While some books have obscure titles, a combination of the title and the subtitle will usually make it plain what the book is about. But I can pretty much guarantee that most readers, seeing Infinitesimal - how a dangerous mathematical theory shaped the modern world would leap to an incorrect conclusion as I did. The dangerous aspect of infinitesimals was surely going to be related in some way to calculus, but I expected it to be about the great priority debate between Newton and Leibniz, where in fact the book concentrates on the precursors to their work that would make the use of infinitesimals - quantities that are vanishingly close to zero - acceptable in mathematics.
The book is in two distinct sections. The first focuses on the history of the Jesuits, from their founding to their weighing into the mathematical debate against those who wanted to use infinitesimals in maths. For the Jesuits, everything was cut and dried, and where Aristotle's view and the geometry of Euclid had an unchanging nature that made them acceptable, the use of infinitesimals was far too redolent of change and rebellion. This was interesting, particularly in the way that the history gave background on Galileo's rise and fall seen from a different viewpoint (as he was in the ascendancy, the Jesuits were temporarily losing power, and vice versa). However, this part goes on far too long and says the same thing pretty much over and over again. This is, I can't help but feel, a fairly small book, trying to look bigger and more important than it is by being padded.
The second section I found considerably more interesting, though this was mostly as a pure history text. I was fairly ignorant about the origins of the civil war and the impact of its outcome, and Amir Alexander lays this out well. He also portrays the mental battle between philosopher Thomas Hobbes and mathematician John Wallis in a very interesting fashion. I knew, for example, that Wallis had been the first to use the lemniscate, the symbol for infinity used in calculus, but wasn't aware how much he was a self-taught mathematician who took an approach to maths that would horrify any modern maths professional, treating it more as an experimental science where induction was key, than a pure discipline where everything has to be proved.
Hobbes, I only really knew as a name, associated with that horrible frontispiece of his 'masterpiece' Leviathan, which seems to the modern eye a work of madness, envisaging a state where the monarch's word is so supreme that the people are more like automata, cells in a body or bees in a hive rather than individual, thinking humans. What I hadn't realised is that Hobbes was also an enthusiastic mathematician who believed it was possible to derive all his philosophy from geometry - and geometry alone, with none of Wallis' cheating little infinitesimals. The pair attacked each other in print for many years, though Hobbes' campaign foundered to some extent on his inability to see that geometry was not capable of everything (he regularly claimed he had worked out how to square the circle, a geometrically impossible task).
Although I enjoyed finding out more about the historical context it's perhaps unfortunate that Alexander is a historian, rather than someone with an eye to modern science, as I felt the first two sections, which effectively described the winning of the war by induction and experimentation over a view that expected mathematics to be a pure predictor of reality, would have benefited hugely from being contrasted with modern physics, where some would argue that far too much depends on starting with mathematics and predicting outcomes, rather than starting with observation and experiment. An interesting book without doubt, but not quite what it could have been.
on 2 May 2014
The story or the origins of the integral calculus in the era of the Reformation and post-reformation with the possibility to align a period of great social and political upheaval with the names and personalities of the great Italian and (later) northern European mathematicians has the potential to be an absorbing thriller for those inclined and less inclined to mathematics. Amir Alexander makes an interesting story boring by repetitive repetition. He badly needed a vigorous editor who would have cut two-thirds of the length without losing anything. The aversion of the Jesuits to the new mathematics and their adherence to Euclidian maths as a bulwark against the threatening social chaos of the protestant ethos is maybe not surprising and the lengths to which they went in maligning our early mathematical heroes (Galileo et al.) might be an analogy for the current obsession of the American administration with Al Quaeda but the story loses all its potential excitement in Infinitesimal through repetitiveness. Again, the battle between Hobbes and Wallis provides new insights to the obsessive mind of the author of Leviathan but why tell the same story again and again page after page in part 2 of Infinitesimal. And why not tell us more of the parts that other greats of the era made of all this hateful to-and fro. Finally an epilogue carrying the story of the new maths forward to Newton and Leibnittz would given some better context to the story for the modern reader.
I will definitely not recommend this book to my friends - even the maths inclined among them. Not even 1 star