November | 2005 | m759's Xanga Site

Month: November 2005

Hobgoblin?

Brian Davies is a professor of mathematics at King's College London. In the December Notices of the American Mathematical
Society, he claims that arithmetic may, for all we know, be inconsistent:

"Gödel taught us that it is not possible to prove that Peano arithmetic
is consistent, but everyone has taken it for granted that in fact it is
indeed consistent.
    Platonistically-inclined mathematicians would
deny the possibility that Peano arithmetic could be flawed. From
Kronecker onwards many consider that they have a direct insight into
the natural numbers, which guarantees their existence. If the natural
numbers exist and Peano’s axioms describe properties that they possess
then, since the axioms can be instantiated, they must be consistent."

"It is not possible to prove that Peano arithmetic is consistent"...?!

Where
did Gödel say this? Gödel proved, in fact, according to a
well-known mathematician at Princeton, that (letting PA stand for Peano
Arithmetic),

"If PA is consistent, the formula expressing 'PA is consistent' is unprovable in PA."

-- Edward Nelson,
   Mathematics and Faith (pdf)

Remarkably, even after he has stated correctly Gödel's result, Nelson, like Davies, concludes that

"The consistency of PA cannot be concretely demonstrated."

I prefer the argument that the existence of a model ensures the consistency of a theory.

For instance, the Toronto philosopher William Seager writes that

"Our
judgement as to the consistency of some system is not dependent upon
that system’s being able to prove its own consistency (i.e. generate a
formula that states, e.g. ‘0=1’ is not provable). For if that was the
sole basis, how could we trust it? If the system was inconsistent, it
could generate this formula as well (see Smullyan, Gödel’s Incompleteness Theorems,
(Oxford, 1992, p. 109)). Furthermore, [George] Boolos allows that we do
know that certain systems, such as Peano Arithmetic, are consistent
even though they cannot prove their own consistency. Presumably, we
know this because we can see that a certain model satisfies the axioms
of the system at issue, hence that they are true in that model and so
must be consistent."

-- Yesterday's Algorithm:
    Penrose and the Gödel Argument

The relationship between consistency and the existence of a model is
brought home by the following weblog entry that neatly summarizes a
fallacious argument offered in the AMS Notices by Davies:

The
following is an interesting example that I came across in the article
"Whither Mathematics?" by Brian Davies in the December issue of Notices
of the American Mathematical Society.

Consider the following list A1 of axioms.

(1) There is a natural number 0.
(2) Every natural number a has a successor, denoted by S(a).
(3) There is no natural number whose successor is 0.
(4) Distinct natural numbers have distinct successors: a = b if and only if S(a) = S(b).
(5) If a property is possessed by 0 and also by the successor of every
natural number which possesses it, then it is possessed by all the
natural numbers.

Now consider the following list A2 of axioms.

(1) G is a set of elements and these elements obey the group axioms.
(2) G is finite but not isomorphic to any known list of finite simple groups.
(3) G is simple, in other words, if N is a subset of G satisfying certain properties then N=G.

We can roughly compare A2 with A1. The second axiom in A2 can be
thought of as analogous to the third axiom of A1. Also the third axiom
of A2 is analogous to the fifth axiom of A1, insofar as it refers to an
unspecified set with cetain properties and concludes that it is equal
to G.

Now, as is generally believed by most group theorists, the system A2 is internally inconsistent and the proof its inconsistency runs for more than 10000 pages.

So who is to deny that the system A1 is also probably internally inconsistent! Particularly since Godel proved
that you can not prove it is consistent (staying inside the system).
May be the shortest proof of its inconsistency is one hundred million
pages long!

-- Posted by Krishna,
   11/29/2005 11:46:00 PM,
   at his weblog,
"Quasi-Coherent Ruminations"

An important
difference between A1 (the set of axioms of Peano arithmetic) and A2 (a
set of axioms that describe a new, unknown, finite simple
group) is that A1 is known to have a model (the nonnegative integers) and A2
is not known to have a model.

Therefore, according to Seager's argument, A1 is consistent and A2 may or may not be consistent.

The degree to which Seager's argument invokes Platonic realism is debatable. Less debatable is the quasireligious faith in nominalism
proclaimed by Davies and Nelson. Nelson's own account of a
religious experience in 1976 at Toronto
is instructive.

I must relate how
I lost my faith in Pythagorean numbers. One morning at the 1976 Summer
Meeting of the American Mathematical Society in Toronto, I woke early.
As I lay meditating about numbers, I felt the momentary overwhelming
presence of one who convicted me of arrogance for my belief in the real
existence of an infinite world of numbers, leaving me like an infant in
a crib reduced to counting on my fingers. Now I live in a world in
which there are no numbers save those that human beings on occasion
construct.

-- Edward Nelson,
   Mathematics and Faith (pdf)

Nelson's "Mathematics and Faith" was written for the Jubilee for Men and Women from the World of Learning held at the Vatican, 23-24 May 2000. It concludes with an invocation of St. Paul:

During
my first stay in Rome I used to play chess with Ernesto Buonaiuti. In
his writings and in his life, Buonaiuti with passionate eloquence
opposed the reification of human abstractions. I close by quoting one
sentence from his Pellegrino di Roma.
"For [St. Paul] abstract truth, absolute laws, do not exist, because
all of our thinking is subordinated to the construction of this holy
temple of the Spirit, whose manifestations are not abstract ideas, but
fruits of goodness, of peace, of charity and forgiveness."

-- Edward Nelson,
   Mathematics and Faith (pdf)

Belief
in the consistency of arithmetic may or may not be foolish, and
therefore an Emersonian hobgoblin of little minds, but bullshit is
bullshit, whether in London, in Princeton, in Toronto, or in Rome.
For St. Andrew's Day

"The miraculous enters....
When we investigate these problems, some fantastic things happen...."

-- John H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, preface to first edition (1988)

The remarkable
Mathieu group M₂₄, a group of permutations on 24 elements, may be
studied by picturing its action on three interchangeable 8-element
"octads," as
in the "Miracle Octad Generator" of R. T. Curtis.

A picture of the Miracle Octad Generator, with my comments, is available online.

Cartoon by S.Harris

Related material:
Mathematics and Narrative.
The Way of the Pilgrim,

Part III:

For the Birthday

of C. S. Lewis

The Tao, Chapter I

"The Chinese... speak of a great thing (the
greatest thing) called the Tao. It is the reality beyond all
predicates, the abyss that was before the Creator Himself. It is
Nature, it is the Way, the Road. It is the Way in which the universe
goes on, the Way in which things everlastingly emerge, stilly and
tranquilly, into space and time. It is also the Way which every man
should tread in imitation of that cosmic and supercosmic progression,
conforming all activities to that great exemplar."

-- C. S. Lewis in The Abolition of Man

"In his preface to That Hideous Strength, Lewis says the novel has a serious point that he has tried to make in this little book, The Abolition of Man. The novel is a work of fantasy or science fiction, while Abolition
is a short philosophical work about moral education, but as we shall
see the two go together; we will understand either book better by
having read and thought about the other."

-- Dale Nelson, Notes on The Abolition of Man

"In Epiphany Term, 1942, C.S. Lewis delivered the Riddell
Memorial Lectures... in....
the University of Durham.... He
delivered three lectures
entitled 'Men without Chests,' 'The Way,' and 'The
Abolition of Man.' In them he set out to attack and
confute what he saw as the errors of his age. He started by
quoting some fashionable lunacy from an educationalists'
textbook, from which he developed a general attack on moral
subjectivism. In his second lecture he argued against
various contemporary isms, which purported to replace
traditional objective morality. His final lecture, 'The
Abolition of Man,' which also provided the title of the
book published the following year, was a sustained attack on
hard-line scientific anti-humanism.

The intervening fifty years have largely vindicated Lewis."

-- J. R. Lucas, The Restoration of Man
The Way of the Pilgrim,
Part II:

Einstein's Orgy

In a recent Edge article, "The Vagaries of Religious Experience," a Harvard psychologist, Daniel Gilbert, quotes Einstein on his own religious vagaries:

"(I
had) a deep religiosity, which, however, found an abrupt ending at
the age of 12. Through the reading of popular scientific books I soon
reached the conviction that much in the stories of the Bible could
not be true. The consequence was a positively fanatic orgy* of freethinking
coupled with the impression that youth is intentionally being deceived
by the state through lies. It was a crushing impression. Suspicion
against every kind of authority grew out of this experience, a skeptical
attitude towards the convictions which were alive in any specific
social environment-- an attitude which has never again left
me." (Autobiographical Notes, 1949)

Gilbert adds,

"Einstein's
orgy* of freethinking forever changed our understanding of space and
time, and the phrase 'Religion for Dummies' became, in the
view of many scientists, a redundancy."

Here is another Einstein quotation, from the paragraph in Autobiographical Notes following the paragraph quoted by Gilbert:

"It is quite clear to me that the religious paradise of youth, which
was thus lost, was a first attempt to free myself from the chains of
the 'merely-personal,' from an existence which is dominated by wishes,
hopes and primitive feelings. Out yonder there was this huge
world, which exists independently of us human beings and which stands
before us like a great, eternal riddle, at least partially accessible to
our inspection and thinking. The contemplation of this world
beckoned like a liberation.... The road to this paradise was not as
comfortable and alluring as the road to the religious paradise; but it
has proved itself as trustworthy, and I have never regretted having
chosen it."

Einstein
describes "the road to the religious paradise" as "comfortable and
alluring." He might therefore have profited by the book saluted
in the previous entry... a book that might be described, to adapt Gilbert's
charming phrase, as "Religion for Dummies like Einstein."

For an approach to the contemptible religion of Scientism that is more subtle than Gilbert's, see "Einstein's Third Paradise," by Gerald Holton, another Harvard savant.

* In the original, the words "orgy of" appear in square
brackets to indicate an interpolation by the editor, Paul A. Schilpp, a Methodist minister (pdf). Einstein's own words were "eine geradezu fanatische Freigeisterei."
Gilbert's omission of the brackets indicates both the moral
slovenliness typical of those embracing Scientism and the current low
standards of scholarship at Harvard. (Related material: The Crimson Passion.)
The Way of the Pilgrim,
Part I:

For John Bunyan's
Birthday

Click on picture to enlarge.

"AS I walk’d through the wilderness of this
world, I lighted on a certain place where was a Den, and I laid me down
in that place to sleep; and as I slept, I dreamed a Dream. I dreamed,
and behold I saw a Man cloathed with Rags, standing in a certain place,
with his face from his own house, a Book in his hand, and a great
Burden upon his back. I looked, and saw him open the Book, and read
therein; and as he read, he wept and trembled; and not being able
longer to contain, he brake out with a lamentable cry, saying What shall I do?"

-- The Pilgrim's Progress, by John Bunyan
Holy Geometry

What was "the holy geometry book" ("das heilige Geometrie-Büchlein," p.
10 in the Schilpp book below) that so impressed the young Albert
Einstein?

"At
the age of 12 I experienced a second wonder of a totally different
nature: in a little book dealing with Euclidian plane geometry, which
came into my hands at the beginning of a schoolyear. Here were
assertions, as for example the intersection of the three altitudes of a
triangle in one point, which-- though by no means evident-- could
nevertheless be proved with such certainty that any doubt appeared to
be out of the question. This lucidity and certainty made an
indescribable impression upon me."

("Im
Alter von 12 Jahren erlebte ich ein zweites Wunder ganz verschiedener
Art: An einem Büchlein über Euklidische Geometrie der Ebene, das ich am
Anfang eines
Schuljahres in die Hand bekam. Da waren Aussagen wie z.B. das
Sich-Schneiden der drei Höhen eines Dreieckes in einem Punkt, die--
obwohl an sich keineswegs evident-- doch mit solcher Sicherheit
bewiesen werden konnten, dass ein Zweifel ausgeschlossen zu sein
schien. Diese Klarheit und Sicherheit machte einen
unbeschreiblichen Eindruck auf mich.")

-- Albert Einstein, Autobiographical Notes, pages 8 and 9 in Albert Einstein: Philosopher-Scientist, ed. by Paul A. Schilpp

From a website by Hans-Josef Küpper:

"Today
it cannot be said with certainty which book is Einstein’s 'holy
geometry book.' There are three different titles that come into
question:

Theodor Spieker,
1890

Lehrbuch der ebenen
Geometrie. Mit Übungsaufgaben für höhere Lehranstalten.

Heinrich Borchert
Lübsen, 1870

Ausführliches
Lehrbuch der ebenen und sphärischen Trigonometrie. Zum
Selbstunterricht. Mit Rücksicht auf die Zwecke
des praktischen Lebens.

Adolf Sickenberger,
1888

Leitfaden der
elementaren Mathematik.

Young
Albert Einstein owned all of these three books. The book by T.
Spieker was given to him by Max Talmud (later: Talmey), a Jewish
medic. The book by H. B. Lübsen was from the library of his uncle
Jakob Einstein and the one of A. Sickenberger was from his parents."

Küpper does not state clearly his source for the geometry-book information.

According to Banesh Hoffman and Helen Dukas in Albert Einstein, Creator and Rebel, the holy geometry book was Lehrbuch der Geometrie zum Gebrauch an höheren Lehranstalten, by Eduard Heis (Catholic astronomer and textbook writer) and Thomas Joseph Eschweiler.

An argument for Sickenberger from The Young Einstein: The Advent of Relativity (pdf), by Lewis Pyenson, published by Adam Hilger Ltd., 1985:

    "Throughout Einstein's five and a half years at
the Luitpold Gymnasium, he was taught mathematics from one or
another edition of the separately published parts of
Sickenberger's Textbook of Elementary Mathematics.
When it first appeared in 1888 the book constituted a major
contribution to reform pedagogy. Sickenberger based his book on
twenty years of experience that in his view necessarily took
precedence over 'theoretical doubts and systematic scruples.' At
the same time Sickenberger made much use of the recent
pedagogical literature, especially that published in the pages of
Immanuel Carl Volkmar Hoffmann’s Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht,
the leading pedagogical mathematics journal of the day. Following
in the tradition of the reform movement, he sought to present
everything in the simplest, most intuitive way possible. He
opposed introducing scientific rigour and higher approaches in an
elementary text. He emphasised that he would follow neither the
synthesis of Euclidean geometry nor the so-called
analytical-genetic approach. He opted for a great deal of freedom in
the form of presentation because he believed that a textbook was
no more than a crutch for oral instruction. The spoken word, in
Sickenberger's view, could infuse life into the dead forms of the
printed text. Too often, he insisted in the preface to his text,
mathematics was seen and valued 'as the pure science of
reason.' In reality, he continued, mathematics was also 'an
essential tool for daily work.' In view of the practical
dimension of mathematics Sickenberger sought most of all to
present basic propositions clearly rather than to arrive at
formal conciseness.   Numerous examples took the place of long,
complicated, and boring generalities. In addition to the
usual rules of arithmetic Sickenberger introduced diophantine
equations. To solve three linear, homogeneous, first-order
equations with three unknowns he specified determinants and
determinant algebra. Then he went on to quadratic equations and
logarithms. In the second part of his book, Sickenberger treated
plane geometry.
     According to a biography of Einstein written
by his step-son-in-law, Rudolf Kayser-- one that the theoretical
physicist described as 'duly accurate'-- when he was twelve years
old Einstein fell into possession of the 'small geometry book'
used in the Luitpold Gymnasium before this subject was formally
presented to him. Einstein corroborated Kayser's passage in
autobiographical notes of 1949, when he described how at the age
of twelve 'a little book dealing with Euclidean plane geometry'
came into his hands 'at the beginning of a school year.' The
'lucidity and certainty' of plane geometry according to this
'holy geometry booklet' made, Einstein wrote, 'an indescribable
impression on me.' Einstein saw here what he found in other
texts that he enjoyed: it was 'not too particular' in logical
rigour but 'made up for this by permitting the main thoughts to
stand out clearly and synoptically.' Upon working his way through
this text, Einstein was then presented with one of the many
editions of Theodor Spieker's geometry by Max Talmey, a medical
student at the University of Munich who dined with the Einsteins
and who was young Einstein’s friend when Einstein was between the
ages of ten and fifteen. We can only infer from Einstein's
retrospective judgment that the first geometry book exerted an
impact greater than that produced by Spieker's treatment, by the
popular science expositions of Aaron Bernstein and Ludwig Büchner
also given to him by Talmey, or by the texts of Heinrich Borchert
Lübsen from which Einstein had by the age of fourteen taught himself differential and integral calculus.
     Which text constituted the 'holy geometry
booklet'? In his will Einstein gave 'all his books' to his
long-time secretary Helen Dukas. Present in this collection are
three bearing the signature 'J Einstein': a logarithmic and
trigonometric handbook, a textbook on analysis, and an
introduction to infinitesimal calculus. The signature is that of
Einstein's father's brother Jakob, a business partner and member
of Einstein's household in Ulm and Munich. He presented the books
to his nephew Albert. A fourth book in Miss Dukas's collection,
which does not bear Jakob Einstein's name, is the second part of
a textbook on geometry, a work of astronomer Eduard Heis's which
was rewritten after his death by the Cologne schoolteacher Thomas
Joseph Eschweiler. Without offering reasons for his choice
Banesh Hoffmann has recently identified Heis and Eschweiler's
text as the geometry book that made such an impression on
Einstein. Yet, assuming that Kayser's unambiguous reporting
is correct, it is far more likely that the geometrical part of
Sickenberger's text was what Einstein referred to in his
autobiographical notes. Sickenberger's exposition was published
seven years after that of Heis and Eschweiler, and unlike the
latter it appeared with a Munich press. Because it was used in
the Luitpold Gymnasium, copies would have been readily available
to Uncle Jakob or to whoever first acquainted Einstein with
Euclidean geometry."

What might be the modern version of a "holy geometry book"?

I suggest the following,

first published in 1940:

Click on picture for details.
Rehearsing Hell

Art critic Michael Kimmelman
in today's New York Times:

The Los Angeles veteran Mike Kelley's latest show is a sprawling,
scabrous spectacle of noisome installations and hilarious videos,
occupying the whole of the cavernous Gagosian Gallery in Chelsea.
Ingratiating Mr. Kelley's work never has been, nor is it now. But
serious it is, in its brainy, abrasive, black-humored way, and this is
by far his most ambitious and perversely entertaining effort, an
attempted Gesamtkunst-werk of satanic rituals and advertising jingles
mingled with allusions to Godard, German Expressionist cinema and
Stockhausen....
    A teenage girl dressed like a hillbilly recounts
a nonsense parable in the manner of H. P. Lovecraft crossed with William Faulkner
as part of a faux-reality show....

Did I mention the church confirmation in
which a plump female communicant morphs into a devil worshiper, and
teenage boys dressed in Nazi outfits suddenly rap about sex with fat
women?....
     ... Mr. Kelley's deep roots are in the performance tradition going back to the Vienna Actionists.

For descriptions of the Vienna Actionists, do a Google search.

From yesterday:

Angels
Even devils too
Wait to show
How far we come
To joy

-- Chris Whitley, "To Joy
(Revolution of the Innocents)" --
mp3 and lyrics.

It seems that Mike Kelley and Michael Kimmelman are among Chris
Whitley's "devils." Let us hope that they enjoy the company of
General Augusto Pinochet (see previous entry) in the afterlife.

Related material: Art Wars and The Crimson Passion.
Buckley and Pinochet

Yesterday, William F. Buckley, Jr., author of God and Man at Yale, turned 80.

Here is an entry from yesterday, postponed until today so it would not
intervene between yesterday's related entries "Crossroads" and "For
Constantine's Angel."
Recommended reading

for William F. Buckley, Jr.

Joyce and Aquinas (Yale Studies in English)

God and Man in Twentieth-Century Fiction

Modern Literature and the Sense of Time

Three Young Men in Rebellion

James Joyce: Unfacts, Fiction, and Facts

Yeats and the Human Body

Poetry and Prayer
These titles are from an Amazon.com
search. All seem to be by the same "William T. Noon," a Jesuit
priest. Except for Joyce and Aquinas and Poetry and Prayer,
little of Noon's work is now remembered.

Thought to accompany the above reading list:

"And now I was beginning to surmise:
Here was the library of Paradise."

-- Hermann Hesse, Magister Ludi

Before he attains to Paradise, Buckley's reading list in Purgatory might include the complete weblog of Andrew Cusack, a young Christian Fascist at the University of St. Andrews.

A related item...

According to "Today in History," by the Associated Press, for Nov. 25, 2005,

"Today's Birthdays: Former Chilean dictator General Augusto Pinochet is 90...."

If, in fact, Hell also has a library, let us pray that it contains, for Pinochet's future edification, the collected works of Pablo Neruda.
Crossroads

In memory of Diego Rivera,

who died on this date in 1957

"... the socialist muralist Diego
Rivera, hired by Nelson Rockefeller to paint a fresco for the newly
constructed Rockefeller Center in New York, inserted a likeness of
Lenin's head into the fresco. Rockefeller insisted that the head be
replaced or removed, and when Rivera refused the fresco was destroyed....
The event... is captured with
great wit in E.B. White's poem...."

-- Harvard Law Review

I Paint What I See
[A Ballad of Artistic Integrity]
by E.B. White
The New Yorker, 20 May 1933

"'What do you paint, when you paint on a wall?'

Said John D.'s grandson Nelson.

'Do you paint just anything there at all?

'Will there be any doves, or a tree in fall?

'Or a hunting scene, like an English hall?'

'I paint what I see,' said Rivera.

'What are the colors you use when you paint?'

Said John D.'s grandson Nelson.

'Do you use any red in the beard of a saint?

'If you do, is it terribly red, or faint?

'Do you use any blue? Is it Prussian?'

'I paint what I paint,' said Rivera.

'Whose is that head that I see on the wall?'

Said John D.'s grandson Nelson.

'Is it anyone's head whom we know, at all?

'A Rensselaer, or a Saltonstall?

'Is it Franklin D.? Is it Mordaunt Hall?

Or is it the head of a Russian?

'I paint what I think,' said Rivera.

'I paint what I paint, I paint what I see,
'I paint what I think,' said Rivera,
'And the thing that is dearest in life to me
'In a bourgeois hall is Integrity;
'However . . .
'I'll take out a couple of people drinkin'
'And put in a picture of Abraham Lincoln;
'I could even give you McCormick's reaper
'And still not make my art much cheaper.
'But the head of Lenin has got to stay
'Or my friends will give the bird today,
'The bird, the bird, forever.'

'It's not good taste in a man like me,'

Said John D.'s grandson Nelson,

'To question an artist's integrity

'Or mention a practical thing like a fee,

'But I know what I like to a large degree,

'Though art I hate to hamper;

'For twenty-one thousand conservative bucks

'You painted a radical. I say shucks,

'I never could rent the offices-----

'The capitalistic offices.

'For this, as you know, is a public hall

'And people want doves, or a tree in fall

'And though your art I dislike to hamper,

'I owe a little to God and Gramper,

'And after all,

'It's my wall . . .'

'We'll see if it is,' said Rivera.

Related material:

Pictures of the Rockefeller Center mural,
"Man at the Crossroads," and
Rivera's re-creation of the mural,
"Man, Controller of the Universe."

See also another treatment of the "Man at the Crossroads" theme--

The Concrete Gospel
of Donald E. Knuth:

In Hoc Signo
(from Feb. 18),
continued --

This holy icon
appeared at
N37°25.638'
W122°09.574'
on August 22, 2003,
at the Stanford campus.

-- Log24, Feb. 19, 2005
For Constantine's Angel

Lyrics by Chris Whitley,
who died on Sunday,
November 20, 2005:

Angels
Even devils too
Wait to show
How far we come
To joy

-- "To Joy
(Revolution of the Innocents)" --
mp3 and lyrics.