Thursday, September 11, 2014

The Artist and the Mathematician: A Revolution of Human Thought

by Drew Martin
A couple years ago I bought a book titled The Artist and the Mathematician: The Story of Nicolas Bourbaki, the Genius Mathematician Who Never Existed. I was hoping it would be something like The Professor and the Madman: A Tale of Murder, Insanity, and the Making of the Oxford English Dictionary but it did not have the same kind of flow or kick-start so I put it down and spurned it every time I saw it on the book shelf.

For some reason I picked it up a couple weeks ago and discovered what a brilliant book it is. The Artist and the Mathematician is about a French mathematics group that..."embarked on the ambitious project of setting the mathematics curriculum for calculus and mathematical analysis offered in all universities in France." They were the best mathematicians in their country whose collective genius changed the course of mathematics around the world.

They published as "Nicolas Bourbaki," a character they not only invented but also built a life around including a baptism, baptismal certificate, godparents, and even invitations to his "daughter's" wedding. There was one member of the collective who most captured the essence of Bourbaki, Alexandre Grothendieck - a brilliant man who survived a horrific childhood during the Holocaust, changed modern mathematics, and then abandoned his academic post, friends, and family to return to the solitude that he credited for the success of his deep understanding of mathematics.

The Bourbaki group enjoyed a special place in society:

é life in Paris would, therefore be dominated not only by the philosophers and the writers and artists, but also by the mathematicians. And perhaps for the first time in modern history, mathematics would play a key role in the general culture - in a way that it did only in the very distant past of ancient Greece. 

Since Sartre and his allies were decidedly non-mathematical in their approach to life, they would inevitably be left behind. Their philosophical theory of existentialism would end its reign as the strictly axiomatic, rigorous, and system-oriented theory called structuralism swept France and the rest of the Western world. The mathematicians would play a key role in the new milieu not only as proponents of a new and widely-used approach to life, but also as 'connectors' among practitioners in different fields: the exact sciences, the social sciences, art, literature, psychology, economics, and philosophy. This would launch a new age for mathematics, one in which the role of the discipline in our culture could not be matched by any other... fact, the ideas developed by these mathematicians sitting in Parisian caf
és would prove to be of crucial importance for society as a whole. The ideas of these mathematicians would constitute nothing less than a revolution in human thought - one whose effect would be felt far and wide.

The art world in this period was driven by Pablo Picasso, Georges Braque, and Marcel Duchamp who absorbed this mathematical and scientific awakening into their work.

Einstein's dramatic discovery in 1905 dealt a fatal blow to our view of the universe. His theory (of relativity) added a fourth dimension to our world: the dimension of time. It was now time for our new view of nature to enter into art. The mathematician Maurice Princet explained Einstein's theory to artists, and they began to mimic these discoveries in physics in their own work.

While Princet did not create cubism, his explanations did influence the new art form. Picasso, Braque, Duchamp, Metzinger, Gris, and others created the new art form by destroying the old perspective and realism and creating the new way of painting. But the fourth dimension itself was now to enter as well. Artists tried both to paint in a way that seemed to reflect a new dimension and to create art in which time itself was the variable. Time was the added dimension. Marcel Duchamp, in particular, explored paintings in which subjects were depicted at different point in time. He even experimented with paintings in which time was accelerating or slowing down. Duchamp's Portrait de joueurs d'echecs, 1911...serves as a good illustration of these ideas...

...He painted a series of studies and canvases of checkers and chess players...The cubist painting (Portrait de joueurs d'echecs) shows his two brothers, Villon on the right and Duchamp-Villon on the left, engrossed in the game. The two figures are fused in the center, creating an illusion from a single point of view, thus doing away with the old spatial perspective. The figures reappear, deformed and at changing angles, higher on the canvass, reflecting the effects as a fourth dimension. The two players, however, are unaware of the fourth dimension and are locked in a static subspace of the dynamic universe around them.

Structuralism is defined by the author, Amir Aczel, as...

...a method of intellectual inquiry that provides a framework for organizing and understanding areas of human study concerned with the production and perception of meaning.

Structuralism is interdisciplinary, and multidisciplinary.

Although this system originated with mathematicians as well as linguists, the French anthropologist Lévi-Strauss is credited as being the father of structuralism because he released it from its breeding grounds to other fields where it flourished.

The fundamental assumption of structuralism is that all of human behavior arises from an innate structuring capability. This structuring ability latent in the human brain, gives rise to language. But the same structures hidden inside the brain also lead to myths, creativity, and various social patterns...

Structuralism deals with the relationships between parts and the whole. Totality takes logical priority over individual parts, and the relationships are more important than the entities they connect. The hidden structure is thus much more important than whit is obvious or apparent in any given situation. It is the symbolism that matters, rather than the entities symbolized. Because of the philosophical components of structure, we find the ideas of structure in areas far beyond the sciences...

Structuralism was a deep method that stripped away all the unnecessary elements of a system.

The linguistic origins of structuralism started with Roman Jakobson, who helped found the Moscow Circle of linguistics in 1915, the Saint Petersburg Circle of linguistics in 1917, and then the Circle of Prague.

In the Theses of 1929 the Prague linguists wrote:

In its social role, one must distinguish language following the existing relationship between it and the extra-linguistic reality. Language has a function in communication, that is, it is aimed at the signified: or it has a poetic function, that is, it is aimed at the symbolism itself.

A literary society founded in 1960 was also modeled after Bourbaki. They were called Oulipo (Ouvroir de littérature potentielle - Workshop of potential literature). 
One of the most prominent members was the Italian writer Italo Calvino. 

Oulipo translated geometrical ideas into linguistic ones:

Point = Word, Line = Sentence, Place = Paragraph

Oulipo attempted to deconstruct literature and rebuild it in a new way, thus bringing the new structuralism from mathematics, anthropology, and art into the foundation of a new literature.

They created systems for producing poetry such as the M+/-n Method, and from that the S+7 Method, which brings to mind the game of juxtapositions by the surrealists that we refer to as Exquisite Corpse. Or perhaps it is even more akin to the word substitution game marketed as Mad Libs.

S+7 Method:

  • choose a text
  • select a dictionary (bilingual was typically used)
  • replace every noun with the seventh noun appearing after it in the dictionary
So a passage such as...

I saw a rabbit in the garden,
eating my cabbage, tomatoes, and carrots.
I tried to throw a stone at it,

but it bolted when it noticed my raised arm.


I saw a radio in the garrison,
eating my cacophony, tongue, and cartwheels.
I tried to throw a stove at it,

but it bolted when it noticed my raised aroma.

Having done this on the spot with the student's dictionary next to me, the process is a kind of random/false creativity.

The way the brain processes information, according to Lé
vi-Strauss, is by using symbolism. The symbolism is what structural analysis  is designed to uncover. Structure is thus a code, consisting of concise symbols. The symbolism inherent in brain function follows mathematical rules that are tantamount to the ideas developed by Bourbaki: the notions of closeness, transformation, groupings, and other of the "mother structures" studied by the group.