21 March 2006

"A Grouping of Units"

That's a curious definition of number which Aristotle refers to as being held 'by some' when, in Metaphysics Z.13, he wishes to criticize the view that a substance could be compounded out of other actual substances.

If the substances that supposedly constitute a substance were fully actual, then, Aristotle remarks, what Democritus had claimed would be correct: you couldn't get a single thing out of two, or two things coming from one. (Two would remain two; and the putative 'one' thing would have had to be two from the start.) But the same would hold of any number (not simply two), because a number of things would, on this view, be no more than a 'grouping of units', as indeed some have alleged:

A substance cannot consist of substances present in it in complete reality; for things that are thus in complete reality two are never in complete reality one, though if they are potentially two, they can be one (e.g. the double line consists of two halves-potentially; for the complete realization of the halves divides them from one another); therefore if the substance is one, it will not consist of substances present in it and present in this way, which Democritus describes rightly; he says one thing cannot be made out of two nor two out of one; for he identifies substances with his indivisible magnitudes. It is clear therefore that the same will hold good of number, if number is a synthesis of units, as is said by some; for two is either not one, or there is no unit present in it in complete reality.

a)du/naton ga\r ou)si/an e)c ou)siw=n ei)=nai e)nuparxousw=n w(j e)ntelexei/a|: ta\ ga\r du/o ou(/twj e)ntelexei/a| ou)de/pote e(\n e)ntelexei/a|, a)ll' e)a\n duna/mei du/o h)=|, e)/stai e(/n (oi(=on h( diplasi/a e)k du/o h(mi/sewn duna/mei ge: h( ga\r e)ntele/xeia xwri/zei), w(/st' ei) h( ou)si/a e(/n, ou)k e)/stai e)c ou)siw=n e)nuparxousw=n kai\ kata\ tou=ton to\n tro/pon, o(\n le/gei Dhmo/kritoj o)rqw=j: a)du/naton ga\r ei)=nai/ fhsin e)k du/o e(\n h)\ e)c e(no\j du/o gene/sqai: ta\ ga\r mege/qh ta\ a)/toma ta\j ou)si/aj poiei=. o(moi/wj toi/nun dh=lon o(/ti kai\ e)p' a)riqmou= e(/cei, ei)/per e)sti\n o( a)riqmo\j su/nqesij mona/dwn, w(/sper le/getai u(po/ tinwn: h)\ ga\r ou)x e(\n h( dua\j h)\ ou)k e)/sti mona\j e)n au)th=| e)ntelexei/a|.

What is the source of this curious definition of number? Ross notes the attribution of a very similar formula to Thales:
This is practically the same as the earliest recorded Greek definition of number, mona/dwn su/sthma, which Thales is said to have borrowed from the Egyptians (Iambl. in Nicom. Ar. Introd. p. 10.8). Cf. D.1o20a13n.
I gather that this attribution is not taken to have much weight. At least, "a number is a grouping of units" isn't included in the standard lists of the three things Thales is credited with having said.

But I wonder whether the Phaedo provides evidence that the definition originates at least with presocratic natural philosophy. What I have in mind is the similarity between Aristotle's discussion, and the passage, in the autobiographical part of the Phaedo, where Socrates says that he no longer accepts a view he had previously accepted, about the nature of the numbers one and two. It is true that Aristotle takes the definition to be inadequate on the grounds that constituents need to be, as it were, 'matter', if they are to be united into a true unity. But it would make sense for Plato to object to the definition on the rather different grounds that it treats a number as if constituted by a mechanical operation, viz. the placing of units in proximity. (Cornford remarked that the definition, mona/dwn su/sthma, is "crude and, so to say, materialistic".) Note that what Socrates objects to is a number regarded as generated by a pro/sqesij.
“By Zeus,” said he, “I am far from thinking that I know the cause of any of these things, I who do not even dare to say, when one is added to one, whether the one to which the addition was made has become two, or the one which was added, or the one which was added and [97a] the one to which it was added became two by the addition of each to the other. I think it is wonderful that when each of them was separate from the other, each was one and they were not then two, and when they were brought near each other this juxtaposition was the cause of their becoming two. And I cannot yet believe that if one is divided, the division causes it to become two; for this is the opposite of [97b] the cause which produced two in the former case; for then two arose because one was brought near and added to another one, and now because one is removed and separated from other. And I no longer believe that I know by this method even how one is generated or, in a word, how anything is generated or is destroyed or exists, and I no longer admit this method, but have another confused way of my own."

4 comments:

Stephen Menn said...

Michael--I'm not sure why you call this a "curious" definition of number. As far as I can see, this or variations on it were the standard Greek way of defining number. Euclid Elements VIIdef2 says that number is "plurality composed out of units," Aristotle in Metaphysics Iota "plurality of units" (Iota 1 1053a30) or
"plurality measured by [a] one" (Iota 6 1057a3-4, "measured" in the technical sense of "evenly divided"). Anyway these phrases, or the one Aristotle cites in Z13, or the one attributed to Thales, seem to accurately reflect the conception of number used by Greek mathematicians, and Aristotle has no wish to quarrel with the mathematicians' conception of number; in a fragment of his De Philosophia quoted by Syrianus, he asks rhetorically, if there is any kind of number besides mathematical number, who of us can understand it? And this corresponds to what Aristotle assumes about numbers in Metaphysics M-N. I don't see any reason to think that he's *criticizing* the account of number he cites in Z13. However, he does think that if a number (or any other whole) is to be actually one thing, its units (or other parts) can exist within it only potentially, in the sense that the whole is capable of being destroyed and becoming these many parts. And one consequence is that there cannot be an essentially eternal and immobile number, since such a number would have no potentiality for perishing and becoming anything other than what it now is. This argument is very important in Metaphysics M-N, and would not work unless Aristotle accepts that a number is a whole containing its units as parts. Julia Annas in her Clarendon commentary on M-N has maintained that Aristotle denied this (indeed that he intended a reductio ad absurdum of the *Platonist* thesis that a number is a whole containing its units as parts), but this is wishful thinking, making Aristotle look respectable by harmonizing him with Frege, and is not supported by anything in the texts.

Michael Pakaluk said...

Stephen,

Yes, it is correct that some phrase of the form, 'compound of numbers' is standard as a definition of number.

So, in response to your question, I suppose I'd want to say that everything hinges on what one means by such a phrase. Nearly everyone would agree that a number is composed of units. I'd want to say that myself, and perhaps even Frege would agree to that. But what does the phrase mean in a particular context?

If monadwn synthesis were to occur in that autobiographical passage in the Phaedo, it would indicate, I take it, some kind of physical process of concretion. It clearly doesn't mean that in Euclid (even if sugkeimenon is even more suggestive of such)--and yet what precisely it means there isn't particularly clear.

What led me to suppose that Aristotle in Z.13 was not deferring to a mathematician's notion as authoritative was his "as is claimed by some" (hwsper legetai hypo tinwn), which I took to indicate at least a withholding of judgment on his part, and also the context, where what is being debated is a particular view of how composite things might be generated. In that context, the phrase would mean something like physical concretion.

But perhaps I've made the mistake of reading the Phaedo critique into the passage, instead of finding it there.

M

Michael Pakaluk said...

Stephen,

It seems to me, on further consideration, that the people who in Z.13 claim that number is a synthesis monadwn, are likely the persons who claim a number is a number "of units" in H.3.1043b24 (hws tines legousi).

Do you find this identification plausible? --Because then in H.3 Aristotle would clearly be going on to reject that definition.

M

Stephen Menn said...

Michael--you mean H3 1043b34, not b24. Yes, there may well be some connection between the texts. But in the H3 text he's not discussing and rejecting a definition of numbers in general, but rather a view of some people who claim that ousiai are numbers: apparently some Platonists, who are in similar difficulty about the unity of the units within a number and about the unity of the genera and differentiae within a definition. (They seem to think, like the targets of Z14, that a species-form is composed of its genus-forms and differentia-forms as constituents.) Aristotle needn't be rejecting the thesis that numbers are collections of units, as long as these numbers aren't substances. You might expect him to reject any claim that numbers are substances, and still more any reduction of substances in general to numbers, but here he seems to be saying, well, "if substances *are* somehow numbers" (b33), then here's how to do it (cp. the summary at the end of H3), by analyzing the ousia of a thing into the components of its definition and noting the similarity of definitions to numbers. This isn't going to get you a number in the more standard sense; but, Aristotle says, reducing the ousia of e.g. horse to a number in the more standard sense, as perhaps the Platonists would like to do, isn't going to work, notably because the Platonists won't be able to explain how these many units are united into a single substance. Of course, they also have trouble explaining how the genera and differentiae are united into a single substance, but Aristotle claims to have shown how to do that, as long as the genus is in potentiality to the differentiae (which it can't be if you insist that the genus is a numerically single substance which is eternally actually whatever it is potentially). Anyway, the H3 text is very interesting and very complicated, in itself and in relation to the larger argument of ZH, but that's a first stab at what seems to me to be going on in it. The implications for your Z13 text are worth pondering, but at the moment I don't see good reason to think that Aristotle is criticizing the account of number as a plurality of units or the like.