Three Ones and Aristotle’s Metaphysics

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Authors

• Adam Crager

Abstract

Aristotle’s Metaphysics defends a number of theses about oneness [to hen]. For interpreting the Metaphysics’ positive henology, two such theses are especially important: (1) to hen and being [to on] are equally general and so intimately connected that there can be no science of the former which isn’t also a science of the latter, and (2) to hen is the foundation [archē] of number qua number. Aristotle decisively commits himself to both (1) and (2). The central goal of this article is to improve our understanding of what the Metaphysics’ endorsement of their conjunction amounts to. To this end we explore three manners of being one which enter into Aristotle’s Metaphysics: I call them unity, uniqueness, and unit-hood. On the view the article defends, it’s unity (and not uniqueness) that’s at issue in Aristotle’s endorsement of (1) and unit-hood (and not uniqueness) that’s at issue in his endorsement of (2). The Metaphysics’ positive henology as whole, I suggest, is best interpreted by positing a theory-internal distinction between unity, uniqueness, and unit-hood.

Submitted on Feb 27, 2018

Accepted on Aug 20, 2018

Published on Nov 23, 2018

Peer Reviewed

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1 A Distinction

Oneness is manifold.1 Words for ‘one’, at any rate, and their derivatives are equivocal across many languages. And there have been a number of different things philosophers have called oneness. Here are three:

• (a) unity (to be united, to hang together as a unified whole)
• (b) uniqueness (to be countable as ‘one’, to be a single _ _ _)
• (c) unit-hood (to be a unit of measure, a standard of measurement)

I add some possible English uses of ‘one’ exemplifying (a)–(c) respectively:

• (a′) Today, Berlin is one.
• (b′) Unlike Sparta, Syracuse had only one king.
• (c′) Ancient construction records consistently specify the length as ‘four long’. But we don’t know how long their one was.

Linguistically, the use of ‘one’ in (c′) is atypical. To refer to a standard of measurement English prefers a term of Latinate rather than Germanic origin: ‘unit’ instead of ‘one’. But other languages more happily employ a single term (or single root) to express each of (a)–(c). Thus we find in Aristotle the following uses of the Greek word hen:2

• (a″) A plot isn’t unified [hen] simply because it’s about one person…
• (b″) Next to consider is whether we should reckon there to be one [hen] kind of constitution or more, and if more: How many? and What are they?…
• (c″) The unit [hen]…in magnitude is a finger or foot…in rhythms the unit is a beat or syllable…in heaviness it’s some definite weight…

Prima facie, (a)–(c) look to present distinct notions. You, the reader, are both united and unique; but unlike the centimeter you are no unit of measure. And if I say that one person wrote the Iliad and Odyssey—‘Homer was one person’ I might say—I’m evidently not making a claim about any poet’s wholeness or affirming that a person hangs together.

There are philosophically important questions concerning (a)–(c). Are any of (a)–(c) interestingly connected with being? How do (a)–(c) relate to number? Are any of them real attributes that combine with objects in the manner of properties? Are there illuminating priority/posteriority relations between (a)–(c)? What should we make of the shared un- in ‘unity’, ‘unique’, and ‘unit’, and the equivocity of ‘one’-vocabulary in so many languages?

It will be helpful to have a way of differentiating ‘ones’ in each of the above senses. Using the resources of English, it’s natural to call a one [hen] in the sense of unity or something unified a unity; a one [hen] in the sense of uniqueness—something viewed as countably ‘one’, or as single X—we can refer to as a singularity; and a one [hen] in the sense of unit-hood, i.e. a unit of measure, I’ll call a unit. To speak about a hen while withholding precisification we reserve the neutral: one, in such contexts rendering to hen as oneness or ‘one’, or even the One, depending on the function of the Greek definite article (to) that’s at issue. Note, finally, that in thus putting to work the everyday metric sense of the English word ‘unit’, we are deviating from the scholarly convention of reserving ‘unit’ to render monas in philosophical/mathematical contexts. To avoid confusion, we ourselves render the latter by simply transliterating it: monad.

2 Some Interpretive Questions

In the Platonist philosophical tradition, two basic intuitions about oneness [to hen] loom large. The first is that to hen has some kind of deep connection with being; the second is that to hen has some kind of deep connection with number. Developing these intuitions—and doing so rather differently than Plato and more orthodox Platonists—Aristotle’s Metaphysics elaborates and defends the following theses about to hen:

Thesis A. being [to on] and to hen are equally general and so intimately connected that there can be no science [epistēmē] of the former that isn’t also a science of the latter and its per se attributes

**Thesis B.**to hen is the foundation [archē] of number [arithmos] qua number

Aristotle decidedly commits himself to both theses. The central goal of this paper is to improve our understanding of what his commitment to their conjunction amounts to.

Suppose we use ontological as a label for the notion of one/oneness [to hen] at issue in Aristotle’s endorsement of Thesis A, and arithmetical for the notion of one/oneness at issue in his endorsement of Thesis B. We can then ask what, on Aristotle’s view, being one in the ontological sense and being one in the arithmetical sense each amount to, whether—and to what extent—he views them as distinct, and how—if he does think them distinct—these varieties of oneness relate to one another.

In the last decades, these questions have not received much sustained attention. But a great many issues in Aristotle, as well as later Aristotelian thought, seem to be tied up with these questions—not least among them, questions about the role of hylomorphism in Aristotle’s metaphysical project. Most obviously, our questions bear on the Aristotelian pedigree of the medieval distinction between ‘transcendental’ and ‘quantitative’ oneness [Ar.: waḥda, Lat.: unitas]. Now, this latter distinction has its own complex history of interpretations and reinterpretations.3 And in the interest of prejudging as little as possible and guarding against the conflation of what may turn out to be distinct philosophical issues, I think it safest to here avoid anachronistic application of such scholastic terminology to Aristotle himself. Thus we speak rather of ‘ontological’ ones/oneness and ‘arithmetical’ ones/oneness in what follows; and we pursue our questions using these labels in the exact interpretive sense specified in the paragraph above.

Now, much of Aristotle’s Metaphysics is dialogical and aporetic. As with other topics, its various discussions of oneness are aimed at—and conducted from—a variety of theoretical perspectives and often involve playing off different intuitions about being one against each other.4 But it turns out to be easy enough to collect from the Metaphysics: (a) a critical mass of texts that clearly pertain to Aristotle’s own endorsement of Thesis A (and thus ontological oneness) and (b) a critical mass of texts that clearly pertain to Aristotle’s own endorsement of Thesis B (and thus arithmetical oneness). What we find in this body of texts, I contend, are two quite incompatible accounts of oneness. I contend, moreover, that in the overall henology of the Metaphysics these two accounts are best viewed as neither competing nor confused, but as intended to conceptualize two different things that Aristotle himself sees as quite distinct. On the interpretation of the Metaphysics’ henology I develop below, ontological oneness in Aristotle is unity (and not uniqueness) and arithmetic oneness in Aristotle is unit-hood (and not uniqueness). On Aristotle’s own view of them, ontological and arithmetical oneness are quite separate.

This account of Aristotle’s henology is far from uncontroversial. It clashes most strikingly with a common interpretation of the Metaphysics’ henology on which Aristotle assimilates ontological to arithmetical oneness and/or effectively identifies the former with the latter. We can call such interpretations Assimilationist, this general line of interpretation Assimilationism about Aristotle’s henology.5 Now, the foundation of Assimilationism is a trio of Metaphysics passages: two from Met. Iota 1, one from Met. Δ.6. The three passages are traditionally interpreted together in a very tight hermeneutic circle. Thus interpreted, they seem to constitute strong evidence for some kind of Assimilationism and strong evidence against my competing account of Aristotle’s henology. Yet all three, and two of them in particular, are attested in remarkably different versions in the Byzantine manuscript tradition—our best evidence for what Aristotle actually wrote. This has not, I think, been taken seriously enough; but fully engaging with Assimilationism will require that we do so. For operative among the (alternately reinforced and reinforcing) assumptions in the hermeneutic circle that’s given rise to Assimilationism are some highly questionable text-critical judgements about these passages. Thinking through what, on the final analysis, the three passages do or do not tell us about Aristotle’s henology will involve consideration of some thorny text-critical and text-genealogical issues. This work will occupy us in the paper’s penultimate section: Section 6. Prerequisite for serious engagement with the text-critical and interpretative questions addressed by Section 6 is philosophical consideration of much else in the Metaphysics’ henology. Among other things, the more purely philosophical work of the five preceding sections is intended to provide such preparation.

The section that succeeds this one (Section 3 below) offers a synoptic account of ontological oneness in Aristotle, drawing centrally on Met. Γ.2 and the closely connected analysis of unity in Met. Δ.6 1015b16-1016b17.6 Section 4 turns to Aristotle’s positive conception of number [arithmos], developing an interpretation of arithmetical oneness and the sense in which Aristotle himself thinks it true to say that to hen is the foundation of number. Naturally, our central focus there is Met. Iota 1 1052b20-1053b8 and N.1 1087b33-1088a14. Building on the previous sections’ work, Section 5 sets out a series of arguments concerning the distinctness of arithmetical and ontological oneness in Aristotle’s thought (and the distinctness of both from uniqueness as Aristotle conceives of it). Finally, Section 6 turns to the three passages that motivate Assimilationism and will prima facie seem to pose a serious challenge—indeed, the main challenge—to my interpretation of Aristotle’s henology: Met. Iota 1 1052b16-19, Met. Iota 1 1053b4-6, and Met. Δ.6 1016b17-18. Attending to the relevant text-critical problems, Section 6 defends interpretations of the three passages on which they can be seen to cohere well with both my account of Aristotle’s henology and their own immediate context. Section 7 is a conclusion.

3 Ontological Oneness: Unity

Aristotle’s Metaphysics develops a conception of wisdom [sophia] as an epistēmē: as the mastery of a certain perfected science. So conceived, wisdom will be like other epistēmai in being a kind of impeccable systematic understanding of a subject-matter—a form of perfected knowledge whose characteristic expression is to give the definitive accounts of (subject-matter specific) inexorable phenomena [anagkaia] in terms of of their causes [aitia]. But in contrast to other epistēmai, the Metaphysics argues that wisdom will be an especially profound epistēmē due to the abstraction and extreme universality of its subject-matter. For Aristotle thinks the epistēmē most deserving of the name sophia will have to be a ‘big picture’ epistēmē of reality as a whole. In particular, his Metaphysics argues that true sophia would be an epistēmē that accounts for the the most general of all truths by explaining them on the basis of their most primitive causes and ultimate foundations [archai].

To attain this epistēmē of sophia is the ultimate goal of the investigative discipline that Aristotle calls First Philosophy [protē philosophia]. The ultimate goal of Aristotle’s Metaphysics is not, of course, to propound any such sophia—Aristotle doesn’t claim to have it—but simply to make progress in First Philosophy (so conceived).

In Met. Γ.1 Aristotle famously teaches that the maximally general truths that First Philosophy takes as explananda concern being as such. They are, that is, inexorable truths concerning what holds of (all or certain types of) beings simply insofar as they manifest their associated ways of being. But as Γ.1–2 develops this line of thought, we soon learn that this science of being qua being is (somehow) also a science of oneness qua oneness and what pertains to it per se (1003b33-6, 1004b5-8). For according to Aristotle, there is a type of oneness [to hen] that is as universal a phenomenon as being is—a type of oneness that’s (non-accidentally) convertible with being. (Aristotle calls two phenomena convertible iff every case of the first is a case of the second and vice versa). Met. Γ.2 argues that being and this type of oneness are in some sense ‘the same single nature’ with the result that this type of oneness is ‘nothing different over-and-above being’ (1003b22-3, 1003b31). The upshot of this is supposed to be that the envisioned epistēmē of wisdom must be thematically concerned with the explication of this particular phenomenon of oneness and must account for its per se attributes (= whatever holds of things insofar as things somehow manifest this type of oneness). At issue in these remarks is ontological oneness in the sense of Section 2. And what Aristotle has in mind here clearly isn’t uniqueness: i.e. being one in the sense of countable as ‘one’. For if it were then the mathematical epistēmē of number theory [arithmētikē] would (by Aristotle’s lights) be part of the metaphysical epistēmē that First Philosophy seeks— and according to Aristotle it most certainly isn’t. Nor can ontological oneness be identified with self-sameness. For, in line with other texts, Met. Γ.2 affirms the priority of ontological oneness to sameness: the latter being among the ‘per se attributes’ pertaining to the former (Γ.2 1004b5-8; cf. Δ.9 1018a7-9). No, for Aristotle ontological oneness is unity.7

Now, according to Aristotle being and ontological oneness—i.e being and unity—are not just convertible but convertible per se.8 On this view, there can be no generation of a being that isn’t as such the generation of a unity nor any generation of a unity that isn’t as such the generation of a being: and mutatis mutandis for destruction. Every conception of a being is a conception of a unity, every conception of a unity a conception of a being. And by necessity: every being is a unity and every unity is a being.

It is important to appreciate that in assenting to such claims, Aristotle isn’t conceiving of unity and being as determinate characteristics or uniform natures in which all things share. For Aristotle thinks it manifest that there are many different ways to be a being and many different ways to be unified. And so ‘being’ and ‘unity’ in the paragraph above need to be interpreted such that: X ‘is a being’ means X exhibits some way of being, and X ‘is a unity’ means X exhibits some way of being unified. In such contexts, Aristotle will think of being and unity as not ‘subjects’ [hupokeimena] but maximally general ‘predicables’ [katēgorēmata]9—albeit predicables of a very peculiar sort. They will not, he thinks, fall into any of the categories [katēgoriai]: calling X a unity or being won’t express what X is, or a quality of X, or quantity of X, or…. Nor do being and unity, thus construed, transcend categories in the manner that per accidens compounds like teenager do; nor are they at all property-like since, pace Avicenna (and perhaps Plato), Aristotle thinks it makes no sense to posit intrinsically being-less and unity-less subjects that underlie being and unity.

So, according to Aristotle there are different ways to be a being (something that is), and different ways to be a unity (something that’s unified). And to say that there are different ways to be X is not simply to say that there are different kinds of X with impressively different essences. Though I’m relabeling it, the distinction I have in mind is Aristotle’s own. To see it contrast (i) the manner in which an isosceles triangle and a scalene triangle are both triangles, with (ii) the manner in which (the horse) Rocinante and a photo of Rocinante are both animals. The former two items are different kinds of triangle, but they are not triangles in different ways because (as Aristotle would put it) what it is for each of them to be a triangle is the same. In contrast, while the sentence ‘This is an animal’ can be truly said of both Rocinante and his photograph, the horse and the photo aren’t different kinds of animal. These two (in contrast to Rocinante and Xanthippe) are animals in different ways since what it is for Rocinante to be an animal differs from what it is for the photo to be an animal.10

A more a illuminating example of this different ways to be X phenomenon, Met. Γ.2 invites us consider the term ‘healthy’ as deployed in medicine. (Here, and in what follows, ‘medicine’ means human medicine). Now, among the things that doctors know to be healthy there are humans, foods, complexions, lungs, and exercise regimens. But what it is for a food to be healthy (≈for its consumption to promote health) differs from what it is for a human to be healthy; what it is for a complexion to be healthy (≈for it to indicate health) differs from both, and what it is for a lung (or exercise regimen) to be healthy differs still. Quite evidently, there isn’t some single property of healthiness that all such healthy things share. What we have here is rather a plurality of different ways to be healthy. And this case is particularly interesting to Aristotle because if we collect together all such ways of being healthy with which medicine is concerned we’ll have network of distinct properties linked together not only by our language but also in extra-linguistic reality. For, as he interprets the case, the complex disposition whose possession constitutes human health enters into the real definitions of all other such ways to be healthy; and they, in turn, are all (in one manner or another) ‘of or related to [human] health’ [pros hugieian].

One of the central proposals of the Metaphysics is that what medicine calls ‘healthy’, in taking as its subject-matter everything that’s healthy, is structurally analogous to what First Philosophy calls ‘being’ in taking as its subject-matter everything that is a being (i.e. everything that is). For, Aristotle thinks it difficult to maintain that (e.g.) humans, deaths, numbers, and pleasures all exist (=are beings) in the same way. And there a great many aporiai that he takes to be best solved using well-motivated distinctions between different ways to be a being. But as with the various ways to be healthy, Aristotle further contends that there’s one particular way to be a being that’s fundamental and definitionally prior to the rest: the way of being enjoyed by substantial-beings [ousiai]. More precisely, if X is an ousia then what it is for X to be something that is is for X to be an ousia; if X isn’t an ousia, it isn’t. In the latter case what it is for X to be something that is can differ for different values of X—but it will always involve some sort of relationship to ousia. Reasonably, Aristotle thinks that it’s by studying of the nature and causes of human health that the field of medicine best advances its understanding of healthy diets, healthy complexions, healthy respirations, etc. And for analogous reasons, he thinks that First Philosophy will best advance its understanding of being in general by privileging foundational studies of the primitive causes and foundations of ousia.

Now, as with being Aristotle thinks that adequate sensitivity to the diversity of real should compel us to admit that there are many ways to be a unity. To see the plausibility of this line, one might note that often (if not always)11 unity is a matter of some parts constituting a whole. But consider (e.g.) this human, this making of a brisket, this episode of pleasure, this number, the plot of this tragedy, water (the natural kind), and color (the universal). And consider what it is for each of these to be unified—what it is for each of them to have its parts constitute the whole it is. Intuitively, these items would appear to have parts in some strikingly different ways. But then why think they constitute wholes in the same way? Or more concretely, compare the unity of this drop of water with the unity of all that water (cf. Met. Δ.26). The unity of the drop will be destroyed if we divide its left side from its right with a barrier, but there is no positional rearrangement [metathesis] of all that water that destroys its unity. She who insists that all unities are unified in the same way will need another way to resolve this and great many other aporiai. But Aristotle responds by distinguishing between two ways of being unified. For the water drop to be unified, he proposes, is for its portions to be continuous. This is why the drop will survive any positional rearrangement [metathesis] that preserves corporeal continuity and none which do not. However, he will add, it’s in a different way that all that water is a unity: for it to be unified is not for its portions to be continuous but simply for its portion to exist as what they essentially are (i.e. water). Here as elsewhere, Aristotle is attracted to well-motivated distinctions between ways of being unified where they prove readily intelligible and explanatorily powerful.

Aristotle often insists that philosophy respect the radical diversity of the real. And among other things, his division of categories is supposed to capture a dimension of this radical diversity. So it is not surprising, that Aristotle is attracted to the view that there are distinct ways of being unified for items in distinct categories. Met. Γ.2 (esp. 1003b33-34) and Iota 2 (1054a14) strongly suggest what Z.4 (1030b10-11) explicitly states—that, with respect to the categories, unity and being are predicated in equally many ways.

Suppose we call a way of being unified derivative iff some other way of being unified enters into its real definition, and otherwise call it primitive. Aristotle quite evidently thinks some ways of being unified are primitive while the rest are derivative. Having previously proposed that being is structured focally [pros hen] in the manner that healthy is, Met. Γ.2 (1004a25-31) goes on to add that the same holds for unity. The immediate lesson Aristotle draws from this is that First Philosophy must not only distinguish between different ways to be a unity, but also account for ‘how [other ways to be a unity] are formulated in relation to the primitive [way]’ (1004a28-30). Aristotle is sometimes read as holding that there’s exactly one primitive way of being a unity—that which is characteristic of ousiai. But Met. Δ.6 actually tries to work out the kind of focal analysis that Met. Γ.2 calls for at 1004a25-31. And from that discussion a somewhat more complex picture emerges (Δ.6 1016b6-9):

Most things are said to be united [hen] because they either do or have or undergo or are related to something else united. But things said to be united in the primitive way are those whose ousia is united—and united either by continuity or by a form or by a definition.

Met. Δ.6 had previously analyzed these three aforementioned ways for X’s ousia to be united as three distinct ways of being undivided [adiaireton]. So, regarding the focal structure of unity, Aristotle’s more considered view would seem to be (1) that every way to be a unity is primitive or derivative, (2) that while there are many derivative ways to be a unity there are several (but perhaps not terribly many) primitive ways, and (3) that the several primitive ways to be a unity are all akin to one another in constituting analogous ways for things’ ousiai to be undivided. With respect to this last point, consider (say) the corporeal undividedness that’s the substantial unity of this drop of water and the formal undividedness that’s the substantial unity of The basic The idea would be that while these two types of undividedness don’t amount to the same thing—and while neither of these two is definitionally prior to the other—that nonetheless the two types constitute abstract (but non-trivial) analogues with respect to one another. Aristotle’s habitual characterization of being unified as a matter of being internally undivided [adiateron] might seem to suggest that that he conceives of unity in fundamentally privative terms. After all, as Aristotle himself notes, the word ‘undivided’ [adiaireton] is certainly privative in its linguistic form.12 But on Aristotle’s considered view, the unity of X’s ousia is always a sort of fulfillment [entelecheia] and always an undividedness that makes X something definite. And on Aristotle’s considered view, it’s not unity but the indefinite manyness to which it’s opposed that’s the privative phenomenon. (More on this in Sections 5–6).

Descendants of Aristotle’s idea that there are different ways to be a unity will be familiar to some readers from its renaissance in contemporary ‘neo-Aristotelian’ metaphysics. Other features of his account of unity have made much less impact on contemporary debates about part/whole. One in particular warrants special emphasis here. Aristotle thinks that unity comes not only in many varieties but also in degrees: that some things are more unified than others. The thought that being a single thing (countably ‘one’) comes in degrees verges on incoherency. Bo the dog is a single thing so is the American Department of Defense. If one counts how many things Obama thought about today, they counted equally—neither is more ‘one’ than the other. But it’s both coherent and prima facie reasonable to contend that any live dog is more unified than the American Defense Department.13

On this kind of basis, Aristotle will insist that a dog’s wholeness constitutes an achievement that not every unity matches. Heaps and collections are not unified to this degree. Aristotle will explain a dog’s high degree of unity by arguing that every (proximate) part of a dog is essentially a part of that dog—that is, dog is prior in definition to all dog-parts. In contrast, Aristotle would claim, the parts of a heap are not prior to the whole they compose. Interestingly, Aristotle thinks animals which can be divided into two animals of the same kind (e.g. worms) are less unified than those which cannot (e.g. humans).14 But Aristotle also thinks a human being less unified than an Unmoved Mover who has neither different parts at different times, different parts at different places, or different parts conceivable through different accounts.15

There are further complexities to Aristotle’s theory, and interesting philosophical questions about all of this that I set aside here. Having briefly treated the one that Aristotle takes to be closely connected with being, I turn now to the one(s) that enter into Aristotle’s account of number.

4 Arithmetical Oneness, Number, and Unit-hood

Number, like other mathematical concepts, has a history. Thus it’s often noted that the mathematicians of Greek antiquity have no notion of negative number or irrational number, that (officially at least) they do not even take fractions to be numbers. But to understand Aristotle’s approach to number we must also take seriously the fact that our present concept of natural number is a fairly recent achievement. For the concept of number [arithmos] one finds in both philosophical and non-philosophical texts of Greek antiquity turns out to be remarkably alien to what now seems the intuitive notion of ‘counting number’.

Consider, for instance, the following exchange from Plato’s Theaetetus (204d: trans. in Burnyeat 1990, lightly revised):

From Plato’s perspective, the exchange dramatizes a straightforward application of a (if not the) ordinary concept of arithmos. The exchange becomes intelligible if one appreciates that the ancients primarily use the noun arithmos to mean a count or countable multitude and that it often meant quantifiable amount. (Thus the arithmos of a mile is the 5,280 feet that compose it. The arithmos of this army is, say, a number of men now laying siege to our village: not the cardinality of a set or any kind of ‘abstract object’). As a foundation for a mathematical theory of countable multitudes, Euclid defines number [arithmos] as ‘plurality consisting of monads’ (Elements VII def. 1). This is a significantly different, and less primitive, mathematical concept than that at issue in Frege’s attempts to define number. The modern number theorist starts with an object she calls ‘0’ and uses a successor function to define a sequence of objects (0, 1, 2, …) whose third member is plausibly interpreted (given the theory) as answering to the description ‘the number two’. The ancient number theorist starts by allowing herself to posit as many mathematical monads (≈position-less mathematical points) as she wants, and develops her ‘number theory’ [arithmētikē] as a theory of finite pluralities of these monads. In this setting, the smallest ‘numbers’ (=countable pluralities) are twos; and to get a four you need distinct twos to compose it. Nothing in the theory is readily interpretable as the number two.

In an attempt to explicate the (then) contemporary concept of arithmos, Met. Δ.13 (cf. Cat. 6) teaches that a number16 is a delimited multitude [peperasmenon plēthos]: an amount [poson] that’s countable [arithmēton] because it admits of some determinate division into finitely many discrete parts. On this conception, anything viewed as a finite plurality of distinct existents will constitute a number. So it’s not surprising that the existence of numbers is something that Aristotle takes to be manifest. Indeed, he takes so much to be manifest by sense-perception: numbers being (as we learn in De Anima II.6) among the per se objects of sense-perception common to all senses. In the context of Aristotle’s psychology, this latter claim means that even non-rational animals can perceptually discern some of the numbers in their sensible environment. A bird, e.g., will perceive a number in her nest when she sees two white bodies and is thereby alerted that one of her eggs has gone missing. Human beings, thinks Aristotle, differ from non-rational animals in having both a perceptual discernment of some sensible numbers (say, 3 goats) but also the ability to count–and thereby gain knowledge of—other sensible numbers that our perceptual capacities do not suffice to grasp (e.g. 133 goats). (NB that with Plato and Aristotle, we use ‘sensible number’ to mean any arithmos that is a plurality of sensible existents. Thus, while Aristotle will contend that animals can–quite literally–see some sensible numbers as the numbers they are, most sensible numbers will be far too large to admit of perceptual discernment).

So, for Plato and Aristotle, some numbers are sensible and corporeal. Now, from Aristotle’s perspective it is of great importance to see that every number is a number of things—that every number is a plurality that’s determinately many somethings.17 And this, he thinks, is no less true for incorporeal numbers than it is for sensible numbers. He will ultimately argue that numbers in general have reality only as amounts [posa] that more basic entities give rise to. And this means that no number is a substantial-being [ousia] per se and that every number exists in the category of quantity. As for numerical ratios, Aristotle contends that they are not strictly speaking numbers at all but are actually relatives [pros ti]: 1092b16–35.

Evaluated as an account of our present concept of natural number, this picture will seem basically a non-starter. But mathematical concepts have histories. And as an account of the (then) contemporary notion of arithmos the view proved highly attractive—and not only to opponents of Platonic metaphysics. For concerning the sensible numbers with which ordinary reasoners are ubiquitously engaged, ancient Platonists could and in some cases explicitly did accept Aristotle’s analysis of them as quantifiable amounts of things: i.e. as non-substances in the category Aristotelian of quantity [poson]. What the ancient Platonist proceeded to insist on was that in addition to these non-substantial numbers [arithmoi] there also exist incorporeal intelligible numbers which are not mere quantities [posa] but separate substantial-beings in their own right. From Aristotle’s testimony, it seems that Plato himself (in his mature metaphysics at any rate) posited two different kinds of intelligible numbers as separate substantial-beings: (1) the objects studied by expert mathematicians in number theory [arithmētikē], and (2) the Forms themselves which Plato now proposed to interpret as numbers that transcend those at issue in number theory.

Aristotle tells us in the Metaphysics that when Plato introduced intelligible numbers as separate substantial-beings he went on to argue that such numbers are causes of being. Aristotle adds that Plato had proposed a substantial-being named ‘the One’ [to hen] as the highest cause and foundation [archē] of being. And we further learn that Plato and his followers had attempted to work out a theory of how to hen functions as a foundation of being by developing the intuition that to hen is a foundation of number. Aristotle spends a good deal of time in the Metaphysics trying to show that this research programme has gone nowhere and is ultimately hopeless. On the final analysis, he contends, neither to hen nor any numbers are separately existing substantial-beings; thus neither can be identified as foundational sources of being in general. Mathematicians, he explains, are methodologically correct to pursue number theory as they in fact do—positing point-like but position-less mathematical monads and speaking about them as if they were non-corporeal substantial individuals. But these theoretical objects are not in fact separate substantial-beings at all. And neither are the pluralities of mathematical monads that get called ‘numbers’ in the context of mathematical number theory—such aggregates being mere amounts [posa] of monads: abstracted quantities of a sort. The truth of number theoretic theorems, he argues, gives us no reason to think otherwise.

Despite his decisive rejection Plato’s henological programme for First Philosophy, Aristotle agrees with his Platonist interlocutors that there’s something importantly right in the thought that to hen constitutes a foundation [archē] for numbers. And in the Metaphysics Aristotle proves eager to explain the sense in which it is indeed true that to hen is the foundation of numbers. Now, in contrast to its English correlate, the Greek verb metrein can mean not only ‘to measure’ but also ‘to count’. Observing that in ordinary Greek, hen sometimes functions in a noun-like way and means metron: i.e. ‘measure’ in the sense of unit-of-measure (unit-of-count), Aristotle’s basic proposal is that what most deserve to be called the ‘foundations’ of numbers [arithmoi] are the measuring-units [metra] that make counting [arithmein, metrein] possible. For a number [arithmos], as he explains, is not simply a multitude but is more specifically a determinately countable multitude. A multitude, however, can only be determinately countable relative to some metron, some unit-of-count, with respect to which the multitude can get correctly or incorrectly counted. It thus seems, to Aristotle at any rate, that a number’s being-the-number-it-is must always be founded upon some hen in this sense of metron. And this suggests that a hen in this specific sense is the kind of hen that constitutes a foundation of numbers qua numbers.

Aristotle’s positive account of arithmetical oneness, of to hen qua foundation of number, gets its most detailed exposition in two texts: Met. Iota 1 (1052b20-1053b8) and N.1 (1087b33-1088a14). Aristotle’s strategy in both texts is to explain his proposal in the context of a kind of abstract theory of measurement and quantitative knowledge (i.e. knowledge of quantity qua quantity).18 For, while Aristotle sometimes deploys a more narrow notion of measurement in which counting and measuring are contrasted and viewed as different kinds of activities (see e.g. Met. Δ.13), he evidently thinks that on a deeper and more abstract level measuring a magnitude and counting a plurality need to be understood as the same kind of activity.

The theory starts from the observation that there are remarkably different kinds of thing that can be understood quantitatively: herds of sheep, harmonic intervals, metric poetry, corporeal magnitudes, weights, speeds, and so on. We develop our quantitive knowledge of the quantities populating these various domains of quantifiables by measuring them. And in every such case this involves deploying one or more domain specific unit-of-measure [metron]. Speaking a bit more precisely, on the abstract theory of measurement Aristotle develops in Met. Iota 1 and N.1 measuring is understood to involve the following ingredients:19

1. a particular quantity q to be measured
2. a genus of quantifiables G to which q belongs
3. one (or more) primitive units-of-measure U1, U2,…, where each such Ui marks off equisized quantity-tokens of type Ui in genus G

If q is the height of the statue, G will be length, there will be (say) just one unit-of-measure U and it will be (say) foot. If q is the size of an octave, G will be harmonic magnitude, and (on the harmonic theory Aristotle has in mind in Met. Iota 1 and Nu 1) our units-of-measure will be two: a diesis-interval of one size and diesis-interval of another size. If q is a number of goats in a pen, G will be goat pluralities, and U will be goat.

To better see how the theory handles the special case of counting, it will be useful quote Met. N.1 1087b33-1088a14 at length. (Bringing out the intended sense of metron I render it below as ‘unit-of-measure’ rather than simply ‘measure’).

It is apparent that to hen signifies a unit-of-measure [metron]; also [that] in every case there is something else underlying it. For instance, in harmonics [the hen is] diesis-interval, in magnitudes [the hen is] finger or foot or something of the sort, in rhythms [the hen is] beat or syllable, and similarly in heaviness [the hen is] is a definite weight. And indeed in all these cases in the same way [to hen signifies a unit-of-measure and there’s something else underlying]…so that to hen per se is not the substantial-being [ousia] of anything. And this accords with reason. For, to hen signifies anything that’s a unit-of-measure for a certain [type of] multitude [plēthos], and number [arithmos] signifies anything that’s a measured multitude and a multitude of units-of-measure.20This is why there’s good reason also [to say that] to hen is not a number: for the unit-of-measure [for a number] is not [several] units-of-measure: it’s rather the foundation [of the number] that the unit-of-measure and to hen is. And in every case, the unit-of-measure must be some same thing that holds of [huparchein] all [the measured things]. For instance, if horse is the unit-of-measure then horses [get counted]; if man [is the unit-of-measure] men [get counted]. If man, horse, god [are to be counted] the unit-of-measure is presumably animal and their number is animals. If man, pale, and walking [are to be counted]… their number will be a number of ‘kinds’ or of some such appellation…

Echoing Met. Iota 1 (1052b21-4, b31–2), our N.1 text explains that where to hen signifies foundation of number, to hen signifies a metron: a measuring-unit. A number, on this analysis, is taken to be a multitude partitioned by an associated measuring-unit into a determinate multiplicity of measured-units.21 For instance, says Aristotle, if a number of horses are to be counted, the unit [to hen] will be horse. In contrast, if human being, horse, god are to be counted, the unit [to hen] is presumably animal, and their number will be a number of animals (1088a9-13). If I am counting ‘man, pale, walking’, the unit [to hen] will be something like kind, and their number will be a number of kinds (1088a13-14). In all such examples, to hen qua foundation of number is the type of thing which the number determined is a number of.22

A comprehensive philosophical examination of these ideas would require a very extended discussion. Here we shall have to pass over more than a few important questions about this theory. But for present purposes, I’d like to emphasize one point in particular made quite manifest by the text above. When the Greek terms hen and metron indicate ‘unit’ and ‘unit of measure’, they exhibit the same type/token ambiguity as their English correlates (‘Yard and meter are two units’ vs. ‘Two units stat’). But in explaining how to hen in the sense of unit functions as a principle of number, N.1 1087b33-1088a14 emphasizes that a unit (a hen) of the relevant sort will always be a repeatable type: ‘in every case, the unit-of-measure [metron] must be some same thing that holds of [huparchein] all [the measured things]’ (1088a8).23 Thus if a census-taker is determining the size of the populace, the hen which measures the plurality and constitutes the foundation of the number can be neither Callias nor Xanthippe. The hen must be something repeatable: e.g. human being. Note that on this picture, the unit [hen] that gets called the ‘foundation’ for an arithmos doesn’t itself get counted when we count that arithmos. And indeed, I fail to count the arithmos of goats in my field if I include in my count the repeatable type goat.

That the foundation of a number—a measuring unit—should thus be external to the numbers it measures, is (I think) a point of some importance for Aristotle in both Iota 1 and Nu 1.24 We noted above that from Aristotle’s testimony, we learn that Plato (and certain of his followers) wanted to posit some particular substantial-being called the One [to hen] as a foundation [archē] of number and of being. As a transcendent ‘one above many’, this One was understood to be a unique substance and separate from the intelligible numbers of which it is the foundation. To better understand what has gone wrong here, Aristotle thinks it important to clear get on the extent to which this picture gets something right about to hen as a foundation of number. He takes it to be importantly correct that a totality can only constitute a particular arithmos–a determinately countable plurality—by being related to a unit of count.25 The unit of count in relation to which a totality constitutes some particular determinate arithmos is a hen. But this kind of hen is not among the items that gets counted as ‘one’ when we use it to count. This hen transcends the numbers of which it is the foundation [archē] as a measure or standard of measurement.

In sum, consider the utterance ‘I’m holding two coins: one in this hand and one in that.’ On Aristotle’s view of the utterance, neither ‘one’ nor ‘one in this hand’ will pick out a unit: i.e. the foundation of this two. The relevant unit here is coin (or coin I’m holding). And what the utterance characterizes as singularities (in the sense of Section 1) are the two coins.26

5 The Distinctness of Ontological and Arithmetic Oneness

In Section 2, we noted Aristotle’s deep commitment to the following two theses:

Thesis A. being [to on] and to hen are equally general and so intimately connected that there can be no science [epistēmē] of the former that isn’t also a science of the latter and its per se attributes

**Thesis B.**to hen is the foundation [archē] of number [arithmos] qua number

We proceeded to introduce ‘ontological’ as a label for the notion of one/oneness [to hen] at issue in Aristotle’s endorsement of the former thesis and ‘arithmetical’ for the notion of one/oneness at issue in his endorsement of the latter thesis. Taking stock of our work in Sections 3–4, it will be useful to collect some arguments concerning the distinctness of ontological and arithmetical oneness in Aristotle’s thought.

First, an ‘extensional’ argument to the effect that Aristotle could not, by his own lights, have coherently identified ontological and arithmetical oneness. As we’ve seen, Aristotle maintains that being and ontological oneness are convertible (and convertible per se). From this it follows that by necessity: X is a being iff X is one in the ontological sense. Now, units—arithmetical ones as conceptualized in Metaphysics N.1 and Iota 1— are always repeatable types. But Aristotle holds that some beings are non-repeatable individuals. So insofar some beings are not units but all beings are ones (in the ontological sense) Aristotle must think that arithmetic oneness is something different from ontological oneness.

Now, unit-hood is not something Aristotle discusses in very many texts. And he’s never (as far as I’m aware) especially concerned to highlight the fact that mundane individuals cannot be units. But that the First Unmoved Mover is not a unit, is a point Aristotle will make. So, in a Met. Λ.7 (1072a31-34) text affirming the simplicity of the First Unmoved Mover, we find Aristotle refusing to call his highest deity hen in the sense of ‘unit’. Aristotle explains (1072a32-34): ‘“unit” [hen] and “simple” [haploun] are not the same; for “unit” indicates metron, but “simple” [indicates] a [thing] itself holding in a certain way [pōs echon auto]’. From Aristotle’s perspective this entails that, contra Plato, the god which is the most fundamental foundation of being is not also a foundation of number. But though the First Unmoved Mover of Metaphysics Λ is not an arithmetical one, Aristotle does maintain that the god is simple and (hence) a unity. Indeed, while Λ.7 1072a32 uses ‘hen’ to mean ‘unit-of-measure’ in indicating something which the First Unmoved Mover is not, Λ.8 uses hen as adjective to mean ‘unified’ in explaining something this god is (1074a35-37): ‘the primary essence [to ti ēn einai] has no matter; for it is fulfillment [entelecheia]; thus the First Unmoved Mover is unified [hen] in account and individual’.27

A second argument concerning the distinctness of ontological and arithmetical oneness has the added benefit of helping us see why ontological oneness in Aristotle cannot be *un