Commentary Aristotle Posterior Analytics
Lecture 15
Demonstration does not skip from one genus to an alien genus a38. It follows that we cannot in demonstrating pass from one genus to another. We cannot, for instance, prove geometrical truths by arithmetic.
a39. For there are three elements in demonstration: (1) what is proved, the conclusion — an attribute inhering essentially in a genus;
(2) the axioms, that is, axioms which are premises of demonstration;
(3) the subject — genus whose attributes, that is, essential properties, are revealed by the demonstration.
b2. The axioms which are premises of demonstration may be identical in two or more sciences: but in the case of two different genera such as arithmetic and geometry you cannot apply arithmetical demonstration to the properties of magnitudes unless the magnitudes in question are numbers. How in certain cases transference is possible I will explain later. Arithmetical demonstration and the other sciences likewise possess, each of them, their own genera;
b8. so that if the demonstration is to pass from one sphere to another, the genus must be either absolutely or to some extent the same.
b10. If this is not so, transference is clearly impossible, because the extreme and the middle terms must be drawn from the same genus: otherwise, as predicated, they will not be essential and will thus be accidents.
b13. That is why it cannot be proved by geometry that opposites fall under one science, nor even that the product of two cubes is a cube. Nor can the theorem of any one science be demonstrated by means of another science, unless these theorems are related as subordinate to superior (e.g., as optical theorems to geometry or harmonic theorems to arithmetic).
b17. Geometry again cannot prove of lines any property which they do not possess qua lines, that is, in virtue of the fundamental truths of their peculiar genus: it cannot show, for example, that the straight line is the most beautiful of lines or the contrary of the circle; for these qualities do not belong to lines in virtue of their peculiar genus, but through some property which it shares with other genera.
It follows that we cannot. After showing that demonstration is from things that are per se, the Philosopher here concludes that demonstration is from principles which are proper and not from extraneous or common principles.
His treatment falls into two parts.
In the first he shows that demonstration proceeds from proper principles.
In the second he establishes which principles are proper and which common, at: it is hard to be sure (76a26; [149]).
In the first he shows that demonstration does not proceed from extraneous principles;
in the second that it does not proceed from common principles, at: it is clear that if the conclusion (75b37; [143]).
In the first, from the preceding, he shows that demonstration is not from extraneous principles.
In the second, also from the preceding, he shows that demonstrations do not bear on corruptible things but on eternal, at: it is also clear that if the premises (75b21; [135]).
Concerning the first he does three things.
First, he states his proposition.
Second, he proves it, at: for there are three elements (75a39).
Third, he concludes to what he intended, at: that is why it cannot be proved (75b13).
He says therefore first (75a38) that inasmuch as demonstration is from things that are 'per se', it is plain that demonstration does not consist in descending or skipping from one genus to another, as geometry, demonstrating from its own principles, does not descend to something in arithmetic.
Then, at: for there are three elements (75a39) he proves this proposition.
About this he does three things.
First, he lays down the things that are necessary for a demonstration, saying that “three things are necessary in demonstrations.
One is that which is demonstrated,” namely, the conclusion which, as a matter of fact, contains within itself that which inheres in its genus per se. For through demonstration a proper attribute is concluded of its proper subject.
Another is the dignities [maxims or axioms] from which demonstration proceeds.
The third is the generic subject whose proper attributes and per se accidents demonstration reveals.
Second, at: the axioms which are premises (75b3), he shows which of the three aforesaid can be common to various sciences and which cannot, saying that one of the three, namely, the dignities, from which demonstration proceeds, happens to be the same in diverse demonstrations and even in diverse sciences. But in those sciences whose respective generic subject is diverse, as in arithmetic which is concerned with numbers, and in geometry which is concerned with magnitudes, it does not occur that a demonstration which starts with the principles of one science, say, of arithmetic, descends to the subjects of another science, say to magnitudes, which pertain to geometry; unless perchance the subject of one science should be contained under the subject of the other, for example, if magnitudes should be contained under numbers. (How this occurs, namely, how the subject of one science may be contained under the subject of another, will be discussed later. For magnitudes are not contained under numbers except perhaps in the sense that magnitudes are numbered.) In any case, the subjects of diverse demonstrations or sciences are diverse. For an arithmetical demonstration always has its proper genus in respect to which it demonstrates, and so do the other sciences.
Third, at: so that if the demonstration (75b8), he proves his proposition.
Here he does two things.
First he brings in the main intent after the manner of a conclusion on the ground that it can be obtained from the aforesaid, saying: Hence it is clear that it is necessary either that the genus with which the principles and conclusions deal be absolutely the same (in which case there is no descent or skipping from one genus to another); or if the demonstration is to descend from one genus to another, the two must be one thus, that is, somehow. For otherwise it is impossible for a conclusion to be demonstrated from principles, since there is not the same genus either absolutely or in a qualified sense.
However, it should be noted that a genus absolutely the same is being taken when on the part of the subject no determinate difference is admitted which is alien to the nature of that genus: for example, if someone using principles verified of triangle should proceed to demonstrate something about isosceles or about any other species of triangle.
But a genus is one in a qualified sense when a difference alien to the nature of the genus is admitted of the subject, as “visual” is alien to the genus of line, and “sound” to the genus of number. Therefore, number absolutely, which is the generic subject of arithmetic, and sonant number, which is the generic subject of music, are not one genus absolutely speaking. The same goes for line absolutely, which geometry considers, and the visual line considered in optics. Hence it is clear that when matters pertaining absolutely to the line are applied to the visual line, a descent is being made to another genus—which is not the case when matters pertaining to triangle are applied to isosceles.
Second, at: if this is not so (75b10), he manifests the proposition in this way. In demonstration the middles and the extremes must belong to the same genus. Now, the extremes are contained in the conclusion: for the major extreme in the conclusion is its predicate, and the minor its subject. But the middle is contained in the premises. It is required, therefore, that the principles and conclusion be taken with respect to the same genus. When we add to this the fact that diverse sciences are of necessity concerned with subjects generically diverse, it follows that from the principles of one science, something in another science not under it may not be concluded. That the middles and extremes in a demonstration must be of one genus he proves thus:
Suppose that a middle belongs to a genus other than that of the extremes which might be, for example, 'triangle' and 'have three angles equal to two right angles.' Now, a proper attribute concluded of triangle is in it per se but is not in "brazen" per se; or, conversely, if the proper attribute is per se in “brazen,” say, “high-sounding” or the like, it is obviously in triangle per accidens. This example makes it plain that if the subject of the conclusion and the middle are in entirely different genera, then the proper attribute is either not in the middle per se or else not in the subject per se. Consequently, it is in one of them per accidens. If it is in the middle per accidens, something will be in the premises per accidens. But if it is in the subject per accidens, something will be per accidens in the conclusion: and this on the part of a proper attribute. But either way something will be per accidens in the premises, inasmuch as the subject is subsumed under the middle, for example, triangle under brazen, or vice versa. However, it has been previously established that in a demonstration both the conclusion and the premises are per se and not per accidens. It is required, therefore, in demonstrations that the middle and the extremes be of the same genus.
Then, at: that is why it cannot be proved (75b13) he infers two conclusions from the foregoing.
The first conclusion is that no science demonstrates anything about the subject of another science, when this other science is more common than or entirely disparate from the first. Thus, geometry does not demonstrate that one and the same science deals with the both of two contraries: for contraries pertain to the common science, namely, to first philosophy or to dialectics. In like manner, geometry does not demonstrate that “two cubes are one cube,” that is, that the product of one cube and another cube is a cube. “Cube” here means the number which results from multiplying a number by itself twice: for example, 8 is a cube, for it is the result of multiplying 2 by itself twice, for 2 times 2 times 2 are 8. Similarly, 27 is a cube, whose root is 3, because 3 times 3 times 3 make 27. Now, if 8 be multiplied by 27 the product is 216, which is a cube whose root is 6, because 6 times 6 times 6 make 216. Therefore, arithmetic, not geometry, has the power to prove this. Similarly, that which pertains to one science cannot be proved by another science unless the one happens to be under the other, as optics is under geometry, and consonance (or harmony, that is, music,) is under arithmetic.
The second conclusion, at: geometry again cannot prove (75b17) states that a science cannot prove just any random accident of its subject, but the accidents proper to its genus. Thus, if something belongs to lines not as lines or not according to the proper principles of lines, geometry does not demonstrate it of lines: for example, that a straight line is the most beautiful of lines, or whether straight line is contrary or not to the curved. For these matters are outside the proper genus of line and belong to something more general. For beauty and contrary transcend the genus of line.