"Jak Arystoteles z Euklidesem...": dowody matematyczne w filozofii oksfordzkiej XIV wieku

Abstract

In his writings, Aristotle forbade taking advantage of mathematical theorems and arguments within the realm of physics many times. The majority of medieval philosophers honoured this restriction. However, one can find strictly geometrical arguments in the works of the fourteenth century Oxford thinkers. A few English philosophers assumed then that all material substances are composed of indivisibles. The theories of „atomists” contradicted commonly accepted opinion of Aristotle that every continuum is divisible into divisible parts, that is. infinitely divisible. In order to prove „atomism” false some Oxford philosophers applied Euclidean arguments and theorems in their works. Their line of reasoning was as follows: if we accepted that any continuum was composed of indivisibles, we should admit that certain theorems of geometry are incorrect. This is impossible, therefore „atomism” must be rejected. Since both „atomists” and the philosophers who used Euclidean theorems against them found the above consequence sound, it means that they all recognized a strict equivalence between theorems of geometry and physical world. In most cases, medieval philosophers used geometrical arguments in order to support the traditional, Aristotelian view. Only one of early fourteenth century Oxford philosophers, Richard Kilvington contrasted Euclidean geometry and Aristotelian opinions about continuum. Yet Kilvington, in his question Utrum continuum sit divisibile in infinitum intended neither to refute, nor to approve „atomism”. Rather he wanted to eliminate all the contradictions between Aristotle’s and Euclid’s theories; and in the above-mentioned question mathematics plays a role similar to logic. This is an intellectual heritage of Ockham, whose reduction of metaphysics caused physics and other sciences to became speculative sciences, where a priori argumentations are recognized as better than any other way of describing the world. Since then, consistence and logical coherence of arguments were found as the criteria of correctness of conclusions. Euclidean geometry, especially the theory of proportions from book V of „Elements” fulfilled those criteria perfectly, therefore Ockham’s followers started to use mathematics as a kind of formal language of philosophical argument. Employing consequently the Euclidean theory of proportions, Richard Kilvington determined new „rules of motion”, more proper — in his opinion — than the Aristotelian ones. This way Kilvington and his contemporary, Thomas Bradwardine, became this way the founders of the so called Oxford Calculators school — the group of the fourteenth century English philosophers, who applied Euclidean theory of proportions to almost all disciplines of Aristotelian philosophy.