Commentary Aristotle Posterior Analytics
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 of 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.