Thursday, December 29, 2016

Amir Alexander - Infinitesimal - How A Dangerous Mathematical Theory Changed The World (Oneworld, 2015) ****

Fascinating! In 17th Century Italy, some mathematicians, first the monk Cavalieri, later, Torricelli and Angeli, came to the conclusion that in order to make correct calculations, "a line should be considered as composed of distinct and limitlessly tiny parts". This would later become the basis for calculus. The concept itself was strongly opposed by the Jesuits, the christian order of the educated and educators themselves, who could not accept this reasoning for theological reasons. They could not accept that their god would have created a universe where ambiguity and lack of precision played a role. The mathematicians themselves, had of course no theological or religious intention, but discovered that their use of "infinitesimals" was the only way to calculate slopes and volumes. What ensued was a real battle to destroy any thought and use of this new mathematics, because they endangered the world view of order as organised by the creator himself.

In the Jesuit view, "divine mathematics, universal and perfectly rational, orders and arranges the physical world to the best possible effect".

"For the Jesuits, the purpose of mathematics was to establish the world as a fixed and externally unchanging place, in which order and hierarchy could never be challenged. That is why each item in the world must be carefully and rationally constructed, and why any hint of contradictions and paradoxes could never be allowed to stand. It was a 'top-down' mathematics, whose purpose was to bring rationality and order in an otherwise chaotic world. For Cavalieri and his fellow indivisiblists, it was the exact reverse: mathematics began with a material intuition of the world, that plane figures were made up of lines and volumes of planes, just as cloth was woven of thread and a book compiled of pages. One does not need to rationally construct such figures, because we all know they already exist in the world. All that is needed, as Cavalieri says, is to assume and imagine them, and then proceed to investigate the inner structure. Ultimately, he continues, nothing contractory can be deduced, because the fact that the figures exist guarantee that they are internally consistent".

The attack by the Jesuits was fierce. Excommunication, joblosses for university mathematicians, angry letters and public denunciation, the abolishment of the monastic order who welcomed the mathematicians ... every trick could be used to bring these mathematicians back to order and old-school Eucledian geometry. And even if they managed to destroy the Italian world leadership in mathematical thinking, the concept of the new mathematics resonated with mathematicians in northern Europe, with again a comparable existential philosophical battle between John Wallis who expanded on the new mathematics and Thomas Hobbes who fiercely opposed them, to become gradually accepted, and part of every secondary school curriculum.

Again: fascinating! Amir Alexander manages to write a book on the history of mathematics that reads like a suspense novel. He goes into sufficient detail in the lives and contexts of each of the various 'dramatis personae' to bring them to life, even illustrating their personal hesitations and uncertainties from a load of well-documented material like personal letters. But it is even stronger that he shows how this - to lay people insignificant - mathematical innovation created a seismic shift in the way the worldview changed, how even well before the enlightment, and at the same time of Galileo, another revolution took place to bring science and factual thinking to a higher level than church doctrine to understand reality.