Aristotle and Plato on Mathematical Forms

I have no idea how to blog.

So here’s part of the first paper I wrote as a doctoral student that was recently published in the Journal of Speculative Philosophy. I figured this post would be a good contrast to the idiotic video I made this morning, in that nobody will like it. Maybe I’ll even add an image for what WordPress calls “an optimal user experience.”

Without going into detail about the difficult set of arguments in A.9, it would be helpful to clarify what, according to Aristotle, are the characteristic attributes that he finds objectionable in the Platonic doctrine of Forms. We have three hints at this in 991a9-991b3: 1) that Forms are causes; 2) that Forms are patterns; and perhaps most importantly, 3) that Forms are “separate,” that is, have existence independent of any particular, a claim is given more attention in M.4.

            Aristotle, in a particularly frustrated tone, asks “what on earth the Forms contribute” to natural beings: “for they cause neither movement nor any change in them.”[i] In Aristotle’s terminology, this means that Forms cannot be efficient causes. Nor can they be Aristotelian formal causes, since he continues shortly after that “they are not even the substance of these, else they would have been in them.”[ii] One might add here that if Forms are not formal causes, neither are they final causes, since these correspond to formal causes in natural beings.[iii] Finally, Forms are not material causes, which thesis Aristotle sees as posited by Anaxagoras and Eudoxus, and is, according to him, “easily upset,” though he does not give details how.[iv] Thus Forms are not causes of any kind – a conclusion he echoes, as we will see, at the end of N.5. 

            The next section of A.9 addresses the claim that Forms are paradigms or patterns, and that particulars share or come “from” them. The latter issue, regarding the senses of “from,” is again both paralleled and clarified more deeply in N.5, which I will discuss shortly. But the main objection Aristotle sees to positing Forms as paradigms is that it is a meaningless position, or at best one that is incomplete without some efficient cause: “to say that [the Forms] are patterns and the other things share them is to use empty words and poetical metaphors. For what is it that works, looking to the Ideas?”[v]

            It seems that separation is the characteristic that most distinguished Plato’s notion of Form from that of Socrates’s definitions or universals, at least according to Aristotle: “Socrates did not make the universals or the definitions exist apart; his successors, however, gave them separate existence, and this was the kind of thing they called Ideas.”[vi] As usual, there is a great deal to debate here, from what Aristotle argued to what Plato himself meant,[vii] but for our purposes is it enough to state that one of the central issues Aristotle sees as disparate between himself and Plato is that his universals (and, as we saw above, his notion of mathematicals) do not have independent existence, while (at least according to him) Platonic Forms do.


[i] 991a11.

[ii] 991a12.

[iii] See, for example, Physics II.8 (199a33).

[iv] 991a16. This is discussed by Nicholas Denyer, “Plato’s Theory of Stuffs,” in Philosophy, 58.225 (Jul., 1983), 315-327.

[v] 991a21.

[vi] 1078b28.

[vii] For a thorough discussion see Gail Fine, “Separation,” in Oxford Studies in Ancient Philosophy 2:31-87 (1984).

2 thoughts on “Aristotle and Plato on Mathematical Forms

  1. Wondering how this has anything to do with mathematics? Maybe their definition of mathematics is different than modern day? Do they mean mathematics in the sense that it portrays to the physical world?

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