## "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.

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