New Approaches to an Ancient Affinity N. Sinclair et al. Mathematical Inelligencer 32, 2010, 77-8
This book is an edited collection of essays from nine contributors-the three editors and six others-most of which appeared as talks at a conference, Beauty and the Mathematical Beast, at Queen’s University in Ontario in 2001. It comprises an introduction, a concluding chapter, and three parts of three chapters each.
The first part, The Mathematician’s Art, starts with “Aesthetics for the Working Mathematician,” by Jonathan Borwein. Embracing Emil Artin’s proclamation, “We believe that mathematics is an art,” Borwein argues that aesthetics comes before utility in mathematical research and that opportunities for invoking questions of economy and stmc tural beauty abound not only in research but throughout mathematical education. In particular, Borwein is an impas sioned advocate for experimental mathematics, for the importance of computational methods and the use of visua lization/manipulation software as a source of intuition and fmitful hunches and a means of generating a “feel” for a problem that eludes purely symbolic analysis. Among his many examples from his own research and others are two gems of visualization: Tom Apostol’s “lovely new geometric proof’ of the irrationality of J2-which, as he points out, delivers a new insight into this ancient result, and the Cox eter/Kelly visual demonstration of Sylvester’s conjecture about noncollinear points in the plane which, like Apostol’s, uses a minimal configuration argument. The next chapter is “Beauty and Tntth in Mathematics,” by Doris Schattschnei der. Observing that though there are certainly formulas and assertions that have been found to be aesthetically pleasing beautiful by mathematicians. She elaborates by listing char acteristics of a proof that prompt mathematicians to designate it as beautiful: “Elegance” (cutting to the essential idea), “ingenuity” (has an unusual or surprising twist), “insight” (offers a revelation of why something is tme), “connections” (enlighten a larger picture) , “paradigm” (provides a widely applicable heuristic). Examples, for the most part accessible to high-school students, include the different Chinese and Western geometrical proofs of Pythagoras’s theorem and (another gem) Polya’s proof of Jab ::: (a + b)/2, which not only demonstrates the arithmetic-geometric inequality, but also shows why it’s tnte and why equality occurs only when a = b. Schattschneicler ends with an insistence on the aesthetic of “doing mathematics, ” which she illustrates by examining different conceptions of a “point” and a “straight” line and focusing on various physical models of the hyperbolic plane (including a crocheted model), as well as introducing certain nineteenth-century machines for draw ing straight lines, they show how concrete models and mechanisms can illuminate otherwise obscure, or at least difficult to conceptualize, mathematical abstractions. They conclude: “We believe that the understanding of meanings in mathematics (often through aesthetic experiences) comes before an understanding of the analytic formalisms.”
The next section, A Sense of Mathematics, offers a triple of increasingly more general perspectives. It starts with “The Aesthetic Sensibilities of Mathematicians,” by Natalie Sin clair, who asks what are “the animating purposes of mathe maticians, why do mathematicians do mathematics? What impulses, what inclinations are responsible for producing the body of knowledge that is mathematics?” Sinclair’s approach is mainly descriptive and classificatory: She gathers quotations from the literature and seeks answers by interviewing a group of mathematicians. These lead her to propose a tripartite categorization of the aesthetic impulses underlying mathematics: The evaluative–the familiar and ever-present judgements mathematicians make about the significance and value of results and proofs ; the generative- the dimension of the aesthetic pertaining to the creation of new ideas and insights into what mathematicians do; the motivational-the role of the aesthetic involved in attracting mathematicians to certain fields and stimulating interest in particular problems. After ranging widely over different areas, time periods and mathematicians, she concludes that the subject “satisfies the basic human impulse to find and describe pattern.” The final chapter in this section, “The Meaning of Pattern,” by Martin Schiralli, again emphasizes mathematicians’ search for and constniction/appreciation of pattern, but looked at in a wider context of how pattern is conceived in biology (the writings of Gregory Bateson) and in visual art (Ernst Gombrich ‘s A11 and !llllSion). Both these thinkers, in Schiralli’s reading of them, point to an idea of pattern beyond a static, immediately apprehensible arrangement, a pattern behind the pattern as it were, what Bateson articulates as “a dance of interacting parts.” The ubiquity of pattern and its importance in diverse fields sug gests that for mathematics the concept needs a more focused view, one that goes back to the roots of the subject, if it’s to be distinguished from its occurrence in biology and art. To this encl, the essay looks at the school of Pythagoras and the understanding of pattern to be found in their notion of number (particularly in relation to arithmos) and the meaning of concept as this plays out in the work of Philolaus of Croton.
The third chapter, “Mathematics, Aesthetics and Being Human,” by William Higginson, is the most ambitious . It asks in relation to mathematics, “What does it mean to be human?” Higginson argues that the aesthetic drive is a “manifestation of a universal human ability to sense what ‘fits’ in a given situation and what does not .” He elaborates this over a wide terrain: Anecdotes and surveys of how mathematics and mathematicians have been (and still are) perceived by schoolchildren, the recent surge in popular interest and artistic production of films, novels and plays featuring mathematics, and claims by cognitive science about embodied origins of mathematics. It culminates in the suggestion that the thinking behind Ellen Dissanayake’s book Homo Aestheticus, on the origins and motives of art, might find an appropriate (and even more fundamental) formulation in a concept he designates as Homo Mathe matico-Aestheticus .
The final section, Mathematical Agency, starts with “Mechanism and Magic in the Psychology of Dynamic Geometry,” by R . Nicholas Jackiw . jackiw is the designer of The Geometer’s Sketchpad, a well-known software program that allows users to create and manipulate mathematical constructions . His essay has two concerns, each with its own aesthetic dimension. One is to confirm the powerful and vivid mathematical experience -in mathematical research no less than education – afforded by what has come to be called “dynamic geometry” that such software and programs like it facilitate. The other seeks to embed the Sketchpad in a wider examination of the nature and importance of mechanical devices in relation to mathematical thought. Arguing against what he sees as the trivialization of devices as merely didactic and inessential aids, he sketches a deeper historical tradition which portrays “machines being conceived and received as embodiments, exemplars, repositories and demonstrations of profound scientific knowledge .” This allows him to conclude that devices such as the Sketchpad operate under a tension between two impulses: A conventional, explanatory, didactic one and a more “magical” mode when their purpose is to “astound and amaze” rather than produce a stabilizing enlightenment. Next in this section is “Drawing on the Image in Mathematics and Art,” by David Pimm . Aesthetics, Pimm declares at the outset, is to be interpreted in sensorial terms as “firmly rooted in the senses by means of which we perceive.” Further, perception here is overwhelmingly visual, involving what]ohn Berger called “ways of seeing.” The combination allows him to put the question of aesthetics equally to (visual) art and mathematics, mobilizing a wealth of comment by artists, critics and art historians as well as by mathematicians. This in n1rn provides a naniral platform for a focus on the visual-diagrammatic dimension of mathematics which leads to predictable issues: The nanire of telling (the letter) ver sus showing (the image) concrete, demonstrative examples versus abstract axiomatic presentation, the significance of the attribution “modern” to mathematics, and the complete and deliberate expunging of all diagrams in the Bourbaki group’s set-theoretical axiomatization of mathematics that exerted such a forceful (and for many baneful) influence on the presentation of the subject for several decades. Pimm quotes Pierre Cartier (himself a member of the group) explaining that the “Bourbaki were Puritans and Puritans are strongly opposed to pictorial representation of truths of their faith.” From this Pimm dilates on iconoclasm and an anthropologi cal understanding of “purity,” which allows him to see a proof as exerting a form of agency . The final essay of the section, “Sensible Objects,” by Dick Tahta, perhaps the most loosely focused contribution here, consists of an associatively linked series of responses, suggestions and remarks provoked by the question of what a sensible object is and how it might or might not relate to the kind of objects mathematicians referto. Mixing and sampling eigth-century Christian iconoclasm, nineteenth-cenn1ry Romantic poetry, several psychoanalytic theories, Renaissance art, mysterious Neolithic stone balls, vibrating strings and resonating ideas, the essay evades any definite conclusions, but instead ends with “the mystery of things.” Connecting all this is an insistence that interpretive practices together with their communities are important to how “sensible” and “object” are to be thought together.
As I mentioned at the beginning, these contributions are sandwiched between an (historical) introductory chapter and a concluding chapter (“Aesthetics and the ‘Mathemat ical Mind’, ” which focuses on psychological themes) . The authors insist that the oft-cited attributes of mathematics detachment, certainty, abstraction and perfection-be seen not as “objective” characteristics or as elements of generic aesthetic theories (such as those of Kant, for example) but as questions about the individual subject, questions of motive, desire, psychological need and pleasure . This leads in two directions . One is the nan1re of the unconscious and the preconscious aspect of mathematical knowledge/inn1- ition and its coming into consciousness (for example, as told famously by Poincare and subsequently systematized by Hadamard into stages of mathematical creation). The other points to the recognition of the “darker” aspects of the mathematical psyche. For example, the desire for detach ment, and so on, can be seen as the mathematician’s fight against what one commentator describes as the “uncer tainty, disorder, imitionality, being out of control” that haunts its practitioners . This is followed by a clutch of dark affects: “The melancholy disposition of the mathematical mind” (Albrecht Dlirer’s engraving Melancolia is repro duced), and then, more extreme-further from Reason aspects of the mathematical psyche in “blindness, solipsism and the ‘mathematical mind”‘ and “autism and the ‘mathe matical brain.”‘
Mathematics and the Aesthetic is a richly varied collection of essays that will supply numerous leads, avenues, openings and provocations to anybody interested in the pleasures and rigors of mathematical thinking.