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Review - Infinite Ascent: a short history of mathematics - David Berlinski
Taking in all of mathematics in a single, accessible sweep is quite a challenge, though David Berlinski is not the first to fall for its siren lure. (Apologies for that touch of flowery language - it rubs off, as you will see later in the review.) It also seems to be difficult to avoid using the word "infinite" in the title of a maths overview, a distinction that might be better left to books dealing with infinity like my own A Brief History of Infinity. For example Ian Stewart does a reasonable job if you want to concentrate on the maths rather than the history in From Here to Infinity and though Robert and Ellen Kaplan are much too heavy on the technical stuff in The Art of the Infinite, they do manage to get across the sheer joy of mathematics for the enthusiast. Berlinski, though, opts to give us a true popular maths exploration, driven more by the people and the history than the practicalities.
On the whole, the book isn't a bad attempt to survey the significant mathematical high ground. Berlinski starts with Pythagoras and Euclid, jumps over the Arab mathematicians (I think on the theory that they did great stuff, but no one took much notice - this is unfair and, well, wrong) and picks up the story again with Descartes. From here there's a magnificent acceleration along the borders of madness (as quite a few mathematicians would also tread that fine line) as we discover calculus, complex numbers, groups, non-Euclidian geometry, sets, incompleteness and a good amount of the aspects of mathematics that make it truly fascinating, rather than just arithmetic on overdrive. Perhaps the best part of the book is the final section, summarising modern mathematical ventures, which is readable and effective.
But there is a significant problem. I don't think it's unfair to suggest that the prose in the rest of the book, perhaps in an attempt to get away from the dull reputation of mathematics, regularly veers between flowery, pretentious and totally obscure, sometimes concealing the meaning and rarely helping to make the content more comprehensible. Take this masterful sentence: "With his vein-ruined hands describing circles in the smoky air, Pythagoras has come to believe in numbers, their unearthly harmonies and strange symmetries." Leaving aside the fact that Pythagoras thought the harmony of numbers was entirely earthly (or at least universal, in the sense that he thought all creation was based on number), what does that mean? "Vein-ruined?" I'd be worried if my hands didn't have veins; they'd be somewhat dead.
It's also a bit worrying that Berlinski seems not to worry too much about checking his facts. He refers to Paul Dirac as a "French mathematician". Unfortunately this is a robustly incorrect statement. Dirac was primarily a physicist, and was British. (Admittedly his father was Swiss, but Dirac wasn't... and even so, French and Swiss are more than subtly different, something someone who lives in Paris like Berlinski ought to be aware of.) Berlinski can also be rather weak on history. He tells us how Descartes' coordinate map differed from a medieval Mappa Mundi without pointing out (without knowing?) that medieval scholars were well aware that practical maps had to use a coordinate system, as described, for instance, by Roger Bacon in his Opus Majus in 1267.
All in all, then, it's a great idea, but the writing style will not appeal to everyone and it could do with a good edit to clear up the factual side. It's worth a try - you may find it delightful. But then again...
Only in hardback
Reviewed by Brian Clegg
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Last update 16 April 2011