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Overview
The year's finest writing on mathematics from around the world
This annual anthology brings together the year's finest mathematics writing from around the world. Featuring promising new voices alongside some of the foremost names in the field, The Best Writing on Mathematics 2014 makes available to a wide audience many articles not easily found anywhere else—and you don’t need to be a mathematician to enjoy them. These writings offer surprising insights into the nature, meaning, and practice of mathematics today. They delve into the history, philosophy, teaching, and everyday occurrences of math, and take readers behind the scenes of today’s hottest mathematical debates. Here John Conway presents examples of arithmetical statements that are almost certainly true but likely unprovable; Carlo Séquin explores, compares, and illustrates distinct types of one-sided surfaces known as Klein bottles; Keith Devlin asks what makes a video game good for learning mathematics and shows why many games fall short of that goal; Jordan Ellenberg reports on a recent breakthrough in the study of prime numbers; Stephen Pollard argues that mathematical practice, thinking, and experience transcend the utilitarian value of mathematics; and much, much more.
In addition to presenting the year’s most memorable writings on mathematics, this must-have anthology includes an introduction by editor Mircea Pitici. This book belongs on the shelf of anyone interested in where math has taken us—and where it is headed.
Product Details
| ISBN-13: | 9781400865307 |
|---|---|
| Publisher: | Princeton University Press |
| Publication date: | 11/23/2014 |
| Series: | The Best Writing on Mathematics , #5 |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 360 |
| File size: | 13 MB |
| Note: | This product may take a few minutes to download. |
About the Author
Read an Excerpt
The Best Writing on Mathematics 2014
By Mircea Pitici
PRINCETON UNIVERSITY PRESS
Copyright © 2015 Princeton University PressAll rights reserved.
ISBN: 978-1-4008-6530-7
CHAPTER 1
Mathematics and the Good Life
Stephen Pollard
Introduction
A full account of mathematics will identify the distinctive contributions mathematics makes to our success as individuals and as a species. That, in turn, requires us to reflect both on what mathematicians do and on what it means for humanity to flourish. After all, we cannot say how mathematics contributes to our success unless we have a good sense of what mathematicians accomplish and a good idea of what it means for us to succeed. To give our discussion some focus, I offer the following proposition:
Mathematics makes one substantial contribution to human prosperity: it enhances our instrumental control over physical and social forces.
I say "offer," not "endorse." In fact, I want to persuade you that this proposition is wrong because it presupposes either too narrow a conception of mathematical activity or too narrow a conception of human success. Mathematicians are not just "devices for turning coffee into theorems" (as Erdos may or may not have said). Furthermore, if they were such devices, humanity would be the worse for it, and this would be so even if the caffeine-fueled theorem mills were more efficient than real mathematicians at disgorging instrumentally useful product.
The theorem-mill model leaves out at least two vital features of mathematical activity: mathematicians achieve deep insights and have intricately meaningful experiences. These features are vital because they are intrinsic goods for human beings: they are characteristically human ways to prosper. This is not news. It is an ancient idea: mathematics provides insights and experiences that, in themselves, ennoble our species. This idea is one of those enduring themes that make the history of philosophy a long, long conversation rather than a series of disconnected episodes. It can be enervating to feed on the sere remains of antique harvests. It can be energizing, though, to advance a conversation begun of old. We hope to do the latter, with John Dewey as our main interlocutor. We begin, however, with some inspiring words from Bertrand Russell that should make vivid the point of view I am promoting and give you some idea of how I mean to promote it.
A Glorious Torment
In the penultimate chapter of Education and the Good Life, Russell considers "the functions of universities in the life of mankind" [1926, p. 311]. He assumes that universities exist for two purposes: "on the one hand, to train men and women for certain professions; on the other hand, to pursue learning and research without regard to immediate utility" [Russell 1926, p. 306]. It is the latter theme, research for the sake of seeking and knowing, that causes Russell's language to soar.
I should not wish the poet, the painter, the composer or the mathematician to be preoccupied with some remote effect of his activities in the world of practice. He should be occupied, rather, in the pursuit of a vision, in capturing and giving permanence to something which he has first seen dimly for a moment, which he has loved with such ardour that the joys of this world have grown pale by comparison. All great art and all great science springs from the passionate desire to embody what was at first an unsubstantial phantom, a beckoning beauty luring men away from safety and ease to a glorious torment. The men in whom this passion exists must not be fettered by the shackles of a utilitarian philosophy, for to their ardour we owe all that makes man great. [Russell 1926, pp. 312–313]
There is a utilitarian reading even of this passage, a reading that Russell himself endorses. The utilitarian pursuit of knowledge, the quest for greater control over physical and social forces, is not "self-sustaining"; it needs to be "fructified by disinterested investigation" [Russell 1926, p. 312]. Our quest for control profits from seekers who take little or no interest in that quest. Our legitimate utilitarian interests can be served by people who do not give a fig for technological prowess, precisely because they do not give a fig. Yes, this is one reason Russell would not fetter mathematicians with the shackles of a utilitarian philosophy. But it is not the only reason. For "even if some splendid theory never has any practical use, it remains of value on its own account; for the understanding of the world is one of the ultimate goods" [Russell 1926, p. 312]. This is one half of the view I am trying to promote: the half that emphasizes the intrinsic value of certain products of inquiry, the insights that crown a successful inquiry. The other half emphasizes the process, reminding us that the whole drama of seeking and finding can be intrinsically good. The hunt itself is one of the glories of human life. It is, to use Russell's phrase, an ultimate good.
Now why might you agree that this is so? Russell's discussion of ultimate goods begins with an argument whose conclusion is not that some particular goods are ultimate, but that there must be ultimate goods.
The essence of what is "useful" is that it ministers to some result which is not merely useful ... life must be capable of some intrinsic value: if life were merely useful as a means to other life, it would not be useful at all ... Somewhere we must get beyond the chain of successive utilities, and find a peg from which the chain is to hang; if not there is no real usefulness in any link of the chain. [Russell 1926, p. 21]
And what are the ultimate goods for human beings? Here Russell offers, not argument, but description. He shows us, in arresting word-pictures, admirable ways for human beings to live, making vivid the features he finds intrinsically valuable and inviting us to find them so as well. Here is a striking example from his discussion of how we should present human history to the young.
I think we should keep in our own minds, as a guiding thread, the conception ... of the human race as a whole, fighting against chaos without and darkness within, the little tiny lamp of reason growing gradually into a great light by which the night is dispelled. The divisions between races, nations and creeds should be treated as follies, distracting us in the battle against Chaos and Old Night, which is our one truly human activity. [Russell 1926, p. 267]
If we find this passage evocative, if we feel tempted to declaim it "in tones vibrant with manly pathos," we might remind ourselves that "The intrusion of emotion and sentimentality is always the mark of a bad case" [Russell 1932a, pp. 109–110]. Then again, bad case is not quite right. There is no case in the sense of an argument. There is a depiction meant to be suggestive. If it strikes us in the way it was intended, if we find ourselves more strongly inclined to value things Russell valued in the way Russell valued them (that is, precisely, without benefit of argument, but also without harm to our critical faculties), then the depiction was successful. Can we really hope to do much better?
Russell and Dewey think not. Russell makes his position particularly clear: "Every attempt to persuade people that something is good (or bad) in itself, and not merely in its effects, depends upon the art of rousing feelings, not upon an appeal to evidence" [Russell 1935, p. 235]. This is so because questions of value "cannot be intellectually decided at all" [Russell 1935, p. 243]. Dewey would not be comfortable with this formulation since he himself has no trouble detecting an intellectual element in the determination of values. Extreme emotivism is no better than "oversimplified rationalism": it is wrong to deny "any efficacy whatever to ideas, to intelligence" in the development and propagation of ultimate purposes [Dewey 1939b, p. 150].7 However, Dewey agrees with Russell that pure ratiocination is unlikely to play the leading role, being much better suited to serve as an adjunct to what Russell calls "preaching." When Russell offers poetry rather than proof, he is, according to Dewey, contributing to the moral progress of our species in the way that has the best chance of success. The "chief instrument of the good" is imagination, not argumentation [Dewey 1934a, p. 348]. "Only imaginative vision elicits the possibilities that are interwoven within the texture of the actual" and, hence, "the sum total of the effect of all reflective treatises on morals is insignificant in comparison with the influence of architecture, novel, drama, on life" [Dewey 1934a, p. 345]. Indeed, "Art has been the means of keeping alive the sense of purposes that outrun evidence and of meanings that transcend indurated habit" [Dewey 1934a, p. 348]. Philosophy, when merely prosaic, can contribute to moral progress by giving verbal form to important currents in aesthetic experience and providing them an "intellectual base" [Dewey 1934a, p. 345]. Make no mistake; a prophetic work of art is, in itself, an intellectual achievement of the highest order [Dewey 1934a, pp. 46, 73–74; 1931, p. 116; 1960, p. 198]. Art does not wait upon philosophical prose to draw it into the arena of thought. However, to make our lives whole and to increase our intelligent reflective control over our lives, we need to ponder and discuss human purposes and aspirations when we are not in the grip of aesthetic experiences. Discursive philosophy provides the intellectual base for that sort of intelligent reflection and inquiry. Dewey himself provides just such a base with his notion of a "balanced experience" whose cultivation is "the essence of morals" [Dewey 1916, p. 417].
To return for a moment to our main interest, we would like Dewey to teach us about mathematics. We might be expected, then, to focus on his remarks about mathematics. In fact, we largely ignore that material. My slantwise strategy is to focus on a notion central to Dewey's aesthetics and philosophy of education: the aforesaid notion of experience. This notion lies at the heart of Dewey's comprehensive vision of human prosperity and so guides us to a broader understanding of how mathematics helps us flourish as human beings. We now consider human experience in general and, more narrowly, the sort of experiences provided by mathematical inquiry.
Mathematics as Experience
I would like to serve on a jury. I think it would be a good experience. Yes, that would be quite an experience. An experience with a definite beginning, a significant internal structure, and an end that is a completion, not just a termination. Experiences, in just this everyday sense, are the building blocks of a human life that is more than an animal existence, a life distinguished by the development and exercise of distinctively human capacities.
The basic conditions of an experience are not uniquely human. "No creature lives merely under its skin" [Dewey 1934a, p. 13]. We animals are all active in an environment that poses problems. How do I escape from the tiger? Where can I find water? Is the defendant guilty? We respond to those problems. If things work out well, we survive, we drink, we learn, we grow. There is a rhythm of problem, activity, resolution, growth in the lives of mollusks and mathematicians.
What is distinctively human is the conscious, reflective, intelligent integration of experiences into a life we can affirm as worthwhile, in which we exercise greater and greater control over a richer and richer array of experiences. If our effort at integration fails spectacularly enough, we will not even be sane [Dewey 1963, p. 44]. If we are unreflective, if we squander opportunities for growth, we will be not only stunted but unfree, "at the mercy of impulses into whose formation intelligent judgment has not entered" [Dewey 1963, p. 65].
So the quality of our experiences and the way those experiences build on one another matters desperately. Why is democracy better than tyranny? Why is kindness better than cruelty? There is, ultimately, a single reason: a democratic arrangement of society and a kindly attitude toward our fellows promote "a higher quality of experience on the part of a greater number" [Dewey 1963, p. 34]. Why should we care about the artworks of distant times and places? Because "the art characteristic of a civilization is the means for entering sympathetically into the deepest elements in the experience of remote and foreign civilizations" and, hence, their arts "effect a broadening and deepening of our own experience" [Dewey 1934a, p. 332]. What is education? "It is that reconstruction or reorganization of experience which adds to the meaning of experience, and which increases ability to direct the course of subsequent experience" [Dewey 1916, pp. 89–90].
Experience, experience, experience. Why keep going on and on about experience when we are supposed to be talking about mathematics? Two reasons. First, if we take at all seriously a view that places experience at the heart of human prosperity, we have good reason to reflect on the quality of mathematical experiences and the capacity of those experiences to enrich subsequent experiences. Second, when we do so reflect, we better appreciate how mathematics contributes to human happiness in ways that are not narrowly instrumental. Here are five properties of mathematical experience we consider in turn.
1. Mathematics provides examples of "total integral experiences" [Dewey 1934a, p. 37; 1960, p. 153] of a particularly pure form.
2. Those experiences yield products that allow others to recreate the experiences.
3. Mathematics provides prime examples of experiences that "live fruitfully and creatively in subsequent experiences" [Dewey 1963, p. 28].
4. There is no necessary incompatibility between the ordered growth of mathematical experiences and other desirable forms of human growth.
5. Mathematical experiences not only allow for breadth of cultivation, but once such breadth is achieved, fit easily into the texture of a variegated life.
We start at the top of the list: the inner structure of a mathematical experience.
Experience
Chimpanzees negotiate environments that offer problems and opportunities. A chimp finds a nut that hides its tasty meat in a hard shell. The chimp finds a stone. The chimp thoughtfully manipulates the stone and the nut. Stone cracks shell. Here "the material experienced runs its course to fulfillment ... a problem receives its solution ... a situation ... is so rounded out that its close is a consummation and not a cessation" [Dewey 1934a, p. 35; 1960, p. 151]. Whether the chimpanzee is a pioneer or just an apprentice, there is growth. "Life grows when a temporary falling out [damn this shell!] is a transition to a more extensive balance of the energies of the organism with those of the conditions under which it lives" [Dewey 1934a, p. 14]. As with the chimpanzee, so with the mathematician.
... every experience is the result of interaction between a live creature and some aspect of the world in which it lives.... The creature operating may be a thinker in his study and the environment with which he interacts may consist of ideas instead of a stone. But interaction of the two constitutes the total experience that is had, and the close which completes it is the institution of a felt harmony. [Dewey 1934a, pp. 43–44; 1960, p. 160]
The drama of experience runs its characteristic course: problem, activity (physical or mental), success, growth. The problem may be a nut on the forest floor, an exercise in a book, or a "beckoning beauty" in some corner of your mind. Chimp or human, if you have any semblance of puzzle drive (to borrow Feynman's phrase), you are hooked. You have to figure it out. And when the nut is especially hard to crack, when it draws deeply on your resources, success is graced with an exultation that probably flows from somewhere deep in our primate nature. "Few joys," says Russell, "are so pure or so useful as this" [Russell 1926, p. 259]. As George Pólya observes, even minor triumphs have their savor.
Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery. Such experiences at a susceptible age may create a taste for mental work and leave their imprint on mind and character for a lifetime. [Pólya 1957, p. v]
Again, a mathematical experience is no mean thing. Though Dewey does not command Russell's and Pólya's insider view of mathematics, he too appreciates this point.
(Continues...)
Excerpted from The Best Writing on Mathematics 2014 by Mircea Pitici. Copyright © 2015 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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Table of Contents
Introduction, Mircea Pitici ixMathematics and the Good Life, Stephen Pollard 1
The Rise of Big Data: How It’s Changing the Way We Think about the World, Kenneth Cukier and Viktor Mayer-Schönberger 20
Conway’s Wizards, Tanya Khovanova 33
On Unsettleable Arithmetical Problems, John H. Conway 39
Color illustration section follows page 48
Crinkly Curves, Brian Hayes 49
Why Do We Perceive Logarithmically? Lav R. Varshney and John Z. Sun 64
The Music of Math Games, Keith Devlin 74
The Fundamental Theorem of Algebra for Artists, Bahman Kalantari and Bruce Torrence 87
The Arts—Digitized, Quantified, and Analyzed, Nicole Lazar 96
On the Number of Klein Bottle Types, Carlo H. Séquin 105
Adventures in Mathematical Knitting, Sarah-Marie Belcastro 128
The Mathematics of Fountain Design: A Multiple-Centers Activity, Marshall Gordon 144
Food for (Mathematical) Thought, Penelope Dunham 156
Wondering about Wonder in Mathematics, Dov Zazkis and Rina Zazkis 165
The Lesson of Grace in Teaching, Francis Edward Su 188
Generic Proving: Reflections on Scope and Method, Uri Leron and Orit Zaslavsky 198
Extreme Proofs I: The Irrationality of 2, John H. Conway and Joseph Shipman 216
Stuck in the Middle: Cauchy’s Intermediate Value Theorem and the History of Analytic Rigor, Michael J. Barany 228
Plato, Poincaré, and the Enchanted Dodecahedron: Is the Universe Shaped Like the Poincaré Homology Sphere? Lawrence Brenton 239
Computing with Real Numbers, from Archimedes to Turing and Beyond, Mark Braverman 251
Chaos at Fifty, Adilson E. Motter and David K. Campbell 270
Twenty-Five Analogies for Explaining Statistical Concepts, Roberto Behar, Pere Grima, and Lluís Marco-Almagro 288
College Admissions and the Stability of Marriage, David Gale and Lloyd S. Shapley 299
The Beauty of Bounded Gaps, Jordan Ellenberg 308
Contributors 315
Notable Writings 325
Acknowledgments 333
Credits 335