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Experimental Mathematics

V. I. Arnold
Publication Date: 
Number of Pages: 
MSRI Mathematical Circles Library 16
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Tushar Das
, on
In this series of lectures I will talk about several new directions in mathematical research. All of these are based on the idea of numerical experimentation. After looking at examples such as \(5 \cdot 5 = 25\) and \(6 \cdot 6 = 36\), we advance an hypothesis, such as \(7 \cdot 7 = 47\). Further experimentation either supports or disproves it.

Thus begins Arnold, as he prepares to take his audience on a a whirlwind romp across four distinct mathematical themes:

  • Lecture 1. The Statistics of Topology and Algebra
  • Lecture 2. Combinatorial Complexity and Randomness
  • Lecture 3. Random Permutations and Young Diagrams of Their Cycles
  • Lecture 4. The Geometry of Frobenius Numbers of Additive Semigroups

Experimental Mathematics is a loose transcription of lectures that Arnold gave at the 2005 Dubna summer camp in Russia. Some video lectures from this extraordinary mathematics sumer camp are available online. Unfortunately, I could not find those among Arnold’s lectures that made it in to Experimental Mathematics, but there are other videos that have him lecturing on similar topics. The vast majority of these lectures are delivered in Russian, and therefore the translators, Mark Saul and Dmitry Fuchs, should be warmly thanked for their labor of love. Saul goes on to describe their contribution:

These lecture notes were gathered in haste from the field, and we have corrected numerous misprints and small errors in notation. We have given several extensions — in Arnold’s own style — to the work, in “editors’ notes”. At the same time, we have striven to deliver intact the style of the work.

The “style” being referred to is truly remarkable, especially to someone who has not experienced an Arnold lecture. There is a sense of great urgency that comes across in the writing, which is miraculously preserved in spite of translation/transcription. One can at times sense the rush that accompanies mathematics being discovered. The reader is offered a view inside the machinations of Arnold’s creative side, one that is possibly tired of giving rigorous (mortified?) proofs in the traditional style, yet one that is hungry for mathematical discovery. Arnold appears always grasping for new theorems and conjectures, which are often based on searching for patterns and geometric meaning in swathes of numerical or arithmetic data. To quote Saul again:

But it is in the chase, in the experimental “phase” of the process of doing mathematics, that Arnold here seems to take the most joy, and offers this joy to a new generation. Mathematical mainstream culture, in which one burns one’s scrap work, discourages this. Few mathematicians — indeed few scientists in any field — open their minds so completely as he has to their students.

All this suggests, to my mind at least, that an entire oeuvre of analogous expository literature is waiting to be deemed worthy of translation.

This is a wonderful book, but I may be allowed a minor quibble or two. The description of the text states that

the book is intended for a wide range of mathematicians, from high school students interested in exploring unusual areas of mathematics on their own, to college and graduate students, to researchers interested in gaining a new, somewhat nontraditional perspective on doing mathematics.

I found the invitation to high school students to be somewhat optimistic, though I would be pleasantly surprised to find a high school student (or teacher) who is able to follow Arnold’s romps. A similar dissonance could be discerned in the preface: “Arnold’s mind leaps from peak to peak, connecting disparate areas of mathematics, all (or most) accessible to the student with an advanced high school education.” This sounded a bit like false advertising. Throughout the lectures Arnold is continually exhorting his audience to prove his “theorems” and to try find mathematical justifications for the patterns that he has discovered. Yes, a few advances have since been made. Though not by high-schoolers nor graduate students, but by senior mathematicians who were sufficiently provoked or piqued by Arnold’s lectures.

Arnold’s style, and the mythology that surrounded him, affords more than a fair share of idiosyncrasy, arrogance and polemics that are at best amusing, at worst grating. The prospective reader can get a taste by accessing the first section of the first lecture made available by the publisher. After reading this excerpt the reader may wonder whether this book really belongs to the MSRI Mathematical Circles Library series. In this reviewer’s opinion, starting the book with mention of Riemann surfaces, Betti numbers, Harnack’s inequality, signatures of intersection forms, etc., does little to endear a high school student and in fact, may do quite the reverse. Therefore, though I do believe these lectures are valuable, their inclusion to the MCL seemed incongruous. They might have been better suited to the AMS Student Mathematical Library, or the MAA Spectrum or Dolciani Mathematical Expositions series; or perhaps even in a new series dedicated to translating Russian texts aimed at exposing mathematics to young audiences. It appears that certain collections of such texts, often aimed at the middle-to-high school level, used to be produced a few decades ago, e.g., the Popular Lectures in Mathematics published by the University of Chicago.

If I were to have edited this book, I would not have tampered with the content of the lectures, but I would have changed their order of appearance. The first lecture makes far too bombastic an entry, and would more profitably serve as a coda. The problem(s) discussed in the remaining three lectures seemed better suited to hook the attention of youngsters. For instance, the problem of computing Frobenius numbers (Lecture 4) is completely accessible to a student with the most basic algebra and geometry skills, and leads very quickly to unknown and interesting territory. Similarly, the material covered in Lecture 2 that starts with the study of finite binary sequences and their differences, or the statistical analysis of permutations via the growth of Young diagrams in Lecture 3 seemed more accessible entry points.

I repeat, however, that these are but minor quibbles. The mathematics described in Experimental Mathematics is beautiful, and Arnold’s inimitable style vividly imparts the complementary flavors of mathematical play and discovery. I strongly recommend these lectures as much for their form (which at parts will doubtless frustrate and provoke certain readers) as for their content. There is much in this slim volume to be enjoyed by research mathematicians, as well as keen graduate students and precocious undergraduates. We are grateful to Arnold’s proselytizers, the translators, to provide us with a glimpse in to the late style of a celebrated 20th century mathematical mind.

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Tushar Das is an Assistant Professor of Mathematics at the University of Wisconsin–La Crosse.

Lecture 1. The Statistics of Topology and Algebra 3
§1. Hilbert’s Sixteenth Problem 4
§2. The Statistics of Smooth Functions 20
§3. Statistics and the Topology of Periodic Functions and
Trigonometric Polynomials 35
§4. Algebraic Geometry of Trigonometric Polynomials 46
Editor’s notes 55

Lecture 2. Combinatorial Complexity and Randomness 59
§1. Binary Sequences 60
§2. Graph of the Operation of Taking Differences 64
§3. Logarithmic Functions and Their Complexity 69
§4. Complexity and Randomness of Tables of Galois Fields 74
Editor’s notes 79

Lecture 3. Random Permutations and Young Diagrams of Their Cycles 83
§1. Statistics of Young Diagrams of Permutations of Small Numbers of Objects 85
§2. Experimentation with Random Permutations of Larger Numbers of Elements 92
§3. Random Permutations of $p^2$ Elements Generated by Galois Fields 96
§4. Statistics of Cycles of Fibonacci Automorphisms 97
Editor’s notes 106

Lecture 4. The Geometry of Frobenius Numbers of Additive Semigroups 111
§1. Sylvester's Theorem and the Frobenius Numbers 112
§2. Trees Blocked by Others in a Forest 115
§3. The Geometry of Numbers 117
§4. Upper Bound Estimate of the Frobenius Number 121
§5. Average Values of the Frobenius Numbers 132
§6. Proof of Sylvester's Theorem 135
§7. The Geometry of Continued Fractions of Frobenius Numbers 137
§8. The Distribution of Points of an Additive Semigroup on the Segment Preceding the Frobenius Number 148 Editor’s notes 154

Bibliography 157