# Mathematics and Physical Science

# Mathematics and Physical Science

# Abstract and Keywords

The chapter concerns the relation between physical science and mathematics. There is no contradiction involved in the assumption that both the physicist and the mathematician study body and magnitudes, but are nonetheless engaged in two distinct sciences. A physicist studies quantities insofar as they are the bodies and magnitudes *of* physical substances. A mathematician studies bodies, lines, and surfaces as if they were separate. Quantities are logically separable from physical substances and mathematics studies the quantitative aspect of physical substances in isolation. Since both the physicist and the mathematician are concerned with the same underlying reality, the physicist can draw on mathematical theorems. What holds of an object insofar as it is a body also holds of the object insofar as it is a movable body which belongs to a substance.

*Keywords:*
mathematics, physical science, Physics II.2, qua, quantity, separation

In the preceding chapter I have attempted to justify the first two claims that I announced.^{1} The study of bodies and magnitudes can be seen as a part of the conceptual underpinnings of physical science. It is a study that the physicist cannot do without. Moreover, this study is not purely mathematical in nature. This supposition does not capture the genuine interest the physicist takes in physical bodies. The physicist studies these insofar as they contribute to his understanding of natural substances. Thus, her study is more concerned with ontological issues arising in the study of magnitudes. A full justification of these claims is this study as a whole.

In the present chapter I will return to the remaining claims. These revolve around the question of the relation between physical science and mathematics. Of course, the general question of how mathematics and physical science are related cannot be dealt with here. For our purposes it is salient to see that there is no contradiction involved in the assumption that both the physicist and the mathematician study body and magnitudes, but are nonetheless engaged in two distinct sciences (claim 3). And despite being engaged in distinct sciences the physicist may draw on results from mathematical results (claim 4).

My view, to put it in a nutshell, is that physical science studies quantities *qua* physical or insofar as they are the bodies and magnitudes *of* physical substances. Mathematics, on the other hand, studies bodies, lines, and surfaces insofar as they are separate. The reason why mathematics is a science distinct from physical science is that quantities are logically separable from physical substances. Mathematics studies the quantitative aspect of physical substances in isolation. To be sure, the mathematician is concerned with objects that are ultimately physical substances, but she studies them not insofar as they are the magnitudes *of* a physical substance. Rather, the mathematician studies them *as if* they were separate. This is something that the physicist does not do, as I argued in the last chapter. She studies bodies, surfaces, and lines not insofar as they are separate, but rather insofar as they belong to a certain substance.

#
(p.27)
4.1. The Theory of *Physics* II.2

Aristotle, I have argued, not only announces the need for an account of bodies and magnitudes for the purposes of physical science, but also discusses problems that may arise for such a project. One such problem is whether this inquiry should rather be the task of the mathematician. Isn’t it the mathematician who studies magnitudes? And if so, should we say that physics is a branch of mathematics? Aristotle picks up these questions explicitly in *Physics* II.2. This chapter, I argue, provides a systematic answer to the question how it should be possible that both the mathematician and the physicist study magnitudes. Thus, Aristotle asks at the beginning of *Physics* II.2:

We must consider how the mathematician differs from the physicist; for natural bodies have surfaces and solids, lines and points, which are treated by the mathematician.

^{2}(

Ph.II.2 193b23–25)

Aristotle observes that the mathematician and the physicist apparently study the same objects. Since a science is individuated by its subject matter and the attributes belonging to this subject matter, the problem arises how there can be two sciences. One could assume that the physicist studies the moon and the mathematician studies the attributes of the moon, e.g. sphericity. One version of this proposal could be that physical objects have a certain extension and are bounded by surfaces, but the study of the extension and the surfaces is part of mathematics. Thus, there is a division of labour between the mathematician and the physicist. I have already argued against such an interpretation, but Aristotle himself makes clear in this passage that such a proposal won’t work. The physicist clearly studies bodies and their per se attributes (*τὰ σνμβεβηκότα καθ’ αὑτὰ*).^{3} As Charlton rightly notes:

By ‘things which supervene’ [

τὰ συμβεβηκότα καθ’ αὑτὰ /CP] here Aristotle probably means not just ‘accidents’, things which are affections of natural things, but features about which it is the business of the student of nature to argue and attempt demonstrations.(Charlton 1992, 93)

What are the per se attributes? Following Charlton we should say that they are those attributes about which the scientist makes demonstration. This is indeed a very plausible suggestion. For, consider the questions whether there is an infinite body or how large the place of an object is. These questions cannot be decided without taking into account the extension of a physical object. Extension and sphericity are not properties that the moon just happens to have and that play no role in a scientific account of the moon. The physical scientist may not be concerned with the question who was the first man on the moon. This is something that should not concern the physicist. But the shape and extension of the moon clearly is an altogether different thing. Thus, Aristotle rightly claims that it is obvious that the physicist is
(p.28)
concerned with the quantitative properties of the moon.^{4} Of course, this does not mean that the physicist is solely concerned with the quantitative properties of the moon. It is equally important for her to know what the moon is made of, or where the moon will be in two weeks’ time.

Be that as it may, it seems generally true that both the mathematician and the physicist study body and magnitudes and questions pertaining to the dimensions and distances of the heavenly bodies was a well-established topic in natural philosophy. Nevertheless, Aristotle equally opposes the suggestion that the mathematician and the physicist do not differ after all. Mathematics is not a branch of physical science, nor is physical science a branch of mathematics. How, then, do they differ? Aristotle explains it in the following lines:

Now the mathematician, though [1] he too treats of these things [i.e. shape of the moon and sun], nevertheless [2a] does not treat of them insofar as they are the limits of a natural body; nor [2b] does she consider the attributes insofar as they are the attributes of such bodies. [3] That is why she separates; [4a] for in thought they are separable from motion, and [4b] it makes no difference, nor does any falsity result, if they are separated. [5] The holders of the theory of Forms do the same, though they are not aware of it; for they separate the natural things, although they are less separable than the mathematical objects. [6] This becomes plain if one tries to state in each of the two cases the definitions of the things and of their attributes. [6a] Odd and even, straight and curved, and likewise number, line, and figure, can be defined without reference to motion; [6b] not so flesh and bone and man—these are said to be like a snub nose, not like a curve.

^{5}(

Ph.II.2 193b31–194a7)

The point of the passage is to explain the difference between the mathematician and the physicist.^{6} Let me paraphrase the course of the argument as I understand it. Aristotle first affirms that [1] the science of nature and mathematics do indeed concern the same things.^{7} But the way in which they study these items is different.
(p.29)
For, the mathematician studies bodies and their limits, but not [2a] insofar as they are physical and not [2b] insofar as the attributes are attributes of a physical body, i.e. a physical substance. In other words, the mathematician studies bodies and their attributes without reference to their physical nature. She studies them *as if they were* pure quantities. By implication, the physicist studies bodies and magnitudes insofar as they belong to physical substances. I take this to be the gist of my argument in Section 3.3, where I discuss the specific mode and interest with which the physicist studies magnitudes. If this is right, we see that Aristotle explicitly recognizes the two distinct ways in which body, surfaces, and lines can be studied: either as bodies, surfaces, and lines *of* physical substances, or as bodies, surfaces, and lines *tout court,* and not insofar as they belong to a physical substance.

Aristotle claims further that [3] the work of the mathematician involves a separation. The reason why the study involves a separation is due to the specific character of the mathematician’s investigation. Because she does not study magnitudes insofar as they belong to a physical substance, she separates. From the sentential connection we can, moreover, read off *what* is separated. The geometer separates quantities (lines, planes, bodies) and their attributes (straight, curved, etc.). In [4] Aristotle justifies this method with two important points. First, he explains *what* they are separated *from*. Quantities are separated from motion. I think that the *‘from what’* and the *‘what’* of separation are simply opposing perspectives on the same method. Either one can ask *what* the mathematician separates—to this the answer is ‘magnitudes’—or one can ask *what* the magnitudes are separated *from*—to this the answer is ‘motion’. By subtracting all properties that imply motion, the mathematician separates magnitudes. Second, Aristotle makes clear that the separation the mathematician performs neither makes a difference nor produces any falsity. Thus, in separating magnitudes from motion or in studying magnitudes, but not insofar as they belong to a physical substance,^{8} the mathematician does not make false claims. If the mathematician considers what belongs to a physical substance solely insofar as it is extended, she does not get false results.

Aristotle compares his method [5] to the method of the Platonists, who unknowingly do the same with regard to the objects of physical science. They separate things *which are less separable* than mathematical objects. This incidental remark is important, since it shows that Aristotle thinks that there must be a *fundamentum in re* for separation. There is a reason why there is no falsity when a mathematical object is separated and there is a falsity involved when a physical object is separated. This reason is grounded in the difference of the two classes of entities. It is a special characteristic of quantities that they can be separated. [6] This special characteristic is explained by the simile of the snub nose. Physical things are defined like the snub,
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whereas quantities are defined like the concave. Quantities are separable because of what they are. The nature of quantities explains why the mathematician can study them the way she does.

This is the argument as I understand it. Of course, this is no more than the sketch of a detailed interpretation and it would be a separate task to flesh out the details. But since our topic is the more limited question how the mathematician and the physicist can both study quantities and yet be engaged in different sciences, I would like to address the following set of questions more specifically:

1. What does it mean that the mathematician studies magnitudes, but not insofar as they are the magnitudes of a physical substance?

2. Why can the physicist draw on results from mathematics?

An answer to the first question will separate a mathematical study from a physical study of magnitudes. In showing how these studies differ, we may show what is distinctive about them. An answer to the second question will bring the two studies closer together again. We may show that a focus on the physical ontology of bodies does not mean that we have to neglect mathematical results.

## 4.1.1. *Studying X but not* qua *Y*

Both the physicist and the mathematician study magnitudes. However, though the mathematician studies physical bodies and their boundaries, she:

[2a] does not treat of them insofar as they are the limits of a natural body; nor [2b] does she consider the attributes insofar as they are the attributes of such bodies.

^{9}(

Ph.II.2 193b32–33)

With this sentence Aristotle opens his argument on why the mathematician and the physicist differ. The physicist studies body and magnitudes insofar as they are the bodies and magnitudes of a physical substance. The mathematician studies them, but not insofar as they are the bodies and magnitudes of a physical substance.

### 4.1.1.1. A Remark on x *qua* y

Let me begin with a brief general remark about the *qua*-locution. What is it to investigate Xs *qua* Y? To prove something of X *qua* Y means that one takes Y as the proper subject and proves a theorem about Y.^{10} The theorem holds of X because it is a Y. Take the brazen triangle as an example. If one proves that the brazen triangle has internal angles equal to two right angles (2R), the proof is about the brazen triangle *qua* triangle. Insofar as the object is a triangle, it has internal angles equal to two right angles. The proper subject for the proof is triangle, not brazen triangle. In this sense
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the property of being brazen is disregarded. The *qua*-locution is intimately connected to Aristotle’s concept of an universal proof.^{11} A universal proof proves something of an arbitrary and primitive case.^{12} That is to say, even though 2R holds of brazen triangle, there is no universal proof of this fact. The reason is that it does not hold primitively of brazen triangle. Insofar as it is a triangle, it is true of the object that it has 2R. 2R holds primitively and universally of triangle. Hence, the proof concerns its being a triangle alone.

The method of subtraction thus understood is not a psychological theory.^{13} To take away or to subtract a property in this context does not refer to a psychological process of imagining the object without the property.^{14} If one takes away the red colour of a triangle, one does not (and how should one?) imagine a colourless triangle. Rather one considers the red triangle only insofar as it is a triangle.

Moreover, the *qua*-locution describes a manner of investigation.^{15} This has an implication for the ontological import of sciences. Sciences are individuated by their manner of investigation which does not imply a separate set of objects with which a given science deals.^{16} In studying X *qua* Y one does not study an object ‘X *qua* Y’. In contrast to modern theories about the *qua*-locution, Aristotle’s use does not introduce a ‘*qua*-object’ different from the original object.^{17} Rather, in studying X *qua* Y one studies X insofar as X is Y.^{18} In studying physical substances insofar as they are moved, one considers what is true of them in virtue of the fact that they are moved. And in studying physical substances insofar as they are extended, one considers what is true of them in virtue of the fact that they are extended.

Following Jonathan Barnes, one can express this by distinguishing between the *domain* and the *focus* of the study:

In phrases of the form ‘to study Fs

quaG,’ the term replacing F fixes thedomainof the study, and the term replacing G fixes the aspect or thefocusof the study.(Barnes 1995, 70)

The metaphysician studies beings insofar as they are beings. She considers beings and studies what belongs to them in virtue of the fact that they are beings. The domain of
(p.32)
metaphysics is beings and the focus is what belongs to them as beings.^{19} Similarly, Aristotle states that the geometer studies physical magnitudes insofar as they are extended in n-dimensions (n=0,1,2,3).^{20}

Clearly it is possible that there should also be both formulae and demonstrations about perceptible magnitudes, not however insofar as they are perceptible, but insofar as they are of such-and-such a kind.

^{21}(

Metaph.XIII.3 1077b20–22)

Thus the domain of the study of the geometer are the bodies and magnitudes of physical substances, but the focus of her study is simply magnitudes. The geometer studies physical substances, but considers only what belongs to them *qua* being body. In the context of *Metaphysics* XIII.3, this distinction between the domain and the focus of a study is used to explain why the work of the mathematician does not involve the assumption that she studies a separate *domain* of objects. Both the mathematician and the physicist are ultimately concerned with the same underlying reality. There is no ontologically separate realm of bodies and magnitudes that the mathematician studies. In doing her science the mathematician is ultimately concerned with the domain of physical substances, though her focus is on them only insofar as they are extended.^{22}

However, it is important to see that the distinctive character of a science, what a science is *about*, is given by the focus of the study. The mathematician, for example, is concerned with physical magnitudes, but the subject matter of mathematics is not physical magnitudes.^{23} Rather, the focus of the science of mathematics is simply magnitudes. In the course of a mathematical deduction no terms are used that reveal the physical character of the objects under investigation. It is not a theorem of mathematics that a *brazen* triangle has internal angles equal to two right angles. The mathematician studies triangles. So when she considers the brazen triangle *qua* triangle, the proper object of her study is a triangle.^{24}

In general, geometry is about extended and continuous objects (plus points). This is the subject matter of geometry. Geometry is *concerned with* physical magnitudes, but it is not *about* them. It is concerned with physical objects because the objects
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geometry is about are ultimately the body and magnitudes of physical substances. Note again that, on my interpretation, the distinction between the physicist and the mathematician is not merely that the former studies physical substances whereas the latter studies magnitudes. I am inclined to take the reference to ‘physical magnitudes (*περὶ τῶν αἰσθητῶν μεγεθῶν*)’ to refer to the study of magnitudes insofar as they are the bodies of physical substances. In other words, the physicist also studies magnitudes, but *qua* physical. In this way, both the physicist and the geometer are concerned with physical magnitudes, the magnitudes *of* physical substances, but mathematics is about magnitudes, whereas physical science is about *physical* magnitudes.

### 4.1.1.2. Magnitudes, but not *qua* Physical

In the larger context of Aristotle’s philosophy of mathematics, the fact that geometry is ultimately concerned with physical substances is undoubtedly a key doctrine. For our present purposes, however, we are not interested in the question why the existence of a separate science of geometry, which seems to be taken for granted in *Metaphysics* XIII.3, does not imply a separate domain of objects over and above physical substances.^{25} Rather, our interest is how both the mathematician and the physicist can study magnitudes and yet be engaged in distinct scientific endeavours. What is the focus or manner of the mathematician’s investigation that justifies there being a separate science of mathematics? It is in this light that the passage in *Physics* II.2 should be read. Aristotle wants to show the *difference* between the physicist and the mathematician by noting that the mathematician’s work involves a separation that the physicist’s work does not involve.^{26}

The physicist studies—and as I argued must study—magnitudes. The mathematician, too, studies them, but not insofar as they are physical. The mathematician studies surfaces, but treats these surfaces not insofar as they are limits of physical substances. The *qua*-locution is combined here with a negation. How should we understand this? The negation applies to the whole *qua*-locution. Aristotle does not say that the mathematician studies surfaces insofar as they are *not* boundaries of a physical substances. This would mean that the mathematician explicitly *denies* that the surfaces she studies are boundaries of a physical substances. If she did this, it would be hard to explain why mathematical theorems should apply to physical substances.^{27} Rather, Aristotle makes clear that the mathematician studies surfaces that are in *fact* boundaries of a physical substance, though in the course of her study she does
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not consider them *qua* boundaries of a physical substance. Thus, Aristotle does not deny that the mathematician considers surfaces of a physical body. In mathematics this simply plays no role. There are no theorems involved that refer to the physical nature of magnitudes. Whether or not these magnitudes are physical magnitudes is not in the *focus* of the mathematician’s study. This might be better expressed in terms of ‘omitting’. The mathematician *omits* that the surfaces are in fact surfaces of a physical substance. However, in omitting that the surfaces are in fact surfaces of a physical substance she does not neglect her duty as a mathematician. As a geometer she should not be studying surfaces insofar as they belong to physical substances. It simply is not her field of study.

## 4.1.2. Separation and falsity

Aristotle expresses this idea in terms of a separation that the mathematician performs. In studying surfaces but not *qua* limits of a natural body the mathematician separates them as follows:

[3] That is why she separates; [4a] for in thought they are separable from motion, and [4b] it makes no difference, nor does any falsity result, if they are separated. [5] The holders of the theory of Forms do the same, though they are not aware of it; for they separate the natural things, although they are less separable than the mathematical objects. [6] This becomes plain if one tries to state in each of the two cases the definitions of the things and of their attributes. [6a] Odd and even, straight and curved, and likewise number, line, and figure, can be defined without reference to motion; [6b] not so flesh and bone and man—these are said to be like a snub nose, not like a curve.

^{28}(

Ph.II.2 193b33–194a7)

Aristotle presents it as an inference: *because* the mathematician studies surfaces, but not *qua* limits of a natural body, she separates. The separation points again to the distinctive method of the mathematician. She treats of surfaces, but—in a sense to be explained in a moment—she separates them. This is, I believe, also the thought behind Aristotle’s remark:

Each thing will be best investigated in this way—by supposing what is not separate to be separate, and this is exactly what the arithmetician and the geometer do.

^{29}(

Metaph.XIII.3 1078a21–22)

The mathematician supposes that surfaces constitute an independent field of study. Again, this neither means that surfaces are in fact part of a realm ontologically distinct (p.35) from physical objects. Aristotle explicitly says that they are not. Nor does it mean that the mathematician explicitly denies that surfaces are boundaries of physical objects. In supposing that they are separate she simply treats of surfaces without taking into account what these surfaces really are (for instance, coloured surfaces of a physical substance). To illustrate the point: when we read Euclid, the theorems simply speak of surfaces and their various properties, but nowhere is a connection to physical substances established. The study proceeds as if these surfaces were separate. In other words, if there were a separate realm of mathematical objects, the work of the mathematician would not look any different. From her study we simply cannot tell. Thus I believe that Netz is correct when he remarks that, according to Aristotle, the geometer may speak as if she were studying a Platonic realm of objects: ‘So there is nothing wrong in mathematicians talking as if they discussed Platonic objects’ (Netz 2006, 24). There is nothing wrong with this as long as one is aware of the fact that the specific manner of investigation does not commit us to the assumption that there are in fact separate objects. The theorems of the mathematician are compatible both with the assumption of a Platonic realm of intelligible mathematical objects and with the assumption that in fact mathematicians speak about physical substances. Again, this is an entirely legitimate method for the working geometer. Her field of study is dimensional objects. She need not be concerned with the question how her theories relate to physical reality.

To be sure, according to Aristotle, there *is* a connection to physical reality. What the mathematician is concerned with are physical surfaces, just like the physicist. But this connection is omitted in the work of the mathematician. It is the task of the *philosophy* of mathematics to establish this connection, not the task of mathematics itself. In the work of the physicist, on the other hand, this connection is explicit. She treats of surfaces because an explanation of surfaces is, for example, needed to explain how physical substances are in a place. The subject matter of the physicist is physical substances which have an internal principle of motion and rest. And in the course of studying these substances, she inevitably has to study magnitudes. But evidently this is not a study that separates the surfaces. On the contrary, it is a study of surfaces *of* physical substances. The physicist studies magnitudes insofar as they are magnitudes *of* a physical substance.

This, then, suggests an answer to the question why mathematics as a science is distinct from physical science. Sciences are individuated by their focus. But the focus may be expressed not only in positive terms, but also in negative terms. That is, not only can one say that a science studies X *qua* Y, but also that a science studies X not *qua* Q. In our present case Q is a placeholder for the specific physical manner of investigation.^{30} The mathematician, then, omits in her investigation the physical focus. She separates magnitudes and considers them not *qua* being physical.
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Despite having the same domain of objects as the physicist, her study is individuated by a focus distinct from the focus of the physicist.

If this is separation, why should the mathematician get away with it? This is not a trivial question. After all, magnitudes are ontologically inseparable attributes of physical objects. Ontologically they have the same status as, for example, colours. Why cannot colours be separated and subsumed under a study distinct from physical science? I shall argue that the reason why separation involves no falsity is because magnitudes are defined like concavity, and involve no motion. Separation leads to no falsity because of the logical status of magnitudes. There must be an objective basis for the success of separation. This objective basis is provided by the comparison to the snub a few lines down.

How plausible is this interpretation? Notice how carefully Aristotle presents his argument in *Physics* II.2. The successive steps of the argument are introduced by particles such as ‘*διό*’ or ‘*γάρ*’ that indicate that the sentences constitute a chain of argumentation: [3] The mathematician separates *because* [4] magnitudes are ‘separable in thought from motion’. But this [4a] separation does not imply [4b] any falsity.^{31} One can see that it does not imply any falsity if one [6] states the definition of these items. For the definition of [6a] magnitudes does not involve motion, whereas the definition of [6b] physical objects does involve motion.

If we read the argument in this way, we see that the emphasis is placed entirely on the *definition* of magnitudes, their peculiar logical standing, most importantly that their definition does not involve motion. The mathematician can separate magnitudes because their definition makes no reference to motion. But it is not up to the mathematician to define her objects any way she wants. The same is true of the physicist. Flesh has a definition that makes reference to motion and, if someone defines flesh otherwise, she makes an objective mistake. This is, in brief, the function of the snub example.

I think that my analysis is further corroborated in [5]. Though I believe that the main point of the whole passage can be understood without [5], it provides an important link between the procedure of the mathematician, which due to the definition of its items does not involve a falsity, and the different snub-like definitions of physical objects. Aristotle has said that the mathematician separates magnitudes from motion. The Platonists, Aristotle remarks, do, without noticing it, the same. However, they separate physical things which are ‘less separable’ than mathematical
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objects. What is the point of Aristotle’s remark? I think that Charlton and Ross are correct in pointing out that Aristotle is not criticizing the Platonists for performing an ontological separation.^{32} This, I propose, is the force of the phrase ‘without noticing it’. The Platonists assume that physical objects are ontologically separate, but their method is based on a logical procedure. It is, no doubt, true that Aristotle believes that logical independence does not imply ontological independence.^{33} But this is, as Aristotle’s recurrent remarks about definability suggest, not the main point here. The line of criticism here is rather that even the logical point that the Platonists want to make is wrong. For, natural things are not like mathematical things. The Platonists work on a paradigm they take from mathematics, but this paradigm fails in the case of natural objects such as man. This, Aristotle argues, becomes obvious when one states the definition of the items. It may be possible to separate magnitudes, but physical objects such as man are on a different footing. Conversely, however, this also shows that Aristotle would probably grant the logical point to the Platonists, if this claim were restricted to magnitudes. The mistake of the Platonists, from Aristotle’s perspective, lies not in their view about the subject matter of mathematics. They are not wrong because their theorems are wrong or because mathematical bodies really ‘look’ different. Their mistake rather lies in thinking that natural objects can be defined and treated in the same way as mathematical objects.

## 4.1.3. The snub example

The comparison to the snub justifies the method of separation. Though as a matter of fact all bodies are bodies of a physical substance and therefore moveable, bodies can, insofar as they are defined by being three-dimensionally extended and continuous, be studied without reference to motion. This, in short, is the claim of separability. The separation of the mathematician is a legitimate method because the items one studies are not defined with reference to their movability. To put it bluntly, look into a treatise on geometry, such as Book I of Euclid’s *Elements* and you see that motion can be excluded. This is the function of the snub example.

Having said this, one might worry that this is wrong. To be true, in Euclid’s *Elements* we find no reference to motion, but what about a treatise such as *On Moving Spheres* by Autolycus of Pitane?^{34} This is a mathematical textbook and it is about moving geometrical objects. This relates to the complicated question of the status of disciplines such as astronomy or optics, a question Aristotle himself addresses in *Physics* II.2.^{35} I cannot discuss these issues here, but merely suggest an answer. Aristotle’s claim is not that there cannot be mathematical sciences that include motion.^{36} His claim merely is that magnitudes in general can be defined and studied without reference to motion.
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In other words, the snub example shows that one can in principle write a treatise on spheres without writing on moving spheres. But one cannot write a treatise on animals without also writing on moving animals. However, this is compatible with the assumption that there is a discipline that studies moving spheres, a science that, in the words of Aristotle, studies ‘a mathematical line, not as mathematical, but as physical’ (*Ph.* II.2 194a11–12).

There is a deeper reason, too, why the objection fails. For in the context of *Physics* II.2 the inclusion of motion implies more than the fact that objects travel from A to B. Rather, motion is a placeholder for the physical nature of the object under consideration. For remember that in *Physics* II.1 Aristotle had stated that ‘nature is a principle or cause of being moved and of being at rest in that to which it belongs primarily, in virtue of itself and not accidentally’ (*Ph.* II.1 192b20–23). To study an object *qua* moved is to study it with recourse to its nature, which is an internal principle of motion and rest. Accordingly, the omission of movement in the study of the mathematician means that the mathematician studies objects without recourse to their physical nature. In the case of the study of body this means that to study body *qua* moved is to study it insofar as it belongs to a physical substance. The inclusion of motion in the study of body and magnitude, therefore, implies that we are interested in the ontology of body and magnitude which is inextricably linked to the nature of the physical substances to which they belong.

It is, I think, helpful to have in mind that Aristotle assigns this specific function to the snub example. For Aristotle’s general use of the example of the snub is puzzling because Aristotle uses the example of the snub both as an example of how some things should be defined and as an example of something that cannot be defined.^{37} The puzzlement is increased if one notes that precisely those features that make the snub a good example of a certain type of definition are the reason why the snub cannot be defined. The snub is a good example of a certain type of definition because it is a ‘this in that’ (*de An.*III.4 429b14), a certain form in some matter. The snub is thus a good example of the definition of natural things because in such definitions some perceptible matter is mentioned. However, in *Metaphysics* VII.5 Aristotle apparently claims that this feature leads to trouble because the snub is defined by an addition.^{38} The basic thought behind this problem seems to be that the snub is a composite of concavity and nose and composites like this cannot be defined.

I shall not attempt to reconcile the two different uses here.^{39} For our purposes only the first usage of the snub is relevant, and it is beyond doubt that, in the context
(p.39)
of *Physics* II.2, Aristotle wants to use the snub as an example of a certain type of definition. This is, of course, compatible with the view that, in other stricter contexts with more demanding criteria of what counts as a definition, the snub cannot be defined.

In *Physics* II.2, then, Aristotle claims that the definition of a surface or a line differs from the definition of flesh in a similar way as the definition of snub differs from the definition of concave. We might help ourselves with the following (dummy) definitions:^{40}

⊳ X is a | = X is a concave nose |

⊳ X is | = X is a surface curved in the shape of the interior of a sphere |

⊳ X is a | = X is continuous and extended in two dimensions |

⊳ X is (a parcel of) | = X is a specific homoemerous mixture of the elements. |

Thus understood the difference between the concave and the snub is that the definition of the snub includes noses. The definition of snub contains ‘concavity’ and ‘nose’ which are related in the definition as a ‘this in that’. What it is to be snub is to be a concave nose. In contrast to the definition of snub, it is possible to define ‘concave’ without reference to noses. Even if we were to make the assumption that all and only noses are concave, the definition of concave could be stated without reference to noses. The definition of concave, we may say, is *logically* independent of noses. You can study concavity without studying noses, but you cannot study snubness without studying noses. And this is true even if we suppose that concavity is *ontologically* dependent on noses, because it cannot exist without noses (there can be no concave things that are not noses). That Aristotle does not make a point about the ontology of the concave or the snub is further corroborated in *Metaphysics* VI.1:

Of things defined, i.e. of essences, some are like snub, and some like concave. And these differ because snub is bound up with matter (for what is snub is a concave nose), while concavity is without perceptible matter. If then all natural things are said to be like the snub, e.g. nose, eye, face, flesh, bone, and, in general, animal (…)—for the definition of these is not without reference to movement, but always contains matter—it is clear how we must seek and define the essence in the case of natural things.

^{42}(

Metaph.VI.1 1025b30–26a3)

(p.40)
When Aristotle says that concavity is independent of perceptible matter, we should not suppose that there is concavity residing in some sort of Platonic realm that is ontologically independent of perceptible matter.^{43} The point rather is that in defining the concave we must not necessarily make reference to perceptible matter. On the other hand, when we define the snub and we state what the snub essentially is, this definition includes perceptible matter. The definition of concavity, on the other hand, differs from the definition of the snub because it does not include perceptible matter. Aristotle makes an observation on the logic of these items.^{44}

If we apply the model of the snub to the general case of magnitudes vis-à-vis physical objects, we can say the following:

Proposition 1.The definition of physical objects differs from the definition of mathematical objects in at least the following way: the definition of physical objects includes or implies movability/perceptible matter, whereas the definition of mathematical objects does not include or imply movability/perceptible matter.

This statement invites several comments and caveats.

First, in Proposition 1, I leave the question open how *being movable* and *having perceptible matter* are related. Since Aristotle says both, the following equivalence must at minimum be true: an object is bound up with perceptible matter if and only if the definiens of it implies movability.^{45} For the present purposes we need not make a definite decision whether one side has priority. Still, I believe that the concept of movability fits the context of the *Physics* better and it is no coincidence that in *Physics* II.2 Aristotle uses this concept.^{46} For the subject matter of physical science is movable things, or, to be more precise, things having an internal principle of motion and rest. Even though, as just noted, all movable things have matter, it requires, as Charlton has rightly noted, an argument.

The study of nature is primarily the study of things subject of change. It is the study of things with matter, only because, according to Aristotle, logical analysis reveals that a material factor is presupposed in any case of change.

(Charlton 1992, 96)

If this is right, one can also see that this fits very well with the function that I ascribe to the whole passage. Aristotle wants to show how the mathematician and the physicist differ. In the context of this question it makes a lot of sense for Aristotle to
(p.41)
emphasize motion in the description of the physicist. For this at once recalls Aristotle’s extended arguments in *Physics* I and II.1 that physical science is about things that have an internal principle of motion and rest. Accordingly, one can immediately see why the work of the mathematician is *not* a physical study. For the mathematician omits the very feature that defines the subject matter of physical science. Thus, when the mathematician separates magnitudes from motion, she at once leaves the realm of physical science behind.

Second, the reader will have noticed that I use the concept of implication to express the relation to motion in the definition of physical things. The reason is that it seems overly strong to assume that the term ‘motion’ occurs in all definitions of physical objects. This, however, raises the question what type of implication is at play here. The basic idea is this: even though ‘elemental body with the primary pair of qualities hot–dry’ could serve as a perfectly fine definition of fire, a fuller understanding of what it is to have these qualities or what these qualities are implies grasping their causal powers.^{47} A full answer to the question would require much more argumentation than can be given here, so I will proceed on the assumption that it is in principle possible get a reasonably clear grasp of what it means that the definition of physical things implies motion.

Third, it is also doubtful whether the definition of magnitudes is completely free from movability. Aristotle sometimes seems to suggest that this is the case, especially when he says that magnitudes are logically prior to physical objects.^{48} However, these passages typically occur within an argument against the Platonists that is designed to show that mathematical objects are ontologically posterior. So it could very well be that logical priority is only granted as a concession to the Platonists.^{49} Moreover, there are, after all, no magnitudes ontologically separate from physical substances. So in saying what surfaces really are and how they exist, it seems quite plausible that one should say that these exist as boundaries of physical substances. In this sense, one could assume that in a full definition which takes into account the ontological status of magnitudes movability or their relation to the physical nature of the object to which they belong comes in at some point. I think that this problem is somewhat analogous to the problem whether the definition of physical objects mentions movability in all cases. It is hard to draw a boundary for what counts as implying motion. But it seems overwhelmingly plausible that in general Aristotle is right on this.

However, if this is true, it only strengthens my case. For it gives a reason for the priority of a physical investigation of magnitudes. Aristotle’s theory of body could, on such an account, in principle not be mathematical. The true account of magnitudes (p.42) mentions their ontological status. This is part of Aristotle’s physics, as I have argued. And this is what the second part of this work tries to achieve.

# 4.2. The Mathematician and the Physicist

## 4.2.1. The most important results reviewed

In concluding the first part of the book let me recapitulate the steps of my argument and present in an abridged form the most important results obtained so far. At the beginning of Part I, I set out to defend the following set of claims:^{50}

1. A study of bodies and magnitudes is central to the framework of Aristotle’s conception of physical science (Chapter 3).

2. This study is not a mathematical investigation. It is a study of body and magnitudes insofar as they are the bodies and magnitudes of physical substances (Section 3.3).

3. The mathematician studies body and magnitudes, too. But she is engaged in a different field of scientific endeavour because she studies magnitudes not insofar as they are physical (Chapter 4).

4. Since both the physicist and the mathematician are concerned with the same underlying reality, the physicist can draw on mathematical theorems. This, however, does not mean that physical science is a branch of mathematics (Section 4.2.2).

I have defended the first claim by pointing out that Aristotle announces the need for such an investigation by arguing that a study of body and magnitudes must be part of the conceptual underpinnings of physical science. In this sense, the study of body and magnitudes is comparable to his accounts of place, time, or the infinite. I have further motivated this idea by reviewing several passages which show that this study is crucial for several branches of physical science. In general, the study of magnitudes is a fragment of a theory. It presents the conceptual tools and important notions needed for an adequate account of physical bodies.

Regarding the second claim I have argued that this study should be seen as genuinely physical. It is a study of magnitudes *qua* physical. The study considers what belongs to body and magnitudes insofar as they are bodies and surfaces of physical substances. It is a study of moved and perceptible magnitudes. I have further argued that this also explains why this study is concerned with ontological issues. Physical substances have a certain nature, and this nature determines their topology. For example, insofar as surfaces are boundaries of physical substances, they depend on them. Furthermore, the question whether a body is continuous or not is determined by the nature of the object. In this sense, the study is ontologically loaded.

(p.43)
However, the study of magnitudes is also the task of the mathematician. Thus, the study of magnitudes differs from the study of place, time, or motion, studies which are firmly rooted within physical science. But, I have argued, Aristotle is aware of this and addresses it in *Physics* II.2. The work of the mathematician is distinct from that of the physicist because it omits any reference to motion. The subject matter of physics is things that change. Though the mathematician is concerned with these things, she studies them not insofar as they are subject to motion. She separates body and magnitudes from motion and thus her study is about a subject matter distinct from physical sciences. The separation, however, does not lead to falsity because of the logical status of body and magnitudes. The definition of quantities does not contain any reference to motion, whereas the definition of physical items contains a reference to motion. Therefore, in studying magnitudes, but not insofar as they are moved, one does not commit a mistake. For, one can, up to a certain extent, say what magnitudes are and study what belongs to them *qua* being a magnitude without taking motion into account. If one were to attempt a study of animals without reference to change, one would be wildly off the mark.

This, then, brings us to the last claim. Though the physicist and the mathematician are engaged in distinct sciences, the physicist can draw on results obtained by the mathematician. Physical science is not a branch of mathematics, and yet the physicist can apply mathematical theorems in her science.

## 4.2.2. Drawing on mathematical results

The question in what way the physicist can draw on mathematical results is, I believe, one of the most difficult issues in Aristotle’s theory of science. One reason is that this question is related to the problem of kind-crossing. Aristotle’s discussion of this topic in *APo* I.7 and I.13 is highly compressed and the strictures on kind-crossing seem overly strong. Kind-crossing is here understood as the claim that a proposition in one science is proved by relying on propositions which belong to another science. Aristotle, for example, claims in *APo* I.7 that it is impossible to demonstrate a geometrical truth by relying on arithmetical propositions.^{51} Another difficulty is the relation of this question to the status of the mixed sciences, such as astronomy and optics.^{52} With respect to the latter issue, we face the additional difficulty that Aristotle never explicitly discusses the structure of the mixed sciences.

For these reasons, I will discuss it only insofar as it is absolutely necessary for our purposes and will leave many details of interpretation to another time. Why then is it necessary? I have argued that the study of magnitudes should be an integral part of a philosophical discussion of the conceptual underpinnings of physical science. Insofar as this study is concerned with the ontology of magnitudes and how they depend on physical substances, one may very well think that mathematical arguments are of (p.44) no help. In this vein, I have argued in Section 3.3 that there is a distinctly physical character to the investigation which cannot be captured within a mathematical approach.

Although I am quite sympathetic to this line of thought, it seems to me that in general the physicist relies on mathematical considerations. Admittedly, it is hard to pin down exactly what is a mathematical consideration or argument, but in general it seems true that the physicist often draws on mathematical results.^{53} For, it is a very plausible principle that:

any argument showing that a specific physical object has a specific mathematical property must be based, at least in part, upon some truth of mathematics.

(Netz 2006, 4)

Thus, even if it belongs to physical science to prove that the heavens are spherical, at least part of the argument must be based on mathematical considerations.^{54} To be sure, this does not make the whole argument a part of mathematical science. After all, in mathematics one does not consider the heavens at all. But it is clear that the physicist must be in some sense allowed to draw on the truths of mathematics. This seems especially true with regard to a study of magnitudes.

Given that the physicist draws on mathematical truths, one might worry that this goes against Aristotle’s restrictions on kind-crossing. In the *Posterior Analytics* Aristotle seems to exclude the possibility that one can prove something by crossing sciences.^{55} Thus, it may seem as if in proving the sphericity of the heavens or in arguing against atomism the physicist employs an illegitimate technique. There is, however, an exception to the strictures on kind-crossing. One science can draw on another science if the former is subordinate to the latter. Insofar as the physicist draws on mathematical truths, physical science is subordinate to mathematics.^{56} This view of subordination, however, is relatively innocent because it does not entail that the subordinate science is a *branch* of the higher science. Physical science is not, as I have argued extensively, a branch of mathematics. It merely implies that physical objects have mathematical properties which are salient to physical science. I think that this is a viable interpretation of Aristotle’s view on subordination.^{57}

(p.45)
Having said that, I readily admit that Aristotle’s remarks on subordination are difficult to interpret and that he never explicitly claims that physical science is subordinated to mathematics.^{58} Connected to this, it has to be said that Aristotle’s examples of subordinate sciences are rather pairs, like harmonics and arithmetic or optics and geometry. Accordingly, one may think that a study of magnitudes, since it draws upon mathematical theorems, is a mixed science, such as astronomy or optics.

**Why the study of body is not a mixed science.** Let me begin by arguing that the study of body is not a mixed science.^{59} Aristotle himself mentions the mixed sciences as further evidence for his general claim in *Ph.* II.2:

The point is made clear by the more natural of the branches of mathematics, such as optics, harmonics, and astronomy, too. These are in a way the converse of geometry. While geometry investigates natural lines but not

quanatural, optics investigates mathematical lines, butquanatural, notquamathematical.^{60}(

Ph.II.2 194a7–11)

The mixed sciences are mathematical sciences, but, in contrast to pure mathematics, they treat mathematical lines as if they were physical lines. An optician, for example, will treat a line between two points as a light ray and the points as the eye and the object seen, respectively. Thus, one could think that the study of magnitudes which I have been outlining is simply a mathematical theory with a physical interpretation.

I think that the theory of bodies should not be seen as a mixed science in this sense. One argument is internal to *Physics* II.2. For, note that in the course of the argument in *Ph.* II.2 the mixed sciences are adduced as *further* evidence. That is to say, that the physicist who is said to study magnitudes is not simply the astronomer or the optician. On the contrary, the astronomer is a mathematician, not a physicist.^{61} All Aristotle maintains is that the case of the mixed mathematical sciences helps us in better understanding how the the physicist and the mathematician both study magnitude, but not *qua* the same. Thus, it remains true that it is the physicist, properly speaking, who will study magnitudes.

The other argument against the assumption that the theory of body is a mixed science is that the theory of body which I am describing here is not a science in the required sense in the first place. It is part of a discussion of the principles or conceptual underpinnings of physical science. In this respect, it is comparable to the theory of place which we find in *Physics* IV.1–4. This also does not constitute a science, but yields principles that may be used in the branches of physical science. Therefore, our question needs to be reformulated: Why should the principles of a theory of body
(p.46)
be part of physical science and not merely be principles of a mixed science, such as optics or astronomy? The answer lies, I believe, in the way these principles are linked to the ontology of physical substances. To use an example from Part II of this work:^{62} if two objects are in contact, there are two boundaries in the same place. The reason for there being two boundaries, rather than one, is due to the fact that boundaries are dependent particulars. In the case of continuity, on the other hand, the parts of an object share a boundary. There is, thus, a topological distinction between contact and continuity. However, this difference is grounded in the ontology of physical objects. It is the nature or form of a physical object which ultimately explains why some parts are continuous and constitute a whole.^{63}

My claim, therefore, is not that the principles concerning bodies which I discuss in this work cannot be applied in the mixed sciences. The difference between the purely mathematical and the optical treatment lies, as I suggested, in the fact that lines are treated as light rays, or a point as an eye. This is, I take it, behind Aristotle’s remark that ‘optics investigates mathematical lines, but *qua* natural, not *qua* mathematical’ (*Ph.* II.2 194a10–11). Insofar as the optician takes the mathematical lines in a diagram to be a representation of actual physical lines, she will have to say that, if the lines are in contact, there are two boundaries in the same place. Crucially, however, the mathematician or the optician are in no position to *justify* the claim that physical objects are continuous due to their nature or that, if two physical objects are in contact, there are two boundaries in the same place. An explanation of why physical substances have the topological structure they in fact have requires an analysis of them *qua* physical bodies. A principle which relies on the *nature* of an object falls outside the realm of mathematical considerations. Thus, my point is that the principles of a theory of bodies are set up in such a way that their main application as well as their justification relies on considerations which are, since they rely on the notion of a nature, genuinely physical.

Related to this discussion is another point I wish to address briefly. In the case of mixed sciences, Aristotle not only claims that the subordinate science can draw on truths of the higher science, but also claims that the subordinate science states the fact, whereas the higher science states the reason why:

For here it is for the empirical scientists to know the fact and for the mathematical to know the reason why; for the latter have the demonstrations of the explanations, and often they do (p.47) not know the fact, just as those who consider the universal often do not know some of the particulars through lack of observation.

^{64}(

APo.I.13 79a2–6)

Consider a work such as Ps.-Euclid’s *Catoptrics*: the optician knows *that* light rays are reflected in a certain way when they hit a mirror, but the mathematician knows *why* they are reflected in a certain way. The mathematician may not know *that* rays of light are reflected in this way. Nothing in her study says anything about light rays. Importantly, however, once this evidence is presented to her, she will be in a position to explain *why* this is the case. For the laws of geometry explain this law of optics.

It is precisely at this point where a genuine physical science differs from optics or the other mixed sciences. For the explanatory ground is not provided by any higher science. An act of seeing, for example, has to be explained by a causal story which involves an account of the reception of a perceptible form. *This* explanation lies outside the realm of optics. In this sense, a philosophical discussion of the principles of physical science and the demonstrations within that science are closely related: in both cases, the fundamental explanatory grounds rely on physical considerations.^{65}

**Why can the physicist draw on mathematical results?** Although the theory of bodies belongs to the foundations of physical science, it remains the case that in a physical study of magnitudes one may, and presumably must, draw on mathematical results. How is that possible? As I said, I am inclined to believe that there is a quite harmless version of subordination in Aristotle, but, since this is quite controversial, I will develop my argument without relying on the notion of subordination. Instead I will rely on the terminology of the *focus* and *domain* of a science, which I developed earlier. In short, I will argue that the *focus* of the physicist and the *focus* of the mathematician overlap. The physicist studies physical substances insofar as they have a nature. The study of nature, in turn, requires a study of magnitudes. The mathematician, on the other hand, studies physical substances insofar as they are magnitudes. This constitutes an overlap between the two sciences. To put it simply, the study of geometry is concerned with physical substances *qua* extended and part of the study of the physicist is concerned with physical substances *qua* extended and movable magnitudes. The reason why the physicist can rely on mathematical truths is based on the thought that what holds of an object insofar as it is a body also holds of the object insofar as it is a movable body. In more general terms, what holds of X *qua* Y also holds of X *qua* Z, if being Z implies being Y.

A simplified model of what I have in mind can be found in *APo.* I.5.^{66} Aristotle argues that to prove that brazen isosceles triangles have interior angles equal to two
(p.48)
right angles one has to prove it of brazen isosceles triangles *qua* triangle. In this sense, the triangle is the proper subject of the proof. To know it *simpliciter* and universally one has to prove it of things *qua* triangle. However, if we were to be engaged in a study of what belongs to brazen isosceles triangles *qua* isosceles triangles, we could draw on results from the study of brazen isosceles triangles *qua* triangles. Thus, what holds of brazen isosceles triangles *qua* triangle also holds of brazen isosceles triangles *qua* isosceles triangle. If having 2R holds of triangles, it also holds of isosceles triangles. Similarly, as Aristotle explains in *APo* I.13 79a13–16, it is the task of the mathematician to prove that circles have the largest area for a given circumference, but in a study of wounds a doctor can draw on this result in his explanation as to why circular wounds heal slower. The point is not that medicine is really a branch a mathematics. No, the point is that insofar as wounds possess geometrical properties, the doctor can draw on mathematical results.^{67}

This, I propose, could be seen as a simplified model. It is true that Aristotle, at least to my knowledge, never considers the case of a study where one investigates what belongs to X *qua* Z (where Z implies Y) and asks how it relates to a study of X *qua* Y. But I believe that we find some support for my claim in *De Caelo*:

For the impossibilities which follow in the case of [atomic lines] will also follow for physical objects, while those which follow for the latter will not all follow in the case of the former, because the former, the mathematical objects, are said on the basis of subtraction, while the physical ones are said on the basis of addition.

^{68}(

Cael.III.1 299a13–17)

Aristotle states in this passage that physical objects are said from addition and mathematical objects are said from subtraction. It seems reasonable to assume that what is added in the case of physical bodies and subtracted in the case of mathematical bodies is ‘movability’ or ‘perceptibility’.^{69} The fact that physical bodies are said from addition and mathematical bodies from subtraction implies an asymmetry: What holds of things said from subtraction also holds of things from addition, but not
(p.49)
necessarily the other way round. Hence, what holds of mathematical bodies also holds of physical bodies, though not necessarily vice versa.^{70}

To be sure, in this passage Aristotle uses the terminology of subtraction and not of the *qua*-locution. Accordingly, it has been interpreted either as an early stage of Aristotle’s philosophical development or as a Platonic remnant.^{71} In contrast to this, I think that it is systematically fruitful to put this passage in the context of the investigation of a science.^{72} The passage should not be read as asserting that a physical body is logically derived from a mathematical body.^{73} It should be read as asserting that the physicist studies objects with more properties, for instance, as having properties in addition to the study of the mathematician. That is to say, the focus of the physicist contains a study of magnitudes, since an analysis of magnitudes is *inter alia* necessary to understand motion. But since the focus of the mathematician is also on magnitudes, the physicist can draw on results from the mathematician.

In this sense, in studying what belongs to a physical substance *qua* physical body one can draw on results from the study of what belongs to a physical substance *qua* body. If bodies generally cannot be composed of planes, physical bodies cannot be composed of planes either because they *are* bodies. And the remarks regarding addition and subtraction explain why.

This, I propose, is an account that *could* work. Spelling it out in detail certainly would require a longer argument. Nonetheless, I think that it provides a plausible model for a more complete account, a model that can do justice to the two basic tenets that physical science is not a branch of mathematical science, and that the physicist can nonetheless less draw on mathematical results.
(p.50)

## Notes:

(^{2})
*θεωρητέον τίνι διαφέρει ὁ μαθηματικὸς τοῦ φνσικοῦ (καὶ γὰρ έπίπεδα καὶ στερεὰ ἔχει τά φυσικὰ σώματα καὶ μήκη καὶ στιγμάς, περὶ ὦν σκοπεῖ ὁ μαθηματικός)*.

(^{3})
*Ph.* II.2 193b27–28.

(^{4})
*Ph.* II.2 193b28–30.

(^{5})
*περὶ τούτων μὲν οὖν πραγματεύεται καὶ ὁ μαθηματικός, ἀλλ’ οὐχ ᾗ φνσικοῦ σώματος πέρας ἕκαστον· οὐδὲ τὰ σνμβεβηκότα θεωρεῖ ᾗ τοιούτοις οὖσι σνμβέβηκεν· διὸ καὶ χωρίζει· χωριστὰ γὰρ τῇ νοήσει κινήσεώς ἐστι, καὶ οὐδὲν διαφέρει, οὐδὲ γίγνεται ψεῦδος χωριζόντων. λανθάνονσι δὲ τοῦτο ποιοῦντες καὶ οἱ τὰς ἰδέας λέγοντες· τὰ γὰρ φνσικὰ χωρίζονσιν ἧττον ὄντα χωριστὰ τῶν μαθηματικῶν. γίγνοιτο δ’ ἂν τοῦτο δῆλον, εἴ τις ἑκατέρων πειρῷτο λέγειν τοὺς ὅρονς, καὶ αὐτῶν καὶ τῶν σνμβεβηκότων. τὸ μὲν γὰρ περιττὸν ἔσται καὶ τὸ ὄρτιον καὶ τὸ εὐθὺ καὶ τὸ καμπύλον, ἔτι δὲ ἐριθμὸς καὶ γραμμὴ καὶ σχῆμα, ἄνεν κινήσεως, σὰρξ δὲ καὶ ὀστοῦν καὶ ἄνθρωπος οὐκέτι, ἀλλὰ ταῦτα ὥσπερ ῥὶς σιμὴ άλλ’ οὐχ ὡς τὸ καμπύλον λέγεται*

(^{6})
In an interesting paper Lennox 2008 argues that the passage has an important function within the context of *Physics* II. According to Lennox, the point of the passage is to make clear that the natural form of physical objects is not identical to mathematical attributes. Since Aristotle has argued in *Physics* II.1 that natural objects have matter and form, he wants to fend off a possible misconception which says that physical science has a material part and a formal part, the latter being mathematics. Though I am sympathetic with the attempt at giving a unified reading of the passage and believe that Lennox’s reading may be correct, I remain non-committal with regard to the question, what is the overall function of this passage. It seems to me that the distinction between the mathematician and the physicist can be studied in isolation from its immediate surroundings.

(^{7})
I follow Hussey 1991, 108 and use the phrase ‘to be concerned with’ here and elsewhere to be neutral with respect to the question, what is the primary subject matter of a science. In a study of X *qua* X the primary subject matter is X. In a study of X *qua* Y it is Y which is the primary subject, but the study is concerned with X.

(^{8})
These two phrases should be taken as equivalent.

(^{9})
*ἀλλ’ οὐχ ᾖ φυσικοῦ σώματος πέρας ἕκαστον· οὐδὲ τὰ συμβεβηκότα θεωρεῖ ᾖ τοιούτοις οὖσι συμβέβηκεν·*

(^{10})
I do not agree with Detel, who thinks that logical abstraction and the *qua*-locution are quite different. Cf. Detel 1993a, 175. It is true that in the special case of mathematics a genus is abstracted. But I think that this is an extension, rather than revision, of the theory. After all, a genus can be seen as a (special kind of) subject.

(^{11})
See *APo* I.5 and the analysis of that chapter in Hasper 2006b.

(^{12})
*APo* I.5 73b32–33.

(^{13})
Cf. Annas 1976; Philippe 1948. ‘Subtraction’ is my translation of ‘*ἀφαίρεστς*’. I prefer it over ‘abstraction’ because, as we will see, the method of *ἀφαίρεστς* relies on *disregarding* or *omitting* certain properties of an object in the study of this object.

(^{14})
As noted by Henry Mendell: ‘By subtracting properties, we are not depriving the object of them’ (Mendell 1986, 62).

(^{15})
As was noted by Barnes 1995, 70.

(^{16})
As is emphasized by Netz 2006, 18: ‘The existence of a separate science does not entail the existence of a separate set of objects.’

(^{17})
Therefore, it is misleading to speak of ‘the *qua*-operator’. See Netz 2006, 11f. For a modern theory in which the *qua*-locution is treated as an operator on objects cf. Fine 1982.

(^{18})
In the words of Netz: ‘An argument about A, QUA G is an argument whose only special presupposition is that A is G’ (Netz 2006, 13).

(^{19})
I assume that what Barnes calls ‘domain’ and ‘focus’ corresponds to what Netz calls ‘object’ and ‘manner’. Cf. Netz 2006, 14.

(^{20})
I include points which are extended in 0-dimensions, i.e. not extended at all. In what follows I will, for reasons of simplicity, use the term ‘extended objects’ in a way that tacitly includes points.

(^{21})
*δῆλον ὅτι ἐνδέχεται καὶ περὶ τῶν αἰσθητῶν μεγεθῶν εἶναι καὶ λόγους καὶ ἀποδείξεις, μὴ ᾖ δὲ αἰσθητὰ ἀλλ’ ᾖ τοιαδὶ*.

(^{22})
For a more complete interpretation I refer the reader to Lear 1982; Detel 1993b; Pettigrew 2009; Hussey 1991.

(^{23})
Again, in speaking of the mathematician I use the phrase ‘is concerned with physical magnitudes’ to capture the Greek ‘*περὶ τῶν αἰσθητῶν μεγεθῶν*’ (*Metaph.* XIII.3 1077b21). In this usage I follow Hussey 1991, 108 fn. 7. If a science is concerned with X, it does not follow that the subject matter of this science is X. Cf. Hussey 1991, 108–9.

(^{24})
Cf. Mueller 1970, 164: ‘To say that the mathematician studies man as solid is not to say that he studies man at all. Rather, it is to say that he studies what is quantitative and continuous in three-dimensions.’

(^{25})
Commentators agree that Aristotle’s major point in *Metaphysics* XIII.3 is a negative one. Cf. Detel 1993b; Lear 1982; Hussey 1991. Aristotle wants to show that the existence of a science like mathematics does not presuppose a distinct ontological realm of mathematical objects.

(^{26})
It has been noted by Mueller that it is difficult to say how the theory of *Physics* II.2 and *Metaphysics* XIII.3 go together: ‘There seems to be a significant difference between separating mathematical objects from physical bodies and treating physical bodies as mathematical objects’ (Mueller 1970, 159). For an attempt to reconcile the two passages cf. Detel 1993b, 189–232.

(^{27})
This distinction in connection with the applicability of mathematical theorems will be discussed further below (see Section 4.2.2).

(^{28})
*διὸ καὶ χωρὶζει· χωριστὰ γὰρ τῇ νοήσει κινήσεώς ἐστι, καὶ οὐδὲν διαφέρει, οὐδὲ γίγνεται ψεῦδος χωριζόντων. λανθάνουσι δὲ τοῦτο ποιοῦντες καὶ οἱ τὰς ἰδέας λέγοντες· τὰ γὰρ φυσικὰ χωρίζουσιν ἧττον ὃντα χωριστὰ τῶν μαθηματικῶν. γίγνοιτο δ’ ὄν τοῦτο δῆλον, εἴ τις ἑκατέρων πειρῷτο λέγειν τοὺς ὅρους, καὶ αὐτῶν καὶ τῶν συμβεβηκότων. τὸ μὲν γὰρ περιττὸν ἕσται καὶ τὸ ὄρτιον καὶ τὸ εὐθὺ καὶ τὸ καμπύλον, ἕτι δὲ ἀριθμὸς καὶ γραμμή καὶ σχῆμα, ἂνευ κινὴσεως, σὰρξ δὲ καὶ ὀστοῦν καὶ ἄνθρωπος οὐκέτι, ἀλλὰ ταῦτα ὥσπερ ῥὶς σιμὴ ἀλλ’ οὐχ ὡς τὸ καμπύλον λέγεται*.

(^{29})
*ἄριστα δ’ ὄν οὓτω θεωρηθείη ἕκαστον, εἴ τις τὸ μὴ κεχωρισμένον θείη χωρίσας, ἄπερ ὁ ἀριθμητικὸς ποιεῖ καὶ ὁ γεωμέτρης*.

(^{30})
The specific physical manner is characterized by the inclusion of motion, as I will argue in Section 4.1.3.

(^{31})
Aristotle also says that ‘it makes no difference if one separates’. There are two ways in which this could be understood: First, one could assume that it makes no difference to the *truth* of what the mathematician does. The *οὐδέ* in line 35 could be understood as explicative: ‘It makes no difference which means that no falsity results in separation’. Second, one could assume that Aristotle is pointing to the fact that the separation makes no *ontological* difference. The separation the mathematician performs is merely a logical separation.

Within the context of the whole chapter both interpretations are possible because Aristotle endorses both points. Therefore, I think that we need not decide the issue. It might very well be that Aristotle wants to allude to both interpretations.

(^{32})
Cf. Ross 1936, *ad loc.*, Charlton 1992, 94.

(^{33})
Cf. *Metaph*. XIII.2 1077a36–77b11.

(^{34})
Cf. Aujac 1979. Thanks to Ben Morison for drawing my attention to this book.

(^{35})
Cf. *Ph.* II.2 194a7–12.

(^{36})
Though this statement itself might be disputed. Ross 1936 believes that astronomy and optics are parts of physical science. Accordingly, I think he is committed to classify *On moving spheres* as a physical treatise. That seems wrong. For a detailed discussion of *Ph.* II.2 194a7–12 and the question of the classification of sciences such as astronomy and optics, see Mueller 2006.

(^{37})
The snub as an example of a type of definition: *Metaph*. VI.1; *de An.* I.1; *Ph.* II.2. The snub as an example of an item that cannot be defined: *Metaph*. VII.5.

(^{38})
*Metaph.* VII.5 1030b15. Although it must be noted that hylomorphism is absent in *Metaphysics* VII.4–6. Thus, the puzzle is more general about the definition of properties whose definitions include the subjects to which they belong.

(^{39})
For some general remarks on snubness see Ferejohn 1994; Balme 1980; Peramatzis 2011, part I.

(^{40})
It is not entirely clear whether Aristotle wants to compare the definition of the *property* of snubness with definition of the *property* of concavity or whether he wants to make a point about the definition of a snub thing and the definition of a concave thing. (The same ambiguity haunts interpreters of *Metaphysics* VII.5 where Aristotle argues that the snub has no definition.) Without settling this question I propose that a minimal requirement is that all the definitions above express the essence of something or, equivalently, answer a what-is-it-question. Cf. *Metaph*. VI.1 1025b30–31.

(^{41})
This is why I speak of dummy definition. It might be possible that lines are also concave. I focus on the case of a surface because a snub nose has a surface that is concave.

(^{42})
*ἔστι δὲ τῶν ὁριζομένων καὶ τῶν τί ἐστι τὰ μὲν ὡς τὸ σιμὸν τὰ δ’ ὡς τὸ κοῖλον. διαφέρει δὲ ταῦτα ὃτι τὸ μὲν σιμὸν συνειλημμένον ἐστὶ μετά τῆς ὓλης (ἔστι γὰρ τὸ σιμὸν κοὶλη ῥὶς), ἡ δὲ κοιλότης ἄνευ ὓλης αἱσθητῆς. εἱ δὴ πάντα τὰ φυσικὰ ὁμοίως τῷ σιμῷ λέγονται, οἷον ῥὶς ὀφθαλμὸς πρόσωπον σὰρξ ὀστοῦν, ὃλως ζῷον, φύλλον ῥίζα φλοιός, ὃλως φυτόν (οὐθενὸς γὰρ ἄνευ κινήσεως ὁ λόγος αὐτῶν, ἀλλ’ ἀεὶ ἔχει ὕλην), δῆλον πῶς δεῖ ἐν τοῖς φυσικοῖς τὸ τί ἐστι ζητεῖν καὶ ὁρίζεσθαι*.

(^{43})
Both the snub and the concave may (or may not) have the same ontological status as accidents of a substance. For Aristotle they in fact have the same ontological status. This is clear from his remarks in *de An.* III.8 432a3–6. However, Aristotle engages in an argument with Platonists about this. For they in fact believe that mathematical objects have ontological priority. Texts relevant for that debate are *Metaph*. III.5; V.8; XIII.1–2.

(^{44})
Cf. *Ph.* I.3 186b21–23.

(^{45})
See also Peramatzis 2011, 139.

(^{46})
In this I agree with Detel 1993a, 190–200. For the opposite conclusion cf. Peramatzis 2011, 140–1.

(^{47})
For an interpretation of the elements as causal powers see Sokolowski 1970.

(^{48})
*Metaph.* XIII.2 1077a36–77b11.

(^{49})
Notice in this respect the phrase ‘Let them be prior in definition’ (*Metaph.* XIII.2 1077a36–77b1). This could be read as a concession to the Platonists.

(^{51})
*APo* I.7 75a37–39.

(^{52})
I want to thank an anonymous referee for OUP for pressing me on this point.

(^{53})
For example, when the biologist reasons that dogs have four legs and humans two legs and, hence, dogs have more legs than humans, is she relying on mathematics? Or, again, is the proof that lines cannot be composed of points part of mathematical science or rather part of a more fundamental and philosophically oriented discussion of mathematics?

(^{55})
*APo.* I.7 75a38–b21. On this chapter see also the commentaries by Detel 1993a; Barnes 1993. On kind-crossing see also Hankinson 2005.

(^{56})
A view that seems to have been suggested already by Barnes 1993, 136. For a general discussion of subordination see McKirahan 1978.

(^{57})
In personal conversation Henry Mendell has suggested that Aristotle’s strictures of kind-crossing should be seen as directed against Plato. For Plato apparently thought that all sciences are *completely* reducible to the first principles, the one and the dyad. Aristotle objects to this radical view of reduction, not necessarily to the occasional use of theorems from other sciences. If this is correct, it would line up well with my argument in the next paragraph that the explanatory ground should not come from the higher science.

(^{58})
This has led some commentators to the bold assertion that ‘there is no evidence that Aristotle has ever thought of the science of nature as a subordinated science’ (Falcon 2005, 28).

(^{59})
In this paragraph I have greatly benefited from my discussions with Henry Mendell.

(^{60})
*δηλοῖ δὲ καὶ τὰ φυσικώτερα τῶν μαθημάτων, οἷον ὀπτικὴ καὶ ἁρμονικὴ καὶ ἀστρολογία· ἀνάπαλιν γὰρ τρόπον τιν’ ἔχουσιν τῇ γεωμετρί. ἡ μὲν γὰρ γεωμετρία περὶ γραμμῆς φυσικῆς σκοπεῖ, ἀλλ’ οὐχ ᾖ φυσική, ἡ δ’ ὀπτικὴ μαθηματικὴν μὲν γραμμήν, ἀλλ’ οὐχ, ᾖ μαθηματικὴ ἀλλ’ ᾖ φυσικὴ*.

(^{61})
On this I agree with Mueller 2006.

(^{62})
I will discuss these issues at length in Sections 7.3 and 7.4. Cf. also Propositions 31 in Section 7.3.2.1, Proposition 32 in Section 7.4.1, and Proposition 33 in Section 7.4.2.1.

(^{63})
Of course, these considerations may not always be relevant to an issue at hand. Most of the time, referring to the topological distinctions is enough as an explanation. As I said in the Introduction, depending on the level of generality, Aristotle may or may not adduce certain considerations or explanations. The theory of natural places is discussed in *Physics* IV, but it is absent from *Physics* VI. It is again very important in *De Caelo.* Does this imply that the notion of place has changed in *Physics* VI? Not necessarily; it may simply be due to the fact that, on the level of generality the discussion in *Physics* VI proceeds, these considerations are not relevant.

(^{64})
*ἐνταῦθα γὰρ τὸ μὲν ὅτι τῶν αἰσθητικῶν εἰδέναι, τὸ δὲ διότι τῶν μαθηματικῶν· οὗτοι γὰρ ἔχονσι τῶν αἰτίων τὰζ ἀποδείξειζ, καὶ πολλάκιζ οὐκ ἵσασι τὸ ὅτι, καθάπερ οἱ τὸ καθόλον θεωροῦντεζ πολλάκιζ ἔνια τῶν καθ’ ἕκαστον οὐκ ἴσασι δἰ ἀνεπισκεψίαν*.

(^{65})
For a discussion of *APo.* I.13 see Patzig 1996.

(^{66})
For a general analysis of this difficult chapter cf. Detel 1993a; Hasper 2006b; Barnes 1993.

(^{67})
Notice that I have not explained in what way wounds have geometrical properties. Thus, I have *not* given an answer to the question ‘How can mathematics be true of the world?’ What I have said is consistent with the assumption that mathematical theorems do not apply straightforwardly to physical substances. Consider the theorem ‘All radii in a sphere are of equal length’. If perfect spheres exist, the theorem is straightforwardly applicable. However, if there are in fact no perfect spheres, it is not straightforwardly applicable (if one is inclined to answer this by pointing to the heavenly sphere, use ‘perfect chiliagon’ instead).I believe that part of the answer to why the theorem is nonetheless applicable must come from the thought that a physical substance, insofar as it is three-dimensional, is potentially a perfect sphere. I think that this is suggested by the difficult thought that ‘Thus, then, geometers speak correctly—they talk about existing things, and their subjects do exist; for being has two forms—it exists not only in fulfillment but also as matter’ (*Metaph.* XIII.3 1078a28–31). Interpretations along this line (though they differ at some points) are Detel 1993b; Netz 2006; Pettigrew 2009; Hussey 1991. Be that as it may, it should be clear that the questions ‘Why can the physicist draw on mathematical truths?’ and ‘How is mathematics true of the world?’ are distinct questions.

(^{68})
*τὰ μὲν γὰρ ἐπ’ ἐκείνων ἀδύνατα σνμβαίνοντα καὶ τοῖς φνσικοῖς ἀκολονθήσει, τὰ δὲ τούτοις ἐπ’ ἐκείνων οὐχ ἅπαντα διὰ τὸ τά μὲν ἐξ ἀφαιρέσεως λέγεσθαι, τὰ μαθηματικά, τὰ δὲ φνσικὰ ἐκ προσθέσεως*.

(^{69})
Cf. *Metaph.* XI.3 1061a28–33 where it is said that the primary qualities, i.e. the qualities that define the elements, are taken away.

(^{70})
Aristotle’s terminology of subtraction in this passage closely resembles a passage from the *Posterior Analytics*. *APo.* I.27 87a31–37. For a more complete account of subtraction in Aristotle see the articles by Cleary 1985; Annas 1987.

(^{71})
The first view is endorsed by Cleary 1985, 31. The second is endorsed by Detel 1993a, 174–5. Detel’s suggestion is plausible if one considers that the argument in *De Caelo* is part of an anti-Platonic argument. Aristotle considers the view of the *Timaeus* that physical bodies are composed of planes. His point simply could be that since it has been shown to be mathematically impossible to construct bodies out of planes, it is also physically impossible because the Platonists *themselves* agree that mathematical impossibility translates into physical impossibility.

(^{72})
The difference between subtraction and the *qua-*locution is that subtraction seems to be exclusively used with respect to mathematical objects. On this see Annas 1987.

(^{73})
In addition to the fact that this view seems inherently flawed, Platonic and Cleary’s attempts notwithstanding, it is also implausible that this procedure could succeed. The procedure, as Cleary makes explicit, is based on a complete *dihairesis*. But it is not a good *dihairesis* to move from triangle to isosceles triangle and then to *brazen* isosceles triangle. Moreover, it also cannot account for parallel subtractions that are not logically ordered. One can study man *qua* indivisible substance, *qua* divisible solid, or *qua* self-mover. But these subtractions are not ordered. For these criticisms see Detel 1993a, 174–5.