Aristotle (384–322 BC)

DOI: 10.4324/9780415249126-A022-1
Version: v1,  Published online: 1998
Retrieved September 18, 2018, from

15. Universals, Platonic Forms, mathematics

These disputes partly concern Aristotle’s attitude to the reality of universals. One-sided concentration on some of his remarks may encourage a nominalist or conceptualist interpretation (see Nominalism §§1, 2). (1) He rejects Plato’s belief (as he understands it) in separated universal Forms (see Plato §§10, 12–16; Forms, Platonic), claiming that only particulars are separable. (2) In Metaphysics VII 13–16 he appears to argue that no universal can be a substance. (3) He claims that the universal as object of knowledge is – in a way – identical to the knowledge of it (On the Soul 417b23).

Other remarks, however, suggest realism about universals. (4) He claims they are better known by nature; this status seems to belong only to things that really exist. (5) He believes that if there is knowledge, then there must be universals to be objects of it; for our knowledge is about external nature, not about the contents of our own minds.

Aristotle’s position is consistent if (1)–(3) are consistent with the realist tendency of (4)–(5). The denial of separation in (1) allows the reality of universals. Similarly, (2) may simply say that no universals are primary substances (which are his main concern in Metaphysics VII). And (3) may simply mean (depending on how we take ‘in a way’) that the mind’s conception of the extra-mental universal has some of the features of the universal (as a map has some of the features of the area that it maps). While Aristotle denies that universals can exist without sensible particulars to embody them, he believes they are real properties of these sensible particulars.

He offers a rather similar defence of the reality, without separability, of mathematical objects (Physics II 2; Metaphysics XIII 3). While agreeing with the Platonist view that there are truths about, for example, numbers or triangles that do not describe the sensible properties of sensible objects, he denies that these truths have to be about independently-existing mathematical objects. He claims that they are truths about certain properties of sensible objects, which we can grasp when we ‘take away’ (or ‘abstract’) the irrelevant properties (for example, the fact that this triangular object is made of bronze). Even though there are no separate objects that have simply mathematical properties, there are real mathematical properties of sensible objects.

Citing this article:
Irwin, T.H.. Universals, Platonic Forms, mathematics. Aristotle (384–322 BC), 1998, doi:10.4324/9780415249126-A022-1. Routledge Encyclopedia of Philosophy, Taylor and Francis,
Copyright © 1998-2018 Routledge.

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