Log24

Friday, November 10, 2017

A Mathematician’s Apology

Filed under: General — m759 @ 4:00 PM

(Click to enlarge.)

For the paper on Steiner systems, see the bibliographic link in
the previous Log24 post.

See as well Cameron's posts before and after his post above:

     .

Tuesday, June 19, 2007

Tuesday June 19, 2007

Filed under: General — m759 @ 3:17 PM
 Faustus is gone:
regard his hellish fall


Marlowe

I have just read, in the New York Times Book Review that arrived in yesterday’s mail, a review of Segre’s Faust in Copenhagen.  The review, on news stands next Sunday, was titled by the Times “Meta Physicists.”

On Faust— today’s noon entry and yesterday’s “Nightmare Lessons.”

On “Meta Physicists“– an entry of June 6, on Cullinane College, has a section titled “Meta Physics.”

On Copenhagen— an entry of Bloomsday Eve, 2004 on a native of that city.

Another Dane:

“Words, words, words.”
Hamlet

Another metaphysics:

“317 is a prime,
not because we think so,
or because our minds
are shaped in one way
rather than another,
but because it is so,
because mathematical
reality is built that way.”

 — G. H. Hardy,
A Mathematician’s Apology

Saturday, June 2, 2007

Saturday June 2, 2007

Filed under: General,Geometry — m759 @ 8:00 AM
The Diamond Theorem
 
Four diamonds in a square
 

“I don’t think the ‘diamond theorem’ is anything serious, so I started with blitzing that.”

— Charles Matthews at Wikipedia, Oct. 2, 2006

“The ‘seriousness’ of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is ‘significant’ if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas.”

— G. H. Hardy, A Mathematician’s Apology

Thursday, October 26, 2006

Thursday October 26, 2006

Filed under: General — m759 @ 12:00 PM
Hardy & Wright 
The image “http://www.log24.com/log/pix06A/061025-Wright.jpg” cannot be displayed, because it contains errors.

“When he was taken to church
he amused himself by factorizing
the numbers of the hymns.”

— C. P. Snow, foreword to
A Mathematician’s Apology,
by G. H. Hardy

An application of
lottery hermeneutics:

420 –> 4/20 –>

Hall of Shame,
Easter Sunday,
April 20, 2003;

145 –> 5*29 –> 5/29 –>

The Shining of May 29.

The Rev. Wright may also
be interested in the following

Related material:

“Shem was a sham….”
(FW I.7, 170 and Log24 Oct. 13),
and The Hebrew Word Shem:

“When I teach introductory Hebrew, the first word I typically teach is the common noun SHEM. It’s pronounced exactly like our English word ‘shame,’ means almost exactly the opposite, and seems to me to be a key….” — Glen Penton

This word occurs, notably, in Psalm (or “hymn”) 145.

See http://scripturetext.com/psalms/145-1.htm:

thy name
shem  (shame)
an appellation, as a mark or memorial of individuality; by implication honor, authority, character — + base, (in-)fame(-ous), named(-d), renown, report.

Update of 12:25 PM 10/26
from the online Crimson:


Related material:
The Crimson Passion

Sunday, October 15, 2006

Sunday October 15, 2006

Filed under: General — m759 @ 2:00 PM

Cleavage Term

Snow is mainly remembered as the author of The Two Cultures and the Scientific Revolution (1959).

According to Orrin Judd, we can now see “how profoundly wrong Snow was in everything except for his initial metaphor, of a divide between science and the rest of the culture.”

For more on that metaphor, see the previous entry, “The Line.”

I prefer a lesser-known work of Snow– his long biographical foreword to G. H. Hardy’s A Mathematician’s Apology. The foreword, like the book itself, is an example of what Robert M. Pirsig calls “Quality.”  It begins with these words:

“It was a perfectly ordinary night at Christ’s high table, except that Hardy was dining as a guest.”

Related material:

Wallace Stevens,
“The Sail of Ulysses,”
Canto V

Friday, August 4, 2006

Friday August 4, 2006

Filed under: General — m759 @ 2:00 PM
The Double Cross

The following symbol
has been associated
with the date
December 1:

The image “http://www.log24.com/log/pix06A/060804-DWA2.gif” cannot be displayed, because it contains errors.

Click on the symbol
for details.

That date is connected
to today’s date since
Dec. 1 is the feast
i.e., the deathday– of
a saint of mathematics:
G. H. Hardy, author of
the classic
A Mathematician’s Apology
(online, pdf, 52 pp. ),
while today is the birthday
of three less saintly
mathematical figures:
Sir William Rowan Hamilton,

For these birthdays, here is
a more cheerful version of
the above symbol:

The image “http://www.log24.com/theory/images/PeirceBox.bmp” cannot be displayed, because it contains errors.

For the significance of
this version, see
Chinese Jar Revisited
(Log24, June 27, 2006),
a memorial to mathematician
Irving Kaplansky
(student of Mac Lane).

This version may be regarded
as a box containing the
cross of St. Andrew.
If we add a Greek cross
(equal-armed) to the box,
we obtain the “spider,”
or “double cross,” figure

The image “http://www.log24.com/theory/images/PeirceSpider.bmp” cannot be displayed, because it contains errors.

of my favorite mythology:
Fritz Leiber’s Changewar.

Friday, December 2, 2005

Friday December 2, 2005

Filed under: General,Geometry — m759 @ 5:55 AM

Proof 101

From a course description:

“This module aims to introduce the student to rigorous university level mathematics….
    Syllabus: The idea of and need for mathematical statements and proofs…. proof by contradiction… proof by induction…. the infinite number of primes….”

In the December Notices of the American Mathematical Society, Brian (E. B.) Davies, a professor of mathematics at King’s College London, questions the consistency of Peano Arithmetic (PA), which has the following axioms:

From BookRags.com

Axiom 1. 0 is a number.

Axiom 2. The successor of any number is a number.

Axiom 3. If a and b are numbers and if their successors are equal, then a and b are equal.

Axiom 4. 0 is not the successor of any number.

Axiom 5. If S is a set of numbers containing 0 and if the successor of any number in S is also in S, then S contains all the numbers.

It should be noted that the word “number” as used in the Peano axioms means “non-negative integer.”  The fifth axiom deserves special comment.  It is the first formal statement of what we now call the “induction axiom” or “the principle of mathematical induction.”

Peano’s fifth axiom particularly troubles Davies, who writes elsewhere:

I contend that our understanding of number should be placed in an historical context, and that the number system is a human invention.  Elementary arithmetic enables one to determine the number of primes less than twenty as certainly as anything we know.  On the other hand Peano arithmetic is a formal system, and its internal consistency is not provable, except within set-theoretic contexts which essentially already assume it, in which case their consistency is also not provable.  The proof that there exists an infinite number of primes does not depend upon counting, but upon the law of induction, which is an abstraction from our everyday experience…. 
… Geometry was a well developed mathematical discipline based upon explicit axioms over one and a half millennia before the law of induction was first formulated.  Even today many university students who have been taught the principle of induction prefer to avoid its use, because they do not feel that it is as natural or as certain as a purely algebraic or geometric proof, if they can find one.  The feelings of university students may not settle questions about what is truly fundamental, but they do give some insight into our native intuitions.

E. B. Davies in
   “Counting in the real world,”
    March 2003 (word format),
    To appear in revised form in
    Brit. J. Phil. Sci. as
   “Some remarks on
    the foundations
    of quantum mechanics”

Exercise:

Discuss Davies’s claim that

The proof that there exists an infinite number of primes does not depend upon counting, but upon the law of induction.

Cite the following passage in your discussion.

It will be clear by now that, if we are to have any chance of making progress, I must produce examples of “real” mathematical theorems, theorems which every mathematician will admit to be first-rate. 

… I can hardly do better than go back to the Greeks.  I will state and prove two of the famous theorems of Greek mathematics.  They are “simple” theorems, simple both in idea and in execution, but there is no doubt at all about their being theorems of the highest class.  Each is as fresh and significant as when it was discovered– two thousand years have not written a wrinkle on either of them.  Finally, both the statements and the proofs can be mastered in an hour by any intelligent reader, however slender his mathematical equipment.

I. The first is Euclid’s proof of the existence of an infinity of prime numbers.

The prime numbers or primes are the numbers

   (A)   2, 3, 5, 7, 11, 13, 17, 19, 23, 29, … 

which cannot be resolved into smaller factors.  Thus 37 and 317 are prime.  The primes are the material out of which all numbers are built up by multiplication: thus

    666 = 2 . 3 . 3 . 37. 

Every number which is not prime itself is divisible by at least one prime (usually, of course, by several).   We have to prove that there are infinitely many primes, i.e. that the series (A) never comes to an end.

Let us suppose that it does, and that

   2, 3, 5, . . . , P
 
is the complete series (so that P is the largest prime); and let us, on this hypothesis, consider the number

   Q = (2 . 3 . 5 . . . . . P) + 1.

It is plain that Q is not divisible by any of

   2, 3, 5, …, P;

for it leaves the remainder 1 when divided by any one of these numbers.  But, if not itself prime, it is divisible by some prime, and therefore there is a prime (which may be Q itself) greater than any of them.   This contradicts our hypothesis, that there is no prime greater than P; and therefore this hypothesis is false.

The proof is by reductio ad absurdum, and reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons.  It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.

— G. H. Hardy,
   A Mathematician’s Apology,
   quoted in the online guide for
   Clear and Simple as the Truth:
   Writing Classic Prose, by
   Francis-Noël Thomas
   and Mark Turner,
   Princeton University Press

In discussing Davies’s claim that the above proof is by induction, you may want to refer to Davies’s statement that

Geometry was a well developed mathematical discipline based upon explicit axioms over one and a half millennia before the law of induction was first formulated

and to Hardy’s statement that the above proof is due to Euclid.

Thursday, July 21, 2005

Thursday July 21, 2005

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

Permanence

“What we do may be small, but it has a certain character of permanence.”

G. H. Hardy, A Mathematician’s Apology

For further details, see
Geometry of the 4×4 Square.

“There is no permanent place in the world for ugly mathematics.”

— Hardy, op. cit.

For further details, see
Four-colour proof claim.

Friday, October 15, 2004

Friday October 15, 2004

Filed under: General — m759 @ 7:11 PM
Snow Jobs

In memory of C. P. Snow,
whose birthday is today

“Without the narrative prop of
High Table dinner conversation
at Cambridge, Snow would be lost.”
— Roger Kimball*

The image “http://www.log24.com/log/pix04A/041015-Sup.jpg” cannot be displayed, because it contains errors.

“It was a perfectly ordinary night
at Christ’s high table, except that
Hardy was dining as a guest.”
— C. P. Snow**

“666=2.3.3.37, and there is
no other decomposition.”
— G. H. Hardy***

* The Two Cultures Today

** Foreword to
A Mathematician’s Apology

*** A Mathematician’s Apology

Oct. 15, 2004, 7:11:37 PM

Sunday, October 10, 2004

Sunday October 10, 2004

Filed under: General — m759 @ 10:35 PM

Introduction to Aesthetics

“Chess problems are the
hymn-tunes of mathematics.”
— G. H. Hardy,
A Mathematician’s Apology

The image “http://www.log24.com/log/pix04A/041010-Hardy2.jpg” cannot be displayed, because it contains errors.
The image “http://www.log24.com/log/pix04A/041010-Mate2.jpg” cannot be displayed, because it contains errors.


G. H. Hardy in
A Mathematician’s Apology:

“We do not want many ‘variations’ in the proof of a mathematical theorem: ‘enumeration of cases,’ indeed, is one of the duller forms of mathematical argument.  A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way.

A chess problem also has unexpectedness, and a certain economy; it is essential that the moves should be surprising, and that every piece on the board should play its part.  But the aesthetic effect is cumulative.  It is essential also (unless the problem is too simple to be really amusing) that the key-move should be followed by a good many variations, each requiring its own individual answer.  ‘If P-B5 then Kt-R6; if …. then …. ; if …. then ….’ — the effect would be spoilt if there were not a good many different replies.  All this is quite genuine mathematics, and has its merits; but it just that ‘proof by enumeration of cases’ (and of cases which do not, at bottom, differ at all profoundly*) which a real mathematician tends to despise.

* I believe that is now regarded as a merit in a problem that there should be many variations of the same type.”

(Cambridge at the University Press.  First edition, 1940.)

Brian Harley in
Mate in Two Moves:

“It is quite true that variation play is, in ninety-nine cases out of a hundred, the soul of a problem, or (to put it more materially) the main course of the solver’s banquet, but the Key is the cocktail that begins the proceedings, and if it fails in piquancy the following dinner is not so satisfactory as it should be.”

(London, Bell & Sons.  First edition, 1931.)

Saturday, August 9, 2003

Saturday August 9, 2003

Filed under: General — m759 @ 6:29 PM

Bibles

Today is the feast day of St. Hermann Hesse.  A quotation from a work by Hesse that is to some a sort of Bible:

“You treat world history as a mathematician does mathematics, in which nothing but laws and formulae exist, no reality, no good and evil, no time, no yesterday, no tomorrow, nothing but an eternal, shallow, mathematical present.”

Father Jacobus, Benedictine priest, in The Glass Bead Game, ch. 4 (1943, translated 1960), by Hermann Hesse

A Benedictine Archbishop’s Apology:

“Archbishop Weakland described his feelings ‘at this moment’ as ‘remorse, contrition, shame and emptiness,’ also noting that ‘much self-pity and pride remain.’ He contended he ‘must leave that pride behind.’ “

A Mathematician’s Apology:

C.P. Snow in his introduction to A Mathematician’s Apology (also a Bible, or at least a book of a Bible, to some) quotes G. H. Hardy on hearing the chimes of Vespers:

“It’s rather unfortunate that some of the happiest hours of my life should have been spent within sound of a Roman Catholic church.”

A Bible for Benedictines:

The Encyclopedic Dictionary of Mathematics,
by the Mathematical Society of Japan,

is suitable reading for those Benedictines in Purgatory who have too lightly used words like “no reality” and “shallow” to describe mathematics.

For other remedial reading in the afterlife, see Midsummer Eve’s Dream and Quine in Purgatory.

Saturday, July 12, 2003

Saturday July 12, 2003

Filed under: General — m759 @ 6:23 PM

Before and After

From Understanding the (Net) Wake:

24

A.

“Its importance in establishing the identities in the writer complexus….will be best appreciated by never forgetting that both before and after the Battle of the Boyne it was a habit not to sign letters always.”(114)

Joyce shows an understanding of the problems that an intertextual book like the Wake poses for the notion of authorship.

G. H. Hardy in A Mathematician’s Apology:

“We do not want many ‘variations’ in the proof of a mathematical theorem: ‘enumeration of cases,’ indeed, is one of the duller forms of mathematical argument.  A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way.

A chess problem also has unexpectedness, and a certain economy; it is essential that the moves should be surprising, and that every piece on the board should play its part.  But the aesthetic effect is cumulative.  It is essential also (unless the problem is too simple to be really amusing) that the key-move should be followed by a good many variations, each requiring its own individual answer.  ‘If P-B5 then Kt-R6; if …. then …. ; if …. then ….’ — the effect would be spoilt if there were not a good many different replies.  All this is quite genuine mathematics, and has its merits; but it just that ‘proof by enumeration of cases’ (and of cases which do not, at bottom, differ at all profoundly*) which a real mathematician tends to despise.

* I believe that is now regarded as a merit in a problem that there should be many variations of the same type.”

(Cambridge at the University Press.  First edition, 1940.)

Brian Harley in Mate in Two Moves:

“It is quite true that variation play is, in ninety-nine cases out of a hundred, the soul of a problem, or (to put it more materially) the main course of the solver’s banquet, but the Key is the cocktail that begins the proceedings, and if it fails in piquancy the following dinner is not so satisfactory as it should be.”

(London, Bell & Sons.  First edition, 1931.)

Thursday, May 22, 2003

Thursday May 22, 2003

Filed under: General — m759 @ 7:29 PM

Seek and Ye Shall Find:

On the Mystical Properties
of the Number 162

On this date in history:

May 22, 1942:  Unabomber Theodore John Kaczynski is born in the Chicago suburb of Evergreen Park, Ill., to Wanda Kaczynski and her husband Theodore R. Kaczynski, a sausage maker. His mother brings him up reading Scientific American.

From the June 2003 Scientific American:

“Seek and ye shall find.” – Michael Shermer

From my note Mark of April 25, 2003:

“Tell me of runes to grave
 That hold the bursting wave,
 Or bastions to design
 For longer date than mine.”

— A. E. Housman, quoted by G. H. Hardy in A Mathematician’s Apology

“Here, as examples, are one rune and one bastion…. (illustrations: the Dagaz rune and the Nike bastion of the Acropolis)…. Neither the rune nor the bastion discussed has any apparent connection with the number 162… But seek and ye shall find.”

Here is a connection to runes:

Mayer, R.M., “Runenstudien,” Beiträge zur Geschichte der deutschen Sprache und Literatur 21 (1896): pp. 162 – 184.

Here is a connection to Athenian bastions from a UN article on Communist educational theorist Dimitri Glinos:

“Educational problems cannot be scientifically solved by theory and reason alone….” (D. Glinos (1882-1943), Dead but not Buried, Athens, Athina, 1925, p. 162)

“Schools are…. not the first but the last bastion to be taken by… reform….”

“…the University of Athens, a bastion of conservatism and counter-reform….”

I offer the above with tongue in cheek as a demonstration that mystical numerology may have a certain heuristic value overlooked by fanatics of the religion of Scientism such as Shermer.

For a more serious discussion of runes at the Acropolis, see the photo on page 16 of the May 15, 2003, New York Review of Books, illustrating the article “Athens in Wartime,” by Brady Kiesling.

Friday, April 18, 2003

Friday April 18, 2003

Filed under: General — m759 @ 3:17 PM

A Red Mass

For G. H. Hardy, who, although he kept a portrait of Lenin in his rooms, knew more of truth than most Christians ever know.

317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way.”
— G. H. Hardy, A Mathematician’s Apology, 1940

Saturday, April 5, 2003

Saturday April 5, 2003

Filed under: General,Geometry — m759 @ 9:49 AM

Art Wars:
Mathematics and the
Emperor’s New Art

From Maureen Dowd’s New York Times column of June 9, 2002: 

“The shape of the government is not as important as the policy of the government. If he makes the policy aggressive and pre-emptive, the president can conduct the war on terror from the National Gallery of Art.”

NY Times, April 5, 2003:
U.S. Tanks Move Into Center of Baghdad
See also today’s
op-ed piece
by Patton’s grandson.

Meanwhile, at the Washington Post, another example of great determination and strength of character:

Donald Coxeter Dies: Leader in Geometry

By Martin Weil
Washington Post Staff Writer
Saturday, April 5, 2003

“Donald Coxeter, 96, a mathematician who was one of the 20th century’s foremost specialists in geometry and a man of great determination and strength of character as well, died March 31 at his home in Toronto.”

From another Coxeter obituary:

In the Second World War, Coxeter was asked by the American government to work in Washington as a code-breaker. He accepted, but then backed out, partly because of his pacifist views and partly for aesthetic reasons: “The work didn’t really appeal to me,” he explained; “it was a different sort of mathematics.”

For a differing account of how geometry is related to code-breaking, see the “Singer 7-cycle” link in yesterday’s entry, “The Eight,” of 3:33 PM.  This leads to a site titled

An Introduction to the
Applications of Geometry in Cryptography
.

“Now I have precisely the right instrument, at precisely the right moment of history, in exactly the right place.”

 — “Patton,”
the film

Quod erat
demonstrandum
.


Added Sunday, April 6, 2003, 3:17 PM:

The New York Times Magazine of April 6
continues this Art Wars theme.


                 (Cover typography revised)

The military nature of our Art Wars theme appears in the Times’s choice of words for its cover headline: “The Greatest Generation.” (This headline appears in the paper, but not the Internet, version.)

Some remarks in today’s Times Magazine article seem especially relevant to my journal entry for Michelangelo’s birthday, March 6.

“…Conceptualism — suddenly art could be nothing more than an idea….

LeWitt moved between his syntax of geometric sculptures and mental propositions for images: concepts he wrote on paper that could be realized by him or someone else or not at all.  Physical things are perishable.  Ideas need not be.”

— Michael Kimmelman, chief art critic of the New York Times, April 6, 2003

Compare this with a mathematician’s aesthetics:

“A mathematician, like a painter or a poet, is a maker of patterns.  If his patterns are more permanent than theirs, it is because they are made with ideas.”

— G. H. Hardy, A Mathematician’s Apology (1940), reprinted 1969, Cambridge U. Press, p. 84 

It seems clear from these two quotations that the real conceptual art is mathematics and that Kimmelman is peddling the emperor’s new clothes.

Sunday, August 4, 2002

Sunday August 4, 2002

Filed under: General,Geometry — m759 @ 2:52 PM

The Story Theory of Truth

versus

The Diamond Theory of Truth

One year ago today, Lorenzo Music, the voice of Carlton the doorman on Rhoda, died.  His eulogy from Valerie Harper:

 “Valerie’s heart is breaking, but Rhoda is certain that Carlton the doorman is giving St. Peter at the gate a run for his money.”

Today’s birthday: Logician John Venn

Appearing for the story theory…

Flannery O’Connor:

“In the long run, a people is known, not by its statements or statistics, but by the stories it tells. Fiction is the most impure and the most modest and the most human of the arts.”

Appearing for the diamond theory…

Mary McCarthy and G. H. Hardy:

From the Hollywood Investigator:

 On October 18, 1979, Mary McCarthy said on PBS’s Dick Cavett Show: “Every word she writes is a lie, including ‘and’ and ‘the.'”

Don’t forget “a,” as in “a people is known” —

“Greek mathematics is permanent, more permanent even than Greek literature.  Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not.”

— G. H. Hardy in A Mathematician’s Apology

And a closing rebuttal from the story theory…

Martin Heidegger and Dean Martin: 

Words of wisdom from Martin Heidegger, Catholic Nazi:

“The nature of art is poetry.  The nature of poetry, in turn, is the founding of truth…. In the work, truth is thrown toward… an historical group of men.”

Poetry, Language, Thought, page 75, translated by Albert Hofstadter, Harper & Row paperback, 1975

And from Dean Martin, avatar of anti-art :

That’s Amore:

– Artist: Dean Martin as sung on “Dean Martin’s Greatest Hits”
– Capitol 4XL-9389
– peak Billboard position # 2 in 1953
– from the movie “the Caddy” starring Dean, Jerry Lewis, and Donna Reed
– Words and Music by Harry Warren and Jack Brooks

(In Napoli where love is King, when boy meets girl, here’s what they say)

When the moon hits your eye like a big-a pizza pie,
That’s amore!
When the world seems to shine like you’ve had too much wine,
That’s amore!

Powered by WordPress