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Aristotle and Mathematics: Aporetic Method in Cosmology and by John J. Cleary

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 procedure via which he offers with philosophical questions about the principles of arithmetic. the 1st chapters think of Plato's mathematical cosmology within the gentle of Aristotle's severe contrast among physics and arithmetic. next chapters study 3 simple aporiae approximately mathematical gadgets which Aristotle himself develops in his technological know-how of first philosophy. What emerges from this dialectical inquiry is a distinct belief of substance and of order within the universe, which provides precedence to physics over arithmetic because the cosmological technological know-how. inside of this diverse world-view, we will be able to greater comprehend what we now name Aristotle's philosophy of arithmetic.

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8 On the purely literary level, this allegory shows the attraction of images for the human soul (one remembers this part of the Republic, while forgetting the arguments), along with the need for hypotheses to focus our attention on the relevant information contained in the image. 9 At 516A-B there is a vague suggestion of a possible correspondence between the lower section of the intelligible realm and the shadows and reflections in water which the prisoner must first observe on coming into the daylight, before he is able to look at the things themselves.

I show that such a solution implicitly depends on his logical theory of subtraction, which legitimates the consideration of sensible things qua numbers for arithmetic or qua continuous quantities for geometry. Contrary to traditional interpretations, however, I argue that Aristotle's position does not involve an epistemological theory of abstraction and that mathematical entities are not 'abstract objects' in the modern sense of entities that exist only in the mind. 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.

THE ACADEMIC BACKGROUND 13 its real purpose of obtaining knowledge of Being and, hence, its true function as a propaideutic discipline for dialectic. 23 Historically, it seems that Plato's proposed reform of the mathematical sciences led to their reorganization and axiomatization. Perhaps inspired by the work of Theaetetus, some mathematicians within the Academy compiled Elements that were later incorporated into the work of Euclid. e. that it relied heavily on constructions and on appeals to perception.

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