Log24

Friday, March 23, 2018

Reciprocity

Filed under: General,Geometry — Tags: — m759 @ 7:00 PM

Copy editing — From Wikipedia

"Copy editing (also copy-editing or copyediting, sometimes abbreviated ce)
is the process of reviewing and correcting written material to improve accuracy,
readability, and fitness for its purpose, and to ensure that it is free of error,
omission, inconsistency, and repetition. . . ."

An example of the need for copy editing:

Related material:  Langlands and Reciprocity in this  journal.

Tuesday, February 18, 2014

Eichler’s Reciprocity Law

Filed under: General,Geometry — m759 @ 11:00 AM

Edward Frenkel on Eichler's reciprocity law
(Love and Math , Kindle edition of 2013-10-01,
page 88, location 1812)—

"It seems nearly unbelievable that there
would be a rule generating these numbers.
And yet, German mathematician Martin
Eichler discovered one in 1954.11 "

"11.   I follow the presentation of this result
given in Richard Taylor, Modular arithmetic:
driven by inherent beauty and human
curiosity 
, The Letter of the Institute for
Advanced Study [IAS], Summer 2012,
pp. 6– 8. I thank Ken Ribet for useful
comments. According to André Weil’s book 
Dirichlet Series and Automorphic Forms ,
Springer-Verlag, 1971 [pp. 143-144], the
cubic equation we are discussing in this
chapter was introduced by John Tate,
following Robert Fricke."

Update of Feb. 19: 

Actually, the cubic equation discussed
by Frenkel and by Taylor (see below) is 

2 + Y = X 3 – X 

whereas the equation given by Weil,
quoting Tate, is

2 – Y = X 3 – X 

Whether this is a misprint in Weil's book,
I do not know.

At any rate, the cubic equation discussed by
Frenkel and earlier by Taylor is the same as
the cubic equation discussed in greater detail
by Henri Darmon in "A Proof of the Full
Shimura-Taniyama-Weil Conjecture Is
Announced
," AMS Notices , Dec.1999.

For further background, see (for instance)
John T. Tate, "The Arithmetic of Elliptic
Curves," in Inventiones Mathematicae
Volume 23 (1974), pp. 179 – 206, esp. pp.
200-201.

Richard Taylor, op. cit. 

One could ask for a similar method that given any number of polynomials in any number of variables helps one to determine the number of solutions to those equations in arithmetic modulo a variable prime number . Such results are referred to as “reciprocity laws.” In the 1920s, Emil Artin gave what was then thought to be the most general reciprocity law possible—his abelian reciprocity law. However, Artin’s reciprocity still only applied to very special equations—equations with only one variable that have “abelian Galois group.”

Stunningly, in 1954, Martin Eichler (former IAS Member) found a totally new reciprocity law, not included in Artin’s theorem. (Such reciprocity laws are often referred to as non-abelian.) More specifically, he found a reciprocality [sic ] law for the two variable equation

2 + Y = X 3 – X 2.

He showed that the number of solutions to this equation in arithmetic modulo a prime number differs from p  [in the negative direction] by the coefficient of qp in the formal (infinite) product

(1 – q 2 )(1 – q 11) 2 (1 – q 2)2
(1 – q 22 )2 (1 – q 3)2 (1 – 33)2
(1 – 4)2 …  =  
q – 2q2q3 + 2q+ q5 + 2q6
– 2q7 – 2q9 – 2q10 ​+ q11 – 2q12 + . . .

For example, you see that the coefficient of q5 is 1, so Eichler’s theorem tells us that

Y 2 + Y = X 3 − X 2

should have 5 − 1 = 4 solutions in arithmetic modulo 5. You can check this by checking the twenty-five possibilities for (X,Y) modulo 5, and indeed you will find exactly four solutions:

(X,Y) ≡ (0,0), (0,4), (1,0), (1,4) mod 5.

Within less than three years, Yutaka Taniyama and Goro Shimura (former IAS Member) proposed a daring generalization of Eichler’s reciprocity law to all cubic equations in two variables. A decade later, André Weil (former IAS Professor) added precision to this conjecture, and found strong heuristic evidence supporting the Shimura-Taniyama reciprocity law. This conjecture completely changed the development of number theory.

With this account and its context, Taylor has
perhaps atoned for his ridiculous remarks
quoted at Log24 in The Proof and the Lie.

Monday, May 6, 2019

In Memoriam Goro Shimura (d. May 3, 2019)

Filed under: General — Tags: , — m759 @ 3:33 PM

From Richard Taylor, "Modular arithmetic:  driven by inherent beauty
and human curiosity
," The Letter of the Institute for Advanced Study  [IAS],
Summer 2012, pp. 6– 8 (links added) :

"Stunningly, in 1954, Martin Eichler (former IAS Member)
found a totally new reciprocity law . . . .

Within less than three years, Yutaka Taniyama and Goro Shimura
(former IAS Member) proposed a daring generalization of Eichler’s
reciprocity law to all cubic equations in two variables. A decade later,
André Weil (former IAS Professor) added precision to this conjecture,
and found strong heuristic evidence supporting the Shimura-Taniyama
reciprocity law. This conjecture completely changed the development of
number theory."

Wednesday, April 3, 2019

Review

Filed under: General — m759 @ 1:44 PM

A later article about this same William Boyd

"In the end, it’s this indifference on the part of the tastemakers
that makes Boyd’s project a worthy one, pointing as it does to
their ability to treat as real whatever they choose, and to deny
the reality of other things simply by redirecting their gaze." 

Also on November 14, 2011 —

Wednesday, April 1, 2015

Math’s Big Lies

Filed under: General,Geometry — Tags: , — m759 @ 11:00 AM

Two mathematicians, Barry Mazur and Edward Frenkel,
have, for rhetorical effect, badly misrepresented the
history of some basic fields of mathematics. Mazur and
Frenkel like to emphasize the importance of new 
research by claiming that it connects fields that previously
had no known connection— when, in fact, the fields were
known to be connected since at least the nineteenth century.

For Mazur, see The Proof and the Lie; for Frenkel, see posts
tagged Frenkel-Metaphors.

See also a story and video on Robert Langlands from the
Toronto Star  on March 27, 2015:

"His conjectures are called functoriality and
reciprocity. They made it possible to link up
three branches of math: harmonic analysis,
number theory, and geometry. 

To mathematicians, this is mind-blowing stuff
because these branches have nothing to do
with each other."

For a much earlier link between these three fields, see the essay
"Why Pi Matters" published in The New Yorker  last month.

Thursday, September 11, 2014

A Class by Itself

Filed under: General — Tags: — m759 @ 9:48 AM

The American Mathematical Society yesterday:

Harvey Cohn (1923-2014)
Wednesday September 10th 2014

Cohn, an AMS Fellow and a Putnam Fellow (1942), died May 16 at the age of 90. He served in the Navy in World War II and following the war received his PhD from Harvard University in 1948 under the direction of Lars Ahlfors. He was a member of the faculty at Wayne State University, Stanford University, Washington University in St. Louis, the University of Arizona, and at City College of New York, where he was a distinguished professor. After retiring from teaching, he also worked for the NSA. Cohn was an AMS member since 1942.

Paid death notice from The New York Times , July 27, 2014:

COHN–Harvey. Fellow of the American Mathematical Society and member of the Society since 1942, died on May 16 at the age of 90. He was a brilliant Mathematician, an adoring husband, father and grandfather, and faithful friend and mentor to his colleagues and students. Born in New York City in 1923, Cohn received his B.S. degree (Mathematics and Physics) from CCNY in 1942. He received his M.S. degree from NYU (1943), and his Ph.D. from Harvard (1948) after service in the Navy (Electronic Technicians Mate, 1944-46). He was a member of Phi Beta Kappa (Sigma Chi), won the William Lowell Putnam Prize in 1942, and was awarded the Townsend Harris Medal in 1972. A pioneer in the intensive use of computers in an innovative way in a large number of classical mathematical problems, Harvey Cohn held faculty positions at Wayne State University, Stanford, Washington University Saint Louis (first Director of the Computing Center 1956-58), University of Arizona (Chairman 1958-1967), University of Copenhagen, and CCNY (Distinguished Professor of Mathematics). After his retirement from teaching, he worked in a variety of capacities for the National Security Agency and its research arm, IDA Center for Computing Sciences. He is survived by his wife of 63 years, Bernice, of Laguna Woods, California and Ft. Lauderdale, FL, his son Anthony, daughter Susan Cohn Boros, three grandchildren and one great-granddaughter.

— Published in The New York Times  on July 27, 2014

See also an autobiographical essay found on the web.

None of the above sources mention the following book, which is apparently by this same Harvey Cohn. (It is dedicated to "Tony and Susan.")

From Google Books:

Advanced Number Theory, by Harvey Cohn
Courier Dover Publications, 1980 – 276 pages
(First published by Wiley in 1962 as A Second Course in Number Theory )

Publisher's description:

" 'A very stimulating book … in a class by itself.'— American Mathematical Monthly

Advanced students, mathematicians and number theorists will welcome this stimulating treatment of advanced number theory, which approaches the complex topic of algebraic number theory from a historical standpoint, taking pains to show the reader how concepts, definitions and theories have evolved during the last two centuries. Moreover, the book abounds with numerical examples and more concrete, specific theorems than are found in most contemporary treatments of the subject.

The book is divided into three parts. Part I is concerned with background material — a synopsis of elementary number theory (including quadratic congruences and the Jacobi symbol), characters of residue class groups via the structure theorem for finite abelian groups, first notions of integral domains, modules and lattices, and such basis theorems as Kronecker's Basis Theorem for Abelian Groups.

Part II discusses ideal theory in quadratic fields, with chapters on unique factorization and units, unique factorization into ideals, norms and ideal classes (in particular, Minkowski's theorem), and class structure in quadratic fields. Applications of this material are made in Part III to class number formulas and primes in arithmetic progression, quadratic reciprocity in the rational domain and the relationship between quadratic forms and ideals, including the theory of composition, orders and genera. In a final concluding survey of more recent developments, Dr. Cohn takes up Cyclotomic Fields and Gaussian Sums, Class Fields and Global and Local Viewpoints.

In addition to numerous helpful diagrams and tables throughout the text, appendices, and an annotated bibliography, Advanced Number Theory  also includes over 200 problems specially designed to stimulate the spirit of experimentation which has traditionally ruled number theory."

User Review –

"In a nutshell, the book serves as an introduction to Gauss' theory of quadratic forms and their composition laws (the cornerstone of his Disquisitiones Arithmeticae) from the modern point of view (ideals in quadratic number fields). I strongly recommend it as a gentle introduction to algebraic number theory (with exclusive emphasis on quadratic number fields and binary quadratic forms). As a bonus, the book includes material on Dirichlet L-functions as well as proofs of Dirichlet's class number formula and Dirichlet's theorem in primes in arithmetic progressions (of course this material requires the reader to have the background of a one-semester course in real analysis; on the other hand, this material is largely independent of the subsequent algebraic developments).

Better titles for this book would be 'A Second Course in Number Theory' or 'Introduction to quadratic forms and quadratic fields'. It is not a very advanced book in the sense that required background is only a one-semester course in number theory. It does not assume prior familiarity with abstract algebra. While exercises are included, they are not particularly interesting or challenging (if probably adequate to keep the reader engaged).

While the exposition is *slightly* dated, it feels fresh enough and is particularly suitable for self-study (I'd be less likely to recommend the book as a formal textbook). Students with a background in abstract algebra might find the pace a bit slow, with a bit too much time spent on algebraic preliminaries (the entire Part I—about 90 pages); however, these preliminaries are essential to paving the road towards Parts II (ideal theory in quadratic fields) and III (applications of ideal theory).

It is almost inevitable to compare this book to Borevich-Shafarevich 'Number Theory'. The latter is a fantastic book which covers a large superset of the material in Cohn's book. Borevich-Shafarevich is, however, a much more demanding read and it is out of print. For gentle self-study (and perhaps as a preparation to later read Borevich-Shafarevich), Cohn's book is a fine read."

Tuesday, November 26, 2013

Edward Frenkel, Your Order Is Ready.

Filed under: General — Tags: — m759 @ 11:00 AM

Backstory: Frenkel's Metaphors and Waitressing for Godot.

In a recent vulgarized presentation of the Langlands program,
Edward Frenkel implied that number theory and harmonic
analysis were, before Langlands came along, quite unrelated.

This is false.

"If we think of different fields of mathematics as continents,
then number theory would be like North America and
harmonic analysis like Europe." 

Edward Frenkel, Love and Math , 2013

For a discussion of pre-Langlands connections between 
these "continents," see

Ding!

"Fourier Analysis in Number Theory, my senior thesis, under the advisory of Patrick Gallagher.

This thesis contains no original research, but is instead a compilation of results from analytic
number theory that involve Fourier analysis. These include quadratic reciprocity (one of 200+
published proofs), Dirichlet's theorem on primes in arithmetic progression, and Weyl's criterion.
There is also a function field analogue of Fermat's Last Theorem. The presentation of the
material is completely self-contained."

Shanshan Ding, University of Pennsylvania graduate student

Saturday, November 23, 2013

Light Years Apart?

Filed under: General,Geometry — Tags: — m759 @ 9:00 PM

From a recent attempt to vulgarize the Langlands program:

"Galois’ work is a great example of the power of a mathematical insight…. 

And then, 150 years later, Langlands took these ideas much farther. In 1967, he came up with revolutionary insights tying together the theory of Galois groups and another area of mathematics called harmonic analysis. These two areas, which seem light years apart, turned out to be closely related."

— Frenkel, Edward (2013-10-01).
     Love and Math: The Heart of Hidden Reality
     (p. 78, Basic Books, Kindle Edition) 

(Links to related Wikipedia articles have been added.)

 

Wikipedia on the Langlands program

The starting point of the program may be seen as Emil Artin's reciprocity law [1924-1930], which generalizes quadratic reciprocity. The Artin reciprocity law applies to a Galois extension of algebraic number fields whose Galois group is abelian, assigns L-functions to the one-dimensional representations of this Galois group; and states that these L-functions are identical to certain Dirichlet L-series or more general series (that is, certain analogues of the Riemann zeta function) constructed from Hecke characters. The precise correspondence between these different kinds of L-functions constitutes Artin's reciprocity law.

For non-abelian Galois groups and higher-dimensional representations of them, one can still define L-functions in a natural way: Artin L-functions.

The insight of Langlands was to find the proper generalization of Dirichlet L-functions, which would allow the formulation of Artin's statement in this more general setting.

 

From "An Elementary Introduction to the Langlands Program," by Stephen Gelbart (Bulletin of the American Mathematical Society, New Series , Vol. 10, No. 2, April 1984, pp. 177-219)

On page 194:

"The use of group representations in systematizing and resolving diverse mathematical problems is certainly not new, and the subject has been ably surveyed in several recent articles, notably [ Gross and Mackey ]. The reader is strongly urged to consult these articles, especially for their reformulation of harmonic analysis as a chapter in the theory of group representations.

In harmonic analysis, as well as in the theory of automorphic forms, the fundamental example of a (unitary) representation is the so-called 'right regular' representation of G….

Our interest here is in the role representation theory has played in the theory of automorphic forms.* We focus on two separate developments, both of which are eventually synthesized in the Langlands program, and both of which derive from the original contributions of Hecke already described."

Gross ]  K. I. Gross, On the evolution of non-commutative harmonic analysis . Amer. Math. Monthly 85 (1978), 525-548.

Mackey ]  G. Mackey, Harmonic analysis as the exploitation of symmetry—a historical survey . Bull. Amer. Math. Soc. (N.S.) 3 (1980), 543-698.

* A link to a related Math Overflow article has been added.

In 2011, Frenkel published a commentary in the A.M.S. Bulletin  
on Gelbart's Langlands article. The commentary, written for
a mathematically sophisticated audience, lacks the bold
(and misleading) "light years apart" rhetoric from his new book 
quoted above.

In the year the Gelbart article was published, Frenkel was
a senior in high school. The year was 1984.

For some remarks of my own that mention
that year, see a search for 1984 in this journal.

Sunday, August 3, 2008

Sunday August 3, 2008

Filed under: General,Geometry — Tags: , — m759 @ 3:00 PM
Kindergarten
Geometry

Preview of a Tom Stoppard play presented at Town Hall in Manhattan on March 14, 2008 (Pi Day and Einstein's birthday):

The play's title, "Every Good Boy Deserves Favour," is a mnemonic for the notes of the treble clef EGBDF.

The place, Town Hall, West 43rd Street. The time, 8 p.m., Friday, March 14. One single performance only, to the tinkle– or the clang?– of a triangle. Echoing perhaps the clang-clack of Warsaw Pact tanks muscling into Prague in August 1968.

The “u” in favour is the British way, the Stoppard way, "EGBDF" being "a Play for Actors and Orchestra" by Tom Stoppard (words) and André Previn (music).

And what a play!– as luminescent as always where Stoppard is concerned. The music component of the one-nighter at Town Hall– a showcase for the Boston University College of Fine Arts– is by a 47-piece live orchestra, the significant instrument being, well, a triangle.

When, in 1974, André Previn, then principal conductor of the London Symphony, invited Stoppard "to write something which had the need of a live full-time orchestra onstage," the 36-year-old playwright jumped at the chance.

One hitch: Stoppard at the time knew "very little about 'serious' music… My qualifications for writing about an orchestra," he says in his introduction to the 1978 Grove Press edition of "EGBDF," "amounted to a spell as a triangle player in a kindergarten percussion band."

Jerry Tallmer in The Villager, March 12-18, 2008

Review of the same play as presented at Chautauqua Institution on July 24, 2008:

"Stoppard's modus operandi– to teasingly introduce numerous clever tidbits designed to challenge the audience."

Jane Vranish, Pittsburgh Post-Gazette, Saturday, August 2, 2008

"The leader of the band is tired
And his eyes are growing old
But his blood runs through
My instrument
And his song is in my soul."

— Dan Fogelberg

"He's watching us all the time."

Lucia Joyce

 

Finnegans Wake,
Book II, Episode 2, pp. 296-297:

I'll make you to see figuratleavely the whome of your eternal geomater. And if you flung her headdress on her from under her highlows you'd wheeze whyse Salmonson set his seel on a hexengown.1 Hissss!, Arrah, go on! Fin for fun!

1 The chape of Doña Speranza of the Nacion.

 

Log 24, Sept. 3, 2003:
 
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, a novel by Michael Kruger, in The 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 4×4 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 4×4 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


FinnegansWiki:

Salmonson set his seel:

"Finn MacCool ate the Salmon of Knowledge."

Wikipedia:

"George Salmon spent his boyhood in Cork City, Ireland. His father was a linen merchant. He graduated from Trinity College Dublin at the age of 19 with exceptionally high honours in mathematics. In 1841 at age 21 he was appointed to a position in the mathematics department at Trinity College Dublin. In 1845 he was appointed concurrently to a position in the theology department at Trinity College Dublin, having been confirmed in that year as an Anglican priest."

Related material:

Kindergarten Theology,

Kindergarten Relativity,

Arrangements for
56 Triangles
.

For more on the
arrangement of
triangles discussed
in Finnegans Wake,
see Log24 on Pi Day,
March 14, 2008.

Happy birthday,
Martin Sheen.
 

Thursday, August 9, 2007

Thursday August 9, 2007

Filed under: General — m759 @ 12:00 PM
“Serious numbers  
will always be heard.”

— Paul Simon

(See St. Luke’s Day, 2005.)  


Bulletin of the American Mathematical Society
,
Volume 31, Number 1, July 1994, Pages 1-14

Selberg’s Conjectures
and Artin L-Functions
(pdf)

M. Ram Murty

Introduction

In its comprehensive form, an identity between an automorphic L-function and a “motivic” L-function is called a reciprocity law. The celebrated Artin reciprocity law is perhaps the fundamental example. The conjecture of Shimura-Taniyama that every elliptic curve over Q is “modular” is certainly the most intriguing reciprocity conjecture of our time. The “Himalayan peaks” that hold the secrets of these nonabelian reciprocity laws challenge humanity, and, with the visionary Langlands program, we have mapped out before us one means of ascent to those lofty peaks. The recent work of Wiles suggests that an important case (the semistable case) of the Shimura-Taniyama conjecture is on the horizon and perhaps this is another means of ascent. In either case, a long journey is predicted…. At the 1989 Amalfi meeting, Selberg [S] announced a series of conjectures which looks like another approach to the summit. Alas, neither path seems the easier climb….

[S] A. Selberg, Old and new
      conjectures and results
      about a class of Dirichlet series,
      Collected Papers, Volume II,
      Springer-Verlag, 1991, pp. 47-63.

Zentralblatt MATH Database
on the above Selberg paper:

“These are notes of lectures presented at the Amalfi Conference on Number Theory, 1989…. There are various stimulating conjectures (which are related to several other conjectures like the Sato-Tate conjecture, Langlands conjectures, Riemann conjecture…)…. Concluding remark of the author: ‘A more complete account with proofs is under preparation and will in time appear elsewhere.'”

Related material: Previous entry.

Sunday, May 8, 2005

Sunday May 8, 2005

Filed under: General,Geometry — m759 @ 12:00 AM

Geometry and Theology

See

the science fiction writer mentioned in a Friday entry.

Mark Olson’s article is at the website of the New England Science Fiction Association, publisher of Ingathering: The Complete People Stories of Zenna Henderson.  This book, by one of my favorite science-fiction authors, was apparently edited by the same Mark Olson.

The following remarks seem relevant to the recurring telepathy theme in Henderson:

From the first article cited above,
David L. Neuhouser,
Higher Dimensions in the Writings of C. S. Lewis (pdf):

“If we are three-dimensional cross-sections of four-dimensional reality, perhaps we are parts of the same body. In fact, we know we are parts of the same body in some way, this four-dimensional idea just may help us to see it more clearly. Remember the preceding comments are mine, not Lewis’s. He puts it this way, ‘That we can die “in” Adam and live “in” Christ seems to me to imply that man as he really is differs a good deal from man as our categories of thought and our three-dimensional imaginations represent him; that the separateness… which we discern between individuals, is balanced, in absolute reality, by some kind of inter-inanimation of which we have no conception at all. It may be that the acts and sufferings of great archetypal individuals such as Adam and Christ are ours, not by legal fiction, metaphor, or causality, but in some much deeper fashion. There is no question, of course, of individuals melting down into a kind of spiritual continuum such as Pantheistic systems believe in; that is excluded by the whole tenor of our faith.'”

From Webster’s Unabridged, 1913 edition:

inanimate
, v. t.

[Pref. in- in (or intensively) + animate.]
 To animate. [Obs.] — Donne.

inanimation, n.

Infusion of life or vigor;
animation; inspiration.
[Obs.]
The inanimation of Christ
living and breathing within us.
Bp. Hall.

Related words…

Also from the 1913 Webster’s:

circumincession, n.

[Pref. circum- + L. incedere, incessum, to walk.]
(Theol.) The reciprocal existence in each other
of the three persons of the Trinity.

From an online essay:

perichoresis
, n.

“The term means mutual indwelling or, better, mutual interpenetration and refers to the understanding of both the Trinity and Christology. In the divine perichoresis, each person has ‘being in each other without coalescence’ (John of Damascus ca. 650). The roots of this doctrine are long and deep.”

—  Bert Waggoner

coinherence, n.

“In our human experience of personhood, at any rate in a fallen world, there is in each person an inevitable element of exclusiveness, of opaqueness and impenetrability.  But with the three divine persons it is not so.  Each is entirely ‘open’ to the others, totally transparent and receptive.  This transparency and receptivity is summed up in the Greek notion of perichoresis, which Gibbon once called ‘the deepest and darkest corner of the whole theological abyss.’  Rendered in Latin as circumincessio and in English usually as ‘coinherence,’ the Greek term means literally, cyclical movement, and so reciprocity, interchange, mutual indwelling.  The prefix peri bears the sense ‘around,’ while choresis is linked with chora, ‘room,’ space,’ ‘place’ or ‘container,’ and with chorein, to ‘go,’ ‘advance,’ ‘make room for’ or ‘contain.’  Some also see a connection with choros, ‘dance,’ and so they take perichoresis to mean ’round dance.’  Applied to Christ, the term signifies that his two natures, the divine and the human, interpenetrate one another without separation and without confusion.  Applied to the Trinity, it signifies that each person ‘contains’ the other two and ‘moves’ within them.  In the words of St Gregory of Nyssa, ‘All that is the Father’s is seen in the Son, and all that is the Son’s belongs also the Father. For the whole Son abides in the Father, and he has in his turn the whole Father abiding in himself.’ 

By virtue of this perichoresis, Father, Son and Holy Spirit ‘coinhere‘ in one another, each dwelling in the other two through an unceasing movement of mutual love – the ’round dance’ of the Trinity.”

— Timothy Ware, Bishop Kallistos of Diokleia,
    The Human Person as an Icon of the Trinity

Wednesday, September 3, 2003

Wednesday September 3, 2003

Filed under: General,Geometry — Tags: , , , — m759 @ 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 4×4 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 4×4 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.

Monday, September 1, 2003

Monday September 1, 2003

Filed under: General — m759 @ 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.

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