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The Adventure of Numbers

American Mathematical Society
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The Adventure of Numbers is a delightful panoramic story that traces the origin of the concept of number from pre-history, through antiquity, the Middle Ages, the Renaissance, and down to modern times. As such, it is a book that discusses a lot of history; however, it is not so much a book about the history of mathematics, but rather more of a mathematical narrative that uses history to enliven its discussion of the evolution of mathematical concepts.

Chapter 1, “Hands, Sticks, and Stones” begins by centering on the origins of counting (e.g., base 10 likely arising from our hands, keeping track of counts with notches on sticks, and using pebbles in separate piles to compute). This chapter summarizes the state of numbers before the advent of writing.

Chapter 2, “By the Waters of Babylon” discusses the evolution of writing down numbers, by concentrating on the example of the cuneiform number system. Thus, Godefroy can expose each of the intellectual hurdles that have to be overcome on the way to devising a number system; in particular, he focuses on the issues of how to create a reasonable number of symbols, and how to deal with ambiguities. The author uses the example of how the Babylonians almost come up with a fully positional number system (they define a symbol that can be used as a zero inside a number, but they never end up using it at the end of a number to, say, distinguish 1 from 60 or 3600), as an example of how to trace the evolution of a mathematical notion from “developing a tool, to defining a notation, and isolating a concept.”

As an illustration of this narrative’s use of history to enrich and contextualize the discussion of mathematical ideas, the author comes to a point where he presents evidence that the Babylonians were able to compute the square root of 2 to 6 decimal places, and concludes that they must have had a way to compute them; then he says it was probably Hero’s algorithm (a precursor to Newton’s method) and then uses that essentially as an excuse presents the reader with a discussion of how to use that algorithm to perform the approximation.

Chapter 3, “Let None but Geometers Enter Here” discusses the advances of the Greeks. Here, the concept of “theorem” arises, illustrated by the Pythagorean Theorem, and Godefroy focuses on the resulting paradigm shift caused by the discovery of the irrationality of the square root of 2.  The author discusses Eudoxus’s theory of magnitudes, noting how it raises the specter of the “potentially infinite.” He thus also raises the question of what it actually means for two quantities to be “equal”, and in doing so, foreshadows its relationship to Dedekind’s theory of cuts and Robinson’s nonstandard real line.

Chapter 4, “Algebra and Algorithms” emphasizes the contributions of the Arabs in the middle ages, noting the role of Baghdad as a center of learning and scholarship, and highlighting that many scientific and mathematical words (e.g., algebra and algorithms) are derived from the Arabic.

Chapter 5, “A New World” studies the solutions of the cubic and quartic equations by radicals, told through the tales of dal Ferro, Tartaglia, and Cardano, and then leads the reader to experience the inevitable appearance of imaginary numbers.

Chapter 6, “Eppur, Si Muove” takes its name from Galileo’s rejoinder (“And yet, it moves”) to the Church, who had forced Galileo to recant his heliocentric theories. This chapter focuses on the rise of calculus, the concept of limit, infinite series, and the debate over infinitesimals, highlighting Newton's, Leibniz', and Euler' achievements. Euler in particular continues to instruct us: some of his manipulations involving divergent series, e.g. 1 + 2 + 4 + 8 + ... =  –1, obtained by taking plugging x = 2 intothe Taylorseries for 1/(1-x), are valid but only in the field of 2-adics.

Chapter 7, “The Century of Revolutions” starts with the French Revolution and focuses on the revolutions of 19th century mathematics, including Cauchy’s and Dedekind’s work on making sense of the real numbers (through Cauchy sequences and Dedekind cuts), as well as extensions of the concept of number (e.g., Hamilton’s quaternions, ideals, p-adic numbers).

Chapter 8, “From the Paradise That Cantor Has Created For Us...” discusses the revolutions in the concept of the infinite, beginning with how Cantor proved that a trigonometric series that converges to zero for all x is in fact identically zero. Then Cantor was able to weaken this convergence hypothesis, but in characterizing the set of points where the series did not have to converge to zero, he stumbled upon set theory.  Thus Cantor developed the concepts of cardinals, and ordinals, and in particular he gave his famous diagonalization argument that shows that there are strictly more real numbers than integers, hence leading to the concepts of hierarchies of infinities. Then we move to the resultant paradoxes (e.g., whether the set of all natural numbers that cannot be defined in fewer than fourteen words contains “the smallest natural number that cannot be defined in fewer than fourteen words”), and Hilbert’s ambitious program to axiomatize arithmetic.

Chapter 9, “The Present Perplexity” discusses how Gödel’s Incompleteness Theorem dashed the ambitions of Hilbert’s program by implying the impossibility of using finite processes to prove the consistency of arithmetic. Godefroy then goes on to discuss the result of Robinson-Matijasevic that establishes that there exist Diophantine equations for which is no computer program that can decide whether they have a solution or not. In an interesting twist, the Fibonacci numbers play a key role in Matijasevic’s proof.

Chapter 10, “And Now?” is a brief survey of some additional topics including the fast Fourier transform, Conway’s construction of numbers from combinatorial games, and Connes’s infinitesimals. (More information on the relationship between chess and combinatorial game theory can be found in Elkies' article "On Numbers and Endgames" at and the references therein). The book ends with appendices on number bases, the Fibonacci sequence, polynomials, quaternions, and set theory.

It seems that a bright undergraduate mathematics student who wants to read this book would be well served to find an experienced mentor with whom to discuss some of the many interesting concepts that are mentioned only in passing. For instance, the very interesting discussion of the solution of the cubic and quartic equations culminates in a brief invocation of an argument involving conic pencils, which may not be familiar to most American mathematics undergraduates. Furthermore, in some places, Godefroy’s narrative takes a whirlwind tour, mentioning flood of related concepts in mere passing (e.g., p. 79, which mentions Grassman’s exterior calculus, the octonions, the Platonic solids, the classification of finite simple groups and the Monster, and alludes to Hopf’s theorem on the parallelizability of the spheres). A professional mathematician is likely to be pleased, but a student unfamiliar with the concepts may be a bit mystified. One recommended source for further reading is the excellent compilation Numbers by Ebbinghaus et. al., which includes a proof of Hopf's theorem.

The English translation from the original French is overall excellent and the occasional translator’s note is used where necessary to explain certain puns and jokes. Overall, we recommend this book to those at the advanced undergraduate level and above who would like some inspirational reading about the unity and evolution of mathematical thought. We end with the author's wishes at the end of the book, in which he hopes that the reader "was able to enjoy this book, as you might like a poem without analyzing all the details".

Francis Fung is senior software engineer/research scientist at Eduworks in Corvallis, OR. His mathematical interests include representation theory, topology and artificial intelligence. Maria G. Fung is an associate professor of mathematics at Western Oregon University. Her interests are the mathematical preparation of K-8 teachers and geometry.

Date Received: 
Wednesday, September 1, 2004
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Gilles Godefroy
Mathematical World
Publication Date: 
Francis Y. C. Fung and Maria G. Fung
* Hands, sticks, and stones
* By the waters of Babylon
* Let none but geometers enter here
* Algebra and algorithms
* A new world
* "Eppur, si muove!"
* The century of revolutions
* "From the paradise that Cantor has created for us"...
* The present perplexity
* And now?
* Number bases
* The Fibonacci sequence
* Polynomials
* Quaternions
* Axioms of set theory and arithmetic
* Glossary
* Bibliography
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Monday, February 11, 2008