Projects of the Creative Imagination
It is noteworthy that as between two proofs of a theorem mathematicians will prefer the one which, as they say, is more “elegant,” a term which has primarily an aesthetic rather than a logical significance. It is a striking fact that creative mathematicians think of their subject as an art as well as a science. Perhaps the best analogy is with architecture, which in its highest forms combines use and beauty. Both art and science on the highest level are projects of the creative imagination, and the likenesses between them becomes more significant.than the differences.—Frank Aydelotte (Director,1939–47), October 1941
Mathematical Infinity
The concept of infinity has been treated in many different ways during the ages, according to Professor Enrico Bombieri, IBM von Neumann Professor in the School of Mathematics, from rejecting it as an irrational absurdity to accepting it as the ultimate, inaccessible, perfection.
"Is the essence of infinity the uncountable, the immeasurable? Or is infinity the ultimate, complete, perfect entity?" asked Bombieri, during his January 31 lecture "The Mathematical Infinity." "Well, this I would leave to the philosophers and theoligians, but my concern would be infinity in mathematics. Is infinity part of mathematics? Is infinity a number, or can it be treated as such? The answer is, well, it depends on how you do things.” Infinity has played, and still plays, a shifting role in mathematics, Bombieri said: from total avoidance in the Pythagorean school, to the Aristotelian partial acceptance of it as a useful convention (but always avoidable), to the Cantorian point of view related to counting and leading to a notion of different types of infinity, all of them acceptable at the same time in a precise model of mathematics.
Today, mathematics has arrived at a pragmatic view of infinity, according to Bombieri, and its acceptance has lead to some counterintuitive paradoxes, as well as some positive results. “Infinity has come back into mathematics in a far more powerful way, to coexist with the finitistic approach of Aristotle,” said Bombieri. “There is no need to coerce every proof into a finite argument. In the view of many mathematicians, the intellectual contortions needed to remain within the realm of the finite indicate that a wholesale rejection of infinity in mathematics is not a good thing. What really matters is the final understanding, coupled with a good foundation. Goodness is guided by ‘Ockham’s Razor’: always choose the simple way. Aesthetics enters into mathematics.”
Computer science is also leading to a new precise concept, Bombieri said, namely the impossibly large in the realm of the finite. “Is it possible that the solution of the basic P versus NP problem, with its finitistic formulation, will require a daring excursion into the realm of infinity? Only time will tell. We have also seen the existence of the incomputable: Functions that no one can write down in standard recursive terms. Certainly, the computer has shown us, in a dramatic way, the distinction between the ‘finite’ in real life and the ‘finite’ beyond our grasp.”
Enrico Bombieri has been a Professor in the School of Mathematics since 1977 and IBM von Neumann Professor since 1984. One of the world’s leading authorities on number theory and analysis, he is a Fields Medalist for his work on the large sieve and its application to the distribution of prime numbers.
The Difficult Task of Erasing Oneself
Some modern artists, in their pursuit of perfection, have tried to erase all traces of him- or herself, according to Yve-Alain Bois, Professor of Art History in the School of Historical Studies, a precept that has helped sustain many different artistic practices during the past century, from Kasimir Malevich’s Black Square of 1915, Jean Arp’s collages “according to the laws of chance” of 1916–18, and Piet Mondrian’s modular grids of 1918–19 to Pop Art, Minimalism, Process art, Conceptual art, and beyond.
In his March 7 lecture “The Difficult Task of Erasing Oneself: Non- Composition in Twentieth-CenturyArt,” Bois pointed to “The Search for the Absolute,” Sartre’s famous text on art written as a preface to an Alberto Giacometti exhibition in 1948. In his text, Sartre noted that the sculptor knows that there is “nothing superfluous in man.” “But what if it is man, man himself, who was deemed superfluous?” asked Bois. “In fact, it is my contention that a good portion of the modernist production, throughout the last century, tried to ask this very question, tried to conceive of itself as a search for the possibility of effacing man and erasing his traces.”
Malevich, according to Bois, was “the first abstract artist to claim that he was searching for the absolute, which he called the ‘zero of form.’ ” In his lecture, Bois mapped the patterns of development in this drive towards impersonality in twentieth-century art. “All these works do have something in common,” Bois stated. “They all manifest an urge to suppress composition as sheer arbitrariness and to motivate the practice of art by way of some ordering principle that would not depend upon subjective choice.”
In the past century, modern artists have engaged in various non-compositional strategies, Bois stated, which can be divided into six categories: chance; the grid; the collapse of image and field; the deductive structure; the monochrome; and the process. Yet, according to Bois, noncomposition ultimately proves unsustainable. “One after the other, the evolution of each non-compositional artist seems to reach a point after which, progressively or abruptly, the wheel turns back on itself,” Bois said, pointing to the oeuvres of Ellsworth Kelly, Sol LeWitt, Piet Mondrian, Alexandr Rodchenko, and Frank Stella.
Even Duchamp, Bois said, accepted defeat. In a 1960 interview, Duchamp stated, “I consider taste—bad or good—the greatest enemy of art. In the case of the ready-mades, I tried to remain aloof from personal taste and to be fully conscious of the problem ... many people can prove I’m wrong by merely pointing out that I choose one object rather than another and thus impose something of my own personal taste. Again, I say man is not perfect, but at least I have tried to remain as aloof as possible, and don’t think or one minute that this hasn’t been a difficult task.”
Yve-Alain Bois, a specialist in twentieth-century European and American art, has been a Professor in the School of Historical Studies since 2005. He is currently working on several longterm projects, including a study of Barnett Newman’s paintings; the catalogue raisonné of Ellsworth Kelly’s paintings and sculptures; and the modern history of axonometric projection.