Log24

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" + "Two-Faced.")

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…"

The video —

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—

(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

See also today's previous post.

 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 two-volume 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 two-volume 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.
 

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.

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 14-25 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):

Related quotations:

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 Beckett-like vocabulary that incorporates the theory and nothing else.'”

J. Peter May, a professor of mathematics at the University of Chicago quoted in the Chicago Tribune:

“There are some ideas you simply could not think without a vocabulary to think them.”

Amen.

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 go-betweens 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

A Mathematical Lie.

“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.”

— Jackie “Moms” Mabley

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 Taniyama-Shimura 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 Shimura-Taniyama 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 j-invariant, 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 [nineteenth-century] theory of elliptic functions.”

The next quote, also in a nineteenth-century 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 nineteenth-century 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 algebraic-arithmetic 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 half-plane 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 Taniyama-Weil 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.)