INFINITESIMAL: How a Dangerous Mathematical Theory Shaped the Modern World, by Amir Alexander. 352 pages, illustrated. Scientific American/Farrar Straus, $27
That there was a dispute at the end of the 17th century between Newton and Liebnitz over priority for calculus is a well-known episode in the history of science, but the strongest word used to describe it is usually “unseemly,” and nothing depended on the outcome.
In “Infinitesimal,” UCLA professor Amir Alexander goes to the beginning of the century when the dispute over the same thing, then called the “method of exhaustion” or “infinitesimals,” was a matter of life and death, both for mathematicians and, so the disputants thought, for the life of their societies. They were not wrong.
As always seems to be the case wherever you look at that wonderful, horrible century at the divide between the Middle Ages and modern times, the characters are all fuller, stranger and more lively than anything we can produce in the 21at century.
Although Alexander cautions several times against applying today’s standards to men whose experience and understanding were so different, Thomas Hobbes had many characteristics that would have qualified him as a villain in any century. And yet, the philosopher was probably the only out-and-out atheist in the cast, which makes him sound modern; and he was a secularist in government at a time when few could even conceive of that.
The dispute over math was in many ways religious, weird as that sounds to us, and it is a measure of the strong feelings involved that Alexander says the Catholic Jesuits wold have burned Hobbes if they could have caught him, while the Anglicans, who had him, called for burning him but were politically unable to.
No wonder Hobbes thought that two of the leading characteristics of life were its brutality and its brevity.
We do not usually think of Hobbes as a mathematician, and his ideas (like squaring the circle) were disdained at the time, but he was at one point considered so highly that he was named math tutor to Charles Stuart, a man who, to say the least of it, grew up to be one of the greatest calculators of his time.
The war on infinitesimals was fought twice, with opposite outcomes.
The first half of “Infinitesimal” recounts the rise and triumph of the Jesuits as they sought to impose doctrinaire, inflexible and -- as we now know -- insanely wrong ideas on Europe. They succeeded in Italy, turning it from the scientific vanguard to a rural backwater. People really did get murdered for math in this war.
Along the way the Jesuits created what Alexander calls the first and still the only worldwide educational system. It is regrettable that most of what they taught was and still is nonsense.
What the Jesuits wanted, after generations of savage religious warfare, was stability, order and religion. If that sounds very modern, it is.
Gunpowder saved northern Europe from the Jesuits. In 1632, the same year Galileo was condemned (and might well have been burned had he not been a celebrity), Gustavus Adolphus defeated the Catholic army at Breitenfeld, and that was that as far as imposing Catholicism by force went, except around the Mediterranean and in Latin America.
The war over infinitesimals began earlier in Italy but the crisis overlapped in the two theaters, coming to its climaxes around 1660. In England, an odd character named John Wallis was the key disputant. His proofs seemed sloppy to, for example, Fermat; but they worked and appeared to ally mathematics with experimentalism and against rigid authoritarianism and Hobbes.
And that proved to be the winning approach, in economic, power political and philosophic arenas.
Alexander spends only a few brief paragraphs in sketching the consequences, which were, of course, dependent on many other things besides an experimental and inquisitive approach to math.
Advanced math is not sufficient to become modern, but it does seem to be a requirement.
Alexander does not speculate on or extend the point, but that does not have to stop his readers. It is a fact, unmentioned by Alexander, that Islam deliberately took itself out of the running for math supremacy in the 13th century and China about a century later. Of the other centers where mathematics was so sophisticated that it is possible to imagine one of them pushing forward into modernity (India, Central America), all were conquered by backward religions and so stultified.