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Simplicity: Ideals of Practice in Mathematics and the Arts Reviewed by Douglas Norton

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Simplicity: Ideals of Practice in Mathematics and the Arts Reviewed by Douglas Norton BOOK REVIEW Simplicity: Ideals of Practice in Mathematics and the Arts Reviewed by Douglas Norton Simplicity: Ideals of Practice tieth-century proof theory, in Mathematics and the Arts model theory, and algorith- Edited by Roman Kossak mic information theory. and Philip Ording The book Simplicity: Ide- als of Practice in Mathematics Simplify this fraction. Simplify the and the Arts addresses ideas expression. Simplify your answer. of simplicity in mathemat- We certainly present simplicity ical proof in a general and to our students as a desired goal, philosophical way that re- sometimes to the extent of con- quires no previous ground- flating in significance the path to ing in the specialty theories Springer, 2017, xx+305 pages. xx+305 2017, Springer, a solution and the form of the of the preceding paragraph solution. On the research side of our mathematical lives, while providing both sub- tle and fascinating insights embedded in our own reference to a proof as “elegant” is Figure 1. David Hilbert, c. 1900. the idea of a proof demonstrating some sort of simplicity. into the questions raised. One hundred years after David Hilbert (Figure 1) presented The volume presents selected lectures and additional con- his famous list of unsolved problems at the International tributions from a conference also titled Simplicity, held at Congress of Mathematicians in 1900 [1], historian of the Graduate Center of the City University of New York mathematics Rüdiger Thiele discovered another problem in April of 2013. (See the conference poster in Figure 2.) buried away in Hilbert’s mathematical notebooks: “The Why mathematics and the arts? Mathematical propor- 24th problem in my Paris lecture was to be: Criteria of tions proposed by the Greek sculptor Polykleitus in the fifth simplicity, or proof of the greatest simplicity of certain century BCE, perspective in Renaissance Italian painting, symmetry in Islamic tilings, geometry in the paintings of proofs” [2]. While Hilbert’s list of problems inspired and Piet Mondrian and the De Stijl school, and tessellations in challenged the mathematical community throughout the M. C. Escher are all examples of specific mathematical tools twentieth century, his 24th problem never appeared in the utilized by artists. The past two decades have found both literature until this relatively recent discovery. Nevertheless, a broadening of the content and a widening of the appeal the ideas appear independently as formal threads in twen- of the crossover between mathematics and the arts. The Bridges Organization works to “foster research, practice, Douglas Norton is an associate professor in the Department of Mathematics & Statistics at Villanova University. His email address is douglas.norton@ and new interest in mathematical connections to art, music, villanova.edu. architecture, education, and culture” through its annual Communicated by Notices Book Review Editor Stephan Ramon Garcia. Bridges Conferences [3]. The Journal of Mathematics and For permission to reprint this article, please contact: reprint-permission the Arts [4] is a peer-reviewed journal that focuses on con- @ams.org. nections between mathematics and the arts. An impressive DOI: https://dx.doi.org/10.1090/noti2027 juried exhibition of mathematical art has become a regular FEBRUARY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 209 Book Review feature at the Joint They see, and the conference participants explore, sim- Mathematics Meet- plicity as the essence of the similarity of method and the ings [5]. Educators at ideal of practice common to twentieth-century Western all levels have begun art and Hilbert’s quest for consistency, efficiency, and to advocate for the rigor in proofs. The papers gathered in this volume pres- inclusion of the arts ent a fascinating peek at what the interactions among the in the push for sci- mathematicians, artists, and philosophers gathered at the ence, technology, en- conference were like. Talks at the conference (as in Figure gineering, and math- 3) were complemented by panel discussions across disci- ematics education, plinary boundaries; see Figure 4. The observations below with STEM evolving are intended to follow a few threads that wend their way into STEAM [6]. through the text rather than providing a sequential stroll Organizers of the through the papers in the collection. Simplicity confer- Juliet Floyd, philosopher of mathematics and of lan- ence were Juliette guage at Boston University, opens her piece “The Fluidity Kennedy (Univer- of Simplicity: Philosophy, Mathematics, Art” with the line: “Simplicity is not simple” [p. 155]. Is there a definition on Figure 2. Poster for the conference sity of Helsinki), Simplicity, held April 3–5, 2013, at the Roman Kossak (the which we can agree in the mathematical context? Is there Graduate Center of the City University Graduate Center of one in the arts? Are they mirror images, funhouse mirror of New York. CUNY), and Philip images, or completely unrelated? Andrés Villaveces, profes- Ording (then at Medgar Evers College of CUNY, now at sor of mathematics at the National University of Colombia, Sarah Lawrence College), all mathematicians with cross- Bogotá, observes in his piece “Simplicity via Complexity: over interests in logic and philosophy, model theory, and Sandboxes, Reading Novalis”: mathematics and the arts, respectively. In their preface to The simplicity question—the quest for the sim- the book, editors Kossak and Ording provide the following: plest proof or the simplest design, line, or That mathematicians attribute aesthetic quali- resolution of architectural space or rhyme ties to theorems or proofs is well known. The or melody…draws a tenuous but intriguing question that interests us here is to what extent connection between mathematics and various aesthetic sensibilities inform mathematical other disciplines (architecture, physics, design, practice itself. When one looks at various as- chemistry, music, etc.) [p. 192]. pects of mathematics from this perspective, it Let us first consider what the authors have to say in the is hard not to notice analogies with other areas mathematical arena. of creative endeavor—in particular, the arts.… Étienne Ghys, mathematician at the École Normale [W]e find that a more profound connection Supérieure in Lyon, presented the first address at the con- between art and mathematics than any formal ference and the first paper in the collection, entitled “Inner similarity is a similarity in method. For this Simplicity vs. Outer Simplicity.” In these, he demonstrates reason the conference emphasized ideals of why he was the inaugural recipient of the Clay Mathemat- practice [pp. viii–ix]. ics Institute Award for Dissemination of Mathematical Figure 4. Panel discussion at the Simplicity conference. Figure 3. Professor Dusa McDuff of Barnard College, presenting Panelists from left: Philip Ording, Amy Baker Sandback, Rachael at the Simplicity conference. DeLue, and Étienne Ghys. 210 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 2 Book Review Kolmogorov complexity of an object is the length of the shortest computer algorithm that produces the object as output. A more general usage would be that the complex- ity of an object is the length of the shortest description of the object. Ghys contrasts high and low Kolmogorov complexity through two pictures. A square with a random distribution of yellow and orange dots (Figure 5) would require a long sentence for a complete dot-by-dot descrip- tion, while just a few short lines of code can generate the Mandelbrot set (Figure 6). This brief description renders the Mandelbrot set “simple” from the outer simplicity per- spective, but Ghys finds this unsatisfactory; it is not simple in terms of inner simplicity. Ghys provides another example with proofs. He presents a single sentence from a number theory book by Jean-Pierre Serre that he recalls and describes as follows: I spent two days on this one sentence. It’s only one sentence, but looking back at this sentence, I see now that it is just perfect. There is nothing to change in it; every single word, even the Figure 5. Étienne Ghys, “…my own art object. This is a totally smallest, is important in its own way.… Serre’s random object.” language is so efficient, so elegant, so simple. Knowledge by posing questions, giving examples, and set- It is so simple that I don’t understand it.… Ev- ting the tone for the conference and the collection. He sees erything, every single word is fundamental. Yet, a basic dichotomy in mathematics: “For me, mathematics is from the Kolmogorov point of view, this is very just about understanding. And understanding is a personal simple.… Finally, at the end of the second day, and private feeling. However, to appreciate and express this all of a sudden, I grasped it and I was so happy feeling, you need to communicate with others…” [p. 3]. For that I could understand it. From Kolmogorov’s him, this dichotomy translates into two kinds of simplicity: point of view, it’s simple, and yet for me—and, inner simplicity, reflecting an ease of personal understand- I imagine many students—it’s not simple [p. 6]. ing that may nonetheless be difficult to communicate, and He provides counterpoint to this with a delightful mean- outer simplicity, in which something may be easy to express der through networks, density, the Internet, and a theorem concisely but difficult to comprehend. by Endre Szemerédi, with the following conclusion: “[I]t’s Because Ghys’s outer simplicity relies on communica- an example of a theorem for which the published proof tion and description, he invokes Kolmogorov complexity is complicated, but nevertheless I understand it. For me as a measure of simplicity. In information theory, the it’s simple. I think I will never forget the proof because I understand it. And this is the exact opposite of the one-line by Jean-Pierre Serre, which was so short that it took me days to understand it” [p.
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