" Infinitesimal-A Dangerous Mathematical Theory " - A review essay (May 2017) (original) (raw)

“Infinitesimal - A Dangerous Mathematical Theory” by Amir Alexander (Scientific American/Farrar – 2014) © A review essay by H. J. Spencer May 2017 This book is about one of the most dangerous ideas ever invented by human beings. It led to the theory that eventually destroyed the monopoly hold on the educated minds of Western Europe by planting one of the principal seeds of modern physics. Award-winning UCLA historian, Alexander has written a fascinating tale of how abstract ideas influence society. This book deserves to be read not just by scientists and engineers, whose lives are dominated by mathematics, but by any educated person who had to suffer through calculus class. This book is about one of the most dangerous ideas ever invented by human beings. It led to the theory that eventually destroyed the monopoly hold on the educated minds of Western Europe by planting one of the principal seeds of modern physics. Award-winning UCLA historian, Amir Alexander has written a fascinating tale of how abstract ideas influence society. This book deserves to be read not just by scientists and engineers, whose lives are dominated by mathematics, but by any educated person who had to suffer through calculus class. This book is much more than an esoteric history of an area of mathematics. It tracks the ancient rivalry between 'rationalists' and 'empiricists'. The dominant rationalists have always believed that human minds (at least those possessed by educated intellectuals) are capable of understanding the world purely by thought alone. The empiricists acknowledge that reality is far too complicated for humans to just guess its detailed structures. This is not simply an esoteric philosophical distinction but the difference in fundamental world-views that have deeply influenced the evolution of western civilization. In fact, rationalist intellectuals have usually looked to the logical perfection of mathematics as a justification for the preservation of religion and hierarchical social structures. In particular, the rationalists have raised the timeless, unchanging mathematical knowledge, represented by Euclidean geometry, as not just the only valid form of symbolic knowledge but as the only valid model of the logic of " proof ". In particular, this book focuses on the battle between the reactionaries (e.g. Jesuits and Hobbes), who needed a model of timeless perfection to preserve their class-based religious and social privileges and reality-driven modernists, like Galileo and Bacon, who were desirous of major changes. In the late Middle-Ages, the new order of Jesuits were the intellectual leaders of the Catholic Church and were formed to defeat the recent Reformation. They not only opposed Protestant theology but also the parallel forces of pluralism, populism and social reform. The Jesuits, like their Church itself and the ancient social structures they supported, were all organized on traditional (militaristic) hierarchical principles. In 1632, the Jesuits convened a major council in Rome and decided to ban the idea of " indivisibles " – the old idea that a line was composed of distinct and an infinity of tiny parts. They correctly anticipated that the threat of this idea to their rational view of the world, as an " orderly place, governed by a strict and unchanging set of rules. " Geometry was their best exemplar of their Catholic theology. The core of the disagreement was over the nature of the continuum, a concept that had surfaced in Ancient Greece. The reality of the idea of 'physical indivisibles' (atoms) was even being disputed by serious scientists as late as 1900. The original idea of continuity was stimulated by the apparent lack of observable 'gaps' in solids or liquid materials and our personal sense of the continuous flow of time (ideally, also endless). This idea became a key concept to many Ancient Greek thinkers; even, Aristotle made the plausible statement that: " No continuous thing is divisible into parts. " This was soon 'cast in concrete' with Euclid's major definition of a line as an infinite number of points. This became one of the core (obvious) assumptions of geometry – the basis of so much of western education. The concept of 'Continuity' became a key Principle of medieval scholastic thought, eagerly latched onto by Aquinas and other theologians in their battle with the atheistic and equally ancient idea of atoms.