Reading Aristotle: Physics 2.2: Physics, Math, and Metaphysics

*Note: Instead of giving line by line commentary as I normally do, for at least the first part of this chapter I’m just going to be giving commentary on the main points with a few quotes added throughout. This is because I think there’s less in the first part that needs to be worked out in detail.

In the first chapter of Book Two of the Physics, Aristotle distinguished between natural and artificial things, where to be “natural” is to have an intrinsic nature; he then argued that things do in fact have such intrinsic natures, and that the nature of a thing is related primarily to its form. The second chapter begins with a discussion on the difference between physics as a science and mathematics as a science. Since physics is the science that studies nature, having in chapter one established what nature is, it makes sense for Aristotle now to consider how physics studies nature in relation to other sciences.

Both physics and mathematics seem to have the same “subject-matter”: they both study physical bodies. This is evident, says Aristotle, because “physical bodies contain surfaces and volumes, lines and points” which are what mathematics study [1]. This raises the question of whether physics and mathematics are actually two distinct sciences or not, if their subject-matter is the same. There are then three apparent options: either physics and mathematics are different sciences, they are identical sciences, or one is a part or “department” of the other.

A similar question, he points out, can also be asked of physics and astronomy. Aristotle saw astronomy as a branch of mathematics (since charting and predicting the movements of celestial bodies was an entirely mathematical enterprise), so knowing where it stands in relation to physics can help determine where mathematics broadly stands in relation to physics. He answers that astronomy must be a part of physics. Physics, he notes, studies natural bodies, which includes the sun, moon, stars, earth, etc. Indeed these are some of the most important natural bodies to be studied in physics, since their activity helps us to understand motion more broadly. Physics, as the study of nature, obviously must study the nature of these natural bodies. But it’s absurd, Aristotle remarks, to think that natural philosophers could study the nature of certain bodies without studying their “essential attributes” [2], those qualities which flow directly and necessarily from their nature. But these essential attributes of celestial bodies are what astronomers study as well. Furthermore, he says, just look at historical examples: natural philosophers have very often discussed celestial bodies, their qualities, their motion, etc. So it seems that astronomy is a department of physics.

So physics and mathematics must also have some relation. But what is it precisely? Aristotle explains that though they both treat of physical bodies, the mathematician does not treat of them in the same way, for he “nevertheless does not treat of them as the limits of a physical body; nor does he consider the attributes indicated [surface, volume, lines, points] as the attributes of such bodies” [3]. In other words, when a natural philosopher considers surface, volume, lines, and points, he considers them insofar as they make up the spatial limits/boundaries of some physical body, i.e. insofar as they are the quantitative attributes of some physical body. The natural philosopher is primarily considering the physical body itself, and its nature, and only secondarily and derivatively its quantitative attributes. If the physical bodies somehow had no quantitative attributes, then the natural philosopher would not consider surface, volume, points, lines, and the like at all. The mathematician, however, studies primarily exactly these things, and studies them in themselves. That is why, Aristotle points out, the mathematician “separates” the quantitative features from the physical realities in its study of them, i.e. abstracts them. For “in thought they are separable from motion, and it makes no difference, nor does any falsity result, if they are separated” [4].

Aquinas comments on abstraction:

“As evidence for this reason we must note that many things are joined in the thing, but the understanding of one of them is not derived from the understanding of another. Thus white and musical are joined in the same subject, nevertheless the understanding of one of these is not derived from an understanding of the other. And so one can be separately understood without the other. And this one is understood as abstracted from the other” (Lectio 3.161) [5].

The subject-matter of mathematics, then, is studied via abstraction. There are no pure “lines” that exist on their own in the world; the concept of a “line” is abstracted from physical things and considered conceptually, apart from the physical things. So where the natural philosopher would study the physical thing itself, which just happens to have quantitative attributes, the mathematician would study the quantitative attributes themselves, which just happen to be conceptually abstracted from physical things.

Interestingly, Aristotle points out that the Platonists essentially do the same thing as the mathematicians, except they derive an error from their practice. The Platonists observed that the mind can separate/abstract concepts from their actual, physical existence (e.g. you can consider the concept of “man” generally apart from any actual, individual man), and hence concluded that the concepts must have some existence on their own, apart from the particular, concrete existents: hence the doctrine of Ideas/Forms. For the Platonists, if something were separable in the mind, this must reflect an actual separation/distinction in reality. Since the mind can conceptually separate the universal from the particular, there must in reality exist an actual separation between universal and particular. While Aristotle (and Aquinas following him) agreed that the mind can indeed separate form from matter and universal from particular, they rejected that this required an actual separation in reality.

To support his claim that mathematics and physics treat of the same subjects but under different considerations, Aristotle says, just look at the ways in which the two state their definitions. A mathematician will discuss things in terms of being “odd”, “even”, “straight”, or “curved”, and none of these definitions imply any “motion” or matter or sensible quality. This is not so for the natural philosopher, on the other hand, who will discuss things in terms of its nature, such as “flesh”, “bone”, “man”, etc. When the natural philosopher gives his definitions, it includes a “sensible subject” — e.g. the definition of “snub nose” includes the nose as sensible subject — but when the mathematician gives his definitions, it will not — e.g. he would only consider the snub nose insofar as it is “curved” [6].

What this shows is that physics and mathematics indeed study the same things but under different aspects/from different points of view. Further evidence of this, Aristotle says, comes from the fact that there are certain “intermediate sciences” between physics and mathematics. These are branches of mathematics but they are “more physical”, which shows that mathematics really does study the same subjects as physics. Such intermediate sciences include “optics, harmonics, and astronomy” [7]. Their intermediacy is due to the fact that in these sciences, mathematics and physics converge. As Aquinas explains, these sciences “take principles abstracted by the purely mathematical sciences and apply them to sensible matter” (Lectio 3.164)[8]. Even though these sciences use mathematical principles, their “terminus” is in natural bodies, which is the same as physics. And so the direction of these sciences is in a way actual the “converse” of mathematics, even though they apply mathematical principles. Aristotle says: “While geometry investigates physical lines but not qua physical, optics investigates mathematical lines, but qua physical, not qua mathematical” [9]. So mathematics takes physical things and abstracts from them their quantitative features. These features exist in reality in physical things as their attributes, but mathematics separates them from these physical things and does not study them insofar as they are physical. These intermediate sciences, on the other hand, start from the pure, abstracted mathematical principles and then apply them to the physical things. So they study those quantitative features, but insofar as they relate to physical reality.

So these are the ways in which the sciences of mathematics and physics relate and differ. In the second part of the chapter, Aristotle turns to consider how exactly physics studies nature and natural things.

“Since ‘nature’ has two senses, the form and the matter, we must investigate its objects as we would the essence of snubness. That is, such things are neither independent of matter nor can be defined in terms of matter only. Here too indeed one might raise a difficulty. Since there are two natures, with which is the physicist concerned? Or should he investigate the combination of the two? But if the combination of the two, then also each severally. Does it belong then to the same or to different sciences to know each severally?” (Physics 2.2, 194a12-19) [10].

Although Aristotle concluded in the first chapter that nature is primarily related to form, he admitted that it is also related to matter, and hence has two senses. So to investigate nature, we must to an extent investigate both form and matter, just as when we consider the essence of “snubness” we consider both its formal aspect (that it is a type of curvature) and its material aspect (that it is the curvature of something, namely of a nose). To know the nature of a snub nose, we cannot just know it independently of its matter, for then we would just be studying “curvature” and would be under the domain of mathematics, not physics. But neither can we know it merely in terms of its matter, for that would just be the nose, and there would be nothing to distinguish the essence of a snub nose from all other types of noses.

So since nature has two senses, matter and form, is physics primarily concerned with the former or the latter, or the composite of the two? And if the composite of both, is physics just one science that studies both, or is there perhaps a division of sciences within physics that studies each?

He begins to answer these questions:

“If we look at the ancients, physics would seem to be concerned with the matter. (It was only very slightly that Empedocles and Democritus touched on the forms and the essence.) But if on the other hand art imitates nature, and it is the part of the same discipline to know the form and the matter up to a point (e. g. the doctor has a knowledge of health and also of bile and phlegm, in which health is realized, and the builder both the form of the house and of the matter, namely that it is bricks and beams, and so forth): if this is so, it would be the part of physics also to know nature in both its senses” (2.2, 194a19-27) [11].

So on the one hand, the ancient physicists seem to have thought of physics as concerned with matter; and indeed only a few of them every touched on form or essence whatsoever at all. But on the other hand, it seems that in many disciplines, such as medicine and architecture/construction, it is necessary to know both form and matter. These disciplines are artificial, i.e. they deal with art (human production) rather than the natural. But art imitates nature, so it is reasonable to assume that if disciplines of art deal with both form and matter, studies of nature will as well. Aquinas extrapolates:

“Art imitates nature. Therefore natural science must be related to natural things as the science of the artificial is related to artificial things. But it belongs to the same science of the artificial to know the matter and the form up to a certain point, as the doctor knows health as a form, and bile and phlegm and such things as the matter in which health is. For health consists in a harmony of humours. And in like manner the builder considers the form of the house and also the bricks and the wood which are the matter of the house. And so it is in all the other arts. Therefore it belongs to the same natural science to know both the matter and the form” (Lectio 4.170) [12].

It would be strange to think of a builder who knows what houses are but does not know much about what houses are made of. In the same way, suggests Aristotle, it would perhaps be strange for physicists to know what natural things are but not know what they materially consist of.

He continues:

“Again, ‘that for the sake of which’, or the end, belongs to the same department of knowledge as the means. But the nature is the end or ‘that for the sake of which’. For if a thing undergoes a continuous change and there is a stage which is last, this stage is the end or ‘that for the sake of which’. (That is why the poet was carried away into making an absurd statement when he said ‘he has the end [i.e. death] for the sake of which he was born’. For not every stage that is last claims to be an end, but only that which is best” (2.2, 194a27-32) [13].

There’s much of significance to unpack here. First, Aristotle asserts that if some department studies an end, it must also study the means. The mean is that which is for the end, that which is for the sake of or directed to the end. If, for instance, one wants to win the lottery, and buying a lottery ticket is a necessary means to the end of winning the lottery, then it is necessary for one to first know how to buy a lottery ticket. But then Aristotle states that “the nature is the end”, and this has several implications. On a fundamental letter, potency is a “means” to act, i.e. potency is always directed towards and is “for the sake of” act. This is because potency is always a potency for some actuality. And form is a type of act and matter a type of potency. Matter is always for the sake of its form; matter is a “means” to the end of its form. Just as one could not win the lottery without first buying a ticket, so one could not have a house without first having the necessary materials (wood, stone, brick, etc.). Nor would one have any desire or use for those materials in themselves if one did not first desire to build a house. So form is the end of matter. But nature is primarily form (again, he admits that nature is in a sense both matter and form, but it is primarily form, and it is only matter in a derivative sense of form. E.g. wood is only considered as the nature of a house because it is that which the form of house is made of); and so nature is the end. In knowing exactly what the end of a certain thing is, the first condition is that it is the “final” state of its development. This however raises an immediate and obvious challenge: the final state of human life is death. And so the poet mocked that the dead man is truly fulfilling his human purpose, that death must be the sake for which all are born, because it is the last stage of our lives. So Aristotle qualifies this condition with another: the end is the last and best stage of a process of motion. It is perhaps more apt to think of the end as the climax/apex/culmination. If one throws a stone into the air, the arc goes upwards, reaches a highest point, and then descends back to the ground. The “end” is perhaps best analogously compared not to the stone as it falls back and hits the ground, but rather at its highest point. You threw the stone so that it would go up into the air; and so its highest point in the air is the “last” point in the process which fulfills the purpose for which the process was undertaken. You knew the stone would eventually come back down, but the coming back down was not the original intent of the throwing. Rather the coming back down as a necessary byproduct of the throwing. In the same way, a builder may construct a house knowing full well that it will one day necessarily decay, crumble, and collapse, but even though this is the “last” point in the process of the house’s existence, it is not the “end” of the house, the that for the sake of which the house is built.

Aristotle continues with a discussion of how matter is related to ends:

“For the arts make their material (some simply ‘make’ it, others make it serviceable), and we use everything as if it was there for our sake. (We also are in a sense an end. ‘That for the sake of which’ has two senses: the distinction is made in our work On Philosophy.) The arts, therefore, which govern the matter and have knowledge are two, namely the art which uses the product and the art which directs the production of it. That is why the using art also is in a sense directive; but it differs in that it knows the form, whereas the art which is directive as being concerned with production knows the matter. For the helmsman knows and prescribes what sort of form a helm should have, the other from what wood it should be made and by means of what operations. In the products of art, however, we make the material with a view to the function, whereas in the products of nature the matter is there all along” (2.2, 194a32-194b9) [14].

The main idea here is that in art, matter is directed to form as to use, i.e. as to an end. There are at least two roles related to an art: the production of the artifact and the use of it. For instance, both the art of a sailer and the art of a shipbuilder have to do with a boat. The shipbuilder produces the boat and the sailer uses it. The “using” art is directive in relation to form, and the productive art is directive in relation to matter. The image Aristotle is thinking of is something like this: A sailor has a desire to sail a boat. He has this purpose in mind, and knows what he needs for that purpose, namely a particular type of boat that will fit his particular purpose in wanting to sail. Thus the sailor knows and judges what “form” of a boat he wants for his purpose. So the sailor goes to a local shipbuilder and tells him the form that he wants, i.e. instructs him as to the particular type of boat he needs and what is particular qualities should be. The shipbuilder, knowing what form the sailor wants, judges what the best materials will be in constructing that form, and what the best method will be to go about arranging those materials, etc. In other words, he chooses the materials based on what he knows that he needs in order to bring about the proper form. The sailor knew the use he had in mind for all these things, and so he knew what form he needed for that use; the shipbuilder is directed by that use and form to choose the right materials that will be best-suited for the end. The form and its use are directive of the material. In artificial things, of course, we choose and manipulate and “make” the material with a view to the function we have in mind; in natural things, the matter is already there. Despite this difference, Aristotle is arguing that since art imitates nature, we can conclude that just as in art material is directed to use, so in nature matter is directed to form as to its natural end. Aquinas comments:

“From this, then, we can conclude that matter is related to form as form is related to use. But use is that for the sake of which the artifact comes to be. Therefore, form also is that for the sake of which matter is in artificial things. And so as in those things which are according to art we make matter for the sake of the work of art, which is the artifact itself, likewise matter is in natural things from nature, and not made by us; nevertheless it has the same ordination to form, i.e., it is for the sake of form” (Lectio 4.173) [15].

So since matter is directed to form as to its end, and since a science that studies some end must also study its means, physics, which studies form as nature, must also study the matter of that form. Thus physics studies both matter and form.

Aristotle goes on to finish the second chapter:

“Again, matter is a relative term: to each form there corresponds a special matter. How far then must the physicist know the form or essence? Up to a point, perhaps, as the doctor must know sinew or the smith bronze (i. e. until he understands the purpose of each): and the physicist is concerned only with things whose forms are separable indeed, but do not exist apart from matter. Man is begotten by man and by the sun as well. The mode of existence and essence of the separable it is the business of the primary type of philosophy to define” (2.2, 194b9-15) [16].

His first comment seems to be that since each form has a corresponding “special” matter, then a science which studies the particular form ought also to study its particular matter.

So natural science studies nature, nature is primarily form but includes in a sense matter as well, and form must anyways be studied in relation to matter under the same science. The final question then becomes to what extent natural science studies form. Doctors, of course, spend years studying anatomy in excruciating detail. Why? Because the purpose of the doctor is to produce health, and it is necessary to produce health to know anatomy in great detail. There might be some science of anatomy which studies anatomy simply for the sake of knowing anatomy, but that cannot be why the doctor specifically studies anatomy. Rather he studies it because it is necessary for the purpose of his science. Since the end of physics is physical reality, the physicist does not study nature/form qua nature/form, but rather qua nature/form as it exists in physical, material reality. The physicist, of course, considers forms as separated/abstracted from their matter (i.e. he can consider the nature of man apart from any individual existing man), but the only forms/natures he considers, even abstracted from matter, are those which in reality do not exist apart from matter. This extends up to the nature of man, which is the “limit” of natural philosophy: for even though man has a rational soul that is immaterial, it exists in man in a material body. (This is what Aquinas takes Aristotle’s curious comment about “man begotten by man and sun” to mean, i.e. that what is begotten by material agents must itself be to some extent material; and since man is begotten in a sense by “the sun”, he must in a sense be material.) So natural philosophy studies all natures/forms insofar as they exist in material reality, up to the nature of man. But to study natures/forms insofar as they are natures/forms, meaning insofar as they might exist entirely apart from all physical reality, is not in the domain of natural philosophy to study, but rather “the primary type of philosophy”, i.e. metaphysics.



[1].McKeon, Richard, editor. The Basic Works of Aristotle. New York: Random House, Inc, 1941. Print, Physics 2.2, 193b23-24.

[2]. Ibid. 193b27-28.

[3]. Ibid. 193b32-34.

[4]. Ibid. 193b33-35.

[5]. Thomas Aquinas. Commentary on Aristotle’s Physics. Books I-II translated by Richard J. Blackwell, Richard J. Spath & W. Edmund Thirlkel Yale U.P., 1963. <>.

The complete Thomistic understanding of abstraction and its relation to mathematics is actually a bit more technical and complex then what is able to be discussed in any depth here. For an overview paper on this topic, see Kanne, Marvin E. Saint Thomas Aquinas’ Division of the Sciences. Found online here: <;.

[6]. McKeon. Aristotle. 194a1-6.

[7]. Ibid. 194a8.

[8]. Aquinas. Commentary.

[9]. McKeon. Aristotle. 194a9-11.

[10]. Ibid.

[11]. Ibid.

[12]. Aquinas. Commentary.

“Art imitates nature” is a common quoted expression from Aristotle but is one that is usually not taken serious qua definitive, scientific principle. Aquinas, however, in Lectio 4.171, gives some of the reasoning behind holding to it as a principle:

“The reason for saying that art imitates nature is as follows. Knowledge is the principle of operation in art. But [our] knowledge is through the senses and taken from sensible, natural things. Hence in artificial things we work to a likeness of natural things. And so imitable natural things are [i.e., are produced] through art, because all nature is ordered to its end by some intellective principle, so that the work of nature thus seems to be the work of intelligence as it proceeds to certain ends through determinate means. And this order is imitated by art in its operation.”

[13]. McKeon. Aristotle.

[14]. Ibid.

[15]. Aquinas. Commentary.

[16]. McKeon. Aristotle.

Header Image: Raphael [Public domain], via Wikimedia Commons.



Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s