discrimina vocum," is from a description of the Elysian Fields in Virgil's Aeneid, Book VI:
His demum exactis, perfecto munere divae, Devenere locos laetos, & amoena vireta Fortunatorum nemorum, sedesque beatas. Largior hic campos aether & lumine vestit Purpureo; solemque suum, sua sidera norunt. Pars in gramineis exercent membra palaestris, Contendunt ludo, & fulva luctanter arena: Pars pedibus plaudunt choreas, & carmina dicunt. Necnon Threicius longa cum veste sacerdos Obloquitur numeris septem discrimina vocum: Jamque eadem digitis, jam pectine pulsat eburno.
PITT: These rites compleat, they reach the flow'ry plains, The verdant groves, where endless pleasure reigns. Here glowing AEther shoots a purple ray, And o'er the region pours a double day. From sky to sky th'unwearied splendour runs, And nobler planets roll round brighter suns. Some wrestle on the sands, and some in play And games heroic pass the hours away. Those raise the song divine, and these advance In measur'd steps to form the solemn dance. There Orpheus graceful in his long attire, In seven divisions strikes the sounding lyre; Across the chords the quivering quill he flings, Or with his flying fingers sweeps the strings. DRYDEN: These holy rites perform'd, they took their way, Where long extended plains of pleasure lay. The verdant fields with those of heav'n may vie; With AEther veiled, and a purple sky: The blissful seats of happy souls below; Stars of their own, and their own suns they know. Their airy limbs in sports they exercise, And on the green contend the wrestlers prize. Some in heroic verse divinely sing, Others in artful measures lead the ring. The Thracian bard surrounded by the rest, There stands conspicuous in his flowing vest. His flying fingers, and harmonious quill, Strike seven distinguish'd notes, and seven at once they fill.
It is perhaps not irrelevant that the late Lorraine Hunt Lieberson's next role would have been that of Orfeo in Gluck's "Orfeo ed Euridice." See today's earlier entries.
The poets among us may like to think of Mazur's own role as that of the lyre:
In memory of
Irving Kaplansky,
who died on
Sunday, June 25, 2006
“Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.”
— T. S. Eliot
Kaplansky received his doctorate in mathematics at Harvard in 1941 as the first Ph.D. student of Saunders Mac Lane.
From the April 25, 2005, Harvard Crimson:
ExMath Prof Mac Lane, 95, Dies
Gade University Professor of Mathematics Barry Mazur, a friend of the late Mac Lane, recalled that [a Mac Lane paper of 1945] had at first been rejected from a lowercaliber mathematical journal because the editor thought that it was “more devoid of content” than any other he had read.“Saunders wrote back and said, ‘That’s the point,'” Mazur said. “And in some ways that’s the genius of it. It’s the barest, most Beckettlike vocabulary that incorporates the theory and nothing else.”
He likened it to a sparse grammar of nouns and verbs and a limited vocabulary that is presented “in such a deft way that it will help you understand any language you wish to understand and any language will fit into it.”
A sparse grammar of lines from Charles Sanders Peirce (Harvard College, class of 1859):
Related entry: Binary Geometry.
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).
“There is an
underlying timelessness
in the basic conversation
that is mathematics.”
— Barry Mazur (pdf)
“The authors of the etiquette book The Art of Civilized Conversation say that conversation’s versatility makes it ‘the Swiss Army knife of social skills.'”
Today is the birthday of Helmut Wielandt (Dec. 19,
"In his speech accepting membership of the Heidelberg Academy in 1960 he said:
It is to one of Schur's seminars that I owe the stimulus to work with permutation groups, my first research area. At that time the theory had nearly died out. It had developed last century, but at about the turn of the century had been so completely superseded by the more generally applicable theory of abstract groups that by 1930 even important results were practically forgotten – to my mind unjustly."
Permutation groups are still not without interest. See today's updates (Notes [01] and [02]) to Pattern Groups.
Mathematics and Metaphor
The current (June/July) issue of the Notices of the American Mathematical Society has two feature articles. The first, on the vulgarizer Martin Gardner, was dealt with here in a June 19 entry, Darkness Visible. The second is related to a letter of André Weil (pdf) that is in turn related to mathematician Barry Mazur’s attempt to rewrite mathematical history and to vulgarize other people’s research by using metaphors drawn, it would seem, from the Weil letter.
A Mathematical Lie conjectures that Mazur’s revising of history was motivated by a desire to dramatize some arcane mathematics, the Taniyama conjecture, that deals with elliptic curves and modular forms, two areas of mathematics that have been known since the nineteenth century to be closely related.
Mazur led author Simon Singh to believe that these two areas of mathematics were, before Taniyama’s conjecture of 1955, completely unrelated —
“Modular forms and elliptic equations live in completely different regions of the mathematical cosmos, and nobody would ever have believed that there was the remotest link between the two subjects.” — Simon Singh, Fermat’s Enigma, 1998 paperback, p. 182
This is false. See Robert P. Langlands, review of Elliptic Curves, by Anthony W. Knapp, Bulletin of the American Mathematical Society, January 1994.
It now appears that Mazur’s claim was in part motivated by a desire to emulate the great mathematician André Weil’s manner of speaking; Mazur parrots Weil’s “bridge” and “Rosetta stone” metaphors —
From Peter Woit’s weblog, Feb. 10, 2005:
“The focus of Weil’s letter is the analogy between number fields and the field of algebraic functions of a complex variable. He describes his ideas about studying this analogy using a third, intermediate subject, that of function fields over a finite field, which he thinks of as a ‘bridge‘ or ‘Rosetta stone.'”
In “A 1940 Letter of André Weil on Analogy in Mathematics,” (pdf), translated by Martin H. Krieger, Notices of the A.M.S., March 2005, Weil writes that
“The purely algebraic theory of algebraic functions in any arbitrary field of constants is not rich enough so that one might draw useful lessons from it. The ‘classical’ theory (that is, Riemannian) of algebraic functions over the field of constants of the complex numbers is infinitely richer; but on the one hand it is too much so, and in the mass of facts some real analogies become lost; and above all, it is too far from the theory of numbers. One would be totally obstructed if there were not a bridge between the two. And just as God defeats the devil: this bridge exists; it is the theory of the field of algebraic functions over a finite field of constants….
On the other hand, between the function fields and the ‘Riemannian’ fields, the distance is not so large that a patient study would not teach us the art of passing from one to the other, and to profit in the study of the first from knowledge acquired about the second, and of the extremely powerful means offered to us, in the study of the latter, from the integral calculus and the theory of analytic functions. That is not to say that at best all will be easy; but one ends up by learning to see something there, although it is still somewhat confused. Intuition makes much of it; I mean by this the faculty of seeing a connection between things that in appearance are completely different; it does not fail to lead us astray quite often. Be that as it may, my work consists in deciphering a trilingual text {[cf. the Rosetta Stone]}; of each of the three columns I have only disparate fragments; I have some ideas about each of the three languages: but I know as well there are great differences in meaning from one column to another, for which nothing has prepared me in advance. In the several years I have worked at it, I have found little pieces of the dictionary. Sometimes I worked on one column, sometimes under another.”
Here is another statement of the Rosettastone metaphor, from Weil’s translator, Martin H. Krieger, in the A.M.S. Notices of November 2004, “Some of What Mathematicians Do” (pdf):
“Weil refers to three columns, in analogy with the Rosetta Stone’s three languages and their arrangement, and the task is to ‘learn to read Riemannian.’ Given an ability to read one column, can you find its translation in the other columns? In the first column are Riemann’s transcendental results and, more generally, work in analysis and geometry. In the second column is algebra, say polynomials with coefficients in the complex numbers or in a finite field. And in the third column is arithmetic or number theory and combinatorial properties.”
For greater clarity, see Armand Borel (pdf) on Weil’s Rosetta stone, where the three columns are referred to as Riemannian (transcendental), Italian (“algebraicogeometric,” over finite fields), and arithmetic (i.e., numbertheoretic).
From Fermat’s Enigma, by Simon Singh, Anchor paperback, Sept. 1998, pp. 190191:
Barry Mazur: “On the one hand you have the elliptic world, and on the other you have the modular world. Both these branches of mathematics had been studied intensively but separately…. Than along comes the TaniyamaShimura conjecture, which is the grand surmise that there’s a bridge between these two completely different worlds. Mathematicians love to build bridges.”
Simon Singh: “The value of mathematical bridges is enormous. They enable communities of mathematicians who have been living on separate islands to exchange ideas and explore each other’s creations…. The great potential of the TaniyamaShimura conjecture was that it would connect two islands and allow them to speak to each other for the first time. Barry Mazur thinks of the TaniyamaShimura conjecture as a translating device similar to the Rosetta stone…. ‘It’s as if you know one language and this Rosetta stone is going to give you an intense understanding of the other language,’ says Mazur. ‘But the TaniyamaShimura conjecture is a Rosetta stone with a certain magical power.'”
If Mazur, who is scheduled to speak at a conference on Mathematics and Narrative this July, wants more material on stones with magical powers, he might consult The Blue Matrix and The Diamond Archetype.
From Log24,
April 28, 2005:
(See also Log24,
April 5, 2005.)
Compare this diagram with that of
Samuel Beckett in Quad (1981):
Barry Mazur on a seminal paper of algebraist Saunders Mac Lane:
The paper was rejected “because the editor thought that it was ‘more devoid of content’ than any other he had read. ‘Saunders wrote back and said, “That’s the point,”‘ Mazur said. ‘And in some ways that’s the genius of it. It’s the barest, most Beckettlike vocabulary that incorporates the theory and nothing else.'”
“M. de Villepin positively worships Napoleon, and models himself after his hero. In a 600page biography, Villepin wrote admiringly about the difference between great men like Napoleon and the ‘common run’ of men. It is worth reading every word carefully.
‘Here we touch on that particular essence of great men, on what distinguishes Napoleon or Alexander, Caesar or de Gaulle, from the common run. It is excess, exaltation, and a taste for risk that forms their genius. It is why they are often better understood in their élan by writers and poets, who are possessed of the same thirst for the absolute, than by those who pray at the altar of facts.’
(New Republic)
And in praise of French nationalism, de Villepin wrote,
‘The Gaullist adventure renewed the élan of [Napoleon’s] Consulate through the restoration of a strong executive and the authority of the State, the same scorn for political parties and for compromise, a common taste for action, and an obsession with the general interest and the grandeur of France.’
Those words come straight from 1800. Napoleon’s ‘genius,’ his ‘thirst for the absolute,’ ‘excess, exaltation, and a taste for risk,’ ‘a strong executive and the authority of the State,’ ‘his ‘scorn for political parties and for compromise,’ and ‘an obsession with the grandeur of France’ — it is all classic national hero worship. But today that kind of thinking is used to promote a new vision of destiny, the European Union.”
Harvard’s Barry Mazur on
one mathematical style:
“It’s the barest, most Beckettlike vocabulary
that incorporates the theory and nothing else.”
Samuel Beckett, Quad (1981):
A Jungian on this sixline logo:
“They are the same six lines
that exist in the I Ching….
Now observe the square more closely:
four of the lines are of equal length,
the other two are longer….
For this reason symmetry
cannot be statically produced
and a dance results.”
— MarieLouise von Franz,
Number and Time (1970),
Northwestern U. Press
paperback, 1979, p. 108
A related logo from
Columbia University’s
Department of Art History
and Archaeology:
Also from that department:
Meyer Schapiro Professor
of Modern Art and Theory:
“There is no painter in the West
who can be unaware of
the symbolic power
of the cruciform shape
and the Pandora’s box
of spiritual reference
that is opened
once one uses it.”
“In the garden of Adding
live Even and Odd…”
— The Midrash Jazz Quartet in
City of God, by E. L. Doctorow
THE GREEK CROSS
Here, for reference, is a Greek cross
within a ninesquare grid:
Related religious meditation for
Doctorow’s “Garden of Adding”…
Types of Greek cross
illustrated in Wikipedia
under “cross“:
THE BAPTISMAL CROSS
Related material:
Fritz Leiber’s “spider”
or “double cross” logo.
See Why Me? and
A Shot at Redemption.
Happy Orthodox Easter.
Harvard's Barry Mazur likes to quote Aristotle's Metaphysics. See 1, 2, 3.
Here, with an introductory remark by Martha Cooley, is more from the Metaphysics:
The central aim of Western religion —
"Each of us has something to offer the Creator... the bridging of masculine and feminine, life and death. It's redemption.... nothing else matters."  Martha Cooley in The Archivist (1998)
The central aim of Western philosophy —
Dualities of Pythagoras as reconstructed by Aristotle: Limited Unlimited Odd Even Male Female Light Dark Straight Curved ... and so on ....
"Of these dualities, the first is the most important; all the others may be seen as different aspects of this fundamental dichotomy. To establish a rational and consistent relationship between the limited [man, etc.] and the unlimited [the cosmos, etc.] is… the central aim of all Western philosophy."
— Jamie James in The Music of the Spheres (1993)
"In the garden of Adding,
Live Even and Odd….
And the song of love's recision
is the music of the spheres."
— The Midrash Jazz Quartet in City of God, by E. L. Doctorow (2000)
Harvard University, Department of English:
• 
Today's birthday: Jerry Seinfeld.
Related material:
Is Nothing Sacred? and Symmetries.
Black Moses
For an explanation of the title, see
the previous entry and
Robert P. Moses and The Algebra Project.
For another algebra project, see
Log24 entries of April 1425 as well as
the following “X in a box” figure
from March 10, 2005 and
April 5, 2005.
Those interested in artistic rather than mathematical figures may compare this diagram with that of Samuel Beckett in Quad (1981):
Barry Mazur on a seminal paper of algebraist Saunders Mac Lane:
The paper was rejected “because the editor thought that it was ‘more devoid of content’ than any other he had read. ‘Saunders wrote back and said, “That’s the point,”‘ Mazur said. ‘And in some ways that’s the genius of it. It’s the barest, most Beckettlike vocabulary that incorporates the theory and nothing else.'”
“There are some ideas you simply could not think without a vocabulary to think them.”
Amen.
Mathematical Style:
Mac Lane Memorial, Part Trois
(See also Part I and Part II.)
“We have seen that there are many diverse styles that lead to success in mathematics. Choose one mathematician… from the ones we studied whose ‘mathematical style’ you find most rewarding for you…. Identify the mathematician and describe his or her mathematical style.”
— Sarah J. Greenwald, takehome exam from Introduction to Mathematics at Appalachian State U., Boone, North Carolina 
From today’s Harvard Crimson:
ExMath Prof Mac Lane, 95, Dies
[Saunders] Mac Lane was most famous for the groundbreaking paper he cowrote with Samuel Eilenberg of Columbia in 1945 which introduced category theory, a framework to show how mathematical structures relate to each other. This branch of algebra has since influenced most mathematical fields and also has functions in philosophy and linguistics, but was first dismissed by many practical mathematicians as too abstract to be useful.
Gade University Professor of Mathematics Barry Mazur, a friend of the late Mac Lane, recalled that the paper had at first been rejected from a lowercaliber mathematical journal because the editor thought that it was “more devoid of content” than any other he had read.
“Saunders wrote back and said, ‘That’s the point,'” Mazur said. “And in some ways that’s the genius of it. It’s the barest, most Beckettlike vocabulary that incorporates the theory and nothing else.”
He likened it to a sparse grammar of nouns and verbs and a limited vocabulary that is presented “in such a deft way that it will help you understand any language you wish to understand and any language will fit into it.”
“In my hour of weakness,that old enemy
tries to steal my soul.
But when he comes
like a flood to surround me
My God will step in
and a standard he’ll raise.”
“Mr. Denker was of the romantic school
of chess – always looking to attack.”
Related material:
From Endgame:
Black the knight upon that ocean,
Bright the sun upon the king.
Dark the queen that stands beside him,
White his castle, threatening.
In the shadows’ see a bishop
Guards his queen of love and hate.
Another move, the game will be up;
Take the queen, her knight will mate.
The knight said “Move, be done. It’s over.”
“Love and resign,” the bishop cried.
“When it’s done you’ll stand forever
By the darkest beauty’s side.”
From Log24.net, Feb. 18, 2003:
Kali, a goddess sometimes depicted
as a dancing girl; Kali is related to kAla,
time, according to one website, as
“the force which governs and stops time.”
See also the novel The Fermata,
by Nicholson Baker.
From an entry of Sunday, Jan. 2,
the day Denker died:
“Time had been canceled.”
— Stephen King, The Shining
From Truth and Style, a tribute
to the late Amy Spindler, style editor
of the New York Times Magazine:
“I don’t believe in truth. I believe in style.”
— Hugh Grant in Vogue magazine, July 1995
From a related page,
The Crimson Passion:
“He takes us to the central activity
of mathematics—which is imagining….”
— Harvard Magazine on
Harvard mathematician
and author Barry Mazur.
For related material on Mazur, see
“The teenagers aren’t all bad.
I love ’em if nobody else does.
There ain’t nothing wrong
with young people.
Jus’ quit lyin’ to ’em.”
At Last, Some Veritas
From the Harvard Crimson, 1/12/04:
College Faces Mental Health Crisis
“An overwhelming majority of Harvard undergraduates struggle with mental health problems, a recent Crimson poll found.”
Related material:
“The people who intermediate between lunatics and the world used to be called alienists; the gobetweens for mathematicians are called teachers. Many a student may rightly have wondered if the terms shouldn’t be reversed.”
— Book review in the current Harvard Magazine; among the authors reviewed is Harvard mathematician and administrator Benedict H. Gross.
“Dean of the College Benedict H. Gross ’71 has said improving mental health services is one of his top priorities in his first year on the job.”
— Harvard Crimson 1/12/04
“He takes us to the central activity of mathematics—which is imagining….”
— Harvard Magazine on Harvard mathematician and author Barry Mazur.
For related material on Mazur, see
“The teenagers aren’t all bad. I love ’em if nobody else does. There ain’t nothing wrong with young people. Jus’ quit lyin’ to ’em.”
The Proof and the Lie
A mathematical lie has been circulating on the Internet.
It concerns the background of Wiles’s recent work on mathematics related to Fermat’s last theorem, which involves the earlier work of a mathematician named Taniyama.
This lie states that at the time of a conjecture by Taniyama in 1955, there was no known relationship between the two areas of mathematics known as “elliptic curves” and “modular forms.”
The lie, due to Harvard mathematician Barry Mazur, was broadcast in a TV program, “The Proof,” in October 1997 and repeated in a book based on the program and in a Scientific American article, “Fermat’s Last Stand,” by Simon Singh and Kenneth Ribet, in November 1997.
“… elliptic curves and modular forms… are from opposite ends of the mathematical spectrum, and had previously been studied in isolation.”
— Site on Simon Singh’s 1997 book Fermat’s Last Theorem
“JOHN CONWAY: What the TaniyamaShimura conjecture says, it says that every rational elliptic curve is modular, and that’s so hard to explain.
BARRY MAZUR: So, let me explain. Over here, you have the elliptic world, the elliptic curves, these doughnuts. And over here, you have the modular world, modular forms with their many, many symmetries. The ShimuraTaniyama conjecture makes a bridge between these two worlds. These worlds live on different planets. It’s a bridge. It’s more than a bridge; it’s really a dictionary, a dictionary where questions, intuitions, insights, theorems in the one world get translated to questions, intuitions in the other world.
KEN RIBET: I think that when Shimura and Taniyama first started talking about the relationship between elliptic curves and modular forms, people were very incredulous….”
— Transcript of NOVA program, “The Proof,” October 1997
The lie spread to other popular accounts, such as the column of Ivars Peterson published by the Mathematical Association of America:
“Elliptic curves and modular forms are mathematically so different that mathematicians initially couldn’t believe that the two are related.”
— Ivars Peterson, “Curving Beyond Fermat,” November 1999
The lie has now contaminated university mathematics courses, as well as popular accounts:
“Elliptic curves and modular forms are completely separate topics in mathematics, and they had never before been studied together.”
— Site on Fermat’s last theorem by undergraduate K. V. Binns
Authors like Singh who wrote about Wiles’s work despite their ignorance of higher mathematics should have consulted the excellent website of Charles Daney on Fermat’s last theorem.
A 1996 page in Daney’s site shows that Mazur, Ribet, Singh, and Peterson were wrong about the history of the known relationships between elliptic curves and modular forms. Singh and Peterson knew no better, but there is no excuse for Mazur and Ribet.
Here is what Daney says:
“Returning to the jinvariant, it is the 1:1 map betweem isomorphism classes of elliptic curves and C*. But by the above it can also be viewed as a 1:1 map j:H/r > C. j is therefore an example of what is called a modular function. We’ll see a lot more of modular functions and the modular group. These facts, which have been known for a long time, are the first hints of the deep relationship between elliptic curves and modular functions.”
“Copyright © 1996 by Charles Daney,
All Rights Reserved.
Last updated: March 28, 1996″
Update of Dec. 2, 2003
For the relationship between modular functions and modular forms, see (for instance) Modular Form in Wikipedia.
Some other relevant quotations:
From J. S. Milne, Modular Functions and Modular Forms:
“The definition of modular form may seem strange, but we have seen that such functions arise naturally in the [nineteenthcentury] theory of elliptic functions.”
The next quote, also in a nineteenthcentury context, relates elliptic functions to elliptic curves.
From Elliptic Functions, a course syllabus:
“Elliptic functions parametrize elliptic curves.”
Putting the quotes together, we have yet another description of the close relationship, well known in the nineteenth century (long before Taniyama’s 1955 conjecture), between elliptic curves and modular forms.
Another quote from Milne, to summarize:
“From this [a discussion of nineteenthcentury mathematics], one sees that arithmetic facts about elliptic curves correspond to arithmetic facts about special values of modular functions and modular forms.”
Serge Lang apparently agrees:
“Elliptic functions parametrize elliptic curves, and the intermingling of the analytic and algebraicarithmetic theory has been at the center of mathematics since the early part of the nineteenth century.”
— Editorial description of Lang’s Elliptic Functions (second edition, 1987)
Update of Dec. 3, 2003
“The theory of modular functions and modular forms, defined on the upper halfplane H and subject to appropriate tranformation laws with respect to the group Gamma = SL(2, Z) of fractional linear transformations, is closely related to the theory of elliptic curves, because the family of all isomorphism classes of elliptic curves over C can be parametrized by the quotient Gamma\H. This is an important, although formal, relation that assures that this and related quotients have a natural structure as algebraic curves X over Q. The relation between these curves and elliptic curves predicted by the TaniyamaWeil conjecture is, on the other hand, far from formal.”
— Robert P. Langlands, review of Elliptic Curves, by Anthony W. Knapp. (The review appeared in Bulletin of the American Mathematical Society, January 1994.)
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