Mathematics and Narrative,
Continued:
The Happy Ending Problem
Continued:
The Happy Ending Problem
From Google News this afternoon--
See also the previous entry.
Month: August 2005
-
Date: Sun, 28 Aug 2005
12:30:40 -0400
From: Alf van der Poorten AM
Subject: Vale George Szekeres and
Esther Klein SzekeresMembers of the Number Theory List will be sad to learn that George
and Esther Szekeres both died this morning. George, 94, had been
quite ill for the last 2-3 days, barely conscious, and died first at 06:30.
Esther, 95, died a half hour later.Both George Szekeres and Esther Klein will be recalled by number
theorists as members of the group of young Hungarian mathematicians of
the 1930s including Turan and Erdos. George and Esther's coming to
Australia in the late 40s played an important role in the invigoration
of Australian Mathematics. George was also an expert in group theory
and relativity; he was my PhD supervisor.Emeritus Professor
Alf van der Poorten AM
Centre for Number Theory Research
1 Bimbil Place, Killara NSWRelated material:AVE
"Hello! Kinch here. Put me on to Edenville. Aleph, alpha: nought, nought, one." "A very short space of time through very short times of space....
Am I walking into eternity along Sandymount strand?"-- James Joyce, Ulysses, Proteus chapter
A very short space of time through very short times of space....
"It is demonstrated that space-time should possess a discrete structure on
Planck scales."-- Peter Szekeres, abstract of Discrete Space-Time
Peter Szekeres is the son of George and Esther Szekeres.
ATQUE"At present, such relationships can at best be heuristically
described in terms that invoke some notion of an 'intelligent user
standing outside the system.'"-- Gian-Carlo Rota in Indiscrete Thoughts, p. 152
-
Diamond Theorem Revisited
This evening I wrote a revised version of my 1979 "diamond theorem" abstract.
-
Analogical
Train of Thought
Part I: The 24-Cell
From S. H. Cullinane,
Visualizing GL(2,p),
March 26, 1985--From John Baez,
"This Week's Finds in
Mathematical Physics (Week 198),"
September 6, 2003:Noam Elkies writes to John Baez:
Hello again,
You write:
[...]
"I'd
like to wrap up with a few small comments about last Week. There
I said a bit about a 24-element group called the 'binary tetrahedral
group', a 24-element group called SL(2,Z/3), and the vertices of a
regular polytope in 4 dimensions called the '24-cell'. The most
important fact is that these are all the same thing! And I've learned a
bit more about this thing from here:"[...]
Here's yet another way to see this: the 24-cell is the subgroup of the
unit quaternions (a.k.a. SU(2)) consisting of the elements of norm 1 in
the Hurwitz quaternions - the ring of quaternions obtained from the
Z-span of {1,i,j,k} by plugging up the holes at (1+i+j+k)/2 and its
<1,i,j,k> translates. Call this ring A. Then this group maps
injectively to A/3A, because for any g,g' in the group |g-g'| is at
most 2 so g-g' is not in 3A unless g=g'. But for any odd prime p the
(Z/pZ)-algebra A/pA is isomorphic with the algebra of 2*2 matrices with
entries in Z/pZ, with the quaternion norm identified with the
determinant. So our 24-element group injects into SL2(Z/3Z) - which is barely large enough to accommodate it. So the injection must be an isomorphism.Continuing a bit longer in this vein: this 24-element group then injects into SL2(Z/pZ)
for any odd prime p, but this injection is not an isomorphism once
p>3. For instance, when p=5 the image has index 5 - which, however,
does give us a map from SL2(Z/5Z) to the symmetric group of order 5, using the action of SL2(Z/5Z) by conjugation on the 5 conjugates of the 24-element group. This turns out to be one way to see the isomorphism of PSL2(Z/5Z) with the alternating group A5.Likewise the octahedral and icosahedral groups S4 and A5 can be found in PSL2(Z/7Z) and PSL2(Z/11Z), which gives the permutation representations of those two groups on 7 and 11 letters respectively; and A5 is also an index-6 subgroup of PSL2(F9), which yields the identification of that group with A6.
NDE
The
enrapturing discoveries of our field systematically conceal, like
footprints erased in the sand, the analogical train of thought that is
the authentic life of mathematics - Gian-Carlo RotaLike footprints erased in the sand....
Part II: Discrete Space"Hello! Kinch here. Put me on to Edenville. Aleph, alpha: nought, nought, one."
"A very short space of time through very short times of space....
Am I walking into eternity along Sandymount strand?"-- James Joyce, Ulysses, Proteus chapter
A very short space of time through very short times of space....
"It is demonstrated that space-time should possess a discrete structure on
Planck scales."-- Peter Szekeres, abstract of Discrete Space-Time
"A theory.... predicts that space and time are indeed made of discrete pieces."
-- Lee Smolin in Atoms of Space and Time (pdf), Scientific American, Jan. 2004
"... a fundamental discreteness of spacetime seems to be a
prediction of the theory...."-- Thomas Thiemann, abstract of Introduction to Modern Canonical Quantum General Relativity
"Theories of discrete space-time structure are being studied from a
variety of perspectives."-- Quantum Gravity and the Foundations of Quantum Mechanics at Imperial College, London
Disclaimer:
The above speculations by physicists
are offered as curiosities.
I have no idea whether
any of them are correct.Related material:
Stephen Wolfram offers a brief
History of Discrete Space.For a discussion of space as discrete
by a non-physicist, see John Bigelow's
Space and Timaeus.Part III: Quaternionsin a Discrete Space
Apart from any considerations of
physics, there are of course many
purely mathematical discrete spaces.
See Visible Mathematics, continued
(Aug. 4, 2005):
-
High Concept, continued:
"In the beginning there was nothing.
And God
said, 'Let there be light!'
And there was still nothing,
but now you
could see it."-- Jim Holt, Big-Bang Theology,
Slate's "High Concept" departmentRelated material:
- On the phrase "verbum mentis"
- From Satan's Rhetoric, by Armando Maggi
(University of Chicago Press, 2001):
Page 110: "In chapter I I explained that devils
first and foremost exist as semioticians of the world's signs.
Devils solely live in their interpretations, in their destructive
syllogisms. As Visconti puts it, devils speak the idiom of the
mind.37 .... The exorcist's healing voice states that Satan has always been absent
from the world, that his disturbing and unclear manifestations in the
possessed person's physicality are really nonexistent occurrences,
nothing but disturbances of the mind, since evil itself is a lack of
being."Footnote 37, page 110:
"It is necessary to distinguish the devils' 'language of the mind' and
Augustine's verbum mentis (word of the mind), as he theorizes it
first of all in On the Trinity (book 15). The devils' language
of the mind disturbs the subject's internal and preverbal
discourse." -
1:06:55
PM
"I had an epiphany."
-- Apostolos DoxiadisRelated material:
Log24 March 11:
Lucas Promises a
Darker Star Wars -
High Concept*
"Concept (scholastics' verbum mentis)--
theological analogy of Son's procession
as Verbum Patris, 111-12"
-- index to Joyce and Aquinas,
by William T. Noon, S.J.,
Yale University Press 1957,
second printing 1963, page 162"So did God cause the big bang? Overcome by metaphysical lassitude, I finally reach over to my bookshelf for The Devil's Bible.
Turning to Genesis I read: 'In the beginning there was nothing. And God
said, 'Let there be light!' And there was still nothing, but now you
could see it.'"-- Jim Holt, Big-Bang Theology, Slate's "High Concept" departmentRelated material:
Nothing Ventured,
The God-Shaped Hole, and
Is Nothing Sacred?* See also John
O'Callaghan, Thomistic Realism and the Linguistic Turn: Toward a More
Perfect Form of Existence,
(University of Notre Dame Press, 2003) and Joshua P. Hochschild, "Does
Mental Language Imply Mental Representationalism? The Case of Aquinas’s
Verbum Mentis," Proceedings of the Society for Medieval Logic and Metaphysics, Volume 4, 2004 (pdf), pp. 12-17. -
The Hole
Part I: Mathematics and Narrative
Apostolos Doxiadis
on last month's conference on "mathematics and narrative"--Doxiadis is
describing how talks by two noted mathematicians were related to"... a sense of a 'general theory bubbling up'
at the meeting... a general theory
of the deeper relationship of mathematics to narrative....
"Doxiadis says both talks had "a big hole in the middle."
"Both began by saying something like: 'I
believe there is an important connection between story and mathematical
thinking. So, my talk has two parts. [In one part] I’ll tell you a few
things about proofs. [And in the other part] I’ll tell you about
stories.' .... And in both talks it was in fact implied by a variation
of the post hoc propter hoc, the principle of consecutiveness implying
causality, that the two parts of the lectures were intimately related,
the one somehow led directly to the other."
"And the hole?"
"This was exactly at the point of the
link... [connecting math and narrative]... There is this very
well-known Sidney Harris cartoon... where two huge arrays of formulas
on a blackboard are connected by the sentence 'THEN A MIRACLE OCCURS.'
And one of the two mathematicians standing before it points at this and
tells the other: 'I think you should be more explicit here at step
two.' Both... talks were one half fascinating expositions of lay
narratology-- in fact, I was exhilarated to hear the two most purely
narratological talks at the meeting coming from number theorists!-- and
one half a discussion of a purely mathematical kind, the two parts
separated by a conjunction roughly synonymous to 'this is very similar
to this.' But the similarity was not clearly explained: the hole, you
see, the 'miracle.' Of course, both [speakers]... are brilliant men,
and honest too, and so they were very clear about the location of the
hole, they did not try to fool us by saying that there was no hole
where there was one."
Part II: Possible Worlds"At times, bullshit can only be countered with superior bullshit."
-- Norman MailerMany Worlds and Possible Worlds in Literature and Art, in Wikipedia:
"The concept of possible worlds dates back to a least Leibniz who in his Théodicée
tries to justify the apparent imperfections of the world by claiming
that it is optimal among all possible worlds. Voltaire satirized
this view in his picaresque novel Candide....
Borges' seminal short story El jardín de senderos que se bifurcan ("The Garden of Forking Paths") is an early example of many worlds in fiction.""Il faut cultiver notre jardin."-- VoltaireBackground:
Modal Logic in Wikipedia
Possible Worlds in Wikipedia
Possible-Worlds Theory, by Marie-Laure Ryan
(entry for The Routledge Encyclopedia of Narrative Theory)Part III: Modal Theology"'What is this Stone?' Chloe asked....
'...It is told that, when the Merciful One made the worlds, first of all He created that
Stone and gave it to the Divine One whom the Jews call Shekinah, and
as she gazed upon it the universes arose and had being.'"-- Many Dimensions, by Charles Williams, 1931 (Eerdmans
paperback, April 1979, pp.
43-44)"The lapis was thought of as a unity and therefore often stands
for the prima materia in general."-- Aion, by C.
G. Jung, 1951 (Princeton paperback, 1979, p. 236)"Its discoverer was of the opinion that he had produced the
equivalent of the primordial protomatter which exploded into the
Universe."-- The Stars My Destination, by Alfred
Bester, 1956 (Vintage hardcover, July 1996, p. 216)"We symbolize
logical necessitywith the box
( 
)and logical possibility
with the diamond
( "
).
"The possibilia that exist,
and out of which
the Universe arose,
are located in
a necessary being...."
-- Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinityat Christ Church College, Oxford
(the home of Lewis Carroll)
-
Truth vs. BullshitBackground:
For an essay on the above topicfrom this week's New Yorker,
click on the box below.
Representing truth:
Rebecca Goldstein
Representing bullshit:
Apostolos Doxiadis
Goldstein's truth: Gödel was a Platonist who believed in objective truth.
See Rothstein's review of Goldstein's new book Incompleteness.
Doxiadis's bullshit: Gödel, along with Darwin, Marx, Nietzsche, Freud, Einstein, and Heisenberg, destroyed
a tradition of certainty that began with Plato and Euclid."Examples are the stained-glass
windows of knowledge." -- Nabokov
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