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## Archive for the ‘mathematics’ Category

### On the liquidity of mountains, and cats

Sunday, November 24th, 2019

[ by Charles Cameron — you might not have thought of cats as liquid, though they flow quite nicely on a decent carpet; and as for mountains — would they flow to Mohammed, or would he have to flow to them? ]
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Somewhere I’d heard that Muhammad said that mountains moved like waves. I’d wondered which direction the Rocky Mountains might be moving in, whether they were aiming for the Pacific or the Atlantic coast, and what would happen to real estate prices and military bases in either case. I used to live in Denver..

And today I discovered the concept of the Deborah Number, defined thus:

The Deborah number (De) is a dimensionless number, often used in rheology to characterize the fluidity of materials under specific flow conditions. It quantifies the observation that given enough time even a solid-like material might flow, or a fluid-like material can act solid when it is deformed rapidly enough.

Reiner, whose paper originated the term, notes:

Deborah knew two things. First, that the mountains flow, as everything flows. But, secondly, that they flowed before the Lord, and not before man, for the simple reason that man in his short lifetime cannot see them flowing, while the time of observation of God is infinite. We may therefore well define a non-dimensional number the Deborah number..

The equation by which the Deborah Number, De, is defined as:

De = λ / T, where T is a characteristic time for the deformation process and λ is still the relaxation time.

**

The Deborah reference here is to the book of Judges chapter 5, verse 5, which Reiner reads as saying “The mountains flowed before the lord” — whereas the KJV has “The mountains melted from before the LORD” and the NIV, “he mountains quaked before the Lord”. Melting at least has a transition from solid to liquid state implied, whereas quaking doesn’t really shift mountains from their solidity, though they shake — like Quakers, perhaps?

It seems there may have been some conflation here, for Isaiah finally provides us with a text that gives mountains complete liquidity — Isaiah 64:3 in the KJV gives us “the mountains flowed down at thy presence..”

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And what of Muhammad? Shakir translates Qur’an 31.32:

And when a wave like mountains covers them they call upon Allah, being sincere to Him in obedience, but when He brings them safe to the land, some of them follow the middle course; and none denies Our signs but every perfidious, ungrateful one.

That has the waves moving, and their size resembling that of mountains, as I read it. Mohsin Khan‘s version often adds the translator’s explanations in brackets, as here:

And when a wave covers them like shades (i.e. like clouds or the mountains of sea­-water), they invoke Allah, making their invocations for Him only. But when He brings them safe to land, there are among them those that stop in the middle, between (Belief and disbelief). But none denies Our Signs except every perfidious ungrateful.

Here, “mountains of sea-water” are compared with “clouds” inside the brackets, but the text itself doesn’t mention either one..

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Maybe the mountains won’t budge. That was the opinion of John Owen, who wrote in 1643:

If the mountaine will not come to Mahomet, Mahomet will goe to the mountaine.

Bakker, Egbert J. 2005. Pointing at the Past: From Formula to Performance in Homeric Poetics. Hellenic Studies Series 12. Washington, DC: Center for Hellenic Studies devotes chapter 9 to the topic of Mohammed and the Mountain, referencing Karl Bühler’s Sprachtheorie: Die Darstellungsfunktion der Sprache (1934):

As Bühler puts it himself in reference to the well-known anecdote: either the mountain comes to Mohammed or Mohammed goes to the mountain … he adds that in real life the mountain is a lot more willing to move than in the legend, since the ease with which any given speech arena can be transformed into an imagined new reality is remarkable, and lies at the basis of any mimetic, theatrical illusion. .

Here, the mountain’s movement depends on imagination, even though Bühler seems to refer to it as moving “in real life” as well as in “any mimetic, theatrical illusion”.

Of course, the very idea concealed within the name Rheology is that of universal flow, espoused by Heraclitus:

παντα ρει : Everything Flows

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And if everything, then cats. It was a tweet by blog-friend Adam Elkus that put me in mind of liquidity in the first place this morning:

Cat physics!

It’s an an obvious field of study once you understand the centrality of cats to the universe, and It’s appropriate enough that a cat physicist, Marc-Antoine Fardin, should have won the Ig Nobel Prize. His definition of liquid is a simple one:

A liquid is traditionally defined as a material that adapts its shape to fit a container.

He proceeds to show two examples which may fit this definition, which I’ve cropped to show you full size. First, a cat demonstrating oval form:

And here’s the rounded rectangle form, adopted by the cat from Adam‘s tweet —

**

Suggested Thinking:

• >Wolfram MathWorld tells us a sausage-form filled rounded rectangle is termed a stadium. Cat stadium, or a stadium cat? Now there’s food for a football (or baseball, or rock music) thought..
• _____

Tweets from Splymoth A. Klavrock supplied us with the SKV translation of Isaiah 64:1-3:

Oh if you would only tear the Heavens apart and descend. In your presence the mountains would melt down like when fire crackles through kindling. The fire makes oceans seethe, so your name is made known to your adversaries, and nations quake in your presence.

You did fearsome things, things we never hoped for, and in the doing of them you descended. The mountains melted down in your presence.

From all eternity no ear has heard, and no eye has seen any god but you, and the things you do for the ones who wait for you.

Klavrock had another suggestion, linking the “motion” verb to something internal, close to an “emotion”:

I think “melted” is the same verb as when the Israelites’ hearts “melt” in fear facing a strong enemy. Which is interesting.

Many thanks, SK!

### A magical tale, in three accelerating acts

Wednesday, November 6th, 2019

[ by Charles Cameron — magic is imagination, see my post Vlahos: violence is the magical substance of civil war earlier today ]
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Jacob’s ladder, on which angels are show ascending and descending, is revealed to Jacob in a dream: sheer magic — and how richly strange to see the ladder emerge from a simple loop of string..

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Act Two: Pavel Tchelitchew, The Concert {via Alabandine]

That the string figure should become a stringed instrument, plucked by the teeth and accompanied by tambourine.. again, we are raised an octave above our grounded selves..

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Act Three: Bob Dylan, Song and Dance Man

And the song and dance musician magician Dylan — his harmonica, making an anthem for us all.

### Three is a general purpose interest of embodied minds

Friday, April 5th, 2019

[ by Charles Cameron — threes in knotting, braiding, math and bell ringing — in service to governance, and the recognition of pattern within complexity ]
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One is one and all alone and ever more shall be so..
Two is both duel and duet..
and three:

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Well, those are knots, of the Celtic variety. I cam across those images because Tony Judge pointed me to the animations in a piece he’d written, Exploring Representation of the Tao in 3D: Virtual reality clues to reconciling radical differences, global and otherwise?

which gets me thinking about thinking in threes —

— which has been an interest of mine for some time, see below —

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And as is always the case with Judge‘s offerings, a plethora of his links called to me, and I wound up taking a look at his paper, Governance as “juggling” — Juggling as “governance”: Dynamics of braiding incommensurable insights for sustainable governance

— incommensurable insights is another topic of considerable interest to me —

— and that in turn brought me to this illustration of two instances of triple thinking about incommensurables from Australia — a triple helix and braiding:

**

Which brings us in turn to Borromean Rings and Knots:

Now the question to consider with each and all of these illustrations of threeness is whether they trigger any thoughts about the juggling and hopefully braiding and balancing of incommensurable forces in governance.. okay?

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You’ll have noted that the braiding illustration from the Australian double illustration above is a representation of a juggling pattern. Wikimedia has dozens of such patterns with various numbers of balls, heights to which they are lobbed, &c, — and they’re fascinatingly eye-catching — mesmerizing, in fact.

Take a look at just three of them:

Selection of animations of 3-ball juggling patterns by one juggler
(derived from juggling patterns in Wikipedia)

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I mean:

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Wow, and okay:

Now if a pattern of juggling can be represented as a pattern of braiding, we have a comparable situation to Ada Countess of Lovelace‘s brilliant cross-disciplinary leap of insight that the logical patterns Charles Babbage used to program for his proto-computing Analytical Engine could be represented in the punched cards used by Jacquard looms in the production of patterned fabrics:

• James Essinger, Jacquard’s Web
• **

Am I — is Tony Judge — are we — out on a limb?

Judge offers documentation of the mathematical side of things here:

As indicated by Burkard Polster (The Mathematics of Juggling [excerpt], Monash University, 2003), the diagram above-right shows what the trajectories of juggling the basic 3-ball pattern look like (viewed from above). The three trajectories form the most basic braid. Braids are recognized as important mathematical objects. It has been shown that every braid can be juggled in that sense (Polster, 2003; Matthew Macauley, Braids and Juggling Patterns, 2003; Satyan Devadoss and John Mugno, Juggling braids and links, The Mathematical Intelligencer, 29, 2007). The implications have been further discussed separately (Potential cognitive implications of toroidal helical movement, 2016; Category juggling reframed through visualization dynamics, 2016).

And again, let’s remember Tony Judge‘s reason for his interest in juggling and braiding in the first place:

“juggling” is widely used as a metaphor to describe the challenge of responding to conflicting priorities in governance

Judge has eighteen bibliographic supports for that assertion, including:

• Trump Forced to Juggle Syria Response, Rage Over Mueller Probe (WSJ, 13 April 2018)
• Trump juggling 75 pending lawsuits with a presidential campaign (CNBC, 27 October 2016)
• The art of juggling political values and Trump (WaPo, 13 April 2018)
• **

Noting the correspondence between juggling — a circus-performer’s art — and braiding — not quite knitting, not quite knotting, and don’t those two words fit well together — an art associated with the decoration of hair and ribbons — I wondered whether there might not be a musical analog in counterpoint, and posted my inquiry on Twitter using this diagram of braiding:

**

I was fortunate: Change-ringing, surely very speedily responded to my inquiry:

Change-ringing, surely

The art of change-ringing in British churches and among hand-bell ringers is indeed the classic example of highly constrained and patterned musical counterpoint, so I happily Googled away in search of a change-ringing pattern comparable to my braiding patternc[left side, below], and came across the pattern [right side] in a page on the Cambridge Surprise Minor changes:

Just Knecht, too, had some interesting observations & questions..

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Metaphor, analogy, parallelism — these are avenues into the creative process in general, and threeness analogies and metaphors interrupt our usual binary cognitive processing in a way that enhances our capacity to comprehend complexity.

I’m therefore offering this post to Ali Minai and Mike Sellers, in the hope that it will serve as a provocation to their already advanced thinking about systems dynamics. Tony Judge, obviously enough, it’s also a tribute to you…

Previous posts of mine with threeness as a topic include

• Of games III: Rock, Paper, Tank
• Numbers by the numbers: three / pt 1
• Spectacularly non-obvious, I: Elkus on strategy & games
• Spectacularly non-obvious, 2: threeness games
• Numbers by the numbers: three .. in Congress
• Spectacular illustration of a game of three
• Threeness games — some back-up materials
• ### Physicists playing Calvinball

Saturday, February 23rd, 2019

[ by Charles Cameron — wishing I was fluent in music, and might as well ad mathematics, Hebrew, Arabic, classical Persian, you know the drill, Sanskrit, Tibetan, Japanese.. and their courtly modes and rituals, and could play badminton, chess, dharma combat, go, eh? ]
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Here’s a wonderful description of a game in which the rules — in this case, mathematical languages — change from move to move:

It happens again and again that, when there are many possible descriptions of a physical situation—all making equivalent predictions, yet all wildly different in premise—one will turn out to be preferable, because it extends to an underlying reality, seeming to account for more of the universe at once. And yet this new description might, in turn, have multiple formulations—and one of those alternatives may apply even more broadly. It’s as though physicists are playing a modified telephone game in which, with each whisper, the message is translated into a different language. The languages describe different scales or domains of the same reality but aren’t always related etymologically. In this modified game, the objective isn’t—or isn’t only—to seek a bedrock equation governing reality’s smallest bits. The existence of this branching, interconnected web of mathematical languages, each with its own associated picture of the world, is what needs to be understood.

That’s from A Different Kind of Theory of Everything in The New Yorker, an intriguing rerad, though as a non-physicist, seeing an equivalence with Calvinball — a game in which the game in play constantly changes — is about as far as I can go.

When I was talking with Ali Minai, I said that both music and math were languages I didn’t speak, and that cut me off from much by way of discourse with mathematicians (Ali himself) and musicians (my nephew the conductor Daniel Harding), and Ali commented that music is at least an embodied abstraction, whereas math is a pure abstraction with no embodied component. I hope I’ve understood and expressed that well enough. Anyway, it was a striking comment, and not one that had ever crossed my mind, on a topic of considerable interest and real regret.

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Calvinball:

Richard Feynman would have enjoyed a Calvinball reference, methinks — but for any sober-sided physicists who don’t play bongos, here’s the philsopher Alasdair MacIntyre to much the same effect:

Not one game is being played, but several, and, if the game metaphor may be stretched further, the problem about real life is that moving one’s knight to QB3 may always be replied to by a lob over the net.

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I’d hoped to have more intriguing math or game quotes to offer here, but no luck so far, so I’m gonna post anyway.

### Remembering mathematician and Glass Bead Gamer Bob de Marrais

Monday, February 11th, 2019

[ by Charles Cameron — this is strictly for the record — you don’t need to read it unless — like Bob — you’re a poly-mathematician, para-biologist, meta-psychiatrist or native-born glass bead gamer ]
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1984. Illustrating for Bob de Marrais‘ article on Computer Graphics,
published in Digital Deli: The Comprehensive, User-Lovable Menu
of Computer Lore, Culture, Lifestyles and Fancy
, ed. Steve Ditlea.

**

My late friend Bob de Marrais wrote a five-part short-book-length essay, Catastrophes, Kaleidoscopes, String Quartets: Deploying the Glass Bead Game, which is so wide-ranging in its scholarship that no single journal had peers sufficient to review it, so witty, subtle, enchanting, and generally impossible that its continued existence on the web and in the time-worn hard drives of a scattering of computers has made of it a sort of samizdat — a secret publication passsed from hand to hand, or in this case memory to memory, and in this post I wish to memorialize both the essay and Bob himself.

Here are five sips, to give you a sense of the work.

**

#### Catastrophes, Kaleidoscopes, String Quartets:

Part I: Ministrations Concerning Silliness, or: Is “Interdisciplinary Thought” an Oxymoron?

We seek deep concepts by silly means. Think of this, for openers at least, as a cerebral equivalent of a well-known Monty Python skit: welcome to the Ministry of Silly Thoughts. [ … ]

Essential to easy generation of the “silliness effect” – as in the frivolous juxtaposing of Kings Arthur and Elvis in the last paragraph – is production of collisions between disparate things, which context makes us associate unexpectedly. No one not on drugs or writing late-night standup material would be likely to seek a link between the latest news from robotic interplanetary exploratory vehicles and political upheaval in the Hispanic community in the general vicinity of Miami. But when Elian Gonzales’ mom fled Castro’s regime on a flimsy makeshift boat and died at sea while getting her son to (what she thought would be his) freedom, Jay Leno noted how scientists had just discovered water on the Red Planet, “and in an unrelated story, a boat of Cuban refugees washed up on Mars this morning.”

Aside from late-night comedic unwinding from the day’s events, there is only one other area where such juxtapositions are hunted down and put to use. (No, not dreams: that’s involuntary; and besides, many people today no longer have any.) This area is largely deemed, regardless of lip services paid, “absurd; trifling; frivolous” in academia – when not, that is, subjected to sober attempts at its production which typically display all these three aspects in spite of themselves. This is the domain of what often passes for an oxymoron in our supremely specialized research establishment: interdisciplinary thought. And this, of course, is what we’re here to talk about.

Compare “the silliness effect .. is production of collisions between disparate things, which context makes us associate unexpectedly” with, from this morning’s diggings:

Brecht:

He [Darko Suvin] cites Brecht as follows: “A representation which estranges is one which allows us to recognize its subject, but at the same time make it seem unfamiliar.” This permits a new cognition of the now and creates a moment which is potentially liberating. Placing familiar objects (or subjects) in unfamiliar settings allows us to see differently. Our old and tired perceptions can thus be revitalized and transformed. — Lucy Sargisson, Fool’s Gold?: Utopianism in the Twenty-First Century

Boulez:

For Boulez, the challenge was to present the borrowed ideas in a new light that could lead to results far removed from the original, which had provided only a single solution. — The Gramophone

Both quotes via JustKnecht, another Glass Bead Game-player of note, discussing his Rattlesnake Games.

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Part II: Canonical Collage-oscopes, or: Claude in Jacques’ Trap? Not What It Sounds Like!

For this section of Bob’s work, I’ll just post a snippet referencing the Catastrophe Theory of René Thom:

Of the many, many ways to frame the two-control Cusp, the most interesting for us is the predator-prey chain, due to Thom himself. Let us frame it mythically: in the Vedic lore of pre-Hindu India, the great god Indra – the Zeus of the Aryan invaders – had (or was trapped in) a magical net. Depending on the story told, and teller’s point of view, Indra is the hunter and the hunted too. According to the mathematics of Catastrophe Theory, this is fundamental, not unusual. The theory’s creator typically focuses on the single Cusp as the basis of all richer models .. Its stable “splits and mergers” mode of yoyo-ing between the Two and the One, he tells us, is “the most fundamental regulatory process” in non­linear dynamics: not only in the abstract, but, under the guise of the “predation loop,” in the ultimate concreteness of animal feeding. At least since the emerging of the amoeba, this is, sim­ply, merging: “fundamentally, engulfing a prey into the organism” … and herein resides an enigma.

It’s a rich broth, you see — connecting perhaps to Ali Minai‘s comment tweeted today:

Polyphony, in an abstract sense, applies not just to human complex systems but to all complex systems. .. One of the most unappreciated facts about natural complexity is that it emerges from interaction of simpler processes, and not from some prior complexity.

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Part III: Grooving on the Sly with Klein Groups

No one knows that this tale is a part of an immense poem: myths communicate with each other by means of men and without men knowing it. … The situation which Le Cru et le cuit describes is analogous to that of musicians per­form­ing a symphony while kept incommunicado and separated from each other in time and space: each one would play his fragment as if it were the complete work. No one among them would be able to hear the concert because in order to hear it one must be outside the circle, far from the orchestra. In the case of American mythology, that concert began millennia ago, and today some few scattered and moribund communities are running through the last chords.

That’s Octavio Paz writing on Claude Lévi-Strauss, and Bob uses it as the epigraph to section III. Just today I was writing of the various friends of mind who are making profound contributions as islands in an archipelago — and how I long for the richness that will emerge when the connections between them are strong, the transmission of ideas between them fluid..

Further, from section III:

Somebody calls you, you answer: “In theory, a twirl of kaleidoscopes” – why?

If you were called to provide a summary of the first two installments preceding this, to someone who’s only just joined us, the perpetual revolution of Sir David Brewster’s famous tube should certainly be the very first image to pop from that jack-in-the-box you keep in your head. For Jacques Derrida, as we saw, lopped off this capstone of Lévi-Strauss’s extended metaphor of how the mythic mind operates: the workings of “bricolage” were like those of a kaleidoscope, as the anthropologist summed it up; but Derri­da’s demolition job didn’t so much as footnote, much less explicitly point to, this motif. [ … ]

… Beat­les’ paean to “the girl with kaleidoscope eyes.” …

Leary and Ralph Metzner meanwhile wrote about, and advocated, the use of low-tech kaleidoscopes, imported from the East, for inner exploration as well: I refer, of course, to mandalas. Mixing scientific and New Age styles, they managed to synthesize, in brief compass and without the “depth psychology,” the gist of what Jung’s approach toward such sacred objects (about which, more in the next installment) is taken to be by those who’d worn bell-bottoms and “love beads” while reading such things:

[As] the mandala is a depiction of the structure of the eye, the center of the man­dala corresponds to the foveal “blind spot.” Since the “blind spot” is the exit from the eye to the visual system of the brain, by going “out” through the center, you are going in to the brain. The Yogin finds the mandala in his own body. The mandala is an instrument for transcending the world of visually perceived phenomena by first centering them and turning them inward.

Note that Leary’s reading of the foveal blind spot is markedly at odds with Derrida’s

**

Part IV: Claude’s Kaleidoscope . . . and Carl’s

As before, note that the epigraps to this section contain doors intonwhat is within:

All the creative power that modern man pours into science and technics the man of antiquity devoted to his myths. This creative urge explains the bewildering confusion, the kaleidoscopic changes and syncretistic regroupings, the continual rejuvenation, of myths in Greek culture.

That’s Carl Jung, in Symbols of Transformation

Here he goes:

For those who’ve tuned in late to this mini-series, the first episode performed a sort of sitcom set­up of the main conundrum: Derrida’s deconstruction launched itself using Lévi-Strauss’ structuralism – as epitomized in his Mr. Fixit figure of the “bricoleur” – as thrust-block . . . the irony being that the latter “failed” analytics of myth proved a harbinger of advanced mathematical toolkits whose utility in linguistic and cultural studies has been burgeoning, while the former “success story” has shown itself to be ever more hollow – intellectually, morally, and spiritually.

In Part Deux, we blowfished the argument, treating the core event – the 1966 Johns Hopkins con­ference where Derrida struck his “deal with the Devil” – as itself a sort of myth requiring structural analy­sis, inspecting it through the lens of Derrida’s 1987 reminiscences about the postmodernist “quotation market” and his own role in fomenting it . . . and then beefed up our discussion of Lévi-Strauss’ own “canonical law of myths” with Catastrophe Theory mathematics and the tasteful injection of celebrity quotes, movie reviews, and porno­graphic movie ads to, um, “flesh out” the argument.

Strike three, though, was where the ubiquitous form-language of the so-called “A,D,E Problem” and its lowly instancing as a new sort of Timaeus-style creation myth – based on kaleido­scopes instead of an odd lot of triangles and things whose names rhyme with Tipi Hedron[1] but don’t look half as fetching – was taken much too seriously, with the limitations in Husserl’s phenomeno­logy shame­lessly con­trasted (unfa­v­or­ably) with the concentric run-out groove at the end of the Beatles’ Sgt. Pepper album. The point being, natural­ly, that the Madhyamika Buddhism of Nagarjuna’s “full void” was allowed to under­write the super­po­si­­tion principal of quantum mechanics in spite of its looking like something Derrida liked to mutter about, while all the while all of this was subsumed in some mare’s nest of compari­sons between the struc­tures of mythical argument, their “reincarnation” in the forms of classical music, and the Glass Beads that Hermann Hesse’s Magister Ludi was known to like to play with when he thought no one was watching.

Of course, if we’re going to keep a load like that down without providing our readers free Pepto-Bismol, it would behoove us to make the people reading this think the linchpins of the argument were some­­how intrinsic. Put another way (which is our specialty here), we could say that it’s all very nice that this “A,D,E Problem” gives us kaleidoscopes as the Meaning of Life and like that there, but wouldn’t it be so much better if we got the same basic mishmash without all the abstraction – if the kaleidoscope could legi­timately be seen as some kind of “archetype” in its own right, which “just happened” to bring in Catas­trophe-type “shock waves” into the argument without all the hand-waving … and all without losing all the rest of our baggage, once the argument has landed?

**

Part V: Spelling the Tree, from Aleph to Tav (While Not Forgetting to Shin)

I didn’t even know there was a fifth part — quint-esseence? — until a couple of days ago, and am very grateful to Steven H. Cullinane for conserving all five for us.

One of the epigraphs for this fifth section comes from Gregory Bateson, Steps to an Ecology of Mind:

“The heart has its reasons which the reason does not at all perceive.” Among Anglo-Saxons, it is rather usual to think of the “reasons” of the heart or of the unconscious as the inchoate forces or pushes or heavings – what Freud called Trieben. To Pascal, a Frenchman, the matter was rather different, and he no doubt thought of the reasons of the heart as a body of logic or computation as precise and complex as the reasons of consciousness. (I have noticed that Anglo-Saxon anthro­po­logists sometimes misunderstand the writings of Claude Lévi-Strauss for precisely this reason. They say he emphasizes too much the intellect and ignores the “feelings.” The truth is that he assumes that the heart has precise algorithms.)

WHat can I tell you? We haven’t delved in any detail into Bob’s mathematical work, but this section contains a footnote — a quotation that delights us with the concept of a perfectly square ship with vertical sides, and offers enough catastrophe-cusp based math to illustrate that central aspect of the whole work:

Tim Poston and Ian Stewart, Catastrophe Theory and its Applications (Boston, London, Melbourne: Pit­man, 1978): “The commonest kind of water-going vessel which is actually built with vertical sides all the way round is a floating oil-platform. These are normally fixed to the ocean floor when on site, but they float during transport. Often they are built square. This symmetry goes through to the buoyancy locus… and the buoyancy locus is a circularly symmetric paraboloid of revolution. The metacentric locus may therefore, apparently, be found by spinning the two-dimensional case, so that the geometry of the perfectly square, vertical-sided ship is remarkably simple. From a catastrophe theory viewpoint this simplicity is thoroughly deceptive, the energy function takes the form (x2 + y2)2. This is not finitely determined … and so has infinite codimension…. Physically, this means that the apparently simple geometry of the ‘ideal’ vessel .. is violently unstable.”

**

Bobert de Marrais was born Nov. 30, 1948, and died April 4, 2011 in Boston, MA. His obit notes he “had a lifelong interest in history, his French heritage, music, history of science, and multidimensional algebras.” He was a remarkable polymath, profoundly loved and deeply admired by the fortunate few who knew him.

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