For him, traditional Euclidean geometry – which, contrary to induction, used deduction to achieve any result from a few basic statements – was the highest expression of an ordered philosophical system and a model for a perfect state. Here, the leading figure of Thomas Hobbes took the stage in the fight against the indivisibles and the inductive method. In that context, the new ideas represented by infinitesimals were not only condemned by the Anglican Church but also opposed by political powers. The second part of the book moves the reader to 17th century England, just after the English Civil War in the years of Cromwell’s republic and the Restoration. The fierce opposition of the Jesuit mathematicians led Galileo and the “infinitesimalists” to damnation, with irreparable consequences for the ancient tradition of Italian mathematicians. To them, the traditional mathematical approach was a solid basis for the absolute truth represented by the Catholic faith and the authority of the Pope. On the opposite side, the religious congregation of the Jesuits used these same mathematical developments in its fight against heresy and the Reformation. This established the basis of inductive reasoning, which makes broad generalisations from specific observations, and led to a new science founded on experience. Here, the prominent figure of Galileo Galilei – together with Evangelista Torricelli, Bonaventura Cavalieri and others – was at the forefront of the new mathematical approach involving the concept of infinitesimals. The first part leads the reader to late 16th century Italy, where the flourishing and creative ideas of the Renaissance had given birth to a prolific number of mathematicians and scientists. This book is divided into two parts, each devoted to a particular geographical area and period in which this battle took place. But even more striking is that the new way of thinking built around the concept of infinitesimals crossed the borders of science and strongly influenced society, up to the point that mathematics became the main focus of the struggle between the old and new orders. The consequences for mathematics and physics were enormous, giving rise to unprecedented scientific progress that continued for the following decades and centuries. The idea of indivisible quantities and their use in geometry and arithmetic, which had already been developed by ancient Greek mathematicians, underwent its own renaissance 500 years ago, at the same time as Martin Luther launched the Reformation. The former were represented by immutable divine order and the principle of authority, the latter by social change and experimentation. Set in the late 16th and early 17th centuries, it describes how the concept of infinitesimals – a quantity that is explicitly nonzero and yet smaller than any measurable quantity – took a central role in the debate between ancient medieval ideas and the new ideas arising from the Renaissance. Lying midway between the history and the philosophy of science, this book illuminates a fascinating period in European history during which mathematics clashed with common thought and religion. One World Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

Arithmetic of infinitesimals