How Mathematics Happened: The First 50,000 Years , Peter S. Rudman , Prometheus Books, Amherst, NY, 2007. $26.00 (314 pp.). ISBN 978-1-59102-477-4
Given the title of Peter S. Rudman’s How Mathematics Happened: The First 50,000 Years, one would expect an explanation of, well, how mathematics happened. But in the introduction, Rudman explains what he’s really writing about: “There is an essential role for judicious guessing in mathematics, and sometimes that is the best we can do…. I, or anyone else who tries to interpret the archeological record in terms of not just what was done, but also in terms of why it was so done, must rely on conjecture” (page 27). Thus his book is intentionally full of conjectures—mostly plausible ones.
Some of Rudman’s conjectures show his romantic side. One example is his explanation behind the purpose of the Old Babylonian tablet known as Plimpton 322, which is more than 3500 years old and on display at Columbia University. It is a table of a few lines of numbers that, with some interpretation by researchers, suggests the Babylonians knew what we now call Pythagorean triples, three positive integers a, b, and c, such that a2 + b2 = c2 . Rudman’s contribution is to explain how Plimpton 322 was used in the construction of Babylonian ziggurats, buildings that had the shape of frustums of square pyramids. He justifies his argument, in part, with conjectured contacts between Babylonia and ancient India, areas where altar construction relied on the Pythagorean theorem. His ziggurat conjecture makes for interesting reading, but its historical likelihood is nil.
Rudman, a retired professor of physics at the Technion—Israel Institute of Technology, also has a skeptical side that is not restricted to mathematics. In his section on “pyramidiots,” he lambastes “wacky Egyptology,” in which many amateurs have made numerological claims about the pyramids. Despite his disdain for such claims, he is not above making some of his own. He also validly criticizes Thor Heyerdahl’s diffusion theory that Egyptian pyramid-building technology influenced Mayan pyramids and Erich von Däniken’s “fabricated nonsense” of extraterrestrial contact with ancient Egyptians. Again, this makes for interesting reading, but it’s not clear what business it has in what’s billed as a book on the history of mathematics.
The subtitle of the book also needs an explanation, and Rudman supplies one: “I shall somewhat arbitrarily choose the era with unambiguous physical evidence of contemplative thinking, roughly 50,000 BCE, to define the beginning of real counting and the birth date of arithmetic” (page 53). But he qualifies his statement by saying the birth date might have been as early as a million years ago or as late as 30000 BCE. In any case, the scope of history under his consideration begins with counting and how it started independently in several regions around the world, and moves to the mathematics of ancient Egypt and Babylonia, primarily in the second and third millennia BCE. He also provides a few pages about early Greek mathematics in the first millennium BCE; more recent mathematics is not mentioned except for comparative purposes.
The book is clearly not a research text on the subject but a popularization. Throughout the text Rudman offers “fun questions” that in other books might be called exercises. It’s hard to see how a problem like “Convert 456 and 567 into hieroglyphic symbols and add them” is considered fun. Yet some of the questions are a bit fun, like the one that begins with “Zorbi, an alien from the planet Geek of the star Gamma Centuri in the Andromeda galaxy, landed in my backyard, of all places,” and ends with “Use the greedy algorithm” (page 47). Plenty of room exists to criticize the book: For instance, Rudman writes, “Women’s intuition is more hindsight than foresight, as expressed by a husband’s whimsical wish, ‘If I only knew yesterday what my wife knows today’” (pages 25 and 26). What can one say to that?
Rudman is an accomplished physicist who has used How Mathematics Happened to share his views, both romantic and skeptical, on why the ancients did mathematics in the particular and the peculiar ways they did. From my knowledge, no otherwise similar book on the history of mathematics covers the same period; several others do include most of the material Rudman treats, though they tend to cover more recent times. The value of Rudman’s book comes from his personality, which shines through every page, and from his enjoyable exposition of the subject.
