Memorial of
St. Thomas Aquinas:

Joyce and Aquinas,
by William T. Noon, S. J.

St. Thomas Aquinas,
by G. K. Chesterton

Log24, Epiphany 2006

Mozart, 2006

Mozart, 1935

Poet, be seated at the piano.
Play the present, its hoo-hoo-hoo,
Its shoo-shoo-shoo, its ric-a-nic,
Its envious cachinnation.

If they throw stones upon the roof
While you practice arpeggios,
It is because they carry down the stairs
A body in rags.
Be seated at the piano.

That lucid souvenir of the past,
The divertimento;
That airy dream of the future,
The unclouded concerto . . .
The snow is falling.
Strike the piercing chord.

Be thou the voice,
Not you. Be thou, be thou
The voice of angry fear,
The voice of this besieging pain.

Be thou that wintry sound
As of the great wind howling,
By which sorrow is released,
Dismissed, absolved
In a starry placating.

We may return to Mozart.
He was young, and we, we are old.
The snow is falling
And the streets are full of cries.
Be seated, thou.

-- Wallace Stevens, Ideas of Order (1936)

From the center:

"'Mozart, 1935' immediately discloses a will to counter complaints of pure
poetry, to refute that charge heard regularly from Stevens's critics, to find
'his particular celebration out of tune today' on his own if necessary;
and, in short, to meet the communist [poet and critic Willard] Maas's 'respect
for his magnificent rhetoric' at least halfway across from right to left."

-- Alan Filreis, Modernism from Right to Left: Wallace Stevens, the
Thirties, and Literary Radicalism (Cambridge: Cambridge University Press, 1994), p. 211

From the left:

Norman Lebrecht on this year's celebration of the 250th anniversary of Mozart's birth (January 27, 1756):

   "... Mozart, it is safe to say, failed to take music one step
forward....
   ... Mozart merely filled the space between staves with chords
that he knew would gratify a pampered audience. He was a provider of
easy listening, a progenitor of Muzak....
   ...  He lacked the rage of justice that pushed Beethoven into
isolation, or any urge to change the world. Mozart wrote a little night
music for the ancien regime. He was not so much reactionary as
regressive....
   ... Little in such a mediocre life gives cause for celebration....
   ... The bandwaggon of Mozart commemorations was invented by the Nazis in 1941....
   ...  In this orgy of simple-mindedness, the concurrent
centenary of Dmitri Shostakovich-- a composer of true courage and
historical significance-- is being shunted to the sidelines, celebrated
by the few.
    Mozart is a menace to
musical progress, a relic of rituals that were losing relevance in his
own time and are meaningless to ours. Beyond a superficial beauty and
structural certainty, Mozart has nothing to give to mind or spirit in
the 21st century. Let him rest. Ignore the commercial onslaught. Play
the Leningrad Symphony. Listen to music that matters."
     
The left seems little changed since 1935.

"Something beautiful fills the mind yet invites the
search for something beyond itself, something larger or something of
the same scale with which it needs to be brought into relation. Beauty,
according to its critics, causes us to gape and suspend all thought.
This complaint is manifestly true: Odysseus does stand marveling before
the palm; Odysseus is similarly incapacitated in front of Nausicaa; and
Odysseus will soon, in Book 7, stand 'gazing,' in much the same way, at
the season-immune orchards of King Alcinous, the pears, apples, and
figs that bud on one branch while ripening on another, so that never
during the cycling year do they cease to be in flower and in fruit. But
simultaneously what is beautiful prompts the mind to move
chronologically back in the search for precedents and parallels, to
move forward into new acts of creation, to move conceptually over, to
bring things into relation, and does all this with a kind of urgency as
though one's life depended on it."

The above symbol of Apollo
suggests, in accordance with
Scarry's remarks, larger structures.  
Two obvious structures are the affine 4-space over GF(3), with 81
points, and the affine plane over GF(3²), also with 81 points. 
Less obvious are some related projective structures.  Joseph Malkevitch has discussed the standard method of constructing GF(3²)
and the affine plane over that field, with 81 points, then
constructing the related Desarguesian projective plane of order 9, with 9² + 9 + 1 = 91 points
and 91 lines.  There are other, non-Desarguesian, projective
planes of order 9.  See Visualizing GL(2,p), which
discusses a spreadset construction of the non-Desarguesian translation plane of order 9.  This plane
may be viewed as illustrating
deeper properties of the
3x3 array shown above.
To view the plane
in a wider context, see
The Non-Desarguesian Translation Plane of Order 9
and a paper on
Affine and Projective Planes (pdf).
(Click to enlarge the excerpt beow).

See also Miniquaternion Geometry: The Four Projective Planes of Order 9 (pdf), by Katie Gorder (Dec. 5, 2003), and a book she cites:

Miniquaternion geometry: An introduction to the study of projective planes,
by
T. G. Room and P. B. Kirkpatrick. Cambridge Tracts in Mathematics and
Mathematical Physics, No. 60. Cambridge University Press, London, 1971. viii+176 pp.

For "miniquaternions" of a different sort, see my
entry on Visible Mathematics for Hamilton's birthday last year:

 

In Defense of Hilbert
(On His Birthday)



Michael Harris (Log24, July 25 and 26, 2003) in a recent essay, Why Mathematics? You Might Ask (pdf), to appear in the forthcoming Princeton Companion to Mathematics:

"Mathematicians can... claim to be the first postmodernists: compare an
art critic's definition of postmodernism-- 'meaning is suspended in
favor of a game involving free-floating signs'-- with Hilbert's
definition of mathematics as 'a game played according to certain simple
rules with meaningless marks on paper.'"

Harris adds in a footnote:

"... the Hilbert quotation is easy to find
but is probably apocryphal, which doesn't make it any less significant."

If the quotation is probably apocryphal, Harris should not have called it "Hilbert's definition."

For a much more scholarly approach to the concepts behind the alleged quotation, see Richard Zach, Hilbert's Program Then and Now (pdf):

[Weyl, 1925] described Hilbert's project as replacing meaningful
mathematics by a meaningless game of formulas. He noted that Hilbert
wanted to 'secure not truth, but the consistency of analysis' and
suggested a criticism that echoes an earlier one by Frege: Why should
we take consistency of a formal system of mathematics as a reason to
believe in the truth of the pre-formal mathematics it codifies? Is
Hilbert's meaningless inventory of formulas not just 'the bloodless
ghost of analysis'?"

Some of Zach's references:

[Ramsey, 1926] Frank P. Ramsey. Mathematical logic. The Mathematical Gazette, 13:185-94, 1926. Reprinted in [Ramsey, 1990, 225-244].

[Ramsey, 1990] Frank P. Ramsey. Philosophical Papers, D. H. Mellor, editor. Cambridge University Press, Cambridge, 1990

From Frank Plumpton Ramsey's Philosophical Papers, as cited above, page 231:

"... I must say something of the system of Hilbert and his
followers.... regarding higher mathematics as the manipulation of
meaningless symbols according to fixed rules....
   Mathematics proper is thus regarded as a sort of game,
played with meaningless marks on paper rather like noughts and crosses;
but besides this there will be another subject called metamathematics,
which is not meaningless, but consists of real assertions about
mathematics, telling us that this or that formula can or cannot be
obtained from the axioms according to the rules of deduction....
   Now, whatever else a mathematician is doing, he is
certainly making marks on paper, and so this point of view consists of
nothing but the truth; but it is hard to suppose it the whole truth."

[Weyl, 1925] Hermann Weyl. Die heutige Erkenntnislage in der Mathematik. Symposion, 1:1-23, 1925. Reprinted in: [Weyl, 1968, 511-42]. English translation in: [Mancosu, 1998a, 123-42]....

[Weyl, 1968] Hermann Weyl. Gesammelte Abhandlungen, volume 1, K. Chandrasekharan, editor. Springer Verlag, Berlin, 1968.

[Mancosu, 1998a] Paolo Mancosu, editor. From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s. Oxford University Press, Oxford, 1998.

From
Hermann Weyl, "Section V: Hilbert's Symbolic Mathematics," in Weyl's
"The Current Epistemogical Situation in Mathematics," pp. 123-142 in
Mancosu, op. cit.:

"What Hilbert wants to secure is not the truth, but the consistency
of the old analysis.  This would, at least, explain that historic
phenomenon of the unanimity amongst all the workers in the vineyard of
analysis.
   To furnish the consistency proof, he has first of all to formalize
mathematics.  In the same way in which the contentual meaning of
concepts such as "point, plane, between," etc. in real space was
unimportant in geometrical axiomatics in which all interest was focused
on the logical connection of the geometrical concepts and statements,
one must eliminate here even more thoroughly any meaning, even the
purely logical one.  The statements become meaningless figures
built up from signs.  Mathematics is no longer knowledge but a game of formulae,
ruled by certain conventions, which is very well comparable to the game
of chess.  Corresponding to the chess pieces we have a limited
stock of signs in mathematics, and an arbitrary configuration of the pieces on the board corresponds to the composition of a formula out of the signs.  One or a few formulae are taken to be axioms;
their counterpart is the prescribed configuration of the pieces at the
beginning of a game of chess.  And in the same way in which here a
configuration occurring in a game is transformed into the next one by
making a move that must satisfy the rules of the game, there, formal rules of inference hold according to which new formulae can be gained, or 'deduced,' from formulae.  By a game-conforming [spielgerecht]
configuration in chess I understand a configuration that is the result
of a match played from the initial position according to the rules of
the game.  The analogue in mathematics is the provable (or, better, the proven) formula,
which follows from the axioms on grounds of the inference rules. 
Certain formulae of intuitively specified character are branded as contradictions;
in chess we understand by contradictions, say, every configuration
which there are 10 queens of the same color.  Formulae of a
different structure tempt players of mathematics, in the way checkmate
configurations tempt chess players, to try to obtain them through
clever combination of moves as the end formula of a correctly played
proof game.  Up to this point everything is a game; nothing is
knowledge; yet, to use Hilbert's terminology, in 'metamathematics,' this game now becomes the object of knowledge.  What is meant to be recognized is that a contradiction can never occur as an end formula of a proof. 
Analogously it is no longer a game, but knowledge, if one shows that in
chess, 10 queens of one color cannot occur in a game-conforming
configuration.  One can see this in the following way: The rules
are teaching us that a move can never increase the sum of the number of
queens and pawns of one color.  In the beginning this sum = 9, and thus-- here we carry out an intuitively finite [anschaulich-finit]
inference through complete induction-- it cannot be more than this
value in any configuration of a game.  It is only to gain this one
piece of knowledge that Hilbert requires contentual and meaningful
thought; his proof of consistency proceeds quite analogously to the one
just carried out for chess, although it is, obviously, much more
complicated.
   It follows from our account that mathematics and logic must be formalized together.  Mathematical logic, much scorned by philosophers, plays an indispensable role in this context." 

Constance Reid says it was not Hilbert himself, but his critics, who
described Hilbert's formalism as reducing mathematics to "a meaningless
game," and quotes the Platonist Hardy as saying that Hilbert was
ultimately concerned not with meaningless marks on paper, but with ideas:

"Hilbert's program... received its share of criticism.  Some
mathematicians objected that in his formalism he had reduced their
science to 'a meaningless game played with meaningless marks on
paper.'  But to those familiar with Hilbert's work this criticism
did not seem valid.
   '... is it really credible that this is a fair account of
Hilbert's view,' Hardy demanded, 'the view of the man who has probably
added to the structure of significant mathematics a richer and more
beautiful aggregate of theorems than any other mathematician of his
time?  I can believe that Hilbert's philosophy is as inadequate as
you please, but not that an ambitious mathematical theory which he has
elaborated is trivial or ridiculous.  It is impossible to suppose
that Hilbert denies the significance and reality of mathematical
concepts, and we have the best of reasons for refusing to believe it:
"The axioms and demonstrable theorems," he says himself, "which arise
in our formalistic game, are the images of the ideas which form the
subject-matter of ordinary mathematics."'"

-- Constance Reid in Hilbert-Courant, Springer-Verlag, 1986 (The Hardy passage is from "Mathematical Proof," Mind 38, 1-25, 1929, reprinted in Ewald, From Kant to Hilbert.)

Harris concludes his essay with a footnote giving an unsourced Weyl quotation he found on a web page of David Corfield:

".. we find ourselves in [mathematics] at exactly that crossing
point of constraint and freedom which is the very essence of man's
nature."

One source for the Weyl quotation is the above-cited book edited by Mancosu, page
136.  The quotation in the English translation given there:

"Mathematics is not the rigid and petrifying schema, as the layman
so much likes to view it; with it, we rather stand precisely at the
point of intersection of restraint and freedom that makes up the
essence of man itself."

Corfield says of this quotation that he'd love to be told the
original German.  He should consult the above references cited by Richard Zach.

For more on the intersection of restraint and freedom and the essence
of man's nature, see the Kierkegaard chapter cited in the previous
entry.

The Case

An entry suggested by today's New York Times story by Tom Zeller Jr., A Million Little Skeptics:

From The Hustler, by Walter Tevis:

The only light in the room was from the lamp over the couch where she was reading.
    He looked at her face.  She was very
drunk.  Her eyes were swollen, pink at the corners.  "What's
the book?" he said, trying to make his voice conversational. But it
sounded loud in the room, and hard.
    She blinked up at him, smiled sleepily, and said nothing.
    "What's the book?"  His voice had an edge now.
    "Oh," she said.  "It's Kierkegaard.  Soren
Kierkegaard."  She pushed her legs out straight on the couch,
stretching her feet.  Her skirt fell back a few inches from her
knees.  He looked away.
    "What's that?" he said.
    "Well, I don't exactly know, myself."  Her voice was soft and thick.
    He turned his face away from her again, not knowing
what he was angry with.  "What does that mean, you don't know,
yourself?"
    She blinked at him.  "It means, Eddie, that I
don't exactly know what the book is about.  Somebody told me to
read it, once, and that's what I'm doing.  Reading it."
    He looked at her, tried to grin at her-- the old,
meaningless, automatic grin, the grin that made everybody like him--
but he could not.  "That's great," he said, and it came out with
more irritation than he had intended.
    She closed the book, tucked it beside her on the
couch.  "I guess this isn't your night, Eddie.  Why don't we
have a drink?"
    "No."  He did not like that, did not want her
being nice to him, forgiving.  Nor did he want a drink.
    Her smile, her drunk, amused smile, did not
change.  "Then let's talk about something else," she said. 
"What about that case you have?  What's in it?"  Her voice
was not prying, only friendly.  "Pencils?"
    "That's it," he said.  "Pencils."
    She raised her eyebrows slightly.  Her voice seemed thick.  "What's in it, Eddie?"
    "Figure it out yourself."  He tossed the case on the couch.

Related material:

Soren Kierkegaard on necessity and possibility
in The Sickness Unto Death, Chapter 3,

The Diamond of Possibility,

the Baseball Almanac,

and this morning's entry, "Natural Hustler."