By Mike Holderness Thomas Hobbes shared the Jesuits’ objection to infinitesimals (Image: Getty Images/DeAgostini) At stake in the fierce 17th-century debate over a mathematical concept was nothing less than our modern world, says Amir Alexander in Infinitesimal DID you know that every time you use your cellphone, you strike a blow for republicanism against both monarchy and a fixed Divine Order? Historian Amir Alexander sets out why this is so in his book Infinitesimal, through two tales of struggles against the fundamental tool that makes almost all modern science and technology possible. This tool is infinitesimals, small slices of continuous measurements that are the basis of calculus. They allow us to discover the areas under curves and to model systems and control them – offering immense power to shape many facets of our modern world, including cellphone signal processing. Infinitesimals offer immense power to shape many facets of our modern world The first of Alexander’s tales is of the Jesuits, a Catholic order of priests in personal allegiance to the Pope, founded to stamp out Protestant rebellion. During the early 17th century they strove to suppress the ideas of Bonaventura Cavalieri, a member of the rival order the Jesuats. His use of infinitesimals prefigured the fully fledged calculus of Gottfried Leibniz and Isaac Newton. To the Jesuits, only a theoretical system built from first principles along logical lines laid down by the ancient Greek mathematician Euclid could provide freedom from paradox and an absolute, approved truth. Pesky new mathematicians like Cavalieri were blatantly assuming the existence of triangles, spheres and so on, then researching them as objects. That way lay free-thinking and challenges to the authority of the Pope. The 1633 prosecution of Galileo by the Inquisition for insisting that Earth goes round the sun has been better recorded in film and on stage. Alexander connects these two fights – and a 1651 list of 65 doctrines banned in the Jesuits’ colleges, including the atomic theory of matter – to the political struggles of the Jesuits within the Catholic Church. His second story is of the dispute between the philosopher Thomas Hobbes and John Wallis, a minister who had been appointed Savilian Professor of Astronomy at the University of Oxford in 1649, mostly because he was a sound Protestant and therefore politically acceptable at the time. Hobbes gained fame for his book Leviathan, in which he argued that the only alternative to a state of nature in which life was “solitary, poor, nasty, brutish and short” was absolute monarchy. Though suspected of atheism, Hobbes shared the Jesuits’ fear of infinitesimals, defending kingly rather than papal authority. His opponent Wallis took “experimental mathematics” to extremes, using methods of proof that were patently barking, depending on division by infinity, which is meaningless. Hobbes, meanwhile, died convinced that he had constructed a square with the same area as a circle using only Euclidian methods: he couldn’t accept that the nature of the number pi makes this impossible. Those against infinitesimals had ammunition, however. Remember how you puzzled at your teacher’s insistence that as the number of slices you cut of an object tends to infinity, the sum of the slices approaches the volume of the object. An infinite number of slices with a thickness surely forms an infinitely tall stack? But if they have zero thickness, isn’t the total zero? And Euclid’s approach also has appeal, well expressed in the jibe that “the proposal is all very well in practice, but it’ll never work in theory”. Infinitesimal is a gripping and thorough history of the ultimate triumph of the mathematical tool. But it is a shame that Alexander adheres to another doctrinal ban: in Western history, it seems, no mention can be made of Islam as a rival to the Catholic Church. What, for example, was the response to ibn al-Haytham’s use of proto-infinitesimals in Cairo and Iraq 500 years earlier? If you are fascinated by numbers, Infinitesimal will inspire you to dig deeper into the implications of the philosophy of mathematics and of knowledge. Infinitesimal: How a dangerous mathematical theory shaped the modern world Amir Alexander Farrar, Straus and Giroux This article appeared in print under the headline “The smallest subversive” More on these topics: