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Simplicity, in and in

Allyn Jackson

Simplicity is as hard to pinpoint in mathematics as possibility of ever establishing for all time an it is in art. Certainly both subjects have their great absolute notion of simplicity. What we think of as exemplars of the . But is there a definition simple emerges from conventions that are deeply of simplicity? A criterion? A measure? Or a sure embedded in how we live and how we see the path to it? world, and they have a long genetic history. “Our These kinds of questions were in the air at thinking occurs within those conventions,” he said. a conference called Simplicity: Ideals of Practice “There is not really a way out of them.” in Mathematics and the , which took place As conventions evolve, so do notions of sim- at the Graduate Center of the City University of plicity. Franks mentioned Gauss’s 1831 paper that New York in early April 2013. Instead of trying established the respectability of complex numbers. to definitively answer such questions—surely a The problem Gauss was working on—concerning doomed prospect anyway—the participants gave quadratic and biquadratic residues—had only un- in to the sheer joy of discussion in the stimulating satisfyingly complicated and piecemeal solutions atmosphere of each other’s company. The confer- over Z. Over C, a far simpler solution emerged. ence featured lectures and panel discussions by an The complex numbers revealed simplicity where eclectic group of twenty-five , architects, art previously there had seemed to be none. historians, , and mathematically Mathematics is not engaged in a straightforward inclined philosophers, as well as a film program. march toward absolute simplicity. Rather, by The audience included academics from nearby discovering simplicity anew, Franks said, “We will institutions and local artists; as the conference be more awake to the changing landscape of offered easy and free online registration, a random mathematical thought.” He noted a parallel in smattering of folks wandered in out of curiosity. art, where something new—like the work of Andy Warhol or —acts as a sort of Not an Absolute Notion “shock treatment” that compels a new . Simplicity often seems to be a timeless, absolute Several conference speakers mentioned the quality, and for good reason. Peter Sarnak, Institute art of Fred Sandback, who used taut lengths of for Advanced Study and Princeton University, yarn to represent outlines of three-dimensional offered ’s proof of the infinitude of primes shapes. In photos, the works look unimpressive; as simplicity par excellence. The stark of as philosopher Juliet Floyd, Boston University, this ancient proof is as striking today as it must noted, they are “unphotographable”. But walking have been to people encountering it through the around and through the constructions, she found millennia. Of course, the proof is an exemplar of them to be “extremely moving objects”. Finnish simplicity, not a definition. Indeed, Curtis Franks, architect Juhani Pallasmaa described how a Sand- University of Notre Dame, argued against the back construction, merely “a few lines stretched in space”, sets off a chain reaction in the viewer’s Allyn Jackson is senior writer and deputy editor of the mind, causing the viewer to see figures of specific Notices. Her email address is [email protected]. material shapes, to feel their weight and texture. DOI: http://dx.doi.org/10.1090/noti1028 “The air inside the imaginary figure seems to get

920 Notices of the AMS Volume 60, Number 7 denser and to have a slightly different consistency the exam, as he took a final glance through the from the air outside,” he said. Simple constructions book, it suddenly occurred to him that he could that hold much complexity and meaning: That’s compress the contents into a single, simple picture. just what mathematicians seek in their work. Moving back and forth across the stage, he used Pallasmaa’s erudite lecture contained many gesticulations to indicate a 2-sphere on one side, a striking quotations, including this one of Balthus: 3-sphere on the other, and a “slinky” curve between “The more anonymous is, the more real it them. This curve, representing the preimage of a is.” The same can be said for , Pallasmaa regular value of a map from the 3-sphere to the 2- stated. Could a similar statement be made for sphere, provided a mental image summarizing the mathematics? Are there mathematical results Pontryagin–Thom construction. If one knows the that are so natural, so pristine that one cannot language of manifolds and transversality, Sullivan perceive the fingerprints of the mathematicians claimed, one can reconstruct the whole theory who discovered them? Perhaps one example would of cobordism in differential just from be the previously mentioned proof of the infinitude the intuition conveyed by his slinky picture. This of primes, its attribution to Euclid notwithstanding. experience made him realize, “That’s what it means Perhaps others are found in what Paul Erd˝os to understand a piece of mathematics.” famously called “proofs from the Book”. The visceral component of mathematical work Pallasmaa also quoted the philosopher Gaston surely evokes strong feelings, but mathematicians Bachelard, who in his book The Philosophy of No: usually do not discuss their feelings about their A Philosophy of the New Scientific Mind, stated that work, at least not in public lectures. In an earlier scientific thought “develops along a predestined panel discussion, Riikka Stewen, Finnish Academy path, from animism through realism, rationalism, of Fine Arts, asked whether mathematicians have and complex rationalism, to dialectical rationalism.” strong love/hate feelings about their work. “Yes, Pallasmaa did not say that mathematics develops very strong feelings,” came the immediate reply in this way; his point rather was that art aspires to from a on the panel, Andrés Villave- develop in the opposite direction, from the rational ces, National University of Colombia. There is a back towards “a unifying, mythical, and animistic loneliness in the work of an , and much math- experience”. Perhaps mathematics shuttles back ematical work shares this quality. Just as a painter and forth between the two endpoints. faces an empty canvas, he said, “Mathematicians are up against the empty page every day.” Visceral Encounters The longing, even desperation, that is implicit Bachelard’s “predestined path” at times echoed in the remarks of Villaveces also emerged in through the conference in comments that seemed Sarnak’s lecture, titled “Is there a place for ‘ugly’ to derive from the misconception, common outside mathematics?”. Sarnak considered the situation of mathematics, that the subject consists entirely where the only known route to a proof is ugly, of proofs, progressing inexorably from one logical in the sense of being strewn with long and step to the next. This misconception was vividly complicated calculations and verifications. The countered at various points during the conference. question then becomes, How desperate are we In an open microphone session, Blaise Heltai for a proof? When Sarnak gave an example of pointed out that mathematics and art are actually an ugly calculation connected with a beautiful very similar in process: When you are thinking result in the theory of automorphic forms, Mikhail about a mathematical object, you are right inside Gromov, Institut des Hautes Études Scientifiques the thing, trying to puzzle out its structure and and New York University, piped up to say: “Maybe secrets. You’re not thinking about how to prove the mathematics is fine, it’s your mind that’s ugly.” anything—that comes later. The puzzling-out Then there was Gromov’s lecture. A fish says: resembles the conceptual part of doing art. Heltai “You want to understand what water is? Jump in has a special perspective, as he is a painter with and find out.” Instead of plunging in, you could a Ph.D. in mathematics; he makes a living as a study the chemical and physical properties of management consultant. water. But without the experience of plunging into The kind of visceral encounter with mathematics water, you have no frame in which to talk about that Heltai referred to emerged at various times, what water really is. Similarly, when the experience such as in the lecture of Dennis Sullivan, CUNY of plunging into mathematics is absent, there is Graduate Center and Stony Brook University. When no frame in which to talk about what mathematics as a graduate student he was preparing for is—much less what simplicity in mathematics is. the preliminary examination, Sullivan studied That’s a verbose description of one moment John Milnor’s book Topology from the Differential that flashed by in an instant in Gromov’s stream-of- Viewpoint. Sullivan knew the book inside and consciousness lecture. He jumped into Descartes’s out, every definition, every proof. The day before timeless statement, “Cogito ergo sum [I think

August 2013 Notices of the AMS 921 therefore I am]”. The important thing here, Gromov and a beamer, or even just a blackboard, in the case said, is the ergo, the therefore. In a sense, dogs of Sullivan. They tried mightily to avoid technical think: Much of what goes on in a human brain is details, with imperfect success. very similar to what goes on in the brain of a dog. Another contrast was socio-economic. As Surely dogs are. But dogs do not understand ergo. Kennedy pointed out in a panel discussion, This ergo is a major source of the kind of thinking the mathematicians and philosophers at the that is characteristic of humans, Gromov said. And conference all work in academia, which provides yet, “it is completely hidden from us. And there is economic security and social acceptability, while a good reason why it is hidden. If it surfaces, you artists often lead far more precarious lives on the die. You will not survive. It’s against survival, it’s fringes of society. She noted the “heroic” efforts against evolution, it’s against [natural] selection.” that many artists must put forth in order to carry So it went. Gromov passed so quickly over out their work. so many topics, diving to the depths, all the What did each group absorb from the other? while leavening the presentation with flashes It’s difficult to say. One participant observed that of subversive humor. The effect was dizzying. mathematicians tend to have a high opinion of Afterward, during the open microphone session, themselves and their own knowledge and are an audience member demanded a one-sentence therefore not so open to new ideas, while artists summary—with an example. An impossible request are pretty much the opposite: Receptiveness to to fulfill. Nevertheless it can be said that one of impressions and influences from a wide variety Gromov’s main messages was: Guard against the of sources is the artist’s lifeblood. One artist who delusion of false simplicity. Many things that we attended Simplicity, Miyuki Tsushima, said she assume at first glance to be simple are in fact didn’t follow all the details of the math lectures. highly complex. She could simply sit and let the impressions wash After seeing Gromov’s effervescent mind bubble over her as she made some sketches for her latest over for thirty minutes, audience member Al Thaler, work. known to many for his long service at the National An inspiration for the conference was the so- Science Foundation and now an adjunct faculty called twenty-fourth problem of David Hilbert. member at CUNY’s Hunter College, commented, “I This problem, which Hilbert considered adding could never live like that.” to his famous list of twenty-three problems that he presented at the International Congress of Contrasting Groups Mathematicians in Paris in 1900, was unearthed by The Simplicity conference was the brainchild of Rüdiger Thiele, University of Leipzig, from papers mathematician Juliette Kennedy, University of at the library of the University of Göttingen. Part of Helsinki, and two CUNY mathematicians, Roman Hilbert’s description of the problem reads: “Criteria Kossak of the Graduate Center and Philip Ording of simplicity, or proof of the greatest simplicity of of Medgar Evers College. The conference was certain proofs. Develop a theory of the method of something of a follow-up to a 2007 symposium proof in mathematics in general. Under a given set called and Mathematics, which took of conditions there can be but one simplest proof” place in Utrecht and was organized by Kennedy (translation by Thiele from his article “Hilbert’s and two University of Utrecht mathematicians, 24th Problem”, American Mathematical Monthly, Rosalie Iemhoff and Albert Visser (Iemhoff was one January 2003). of the lecturers at Simplicity). Participants in the Etienne Ghys, École Normale Supérieure de 2007 symposium could drop in at an art exhibition Lyon, pointed out the naiveté of imagining that at the Mondriaanhuis, Logic Unfettered—European such ultimate simplicity is possible. Yet, as the and American Abstraction Now, which was curated conference highlighted, simplicity as a dream, by Kennedy. as an ideal, remains a powerful guiding light in In addition to the film program at the Simplicity mathematics and . As Franks said, there conference, there was an installation of a few works are no absolute notions of simplicity. But do not by artist Kate Shepherd in the lobby outside the relinquish the quest. “On the contrary, I want to hall where the lectures were given (Shepherd also say yes, find criteria for simplicity, continue to do participated in one of the panel discussions). But so,” said Franks. Don’t imagine that the matter will space constraints there, as well as the difficulty of ever be settled definitively; rather, “return to the securing exhibit space in New York City, meant that task often.” Simplicity offered few opportunities to experience Materials from some of the lectures are on the art. As a result, art was represented mainly through Simplicity conference website, http://s-i-m-p-l- the presence and words of the artists themselves. i-c-i-t-y.org, and videos of some of the lectures By contrast, the mathematicians could actually will be posted soon. present pieces of mathematics by using a computer

922 Notices of the AMS Volume 60, Number 7