I had just started high school when I found The Art of Mathematics on the “New Arrivals” shelf of my local public library. I found it engaging and inspirational. While at college, I searched (in vain) for a copy of this book to add to my personal collection. Thus I am overjoyed that Dover Publications is reprinting this book. After reading it from my new point of view — as an assistant professor of mathematics — I still find it inspirational, in both familiar and new ways. My personal philosophy as an educator has been greatly shaped by the philosophy of this book.
As the reader may guess, this book is about art, mathematics, and connections between the two, but it is much more than just that. It is also about educational philosophy, about mathematicians and their motivation, about the way different groups interact in society, and a personal reflection. In his own words, King has written this book because “aesthetic and intellectual fulfillment requires that you know about mathematics,” and to answer the question “why a collection of strange and peculiar people write 25,000 research papers each year on an esoteric subject called mathematics.” King states that this book is not written for mathematicians (whom he defines specifically as those holding tenured or tenure-track positions in mathematics departments at research universities), although they and all who teach mathematics could benefit from internalizing the philosophical ideas presented in this book.
As King states right away, he will be writing about mathematics, but “without apology” (a reference to G. H. Hardy's classic book, A Mathematician's Apology). The Art of Mathematics discusses a large number of topics one might expect to appear in a “math for the masses”-style book, including: the construction of number systems from the natural numbers to the complex numbers, the pigeonhole principle, the irrationality of the square root of 2, the infinitude of primes, Euler's equation, Fermat's last theorem, the Four-Color theorem, and a variety of logic puzzles for the reader's amusement. There is commentary on the “unreasonable effectiveness of mathematics,” discourse on the distinction between pure and applied mathematics, and exposition on the cyclic mathematical modeling process (abstraction, analysis, interpretation, and refinement). King discusses the evolution of theoretical mathematics into useful mathematics, such as the conic sections of ancient Greece finding application in Kepler's laws of planetary motion. He touches on the debate between discovery and creation in math research. He explains the distinction between scientific truth and mathematical truth: scientific truth is based on experience from observation, and thus the “true laws” of science have undergone changes from Aristotle's time to Galileo's time to the present; mathematical truth is “what you find at the end of a correct chain of mathematical argument.”
However, these topics are not presented as a grab-bag. There are four recurring themes discussed in The Art of Mathematics: truth (the goal of mathematics), reality (the applications of mathematics), ethics (the teaching of mathematics, why it sometimes fails, and what can be done to improve the situation), and beauty (the motivation for doing mathematics). It is not a coincidence, King points out, that these four themes are also those encountered in a first course in philosophy. The ideas encountered throughout the book are woven together using these themes, with an emphasis on beauty and aesthetics.
King not only writes about mathematical topics but also about mathematical culture, the people and the institutions behind the subject, giving a human quality to this often austerely-perceived endeavor. In these discussions, I particularly enjoy the straightforward “question-and-answer” format. For example: what do mathematicians understand that non-mathematicians do not? King lists three items (and provides examples of each): recognizing the importance of implications, restricting oneself to the use of axioms and previously proven results, and being completely precise. Other questions King addresses include identifying what makes mathematics difficult and what constitutes good versus mediocre mathematical research.
At times, King is sharply critical. He views a large portion of mathematics research as mediocre: too many mathematicians, as they age, become better at the “mechanics of the research game,” while losing enthusiasm and the ability to do creative mathematics. Specializations become sharper and more time is spent concentrating on less and less… Mathematics is produced “the way a carpenter makes kitchen cabinets… adequately.” King bemoans the existence of teachers who blame their inability to teach on students who “lack the gift,” and the reduction of calculus to “a grab bag of techniques and applications.” In particular, he believes that the educational system is responsible for widening the gap between those who like math and those who dislike math. As a partial solution, he believes, we should present mathematics as an art.
King advocates leading students to math by showing them the poetry within it. What is the major obstacle to the recognition of mathematics as art? To address this question, King introduces the concept of the “artworld,” a system consisting of artists, works of art, an artworld public, etc., and an analogous “mathworld.” He tries to draw a correspondence between the two, and concludes that the mathworld comes up short in terms of critics writing for the general public. An accessible literature exists to determine the relative merits of the music of classical composers, for example, but no such critical literature exists for mathematics. This is at least partly responsible for current perceptions of mathematics by the public. Mathematicians' only critics are other mathematicians, and the majority of these (such as journal referees or tenure and promotion committee members) are only evaluators of correctness or importance, not liaisons to the public. A true embedding of the mathworld in the artworld, necessary for the recognition of mathematics as art, requires three things: an increase in the number of mathematicians talking about mathematics, a beneficial change in the way young people are taught mathematics, and a positive change in the attitudes of non-mathematically educated adults towards mathematics. King muses that with these changes “mathematical ignorance, like public smoking, will become unacceptable.”
King puts forth some elements of an aesthetic theory of mathematics. He introduces two principles that could be used to define a beautiful mathematical result: “Minimal Completeness” (the proof does not require extraneous results) and “Maximal Applicability” (the proof contains ideas which are applicable to notions other than itself). He also introduces the metaphorical notion of “aesthetic distance”: failure to appreciate mathematics as an art may be due to great distance (as in the case of humanists who lack the ability to read mathematics) or extreme closeness (as in the case of engineers or scientists, to whom mathematics is merely a tool). Mathematicians occupy an ideal distance between the two extremes. King presents a convincing argument that as a group, mathematicians are well described as an aristocracy (theocracy being a close second), as they are cloistered by walls of mystery and yet simultaneously supported by others. The basic obligation of this privileged position, according to King, should be to bring more people into the region from which mathematics may be aesthetically appreciated — something which is already in the mathematicians' best interests!
This book also discusses C.P. Snow's “two cultures,” the “humanists” and the “scientists.” The individuals in Snow's scientist category are unable to enjoy the finer points of art, but at least they are aware it exists. Even more regrettable is the situation of the humanists, who not only do not understand science, but also fail to realize that it is a form of art. The bridge between these two groups is mathematics, the language of the sciences. King introduces a refinement Snow's two cultures: “Type M” (those who understand Mathematical topics up to and including calculus), and “Type N” (those who are Not “Type M”). The advantage of this refinement is threefold: it eliminates overlap between Snow's groups, it applies to all of civilization (rather than just those with intellectual occupations), and it brings front and center what must be done to remove the separation: increase the general level at which mathematics is understood. Before attempting to motivate people to attain this level of understanding, it must be realized that the methods of promoting ideas to the two cultures are disjoint. You can not use applications to promote mathematics to the humanists; that method has been tried and has failed. This failure is the greatest shortcoming of mathematicians — having not yet helped the humanists find their mathematical voice, to communicate ideas they may have but can not articulate.
The Art of Mathematics brings to light a number of conversations that need to take place, both more frequently and more publicly. Viewing mathematics as art is a worthy idea. To bring about this (or any other) radical change in the public perception of mathematics, there must be correspondingly radical change within the mathematical community, especially in the training of future mathematicians and mathematics teachers at all levels. There are many questions that must be asked and discussed. Does the experience of obtaining an advanced degree in mathematics actively and adequately prepare individuals to be excellent educators as well as quality researchers? Should the ability to communicate to an audience of non-mathematicians be cultivated with the same emphasis as the ability to give a technical presentation understandable by few? Are mathematicians who excel at exposition and communication valued as highly as those producing advanced research? If so, are these skills similarly rewarded? For example, is such a balance of values reflected in the criteria for tenure and promotion at the college level: is value given to expository articles as it is to research articles?
Whether or not you agree with the highly opinionated and personal reflections of King, The Art of Mathematics is an important and a refreshing read. King is on a quest, and he feels time closing in about him; this is most evident when he shares with the reader his inner thoughts: “…[my] courses are running out. Soon I will be down to just one. Just one more course and I'm done. Make it classical complex variables. Let me do it once more… I'll do it truly… the students will see the art of mathematics. And they will never care for anything half as much.” The Art of Mathematics is a book whose ideas are both timely and timeless. Recommend this book to those you know, be they “Type M” or “Type N”. And may all of us who teach not wait until our final class to heed the advice in this elegant, insightful, and artful book.
Lee Stemkoski is Assistant Professor of Mathematics at Adelphi University.