Leesfragment: The Best Writing on Mathematics 2015

In januari verschijnt The Best Writing on Mathematics 2015 van Mircea Pitici. Wij publiceren voor. 

'In this volume a greater number of contributions than in the previous volumes concern mathematical games and puzzles. For many centuries and in many cultures, recreational mathematics used to be seen as a benign amusement of no immediate utility. That enduring but now old-fashioned perception has gradually changed over the past century because of at least two broad phenomena.'

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 2015 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 David Hand explains why we should actually expect unlikely coincidences to happen; Arthur Benjamin and Ethan Brown unveil techniques for improvising custom-made magic number squares; Dana Mackenzie describes how mathematicians are making essential contributions to the development of synthetic biology; Steven Strogatz tells us why it’s worth writing about math for people who are alienated from it; Lisa Rougetet traces the earliest written descriptions of Nim, a popular game of mathematical strategy; Scott Aaronson looks at the unexpected implications of testing numbers for randomness; and much, much more.

In addition to presenting the year’s most memorable writings on mathematics, this must-have anthology includes a bibliography of other notable writings and an introduction by the editor, Mircea Pitici. This book belongs on the shelf of anyone interested in where math has taken us—and where it is headed.

Mircea Pitici holds a PhD in mathematics education from Cornell University, where he teaches math and writing. He has edited The Best Writing on Mathematics since 2010.

Introduction

This is the sixth anthology in our series of recent writings on mathematics selected from professional journals, general interest publications, and Internet sources. All pieces were first published in 2014, roughly in the form we reproduce (with one exception). Most of the volume is accessible to readers who do not have advanced training in mathematics but are curious to read well-informed commentaries about it.
What do I want by sending this book into the world? What kind of experience I want the readers to have? On previous occasions I answered these questions in detail. To summarize anew my intended goal and my vision underlying this series, I use an extension of Lev Vygotsky’s concept of “zone of proximate development.” Vygotsky thought that a child learns optimally in the twilight zone where knowing and not knowing meet—where she builds on already acquired knowledge and skills, through social interaction with adults who impart new knowledge and assist in honing new skills. Adapting this idea, I can say that I aspire to make the volumes in this series ripe for an optimal impact in the imaginary zone of proximal reception of their prospective audience. This means that the topics of some contributions included in these books might be familiar to some readers but novel and instructive for others. Every reader will find intriguing pieces here.
Besides offering a curated collection of articles, each book in this series doubles into a reference work of sorts, for the recent nontechnical writings on mathematics—with the caveat that I decline any claim to being comprehensive in this attempt. The list of book titles I give at the end of the introduction and the list of notable writings at the end of the volume contain a few entries published prior to the 2014 calendar year, in an acknowledgment that in previous volumes I overlooked materials worth mentioning. The same is surely the case for this year. The fast pace of the series, the immense quantity of literature I survey, and the convention subtly ensconced in calling a “year” the interval from January 1 to December 31 not only make such lapses inevitable but to a high degree determine the content of the book(s). Were we to look at the same literature from July 1 of one year until June 30 of the next year, the books in this series would look very different from what you can read between these covers. That is why each volume should be seen in conjunction with the others, part of a serialized enterprise meant to facilitate the access to and exchange of ideas concerning diverse aspects of the mathematical experience.
In this volume a greater number of contributions than in the previous volumes concern mathematical games and puzzles. For many centuries and in many cultures, recreational mathematics used to be seen as a benign amusement of no immediate utility. That enduring but now old-fashioned perception has gradually changed over the past century because of at least two broad phenomena. First, the history of the most salient branches of contemporary mathematics (algebra, modern algebra, geometry, probability, number theory, graph theory, knot theory, topology, combinatorics, and even calculus) has been either rooted into or decisively influenced by “recreational” problems. Second, talented writers and popularizers of recreational mathematics (the most famous of whom was Martin Gardner) found a large audience in the public, enjoyed appreciation from select but remarkable mathematicians, and built a devoted following of like-minded authors who carry on working in the same vein, encouraged by the lasting impact of their predecessors. Recreational mathematics has a rich and sophisticated history studied in the past by a few authors who contributed brief works (notably David Singmaster); recently the scholarship is growing rapidly, as illustrated by a special issue of the journal Historia Mathematica dedicated entirely to recreational mathematics. Nowadays good recreational mathematics is placed midway between the intelligent but mathematically untrained public and the mathematics professionals, by virtue of linking easy-to-understand problems to serious mathematics. In other words the problems posed in high-quality recreational mathematics are comprehensible to the layperson, while pursuing and understanding the ideas developed in the solutions occasioned by the problems might require an independent learning effort the reader is free to undertake or not. Thus the context of good recreational mathematics straddles the popular and the pedagogical, having the dual value of intellectual entertainment and optional instructional use. At its best, recreational mathematics illustrates the synergetic encounter between the ludic and the serious aspects of mathematics, as well as the one between the amateur and the professional strands of doing mathematics—just two of the many polarities of contrary/complementary features that meet and mesh in mathematical thinking and practice. I am glad that this year “recreational” mathematics is well represented in the anthology. There is much more in this book. Some problems of recreational mathematics are at least two millennia old, representing well the nature of mathematics as we have known it—a type of thinking that endures forever, the timeless and unchanging mathematics of a realm at rest, in equilibrium. In that endurance we feel a flavor of the times when only a few million people lived on this planet, scattered over the Earth, and few of them thought mathematically—and those who did lived far apart, isolated from each other. But now we are in the billions, we instantly communicate with each other, and we put numbers on more things than can be measured. The mathematics adequate to this dynamic world in ceaseless disequilibrium has to have a basis different from the old mathematics, at least in some respects. It has to be a mathematics that treats the dynamic phenomena from the inside, not from the outside (just as Albert Einstein thought up relativity of motion by asking himself what would happen if one traveled as fast as light does). Glimpses of a more statistical bend in mathematical thinking are obvious in some contributions to this volume and attest not only to the ever-changing nature of mathematics but also to the merits of interpreting mathematics in broad personal and societal contexts. Interpreting mathematics is a further stage of thinking mathematically—not a “higher” stage or a “lower” stage, just an essentially different one. Interpreting mathematics is not about mathematical truth (or any other truth); it is a personal take on mathematical facts, and in that it can be true or untrue, or it can even be fiction; it is vision, or it is rigorous reasoning, or it is pure speculation, all occasioned by mathematics; it is imagination on a mathematical theme; it goes back several millennia and it is flourishing today, as I hope this series of books lays clear. Fragments in Plato’s philosophical dialogues qualify as interpreting mathematics, and so does Edwin Abbott’s Flatland, and (for example) the work of our contemporaries Ian Stewart, Steven Strogatz, Edward Frenkel, Jordan Ellenberg, and many others, including the contributors to this volume. Interpreting mathematics is a creative domain, inclusive but also competitive—and certainly potent, despite its neglect in academic circles. Interpreting mathematics points toward protean qualities of mathematics not immediately obvious in doing mathematics per se. An accepted mathematical result is merely the egalitarian premise from which each of us can part with the commonly shared view by interpreting it idiosyncratically, as we please or even as it suits us. While mathematics is a “great equalizer” in a sociological sense (as Jaime Escalante famously proclaims in the 1988 movie Stand and Deliver), interpreting mathematics is a subtle ability, as much competitive as it is differential. Interpretation sets each of us apart, even when we speak about the same mathematical fact. Interpreting mathematics applies to a part of human affairs where opportunities rule, not constraining mathematical rules. To speak about interpreting mathematics sounds odd, but it seems so only because the customary indoctrination served by our school system pervades the common views of mathematics, both among mathematicians and the lay public. For decades the penury of talented authors able to interpret mathematics in original ways has affected the interaction between mathematics and domains in which mathematical methods were coopted. Lack of reflection on the proper context of applied mathematical thinking perverted the humanities, the social sciences, and even the study and the practice of law—to name just a few areas. This state of affairs is changing and The Best Writing on Mathematics series takes notice of the change. Criticisms of (ab)using mathematics and statistics crop up all the time, with a few included in this book. For me, this is payback of sorts. I once made the naive misstep to suggest, at a well-regarded business school, that mathematics relates to reality in subtler ways than is immediately apparent and that I would rather pursue my hunches than submit to the expensive dogma taught in the “mathematical finance” courses. For that I was not only disparaged but shown the door. Out I went, never to regret it—although, for a long while after that, I “pursued” only survival, as a lesson in what can happen if I take initiative in a place that nominally encourages it. I had been fooled by my preconceptions concerning the abstract notion of free inquiry; I had set up myself for the misadventures that come with resisting the enforcement of dogmatism. Those blunders did not shake my passion for inquiry, but it cured me of the tendency to speak my mind. (Other cures followed, also in the name of noble-sounding ideals; my favorite is the one administered by courts under the slogan “in the best interest of the child.”) Since then I became a lot more cautions with my suggestions concerning mathematics; I learn from my errors, the hard way. Yet I still venture a bit in talking about mathematics, here and there—now prudent, aware that attempting to crack the thought establishments is fraught with dangers. As an example, I can say that the process of editing each volume in this series is a lesson in working with uncertainty while at the same time interpreting mathematics. The contrast between my limited knowledge and the limitless possibilities available to all the people who gloss on mathematics offers me palpable practice for a general mnemonic that serves well in other endeavors. I generalized it into a theoretical and practical principle, which I call “the paradox of reward.” The paradox of reward says that in a competitive, fair, unpredictable, and infinitely complex environment, the most valuable knowledge is to know how to be rewarded for ignorance; in other words, more reward is available for taking advantage of ignorance (if one finds such a path to reward) than it is for taking advantage of knowledge. Of course I am not saying that ignorance is preferable to knowledge; it is not. I am saying that in certain environments harvesting rewards off ignorance is (by far) more valuable than seeking rewards for knowledge. This might seem to have little to do with mathematics; yet to my mind it is nothing else but interpreting mathematics, in a world so complex that ignorance is unavoidable but ignoring its benefits is avoidable. This subject is a lot vaster than I sketched in these few sentences, and it has consequences not only for learning and teaching mathematics but also for incorporating the private interpretation of mathematics in strategic thinking. Yet I am mindful of the dangers of venturing too far in speaking unconventionally about mathematics and interpreting mathematics, so I leave it for another occasion.

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