From the web journal of Steven H. Cullinane

Monday, September 1, 2003  3:33 PM

The Unity of Mathematics,

or "Shema, Israel"

A conference to honor the 90th birthday (Sept. 2) of Israel Gelfand is currently underway in Cambridge, Massachusetts.

The following note from 2001 gives one view of the conference's title topic, "The Unity of Mathematics."

Reciprocity in 2001

by Steven H. Cullinane
(May 30, 2001)

From 2001: A Space Odyssey, by Arthur C. Clarke, New American Library, 1968:

The glimmering rectangular shape that had once seemed no more than a slab of crystal still floated before him....  It encapsulated yet unfathomed secrets of space and time, but some at least he now understood and was able to command.

How obvious -- how necessary -- was that mathematical ratio of its sides, the quadratic sequence 1: 4: 9!  And how naive to have imagined that the series ended at this point, in only three dimensions!

-- Chapter 46, "Transformation"

From a review of Himmelfarb, by Michael Krüger, New York, George Braziller, 1994:

As a diffident, unsure young man, an inexperienced ethnologist, Richard was unable to travel through the Amazonian jungles unaided. His professor at Leipzig, a Nazi Party member (a bigot and a fool), suggested he recruit an experienced guide and companion, but warned him against collaborating with any Communists or Jews, since the objectivity of research would inevitably be tainted by such contact. Unfortunately, the only potential associate Richard can find in Sao Paulo is a man called Leo Himmelfarb, both a Communist (who fought in the Spanish Civil War) and a self-exiled Jew from Galicia, but someone who knows the forests intimately and can speak several of the native dialects.

"... Leo followed the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity, which I could not even imitate."

... E. M. Forster famously advised his readers, "Only connect." "Reciprocity" would be Michael Kruger's succinct philosophy, with all that the word implies.

-- William Boyd, New York Times Book Review, October 30, 1994

Reciprocity and Euler

Applying the above philosophy of reciprocity to the Arthur C. Clarke sequence

1, 4, 9, ....

we obtain the rather more interesting sequence

1/1, 1/4, 1/9, .....

This leads to the following problem (adapted from the St. Andrews biography of Euler):

Perhaps the result that brought Euler the most fame in his young days was his solution of what had become known as the Basel problem. This was to find a closed form for the sum of the infinite series

1/1 + 1/4 + 1/9 + 1/16 + 1/25 + ...

-- a problem which had defeated many of the top mathematicians including Jacob Bernoulli, Johann Bernoulli and Daniel Bernoulli. The problem had also been studied unsuccessfully by Leibniz, Stirling, de Moivre and others. Euler showed in 1735 that the series sums to (pi squared)/6. He generalized this series, now called zeta(2), to zeta functions of even numbers larger than two.

Related Reading

For four different proofs of Euler's result, see the inexpensive paperback classic by Konrad Knopp, Theory and Application of Infinite Series (Dover Publications).

Related Websites

Evaluating Zeta(2), by Robin Chapman (PDF article) Fourteen proofs!

Zeta Functions for Undergraduates

The Riemann Zeta Function

Reciprocity Laws
Reciprocity Laws II

The Langlands Program

Recent Progress on the Langlands Conjectures

For more on
the theme of unity,
see

Monolithic Form
and
ART WARS.

Tuesday, September 2, 2003  1:11 PM

One Ring to Rule Them All

In memory of J. R. R. Tolkien, who died on this date, and in honor of Israel Gelfand, who was born on this date.

Leonard Gillman on his collaboration with Meyer Jerison and Melvin Henriksen in studying rings of continuous functions:

"The triple papers that Mel and I wrote deserve comment. Jerry had conjectured a characterization of beta X (the Stone-Cech compactification of X) and the three of us had proved that it was true. Then he dug up a 1939 paper by Gelfand and Kolmogoroff that Hewitt, in his big paper, had referred to but apparently not appreciated, and there we found Jerry's characterization. The three of us sat around to decide what to do; we called it the 'wake.'  Since the authors had not furnished a proof, we decided to publish ours. When the referee expressed himself strongly that a title should be informative, we came up with On a theorem of Gelfand and Kolmogoroff concerning maximal ideals in rings of continuous functions. (This proved to be my second-longest title, and a nuisance to refer to.) Kolmogoroff died many years ago, but Gelfand is still living, a vigorous octogenarian now at Rutgers. A year or so ago, I met him at a dinner party in Austin and mentioned the 1939 paper. He remembered it very well and proceeded to complain that the only contribution Kolmogoroff had made was to point out that a certain result was valid for the complex case as well. I was intrigued to see how the giants grouse about each other just as we do."

-- Leonard Gillman: An Interview

This clears up a question I asked earlier in this journal....

Wednesday, May 14, 2003

Common Sense

On the mathematician Kolmogorov:

"It turns out that he DID prove one basic theorem that I take for granted, that a compact hausdorff space is determined by its ring of continuous functions (this ring being considered without any topology) -- basic discoveries like this are the ones most likely to have their origins obscured, for they eventually come to be seen as mere common sense, and not even a theorem."

-- Richard Cudney, Harvard '03, writing at Xanga.com as rcudney on May 14, 2003

That this theorem is Kolmogorov's is news to me.

See

The above references establish that Gelfand is usually cited as the source of the theorem Cudney discusses.  Gelfand was a student of Kolmogorov's in the 1930's, so who discovered what when may be a touchy question in this case.  A reference that seems relevant: I. M. Gelfand and A. Kolmogoroff, "On rings of continuous functions on topological spaces," Doklady Akad. Nauk SSSR 22 (1939), 11-15.  This is cited by Gillman and Jerison in the classic Rings of Continuous Functions.

There ARE some references that indicate Kolmogorov may have done some work of his own in this area.  See here ("quite a few duality theorems... including those of Banaschewski, Morita, Gel'fand-Kolmogorov and Gel'fand-Naimark") and here  ("the classical theorems of M. H. Stone, Gelfand & Kolmogorov").

Any other references to Kolmogorov's work in this area would be of interest.

Naturally, any discussion of this area should include a reference to the pioneering work of M. H. Stone.  I recommend the autobiographical article on Stone in McGraw-Hill Modern Men of Science, Volume II, 1968.

A response by Richard Cudney:

"In regard to your entry, it is largely correct.  The paper by Kolmogorov and Gelfand that you refer to is the one that I just read in his collected works.  So, I suppose my entry was unfair to Gelfand.  You're right, the issue of credit is a bit touchy since Gelfand was his student.  In a somewhat recent essay, Arnol'd makes the claim that this whole thread of early work by Gelfand may have been properly due to Kolmogorov, however he has no concrete proof, having been but a child at the time, and makes this inference based only on his own later experience as Kolmogorov's student.  At any rate, I had known about Gelfand's representation theorem, but had not known that Kolmogorov had done any work of this sort, or that this theorem in particular was due to either of them. 

And to clarify-where I speak of the credit for this theorem being obscured, I speak of my own experience as an algebraic geometer and not a functional analyst.  In the textbooks on algebraic geometry, one sees no explanation of why we use Spec A to denote the scheme corresponding to a ring A.  That question was answered when I took functional analysis and learned about Gelfand's theorem, but even there, Kolmogorov's name did not come up.

This result is different from the Gelfand representation theorem that you mention-this result concerns algebras considered without any topology(or norm)-whereas his representation theorem is a result on Banach algebras.  In historical terms, this result precedes Gelfand's theorem and is the foundation for it-he starts with a general commutative Banach algebra and reconstructs a space from it-thus establishing in what sense that the space to algebra correspondence is surjective, and hence by the aforementioned theorem, bi-unique.  That is to say, this whole vein of Gelfand's work started in this joint paper.

Of course, to be even more fair, I should say that Stone was the very first to prove a theorem like this, a debt which Kolmogorov and Gelfand acknowledge.  Stone's paper is the true starting point of these ideas, but this paper of Kolmogorov and Gelfand is the second landmark on the path that led to Grothendieck's concept of a scheme(with Gelfand's representation theorem probably as the third).

As an aside, this paper was not Kolmogorov's first foray into topological algebra-earlier he conjectured the possibility of a classification of locally compact fields, a problem which was solved by Pontryagin.  The point of all this is that I had been making use of ideas due to Kolmogorov for many years without having had any inkling of it."

Posted 5/14/2003 at 8:44 PM by rcudney


Wednesday, September 3, 2003  3:00 PM

Reciprocity

From my entry of Sept. 1, 2003:

"...the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity....

... E. M. Forster famously advised his readers, 'Only connect.' 'Reciprocity' would be Michael Kruger's succinct philosophy, with all that the word implies."

-- William Boyd, review of Himmelfarb, New York Times Book Review, October 30, 1994

Last year's entry on this date: 

Today's birthday:
James Joseph Sylvester

"Mathematics is the music of reason."
-- J. J. Sylvester

Sylvester, a nineteenth-century mathematician, coined the phrase "synthematic totals" to describe some structures based on 6-element sets that R. T. Curtis has called "rather unwieldy objects." See Curtis's abstract, Symmetric Generation of Finite Groups, John Baez's essay, Some Thoughts on the Number 6, and my website, Diamond Theory.

The picture above is of the complete graph K6 ...  Six points with an edge connecting every pair of points... Fifteen edges in all.

Diamond theory describes how the 15 two-element subsets of a six-element set (represented by edges in the picture above) may be arranged as 15 of the 16 parts of a 4x4 array, and how such an array relates to group-theoretic concepts, including Sylvester's synthematic totals as they relate to constructions of the Mathieu group M24.

If diamond theory illustrates any general philosophical principle, it is probably the interplay of opposites....  "Reciprocity" in the sense of Lao Tzu.  See

Reciprocity and Reversal in Lao Tzu.

For a sense of "reciprocity" more closely related to Michael Kruger's alleged philosophy, see the Confucian concept of Shu (Analects 15:23 or 24) described in

Shu: Reciprocity.

Kruger's novel is in part about a Jew: the quintessential Jewish symbol, the star of David, embedded in the K6 graph above, expresses the reciprocity of male and female, as my May 2003 archives illustrate.  The star of David also appears as part of a graphic design for cubes that illustrate the concepts of diamond theory:

Click on the design for details.

Those who prefer a Jewish approach to physics can find the star of David, in the form of K6, applied to the sixteen 4x4 Dirac matrices, in

A Graphical Representation
of the Dirac Algebra.

The star of David also appears, if only as a heuristic arrangement, in a note that shows generating partitions of the affine group on 64 points arranged in two opposing triplets.

Having thus, as the New York Times advises, paid tribute to a Jewish symbol, we may note, in closing, a much more sophisticated and subtle concept of reciprocity due to Euler, Legendre, and Gauss.  See

The Jewel of Arithmetic and

The Golden Theorem.

Thursday, September 4, 2003  2:42 AM


Monolith

"Music can name the unnameable
and communicate the unknowable."

— Quotation attributed to Leonard Bernstein

"Finally we get to Kubrick's ultimate trick....  His secret is in plain sight.... The film is the monolith. In a secret that seems to never have been seen by anyone: the monolith in the film has the same exact dimensions as the movie screen on which 2001 was projected."

—  Alchemical Kubrick 2001, by Jay Weidner 

My entry of Saturday, August 30,
included the following illustration:

My entry of Monday, September 1,
concluded with the black monolith.

"There is little doubt that the black monolith
in 2001 is the Philosopher's Stone."

—  Alchemical Kubrick 2001, by Jay Weidner 

 The philosopher Donald Davidson
died on Saturday, August 30.

The New York Times says that as an undergraduate, Davidson "persuaded Harvard to let him put on 'The Birds' by Aristophanes and played the lead, Peisthetairos, which meant memorizing 700 lines of Greek. His friend and classmate Leonard Bernstein, with whom he played four-handed piano, wrote an original score for the production."

Perhaps they are still making music together.

Thursday, September 4, 2003  4:23 PM

Epitaphs

The late philosopher Donald Davidson (see previous entry) had a gift for titles. For example:

"The Folly of Trying to Define Truth"
(Journal of Philosophy June 1996, pp. 263-278)  and

"A Nice Derangement of Epitaphs"
(In R. Grandy and R. Warner (eds.), Philosophical Grounds of Rationality, Oxford: Oxford University Press, 1986).

For my thoughts on the former, see 

Pilate, Truth, and Friday the Thirteenth,

The Diamond Theory of Truth, and

Sept. 2, 2002 (Laurindo Almeida's Birthday).

For my thoughts on the latter, see

Happy Birthday, Mary Shelley (2003),

For Mary Shelley's Birthday (2002),

and, in honor of J. R. R. Tolkien, who died on the date September 2,

The Article on Epitaphs

at Wikipedia Encyclopedia, which contains the following:

"J. R. R. Tolkien is buried next to his wife, and on their tombstone the names 'Beren' and 'Luthien' are engraved, a fact that sheds light on the love story of Beren and Luthien which is recorded in several versions in his works."  

A nice derangement, indeed.