Log24

Mathematics —

See (for instance) a research group at Ghent University.

For those who prefer narrative . . .

See also . . .

Mathematics —

(Some background for the Galois tesseract )

(Click to enlarge)

Narrative —

An essay on science and philosophy in the January 2012
Notices of the American Mathematical Society .

Note particularly the narrative explanation of the double-slit experiment—

"The assertion that elementary particles have
free will and follow Quality very closely leads to
some startling consequences. For instance, the
wave-particle duality paradox, in particular the baffling
results of the famous double slit experiment,
may now be reconsidered. In that experiment, first
conducted by Thomas Young at the beginning
of the nineteenth century, a point light source
illuminated a thin plate with two adjacent parallel
slits in it. The light passing through the slits
was projected on a screen behind the plate, and a
pattern of bright and dark bands on the screen was
observed. It was precisely the interference pattern
caused by the diffraction patterns of waves passing
through adjacent holes in an obstruction. However,
when the same experiment was carried out much
later, only this time with photons being shot at
the screen one at a time—the same interference
pattern resulted! But the Metaphysics of Quality
can offer an explanation: the photons each follow
Quality in their actions, and so either individually
or en masse (i.e., from a light source) will do the
same thing, that is, create the same interference
pattern on the screen."

This is from "a Ph.D. candidate in mathematics at the University of Calgary."
His essay is titled "A Perspective on Wigner’s 'Unreasonable Effectiveness
of Mathematics.'" It might better be titled "Ineffective Metaphysics."

Indiana Jones and the Magical Oracle

Mathematician Ken Ono in the December 2010 American Mathematical Society Notices—

The "dying genius" here is Ramanujan, not Galois. The story now continues at the AMS website—

      (Excerpt from Jan. 27 screenshot;
      the partitions story has been the top
      news item at the site all week.)

From a Jan. 20, 2011, Emory University press release —
"Finite formula found for partition numbers" —

"We found a function, that we call P, that is like a magical oracle," Ono says. "I can take any number, plug it into P, and instantly calculate the partitions of that number. P does not return gruesome numbers with infinitely many decimal places. It's the finite, algebraic formula that we have all been looking for."

For an introduction to the magical oracle, see a preprint, "Bruinier-Ono," at the American Institute of Mathematics website.

Ono also discussed the oracle in a video (see minute 25) recorded Jan. 21 and placed online today.

See as well "Exact formulas for the partition function?" at mathoverflow.net.

A Nov. 29, 2010, remark by Thomas Bloom on that page leads to a 2006 preprint by Ono and Kathrin Bringmann, "An Arithmetic Formula for the Partition Function*," that seems not unrelated to Ono's new "magical oracle" formula—

Click to enlarge

The Bruinier-Ono paper does not mention the earlier Bringmann-Ono work.

(Both the 2011 Bruinier-Ono paper and the 2006 Bringmann-Ono paper mention their debt to a 2002 work by Zagier—  Don Zagier, "Traces of singular moduli," in Motives, Polylogarithms and Hodge theory, Part II  (Irvine, CA, 1998), International Press Lecture Series 3 (International Press, Somerville, MA, 2002),   pages 211-244.)

Some background for those who prefer mathematics to narrative—
The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series ,
by Ken Ono, American Mathematical Society CBMS Series, 2004.

* Proc. Amer. Math. Soc. 135 (2007), 3507-3514.

Apollo's 13: A Group Theory Narrative —

I. At Wikipedia —

II. Here —

See Cube Spaces and Cubist Geometries.

The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
subcubes in the 27-part (Galois) 3×3×3 cube–

A note from 1985 describing group actions on a 3×3 plane array—

Undated software by Ed Pegg Jr. displays
group actions on a 3×3×3 cube that extend the
3×3 group actions from 1985 described above—

Pegg gives no reference to the 1985 work on group actions.

The novel Blood on Snow , set in Oslo, was published
by Knopf on April 7, 2015.  This journal on that date —

A related image —

Continued.

The order-3 affine plane:

Detail from the video in the previous post:

For other permutations of points in the
order-3 affine plane—

See Quaternions in an Affine Galois Plane
and Group Actions, 1984-2009.

See, too, the Mathematics and Narrative post 
from April 28, 2013, and last night's
For Indiana Spielberg.

Related mathematics and narrative —

Talkin' 'bout my generation.

Today's previous post, "For Odin's Day," discussed 
a mathematical object, the tesseract, from a strictly
narrative point of view.

In honor of George Balanchine, Odin might yield the
floor this evening to Apollo.

From a piece in today's online New York Times  titled
"How a God Finds Art (the Abridged Version)"—

"… the newness at the heart of this story,
in which art is happening for the first time…."

Some related art—

and, more recently—

This more recent figure is from Ian Stewart's 1996 revision 
of a 1941 classic, What Is Mathematics? , by Richard Courant
and Herbert Robbins.

Apollo might discuss with Socrates how the confused slave boy
of Plato's Meno  would react to Stewart's remark that

"The number of copies required to double an
 object's size depends on its dimension."

Apollo might also note an application of Socrates' Meno  diagram
to the tesseract of this afternoon's Odin post—

.

From Margaret Wertheim's "Outsider Physicists and the Oh-My-God Particle," New Scientist , Dec. 24, 2011—

For the past 18 years I have been collecting the works of what I have come to call "outsider physicists". I now have more than 100 such theories on my shelves. Most of them are single papers, but a number are fully fledged books, often filled with equations and technical diagrams (though I do have one that is couched as a series of poems and another that is written as a fairy tale)….

The mainstream science world has a way of dealing with people like this— dismiss them as cranks and dump their letters in the bin. While I do not believe any outsider I have encountered has done any work that challenges mainstream physics, I have come to believe that they should not be so summarily ignored.

Consider the sheer numbers. Outsider physicists have their own organisation, the Natural Philosophy Alliance, whose database lists more than 2100 theorists, 5800 papers and over 1300 books worldwide. They have annual conferences, with this year's proceedings running to 735 pages. In the time I have been observing the organisation, the NPA has grown from a tiny seed whose founder photocopied his newsletter onto pastel-coloured paper to a thriving international association with video-streamed events.

The NPA's website tells us that the group is devoted "to broad-ranging, fully open-minded criticism, at the most fundamental levels, of the often irrational and unrealistic doctrines of modern physics and cosmology; and to the ultimate replacement of these doctrines by much sounder ideas".

Very little unites this disparate group of amateurs— there are as many theories as members— except for a common belief that "something is drastically wrong in contemporary physics and cosmology, and that a new spirit of open-mindedness is desperately needed". They are unanimous in the view that mainstream physics has been hijacked by a kind of priestly caste who speak a secret language— in other words, mathematics— that is incomprehensible to most human beings. They claim that the natural world speaks a language which all of us can, or should be able to, understand.

"…a secret language— in other words, mathematics— that is incomprehensible…."

For instance, the "secret language" of Dr. Garret Sobczyk?

See a brief paper by Sobczyk in the NPA Proceedings  described above.

See also Sobczyk in the February 2012 Notices of the American Mathematical Society —

"Conformal Mappings in Geometric Algebra"—

This AMS article, together with Sobczyk's list of previous publications,
indicates that, despite his appearance in the NPA Proceedings , he is definitely not  a crank.

Unfortunately, publication in the Notices  does not by itself guarantee respectability.

For an example, see the Mathematics and Narrative post of Dec. 13, 2011.

Mathematics and Narrative, continued

"… a vision invisible, even ineffable, as ineffable as the Angels and the Universal Souls"

— Tom Wolfe, The Painted Word , 1975, quoted here on October 30th

"… our laughable abstractions, our wryly ironic po-mo angels dancing on the heads of so many mis-imagined quantum pins."

— Dan Conover on September 1st, 2011

"Recently I happened to be talking to a prominent California geologist, and she told me: 'When I first went into geology, we all thought that in science you create a solid layer of findings, through experiment and careful investigation, and then you add a second layer, like a second layer of bricks, all very carefully, and so on. Occasionally some adventurous scientist stacks the bricks up in towers, and these towers turn out to be insubstantial and they get torn down, and you proceed again with the careful layers. But we now realize that the very first layers aren't even resting on solid ground. They are balanced on bubbles, on concepts that are full of air, and those bubbles are being burst today, one after the other.'

I suddenly had a picture of the entire astonishing edifice collapsing and modern man plunging headlong back into the primordial ooze. He's floundering, sloshing about, gulping for air, frantically treading ooze, when he feels something huge and smooth swim beneath him and boost him up, like some almighty dolphin. He can't see it, but he's much impressed. He names it God."

— Tom Wolfe, "Sorry, but Your Soul Just Died," Forbes , 1996

"… Ockham's idea implies that we probably have the ability to do something now such that if we were to do it, then the past would have been different…"

— Stanford Encyclopedia of Philosophy

"Today is February 28, 2008, and we are privileged to begin a conversation with Mr. Tom Wolfe."

— Interviewer for the National Association of Scholars

From that conversation—

Wolfe : "People in academia should start insisting on objective scholarship, insisting  on it, relentlessly, driving the point home, ramming it down the gullets of the politically correct, making noise! naming names! citing egregious examples! showing contempt to the brink of brutality!"

As for "mis-imagined quantum pins"…
This  journal on the date of the above interview— February 28, 2008—

Illustration from a Perimeter Institute talk given on July 20, 2005

The date of Conover's "quantum pins" remark above (together with Ockham's remark above and the above image) suggests a story by  Conover, "The Last Epiphany," and four posts from September 1st, 2011—

Boundary,  How It Works,  For Thor's Day,  and The Galois Tesseract.

Those four posts may be viewed as either an exploration or a parody of the boundary between mathematics and narrative.

"There is  such a thing as a tesseract." —A Wrinkle in Time

Suggested by an Oct. 18 piece in the Book Bench section
of the online New Yorker  magazine—

Related material suggested by the "Shouts and Murmurs" piece
in The New Yorker , issue dated Oct. 24, 2011—

"a series of e-mails from a preschool teacher planning to celebrate
the Day of the Dead instead of Halloween…"

A search for Coxeter + Graveyard in this journal yields…

Here the tombstone says "GEOMETRY… 600 BC — 1900 AD… R.I.P."

A related search for Plato + Tombstone yields an image from July 6, 2007…

Here Plato's poems to Aster suggested
the "Star and Diamond" tombstone.

The eight-rayed star is an ancient symbol of Venus
and the diamond is from Plato's Meno .

The star and diamond are combined in a figure from
12 AM on September 6th, 2011—

Sarah Tomlin in a Nature  article on the July 12-15 2005 Mykonos meeting on Mathematics and Narrative—

"Today, Mazur says he has woken up to the power of narrative, and in Mykonos gave an example of a 20-year unsolved puzzle in number theory which he described as a cliff-hanger. 'I don’t think I personally understood the problem until I expressed it in narrative terms,' Mazur told the meeting. He argues that similar narrative devices may be especially helpful to young mathematicians…."

Michel Chaouli in "How Interactive Can Fiction Be?" (Critical Inquiry  31, Spring 2005), pages 613-614—

"…a simple thought experiment….*

… If the cliffhanger is done well, it will not simply introduce a wholly unprepared turn into the narrative (a random death, a new character, an entirely unanticipated obstacle) but rather tighten the configuration of known elements to such a degree that the next step appears both inevitable and impossible. We feel the pressure rising to a breaking point, but we simply cannot foresee where the complex narrative structure will give way. This interplay of necessity and contingency produces our anxious— and highly pleasurable— speculation about the future path of the story. But if we could determine that path even slightly, we would narrow the range of possible outcomes and thus the uncertainty in the play of necessity and contingency. The world of the fiction would feel, not open, but rigged."

* The idea of the thought experiment emerged in a conversation with Barry Mazur.

Barry Mazur in the preface to his 2003 book Imagining Numbers—

"But the telltale adjective real  suggests two things: that these numbers are somehow real to us and that, in contrast, there are unreal  numbers in the offing. These are the imaginary numbers . 

The imaginary  numbers are well named, for there is some imaginative work to do to make them as much a part of us as the real numbers we use all the time to measure for bookshelves. 

This book began as a letter to my friend Michel Chaouli. The two of us had been musing about whether or not one could 'feel' the workings of the imagination in its various labors. Michel had also mentioned that he wanted to 'imagine imaginary numbers.' That very (rainy) evening, I tried to work up an explanation of the idea of these numbers, still in the mood of our conversation."

See also The Galois Quaternion and 2/19.

New York Lottery last evening