Infinitesimal. How a dangerous mathematical theory shaped the modern world (Q2834589)

The book under review is concerned with important developments in the analysis of indivisibles and infinitesimals. It centres on two fairly different events, namely the struggle of the Jesuits against indivisibles and the quarrel between Hobbes and Wallis; both taking place in the 17th century. The form the author has chosen for his book is closer to that of a novel than to a historically correct report. Therefore, it is very readable and written in a light style and meant for a broader audience.NEWLINENEWLINEHowever, this broader audience would have deserved an unflawed view of the mathematics and disputes involved, but some important flaws diminish the joy of reading. The author's general view is a political, sociological, and religious one. As the main reasons for the resistance of the Jesuits against the mathematics of infinitesimals the fear of undermining religious beliefs and introducing disorder in society are given. This view disregards the fact that there were notable Jesuits who worked with indivisibles themselves. Although Cavalieri was a Jesuati (not a Jesuit), the methods of indivisibles fell on fruitful ground with the Jesuits. The names of Guldin, Torricelli (who was at least educated in a Jesuit school), and de Saint-Vincent have to be mentioned here and there were others, too. Jesuits in general did not criticise indivisibles because they were in fear that social order or religious beliefs were at stake, but because they clearly saw the danger of easily arriving at wrong mathematical results. At least, indivisibles violated principles of the ancients mathematicians as in the \textit{Elements} of Euclid. Some first-class mathematicians of the 17th century therefore (and not of fear for disorder) tried hard to avoid indivisibles in their mathematics, for example Descartes [the reviewer, 3000 Jahre Analysis. Geschichte, Kulturen, Menschen. 2nd corrected edition. Berlin: Springer (2016; Zbl 1352.01004)].NEWLINENEWLINEIn the struggle between Hobbes and Wallis, the author emphasises the political dimension of infinitesimals where we have to ask ourselves whether such a strong political dimension really was in this quarrel. It is worthwhile here to read \textit{D. M. Jesseph}'s thorough and scientifically correct book [Squaring the circle. The war between Hobbes and Wallis. Chicago, IL: The University of Chicago Press (1999; Zbl 0938.01001)] to get a deeper and less biased look at the dispute.NEWLINENEWLINEBesides these flaws the book suffers from some mathematical inaccuracies. The author uses `indivisibles' and `infinitesimals' interchangeably and sometimes mixes them up, although very different things are meant. The infinite product of Wallis giving the value of \(\pi/4\) is called an infinite series. The author's description of the technique of mathematical induction is at least sloppy, if not wrong.NEWLINENEWLINEThe book shows a point of view which, by some, may be regarded as a ``modern'' point of view in the history of maths, namely emphasising the social context in which mathematics was developed. However, even in this regard the book cannot convince.