Abstract
Abstract In Metaphysics M.2, 1077a9-14, Aristotle appears to argue against the existence of Platonic Forms on the basis of there being certain universal mathematical proofs which are about things that are ‘beyond’ the ordinary objects of mathematics and that cannot be identified with any of these. It is a very effective argument against Platonism, because it provides a counter-example to the core Platonic idea that there are Forms in order to serve as the object of scientific knowledge: the universal of which theorems of universal mathematics are proven in Greek mathematics is neither Quantity in general nor any of the specific quantities, but Quantity-of-type-x. This universal cannot be a Platonic Form, for it is dependent on the types of quantity over which the variable ranges. Since for both Plato and Aristotle the object of scientific knowledge is that F which explains why G holds, as shown in a ‘direct’ proof about an arbitrary F (they merely disagree about the ontological status of this arbitrary F, whether a Form or a particular used in so far as it is F), Plato cannot maintain that Forms must be there as objects of scientific knowledge - unless the mathematics is changed.
Highlights
Throughout his works, and in the saved fragments of De Ideis in particular, Aristotle discusses Platonist arguments for the existence of Platonic Forms, as well as arguments against these arguments and arguments against their existence
That we know why Aristotle can claim, in (4) of the argument from Metaphysics M.2, that there cannot be a Form for universal mathematics to be true of, we must turn to the more difficult question why Aristotle feels justified to draw the conclusion, in (6), that there cannot be any Form for mathematics to be true of
That it has been shown that the impossibility Aristotle refers to in (4) of the argument from Metaphysics M.2 is of a mathematical kind, and that Aristotle is entitled to his conclusion (6) that there cannot be any Form for mathematics to be true of, given the conception of proofs and explanation shared between him and the Platonist, I want to conclude this paper with some more general remarks
Summary
Throughout his works, and in the saved fragments of De Ideis in particular, Aristotle discusses Platonist arguments for the existence of Platonic Forms, as well as arguments against these arguments and arguments against their existence. Lear seems to hold that the impossibility of (4) is not of a mathematical kind, and that the Platonist, even if he knew about Euclidean magnitude, must have had peculiar reasons to deny that it could serve as the universal object and as the Form for the theorems of universal mathematics to be true of.4 Needless to say, such interpretations make for a rather weak argument on Aristotle’s part. For if it is impossible for there to be a Form for theorems belonging to universal mathematics because of reasons peculiar to Platonists of Aristotle’s days, these Platonists can resist the general conclusion Aristotle wants to draw, that the normal mathematical objects, like numbers, lines, planes and solids cannot be Forms either: they can claim that these peculiar considerations just do not hold for them. The universal which proofs in universal mathematics are true of, does not meet this requirements for being a Form: it is not separated from the types of quantity it ranges over and it is not prior in account to less general universal subsumed under it; it does not have a single nature by its own
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