Just in case you thought the musings in this place were a bit run-of-the-mill ;) – here are some posts that take the thinking further afield. Here we honour resonance over and above relevance…

The virtual museum is not simply a museum in virtual space

[ by Charles Cameron — on the infinite juxtaposition of similars, opposites and related modes of scholarship — in gallery and museum, catalog and library ]

twins & sunflowers

Van Gogh’s Sunflowers: the Amsterdam and National Gallery Sunflowers side by side, Jan 2014. Photo credit: Julian Simmonds, Telegraph UK

Please note that what I term here the “virtual museum” is intended to cover both a physical museum or gallery space with available or built in digital affordances and the museum as a completely portable function of the digital network and its devices alone.

I originally wrote this set of notes on February 10, 1997, and have made only tiny changes in the text as presented here — removing one paragraph that was left incomplete, switching the last two bullet points, and placing one “spare” sentence in a suitable context.

As I look back to those days of the Magister Ludi list, and forward to Cath’s progress with Sembl, I have a sense that this document was prescient, the seed of much that is coming into being now, as we speak. Like all such visions, the manifestation has developed over time, but the idea of the ready, multiple comparison of museum or gallery objects, together with supporting documentation, is still fresh: over time the invisible becomes cutting-edge.

To set the scene, here is a quote from Sven Birkerts that had long inspired me:

There are tremendous opportunities, and we are probably on the brink of the birth of whole new genres of art which will work through electronic systems. These genres will likely be multi-media in ways we can’t imagine. Digitalization, the idea that the same string of digits can bring image, music, or text, is a huge revolution in and of itself. When artists begin to grasp the creative possibilities of works that are neither literary, visual, or musical, but exist using all three forms in a synthetic collage fashion, an enormous artistic boom will occur.

With that insight in mind, here’s a glimpse of my early thoughts about the glass bead game and the museum:

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That’s right — the virtual museum is not simply a museum in virtual space

I

What’s going on here is that we’re dealing with a multidimensional space rather than the flat space of a wall or the three dimensional space of a room.

  • Walk-through “real-life” museums necessarily organize their collections in such a way that one work of art is sequentially related to the next. The visitor walks up a corridor, or through a room, and takes in each work in sequence, carrying a little of the previous work trailing in memory — and on occasion stepping back to view two works placed next to one another in a comparative way.
  • In her hand or in his ear, a textual commentary is available: the catalogue. And this is typically consulted in a one-to-one relation, such that picture 63 is viewed and the text for picture 63 heard or read.

The museum is a collection of physical objects with stories which explain them: virtual space is a space of virtual objects with linkages between them.

  • It follows that the virtual museum is a collection of virtual objects and the linkages between them.
  • But what are those objects?
  • We cannot assume the objects in the virtual museum are limited to the objects in the physical museum: if nothing else, the stories which explain those objects will themselves be objects in the virtual museum.
  • Both “collection objects” and “catalogue entries” are represented in the same digital fashion. The catalogue entries, in other words, are objects in the virtual museum.
  • We do not carry a catalog as we browse the virtual museum… “collection” and “catalog” merge.

The virtual museum is its own virtual catalog.

  • And this is because the digital democratization of information which obtains on the web renders the “art object” and the “art-historical text” functionally equal.
  • In fact, “digital democratization” allows for the expansion of presentable content to include not only visual and art historical materials on an equal footing, but also all manner of other texts, the world of literature and drama, architectural renderings, mathematical analogs and explanations, sounds and musical items…
  • Thus the virtual museum need not and should not limit itself to physical objects [eg pictures, sculptures] and associated texts, but can and should contain linkages to other arts and modes of representation [eg musical, literary, historical, scientific and mathematical expressions].
  • Furthermore, the virtual museum need not limit itself to the objects in its sponsor museum’s holdings, but may also contain linkages to the holdings of other museums: indeed — and importantly — web-based “frames” make this possible without the viewer leaving the originating web site.
  • Not just the museum catalogue and reference library, but also the world’s other museums, private collections, text libraries, record libraries and databases are all available as reference points for the items in the collection.
  • Linkage, in other words, is the “new” in our context, while objects and their stories are the given.
  • We do not move from room to room but from link to link as we browse the virtual museum.

The virtual museum can be conceived as an ellipse with one focus in the originating collection and the other in world cultural history…

  • The “virtual proximity” of other bodies of knowledge on the net and web invites the inclusion of multiple reference points outside the collection: effectively, the museum as we know it transforms into a repository of world culture whose special focus is the collection:
  • The virtual museum is thus no longer archeologically or artistically based: it encompasses all forms of expression.
  • The museum becomes an expression of cultural totality.

The floorplan of the virtual museum is an n-dimensional graph of nodes and links.

  • The essence of the difference between the museum and the virtual museum is this: objects in the virtual museum are “next to” a far larger number of other objects than objects in the physical museum.
  • The system of linkages inherent in the structure of the Internet and the World Wide Web expands our concept of the museum by making possible a bewildering variety of new “throughways” between and among the items displayed, and “outside” the museum: thus raising new problems and possibilities in sequencing the experience of the “visitor”.
  • What happens as a result is that linkage itself blossoms from a narrow and largely sequential business into a multiplex affair.
  • The juxtaposition of one artefact with another explodes in an unimaginable freedom, and a system of constraints must therefore be imagined to limit and lead the viewer — through a “garden of forking paths” — to a desired and appropriate outcome.

To understand this is to make a virtue of the virtual … and a cathedral of the museum.

II

The virtual museum is not simply a museum in virtual space, but the virtual presentation of whatever the museum-as-archetype has been or will be in the labyrinth of human vision.

  • The sequencing the visitor’s experience in virtual space will thus inevitably reflect the topology not only of the collection, but also of the catalog and of the web itself.
  • And this topography brings a new feature to the foreground: linkage. The links between items themselves begin to assume considerable esthetic importance.
  • The museum and the library can no longer be separated, since their contents are intermingled: and the result is that the virtual museum, like the cathedral before it, becomes a speculum mundi or”mirror of the world”.

We live in secular times, and the museum is our cathedral.

  • This could mean, minimally, that the museum has replaced the cathedral as the central space where people congregate in a culturally rich environment. Maximally, and thus potentially, it means that whatever the cathedral was for us — master artwork of combined artworks in many media, ritual space, hub of the city, mirror of worlds — the museum can be.
  • The secular does not lack for a sacred dimension, but offers access to it in a manner that does not demand a specific, local belief or practice.
  • The virtual museum as secular cathedral is the place where all the world’s imaginal trasures come together as offerings, and from which all the world departs imaginally enriched.
  • The museum is thus heir to the phenomenology of shamans, saints and mystics, as well as of artists and their patrons, teachers and students — for it is visited by crowds in which each individual carries a different cultural inheritance, now Italian, now Congolese, now Navaho, now Santeria…

The test of the museum is its cathedral-effectiveness: its capacity to invoke wonder.

  • The virtual museum is thus a special case of the “art form” described by Hermann Hesse in his novel The Glass Bead Game:
  • The elevation of the virtual museum is a sacramental elevation.

18 years, and what a difference!

Eighteen years ago, in 1996, we used to play HipBone Games by email (“PBEM”). The boards were literally typed out in ASCII characters — to give you the idea, here’s the WaterBird Board as we played on it back then…

WaterBird in 1996 ASCII 601 Sembl

Eighteen years have passed, and the latest board in the genre, designed for the Museum Game, looks like this:

Lotus board 601 for Sembl

Simply beautiful, Cath!

We’ve come a long way… and I’m eager to see where we’ll be heading next..

The White House, Games, and HipBone/Sembl — today

[ cross-posted from Zenpundit — a project of keen interest to me, and a request for your support ]
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Something is going on in one corner of the White House that has me agog in a pleasant way.


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Mark DeLoura, Senior Advisor for Digital Media at the WH Office of Science and Technology Policy is soliciting ideas about Games that Can Change the World. I’ve jumped in, and so have some old friends, one auld acquaintance and one new…

*****

The home page for this project is hosted on its own Games for Impact site, and I’d invite you to take a look, and note in particular…

  • Games where ideas collide (& create new ideas)
  • This is my own page, for the HipBone / Sembl games and DoubleQuotes — and if you have found my style of analysis valuable, you may want to go there, (take the trouble to) log in, and upvote my idea — making a comment too, should you so wish.

  • Positive Impact and Game Evangelism
  • Similarly, you can log in and upvote the whole idea by supporting this proposal, the current “leading” concept…

    Part of what makes this entry so interesting is the fact that Chris Crawford, game designer and thinker non pareil, is discussing his own long-hoped-for paradigm shift in game design in this thread. Chris is the “auld acquaintance” I mentioned, and I met him via the good services of my old friend Mike Sellers late in the last century. It is good to read him again in the new millennium.

  • Games to increase understanding about emergent social systems
  • Mike’s own offering is this one, which I also highly recommend. Mike is one of the founding fathers of multiplayer games with graphical architecture, and has more recently been working to bring human psychology into gameplay with increasing subtlety. By all means give him a vote up if that sounds good.

  • Knecht/Connect – a playable version of the Glass Bead Game
  • As you know, my own games attempt to bring the game concept embedded in Hermann Hesse’s great novel, The Glass Bead Game / Magister Ludi into playable form, and my friend Paul Pilkington has been doing the same in a series of books [1, 2, 3] and a Twitter stream. Let’s help him get some recognition, too…

  • Try the Poietic Generator
  • This one’s a game concept I like, too — it’s based on Conway‘s Game of Life… and brings it alive!

    It was submitted by Olivier Auber, whom I hadn’t previously met — so he’s my new acquaintance, and I’m hoping his game ideas will flourish and that acquaintance will grow into friendship in as things unfold…

    *****

    So that’s the overall project, along with a sampling of specific ideas that I admire and would invite you to support. I hope you’ll find (and support) some other game concepts of interest, too.

    In a follow up post honoring Chris Crawford<– which may still take a while to write and post — I’ll be looking at some of the historical background of “serious games” — and of the HipBone / Sembl style of thinking in particular.

    Leap Worlds: my 3QD attempt

    [ cross-posted from zenpundit.com — another runup to the glass bead game ]

    Below you can read my submission to the wonderful 3 Quarks Daily “web aggregator” site as a candidate to join their regular Monday blogging team — it didn’t even make their “close but no cigar” list, but I wrote it and I like it, so I’m posting it here.


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    They are both fine American authors, stylists of high repute yet little known, Annie Dillard and Haniel Long, but I hadn’t associated the two of them particularly closely until that day. I’d snarfed up a copy of Long’s Letter to Saint Augustine from the dollar box at Pasadena’s magnificent Archives theological bookstore, and on my way home to nearby Eagle Rock, stumbled on a passage that seemed hazily familiar.

    My friend Jens Jensen, who is an ornithologist, tells me that when he was a boy in Denmark he caught a big carp embedded in which, across the spinal vertebrae, were the talons of an osprey. Apparently years before, the fish hawk had dived for its prey, but had misjudged its size. The carp was too heavy for it to lift up out of the water, and so after a struggle the bird of prey was pulled under and drowned. The fish then lived as best it could with the great bird clamped to it, till time disintegrated the carcass, and freed it, all but the bony structure of the talon.

    By the time I arrived back at my books, I knew it was Annie Dillard I’d been reminded of, and a quick, no, excited but fumbling search turned up this passage from her Teaching a Stone to Talk:

    And once, says Ernest Seton Thompson–once, a man shot an eagle out of the sky. He examined the eagle and found the dry skull of a weasel fixed by the jaws to his throat. The supposition is that the eagle had pounced on the weasel and the weasel swiveled and bit as instinct taught him, tooth to neck, and nearly won. I would like to have seen that eagle from the air a few weeks or months before he was shot: was the whole weasel still attached to his feathered throat, a fur pendant? Or did the eagle eat what he could reach, gutting the living weasel with his talons before his breast, bending his beak, cleaning the beautiful airborne bones?

    They’re emblematic, those two quotes, the way I see them: emblematic in the sense that each could be the basis for a heraldic shield, the eagle stooping to carry off the weasel, air creature triumphant over creature of earth in the one, the fish dragging the bird down in the other — creature of water gathering the air creature into its own fatal realm.

    Emblematic too, I see them, of the possibility of a rhyme between thoughts — and I have quoted them as such in previous essays, alongside rhymes of sound and rhymes of meaning, womb and tomb being the classic examples of choice, but also rhymes of image, the fan rotors and helicopter rotors in the opening sequences of Coppola‘s Apocalypse Now, rhymes of melody in counterpoint, in fugue… and rhymes, yes, in history, as when Sir Frederick Stanley Maude told the people of Baghdad in 1917, “Our armies do not come into your cities and lands as conquerors or enemies, but as liberators…” and Donald Rumsfeld chimed in, telling his own troops in that same weary city, almost a century later, “Unlike many armies in the world, you came not to conquer, not to occupy, but to liberate and the Iraqi people know this.”

    Oh yeah?

    *****

    The world is woven, Jung tells us in one of his most original and illuminating insights, of woof and warp, causaland acausal principles , it is at once at every point synchronic and diachronic.

    Moving through time, we have the causal, diachronic principle, one things leads to another: let us sort and analyze the actions of one thing on another until psychology becomes sociology, sociology turns into evolutionary biology, biology into genetics, genetics a form of chemistry that is essentially physics — and physics, at the quark-level at last, a matter of statistics, mathematics unfettered even by the dualism of wave and particle.

    Let me be clear about this. I do not begrudge Higgs his boson or his Nobel, nor Chandrasekhar his Nobel or his limit. But both men followed the warp, the causal, time-bound length-wise threads of discovery to their fraying edges. And the causal threading of events, time’s warp in our lived universe, is the mode best suited to quantity, to the determinable, and knows little of mystery unless magnitude alone — the infinities of astronomy, the infinitessimals of subatomics, alone will qualify.

    Oh, scale is a marvel, true enough — but it is quality, not quantity, where the mystery and the greater meaning resides.

    *****

    And so we come to the mind’s other faculty, the other manner in which the world is woven, the manner of rhyme and repetition, synchronicities and semblances, of patterns recognized within and across disciplines. The cross-weave.

    This has been the step-child of cognition for too long — but with the rise of cybernetics, feedback loops, complexity theory, network thinking and multi-causality, we can no longer think only in linear progressions, but must also cultivate associative, lateral, sideways thinking — in short, creative leaps.

    Creative leaps occur when we recognize commonalities across conceptual distances — theme and variations, as musicians would say, rhymes in the nature of things, multiple perspectives and voices in counterpoint. So the nature of our current world, in all its complexity and variegation, calls for what I would call a music of ideas.

    I’m not the first to have this idea, Glenn Gould was pursuing it in his work for radio, blending the many voices and conversations in a train compartment, or at the different tables in a truck-stop café, to form an interwoven whole that comprehended all of its voices, all of its parts in a greater music. But it is Edward Said — another musicians, when he was not occupied with Israeli-Palestinian politics or literature — who observed in an essay in Power, Politics and Culture, p. 447:

    When you think about it, when you think about Jew and Palestinian not separately, but as part of a symphony, there is something magnificently imposing about it. A very rich, also very tragic, also in many ways desperate history of extremes — opposites in the Hegelian sense — that is yet to receive its due. So what you are faced with is a kind of sublime grandeur of a series of tragedies, of losses, of sacrifices, of pain that would take the brain of a Bach to figure out. It would require the imagination of someone like Edmund Burke to fathom.

    So that’s the symphonic scope of the thing, seeing the whole with all its fractures and dissonances as a music — a music that calls for harmonization, but remains in complex counterpoint.

    *****

    Hermann Hesse was the first great proponent of the music of ideas, at least in modern western times, and his glass bead game — evoked but never defined in his Nobel-winning novel of that name — offers us a glimpse both of the nature of moves and of the possible grandeur of an implicit world-architecture, formed of resonances and semblances, rather than of causes and effects.

    Hesse gives us an insight into how the game is played at the level of moves and themes when he says a given game might have explored “the rhythmic structure of Julius Caesar’s Latin and discovered the most striking congruences with the results of well-known studies of the intervals in Byzantine hymns” — but he could hardly have known, back in the 1940s, that in 1978 the University of Wisconsin Press would publish Jane-Marie Luecke OSB’s monograph, Measuring Old English Rhythm: an Application of the Principles of Gregorian Chant Rhythm to the Meter of Beowulf. Intentionally or unintentionally, his game is played by all whose minds play with meanings. And there’s a game move right there, in the conviviality between the fictional move of Hesse himself, and the monograph, years later, of a Benedictine nun.

    Tiny, you say, a tiny move — but a move in what Hesse termed the “hundred-gated Cathedral of Mind.”

    And the scope of that cathedral, in its gradual entirety, is vast — encompassing all the vast quantities of the sciences, with the qualitative depths and heights of the arts and humanities too:

    The Glass Bead Game is thus a mode of playing with the total contents and values of our culture; it plays with them as, say, in the great age of the arts a painter might have played with the colors on his palette. All the insights, noble thoughts, and works of art that the human race has produced in its creative eras, all that subsequent periods of scholarly study have reduced to concepts and converted into intellectual values the Glass Bead Game player plays like the organist on an organ. And this organ has attained an almost unimaginable perfection; its manuals and pedals range over the entire intellectual cosmos; its stops are almost beyond number. Theoretically this instrument is capable of reproducing in the Game the entire intellectual content of the universe.

    We are drawing close, in a humanly possible way and without making any predications of the being or non-being of such a supposed entity, to the very mind of god.

    *****

    And with immediate real world application, as when Maxwell sees the commonality between electricity and magnetism, Kekulé the serpent-biting-it’s-tail like form of the benzene ring — or Taniyama‘s 1955 “surmise” as Barry Mazur puts it, that “every elliptic equation is associated with a modular form” — an insight that was to bear rich fruit forty years later, in Andrew Wiles‘ proof of Fermat’s Last Theorem.

    Hesse’s vision of the game was lucid, elegant, intellectual — but lacked real world application. Viewed as a method of scoring the music of ideas, it offers illuminations from the most abstract and theoretical of mathematics to the most complex of opposed political intrangencies — from Tamiyama to Edward Said, and from Christopher Alexander‘s Pattern Language to John Holland‘s “genetic algorithms“. Alexander and Holland each indicate their debt to Hesse’s fictional game in their own respective works.

    How, then, to notate this game, these moves that leap side-wise, pattern-wise, semblance-wise across boundaries and boxes, limitations and disciplines — for it is these leaps, as Arthur Koestler notes in his The Act of Creation, that give us the surprised aha! of discovery, the unexpected ha! of laughter, the gut-wrenching wail aiiyeee! when tragedy strikes.

    My own inclinations favor play — solo or with a friend — at a coffee house, with pencil and paper napkin, with a small graph for a board, ideas played at its nodes, connections and resonances represented by its edges. I’m thinking of network mapping on a human scale, between seven and a dozen nodes, each one rich in meaning, anecdote, quote, statistic, image, snatch of song…

    But the essence is the single move, the single resemblance. And for this I have a form which I would like to offer you, downloadable, for your own experiments:

    Download the image and fill the two spaces with what you will — whatever you can sketch, whistle, count out or scribble —

    I think you’ll find the greatest reward comes when the two ideas, visuals, verbals, aurals you juxtapose are closely related yet drawn from distantly separated regions of thought.

    At their best, such juxtapositions cross galaxies. My own most cherished example to date compares a night sky by Van Gogh with a von Kármán diagram of turbulent flow…

    *****

    And where does this lead us? What becomes of CP Snow‘s famed Two Cultures?

    Two great rows of pillars in Hesse’s hundred-gated cathedral, perhaps — best appreciated when one looks up, and sees the great vaulted roof, the magnificence of the arches between them.

    In future Monday columns here on 3QD, I hope to bring you some further moves in the great game of correspondences, weaving together topics that have caught my attention in the preceding month — now in poetry, now on the morning news.

    The leaps.

    ___________________________________________________________________________________________________


    I’m happy to report that friend Bill Benzon made the 3 WQuarks Daily cut and will be blogging there, and that friend Omar Ali is already one of their regulars.

    Iron fists in velvet gloves and the like

    My friend William Benzon posted a photo on his New Savannah site today, and it prompted me to make this post. Here’s Bill’s photo, under his post’s title:

    Cruise with Moss

    IMGP3577

    …..

    The moss is a bit untidy, and so is the cruise ship, as you can see more clearly by clicking on the image above and viewing it at higher scale on Flickr — perhaps we could even say the levels of untidiness (entropy?) are somehow parallel. But it’s the juxtaposition that interests me. Why does the good doctor Benzon chose to phosograph this scene, and present it on his blog? And I think the title spells out what the picture invites us to see: a contrast between “opposites”.

    Opposites? If I’d asked you what the opposite of a ship was, “moss” would be unlikely to be the first word that sprang to mind. And yet they are opposites, as our response to the image may suggest, and as the title Bill gave the picture suggests in a different “cognitive octave”.

    The opposition “water craft, land cover” would describe many a dockyard or seaside photo, of course. But it’s the softness of moss set against the harshness of iron and steel, I’d suggest, that is the real opposition we sense in looking at Benzon’s image. And that’s effectively the same contrast that powers the phrase “an iron fist in a velvet glove”…

    …..

    As many of you may know, President Valdimir Putin of Russia took to the air in a motorized hang glider a year agao, “as part of a project to teach the endangered birds who were raised in captivity to later follow the aircraft on their migration south to Central Asia.” Well, one of the birds went astray, was discovered and returned to Moscow, and is now safely ensconced in a wildlife reserve.

    What interested me, though, was the name of this particular bird, as reported by the Associated Press:

    The white bird, called Raven…

    Ha! Ravens are preeminently black birds!

    All of which is simply to say that dissimilars and oppositions can be as fascinating, to the artist’s eye, as parallelisms and semblances.

    GMTA: Temple Grandin

    [ cross-posted from Zenpundit — here’s today’s windfall apple from the tree of creative delight ]

    On March 31st, 2012 (or very likely the evening of the day before), I posted a graphic on Zenpundit, where I often blog:

    The upper image illustrates Theodore von Kármán‘s mathematics of turbulent flow, the lower image Vincent van Gogh‘s view of the night sky, and I juxtaposed them using my “DoubleQuotes” format to illustrate the underlying unity of the arts and sciences, and the breathtaking beauty and insight we can derive when we recognize a “semblance” — a rich commonality that transcends our usual division of concepts into separate and un-mutually-communicative “disciplines” and “silos”.

    Apparently, this kind of cognition — the basis of every DoubleQuote, and of every move in one of the Hipbone / Sembl games — has now been termed “pattern thinking”.

    …..

    According to Amazon, Temple Grandin and Richard Panek‘s book The Autistic Brain: Thinking Across the Spectrum was released April 30, 2013 although books are often available a couple of weeks ahead of release date, and galleys and proofs earlier still).

    I read about it for the first time today, in Grandin & Panek’s piece, How an Entirely New, Autistic Way of Thinking Powers Silicon Valley in Wired. That article begins with a pull-quote from Grandin’s book:

    I’ve given a great deal of thought to the topic of different ways of thinking. In fact, my pursuit of this topic has led me to propose a new category of thinker in addition to the traditional visual and verbal: pattern thinkers.

    Obviously, that’s something i’d want to find out more about, so I read on into the article, expecting good things. Imagine my surprise when I read this paragraph, though:

    Vincent van Gogh’s later paintings had all sorts of swirling, churning patterns in the sky — clouds and stars that he painted as if they were whirlpools of air and light. And, it turns out, that’s what they were! In 2006, physicists compared van Gogh’s patterns of turbulence with the mathematical formula for turbulence in liquids. The paintings date to the 1880s. The mathematical formula dates to the 1930s. Yet van Gogh’s turbulence in the sky provided an almost identical match for turbulence in liquid.

    Boom!

    …..

    Okay, I just received my review copy of Hofstadter and Sander, Surfaces and Essences: Analogy as the Fuel and Fire of Thinking — I guess I’ll have to review Grandin and Panke here, too.

    Wiles’ Proof of Fermat’s Last Theorem viewed as a Glass Bead Game

    In this piece — written in 2003 or perhaps earlier — I offer an exploration of Andrew Wiles’ proof as described in the book, Fermat’s Last Theorem by Simon Singh, in the light of the Glass Bead Game posited by Hermann Hesse in his Nobel-winning novel Das Glasperlenspiel, together with my own suggestion that the richest “move” in such a game would consist of a rich isomorphism between rich chunks of knowledge in widely separated disciplines… — essentially, that’s a Sembl move.

    …..

    wiles3Andrew Wiles

    …..

    The mathematician Pierre de Fermat scribbled a note in the margin of his copy of DiophantusArithmetica in 1637 or thereabouts claiming that there were no solutions to the equation

    x^n + y^n = z^n

    where n is greater than 2, along with a note saying “I have a marvellous demonstration of this proposition which this margin is too narrow to contain”.

    Mathematicians as great as Leonhard Euler strove to prove or disprove the theorem without success for three and a half centuries:

    Consider the leaps in understanding in physics, chemistry, biology, medicine and engineering that have occurred since the seventeenth century. We have progressed from ‘humours’ in medicine to gene-splicing, we have identified the fundamental atomic particles, and we have placed men on the moon, but in number theory Fermat’s Last Theorem remained inviolate. [Singh, p. xi]

    Andrew Wiles‘ proof of Fermat’s Last Theorem appeared in the May 1995 issue of the “Annals of Mathematics”, and now there’s this fascinating book by Simon Singh, which talks the layman through the process by which Wiles arrived at it.

    …..

    From a mathematical point of view, what is interesting about Wiles’ proof — beyond the fact that it lays to rest the great mathematical puzzle contained in Fermat’s marginal note — is that it consisted in the proof of the “Taniyama-Shimura conjecture”, a brilliant mathematical guess, in effect, which suggested that there was an exact correspondence between two areas at “opposite ends” of mathematics, which nobody would otherwise suppose had anything to do with one another: “modular forms” and “elliptic equations”.

    Yutaka Taniyama made this suggestion in a rather roundabout way at a symposium in Tokyo in 1955, and his work was continued after his suicide by his friend and colleague Goro Shimura.

    Barry Mazur of Harvard describes the conjecture thus:

    It was a wonderful conjecture — the surmise that every elliptic equation is associated with a modular form — but to begin with it was ignored because was so ahead of its time. When it was first proposed it was not taken up because it was so astounding. On the one hand you have the elliptic world, and on the other you have the modular world. Bothg these branches of mathematics have been studied intensively but separately. Mathematicians studying elliptic equations might not be well versed in things modular, and conversely. Then along comes the Taniyama- Shimura conjecture which is the grand surmise that there’s a bridge between these two completely different worlds. Mathematicians love to build bridges. [211-12]

    …..

    In 1984, Gerhard Frey showed a strong connection between Fermat’s Theorem and the Taniyama-Shimura conjecture, and Ken Ribet added a crucial piece to the puzzle, finally proving that *if* the conjecture could be proved, that would be enough to prove Fermat’s Theorem — but the conjecture remained a conjecture, and Fermat’s Theorem remained unproven. Until Wiles proved the Taniyama-Shimura conjecture, and thus Fermat’s Last Theorem as well.

    Wiles, in other words, had not only solved Fermat’s puzzle, but along the way had definitively linked two widely separate areas of mathematics.

    The value of mathematical bridges is enormous. They enable communities of mathematicians who have been living on separate islands to exchange ideas and explore each others’ creations. Mathematics consists of islands of knowledge in a sea of ignorance. For example, there is an island occupied by geometers who study shape and form, and then there is an island of probability where mathematicians discuss risk and chance. There are dozens of such islands, each one with its own unique language…

    Barry Mazur thinks of the Taniyama-Shimura conjecture as a translating device similar to the Rosetta stone… [212]

    …..

    And here’s where the Glass Bead Game comes in. Not surprisingly, Wiles’ proof of the Taniyama-Shimura conjecture had a profound impact on mathematics. In Ken Ribet’s words:

    The landscape is different, in that you know that all elliptic equations are modular and therefore when you prove a theorem for elliptic equations you’re also attacking modular forms and vice versa. [305]

    In Mazur’s:

    It’s as if you know one language and this Rosetta stone is going to give you an intense understanding of the other language… But the Taniyama-Shimura conjecture is a Rosetta stone with a certain magical power. The conjecture has the very pleasant feature that simple intuitions in the modular world translate into deep truths in the elliptic world, and conversely. What’s more, very profound problems in the elliptic world can get solved sometimes by translating them using this Rosetta stone into the modular world, and discovering that we have the insights and tools in the modular world to treat the translated problem. Back in the elliptic world we would have been at a loss. [212-13]

    And in Singh’s:

    Via the Taniyama-Shimura conjecture Wiles had unified the elliptic and modular worlds, and in so doing provided mathematics with a short cut to many other proofs — problems in one domain could be solved by analogy with problems in the parallel domain. [305]

    …..

    The analogy, in other words, illuminates both the fields which it joins. And this kind of deep analogical thinking across disciplinary boundaries lies at the very heart of the Bead Game, and is a hallmark of creativity in general:

    Relationships between apparently different subjects are as creatively important in mathematics as they are in any discipline. The relationship hints at some underlying truth which enriches both subjects. For instance, originally scientists had studied electricity and magnetism as two completely separate phenomena. Then, in the nineteenth century, theorists and experimentalists realised that electricity and magnetism were intimately related. This resulted in a deeper understanding of both of them. Electric currents generate magnetic fields, and magnets can induce electricity in wires passing close to them. This led to the invention of dynamos and electric motors, and ultimately the discovery that light itself is the result of magnetic and electric fields oscillating in harmony. [204-5]

    …..

    But proving the analogy which Taniyama and Shimura conjectured between the two fields of modular forms and elliptic equations involved Wiles in a very wide ranging process:

    During Wiles’s eight-year ordeal he had brought together virtually all the breakthroughs in twentieth-century number theory and incorporated them into one almighty proof. He had created completely new mathematical techniques and combined them with traditional ones in ways that had never been considered possible. In doing so he had opened up new lines of attack on a whole host of other problems. According to Ken Ribet the proof is a perfect synthesis of modern mathematics and an inspiration for the future: ‘I think that if you were on a desert island and you had only this manuscript then you would have a lot of food for thought. You would see all of the current ideas of number theory. You turn to a page and there’s a brief appearance of some fundamental theorem by Deligne and then you turn to another page and in some incidental way there’s a theorem by Hellegouarch — all of these things are just called into play and used for a moment before going on to the next idea. [304]

    Wiles’ work, in other words, is not only a rigorous analogical bridge between two distant branches of mathematics, but also a *symphonic* work.

    …..

    Looking to the future, Wiles’ work can be seen as a first major contribution — and booster — to Robert Langlands‘ proposal for a grand unified scheme which will embrace all of mathematics by means of other “bridging” conjectures and proofs…

    During the 1960s Robert Langlands, at the Institute for Advanced Study, Princeton, was struck by the potency of the Taniyama-Shimura conjecture. Even though the conjecture had not been proved, Langlands believed it was just one element of a much grander scheme of unification. He was confident that there were links between all the main mathematical topics and began to look for these unifications. Within a few years a number of links began to emerge. …

    Langlands’ dream was to see each of these conjectures proved one by one, leading to a grand unified mathematics.

    …..

    And that’s about as far as my layman’s brain can go…

    The Abel Prize for a great Sembl move

    You don’t have to be playing a Sembl or Hipbone game to make a great Sembl move — you just have to see a rich semblance between two concepts in (previously) widely separated fields of thought. Thus Pierre Deligne of Princeton’s Institute for Advanced Study, who won the Abel Prize in mathematics this year, did so by working on a rich Sembl-style insight from André Weil. As Scientific American reports today:

    Deligne’s most spectacular results are on the interface of two areas of mathematics: number theory and geometry. At first glance, the two subjects appear to be light-years apart. As the name suggests, number theory is the study of numbers, such as the familiar natural numbers (1, 2, 3, and so on) and fractions, or more exotic ones, such as the square root of two. Geometry, on the other hand, studies shapes, such as the sphere or the surface of a donut. But French mathematician André Weil had a penetrating insight that the two subjects are in fact closely related. In 1940, while Weil was imprisoned for refusing to serve in the army during World War II, he sent a letter to his sister Simone Weil, a noted philosopher, in which he articulated his vision of a mathematical Rosetta stone. Weil suggested that sentences written in the language of number theory could be translated into the language of geometry, and vice versa. “Nothing is more fertile than these illicit liaisons,” he wrote to his sister about the unexpected links he uncovered between the two subjects; “nothing gives more pleasure to the connoisseur.”

    …..

    While I was still a schoolboy, my favorite place to visit on vacation was the great Abbaye St. Pierre de Solesmes, celebrated for its central part in the renewal of Catholic liturgy and of the Gregorian Chant in particular. Two of my fondest memories are of the terrific bowls of coffee served in the monastic refectory at breakfast, and of my opportunity to take a class in chant under the chironomic hand of Dom Joseph Gajard, then Choirmaster at Solesmes. The liturgy and the chant were sublime.

    I was an Anglican (“Episcopalian”) at the time, and just a wee bit concerned that the monks might want to convert me to the One Holy [Roman] Catholic and Apostolic version of the faith — but when I expressed my concern to one of the monks, I was reassured: they had had an earlier guest at the abbey, one Simone Weil, and she too had been unready to convert, though deeply moved by the liturgy…

    So I’ve felt a quiet kinship with Simone Weil ever since, and try to keep a copy of her Letter to a Priest nearby me at all times. She begins:

    When I read the catechism of the Council of Trent, it seems as though I had nothing in common with the religion there set forth. When I read the New Testament, the mystics, the liturgy, when I watch the celebration of the mass, I feel with a sort of conviction that this faith is mine or, to be more precise, would be mine without the distance placed between it and me by my imperfection.

    I love her for that — and I love, too, that her brother should make such a splendid Sembl move.

    …..

    I suppose I’d better post my reading of Wiles’ Proof of Fermat’s Last Theorem viewed as a Glass Bead Game as a follow up.

    In between Big Bang and Heat Death

    [ cross-posted frpom Zenpundit — a terrific example of the DoubleQuote form, essentially showing a single Sembl resemblance, and thus the smallest form of Sembl game — and why the form is useful ]
    .

    Some time in between t-zero and t-aleph-null, some time between the First Day of Creation and Judgment Day, some time in between the Big Bang and the Heat Death of the Universe, there’s a stretch of time known as always.


    …..

    What does juxtaposing the two statements allow us to understand?

    • That times have changed?
    • That what you tell a foreign government is not what you tell your own?
    • That Brits are more understated and Americans more plainspoken?
    • That President Obama is showing specific support for Israel, while Palmerston was expressing the general rule which covers all such utterances?
    • That the word “always” doesn’t necessarily mean “for ever”, “unto the ages of ages” as the Eastern Church has it?

    Perhaps “for the foreseeable future” would be a better phrase to use, if it didn’t sound so iffy. I’d say it means something closer to “in continuity” than to “in perpetuity”.

    …..

    The great thing about DoubleQuotes as a form is that they jump-start you into thinking about samenesses and differences, without demanding which particular implications you will select, thus giving rise to multiple possibilities and enlarging the scope of narrative or discussion.

    And while I’ve sharpened the pairing of quotes — or graphics — into a tool for repeated use, it’s already a habitual form of thinking, as we can see from the fact that these two particular quotes were juxtaposed by Sam Roggeveen in his post, America’s BFF: Obama calls it, in the Lowy Institute’s Interpreter today.

    …..

    We naturally pair similars to contrast and compare them: it may be the most basic device that human memory affords us — this reminds me of that.

    Here are a few of my own old favorites…


    …..

    You can read Obama’s speech, from which the excerpt above was taken, on this Israel Times page.