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Grothendieck’s mathematics and Child Born of Water

Saturday, December 13th, 2014

[ by Charles Cameron — two approaches to mathematics, two types of heroism, and their respective complementarities ]

I wish to propose a clear analogy between the mathematician Grothendieck‘s two styles of approach to a problem in mathematics, and the Navajo Twin Gods, Monster-Slayer and Child-Born-of-Water.




Steve Landsburg‘s post, The Generalist, compares two approaches to mathematics, as practiced by two eminent mathematicians:

If there was a nut to be opened, Grothendieck suggested, Serre would find just the right spot to insert a chisel, he’d strike hard and deftly, and if necessary, he’d repeat the process until the nut cracked open. Grothendieck, by contrast, preferred to immerse the nut in the ocean and let time pass. “The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough.”


In the paras leading up to this one, Landsburg gives us the insight that these two approaches can be generalized as “zooming in” and “zooming out”:

Imagine a clockmaker, who somehow has been oblivious all his life to many of the simple rules of physics. One day he accidentally drops a clock, which, to his surprise, falls to the ground. Curious, he tries it again—this time on purpose. He drops another clock. It falls to the ground. And another.

Well, this is a wondrous thing indeed. What is it about clocks, he wonders, that makes them fall to the ground? He had thought he’d understood quite a bit about the workings of clocks, but apparently he doesn’t understand them quite as well as he thought he did, because he’s quite unable to explain this whole falling thing. So he plunges himself into a deeper study of the minutiae of gears, springs and winding mechanisms, looking for the key feature that causes clocks to fall.

It should go without saying that our clockmaker is on the wrong track. A better strategy, for this problem anyway, would be to forget all about the inner workings of clocks and ask “What else falls when you drop it?”. A little observation will then reveal that the answer is “pretty much everything”, or better yet “everything that’s heavier than air”. Armed with this knowledge, our clockmaker is poised to discover something about the laws of gravity.

Now imagine a mathematician who stumbles on the curious fact that if you double a prime number and then halve the result, you get back the number you started with. It works for the prime number 2, for 3, for 5, for 7, for 11…. . What is it about primes, the mathematician wonders, that yields this pattern? He begins delving deeper into the properties of prime numbers…

Like our clockmaker, the mathematician is zooming in when he should be zooming out. The right question is not “Why do primes behave this way?” but “What other numbers behave this way?”. Once you notice that the answer is all numbers, you’ve got a good chance of figuring out why they behave this way. As long as you’re focused on the red herring of primeness, you’ve got no chance.

Now, not all problems are like that. Some problems benefit from zooming in, others from zooming out. Grothendieck was the messiah of zooming out — zooming out farther and faster and grander than anyone else would have dared to, always and everywhere. And by luck or by shrewdness, the problems he threw himself into were, time after time, precisely the problems where the zooming-out strategy, pursued apparently past the point of ridiculousness, led to spectacular, unprecedented, indescribable success. As a result, mathematicians today routinely zoom out farther and faster than anyone prior to Grothendieck would have deemed sensible. And sometimes it pays off big.


I no longer have — alas — a copy of Where the Two Came to their Father, the first volume in the Bollingen Series, with its suite of 18 sand paintings beautifully rendered in silkscreen by Maud Oakes, but their respective black and blue colorations lead me to suppose that the illustration at the head of this post, taken rom that series, shows the twin heroes, Monster Slayer (black) and Child Born of Water (blue) whose journeys and initiation are the subject of the rituasl “sing” recorded in that book.

The theme of two male hero twins is central to the mythologies of the American continent, according to Jospeh Campbell, who contributed a commentary to Oakes’ recording of Jeff King‘s performance of this ceremony, and lacking both the King > Oakes > Campbell book and Gladys Reichard‘s two volumes on Navaho Religion, I must draw on brief quotes from miscellaneous web sources to dramatize the differences between the twins.

Monster Slayer is the doer of deeds, similar in nature to other masculine, not to say macho, heroes — while Child Born of Water is the contemplative of the pair:

The Sun [Jóhonaa’éí] gave them prayersticks and then told them that the younger of the two (Born for Water) would sit watching these prayersticks while the older (Monster Slayer) went out to kill the monsters. If these prayersticks began to burn, this would signal that his brother was in danger and that he should go to him to help.

Reichard explains:

Monster Slayer (na’ye’ ne’zyani) (I) represents impulsive aggression, whereas Child-of-the-water represents reserve, caution, and thoughtful preparation.

A measure of their respective strategies, and of the ways in which the insights of Child Born of Water can succeed where the brute force tactics of Monster SLayer fail, can be gleaned from this section of their story, also I believe taken from Reichard:

When The Twins visited Sun the second time, he said he was willing to help them, but this time he wanted them to return the favor: “I wish you to send your mother to the west that she may make a new home for me.” Whereupon Monster Slayer, believing himself equal to any task, replied, “I will do so.I will send her there.” Then Child-of-the-water reminded them both: “No, Changing Woman is subject to no one? we cannot make promises for her. She must speak for herself? she is her own mistress. But I shall tell her your wishes and plead for you.”


One commentator glibly suggests that the joint presentation of the hero as twins is “a clever reminder that progress depends upon cooperation between our mind and our heart” — but the psychologist Dr Howard Teich offers a far more depthful interpretation: that the two twins represent two forms of masculine heroism, one the familiar macho hero of war movies, and the other wiser and subtler, the possessor of traits commonly attributed to the feminine — and hugely undervalued — in our culture.

Dr Teich suggests we must (urgently) abandon the division of virtues into “male” and “female” types, reognize that these types are complementary rather than rivalrous, that both are necessary functions of both males’ and females’ psyches, and begin to integrate the wholeness that both strategies together represent, in our own approaches to our lives in general, to the natural world around us, and indeed to warfare — unsurprisingly, since we first encounter the twins in the ceremonial specifically devised by the Navajo to protect young warriors on their way to battle, and to reintegrate them in harmony and balance on their return.

As Teich puts it:

Monster Slayer and Child Born of Water, as these Twin Heroes are called, are the most sacred of all the legendary heroes in Navaho mythology. It is rare for the Navaho even to speak of the twins; their presence is to be felt rather than observed, and their lessons absorbed rather than applied. Although the lessons the twins hold may be countless, their particular manifestation of a deeper, more complex image of masculinity deserves the reader’s especial attention.


I’d like to suggest that in the same way that there are “zooming in” and “zooming out” styles in mathematics, and “monster-slayer” and “born of water” styles of heroism, there are in fact twin traditions of understanding the world which we might term scientific and poetic, or in Teich’s terms — and those of the alchemists — solar and lunar.

A unified or “solunary” vision will encompass the virtues of both.


Dr Teich’s review of the King > Oakes > Campbell book under the title A Dual Masculinity was irst piublished in The San Francisco Jung Institute Library Journal, Vol. 13, No. 4, 1995. He now has a book out treating these themes: Solar Light, Lunar Light.

Oh, and please don’t expect me to know anything more about Grothendieck’s mathematics than I read in Landsburg’s article.

An English game

Wednesday, December 10th, 2014

[ by Charles Cameron — image and Sotheby’s cataloguing via Michael Robinson ]

Carl Jung told his friend Sir Laurens van der Post:

One of the most striking testimonies to the quality of the English spirit is the English love of sport and games in a classical sense and their genius for inventing games.


Exempli gratia:

Pooh Stcks


Catalog entry:

Shepard, E.H.

Illustration for Chapter six of ‘The House at Pooh Corner,’ the episode “in which Pooh invents a new game and Eeyore joins in.” The game is ‘Poohsticks.’ Sold this morning: £314,500 (US$492,727)

Shepard, Ernest H. (10 December 1879 – 24 March 1976)
188 by 148mm, original ink drawing,
signed “EHShepard” lower left.


For those who don’t know the game, here’s the relevant excerpt from The House at Pooh Corner:

Pooh had just come to the bridge; and not looking where he was going, he tripped over something, and the fir-cone jerked out of his paw into the river. ‘Bother,’ said Pooh, as it floated slowly under the bridge, and he went back to get another fir-cone which had a rhyme to it. But then he thought that he would just look at the river instead, because it was a peaceful sort of day, so he lay down and looked at it, and it slipped slowly away beneath him, and suddenly, there was his fir-cone slipping away too.

‘That’s funny,’ said Pooh. ‘I dropped it on the other side,’ said Pooh, ‘and it came out on this side! I wonder if it would do it again?’ And he went back for some more fir-cones. It did. It kept on doing it. Then he dropped two in at once, and leant over the bridge to see which of them would come out first; and one of them did; but as they were both the same size, he didn’t know if it was the one which he wanted to win, or the other one. So the next time he dropped one big one and one little one, and the big one came out first, which was what he had said it would do, and the little one came out last, which was what he had said it would do, so he had won twice … and when he went home for tea, he had won thirty-six and lost twenty-eight, which meant that he was – that he had – well, you take twenty-eight from thirty-six, and that’s what he was. Instead of the other way round.

And that was the beginning of the game called Poohsticks, which Pooh invented, and which he and his friends used to play on the edge of the Forest. But they played with sticks instead of fir-cones, because they were easier to mark.’

On the history of the selfie

Monday, February 3rd, 2014

[ by Charles Cameron — on self representation, avatars, and what we may be missing ]

Caravaggio, Martha and Mary Magdalen, aka The Conversion of the Magdalen


Where to begin?

The Washington Post doesn’t like selfies much, according to Galen Guengerich in the Religion, yes, the Religion section — in a post titled ‘Selfie’ culture promotes a degraded worldview he writes:

The 2013 word of the year, according to the Oxford Dictionaries, was “selfie,” which Oxford defines as “a photograph that one has taken of oneself, typically one taken with a smartphone or webcam and uploaded to a social media website.” The first use of the term, according to Oxford, occurred when a young Australian got drunk at a friend’s 21st birthday party and fell down the stairs. He hit lip-first and his front teeth punched a hole in his bottom lip. His response was to take a photo of himself and post it online for his friends to see. “Sorry about the focus,” he wrote, “It was a selfie.”


As usual, the Kierkegaard / Kardashian combo that tweets as @KimKierkegaard manages to straddle the worlds material (in the Madonna sense) and spiritual (in the sense of the Madonna):


I wanted to dig deeper — the WashPost Religion section, Kierkegaard, how could I not? I often want to dig deeper, and today I was driven to do so because today — not or the first time — I ran across a terrorism analyst and blogger named Cristina Caravaggio Giancchini, who uses a detail from her namesake Michelangelo Merisi da Caravaggio‘s Martha and Mary Magdalen (above) as her avatar…

Avatars are a kind of selfie, aren’t they?

In any case, I found myself looking for the particular Caravaggio that contains that detail, discovering it was the Martha and Mary Magdalen, which you see that the top of this post — then kept on digging via Google to learn a little more.


Here’s what I found in a blog post titled Fingers and Mirrors: Caravaggio and the Conversion of Mary Magdalene in Renaissance Rome:

The inclusion of the mirror asks viewers to enter into a dynamic conversation about their own delight in the rich textures of the picture; alongside a powder puff and comb, it points us to Mary’s vanity, and her concern with the things of this world. Rather than showing Mary to herself, however, the mirror captures a diamond of light — a visual representation of the divine grace that inspires Mary to look beyond her earthly passions. The flower that Mary clutches to her chest is an orange blossom: symbol of purity.

As Debora Shuger realises, in a stimulating essay on early modern mirrors, for Renaissance viewers ‘the object viewed in the mirror is almost never the self’ (22). Such mirrors are, Shuger suggests, if not totally Platonic (reflected an absolute ideal), at least ‘platonically angled, titled upwards in order to reflect paradigms rather than the perceiving eye’ (26). Renaissance mirrors, she concludes, ask us to think differently about the mental worlds and self-awareness of people living in this period: ‘they reflect a selfhood that … is beheld, and beholds itself, in relation to God’ (38).

Pilgrims who travelled to Aachen in the fifteenth-century appear to have purchased small convex mirrors as souvenirs: as relics were carried through the thronging crowds, travellers held up the mirrors to catch a glimpse of them, and then preserved the mirrors as objects which, according to Rayna Kalas, ‘betokened that moment when the pilgrim had a vision of and was visible before the sacred relic. … Every subsequent glance at this mirror memento might serve to remind the believer of that glimpse of sacred divinity’. In Caravaggio’s painting, though, Mary looks away from the mirror which might capture her reflection (the ‘dark glass’ of Corinthians?), and towards her shadowed but persuasive sister.


We began this post with the idea that our 21st century ‘Selfie’ culture “promotes a degraded worldview” — and here by way of contrast, in the use of hand-held mirrors in 15th century Aachen, we see what we are missing…

… a glimpse of the sacred, in which the sacred glimpses us in transcendent return.

The virtual museum is not simply a museum in virtual space

Sunday, February 2nd, 2014

[ by Charles Cameron — cross-posted from Sembl — on the infinite possibilities of juxtaposition in gallery & museum, catalog & library — creativity & the Sembl game ]

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 Styles‘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:


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


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.


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.

In good, really good company

Friday, January 10th, 2014

[ by Charles Cameronmildly NSFW if your office can’t handle Leonardo, which IMNSHO we should be able to manage now in this 21st century CE — and besides, it’s the weekend ]

Well, we here at Zenpundit have a particular interest in creative thinking, and this last evening I unexpectedly found myself in excellent creative company…

…in a months-old blog-post by an old friend, an astrophysicist by profession who goes by the name Cygnus on the web — presumably after the constellation that harbors Deneb, and also Kepler-22b, the “first known transiting planet to orbit within the habitable zone of a Sun-like star” (WikiP, since I know no better). Cygnus means “swan” in Greek, and Zeus became a swan for his own imperious purposes when he saw LedaHelen of Troy being one of their offspring (see eggs in Da Vinci‘s image below), with the Trojan War ensuing.


Here’s then, is the A-Z of creative folk, as Cygnus pulled it together last April as part of an “A-Z- Challenge” — I’m honored and awed to be named in the company of such as Andre Breton, Donald Knuth, George Carlin, Octavia Butler, Samuel R Delany, Dame Frances Yates and the rest:


For April 2013, my theme for the Blogging from A to Z Challenge was “An A to Z of Masters of the Imagination that You Oughtta Know About.”  In other words, on each day I profiled a person whose brains were just overflowing with weirdness and creativity.  Here’s a list of the posts:


So that’s Cygnus’ list — quite a dinner party! You’ll recognise some members of your own constellation of creatives here, perhaps — feast on some of those you’re not yet familar with! Cygnus blogs about games and such at Servitor Ludi.

As for me, I’ll simply offer you William Bulter Yeats‘ great poem Leda and the Swan, to celebrate the company I just found myself in, and close out a memorable evening:

Leda and the Swan

A sudden blow: the great wings beating still
Above the staggering girl, her thighs caressed
By the dark webs, her nape caught in his bill,
He holds her helpless breast upon his breast.

How can those terrified vague fingers push
The feathered glory from her loosening thighs?
And how can body, laid in that white rush,
But feel the strange heart beating where it lies?

A shudder in the loins engenders there
The broken wall, the burning roof and tower
And Agamemnon dead.
                                 Being so caught up,
So mastered by the brute blood of the air,
Did she put on his knowledge with his power
Before the indifferent beak could let her drop?

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