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The Ontological Status of Mathematical Objects in Aristotle's Metaphysics M Subject Option B – Part 1

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The Ontological Status of Mathematical Objects in AristotleÕs
Metaphysics
M
Subject Option B Ð Part 1 Vikram Kumar Word Count: 4,959
Kumar 2
Abstract:
In this paper, I will study AristotleÕs account of the ontological status of mathematical objects in
Metaphysics
M.3 and briefly in
Physics
B.2 and
Metaphysics
M.1-2. In M.3, Aristotle posits that entities, including mathematical objects, exist in two ways: in actuality and in potentiality. But potentially and actually what? Within the study of mathematics, mathematical objects are said to exist Ôin actuality,Õ whereas outside of mathematics, for the Ôfirst philosopher,Õ they have a Ômatter-like,Õ potential existence. This difference is due to the fact that within the study of mathematics, mathematical objects are change-independent due to their ÔseparationÕ from sensible matter. ÔOutsideÕ of mathematics, these mathematical objects are viewed as existentially dependent upon natural objects instantiated with mathematical features. I argue that Aristotle is offering different internal and external ontological statuses for mathematical objects based on this Ôseparation.Õ
Introduction: Terminology and Relata of Mathematical Separation
Aristotle begins
Metaphysics
Book M with an account of two distinct views on mathematics. One is the Platonist view, which states that mathematical objects are distinct, separate, self-subsistent entities. The other is what I call the Ônatural view,Õ which posits that mathematical objects are ÔinÕ natural objects: ÒIf mathematical objects exist, some say that they must exist in perceptible objects; or separate from them (some say this too).Ó
1
Aristotle has a novel solution to this problem and takes a third route. One the one hand, mathematical objects are not, as the Platonists say, entities existing separately from the perceptible world. On the other hand, Aristotle also denies that mathematical objects can possibly exist ÔinÕ natural objects. What Aristotle concludes is that natural objects instantiate certain mathematical features, but not the mathematical forms found in mathematical objects. At the same time, because of this link to natural objects, mathematical objects share a causal history with natural objects because of these shared mathematical features. Essentially, Aristotle fits his view in-between two opposing camps. In Part 1 of this paper, I will examine how Aristotle understands mathematical separation in
Metaphysics
M.3. Next, in Part 2, I will then explain the essential features of mathematical objects. Finally, in Part 3, I will study the internal/external ontological statuses of mathematical objects in potentiality or actuality.
1
Translations for
Metaphysics
M are by Annas (1976); 1076a32 Ð a35: Ò
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Kumar 3 Although Aristotle has no specific treatise on mathematics, his treatment of the subject is scattered throughout his works. In
Physics
B.2, Aristotle lays out the following difference between the objects of study in both mathematics and natural science:
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Both the student of nature and the mathematician deal with these things; but the mathematician does not consider as boundaries of natural bodies. Nor does he consider things which supervene as supervening on such bodies. That is why he separates them; for they are separable in thought from change, and it makes no difference; no error results.
3
Unlike natural science, mathematics studies mathematical objects as ÔseparateÕ (
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) from natural objects.
4
What Aristotle means by ÔseparateÕ is not immediately clear. Annas suggests that what Aristotle intends to convey is that mathematics is concerned with certain formal properties of natural bodies.
5
These properties of natural bodies would be mathematical features. I think this is the most plausible interpretation Ð mathematical objects appear to be derived from certain formal features and properties of natural objects as opposed to the matter of these objects. Furthermore, Aristotle is adamant about the different treatment mathematicians give to natural bodies as opposed to natural scientists Ð ÔboundariesÕ in nature, says Aristotle, are not ÔboundariesÕ in mathematics. Later,
Metaphysics
M.3 picks up this same theme. Here, I will clarify what Aristotle means by Ôseparation.Õ I do not take separation to be local or existential separation. In fact, I will explain the existential dependence mathematical objects have on their natural instantiations in Part 2 of this paper.
6
Aristotle says that mathematical objects are separated from their natural instantiations in thought (
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). This is, I believe, a kind of essential separation. Mathematical objects are separate from their natural instantiations in essence because of the material and formal differences between the two. Natural objects exist in a natural, sensible, change-involving environment while mathematical objects exist in a solely mathematical, immaterial, and changeless one.
2
193b31 Ð 193b35
3
The translation of
Physics
B is that of Charlton (1970)
4
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can also mean ÔseparableÕ in addition to Ôseparate,Õ which leaves the modal force unclear.
5
Annas (1976): 30
6
Fine (1983): 35 argues that Aristotle usually uses ÔseparationÕ to denote independent existence. I do not think this is the case for mathematical objects as abstracted from their natural instantiations, which I shall come to shortly.
Kumar 4 Separation abstracts away the change-related, sensible features of natural objects, which how we arrive at mathematical objects. Moreover, I will investigate AristotleÕs unique conception of mathematics and its objects. As Aristotle maintains, in separation, Ôno errorÕ (
6;'C
T)M'68
) results.
7
Hence, mathematicians are completely justified in their separation of certain properties from natural objects. Separation merely allows mathematical objects to be studied in isolation from their natural instantiations. I argue that mathematical objects have different ontological statuses from ÔinternalÕ and ÔexternalÕ mathematical perspectives. The internal view relates to the study of mathematical objects as separate from natural objects. The external view, on the other hand, looks at mathematical objects from the perspective of certain Ôcommon axioms.Õ This distinction can be found in
Metaphysics
K.4.
8
There, Aristotle presents Ôfirst philosophyÕ as a more basic science than mathematics or even natural philosophy and studies the Ôcommon axiomsÕ of both natural philosophy and mathematics and how the two fit into the larger scheme of Ôbeing.Õ The internal perspective, however, studies only particular (mathematical) qualities or aspects of entities.
9
It is because of this Ôinternal/externalÕ distinction that mathematical objects have a different ontological status based on scientific perspective. I shall call mathematical objects as studied from the internal perspective: [OIM] = mathematical objects as representative objects
10
, i.e. as studied by mathematicians Natural objects instantiated with mathematical features will be referred to as the following: [OIN] = natural objects with mathematical features, i.e. natural objects which instantiate certain mathematical features I will argue that [OIM] and [OIN] are essentially different due to [OIM]Õs separation from sensible matter. Furthermore, the internal and external viewer would regard [OIM] as having different ontological statuses where [OIM] can exist either in actuality or in potentiality dependent upon perspective. Aristotle explains this ÔsplitÕ view in
Metaphysics
U
.7.
11
There,
7
193b35
8
1061b19 Ð b26: Ò
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9
Mathematics, then, is akin to what Aristotle calls a Ôdemonstrative scienceÕ in
Posterior Analytics
I.10-11 that studies only aspects of entities that are useful for that particular science
10
cf. Hussey (1991): 117; I will return to what Hussey means by Ôrepresentative objectsÕ later
11
1017b1 Ð b5
Kumar 5 Aristotle differentiates between being ÔactuallyÕ and ÔpotentiallyÕ; based on perspective any entity can be said Ôto beÕ in either one of these two senses. I spell this out in terms of actuality and potentiality as delineated by that passage: Internal:
x
is actually a mathematical object
12
External:
x
is potentially a mathematical object
13
x
is actually a natural object that instantiates mathematical features
14
An internal perspective, would be akin to that of an unreflective, practising mathematician whereas the external perspective would be like that of the first philosopher. I will return to this distinction in Part 3 of the paper. Before going any further, I also need to also clarify the relevant
relata
involved in this discussion of mathematical separation. Firstly, both [OIM] and [OIN], I argue, are compounds Ð combinations of form (ÔFÕ) and matter (ÔMÕ).
def
[OIN] = F
n
+ M
s
(ÔnÕ being natural
15
and ÔsÕ being sensible)
def
[OIM] = F
m
+ M
i
(ÔmÕ being mathematical and ÔiÕ being intelligible)
16
This fundamental difference between these two objects, I argue, is one of essence. The fact that [OIM] does not have sensible matter
makes it essentially ÔseparateÕ from [OIN]. This essential difference also means that without sensible matter, [OIM] are change-free whereas [OIN] are change-involving. However, [OIM] are existentially posterior to [OIN] because [OIM] are enformed with mathematical features that are instantiated in [OIN]Õs natural form. For example, take a triangular rock.
17
There is no geometrical triangle instantiating the rock, but nevertheless, the rock possesses certain triangular features. It is from these features that we arrive at the geometrical triangle
18
Ð the geometrical triangleÕs existence depends on the triangular rock, but the two are composed of different essences.
19
[OIM] is enmattered with
12
Where
x
is a representative object, e.g. a right triangle or a cube.
13
Where
x
reflects the mathematical features of a natural object without sensible matter
14
Where
x
is a natural object, e.g. a triangular rock or a spherical planet.
15
Yet, natural form instantiated with mathematical features
16
Mueller (1970): 168; ÒÉgeometric objects are none the less compounds of these properties and intelligible matter.Ó
17
It is key that I use the term Ôtriangular.Õ Just like material descriptors of objects, like ÔwoodenÕ cannot exist on their own, nor can mathematical formal descriptors, like Ôtriangular.Õ ÔTriangularÕ has to be instantiated in sensible matter, so, for example, it becomes a Ôtriangular rock.Õ
18
i.e. a plane figure with three straight sides and three angles adding up to 180
o
19
This process of abstraction would be harder with mathematical features that do not instantiate natural objects, such as complex numbers. Through certain explanatory factors and constraints internal to math, different

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