For a decade now, John Allen Paulos has been on a self-appointed mission to present the human face of mathematics. His motto would seem to be that mathematics is too important to be left to the mathematicians; too important, too interesting, too much fun, and too much like literature, history, art, or music to be locked up in a cupboard of pure logic and formal techniques. It’s not that logic and technique aren’t important, the subject would collapse without them, but that mathematicians are rigorously schooled to omit everything else – intuitions, ideas, stories, metaphors, connections, puzzles – from their descriptions of what they do.
Paulos is hardly alone. There is a rich, varied and much-worked tradition of popularizing and explicating the subject for a general, non-specialist audience – from Lewis Carroll to Martin Gardner, with many in between – that has produced any number of excellent and memorable books. What distinguishes him perhaps is his understanding of fear – the abject fear of symbols that turn the minds of otherwise clever and articulate people into a kind of inert and hopeless jelly. In his previous book, Innumeracy, Paulos successfully created a niche in the large and potentially lucrative market for mathematical enlightenment by deftly and with great skill catering to this aspect of mathematics phobia.
Beyond Numeracy is a continuation. But – perhaps because he thinks his readers are now ready for them or perhaps because there is a limit even to Paulos’s ability to render mathematical ideas purely in words – the symbol count is higher, and the topics covered more technical. Organized alphabetically as a dictionary of short, self-contained entries – from “A Mathematical Accent” to “Zeno and Motion” – the book is full of entertaining facts, educative snippets, stories, explications and anecdotes culled from across the mathematical world.
“O” introduces the acronym Oulipo, the “ouvroir de Littérature Potentielle”, a group started in Paris in the 1960s dedicated to exploring the insertion of mathematical techniques into literature. A founding member, Raymond Queneau, claims to have composed ten trillion sonnets by writing ten sonnets which are such that any of the ten first lines can go with any of the second ten lines…up to the fourteenth line, making 10¹⁴ (1 followed by 14 zeroes) sonnets in all. Can it really be true that each possibility created by Queneau is an intelligible sonnet? One would have to read them all to know for sure – and that would take something like the age of the universe. “H” gives us “Human Consciousness, its Fractal Nature”. This kicks off with the suggestion that human consciousness is a higher-dimensional fractal, and the entry, intended to elaborate the suggestion, takes the form of an amusingly written review of a Borges-like imaginary book, a sort of computerized hypertext, which can be read backwards, forwards and internally through its footnotes (as it were) and which is somehow the same at all levels of magnification. The review, better than the somewhat silly suggestion it serves, shows Paulos at his lightest, most inventive and most diverting.
Almost all the entries are a good deal more straightforward and earthbound that the two I have cited. Most are good, sometimes very good two or three-page summaries and appreciations of specific mathematical concepts or fields such as Topology, Golden Rectangles, Non-Euclidean Geometry, Complexity of programs. Others, such as Arabic Numerals, Partial Orderings and Calculus are merely serviceable and not very inspired renderings of their titles. And some, like that on Notation (a fundamental aspect of mathematics and surely of interest to the self-confessed symbol avoiders Paulos has focused upon), are disappointingly thin.
But these are the opinions of a mathematician reviewer with his own ideas about the subject. And though I imagine Paulos wants his treatment of mathematics to be acceptable to professionals, he is not writing for mathematicians. There is now, more than at any other time it seems (no doubt related to the advent of the computer), a great and unsatisfied hunger to master mathematical abstraction. For those so moved there is well-prepared food on offer here. Beyond Numeracy will undoubtedly entertain and educate many people and open up for them the mysterious and closed book – the book of the universe, as Galileo had it – of mathematics.
TLS March 20, 1992