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

 

 

Click here for Part I - Part II - Part III

 

 

Ideas have a radiation and development, an ancestry and posterity of their own, in which men play the part of god­fathers and godmothers more than that of legitimate parents.

– Lord Acton, cited by Gertrude Himmelfarb in

Darwin and the Darwinian Revolution       

 

Who can marvel sufficiently that the most distinguished minds in each branch of human achievement have happened to adopt the same form of effort, and to have fallen within the same narrow space of time?

– Velleius Paterculus, Res Gestae divi Augusti

 

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.

– Carl Jung, Symbols of Transformation

 

One, two, three – but, my dear Timaeus, of those who yesterday were banqueters and today are the banquet-givers, where is the fourth?

-- Socrates, in Plato’s Timaeus

 

Robert de Marrais

 

 

Transcript of the Author Caught Muttering to Himself Before Actually Writing This Up

 

                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?  (Something like that . . . Time to throw out a long quote while I figure out what to twist and shout about this time . . . )

 

*              *              *              *              *              *              *

 

It was in the year 1811, while writing an article on “Burning Instruments,” that Brewster was led to inves­ti­gate a theo­ry of Buffon, which was to construct a lens of great diameter out of one piece of glass by cutting out the central parts in suc­cess­ive ridges like stair steps.  Brewster did not consider Buffon’s proposal practi­cable.  However, it sparked an idea which pro­­duced awesome scientific re­sults.  Thus was born an apparatus of then un­equaled power – the construc­tion of a lens by build­ing it upon several circular segments.  Here was a useful invention, later perfected, which pro­duced the lighthouse as we know it, creating light-stabs of bril­li­ance that pierced far into the night to guide mariners.

 

This breakthrough was followed by yet other honors.  Brewster was admitted to the Royal Society of London and was later awarded the Rumford gold and silver medal for his theory on the polar­ization of light, which states that light re­flected from a glass sur­face is completely polarized when the reflected and refracted rays are perpendicular to one ano­ther.  Success fol­lowed success; and in 1816, the Institute of France adjudged him 3,000 francs – half the prizes – for the two most impor­tant scientific discoveries to have been made in the two previous years.

 

Then, as an added jewel to his already glittering optics crown, Brewster invented the kalei­do­scope!  This was the year 1816.  Brewster was 35 years of age, and was already an established philosopher, writer, scientist, and inventor.

 

Brewster’s kaleidoscope created unprecedented clamor.  In  a history of Brewster’s Kaleidoscope, found in the June 1818 volume of Blackwood’s Magazine, Dr. Roget said:  “In the memory of man, no invention, and no work, whether addressed to the imagination or to the under­standing, ever produced such an effect.  A universal mania for the instru­ment seized all classes, from the lowest to the highest, from the most ignorant to the most learned, and every person not only felt, but ex­pressed the feeling that a new pleasure had been added to their existence.”[2]

 

            A mania like that engendered by the kaleidoscope is not unknown to us:  I can recall, from my college days, the amazingly widespread obsession with Rubik’s Cube, which even infected the puckish “Metamagical Themas” columnist of Scientific American, Gödel, Escher, Bach author Douglas Hofstadter (with his eleven pages on the topic even inspiring the “Magic Cubology” cover art  for the March, 1981 issue of that learned magazine).  But Dr. Rubik never won the Nobel Prize in Physics, much less  for any research having the puzzle as its logical spin-off – whereas Brewster (by winning, alongside those other honors mentioned by Cozy Baker, the Copley Medal) did the equivalent for his own time and place. 

 

As Hofstadter’s column indicates, the symmetries of the Cube lend themselves to algebraic ana­lysis of a sort that would have made its inven­tion thinkable as a spin-off, say, of Murray Gell-Mann’s Nobel-winning efforts leading to the “Eight­fold Way” of particle classification; only if Gell-Mann, Ne’e­man, Glashow or maybe Feynman had created the Cube en route to their greatest achievements, would we have a truly comparable situation!  But even then, it wouldn’t be fully comparable, since – as Dr. Roget pointed out long ago – the “universal mania” caused by Brewster’s tube was the first ever for a techno-toy.

What we have, then, is a singular sociological phenomenon as well – a sort of “crowd psycho­logy” effect emerging in the very early days of “crowd psychology” as such!  And maybe this isn’t mere coincidence (“maybe,” indeed, is too tame a word choice, as we’ll see shortly when we look at the writings of Walter Benjamin).   Here, we might point to another modern phenomenon – the sudden, massive popu­larity in the States of the musical Beatles in the immediate wake of the first Kennedy assassination – for guidance.  For the immediate context of the St. Vitus’ Dance-like mania for tube-twirling in the Teens of the Nineteenth Century was the sudden lifting of a pan-European state of war, ongoing for over a genera­tion, marked by the final capitulation of Napoleon.  Trade revived, the key cities surged in size, and people were able to get on with their lives in ways they hadn’t been able to contemplate since the falling of the Bastille some three decades prior.  (In their first three months on the market, some 200,000 kaleidoscopes were sold in Paris and London alone – and these were not the cheapo dime-store instruments of today, either.  As Dr. Roget elaborated, “Large cargoes of them were sent abroad, particularly to the East Indies.  They very soon became known throughout Europe, and have been met with by travelers even in the most obscure and retired villages in Switzerland.”  Clearly, something was up!)[3]

 

Indeed, if someone had done a “Rip Van Winkle” and gone to sleep the day before the French Revolution’s inaugural event, and then awakened a quarter century later, he or she would have suffered an analogous shock upon seeing how totally things had changed in the interim.  Such shocks were indeed experienced by many (among them, many great artists and writers) who hadn’t had the luxury of such a long nap, and this is where Walter Benjamin’s researches into the newly “modern” age of Balzac, Edgar Allen Poe, and Baudelaire prove pertinent.  For this was the “boom time” marking the true irruption of mass production and industrialization into general consciousness – just as the last quarter century’s “bull market” sensibility has marked the true irruption of computerization into the worldview of our own time.[4]

 

Indeed, the Rip Van Winkle-like “shock effect” is probably the most frequently elicited aspect of the kaleidoscope’s workings in analogies.  And such analogies frequently relate the “twirling” to the work­ings of the historical process whose non-stop “Progress” was just becoming evident at the time of its emer­gence.  The twirl of kaleidoscopes; revolutionary upheaval:  the one often stands in  for the other when con­templat­ing history, and the late 1960’s and 1810’s were not the only periods of radical change during which kaleidoscope images popped up.  In his monumental socio-  and psycho- logical portrait of fin de siècle France, Marcel Proust pro­vided a stunning usage of just this conjunction of imagery while describing the shake-ups in the upper strata of society  just prior to the Dreyfus Affair.  His remarks are sufficiently gen­er­al in their applicable range, as well as archly insightful and insidiously funny, as to bear citing here in full:

 

In the days of my early childhood, everything that pertained to conservative society was worldly, and no respectable salon would ever have opened its doors to a Republican.  The people who lived in such an atmosphere imagined that the impossibility of ever inviting an “opportunist” – still more, a “horrid radical” – was something that would endure for ever, like oil-lamps and horse-drawn omnibuses.  But, like a kaleidoscope which is every now and then given a turn, society arranges successively in different orders elements which one would have supposed immutable, and composes a new pattern.  Before I had made my first Communion, right-minded ladies had had the stupefying experience of meeting an elegant Jewess while paying a social call.  These new arrangements of the kaleidoscope are produced by what a philosopher would call a “change of criterion.”  The Dreyfus case brought about another, at a period rather later than that in which I began to go to Mme Swann’s, and the kaleidoscope once more reversed its coloured lozenges.  Everything Jewish, even the elegant lady herself, went down, and various obscure nationalists rose to take its place.  The most brilliant salon in Paris was that of an ultra-Catholic Austrian prince.  If instead of the Dreyfus case there had come a war with Germany, the pattern of the kaleidoscope would have taken a turn in the other direction.  The Jews having shown, to the general astonishment, that they were patriots, would have kept their position, and no one would any longer have cared to go, or even to admit that he had ever gone any longer to the Austrian prince’s.  None of this alters the fact, however, that whenever society is momentarily stationary, the people who live in it imagine that no further change will occur, just as, in spite of having witnessed the birth of the telephone, they decline to believe in the aeroplane.[5]

The Proustian imagery implies a curious paradox:  formally, it is the symmetric arrangement of mirrors in the kaleidoscope which bears an exact “A,D,E”-style analogy to the taxonomy of Catastrophic unfoldings; yet it is in fact the twirling of the tube (a feature independent of the mirrors’ arrangements) which most readily bears analogy to Catastrophe Theory’s trademark sudden replacements of one stable regime with another.   Moreover, this twirling effects its changes on the “accidental” aspect of the instru­ment – the bits of colored glass and other oddments whose exact nature vis à vis the kaleidoscope’s “type” is irrelevant, and in fact gratu­i­tous (since the instrument will impose its symmetries on an empty scene just as readily as it will on object-box detritus).  The way toward resolving this “paradox” is through the realiza­tion that the kaleidoscope operates, as we’ve seen (Part II, Note 18), on two levels:  as formal cause of an invariant, hence “ideal” pat­tern, and as efficient cause of the jostling of “these odds and ends,” which Lévi-Strauss, in his contemplation of Brewster’s toy, told us “appear as such only in relation to the history which produced them and not from the point of view of the logic for which they are used.”

 

Such a description obtains, as well, for the behavior of crowds on sidewalks as it does for bits of glass:  the heterogeneous nature of their members, as well as the relative randomness of their motions, are at odds with the directedness of the individuals comprising them when considered each in isolation.  If we in fact remove the kaleidoscope’s features which are irrelevant to Proust’s image, we obtain a highly influ­ential Victorian notion of “stability” which was an explicit harbinger of Catastrophe Theory in evolutionary thinking (the route through population genetics via C. H. Waddington was touched upon in Part II), as well as involving a “twirling” of a polygonal (or polyhedral) framework operating on just such crowd scenes. 

 

In his 1869 Hereditary Genius, Darwin’s smarter cousin, Sir Francis Galton, introduced the motif (frequently reverted to in later writings, of his own as well as of his geneticist followers) of a “polygon of stability.”   This model served to indicate a way to “save the phenomenon” of  natural selection’s operating smoothly while yielding discontinuous results – “how is the law of continuity to be satisfied by a series of changes in jerks?”  He answered his own question in this manner:  “The mechanical conception would be that of a rough stone, having in consequence of its roughness, a vast number of natural facets, on any one of which it might rest in ‘stable’ equilibrium.”  When the stone is pushed, it will totter on an edge, but will regress to its first position when pressure is withdrawn.  However, if a very strong push is given, the stone will be forced

 

to overpass the limits of the facet on which it has hitherto found rest, it will tumble over into a new position of stability, whence just the same proceedings must be gone through as before, before it can be dislodged and rolled another step onwards.  The various posi­tions of stable equilibrium may be looked upon as so many typical attitudes of the stone, the type being more durable as the limits of its stability are wide.  We also see clearly that there is no violation of the law of continuity in the movements of the stone, though it can only repose in certain widely separated positions. [6]

 

                Some twenty pages later in the same work, Galton recasts his “polygonal slab” analogy,  the better to bring it to bear explicitly on the behavior of congested crowds struggling to push and shove their way through narrow passageways:

 

If by a great effort, a man drives those in front of him but a few inches forwards, a recoil is pretty sure to follow, and there is no ultimate advance.  At length, by some accidental unison of effort, the dead lock yields, a forward movement is made, the elements of the crowd fall into slightly varied combinations, but in a few seconds there is another dead lock, which is relieved, after a while, through just the same processes as before.  Each of these formations of the crowd, in which they have found themselves in a dead lock, is a position of stable equilibrium, and represents a typical attitude.[7]

 

                In a landmark paper on “A Theory of Heredity” published in between the two editions overseen by Galton of Hereditary Genius, he presents the third stage of the model – direct application is made to the dynamic pro­­cess of inheritance as it was then understood (i.e., as a “struggle for survival” among animated, self-replicating particles Darwin had dubbed “gemmules”).  Seething multitudes of these germinal beings are assumed to

act on many sides, in a space of three dimensions, just as the personal likings and dislik­ings of an individual in a flying swarm may be supposed to determine the position that it occupies in it. . . . the germs must be affected by numerous forces on all sides, varying with their change of place, and . . . they must fall into many positions of temporary and transient equilibrium, and undergo a long period of restless unsettle­ment, before they severally attain the positions for which they are finally best suited.[8]

 

                It must be noted that such crowd-scene imagery is as typical of Victorian science as it is atypical of the science that went before it:  indeed, the creation of a “statistical mechanics” where vast ensembles of nondescript molecules flit about like brickbats on “drunken walks” was one of the great achievements of that era, leading to the creation of a whole new vocabulary of “mean free paths,” “entropy of interaction,” and so forth.  Major insights were catalyzed by stepping into (Poincaré’s aha! about Fuchsian groups) or just sitting in (Kekulé’s presentiment leading to the famous “dragon biting its tail” dream in which he saw the structure of the benzene ring) that epitome of impersonal crowding, the bus: as Georg Simmel noted,

 

The interpersonal relationships of people in big cities are characterized by a markedly greater emphasis on the use of the eyes than on that of the ears.  This can be attributed chiefly to the institution of public conveyances.  Before buses, railroads, and streetcars became fully established during the nineteenth century, people were never put in a posi­tion of having to stare at one another for minutes or even hours on end without exchang­ing a word.[9]

 

                “That the eye of the city dweller is overburdened with protective functions,” says Walter Benja­min by way of introducing this quote, “is obvious.”  What is no longer so obvious to us today, however, is just how oppressive – even frightening – this burden was felt to be in the early days of urban crowding, when the new trend toward industrialization, among other contemporary factors, brought droves of new­com­ers off the farms to toil in sweatshops and factories.  “Fear, revulsion, and horror were the emotions which the big-city crowd aroused in those who first observed it.  For Poe it has something bar­baric; disci­pline just barely manages to tame it.  Later,” Benjamin tells us, “James Ensor tirelessly con­fronted its dis­cipline with its wildness; he liked to put military groups in his carnival mobs, and both got along splendidly – as the prototype of totali­tarian states, in which the police make common cause with the looters.”[10]  From the vantage of a century’s distance, the most telling innovation of the period was the safety match; to those who lived in the period, revolutionary significance was seen embodied in the kaleidoscope:

 

The invention of the match around the middle of the nineteenth century brought forth a num­ber of innova­tions which have one thing in common:  one abrupt movement of the hand triggers a process of many steps.  This development is taking place in many areas.  One case in point is the telephone, where the lifting of a receiver has taken the place of the steady movement that used to be required to crank the older models.  Of the countless movements of switching, inserting, pressing, and the like, the “snapping” of the photog­rapher has had the greatest consequences.  A touch of the finger now suffices to fix an event for an unlimited period of time.  The camera gave the moment a posthumous shock, as it were.  Haptic experiences of this kind were joined by optic ones, such as are sup­plied by the advertising pages of a newspaper or the traffic of a big city.  Moving through this traffic involves the individual in a series of shocks and collisions.  At dangerous in­ter­sections, nervous impulses flow through him in rapid succession, like the energy from a battery.  Baudelaire speaks of a man who plunges into the crowd as into a reser­voir of electric energy.  Circumscribing the experience of the shock, he calls this man “a kaleido­scope equipped with conscious­ness.”  Whereas Poe’s passers-by cast glances in all direc­tions which still appeared to be aimless, today’s pedestrians are obliged to do so in order to keep abreast of traffic signals.  Thus technology has subjected the human sen­sorium to a complex kind of training.  There came a day when a new and urgent need for stimuli was met by the film.  In a film, perception in the form of shocks was established as a formal prin­ciple.  That which determines the rhythm of production on a conveyor belt is the basis of the rhythm of reception in the film.[11]

 

                The above kaleidoscope image appeared in Baudelaire’s famous 1860 essay “The Painter of Mod­ern Life”[12]; he has one other, appearing in the second draft of his dedication to Arsène Houssaye for Le Spleen de Paris (cited, oddly enough, by Derrida in his turn on a Baudelaire theme called Given Time:  I. Counterfeit Money):    “I have sought titles. The 66.  Although  however this work resembling the screw and the Kaleidoscope could be pushed as far as the Cabalistic 666 and even 6666….”[13]  What is “kaleido­scopic” to Baudelaire here (aside from the Abulafian meditative practice of the “whirling of letters” – the closest thing in Cabalistic lore to the twirling of object-box contents – that one may assume he’s referring to) is revealed a few lines further along in his dedication.  As rendered in Arendt’s translation of Benja­min’s text (which also snips this Baudelairean bit, but with better and more complete English rendering), the dream of this last of the popular lyric poets is of “the miracle of a poetic prose”:  a prose which is

 

musical without rhythm and rhyme, supple and resistant enough to ad­apt itself to the lyri­cal stirrings of the soul, the wave motions of dreaming, the shocks of consciousness. This ideal, which can turn into an idée fixe, will grip especially those who are at home in the giant cities and the web of their numberless interconnecting rela­tion­ships.[14]

 

                One  can be “at home in the giant cities” in different ways.  The poet Nerval walked the boule­vards behind a slow-moving lobster attached to a long colored ribbon.  Around 1840, there was a fad among such “Resistance figures”  (those so-called flâneurs – dandies and artistes –  who refused to suc­cumb to the ano­ny­mous mentalité of the crowd) to take turtles for prome­n­ades in the arcades, and let them set the pace.  “If they had had their way, progress would have been obliged to accommodate itself to this pace.  But this atti­tude did not prevail; Taylor, who popularized the watch­word ‘Down with dawdling!,’ carried the day.”[15]  One can see in this nonchalance a sophisticated deployment of a “shock defense” – but for most people, less self-conscious and compelled to rush, the basic defense entailed a more passive mode of withdrawal, and a more active mode of sensorial rebalance.

 

                The active mode of sensorial rebalance was announced by the kaleidoscope itself:  the century spanning the mania which greeted it and people lining up for movie tickets  saw a radical distancing of people from their own sensory apparatus – a coming, in fact, not to trust its unaided evidence.  This was the age when psychologists like Purkinje began to study “optical illusions,” when the indirection of “point of view” optics and camera ob­scu­ra representation was traded in – the better to study the persistence of retinal images – by scientists  (as well as proto-impressionist painters like J.M.W. Turner), for direct viewing of the sun (with permanent blindness, viz. Joseph Plateau, or at least serious visual impairment, as with Gus­tav Fechner and Sir David Brewster himself, as a side-effect).  The aim was to attain to “an abstract optical experience, that is of a vision that did not represent or refer to objects in the world.”  The key revelation was of the body as “site and producer of chromatic events” – and the seminal byproduct of such en­gage­ment was an ever-increasing “mechanization and formalization of vision.”[16]

 

                Baudelaire not only wrote about the kaleidoscope; in his 1853 essay “Morale du joujou,” he dis­cus­ses two later optical devices which enjoyed vast popularity:  the stereoscope (perfected, as well, by Brewster, and basing its illusionary suasiveness on the ex­ploiting of binocular disparity); and, Plateau’s phena­kistiscope (creating the illusion of continuous motion via rotation of a wheel of static images, the illusion’s effectiveness depending upon retinal persistence).[17]

 

The phenakistiscope substantiates Walter Benjamin’s claim that in the nineteenth century “technology has subjected the human sensorium to a complex kind of training.”  At the same time, it would be a mistake to accord new industrial techniques primacy in shaping or determining a new kind of observer…. In fact, the very physical position required of the observer by the phenakistiscope bespeaks a confounding of three modes:  an individ­ual body that is at once a spectator, a subject of empirical research and observation, and an element of machine production…. In each [mode], it is a question of a body aligned with and operating an assemblage of turning and regularly moving wheeled parts.  The imperatives that generated a rational organization of time and movement in production simultaneously pervaded diverse spheres of social activity.  A need for knowledge of the capacities of the eye and its regimentation dominated many of them.[18] 

 

                The passive mode of withdrawal primarily took the form, as subjected to deep and subtle scrutiny by Richard Sen­nett in The Fall of Public Man, of a retreat from playacting (with concomitant overvaloriza­tion of those few who playacted for a living:  this is the age of Paganini and Liszt, the first “superstars,” and the attain­ing to high status of actors and performers generally).[19]  In the prior century, public attendance at thea­trical events was an occasion for frequently wild engaging of the audience, who might pay to sit on the stage and enter into the action, or at the least exhibit public flights of passion and demands for repetition of mo­ments which affect­ed them:  attendance was as cathartic, that is, in the ancient regime, as it tended to be silent and unde­mon­strative in the Victorian Era.[20] 

 

In the earlier period, new institutions arose, like the coffee house, to faci­litate the intermingling of classes in the new entrepreneurial climate that fomented the innovativeness of the Industrial Age.  Self-presentation on the stage-set of everyday life was, at times, corollarily extreme:  for those with the where­withal, clothing prior to Waterloo was costume drama, while Victorian sensibilities aimed for the total sup­pression of self-expression in dress[21].  While aristocratic values held sway, the highest form of wit was on display in the highly interactive venue of the urban salon; but the following bourgeois century stressed the “domestic virtues” of stay-at-home privacy, romanticizing “family values” as yet another part of “shock defense.”  As that greatest of the era’s social observers, Alexis de Tocqueville, described it,

 

Each person, withdrawn into himself, behaves as though he is a stranger to the destiny of all the others.  His children and his good friends constitute for him the whole of the hu­man species.  As for his transactions with his fellow citizens, he may mix among them, but does not feel them; he exists only in himself and for himself alone.  And if on these terms there remains in his mind a sense of family, there no longer remains a sense of society.[22]

 

                Beneath active and passive modes both, there was something more abstract at work:  a century after Newton, there was a new “force model” emerging – one quite different from the attraction of “central forces” over vast empty spaces of the classical mechanical vision . . . one which came to be referred to as “field theory.”  But as different as this might seem from what preceded it, there is also a deep continuity connecting them.  In order to see it and follow its thread through both ages (and, ultimately, up to our own) we need a more general comprehension of two features evinced in Galton’s stability – features which will let us rotate the “polygonal slab” of our argument not merely from our own “A,D,E Problem” epoch to Galton’s own outlook, but back to the innovative thought-world of a once-celebrated, now largely forgotten Enlightenment Jesuit who hailed from Croatia, and found his way toward an abstraction of the “Rococo curve” that guided the thinking of Faraday, Maxwell, Kelvin, and in fact the most innovative scientific visionaries of the century that came after him.

 

*              *              *              *              *              *              *

 

In “My Chances/Mes Chances:  A Rendezvous with some Epicurean Stereophonies,”[23] Derrida managed to conjoin both the themes we’re after here:  the atomism of Epicurus and Lucretius, and the theme of the aleatoric or “random” (relating it in an interesting way to Heidegger’s dice-like “thrown-ness”).  Curiously, the theme of the random at the origin of things – even as epitomized in the “throw­ing” of dice – is not merely paramount in modern physics (Einstein’s famous objection to quantum uncertainty,   “God doesn’t throw dice!”, comes to mind), but central to such non-Eurocentric texts as the Mahabha­rata, where the role of chance at the root of things is hardly taken as an argument for existential angst or aethe­ism.  (And, until Galton’s withering statistical analysis of “The Efficacy of Prayer” in 1872[24] , this was the case in the West as well!)  More curiouser still, the theme of atomism among the ancient Greeks seems, if anything, remote from notions of “atomic theory” as we know it today – whereas the atomism of Derri­da’s favorite Eighteenth Century, which captivated the imaginations of the following century’s deep­est thinkers, and does have profound bearing on “atomic theory” as we think it presently, seems thoroughly unknown to him.   Moreover, in the Eighteenth Century, arguments in favor of Divine Provi­dence based on the evi­dence of Chance’s working were commonplace[25] – and often conjoined with a mathematical notion of atom­ism which assumed “point particles” as basic, the better to suggest the regulatory priority of the  Soul over mere material embodiment.[26]  All of which suggests some divergent attitudes to take toward the kaleidoscope’s colored lozenges – or Galton’s competing swarms of “gemmules,” for that matter.

 

It’s worth noting that Derrida considers the “clinamen” of Lucretius as well as simple atomism, and its “first swirling” sort of notion has had a modern vogue among the chaotically inclined, especially the great pioneer of far-from-equilibrium thermodynamical systems, chemistry Nobelist Ilya Prigogine:  In Order Out of Chaos, which he co-authored with Isabelle Stengers, he remarks how

 

the early atomists were so concerned about turbulent flow that it seems legitimate to con­sider turbulence as a basic source of inspiration of Lucretian physics.  Sometimes, wrote Lucretius, at uncertain times and places, the eternal, universal fall of the atoms is dis­turbed by a very slight deviation – the “clinamen.”   The resulting vortex gives rise to the world, to all natural things.  The clinamen, this spontaneous, unpredictable deviation, has often been criticized as one of the main weaknesses of Lucretian physics, as being some­thing introduced ad hoc.  In fact, the contrary is true – the clinamen attempts to explain events such as laminar flow ceasing to be stable and spontaneously turning into turbulent flow.  Today hydrodynamic experts test the stability of fluid flow by intro­duc­ing a pertur­bation that expresses the effect of molecular disorder added to the average flow.  We are not so far from the clinamen of Lucretius![27]

 

                The eighteenth century’s greatest scientists tried, and ultimately failed, to make sense out of fluid flow – an insuperably difficult problem, which even the most advanced computer-graphics simulations can­not totally clarify.[28]  In fact, all things they couldn’t comprehend yet they tended to assume were “impon­der­able fluids,” vaguely forming little oceans around the microworlds of atoms.  According to the highly influ­ential chemistry texts of Thomas Thompson, which held sway in the last years of the eighteenth, and first years of the nineteenth, centuries,  there were maybe half a dozen such “fluids”:  one or two each for elec­tricity and magnetism, depending on which theory you believed (Oer­sted’s 1820 one-page announce­ment of the results of playing with a compass in the vicinity of a wire joining two batteries hadn’t shown their shockingly simple unity yet); at least, by some accounts,  two modes of light (Brewster et al. hadn’t unra­veled polarization yet, and anomalies like the “birefringence” of Iceland spar had to be explained away somehow); heat (whose “fluid” was called “phlogiston” until Lavoisier’s care­ful mea­sure­ments indicated contemporary theories of same would require it to have a negative weight, after which folks spoke of “caloric”) –  and perhaps a couple others I can’t remember anymore. 

 

The turbulence nascent in such “fluids” was often put on display in a near-carnival atmosphere:  in one grandstanding set-up, thin magnets would be floated in a tank and brought close enough together to start attracting and repelling each other with wildly mysterious inter­mit­tency.  Ironically, this made espe­cially effective theater precisely because the basic picture assumed by all viewers was one of extreme stasis at the foundations of matter.  The opinions of the first great atomist among modern chemists, John Dalton (1766-1844), were typical, and deemed the voice of orthodoxy:

 

The Daltonian atomic theory had endowed atoms with atmospheres of heat or caloric.  It was these atmospheres which kept the atoms from touching and, since caloric repelled caloric, the atoms of gases were surrounded by quite extensive atmospheres.  What the person who is accustomed to the modern kinetic theory of gases must realize is that this was a totally static picture.  If a gas could be magnified so that its atoms became visible, it would resemble a bowl of gelatine (caloric) in which the gas atoms were imbedded like raisins.  It was in this sense that caloric exercised repulsive powers, just as the gelatine resists compression.[29]

 

                The typical viewpoint of the time was that these fluids obeyed Newtonian inverse-square laws (Coulomb’s law of static electricity is a good for-instance, as well as the law concerning the drop-off with distance of luminosity, first formulated by Newton’s illustrious precedessor, Johannes Kepler); but other­wise (like contemporary physicists’ neutrinos), they did not interact much, if at all, with ordinary matter, which acted like the “flies” in their “ointment.”  And the imponderables, like our contemporary “particles,” tended to proliferate as well:  that celebrated kite-flying experimenter, Ben Franklin, made a compelling case for there being but one electrical fluid instead of one per each charge . . . only to have Luigi Galvani lay claim to having discovered a whole new category of electrical impon­derable.  In 1791, he

 

announced the discovery of animal electricity produced by living tissues.  This fluid, secreted by the nerves, was completely analogous to all the other imponderables except for its connection with life.  The wave of excitement over this epoch-making discovery was accompanied by a ripple of skepticism from those who had recently seen the exis­tence of Anton Mesmer’s animal magnetism denied by the Paris Academy of Science.  Among the skeptics was Alessandro Volta of the University of Pavia who, as a physicist, was convinced that the undoubted signs of electricity could be explained in purely physi­cal terms with no need to call upon vital or other mysterious forces.[30]

 

                It would be decades before Eadweard Muybridge’s horses in motion would startle the world with still-shot photo­graphs able to demonstrate the evanescent happenings that literally came and went in the blink of an eye; yet it was on the basis of a mind’s-eye visualization of this variety (one, in fact, more like those famous mid-twentieth-century Harold Edgerton snaps of, say,  bullets deeply dimpling the balloons they will explode in another milli­second or so, that grace the walls of M.I.T.’s Infinite Corridor) that Roger Joseph Boscovich saw his way to a subtle contradiction in the Newtonian scheme.  This led him to dispense with “imponderable fluids” altogether, replacing them with a purely mathematical notion of alternating zones of attraction and repulsion emanating from infinitesimal points – which just happened to reduce Newton’s laws to rough approximations of subtler things.  While writing his De Viribus vivis of 1745,

 

he investigated the production and destruction of velocity in the case of impulsive action, such as occurs in direct collision.  In this, where it is to be noted that bodies of sensible size are under consideration, Boscovich was led to the study of the distortion and recov­ery of shape which occurs on impact; he came to the conclusion that, owing to this dis­tortion and recovery of shape, there was produced by the impact a continuous retardation of the relative velocity during the whole time of impact, which was finite; in other words, the Law of Continuity, as enunciated by Leibniz, was observed.[31]

 

                The link to Leibniz is important, as it was this philosopher’s notion of “monads” – fundamental unities which were without parts, extension or figure – which would serve to inspire Boscovich’s next advance, when he came to reflect upon the facts determined in De Viribus, and came to enquire as to their causes when his future editor, one Father Scherffer, suggested he investigate the center of oscillation.  This should already sound suggestive of Galton’s “rough stone” which can be stably rocked or, if sufficiently jost­led, caused to rest upon different of its multiple facets; it also suggests the full-blown “competing swarms of gemmules” vision, too, since Leibniz  ascribed to his monads perception and appetition in addi­tion to an equivalent of inertia.  Prior to Galton, heredity was seen as the direct result of quasi-physical forces; his own model, then, was merely more of the same “monadological” thinking:

 

Romanes seems to have envisioned an inertial law of heredity; Spencer speculated that there might be polarized “physiological units” which could carry the hereditary force in much the same way that glass balls carry electric charge; Darwin analogized inheritance to powers which can neutralize or counterbalance each other.  Whether the force model was explicit or implied it was there; in light of the success of nineteenth-century physical science it would have been remarkable if such a model had not presented itself.[32]

 

                The result of Scherffer’s suggestion was the Theoria Philosophiae Naturalis of 1758, which Scherffer edited, and in which Boscovich presented his stripped-down picture of physical law in which the “Newtonian idea of mass is replaced by something totally different; it is a mere number, without ‘dimen­sion’; the ‘mass’ of a body is simply the number of points that are combined to ‘form’ the body”[33] – with every pair of points, no matter how distant, exerting a mutual accelerative force upon every other, the mag­ni­tude of this force depending only on their degree of separation.  At all but the tiniest of distances, this force would obey the Newtonian inverse-square law; but down at the scale where molecules form, spon­ta­neous combustion can be triggered, and atoms would approach each other, the force law takes the form of a Rococo curve:  as with a piece of driftwood battened about by tides until, barkless and twigless, it offers to the eye a purity of smoothness in its twistings, the curve would alternately slope upwards (indica­tive of re­pul­­sion) or down (attraction), with crossings of the shortest path between them indicating relative repose.

 

                Arriving at such a conclusion may seem an unwarranted jump based upon the evidence off­ered; but, if one thought deep and hard on the key matters as Boscovich did, it could appear as an inescap­able conclusion, if one’s mental proclivities were sufficiently abstract and philosophical:

 

To most natural philosophers of the eighteenth century, the collision of two atoms was exactly analogous to the collision of two billiard balls.  Boscovich real­ized that the situ­a­tion was more complicated.  When two billiard balls collide, the result is elastic defor­ma­tion and recovery; the deformation is made possible by the fact that the molecules of the billiard balls can move relative to one another.  It is the displacement of these mole­cules and their subsequent return to normal position that leads to elastic re­bound.  With atoms, such a process is clearly impossible.  Since atoms have no parts and are perfectly hard, deformation is impossible.  What happens, then, when two moving atoms meet?  If they are little billiard balls, Boscovich argued, and if deformation is im­pos­sible, no time can elapse between contact and rebound.  Thus, at the moment of im­pact, the atoms will have two velocities at the same time.  They will have their original velocity and the velo­city of rebound.  To Boscovich this was logically and physically absurd yet it was a neces­sary consequence of the theory of material atoms.  Boscovich’s solution was a radical one; he eliminated matter completely as a separate entity and replaced it with attractive and re­pulsive forces.  An atom was to be considered as a dimensionless mathematical point.  Surrounding this point were forces which were alternately attractive and repul­sive…. With this model, Boscovich was able to account for atomic impact with no difficulty.  Since the curve of atomic forces was continuous, the repulsion between two colliding atoms increased continuously with time and the absurdity of the Newtonian case was avoided.[34]

               

                Boscovich’s model allowed him to give satisfactory explanations – albeit in a qualitative way only – of many phenomena that had proved baffling:  why, when heat was added to a substance, did it suddenly change its physical form at some specific “freezing” or “boiling” point?  It was here that the logic was just like Galton’s, with each “zero” on the curve corresponding to a “typical attitude”:  at such stable points, heat – in the form of atomic or molecular motion – was added to a body composed of these atoms, and “their oscillations would eventually push them over the ‘hump’ of repulsion to a new stable position at a great distance from one another.”[35]  Ditto for crystalline structure, where arrangements analogous to “poly­gonal slabs” (or polyhedral vertex figures) became practicable.  (And, by introducing asymptotic “breaks” in a particular ensemble’s “force curve,” spontaneous combustion and other mysteries became easily com­prehended.)  It is not surprising, then, that this vision captured the imagination of chemists – not just Dalton (and his amateur-chemist close friend, the poet Coleridge), but Humphrey Davy and his protégé Michael Fara­day, were among the many to draw inspiration from it.  In our own time, the one-line boil-down of all sci­entific knowledge which Richard Feynman suggested be passed on to future generations, cited in Part II of these ramblings (and attaching to Note 34 there), is worth repeat­ing:  “all things are made of atoms – little parti­cles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another.”

 

                It is easy to make the link from Boscovich’s curve to the “behavior spaces” of Catastrophe Theory too:  the number of “humps” separating atoms in their vertex figure would demarcate which “cuspoid” figure was stably (if, say, it were crystalline) regulating the “unfolding” of their mutual behavior.  And we can come awfully close to the as-yet-uninvented kaleidoscope as well:  we merely need to place Boscovich in context, and improvise.  For the good Jesuit was lionized for his brilliant researches throughout Europe, invited to join all the most prestigious learned societies, and even lived for some years with a grant of full citizenship in France.  His prominence was such that some of the greatest figures of the age deigned to make him their friend or enemy (the mathematician d’Alembert, for instance, whose “dream” was made famous by his co-Encyclopedist Diderot, loathed the Jesuits, who had recently been dissolved anyway, and ergo, by extension, could not tolerate Boscovich; there was a priority dispute with Laplace, too, concerning a method for deriving the orbits of comets, but the astronomer Lalande was devoted to him, and Louis XV, then XVI, protected him). 

 

I imagine him being received and officiated over in the most fabulous architectural extravagance of that “Age of the Baroque” which had only just recently ended – by which I mean the near-exact contem­po­rary of that gargantuan British edifice, Newton’s Principia, which the Sun King had erected at Versailles.  And I imagine him – renowned optics researcher, and newly appointed director of “Optique Marine” upon his 1773 visit to Paris and incipient citizenship change – vastly en­joy­ing himself in what for him would doubtless have been the most enticing of  “tourist traps” there, the 243-foot-long Galerie des Glace design­ed by Jules Mansart and decorated by Charles Le Brun, illuminated by seventeen great windows.  (At night, in the Sun King’s time, the murals, mirrors, and gilt blazed with the light of three thousand candles.)[36]

 

We find yet another type of perspective in the “hall of mirrors.”  Here visual space and auditory space approach as nearly as possible.  As in auditory space every tone is repeat­ed from octave to octave, here every object is repeated from mirror image to mirror im­age.  As, for the center of the dynamic field of tones, presence at one point in auditory space always means presence in so and so many octave repetitions as well, so in the hall of mirrors no object can be present only once; every appearance of an object at one point is an appearance of it at many places in the mirror images.  As the entire dynamic field is repeated from octave to octave, so the whole space of the hall of mirrors is repeated from one reflection to the other.  And as in auditory space going away from a tone is always also going toward the same tone in its octave, in the hall of mirrors too there is no going away from a place that is not at the same time a going toward the same place in its mirror image.[37]

 

                What might seem somehow forced about Boscovich’s solution when contemplated visually, at­tains to naturality when thought of as an auditory construct:  the alternating zones of attraction and repul­sion become just so many octaves.  The implicit “haptic balance” of this Age of Reason (if you were not a sans culotte, that is) has seemed intoxicating to some McLuhanites, such as Neil Postman, whose recent Building a Bridge to the 18^th Century would tell us, at its subtitle suggests, “How the Past Can Im­prove Our Future.”  What is clear is that the balance was mostly mediated by the intellect, not the heart, and Revolu­tion would soon put an end to it anyhow.  (Boscovich, detained by ill health in Milan, was obliged to ask the French government for an extended leave, only to succumb to melanchony and die on February 13, 1787 – thereby just missing getting caught up in the events celebrated on Bastille Day by less than two and a half years.)

 

                We just miss getting the kaleidoscope here, too, because the public amplification of images in the Hall of Mirrors includes the viewer – but “privatizing,” by excluding the latter from visibility, is what the kaleidoscope (and the increasingly bourgeois tenor of its times) requires.[38]  The dis­appearance of the public persona becomes exa­cer­bated by the creation of a new kind of contemplative object – a personification of the Crowd.  Prigogine, in the work cited earlier, specifies the source of this in scientific thinking, and in­deed goes on (in pages I won’t cite here) to show how the new physics of statis­tical modeling (as exempli­fied in the thermodyna­mics of Maxwell and Boltzmann) drew inspiration from this new abstract notion of an idealized, disembo­died Man, discernible in crowds but not to be found in salons:

 

Maxwell himself appears to have been influenced by the work of Quetelet, the inventor of the “average” man in sociology.  The innovation was to introduce probability to phy­sics not as a means of approximation but rather as an explanatory principle, to use it to show that a system could display a new type of behavior by virtue of its being composed of a large population to which the laws of probability could be applied.[39]

               

                Quetelet was the main inspiration for Galton’s own statistical approach to things evolutionary, and Boltzmann was deeply attracted by the idea of evolution:  indeed, “his ambition was to become the ‘Dar­win’ of the evolution of matter.”[40]  Like the inverted images cast on walls by a camera obscura, the linkage of randomness to Divine action needed to go through the Darwinian pin-hole which Quetelet opened before twentieth-century thinkers could blithely assume (or not so blithely, in Einstein’s case) the “naturalness” of the semantic link between godlessness and Chance so peculiar to our narcissistic age.

 

Benjamin says Baudelaire indulged in the illusion of endowing the crowd with a soul.  Just so, Quetelet and Galton treated the “ideal types” of different populations, revealed by their conforming to the Gaussian normal curve, as stable and palpable entities, and even amenable to hierarchical stratifica­tion.  On the latter, Galton in fact enhanced his technique of “composite portraiture,” not merely to create such “ideal types” from samp­lings of individual representatives, but to generate higher-order montages, whose Jacob’s-Ladder-like navi­gating up to some phylum-level abstraction, then down again to actual humans, he man­aged via judi­cious use of photonegative overlays in his so-called “Theory of Trans­for­mers.” 

 

                In the early days of photographs, poses had to be held for perhaps two min­utes, since exposure of the plate required that much time to complete.  (This is also why nobody smiled in those photos!)  Galton, wishing to generate an atlas of gene­ric images of criminal types, say, or family resemblance among close relatives, conceived the plan of rotat­ing sitters during an exposure.  If each of a dozen family members were given ten seconds of posing time in front of the same lens, the resulting plate would show a blend of their visages that would highlight the re­sem­b­lances and blur out the differences.  He saw this as offering the first method for empirically determin­ing Hume’s “general ideas” (the extrovert’s pre-emptive strike against Jung’s “archetypes”).  And, he saw it as extensible to ever more generalized “ideal types,” whence his “The­­o­ry of Transformers,” which allowed a photographic embodiment of a march up and down what today, in programming circles, might be called an “object hierarchy.”[41] 

 

                Such structures of population groups bear a strong resemblance to the structures of abstract groups which were one of the most profound innovations of that or any age’s mathematics.  The “complex kind of training” to which technology subjected the human sensorium in the Nineteenth Century reduced human response, as much as possible, to a few simple, repeatable motions.  The translations, reflections, and rotations at the basis of all ornamental art (see the comments in Note 6 of Part I) are also the basis of a universal vocabulary of such simple motions – and it is these, rather than the concrete geometric patterns they could be used to make wallpaper out of, which became the focal point of the new mathematics of “Groups” pioneered by Galois before his heavily romanticized death at the age of 20.[42] 

 

From a purely formal vantage, Galois’ approach could be seen (and I apologize in advance for the anachronism) as a “decon­­struction” of the Cartesian idea of an equation.  The central concern was to study the permutations of solutions of algebraic expressions, and show their analogy to the permutations of geo­metrical objects.  Most spectacularly, Galois showed that the performance of sequences of motions belong­ing to the 5-, 4-, and 3- fold symmetric substructures of the icosahedron’s group of (5 * 4 * 3 = ) 60 rota­tions, displayed a peculiar kind of sensitivity to their hierar­chical ordering.[43]  This so-called “absence of nor­mal subgroups” – an absence in the “simple” groups displaying it akin to a prime number’s lack of non-trivial factors – magically indicated that the longstanding problem of exactly resolving fifth-order algeb­raic equations  was insoluble.  The details won’t concern us here; but the fact that the modu­lar­ization of mecha­nical components and motions, culminating in Henry Ford’s Model T assembly lines,  had its parallel in the alleged “pure abstractions” of higher mathematics, is no coincidence.[44]  Of greatest importance to our pre­sent concerns, such a parallel indicates the incredible ubiquity with which the “laws of form” we’ve been focus­ing on were disseminated throughout all levels, no matter how crass or abstract, of general culture.  It is worth considering, in this light, that at the height of the Kaleidoscope Mania, Galois was a little boy of six or so, and that his father – well-off mayor of a prosperous Parisian suburb, soon driven to take his own life in the wake of the political recrimina­tions following Napoleon’s fall – would certainly have bought one of these symmetry-making toys for his young and precocious (and highly impressionable) son!

 

The hierarchical “level-sensitivity” peculiar to the age can be seen as its most envelop­ing feature:  in classical rhetoric, a figure of speech which indicates such sensitivity, by substitution of part for whole, genus for species, or the reverse, is called a “synecdoche” (sounds like the upstate New York city of Sche­nectady).  In Giambattista Vico’s famous rhetoric-driven scheme for appreciating the cyclica­lity of histori­cal change, the third of his “Four Ages” – coming after the Metaphorical Age of Gods and Metony­mi­cal Age of Heroes, and before the Ironic Age of Dissolution and Decay – is called the Synecdo­chic Age of (Quetelet’s Average) Man.  Research in “metahistory” of recent decades, both building on and react­ing to postmodern­ist thinkers like Michel Foucault, will lead us to take such hoary constructs seri­ously (and find them, in fact, lurking within the logical progressions of Lévi-Strauss’ own Mytholo­giques) in the pen­ul­ti­mate in­stallment of this harangue.  For now, assuming it significant, let’s dwell on it as backdrop.

 

The mere use of a term like “backdrop” tells us we’re assuming a stable context in which more vo­la­tile matters can struggle, develop, and change:  we are confined, in some sense whose depth requires spe­ci­fying, to some “typical attitude” of the oscillating rock of the historical unfolding of thought.  Toward the most fine-grained scale in the study of science’s history that isn’t just “data collecting,” such a back­drop would be called a “law,” as when Catastrophist Chris Zeeman says:  “A scientific law is an intellectual rest­ing point.  It is a landing that needs being approached by a staircase, upon which the mind can pause, be­fore climbing further to seek modifications.”[45]  Toward the other extreme, Vico and “meta­his­tori­ans” like Hayden White speak of a succession of four “Ages” that sufficiently complex cultures all go through. 

 

This is to suggest a se­quence of back­drops akin to what thermodynamicists – or histo­rians like Henry Ad­ams[46] who were inspired by them – would call “phases.”  (Or pre-So­cratics would call Ele­ments; or, par­ticle physicists might relate to the three “flavor pairings” of quarks indigenous to dif­ferent strata of pri­mor­­dial “symmetry breaking,” with a fourth “break” moving one toward near-Big-Bang levels of ambi­ent energy where all such material distinctnesses dissolve).  The “ice” phase of water, heated suffici­ent­ly, will more or less suddenly change drastically into a “water” phase, and this, after more heating, to “steam.”  Ob­vi­ous­ly, many levels of hierarchy are thinkable as residing in between these extremes – in fact, Catastro­phe Theory is fundamental­ly a theory of and comprising just such hierarchies, concerning itself before all else with how entities it describes can be modeled by, or moved between, such zones of relative stability.

 

The use of the term “backdrop” here also suggests, however, that it will not be of present interest to us as such:  for now, its stable, “I know what that’s about” quality will let it do service as a projecting screen, on which the normal sorts of movies historians narrate can be seen.  And we know there are at least three screens in the Cineplex – Boscovich, coming before the great giving off of heat indicating the “state change” of the Napoleonic Wars, is one screen down, while Galton (dying three years before the Zimmer­man Telegram launching the next comparable conflagration) has his story unfolding a screen away from “A,D,E” and Thom.  But unless we have come to view that greatest of the experimental silent-film epics, Abel Gance’s Napoleon, where the sizes, shapes, and colors of the screens themselves are ever-shifting, we must wonder as to the “historical” nature of the connections made above, between Boscovich’s curve, the twirl of Brewster’s toy, and Galton’s polygonal slab. 

 

There is surely a profound sense – and a surprising one, as well – in which these radically dispa­rate, historically distinct manifestations are reflections upon the same thematic; but, though separated in time, this sameness is not progressive – rather, there is an aura of inter­mi­ttency:  like Wagner’s leitmotifs announcing key characters’ entrances, Vinteuil’s “petite phrase” in Proust’s Swann’s Way, Apocalyptic omens, or the swelling of arthritic joints presaging rain, their irruption into awareness signifies the reaching of some sort of temporally constrained, change-portending threshold.  But the signifier itself has a radiation and development in the psyche that is timeless, and in some ways never changing.  One physicist who also became an eminent historian of his field, Gerald Holton, has referred to such contents as “themata”; given their typically large subjective loading, he first warns us what they are not: 

 

They are certainly not unapproachably synthetic a priori, in the eighteenth-century sense; nor is it necessary to associate them with Platonic, Keplerian or Jungian archetypes, or with images, or with myths (in the non­derogatory sense, so rarely used in the English language), or with irreducibly intuitive apprehensions.

 

He then suggests that their origin will likely best be unraveled through perception studies, espe­ci­al­ly “of the psychological development of concepts in young children.”  Or, perhaps, from “building upon Kurt Lewin’s dynamic theory of personality” – a theory which attempted, long before Catastrophe Theory could offer a toolkit adequate to the task, a “topological field” approach to psychic development.[47]  He then opines that the best way to go is, basically, that chosen here!

 

But, pending reliable results, the most fruitful stance to take for the moment seems to me akin to that of a folklorist or anthropologist, namely, to look for and identify recurring general themes in the preoccupation of individual scientists and of the profession as a whole, and to identify their role in the development of science.[48]

In an article published while Lévi-Strauss was still completing his Mythologiques, fellow anthro­pologist Edmund Leach notes that, based on the arguments developed in the last chapter of The Savage Mind,  he suspects the French thinker feels “that the only really satisfactory way to make sense of history would be to apply to it the method of myth analysis that Lévi-Strauss is exhibiting in his study of South American mythology!”  He then cautions that “Whether such an argument could possibly have any appeal to professional historians or philosophers of history it is not for me to say.”[49]  But I agree with him anyway:  for all the reasons implicit in the analyses just conducted, that is certainly part of what I’m attempting here! 

 

In Part II, Note 28 we heard Lévi-Strauss contemplate making “an inventory of all the customs which have been observed” in personal relations, in myths, in social customs, in dreams; and express the conviction that the ensemble should ultimately prove reducible to a finite system of types, allowing one to “eventually establish a sort of periodic chart of chemical elements, analogous to that devised by Mende­leiev” in which “all customs, whether real or merely possible, would be grouped by families.”  In the in­tro­­­­­duction to Thematic Origins, Holton notes “the remarkable fact that the range of scale of recent theory, experience, and experimental means have multiplied vastly over the centuries while the number and kind of chief thematic elements have changed little.  Since Parmenides and Heraclitus, the members of the thematic dyad of constancy and change have vied for loyalty,” and likewise for themes like structural stability vs. chaos, hierarchical structure vs. monadic unity.  A total, Holton guestimates, “of fewer than 50 couples or triads” have apparently “sufficed for negotiating the great variety of discoveries.  Both nature and our pool of imagina­tive tools are characterized by a remarkable parsimony at the fundamental level, joined by fruit­ful­ness and flexibility in actual practice.”[50]

 

Holton’s own researches – on the development of Einstein’s thought processes, for instance – hold great interest; but his focus is primarily on the subjective role of his “themata” in the private incubation of publicly enshrined scientific notions.  Likewise, he has, for pragmatic reasons (e.g., the “nonlinear revolu­tion” was barely afoot when he wrote his book), not seen fit to subject the themata themselves to a scien­ti­fic analysis – to study their behaviors and contents from the mathematical model­ing viewpoint.  So while his work certainly paved the way for my own in these pages, a certain parting of the ways is called for.  To demarcate this difference, the term “themata” will not be stressed; instead, I’ll refer to “skim­ming stones.”

 

Used with “stones,” the verb “skim” connotes throwing in a gliding path, so as to ricochet along the surface of water.  (Of course, “skim” by its lonesome has numerous relatable connotations, such as removing the best or most readily obtained contents from something, or scanning a book quickly for the chief ideas or plot elements.  These senses are implicit here as well.)  Another attraction of the phrase is its verb vs. noun ambiguity:  “skimming stones” can be a process description (in response to a question like, “What are you doing?”), or “skimming” can be frozen into adjective status, modifying a most solid noun.  This natural case of something like a “wave/particle duality” suggests, in its turn, a well-turned phrase of Alfred North Whitehead concerning milestones, penned in the first days of quantum mechanics: 

 

one of the most hopeful lines of explanation is to assume that an electron does not contin­u­ously traverse its path in space.  The alternative notion as to its mode of existence is that it appears at a series of discrete positions in space which it occupies for successive dura­tions of time.  It is as though an automobile, moving at the average rate of thirty miles an hour along a road, did not traverse the road continuously; but appeared successively at the successive milestones, remaining for two minutes at each milestone.[51]

 

If the road be considered a time-line, the image of “reincarnation” springs to mind – an image we found used tellingly by Lévi-Strauss at the close of the last installment.  If the road be thought, though, as a spatial extent, then the other half of the last installment’s closing imagery suggests itself:  the skimming stones are, in fact, semiprecious baubles, even gemstones – strung on the lanyard of a cross-disciplinary argument as a Glass Bead Game unfolds.  Holton speaks of his “themata” as coming in two’s or three’s:  I suppose “skimming stones” to come in strands of arbitrary length, and amenable – as on a charm bracelet or array of neck chains – to clustering in other than linear patterns.  Our next step will be to examine some of those which can be un­folded from the discussion above, removing or adding a Bead here and there and seeing where this leads.

 

*              *              *              *              *              *              *

 

                Our three “skimming stones” – or better, three “splash diadems” triggered by successive ricochets of one such stone off the moving stream of History  – all vibrate sympathetically with Catastrophe Theory’s key urgen­cy:  symbolic miming of the dynamic oscillation between “typical attitudes.”  Manifesting in dif­ferent times and places, in disparate fields of contemplation, their com­mona­li­ty is, in Hol­ton’s sense, “the­ma­tic,” hence psy­chic or internal.  We wish to see their ensemble inter­acting with a domain whose com­mon­ality is, on the contrary, determined by the “syn­ec­dochic” back­drop just discussed – by factors, that is, which are external, hence dependent upon historical context. 

 

At a minimum, we require semantic traf­fic with some one content in the backdrop.  The elements of such a fourfold ensemble, in order to “playact” appropriately in suitable roles, would, like the “food ta­boos, good manners and utensils used for eating or for personal hy­giene” of Lévi-Strauss’ “Bos­co­vichian reverie” (or of comparable fetishizing in modern advertising),  perform as “mediatory agents ful­fill­ing a dual func­tion”:  they act, on the symbolic plane, as “insulators or transformers” regulating various psychic “charges”; but, mundanely, they provide pragmatic models and measures of appropriate consumption and behavior.  This dual functionality is of the essence:  like conic sections and the “umbilic” catastro­phes named for them (whose collective cycling-through generates the “splash diadems” captured by Edger­ton’s instan­taneous photographs), their coordination depends upon the interactivity of two foci of attention.

 

The splash diadem triggered by a skimming stone’s transit can take numerous forms, depending upon the height, force and angle of the stone, the viscosity of the fluid involved, and so on.  Two of Edger­ton’s photos of milk splashes are showcased in Bonner’s abridgement of D’Arcy Wentworth Thompson’s  classic tome On Growth and Form,[52] cited as a major source of inspiration by Thom and Lévi-Strauss both.  The sequence of a droplet inducing a circular wavefront, whose undulating rim degenerates into a corona of spikes, the spikes breaking off into further droplets, engendering – if conditions be right – further iterations of the process, suggests a “cas­cade effect” of skimming stone sequences, whose progressive “beading” evokes still-vague imagery in need of sharpening, of “Glass Bead Game” play with the data of History. 

 

Toward the end of the last installment, we saw Lévi-Strauss describing his deploying of the four­fold Klein Group “including a theme, the contrary of the theme and their opposites” as a framing device in organizing his long caravans of mythic evidence in the formulation of his arguments.  He deployed, specifi­cally, “sets of interlocking four-term structures, retaining a relation­ship of homology with each other.”  Soon thereafter, we observed the far less focused “semiotic Sisyphus” approach of Derrida, who would form fickle relationships with various manifestations of “3+1” structures, none of which would sustain his interest for more than an essay or two.  Let’s take one of these latter – the “passe partout” picture frame favored by art galleries, one of whose four sides is open for rapid removals and insertions – and assume we’re on the prowl for an appropriate image to showcase.  Then, let’s constrain our freedom of choice in selecting it, so that we get pairings of themes and oppositions per Lévi-Strauss’ tactic.

 

Boscovich and Galton are at opposite ends of the time-line, and both are intrinsically linked to the thematic notion of “structural stability” which unfolds in Catastrophe Theory; Brewster’s Kaleidoscope is only accidentally related in this way, but bears a deep (in fact exact) analogy to the superstruc­ture of the “A,D,E Problem,” of which Catastrophe Theory itself is but an “accidental” instance.  A natural fourth term would contrast Brewster’s toy with something intrinsically linked to “synecdochic” history, and con­tempo­rary with Brewster’s work.  And, for maximal sharpness, this term would also oppose Brew­ster’s status in some “our stuff makes their Brand X look like the opposite of what they both claim to offer” formula. 

 

As the Kaleidoscope, and Brewster’s own exemplary scientific reputation, were both side-effects of his award-winning work on polarization, the best conceivable “fourth” would be some scien­tific revela­tion accidentally related, like the Kaleidoscope, to the Cata­s­trophic form-lan­guage, while embo­dy­ing some key insight into the still-evolving, soon-to-dominate “field theory” mentality – some­thing which would ma­gically render Brewster’s own contribution to said evolution in a positively reaction­ary light.  And while these specifications may seem overly ab­stract, hence unamenable to fulfillment, save by an act of brute force, they point to palpable fact in spite of that:  for, like a rock star with one chart-busting hit after ano­ther who suddenly loses his touch, Brewster too had a plummet (“fall” is too mild a term for it) from grace.

 

What images are evoked for you by the word “field”?  With no scientific context provided or assumed, you’ll likely see something like sheaves of wheat being combed by mind’s-eye winds.  Iron filings placed on a thin surface beneath which magnets dance provide more such imagery of “field lines” and their swirling patterns.  But there is a fundamental flaw in such depictions, which Brewster – like many other scientists both great and small of his era – could not see his way around.  This “blind spot” in his vision concerned the essence of synecdochic thinking:  the fact that the “field” itself displayed “Ges­talt”-like properties, both conceptually and actually, making it hierarchically distinct from the “par­ticle” level of description:  Quete­let’s “Average Man” was not to be conflated with an ensemble of individual people, nor were the properties of the statistical ensembles studied by Maxwell and Boltzmann at all redu­cible to some massive summing of quantitative attributes of the quadrillions of molecules which comprised them:  the “entropy,” “mean free path,” “phase,” or even “temperature” of a molecule, were neither useful nor mean­ingful constructs, except within the context of the ensemble.

 

If one insisted on thinking of a “beam” of light as behaving like the simple additive sum of myri­ad  corpuscles of illumination (“rays”), one could not see the “wave.” Yet it was ba­sic to Eighteenth Century me­cha­­nics – and indeed, one of its great achievements – to reduce all measura­bles to linear quantities:  pres­­sure could be measured by the inches of mercury in a barometer, and tempera­ture analogously by the newly invented thermometer; even potential energy had a “line” used to measure it – the height to which some stan­dard weight would need to be raised (against gravity’s resistance) for it to attain, upon release, the same degree of kinetic energy at impact.   The wave theory, though, as developed by Au­guste Fresnel in France, assumed the wavefront as the fundamental entity, whose properties could not be reduced to any simple summing of ray properties.  And while Brewster was winning accolades with his various optical researches in the British Isles, another Frenchman, Joseph Fourier, was developing a mathe­matical theory of Heat which replaced such “line” thinking in a thoroughgoing manner with super­posable “harmonics” of circu­lar motions:  a proliferation of sine and cosine waves, whose equations were written with the Imagi­nary numbers whose powers did not increase “exponen­tially” like compound interest, but traced out spirals in the complex plane, the powers of their basic “unit” moving for­ever in a circle.[53]

 

                Even on the question of polarization, the solution of whose mysteries seemed Brewster’s greatest triumph, the new approaches, while accepting (and re-interpreting) his results, presented new questions and depended upon new attitudes which made Brewster’s analytic tools seem antediluvian, and his inability to see things in the “new, improved” way made him progressively marginalized – so much so, that by the 1830’s he could no longer publish in the mainstream journals his researches had once dominated! 

 

 The signal importance of polarization to the historical events arises primarily from its close association with deep-seated views on the nature of rays and beams of light.  As understood … by most optical scientists before about 1830, a beam of light consists of a collection of objects called rays.  These objects, it was assumed, possess individual iden­tities and so can be counted, the intensity of a beam being measured nu­mer­ically by the number of rays it contains.  The ray, it was further thought, has an inher­ent asymmetry about its length.  Think of it as a stick with a crosspiece nailed to it at right angles.  Given the direction of the ray, the orientation of the crosspiece in a plane at right angles to the ray determines the ray’s asymmetry.  

 

According to this way of thinking polarization,  the central topic of experiment and theo­ry in the 1810s, does not characterize the individual rays in a beam but only the beam as a collection of rays.  One cannot, that is, speak of the polarization of a single ray.  This has no meaning because a single ray, as it were, has its crosspiece permanently nailed on and so is always just as asymmetric as it can ever be.  Instead, polarization as a category ap­plied only to collections of rays, to beams….  Accordingly, the polarization of a beam is altogether a mat­ter of degree, of the number of rays that are aligned in various directions.  And so this property of light is very closely tied to instrumental techniques:  if a polariza­tion detector is comparatively insensitive, a particular beam may prove to be unpolarized; but if the de­vice is very sensitive, the same beam may just as reasonably prove to be par­tially polar­ized. [54]

 

While Brewster continued to score successes with his writing and inventions (his perfected stereo­scope, and book about it, both date after the 1831 “break boundary,” beyond which his purely scientific work became ever more “crank”), he never wrapped his head around “waves” successfully.  The new gene­ra­tion of British wave theorists, meanwhile, collected around George Biddell Airy, Lucasian Professor (New­ton’s old job; Steven Hawking holds it now) at Cambridge since 1828.  The second edition of his Tracts, in 1831, marked his com­plete acceptance of the wave theory, and John Herschel and William Whe­well respectively climbed on board just before and after him.  In 1833, William Rowan Hamilton’s predic­tion of the new phenome­non of conical refraction from Fresnel’s biaxial wave surface (and its almost im­me­diate confir­mation in experi­ments by Humphrey Lloyd) marked the complete acceptance of the theory by everybody else.[55]  (Almost every­body, that is:  in that year, the first meeting of the British Association for the Advance­ment of Science was riven by factional strife between the wave theory’s adherents and Brewster’s “selec­tionists”; the latter were routed, and “Brewster was reduced to reporting experiments anomalous to the wave theory, while sniping at Whewell from the columns of the Edinburgh Review.”)[56]

 

The great triumph of the new regime, though, came in 1836:  this marked the publication of the Lucasian Professor’s great study of diffraction (and one of the great early success stories of the new wave-theoretic approach); and, better yet, it contained one of the most wonderful pieces of accidental naming in all of physics (even better, for my money, than the “Poynting vector”) –  the “Airy Integral” giving the complete solution of the optics of . . . the rainbow!

 

And what Fresnel had done for optics in general, Airy did for the rainbow in particular.  No longer could one be satisfied with relatively simple and easily understood geometrical diagrams…. What was essential was a precise analytic expression for the intensity of illu­mination at each and every point of the area brigh­tened by the bow.  Extraordinary ma­th­e­­matical powers were called for, and these were possessed by Airy, who had carried off the prizes in mathematics during his student days at Cambridge…. His computa­tions are too forbidding to include here, for they led him to an integral which can not be evaluated in terms of the elementary functions.  He found that the intensity of light is given by the square of an integral which since has come to be known as “Airy’s rainbow integral.” This he wrote as ∫_w cos π/2 (w³ – mw) [integrated over the whole range of w].  The para­meter m determines the angular departure of the ray from the Cartesian ray, and Airy tab­u­lated the values of his integral (and of its square) to seven decimal places for intervals of 0.2 from m = -4 to m = +4.  The methods of computation, in which use is made of mecha­ni­cal quadratures or approximate integration, are tedious and involved . . . [57]

 

                Three remarks worth making:  first, while this integral, for its time, was a triumph of technical mas­tery and calculational persistence, in fact the basic mathematics can be cranked out using MathCad, Macsyma, Maple, or Mathematica in an eye blink ; secondly, note the form of the content of the equation in parentheses:  it is the simplest possible instance of a Catastrophic unfolding!  (A fully wave-theoretic ren­dering of all Catastrophic forms – which has led, in the hands of Michael Berry, to the first viable theory of the “twinkle twinkle” effect displayed by “little stars” – was created under the name of “oscillatory inte­g­rals” by no less than Vladimir Arnol’d, creator of the “A,D,E Problem” itself.)[58]  Thirdly, the reference to the “Cartesian ray” points to something else:  the “Airy Integral” is “in the splash diadem” of yet another “skimming stone,” whose sequence takes us back to Book VIII of Descartes’ Les Météores (but one com­po­nent of the massive publication package he launched on the world in 1637, including his Géométrie and Discourse sur la Méthode, to which it was the third appendix).  Descartes believed, like his contempo­ra­ries, that this was the first attempt at studying the bow experimentally (although later researchers would find that Dietrich of Freiberg and Kamal al-Din had covered that ground before him); it was also virtually the only instance of an application of his mathematical “method” to an actual, palpable phenomenon.[59]

 

                But there were prior splash diadems:  contemporary with the first inklings of perspective, Dietrich of Freiberg (who, unbeknownst to Descartes, had experimentally contemplated the rainbow before him) had the crucial insight that the way to think about it was not as a phenomenon to be modeled upon the study of a globe of water; rather, one should treat the bow as the artifact of light’s play upon an indefinite num­ber of globes – the “droplets” whose condensate makes clouds.[60]  And we can go back yet one more step . . .

 

                Upon reading the first volume of Lévi-Strauss’ Mythologiques,  Octavio Paz tells us that

 

Three symbols caught my attention:  the rainbow, the opossum, and fishing poison.  The three are mediators between nature and culture, the continuous and the discontinuous, life and death, the raw and the decayed.  The rainbow means the end of rain and the origin of illness; in both these ways it is a mediator:  in the first instance because it is an emblem of the beneficent conjunction between the sky and earth and in the second because it em­bodies the fatal transition between life and death…. The rainbow is a homologue of the opossum, a lecherous and foul-smelling animal:  one attribute ties it to life and the other to death (putrefaction).  “Timbo” is a poison which the Indians use for fishing and thus is a natural substance used in an ambiguous cultural activity (fishing and hunting are trans­for­mations of war).  In all three symbols the essential rupture or discontinuity between nature and culture, whose chief and central example is cooking, becomes thin and attenu­ated.  Their equivocal character does not come solely from their being receptacles of con­tra­dictory properties, but rather from their being logical categories which are difficult to think about:  in them the dialectic of oppositions is at the vanishing point.[61]

 

                The dangerously “narrow straits”[62] which separate Nature and Culture in all three cases are dwelt upon by Lévi-Strauss in his text:  the rainbow is an explicitly “Catastrophic” symbol for the reasons just discussed above; but the opossum and fishing poison play analogous roles in the myths.  Note, too, how the three function as a connected triad, which a close reading of The Raw and the Cooked makes clear is not something Paz has hallucinated.  We will see in the seventh installment how the four volumes of the Myth­ologiques create progressively more complex “Catastrophic” organizing structures, based on ever higher order forms of  (math argot here) “universal unfolding,” or (to use the anthropologist’s term) “logics.” 

 

                We’ve just presented a second “skimming stone sequence,” some of whose “splash diadems” are clearly synchronous with those we started with.  Let’s now introduce a third, which will engage in “syn­chro­nized swimming” with both of these.  Dietrich of Freiberg’s “splashdown” of the rainbow was contem­porary with Giotto’s proto-perspectivist frescoes in Italy.  A brief but terrible interrupt called the Plague (which suffered an especially virulent outbreak we might think of as demarcating the end of the Medieval period) separated both Dietrich and Giotto from the “main line” of Renaissance developments.  Assuming this “lag” be non-problematic, let’s take 1453 as the year “officially” beginning the Renaissance, and reca­pitulate  the initial “splashdown” of Perspective’s “skimming stone” there.  This was a big year in many senses:  it marked the authoring of De Visione Dei sive De icona; the end of the Hundred Years’ War be­tween England and France; and, even more sym­bolic for those who lived through it, the fall of Constanti­nople to the Turks.

 

Nicholas of Cusa, who had been there in 1437, has just brought the frightful news back from Rome, and amidst the rumors of horrors, violence and blood everywhere, he wrote, a month before Icona, his De pace fidei (faith as the basis for peace), an anti-Babelian “vision” of a heavenly “theater” in which, one after another, a delegate from each nation gets up to bear witness to the movement which supports it… each [attesting] in the lan­guage of his own tradition to the truth which is one:  this harmony of “free spirits” an­swers for the furies of fanaticism…. One history is dying.  Another is to be born with the utopian dawn­ing of this new international.  These are the years when printing makes its debut (1450); Leon Battista Al­berti is perfecting his De re aedificatoria (1452); Piero della Francesca is painting his Legend of the Holy Cross in San Francesco D’Arezzo (around 1453).  A new way of seeing is giving rise to a way of con­struct­ing.  Such is the question Nicholas of Cusa poses in Icona:  what does it mean to “see”?  how can a “visi­on” bring a new world into being?[63]

 

The work known as “The Vision of God” in English served as a dedicatory text accompanying the gift to the monks of Tegernsee of one of the newfangled “trompe d’oeil” paintings by which the Cardinal, ever in the vanguard, had been profoundly moved.  The eyes of the painting, he said, like the “Vision” of his lecture’s title, seem to follow you, no matter where you stand in relation to them. 

If you let your ego delude you, you may come to think that God has singled you out and succumb to spiritual inflation; if you transcend this snare, however, you will get a glimpse of the higher truth that all may singly feel His gaze upon them in just the same manner, hence implying the omnivoyance of the Crea­tor.  Five and a half centuries after Alberti and company, this illusion is so well-known as to be a cliché, and is entertainingly discussed, bereft of all meta­physics, in a page of Steinhaus’ delightful classic, Mathe­ma­tical Snapshots; but in Cusa’s day, its effect was a novelty – and a mystery – of a very high degree.

               

To make a plane picture of the three-dimensional object, we appeal to geometrical per­spec­tive.  A camera furnishes it automatically but the ancient masters employed the same means to obtain the impression of perspective depth.  Horizontal parallels always meet on the “horizon-line” of the picture; if they are per­pendicular to the background, their apex is the “principal  [or “vanishing”] point”… Only by placing the eye on the perpendicular to the picture issuing from the principal point does one get, without deformation, the vi­su­al impression corresponding to the three-dimensional reality.

 

The optical illusion of a portrait following with its eyes a spectator walking along is easy to explain.  In the case of a living immobile model the view changes as we walk along:  first one ear disappears behind the head, then one eye begins to hide behind the nose, and so on.  Only if the model turns its head to watch us frontally, do we continue to see both ears, both eyes, and so on.  Now, with a picture we always see both eyes and both ears, whatever our point of view; thus the portrait makes the impression of a person’s turning his head to look at us.[64]

 

                The projective invariance of all “points of view” on one scene is what makes aerial recon­nais­sance photography possible:  by isolating a suitable “benchmark” (the sequence of equal­ly spaced slats making a railroad track, say), pictures taken from arbitrary angles by spy satellites can be computa­tion­ally integrated to give a “virtual viewpoint” from any angle.  Just as sailors in the ancient world could determine a ship’s position by triangulation (a key inspiration for trigonometry), ana­lysis of re­con­naissance photos makes use of a relationship between four points, most readily envisioned as the “point at infin­ity,” the 0 and 1 of one’s “mea­suring stick,” and the position of some object of interest.  The rela­tion­ship can be found treated geo­met­rically in Hellenistic times by Pappus; but its full significance was not suspected or fully ex­ploited until J. V. Poncelet, a French military engineer captured during Napo­leon’s Moscow retreat, made use of the “lux­ury” of internment in a Russian prison to carry out his geomet­ri­cal investiga­tions.  (His Treatise on the Projective Properties of Figures appeared in 1822 – the same year as Fourier’s Analytic Theory of Heat.)

 

 

 

It might seem that there are only a few, very primitive properties that are preserved under the arbitrary projective transformations, but this is by no means so.  For example, we do not notice immediately that the theorem stating that the points of intersection of opposite sides (produced) of a hexagon inscribed on a circle lie on a straight line also holds for an ellipse, parabola, and hyperbola.  The theorem only speaks of projective properties, and these curves can be obtained from the circle by projection.  It is even less obvious that the theorem to the effect that the diagonals of a circumscribed hexagon meet in a point is a peculiar analogue of the theorem just mentioned; the deep connection between them is revealed only in projective geometry.  Also it is not obvious that under a projection, irres­pective of the distortion of distances, for any four points A, B, C, D lying on a straight line the cross ratio AC/CB: AD/DB remains unaltered

 

AC : AD = A’C’ : A’D’

CB   DB    C’B’    D’B’

 

This implies that many relations are maintained in perspective.  For example, by using this fact it is easy to determine the distance of the telegraph poles A, B, C from the point D on a photograph of the road leading into the distance, when their spacing is known…. It stands to reason that its laws are used in architecture, in the construction of panoramas, in decorating, etc.[65]

 

                The successive splashdowns of this “skimming stone” are implicit in the text just cited:  Desargues was one of the few people whose mental equipment the insufferably arrogant Descartes actually respected and admired;[66] the “mystic hexagram” theorem was the creation of their sixteen-year-old mutual protégé, Blaise Pascal.[67]  Poncelet, of course, is “synchronous” with the synecdochic splashdowns of our other two skimming stones.  Next up is the multiplicity of explicit perspectives presented in Cubist paintings and quantum mechanics (which latter sees the region of material description we live in as the result of “projec­tion” from an infinite number of “Hilbert Space” dimensions).[68]  And finally, in the A-D-E Problem, we have the “theory of buildings” due to the unfortunately named Belgian, Jacques Tits.  But the “accidental” con­nec­tion of perspective invariance to Catastrophe Theory is what’s most compelling here.  For of the numer­ous “controls” with which one works the strings of the Double Cusp’s “marionette,” there is one not like the others, which – in the typically “trifling” style mathematicians are so apt to fall prey to – has all its mystery rubbed out by being dubbed a mere “parameter.” 

 

But we sense the possibilities for re-enchantment when we realize that this peculiar control in fact provides a measure of the “cross-ratio,” and that each of its infinity of possible values is associa­ted with a distinct manifold:  like Cusa’s painting, myriad differentiably distinct viewpoints (one “Tegern­see monk” per “differentiable manifold”) are underwritten in a unitary manner from the topological “God’s-eye view” perspective.   The history of this revelation is subtle and arcane, going back to exotic discoveries in the 1950s showing that, in certain higher-dimensional contexts, the protean stretchability of the topologist’s “rubber sheet geometry” no longer implied a similar kink-free “smoothness” in the cascade of rates (and rates of rates) of change.  (Technically speaking, “bouquets of spheres” were discovered for which homeo­-  no longer implied diffeo- morphism.)  But there’s a simple way to envision this profoundly inscru­­table result in the context of Catastrophic unfoldings.  For the Double Cusp, generated by two fourth-order “germs” working in tandem, could thereby be seen as the result of perturbing the relative crossings of four lines dropped like Pick-Up Sticks into “general position” on a tabletop. 

 

Why?  Projective geometry’s key feature, first discerned by Poncelet and since called “dual­ity,”  is that dimensional reduction – like a photo’s 2-D image of the 3-D it’s a “projection” from – cuts both ways:  any theory about lines, say, intersecting in points in a 2-D projec­tion from a 3-D space, will have an exactly analogous result concerning points determining lines.  (Return to the prior long quote, and contem­plate the sentence beginning “It is even less obvious…”:  the theorem limned in the prior sentence is Pas­cal’s “mys­tic hexagram”; that sketched in the sentence just referenced was found by Brian­chon a century later; but Pon­celet showed that, from the “projective” vantage, they are exactly equivalent expressions!  Put another way, “vanishing points” in a picture plane imply “horizon lines at infinity.”)  One can, therefore, consider the “cross-ratio invariance” not just of four points between which linear propor­tions can be mea­sured and related, but between four line segments which intersect each other in an arbitrary manner. 

 

 

                            

 

The "controls" of the Double Cusp attach to the monomials which aren't in shadow:  all combinations of X and Y with no exponent greater than 2.  The terms of the "germ" of the unfolding, the monomials X4 and Y4, are underlined, and it is their first derivatives which "cast the shadows."  In the usual "Mendel's Ratios" usage of Pascal's Triangle, coefficients of the monomials (which always sum to a power of 2 on any given line) matter most; here, the algebraic content does, since each monomial attaches to a variable ("control dimension"):  imagine a zero outside the drawing, with "marionette strings" leading to each control's monomial; then imagine these strings are the axes of a multidimensional "control space" (to which the "behavior dimensions" X and Y also attach).  We'll learn how to navigate this space in the next installment.  From Poston & Stewart, Catastrophe Theory and Its Applications, p. 163. 

 

 

As high-school algebra plus a pencil and a piece of graph paper can show us, each of these four lines can be writ­ten in the form a_iX + b_iY, 1 < i < 4, where the a’s and b’s are constants for a fixed arrange­ment, or variable “controls” if the lines are mobile; in the latter case, their general product would then have the form of X⁴ + Y⁴, plus lower-order terms in X and/or Y on Pascal’s Triangle.[69]  Catastrophe Theory tells us this mess of mono­mials can be reduced, by  a hand-waving “change of variables” technique known as “Siersma’s Trick,”[70] to the 3 by 3 rhombus on the Triangle which has a “1” at the top and “X²Y²” (the “cross-ratio” term) at the bottom, and X² and Y² at the extreme posi­tions on the outer slopes.  (These latter two, you should recall, are just the “splitting factors” of two Cusp catastrophes, one in X, the other in Y, whose first powers are “controlled” by their respective “loading factors”:  hence, the “Double Cusp” no­men­cla­ture.)  Forgetting the “1” (which, from the topolo­gist’s vantage, is merely a displacement factor), we thereby get seven stan­dard “controls” plus the point-of-view-fixing “parameter.” 

 

 

 

 

From Sir David Brewster's The Kaleidoscope, Ch. XIII:  On the Construction and Use of Polycentral Kaleidoscopes.  Basic "wallpaper pattern" for each of the 3-mirror prismatics, each with "Timaeus triangle" cross-section.  The numbers in the triangles indicate how many reflections are involved in painting its image (an index of its brightness); each such scope has one un-numbered triangle, which is the aperture where the viewer looks in at the honeycomb of images.

 

 

 

                What does this parameter mean in terms of our general theme?  Something very concrete:  by another “trick” fundamental to Catastrophists, the so-called “Splitting Lemma,” an indefinite number of behavior variables can co-exist and be dispensed with, provided none have terms of higher than second order:  hence, to the X and Y “behaviors,” we can always assume there’s a third whose “germ” is Z².  What does this get you, you ask?  Well, since the days of Fourier at least, it has become “natural” for mathemati­cians to write equations in the complex plane, with “real” expressions as mere “projec­tions” on the walls of Plato’s Cave.  And this quickly gives us Plato’s “Timaeus Triangles” – and 3-mirror kaleidoscopes!

 

                Pictures of these scopes – related to the 3 infinite families of “non-elementary” catastrophes begun by the three simplest forms with one parameter – can be found in Brewster’s book.  The “triangle,” in each case, is the cross-section of the prism formed by the three mirrors.  If the exponents of the “germ terms” are related to aliquot parts of semicircular[71] orbits in complex space, the Z² reads as 90^o (a right angle, in other words), and the X⁴ and Y⁴ of the double cusp then span 45^o each, yielding the right isosceles triangle.  (I have a scope based on the equilateral triangle – X, Y, and Z terms are all cubes – and ones based on the 30-60-90 triangle also abound.)  What all three of these “Timaeus” kaleidoscopes have in common is not one, but an infinite number, of centers of symmetry – exactly corresponding to the “Vision of God”-like “para­meter effect.”

 

                What does this parameter mean in modeling real things?  Well, it depends upon what you mean by “real.”  As we will explore in some detail in the next installment of this eight-part argument, psychological phenomena like trance induction can be modeled using the “full” Double Cusp; when one does so, the “pa­rameter” measures “susceptibility,” in two senses:  first, to paraphrase the Gospel admonition, “many are called, but few are chosen”; secondly, only a “narrow strait” of values, symmetrically arrayed around the zero of the parameter, allow for merger of the distinct individual in the One which transcends the ego.

 

                The thematic foci of both this current essay and its just-alluded-to sequel have been wonderfully indicated in a singular work of Carl Jung’s – the only in his opus which he ever went back to and totally overhauled.  (Given that it marked his break with Freud, this is not surprising; and, as we shall see next time, and have already foreshadowed in the prior installment, this was also the work where a bevy of kalei­doscope images portended his upcoming shift to mandala symbolism.)  On the subjects of crowd psycho­logy (wherein one is ever at risk of “losing oneself” in a negative way), and the transcendence of the per­sonal in the ineffable (where one “loses oneself” in quite another sense), Symbols of Transformation has this to say:

 

I have often noticed the symbol of the crowd, and particularly of a streaming mass of peo­­­ple in motion, expresses violent motions of the unconscious.  Such symbols always indicate an activation of the unconscious and an incipient dissociation between it and the ego.[72]

 

Individual consciousness is only the flower and the fruit of a season, sprung from the per­ennial rhizome beneath the earth; and it would find itself in better accord with the truth if it took the existence of the rhizome into its calculations.  For the root matter is the mother of all things.[73]

 

                In the pages preceding, we have come to see the possibility of “skimming stone sequences” being treated as having, to paraphrase Lord Acton, “a radiation and development, an ancestry and posterity of their own, in which men play the part of god­fathers and godmothers more than that of legitimate parents.”  A mode of history which would contemplate their synchronous (and eventually, syncopated) deployments across and within develop­mental contexts (like the “Age of Synecdoche” we’ve been focused on herein) has been, perhaps, adum­brated, but clearly calls for further unfolding, framing, and refining.  As a final gesture in these present considerations, it would do us well to reflect on the construction which has held our attention (and, indeed, fed it its contents) over the last few pages.   Rather than leave the four-termed frame­work that has guided our thoughts in a limbo of incoherence – is it meant to embody Lévi-Strauss’ Klein Group only?  Or his Klein Group only as an instance of his “canonical law of myths”?  Or is the instantiation of Derrida’s “passe partout” to be given pride of place, with the Klein Group serving as a constraint to stabilize its protean tendencies? – let’s give it an identity of its own. 

 

                At the time Boscovich’s return trip to France was interrupted by what proved to be his final ill­ness, Haydn’s six “Paris symphonies” scored a major triumph in the French capital.  In 1772 – the year before the Jesuits were disbanded, and Boscovich relocated to Paris – Haydn’s Opus 20 “Sun Quartets” did for the classical form they made paramount what Elvis’ “Sun Tapes” did for rock ‘n’ roll.  The year be­fore the Bastille fell, Haydn’s Opus 54 and 55 string quartets were published (the second of the latter set, called the “Razor,” inspiring the architecture of this discourse, as I’ll explain in my final installment).

 

                In his National Book Award-winning study of The Classical Style:  Haydn, Mozart, Beethoven, pian­ist Charles Rosen tells us that the string quartet’s pre-eminence in chamber music, from Haydn’s first triumphs to the death of Schubert,

 

is not the accidental result of a handful of masterpieces:  it is directly related to the nature of tonality, particularly to its development throughout the eighteenth century.  A hundred years earlier, music had not yet shaken off the last traces of its dependence on the inter­val:  in spite of the central importance of the chord – the triad in particular – dissonance was still conceived in intervallic terms, and the resolution of dissonance, consequently, even in late seventeenth-century music, very often satisfies the aesthetic of two-part coun­terpoint and ignores the tonal implications.  By the eighteenth century, dissonance is always dissonance to a triad, stated or implied...[74]

 

                At the end of the last installment, we heard Lévi-Strauss argue that “the point at which music and mythology began to appear as reversed images of each other coincided with the invention of the fugue.”  But this process of myth’s “reincarnation in music” attained its apotheosis in the string quartet:  for it was there that the shape of the “canonical law of myths” itself attains to splendid self-reflection:

 

The string quartet – four-voice polyphony in its clearest non-vocal state – is the natural con­sequence of a musical language in which expression is entirely based on dissonance to a triad.  When there are fewer than four voices, one of the non-dissonant voices simply must play two notes of the triad, either by a double stop or by moving quickly from one note to the other…  (The resolution of certain dissonances will, of course, itself create a triad and therefore demand no more than three-part writing, but some of the basic disson­ances of late eighteenth-century harmony, like the dominant seventh, require four voices.)  More than four voices gave rise to questions of doubling and spacing, and the woodwind quartet created problems of the blending of tone-color (and, in the eighteenth century, of intonation as well).  Therefore only the string quartet and the keyboard instrument allowed the composer to speak the language of classical tonality with ease and freedom, and the keyboard had the disadvantage (and the advantage!) of less striking linear clarity than the string quartet.[75]

 

                With the death of Schubert, the pre-eminence of the string quartet ends, and the muddying of the tritone’s chordal logic in the chromaticist excesses of the Romantics is well under way.  And as the string quartet itself fades from center stage, what other symptoms of malaise are manifest in the general sensibili­ty?  I have tried to give at least a partial answer to such questions here.  With the fall of public man, the capacity for playacting atrophied, and the age of public spaces as mute wastelands was initiated.  And per­haps the great­est of the public arts, flourishing with such spontaneity in the salons of the ancient regime, was soon lost  from view.  Taking its cue from the rhythms of lived emotion as actual­ly spoken, then ideal­ized in opera buffo, where its acme was achieved in the joyful vocalic interweaves of Haydn’s inti­mate friend and fre­quent part­ner in string quartet sessions, Wolfgang Mozart, it attained to a life freed from the con­straints of embodi­ment, liberated in the polished surfaces of stringed instruments’ impassioned dialec­tics.  Today, it is totally cari­ca­tured and de­based in such dismal proofs of its morbidity as “talk radio”; but its grandiloquence has been captured for­ever in the “rehearsed impromp­tus” of the Vienna classicists’ sonorous crystals of mythi­cal vivacity and charm:

 

Eighteenth-century prose in England, Germany, and France had become, in comparison with the previous age, much more syntactic, relying more exclusively on balance, propor­tion, shape, and the order of the words than did the heavier cumulative technique of the Renaissance.  The eighteenth century was cultivatedly self-conscious about the art of con­versation:  among its greatest triumphs are the quartets of Haydn.[76]