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

By John J. Cleary

John Cleary right here explores the position which the mathematical sciences play in Aristotle's philosophical notion, particularly in his cosmology, metaphysics, and epistemology. He additionally thematizes the aporetic strategy through which he bargains with philosophical questions on the principles of arithmetic. the 1st chapters examine Plato's mathematical cosmology within the mild of Aristotle's severe contrast among physics and arithmetic. next chapters learn 3 uncomplicated 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 perception of substance and of order within the universe, which supplies precedence to physics over arithmetic because the cosmological technology. inside of this diverse world-view, we will greater comprehend what we now name Aristotle's philosophy of arithmetic.

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4 See Richard Martin's (1985) presidential address to the Metaphysical Society. 5 Cf. 'Introduction' in Annas (1976). 6 Cf. Apostle (1952, 1978-9 & 1991). 7 Regretfully, I must fault the method of Apostle (1952) vii when he reverses Aristotle's usual order of inquiry, which involves first outlining and criticizing the views of others before proceeding to expound his own. By way of justification for his reversal, Apostle claims to be following Aristotle's scientific order of presentation, which proceeds from causes to effects.

Methodologically, this little treatise is a perfect example of Aristotle's dialectical treatment of a metaphysical problem; while on the philosophical level it deals with the specific aporia about whether or not mathematical objects are substances. With respect to his method I claim that, as distinct from the evenhanded review of opinions such as we find in Metaphysics III, we have here a partisan refutation of opponents for the purpose of clearing the way to Aristotle's own solution. 2, where he systematically refutes both possibilities for mathematical objects as independent substances either in sensible things or separate from them.

Socrates insists here (525D) that the guardians should use arithmetic purely for theoretical purposes since in that mode it actively leads the soul upwards and forces (avayKa/;Et) it to consider numbers themselves, and not as embodied in visible things. In this connection, Socrates refers to the behavior of contemporary experts in arithmetic who ridicule anyone who thinks that the unit can be divided. Thus, he says (525D-E), if you divide their unit they will multiply it, thereby rejecting any suggestion that it may have parts.

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