How, then is one to account for the visible evidence of a world of stable, recognizable features, so many of which remain in that state from day to day? Most people would be hard put to find a question of less importance to them. For a small enclave within the human race, including mathematicians like René Thom (Catastrophe Theory) , physicists like Elliot Lieb ( stability of matter) , and philosophers likeIlya Prigogine (far from equilibrium physics ), this question is an endless fount of fascination.
Take the French nation, for example : French history, French politics, social life, culture: what could be more baffling? What about the Structural Stability and Morphogenesis of France? What about its Structuralism, Instability and ORPHOgenesis? ( Respectively, its arid academic fads, its anarchistic politics, and its magnificent traditions in all the arts?)
One may arbitrarily fix the origin for the chronometry of French political metamorphism at the Hundred Years War. Tourists may visit this country at any time, knowing there will be a centenary somewhere for some battle in this war. After its re-direction by Jeanne d'Arc , the hydrodynamics of this self-generating quagmire converged, after 300 years, to a stable regime, a strange attractor known as Le Roi Soleil. Exceedingly strange indeed, given that its stable configuration was as chaotic as the chaos it stabilized.
Onward to the Revolution of 1789, a catastrophe in many acts, which may respectively be called, perhaps, the cusp revolution, the fold revolution, the butterfly revolution, the purse revolution, the parabolic umbilical revolution, the hyperbolic umbilical revolution . . . . . . the primordial Big Bang of modern history.
As with the standard cosmological model, all of the stable elements were created in the first 3 minutes : Democracy, the Rights of Man, the People, the Masses, the Mob, the Demagogue, the Nation State, the Police State, the coup d'état, genocide, the existential dilemma, le sujet, the metric system...
Also a new life-form: l'homme machine It's mind works by scientific method, its knowledge is patched together from scattered remnants of mental junk food,its appetites are predominantly popular that is to say, generic. Its heart is sentimental, its culture shallow, its politics imperialist.
Enter Napolean, a Corsica castaway - another strange attractor, stabilizing the country through the export of its chaos abroad. The Congress of Vienna, convened soon after his defeat, seeks to establish a global order that will deprive almost everyone of any reason for being born. This in turn engenders a long chain of catastrophes: 1830 - 1848 - 1870 - 1914 - 1939
. . . . . . . . . . . . . . . . .1968 (?)
Hardly. I was there. "Nineteen Sixty-Eight" was but a self-inflated echo of the convulsions of earlier times; which may not be so bad.The sublimation of 2 centuries of revolution violence into the inane posturings of 1968 may indicate a real advance in civilization's strenuous hike. The screaming of Maoist-Leninist slogans at close range replaced the shriek of the guillotine blade. When the cops beat the imprisioned militants of '68, it was an atrocity to be sure, yet not in the sense that the massacre of tens of thousands of communards by the Versaillais troops in 1870 was an atrocity.
This swollen womb of French Chaos has spawned hosts of intellectual progeny : Girondisme , Jacobinisme, Fourierisme ,Proudhonisme, Lamartinisme, Sadisme, Positivisme, Impressionisme, Pointillisme , Symbolisme, Fauvisme, Cubisme, Dadaisme, Surrealisme, Personalisme, Existentialisme, Phenomenologie, Structuralisme, Nouvel Romanisme, De-Constructionisme, Gauchisme , and . . . . . . . . . . . . . . . Catastrophe Theory!
Existentialism will not concern us in this article, save to the extent that since Existentialism is concerned with "concern", we will be concerned with Existentialism whenever we use the word "concern". As for Structuralism, both it and Catastrophe Theory have roots in French history. Both are schemes for the description of just about everything; thus, although they have nothing whatsoever to do with each other, their intimate connections are obvious.
Structuralism may be defined as "the search for Eternally Stable Structures." While asserting that such must assuredly exists, the Structuralists don't bother to actually find any: they're there for all time, so why hurry? The extreme left of this charade is in the celebrated linguistic system of Noam Chomsky, which I have written about in several previous issues of Ferment.
Catastrophe Theory is more subtle. It seeks to uncover, " Eternally Stable Unstable Structures" ! These are called generic catastrophes, and there are supposed to be 7 of them, that is to say, seven ways in which most things are prone to collapse. This famous result, known as the "Classification" or "Unfolding" Theorem, was proven by the mathematician René Thom. Thom was already famous to mathematicians before he did this, through his invention of an important tool in differential topology : The Co-bordism Group .
Thom is a cautious and level-headed thinker.Carl Zeeman, his most diligent disciple, is quite the opposite, a zealot in the service of The Catastrophe : it was his article in the Scientific American that put the phrase, "Catastrophe Theory" into the general vocabulary. He is also the editor/author of a huge book on the subject: in this tome (sic?) he reveals Catastrophe Surfaces in creativity, mystical experience, panic-stricken dogs, linguistics, psychology, anthropology, sociology, evolution... Strangely enough, it is often in precisely these areas that the Structuralists claim to have located, yet never uncovered, their cherished "Eternally Stable Structures".
Only history will tell whether eternally stable , or eternally stable unstable structures, have a better chance of survival, or even of existence!
So that a disciple of Claude Levi-Strauss may publish an article in a structuralist journal, in which he will claim to have uncovered that "eternal structure in the mind", which makes the incest taboo universal in the human race. A disciple of René Thom will then publish an article in a catastrophist journal in which he describes the 7 different ways in which the incest taboo is likely to be violated. An enervated Chomskyite will assert the existence of a new permanently undetectable deep structure in all human languages, on the very day that a catastrophist lists those situations in which visceral grunts and cries are more effective than spoken language.
Not unlike the way in which , on a certain day, Danton would proclaim the existence of absolute, inalienable human rights, while Robespierre,tommorow, would urge the necessity to violate all of them, specifically those of Danton, on the grounds of national emergency.
Anticipated momently, Dr. Thom was expected to appear somewhere between the small square with fountains where children tossed a soccerball, the ubiquitous urban pigeons, and the future of Socialism in France. I did not have to wait long. He arrived from a great distance as if clearing a path through the debris of a crumbling and decadent society, transporting many messages for mankind and bearing , upon his back and shoulders, the destiny of science.
He recognized me immediately: I was the only person standing in the square with the mathematician squint in his eye. He approached me, we shook hands:
" A static representation of time", he said, " is required for its study."
" Spatial isolation is irrelevant.", I replied, "It is causal isolation that matters."
" Prenez-vous un verre avec moi?" We crossed over the square to the Café Descartes , then past a few tables towards the interior. As we crossed the threshhold, a Butterfly-type Catastrophe arose spontaneously, through abrupt collision with a mathematics graduate student!
What else could he be: fuzzy- bearded, draped in a blue-grey suit from the Marche Aux Puces at Métro Clignancourt , manifold books under his arm, stuffed into his trousers and between his legs , scribbled pages flying from the top of his vest?
With a desperate eagerness that I have known all too well, ( having been myself, and still being, a student ) he cried out, " Dr. Thom! I have an exciting new proof of an old theorem!"
" Later , young man! This is Dr. Lisker from California!"
A brief, surly nod: what has California to do with the implacable advance of Truth? "It'll just be a minute, Professor Thom ; nothing more! "
" Later, young man! Later! There is no now; there is only when!" We pushed our way past him to the back of the café .
Given his celebrity, I was very surprised and flattered when Thom opened our conversation by discussing specific ideas from a long article I'd sent him, which I would be delivering at the 11th General Relativity and Gravitation Conference, in Stockholm, Sweden in a few months : "Causal Algebras" (See Papers in Physics)
"'The existence of a system in isolation must always be associated with a rupture in causality'", he said, quoting from my article: "That is an important idea."
This reception was contrary to my experience. American mathematicians, famous or otherwise, almost never read anything I send them. The reasons for this are quite simple. In contrast to the French, few Americans value literacy. It doesn't help you get ahead.
After we had spoken for awhile, he invited me to attend a lecture he was about to deliver at the Ecole Polytechnique to an organization called CESTA, an odd collectivity of intellectuals. CESTA, a kind of philosophical society, had been established by the Mitterand government for the disposal of weighty issues through the raising of ponderous questions.
For the next half hour or so, as we waited to cross the street once again , the two of us passed the time in congenial banter, amiable mathogenic discussion, extenuated technical irrelevancies.
Entering the lobby before the main auditorium of the Polytechnique I found myself in the midsts of an extraordinary conclave of assorted minds. The eyes of certain persons made it appear as if a broken sliver of insight had,at one time or another, embedded itself into their brains, then frozen solid. Others seemed to chase motes of sunlight on crutches. I observed academic bulemics, gorging themselves at mammoth culture banquets to be vomited out, and academic aneroxics, persons who, through having with such intensity doubted the certainty of all knowledge for so long, had finally ended up knowing nothing at all. I witnessed a soul that, shaped as a giant question mark, waltzed gaily across the lobby. Others stood about in groups, mumbling paradigms, while some stood alone, guarding immense revelations in stingy secrecy.
Contemporary science lacks adequate methods for causal analysis. However, the Principle of Sufficient Reason is not ad hoc, but corresponds to a deep requirement of the human spirit ( Esprit Humaine, which is much larger than our term 'spirit', and comprises both intellect and emotion ). The fact that "Causality" cannot be determined experimentally does not mean that it is possible to dismiss it.
The way to deal with modern problems arising from the Quantum Theory and similar approaches, is to develop a theory of "local determinism" which may not alway have global implications. For example, the fundamental weakness of the theory of "hidden variables" in Quantum Mechanics is that they are globally, not locally defined. We may begin, perhaps, by replacing the Laplacian paradigm of a system moving irrevocably from minus infinity to plus infinity by a modification suggested by Dr.Lisker , who is with us here today ,( Thom then referred to my paper) : " Initial Conditions are defined as a rupture in causality, a singularity." The simple act of stating initial conditions already creates a state of isolation. We can then posit two stages:
This leads to methods of mathematical modelling based on 'initial condition spaces'. After all, the very possibility of locating a point in space-time implies the existence of human intervention, which is equivalent to a causal singularity.
A total "arbitrariness of choice" is therefore built into the very system of representation. One thinks, for example, of the indeterminates employed in Algebraic Geometry.
The position I am developing is quite opposed to a contemporary trend that I see in modern physics to which one can give the name of 'chaotic indeterminism', the very expression being a kind of 'oxymoron' that I find completely unacceptable. "
( As a final observation, I can state with confidence that there were at most 3 persons (myself among them) in the audience who understood his talk. 'Deep intellectuals' of the CESTA sort rarely feel comfortable with the 'exact sciences'.)
Down the hillside slopes are dormitories for students at the extension of the science faculties of the Sorbonne at Orsay, a few miles away. Across the railroad tracks , and another mile down the road, one finds the "Institut des Hautes Etudes Scientifiques" , the French Institute of Advanced Study, filled with hunched beings who devote their days entirely to mathematical research. In 1986, when I first visited it, René Thom was its director.
Apart from the Institute, there is nothing else in Burres-sur-Yvette to match the excitement generated by the shapes of the clouds coasting over the lanes, roads and highways and piling up in the woods ; fabulous bestiaries these, like mountain ranges of dream creatures, or spice-laden vessels of the tropospheric realms. We allowed the spell of these clouds to carry us in our wanderings over the hills, down unfamiliar roads and through unretracable paths until, at last, we came to rest, bewitched, before the restaurant" Le Boeuf a Six Pattes ".
A glance at the menu will be enough to convince anyone that its is the proud possessor of that combination of charm, cuisine, and costliness found only in France, and there rarely in the big cities. The few remaining examples are the last outposts of civilization in an encroaching wilderness: elsewhere, McDonalds' tramp across the land.
Altogether we are 5. In a corner sits Yannick Kergosian; the name is Breton, not, as I believed, Armenian. He is a historian, and talks very little through the meal. Beside him sits Arnold Marvel and his wife, Noel. She seems never to want to stop talking about her extraordinary husband. He first, she doted, claimed the world's attention as a musical prodigy. Then he gaind renown in medical research. Now he is a mathematical biologist.
As the bottle of white Sancerre goes the rounds, and appetizers selected from one of those extraordinary menus holding names of villages from everywhere, the group of us argue for and against the premise that the subject of "Mathematical Biology" did not properly exist until a few years before, and that its modern form, was created by René Thom himself. I finally realize that what they are talking about is the "qualitative theory" of mathematical biology; the "quantitative theory" goes back to Rashevesky's books on mathematical biophysics in the 50's.
I am seated to Thom's right. I ask him for the best reference for the proof of his "Classification Theorem". He refers me to a standard work by the Russian mathematician V.S. Arnauld.
Between the appetizers and the entrée , a sherbet . Arrival of the Entrecote Bordelaise . As we eat, Thom, Marvel and I gleefully denigrate "Renormalization", the panacea of particle physics. Mathematicians get a great kick out of denigrating the way mathematics is used by physicists, as if physics were just another branch of mathematics. It's an attitude totally lacking in humility, quite ignorant in fact, but lots of fun. Its like classically trained musicians running down jazz musicians because they make things up; and vice versa.
Arrival of the Mystere Chocolat. A conversation develops between Marvel and Thom over the role of light caustics ( bundles of light rays reflecting off surfaces ), in the history of the development of Catastrophe Theory. Thom talks about his retirement from the I.H.E.S., slated for 1989
Figs; nuts; cheeses; coffee.
Yannick Kergosian talks to me about his specialty: the French region of the Vendée. The ravages of the French Revolution, which he depicts as outright genocide, have left their marks on the region up to the present , much like the unassimilated bitterness of the American South owing to the Civil War.
A feast for body and mind ! Had Omar Khayyam, himself a mathematician , been present, he might well have written:
" A text on Singularities of Differentiable Mappings
A hunk of Entrecote Bordelaise
Several bottles of Sancerre
And a historian, a famous mathematician,
a mathematical biologist who used to be a music prodigy,
and his adoring wife
Beside me in the Boeuf a Six Pattes
Ah! 'Boeuf a Six Pattes' were Paradise enow! "