By John J. Cleary

John Cleary the following explores the position which the mathematical sciences play in Aristotle's philosophical concept, in particular in his cosmology, metaphysics, and epistemology. He additionally thematizes the aporetic process through which he bargains with philosophical questions about the principles of arithmetic. the 1st chapters give some thought to Plato's mathematical cosmology within the gentle of Aristotle's severe contrast among physics and arithmetic. next chapters study 3 easy aporiae approximately mathematical items which Aristotle himself develops in his technology of first philosophy. What emerges from this dialectical inquiry is a special perception of substance and of order within the universe, which supplies precedence to physics over arithmetic because the cosmological technological know-how. inside of this varied world-view, we will larger comprehend what we now name Aristotle's philosophy of arithmetic.

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Everything hinges on the answer to this question because if the unit "in and by itself" is adequately perceived by the senses then it would not serve as a track 14 towards intelligible reality. e. the many) then the soul would be undecided and would be forced (avayKa~ot1:o) to wonder (anopE'iv) and inquire (~rrrE'iv) by stirring up intelligence to ask what is the unit itself. e. calculation or arithmetic) is a discipline that 'converts' the soul to reality. Having decided that arithmetic belongs among the studies sought for the 'conversion' of the soul, Socrates proposes (525C) that those guardians who are to rule should study arithmetic until they are able to contemplate the nature of numbers in themselves.

In an attempt to place these views within a broader cosmological context, my sixth chapter considers the reasons why Aristotle rejects a mathematical ordering of the universe in favor of a hierarchy with the Prime Mover at its apex. 6-9 through a dialectical inquiry into the view that Forms are identical with certain non-combinable numbers. This view is systematically refuted when Aristotle shows that it conflicts with our intuitions about numbers INTRODUCTION XXXV and that it fails to provide any satisfactory account of their causal role in respect to sensible things.

29 This fits with the reputedly divine origin of the Delian problem, which gave some historical impetus to research in stereometry; cf. Heath (1921) & van der Waerden (1961). THE ACADEMIC BACKGROUND 15 purpose and subject-matter of astronomy. Mindful of Socrates' previous reproaches, Glaucon hastens to assure him that this discipline obviously forces (ava)'Ka~n) the soul to look upwards, by contrast with other disciplines. On the contrary, says Socrates, those who try to lead us up to philosophy make us look downwards on account of the way they practice astronomy.

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