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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 ]
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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.

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Twins

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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.”

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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.

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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.”

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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.

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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.

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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.

Making Historical Analogies about 1914

Friday, January 10th, 2014

[by Mark Safranski, a.k.a. “zen“]

The Independent has a short, quasi-sensationalist, article featuring historian Margaret MacMillan discussing what is likely to become the first pop academic cottage industry of 2014….making historical analogies about 1914 and World War I! MacMillan is a senior scholar of international relations and administrator at Oxford ( where she is Warden of St Antony’s College)  with a wide range of research interests, including the First World War on which she has published two books.  I am just going to excerpt and comment on the historical analogies MacMillan made – or at least the ones filtered by the reporter and editor – she’s more eloquent in her own writing where each of these points are treated at greater length:

Is it 1914 all over again? We are in danger of repeating the mistakes that started WWI, says a leading historian 

Professor Margaret MacMillan, of the University of Cambridge, argues that the Middle East could be viewed as the modern-day equivalent of this turbulent region. A nuclear arms race that would be likely to start if Iran developed a bomb “would make for a very dangerous world indeed, which could lead to a recreation of the kind of tinderbox that exploded in the Balkans 100 years ago – only this time with mushroom clouds,”

…..While history does not repeat itself precisely, the Middle East today bears a worrying resemblance to the Balkans then,” she says. “A similar mix of toxic nationalisms threatens to draw in outside powers as the US, Turkey, Russia, and Iran look to protect their interests and clients. 

Several comments here. There is a similarity in that like the unstable Balkan states of the early 20th century, many of the Mideastern countries are young, autocratic, states with ancient cultures that are relatively weak  and measure their full independence from imperial rule only in decades.  The Mideast is also like the Balkans, divided internally along ethnic, tribal, religious, sectarian and linguistic lines.

The differences though, are substantial. The world may be more polycentric now than in 1954 or 1994 but the relative and absolute preponderance of American power versus all possible rivals, even while war-weary and economically dolorous, is not comparable to Great Britain’s position in 1914.  The outside great powers MacMillan points to are far from co-equal and there is no alliance system today that would guarantee escalation of a local conflict to a general war. Unlike Russia facing Austria-Hungary over Serbia there is no chance that Iran or Russia would court a full-scale war with the United States over Syria.

On the negative side of the ledger, the real problem  is not possible imperial conquest but the danger of regional collapse. “Toxic nationalism” is less the problem than the fact that the scale of a Mideastern Sunni-Shia sectarian conflict is so enormous, as are the implications . Nothing in the Balkans after the turn of the century compares to Syria, then Iraq and then other states sliding into a Muslim version of the Thirty Year’s War. An arc of failed states from Beirut to Islamabad is likelier than, say, a new Persian empire run by Tehran’s mullahs.

Modern-day Islamist terrorists mirror the revolutionary communists and anarchists who carried out a string of assassinations in the name of a philosophy that sanctioned murder to achieve their vision of a better world

Agree here. The analogy between 21st revolutionary Islamists and the 19th century revolutionary anarchists is sound.

And in 1914, Germany was a rising force that sought to challenge the pre-eminent power of the time, the UK. Today, the growing power of China is perceived as a threat by some in the US.

Transitions from one world power to another are always seen as dangerous times. In the late 1920s, the US drew up plans for a war with the British Empire that would have seen the invasion of Canada, partly because it was assumed conflict would break out as America took over as the world’s main superpower.

Imperial Germany’s growing power was less troublesome to Edwardian British statesmen than the strategic error of the Kaiser and von Tirpitz to pursue a naval arms race with Great Britain that did not give Germany’even the ability to break a naval blockade but needlessly antagonized the British with an existential threat that pushed London into the French camp.

As to military plans for invading Canada (or anywhere else), the job of military planning staffs are to create war plans to cover hypothetical contingencies so that if a crisis breaks out, there is at least a feasible starting point on the drawing board from which to begin organizing a campaign. This is what staff officers do be they American, French, Russian, German, Chinese and even British. This is not to be taken as serious evidence that the Coolidge or Hoover administrations were hatching schemes to occupy Quebec.

More importantly, nuclear weapons create an impediment to Sino-American rivalry ending in an “August 1914″ moment ( though not, arguably, an accidental or peripheral clash at sea or a nasty proxy conflict). Even bullying Japan ultimately carries a risk that at a certain point, the Japanese will get fed-up with Beijing, decide they need parity with China, and become a nuclear weapons state.

Professor MacMillan, whose book The War That Ended Peace was published last year, said right-wing and nationalist sentiments were rising across the world and had also been a factor before the First World War

In China and Japan, patriotic passions have been inflamed by the dispute over a string of islands in the East China Sea, known as the Senkakus in Japan and Diaoyus in China. “Increased Chinese military spending and the build-up of its naval capacity suggest to many American strategists that China intends to challenge the US as a Pacific power, and we are now seeing an arms race between the two countries in that region,” she writes in her essay. “The Wall Street Journal has authoritative reports that the Pentagon is preparing war plans against China – just in case.” 

“It is tempting – and sobering –to compare today’s relationship between China and the US with that between Germany and England a century ago,” Professor MacMillan writes. She points to the growing disquiet in the US over Chinese investment in America while “the Chinese complain that the US treats them as a second-rate power”.

The “dispute” of the Senkakus has been intentionally and wholly created by Beijing in much the same way Chinese leaders had PLA troops provocatively infringe on Indian territory, claim the South China Sea as sovereign territory and bully ships of all nearby nations other than Russia in international or foreign national waters. This is, as Edward Luttwak recently pointed out, not an especially smart execution of strategy. China’s recent burst of nationalistic bluffing, intimidation and paranoia about encirclement are working along the path of self-fulfilling prophecy.

Another similarity highlighted by the historian is the belief that a full-scale war between the major powers is unthinkable after such a prolonged period of peace. “Now, as then, the march of globalisation has lulled us into a false sense of safety,” she says. “The 100th anniversary of 1914 should make us reflect anew on our vulnerability to human error, sudden catastrophes, and sheer accident.

Agree that globalization is no guarantee against human folly, ambition or the caprice of chance.

What are your thoughts?

A DoubleQuote in the (Arctic) Wild

Friday, January 3rd, 2014

[ by Charles Cameron — always on the lookout for intriguing double-images ]
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There’s an implied “this is to that as this is to that” double analogy here. Just how well or ill it teaches coordinate systems I leave to others to decide — even without the analogical joking though, it’s an intriguing visual juxtaposition.

Happy New “Creative Leap” Year

Wednesday, January 1st, 2014

[ by Charles Cameron — wondering whether a von Kármán vortex street might be a good place to take a Paul Lévy walk one of these days — when I’m out and about, foraging for new ideas ]
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"Named after French mathematician Paul Lévy, a Lévy walk is characterized by many small moves combined with a few longer trajectories."

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M’friend Bill Benzon of the New Savanna blog posted two paras out of an NYT blog piece, Navigating Our World Like Birds and Bees, today:

What they have found is that when moving with a purpose such as foraging for food, many creatures follow a particular and shared pattern. They walk (or wing or lope) for a short time in one direction, scouring the ground for edibles, then turn and start moving in another direction for a short while, before turning and strolling or flying in another direction yet again. This is a useful strategy for finding tubers and such, but if maintained indefinitely brings creatures back to the same starting point over and over; they essentially move in circles.

So most foragers and predators occasionally throw in a longer-distance walk (or flight), which researchers refer to as a “long step,” bringing them into new territory, where they then return to short walks and frequent turns as they explore the new place.

I can’t help but think that this may give us a closer approximation to the way minds can think than our usual terms, linear and lateral, or on a wider scale, disciplinary and interdisciplinary thinking, with the short walks involving thoughts that require investigation but not analogy, and the long steps being leaps by analogy into new territory — the familiar hop, skip and jumps we also call creative leaps.

From my POV, seeing both linear and leaping thoughts this way allows for the fact that what we’ve been calling linear thoughts aren’t so much linear as local, while analogical thoughts by their very nature take us from one thought domain to another — via parallelism or opposition — leaping conceptual distances.

Which is why I can wish you a Happy New “Creative Leap” Year! — even though 2014 isn’t divisible by 4 and there will still only be 28 days this February.

DoubleBurn: mosque and synagogue

Sunday, October 20th, 2013

[ by Charles Cameron — tipping my hat to a moving interfaith gesture ]
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This past Wednesday, a person or persons unknown torched a mosque in Gdansk, Poland, known to me as the place where Lech Walesa founded the Solidarnosc movement.

Thankfully, the physical damage doesn’t appear to have been complete [upper panel, below]:

What brings this particular event to our attention is the response from the city’s Jewish community [lower panel, above: an image of a current Gdansk synagogue].

The Jewish Telegraphic Agency [JTA] gives us the story:

Polish Jews say mosque torching reminiscent of Kristallnacht

Representatives of the Jewish community of Gdansk, Poland, said the torching of a mosque had “frightening connotations” of the Nazi-inspired Kristallnacht pogroms against Jews.

The association was inescapable, three of the city’s Jewish leaders wrote in a statement Thursday.

“On the eve of the 75th anniversary of Kristallnacht, during which synagogues were burned in the Free City of Gdansk, the burning of the mosque must bear frightening connotations,” the statement said.

Unidentified individuals started the fire early Wednesday morning. It consumed the mosque’s door and some of the equipment, resulting in damages to the tune of $16,000.

“In the face of this cowardly act of barbarism, Jews of Gdansk cannot stand idly by,” wrote the authors of the statement, Michal Samet, Michal Rucki and Mieczyslaw Abramowicz. “We express our deep indignation against the attack on the temple and the sadness of the fact that it took place in Gdansk.”

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An echo in time.

A powerful analogy, deeply felt.


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