Sunday, January 25, 2015

In Defense of Plato on Mathematical Ideas: A Commentary on Aristotle's Metaphysics XIII 6-8

In The Metaphysics (1080a10-1086b12) Aristotle delivers a battery of criticisms against Plato’s theory of Mathematical Ideas.  Mathematical Ideas are a class of Platonic Ideas of mathematical objects, of which the primary members are ideal or eidetic numbers.  The place of mathematical objects was fiercely debated in ancient Athens.  Where the Pythagoreans conceived of numbers as the extended magnitudes constituting the intrinsic limits within all natural bodies, Plato had, in reaction to Heraclitean Flux, separated the Idea of numbers from the realm of sensible objects. Speucippus, the second scholararch of the Platonic Academy, rejected the separateness of Platonic Ideas and alternatively embedded the concepts of numbers in the World-Soul of Nature.  Xenocrates, the third scholararch, re-affirmed the generation of the Ideas of numbers from the mixture of supreme Ideas, or Principles.  Aristotle rejects Plato's Mathematical Ideas of eidetic numbers in favor of the abstract concepts of mathematical numbers, which have been separated in thought from their original grounding in numerically distinct substances.  He depicts Plato’s theory of Mathematical Ideas as an unnecessary hypothesis by purportedly reducing the various consequences of eidetic numbers to absurdity (reductio ad absurdum).  After summarizing the views of all previous known theorists of number, Aristotle presents objections against each view.  The most sustained series of arguments are reserved for the theory of Mathematical Ideas of Plato and the Platonists.  Aristotle argues that the theory of the Mathematical Ideas is a “bizarre and fantastic” hypothesis that, not only results in many absurd consequences, but – most devastatingly - renders arithmetic impossible by assimilating numbers to Ideas.  Since, however, arithmetic is evidently possible, he concludes the theory of Mathematical Ideas must be rejected. 

Plato’s dialogues present different classes of Ideas: the Republic (596a) presents universal Ideas of all predicable particular instances; the Sophist (254d) presents a higher-order of ‘Meta-Ideas’, such as sameness and difference, that are the universal Ideas over the Ideas; and the Philebus (16c) gestures towards the summit of the eidetic hierarchy in the supreme Principles of the limiting One and the unlimited Dyad.  Ancient testimony further reports that Plato taught that Mathematical Ideas were eternally generated by the limiting ‘equalization’ of the unlimited manifold of Being in the Indefinite Dyad under the active influence of the One.  The original fountain of the being and truth of Mathematical Ideas thus lies in the supreme Principles rather than in any particular substances.  Rather than subsisting in their own self-enclosed particularity, Plato evidently believed eidetic numbers to subsist in an eternal emanation from the original transcendent mixture of the absolutely generic Principles of the One and the Indefinite Dyad.[1]

Aristotle deliberately re-conceives Plato’s Ideas as one among many particulate substances, like the substances of natural bodies (e.g. horse), but separated to another realm.  Mathematical Ideas are likewise re-cast as abstract concepts separated in thought from particular substances by the routine activity of counting.  Where Plato had robustly conceived of eidetic numbers as Ideas generated by the supreme Principles and multiply instantiated in numerically distinct sensible objects, Aristotle rejects Mathematical Ideas and thinly re-conceives of numbers as no more than abstract concepts generalized by the intellect from quantities of numerical distinct sensible substances.  Aristotle’s many criticisms of Plato’s theory of Mathematical Ideas are, however, an ignorant argument (ignorationes elenchi) that, not only disregards the eidetic generation of numbers from the supreme Principles, but may only plausibly succeed against his own forced re-conception of eidetic numbers as mathematical numbers.  J. N. Findlay comments that “Aristotle cannot make the [Platonic] theory of eidetic numbers work because he is determined to hold that they consist of component units.”[2]  The many absurdities that Aristotle purports to derive from Plato's theory of Mathematical Ideas are thus the consequence of his own, rather than Plato's, conception of mathematical objects.  The following commentary will (§I) describe how Aristotle re-conceives of Plato's Mathematical Ideas of eidetic numbers; (§II) defend Plato's theory of Mathematical Ideas against Aristotle's criticisms in Metaphysics XIII 6-8; and (§III) prosecute the case for Plato's transcendental argument for eidetic numbers against Aristotle's abstraction theory for mathematical numbers.

Read the Full Commentary Here:

[1] For more information on Plato's unwritten ontology, see the re-constructions of Plato's 'unwritten doctrines' by the Tübingen School, especially as described in Hans Joachim Krämer's Plato and the Foundations of Metaphysics (1990) and Giovanni Reale’s Toward a New Interpretation of Plato (1997).
[2] Plato: The Written and the Unwritten Doctrines, p. 446.

No comments: