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A Deeply Divisive Debate about Reality
on August 30, 2014
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. The core of the disagreement was over the nature of the continuum, which was based on Euclid’s definition of a line as an infinite number of points. This intellectual argument implicitly links back to reality: is matter made of distinct atoms with empty space between them or are there no gaps between continuous matter? Although the model of the reactionaries was always Euclid's geometry, they never recognized they were only dealing with unreal definitions, as they faked out their arguments with appeals to 'real' lines etc. As such, they vigorously rejected the new concept of "indivisibles" (or "infinitesimals", the roots of calculus) and all ideas that were grounded in empirical studies of reality (like physics and the atomic hypothesis). Failure to admit debate about reality led Italy back into the Dark Ages while Northern Europe set off on the course of modernism.
As other reviewers have noted, this book would have benefitted quite a bit by including the story of the rivalry between Leibniz and Newton, who are usually credited with the invention of the calculus. As this book shows, this 17th Century rivalry had much older roots. Indeed, the book could also have been improved by establishing this acrimonious debate back in Classical Greece, where the atomic model, first proposed by Democritus, was immediately seen as an atheistic proposal that threatened traditional religion. The modern reader might assume that science has now firmly voted for the atomic model but the extensive use of the calculus embedded in Quantum Physics has preserved the conceptual features of the continuum advocates, so that we are now faced with the paradox of waves and particles. None-the-less, even readers with minimal competence in mathematics will enjoy discovering how this tiny idea of the infinitely small punctured an ancient dream: that the world is a perfectly rational place that is governed by strict mathematical rules.