Lecture 41 — Comparison of science to science from standpoint of certainty and of unity and diversity

a31. The science which is knowledge at once of the fact and of the reasoned fact, not of the fact by itself without the reasoned fact, is the more exact and the prior science. A science such as arithmetic, which is not a science of properties qua inhering in a substratum, is more exact than and prior to a science like harmonics, which is a science of properties inhering in a substratum; and similarly a science like arithmetic, which is constituted of fewer basic elements, is more exact than and prior to geometry, which requires additional elements. What I mean by ‘additional elements’ is this: a unit is substance without position, while a point is substance with position; the latter contains an additional element.

a38. A single science is one whose domain is a single genus,

a38. namely, all the subjects constituted out of the primary entities of the genus—that is, the parts of this total subject—and their essential properties.

a41. One science differs from another when their basic truths have neither a common source nor are derived those of the one science from those the other.

b1. This is verified when we reach the indemonstrable premises of a science, for they must be within one genus with its conclusions: and this again is verified if the conclusions proved by means of them fall within one genus — that is, are homogeneous.

b5. One can have several demonstrations of the same connexion not only by taking from the same series of predication middles which are other than the immediately cohering term e.g., by taking C, D, and Z severally to prove A-B — but also by taking a middle from another series. Thus let A be change, D alteration of a property, B feeling pleasure, and I relaxation. We can then without falsehood predicate D of B and A of D, for he who is pleased suffers alteration of a property, and that which alters a property changes. Again, we can predicate A of I without falsehood, and I of B; for to feel pleasure is to relax, and to relax is to change. So the conclusion can be drawn through middles which are different, that is, not in the same series — yet not so that neither of these middles is predicable of the other, for they must both be attributable to some one subject.

A further point worth investigating is how many ways of proving the same conclusion can be obtained by varying the figure.

The science which is knowledge. After comparing demonstrations one with another, the Philosopher here treats of the comparison of science, which is the effect of demonstration.

And his treatment falls into two parts.

In the first he compares science to science.

In the second he compares science to other modes of knowing, at: scientific knowledge and its object (89b30; [395]).

Concerning the first he does two things.

First, he compares science to science as to certitude.

Second, as to unity and multiplicity, at: a single science is one (87a38; [361]).

Concerning the first (87a31) he lays down three modes whereby one science is more certain than another.

Laying down the first mode, he says that that science is prior and more certain than another, which, namely, makes one know the same things both 'quia' and 'propter quid'. However, that science is not more certain which knows only the 'quia' apart from one which knows 'propter quid'. But this is the relation of subalternating science to subalternate, as has been said above, namely, that the subalternate science in isolation knows 'quia' without knowing 'propter quid': thus a surgeon knows that circular wounds are healed more slowly, but he does not know 'why'. But such knowledge pertains to the geometer who considers that characteristic of a circle according to which its parts do not lie close enough to form an angle, the nearness of whose sides makes triangular wounds heal more quickly.

He lays down the second mode when he says that a science which is not concerned with a subject is more certain than one which is. Here 'subject' is taken to mean sensible matter because, as the Philosopher teaches in Physics II, some sciences are purely mathematical, those, namely, which abstract according to reason from sensible matter, as geometry and arithmetic; but other sciences are intermediate, namely, those which apply mathematical principles to sensible matter, as optics applies the principles of geometry to the visual line, and harmony, that is, music, applies the principles of arithmetic to sensible sounds. Hence he says here that arithmetic is both more certain and prior to music: it is prior, because music uses its principles for something non-mathematical; it is more certain, because lack of certitude arises from matter’s changes. Hence the closer one gets to matter, the less certain the science.

He lays down the third mode when he says that a science which arises from fewer things is prior and more certain than one which arises from an addition, that is, than one which results from that addition. And he gives the example that geometry is posterior to and less certain than arithmetic: for the things of geometry are the result of adding to the things which pertain to arithmetic. This is easy to see if one admits the postulates of Plato, according to which Aristotle is proceeding here, using them to prove his point (as he frequently uses the opinions of other philosophers as examples in his logical works in order to explain a point).

Now Plato laid it down that “one” is the substance of each thing, because he did not distinguish between “one” which is converted with being and signifies the substance of a thing, and the “one” which is the principle of number and which arithmetic considers. Accordingly, this “one,” as receiving the added characteristic of occupying a position in a continuum, takes on the guise of a point. Hence he said that “one” is a substance not occupying a position; but a point is a substance that does have a position. Consequently, a point adds something to the notion of “one,” namely, position. And just as all numbers not having position are caused from “one,” so from the point, according to the Platonists, all continuous quantities are caused. For a point in motion makes a line; a line in motion makes a plane, and a plane in motion, makes a body. According to this, then, continuous quantities (which are treated in geometry) are the result of additions made to numbers (which are the concern of arithmetic). Hence the Platonists laid it down that numbers are the forms of magnitudes, saying that the point’s form is “one,” and the line’s form is “two,” because it has two endpoints, while a surface’s form is “three,” owing to the first surface’s being the triangle which is terminated by three angles; but the form of bodies is “four,” on the ground that the first solid figure is the triangular pyramid, which is composed of four (non-plane) solid angles, namely, one at the apex and three at the base.

According to this it is obvious that the comparison of the certitude of sciences is here based on two things: for the first mode is taken according to the cause as it is prior to and more certain than the effect; but the other two are taken according to the form as this is more certain than matter, inasmuch as the form is the principle of knowing the matter. But, as it is stated in Metaphysics VII, matter is twofold: one is sensible, according to which the second mode is taken; the other is intelligible, that is, its continuity, according to which the third mode is taken.

And although this third mode was explained according to Plato’s theory, yet even according to Aristotle’s theory, a point results from some addition to “one”: for a point is an indivisible unity in a continuum, abstracting according to reason from sensible matter; but unity abstracts from both sensible and intelligible matter.

Then, at: a single science is one (87a38) he compares sciences one to another according to unity and diversity.

Concerning this he does two things.

First, he shows that there is unity and diversity among sciences both according to subject and according to principles.

Second, he treats concerning both the subjects and principles, at: there is no knowledge by demonstration (88a5; [372]).

Concerning the first he does two things.

First, he shows what makes for unity and diversity of sciences.

Second, he explains something which is needed for understanding what makes for multiplicity of sciences, at: one can have several demonstrations (87b5; [369]).

Concerning the first he does two things.

First, he shows what makes for the unity of a science.

Second, what makes for diversity of sciences, at: one science differs from another (87a41; [365]).

Concerning the first he does two things.

First, he lays it down that the unity of a science is considered from the unity of its generic subject.

Second, he describes the genus which can be the subject of a science, at: namely, all the subjects (87a38; [363]).

He says therefore first (87a38) that a science is said to be one from the fact that it is concerned with one generic subject.

The reason for this is that the process of science of any given thing is, as it were, a movement of reason. Now, the unity of any motion is judged principally from its terminus, as is clear in Physics V. Consequently, the unity of any science must be judged from its end or terminus. But the end or terminus of a science is the genus concerning which the science treats: because in speculative sciences nothing else is sought except a knowledge of some generic subject; in practical sciences what is intended as the end is the construction of its subject. Thus, in geometry the end intended is knowledge of magnitude, which is the subject of geometry; but in the science of building that which is intended as the end is the construction of a house, which is the subject of this art. Therefore, the unity of each science must be considered in terms of the unity of its subject. But just as the unity of one generic subject is more universal than another, for example, being or substance is more common than mobile being, so one science is more general than another. Thus, metaphysics, which treats of being or substance, is more general in scope than physics, which treats of mobile body.

Then, at: namely, all the subjects (87a38) he describes the marks of those genera concerning which there can be sciences.

He lays down two marks.

The first of these is stated when he says: all the subjects constituted out of primary entities, that is, of those subjects of which there is one genus, there is one science.

To understand this it should be noted that, as has been said, the process of science consists in a certain movement of reason passing from one thing to another. But all movement starts from some principle or beginning and is terminated at something definite; hence in the progress of a science, reason must proceed from certain first principles.

Therefore, if there be anything which does not have prior principles from which reason can proceed, there cannot be science of such things, if we take science to mean the effect of demonstration, as we do here. Thus, speculative sciences are not concerned with the very essences of separated substances: for we cannot, through demonstrative sciences, know the essences in them, because the essences of such substances are intelligible of themselves to an intellect proportionate to such intelligibility, and a grasp of the essences of such substances is not obtained by any prior conceptions. The only thing that can be known through speculative sciences about these substances is whether they exist and what they are not; anything else about them is known in terms of likenesses to lower things. But in that case we are really using subsequent things as though they were prior in order to understand them, because things that are subsequent according to nature are prior and better known to us.

And so it is clear that those things concerning which we have science through what is absolutely prior, are of themselves composed of prior items; but things which are known through subsequent items (but prior in reference to us), even though they be simple in themselves, are nevertheless in our knowledge of them composed of things that are first to us.

He lays down the second mark when he says: the parts of this total subject and their per se properties (87a38).

Here it should be noted that the subject of a science can have two types of parts: first, the parts out of which, as out of first things, it is composed, that is, the very principles of the subject; and second, the subjective parts. And although what is stated here can be applied to either of these types of parts, yet it seems to truer of the first type of parts. For in every science there are the principles of its subject, and these must be considered before all else: for example, in natural science the first consideration is about matter and form, and in grammar about the alphabet. But in every science there is also something ultimate, at which the study of that science terminates, namely, that the properties of the subject be manifested. But each of these, namely, the first parts and the properties, can be attributed to something either per se or not per se. For the per se principles and properties of a triangle are not the per se principles and properties of an isosceles triangle precisely as isosceles, but precisely as triangle. Neither are they the per se principles and properties of brass or white, even though the triangle happens to be of brass or be white. Hence, if there were a science which manifests the properties of triangle from the principles of the triangle, the subject of that science would not be isosceles or white or brass, but triangle—whose per se subjective parts would be isosceles equilateral and scalene.

Then, at: one science differs from another (87a41) he shows the reason for sciences being diverse.

First, he lays down this reason.

Second, he explains it, at: this is verified (87b1; [368]).

It should be noted in regard to the first point that although he took the reason for the oneness of a science from the oneness of its generic subject, he does not take the reason for their diversity from the diversity of their subject, but from the diversity of principles. For he says (87a41) that one science is distinct from another when their principles are diverse, in the sense that the principles of the two sciences do not proceed from any prior principles, nor the principles of the one science from those of the other; because if both proceed from the same principles, or if one proceeds from those of another, they would not be diverse sciences.

To understand this it should be noted that a material diversity of objects does not diversity habits, but only a formal diversity. Therefore, since something scientifically knowable is the proper object of a science, the sciences will not be diversified according to a material diversity of, their scientifically knowable objects, but according to their formal diversity. Now, just as the formality of visible is taken from light, through which color is seen, so the formal aspect of a scientifically knowable object is taken according to the principles from which something is scientifically known. Therefore, no matter how diverse certain scientifically knowable objects may be in their nature, so long as they are known through the same principles, they pertain to one science, because they will not differ precisely as scientifically knowable. For they are scientifically knowable in virtue of their own principles. This is made clear by an example, namely, that human voices differ a great deal according to their nature from the sounds of inanimate bodies; but because the consonance of human voices and of the sounds of inanimate bodies is considered according to the same principles, the science of music, which considers both, is one science. On the other hand, if there are things which have the same nature but are considered according to diverse principles, it is obvious that they pertain to diverse sciences. Thus, the mathematical body is never really distinct from a natural body; yet because the mathematical body is known through the principles of quantity, but a natural body through the principles of motion, the science of geometry and the science of nature are not the same. It is clear, therefore, that for sciences to be diverse it is enough that the principles be diverse, this diversity of principles being accompanied by a diversity of scientifically knowable objects. But in order to have one science absolutely, both are required, namely, unity of subject and unity of principles. (That is why above he made mention of unity of subject when he said, “whose domain is one genus,” but made mention of principles when he said, “all the subjects constituted out of the primary entities of the genus.”)

However, it should be further noted that second principles derive their force from the first principles. Hence for diversity of sciences, diversity of first principles is required. But this will not be verified if the principles of diverse things flow from the same principles, as the principles of triangle and square are derived from the principles of figure, or the principles of one are derived from the principles of another, as the principles of isosceles depend on the principles of triangle. Yet it should not be supposed that for the unity of a science it is enough that there be unity of first principles absolutely, but unity of first principles in some scientifically knowable genus. Now, the genera of the scientifically knowable are distinguished according to the diverse modes of knowing: thus things that are defined with matter are known in one mode, and those defined without matter are known in another. Hence the natural body is one genus of the scientifically knowable, and mathematical body is another genus. Hence there are diverse first principles for each of these genera and, consequently, diverse sciences. Furthermore, each of these genera is distinguished into diverse species according to diverse modes and aspects of intelligibility.

Then, at: this is verified (87b1) he explains the reason he laid down.

He says that a sign of the fact that sciences are diversified according to their principles is obtained when one resolves a science into principles that are indemonstrable, for they must be of the same genus as the things demonstrated; because, as stated above, one does not demonstrate by proceeding from an alien genus. Now, the reason why the reaching of indemonstrable principles of one genus is taken as a sign is that the facts demonstrated by them are in the same genus and are co-generic, that is, connatural, or generically proximate to them: for these have the same principles. And so it is clear that the unity of a scientifically knowable genus, precisely as it is the scientifically knowable genus from which was taken the oneness of the science, and the unity of the principles, according to which was taken th diversity among the sciences, mutually correspond.

Then, at: one can have several demonstrations (87b5) he shows how one conclusion can be demonstrated through several principles.

And this can happen in two ways.

In one way, when a number of middles are laid down in the same coordination, and one of these middles is used in one demonstration and another one in another, both leading to the same conclusion. This, of course, requires that the middles be non-continuous. Thus, if the two extremes, A and B, are 'to have three' and 'isosceles' respectively, but the middles, C and D are coordinate, as 'triangle' and 'this type of figure', A could be demonstrated of B with two demonstrations. In the first one, C could be taken as middle, and D in the other, so that in neither demonstration will, there be a middle continuous with the extremes, because in one demonstration a middle continuous with one extreme and not continuous with the other would be taken, and in the other demonstration the converse would be taken.

Or this can happen in a second way, namely, when diverse middles are taken from diverse coordinations; for example, if A, which is the major extreme, were to be altered, and B, which is the minor extreme, were “to be pleased,” and the middles taken were independent, say E would be “to be at rest,” and D “to be in motion.” In this case the same conclusion can be reached through diverse middles that are not of one order. Thus one demonstration would be this: “Whatever is at rest is altered, because it belongs to the same thing to be at rest and to be altered; but whatever is pleased is at rest, because rest in the desired good causes pleasure: therefore, whatever is pleased is altered.” The other demonstration would be: “Whatever is in motion is altered; but whatever is pleased is in motion, because pleasure is a movement of the appetitive faculty [power]: therefore, whatever is pleased is altered.”

Now the statement, “Whatever is pleased is in motion," is Plato’s opinion and is verified in sensible pleasures which involve movement. The other statement, “Whatever is pleased is at rest,” is true according to Aristotle’s opinion, as is clear from Ethics VII and X. And this is verified particularly in intellectual pleasures.

Then he says that just as this has been proved in the first figure, so in the other figures it is easy to see that the same conclusion can be syllogized with diverse middles. Now, the Philosopher mentioned this to show that diverse middles of demonstration sometimes pertain to the same science, as when they are taken from the same coordination, and sometimes to diverse sciences, when they are taken from a different coordination. Thus, astronomy demonstrates that the earth is round, using one middle, namely, the eclipse of the sun and moon; and natural science uses another middle, namely, the motion of heavy objects tending toward the center, as it is stated in Physics II.

Then, when he says: a further point (87b5), he reviews what has been said.

And he says that what has here been shown in the first figure can also be easily considered in the other figures, since the same conclusion follows from the diverse syllogisms.