Commentaria in Primam Partem Summae Theologiae

In the body, one conclusion, answering the question affirmatively: Every being, insofar as it is a being, is good. And there is added ly insofar as in the conclusion, to show that to be good does not belong to every being per accidens, but per se. – It is proved thus. Every being, as such, is in act: therefore perfect: therefore desirable and good. – All things are clear. The first consequence is proved: because every act is a certain perfection.

II. In response to the fourth, note that the text says two things. First it verifies, assigning the reason why mathematical things are not good; second, it answers the argument¹. – The first indeed is founded on this, that things are good, as will be clear below², either because they exist (sunt), or because they are such, or because they are ordered to the good. But mathematics, as such, abstract from all of these: from being (esse) indeed, because as such they do not exist; but from being such, for the same reason, since being such presupposes being (esse); and from order to the good, because they abstract from an end. It is proved: the end has the character of a mover, but these abstract from matter and motion.

III. Concerning this part two doubts arise. First concerns that argument, namely that mathematical things are not good by primary goodness because they do not subsist separately. For if this argument holds, it follows that no thing taken universally is good, since no thing taken universally subsists, according to the Peripatetics. The consequent is false: therefore.

According to there is a doubt concerning that argument, namely, that the mathematical sciences abstract from the end because they abstract more from matter and motion, since the end has the character of a mover. For this argument seems to fail in two ways. First, because it would follow that metaphysical things abstract from the end: for they abstract more from matter and motion than the mathematical sciences do. Second, because a equivocation occurs with respect to motion: since the end does not have the character of a mover by motion properly so called, from which the mathematician abstracts; but by motion spoken of metaphorically. Therefore, although the conclusion is true, the argument is nonetheless worthless.

IV. To the first doubt it is said that mathematical things can be taken in two ways. In one way, according to what they are absolutely. And thus the judgment concerning them and natural things is the same with respect to being (esse) and goodness: for a triangle is no more abstracted from being (ens) than whiteness is. And thus there is no question here about them in this way; but it can be said of them that they are good in the manner in which beings (entia) are good, namely according to their individuals existing in the nature of things, etc.

They are considered in another way as mathematical things, insofar as they are subject to that mode of abstraction. And in this way it is denied that the judgment about them is the same as about other things taken universally. – For the clarification of this, know that although all universals agree in this, that they do not subsist in their universality; nevertheless among physical, mathematical, and metaphysical universals there is a great difference with respect to subsisting in their proper abstractions. Metaphysical things indeed, when taken according to their proper abstraction, do subsist: since they have in the nature of things individuals abstracting from all sensible and intelligible matter, as is clear in the case of intelligences. Mathematical things, however, when taken according to their proper abstraction, do not exist: since they have no individual in the nature of things abstracting from sensible matter; for this line is not found except in the termination of a sensible body. Natural things, however, since they have no proper abstraction of their own, but only that which is common to all sensibles, namely by which the universal is abstracted from the particular; it is manifestly clear that they subsist in the nature of things, while they have individuals subsisting with sensible matter and the other conditions known to you universally. – Therefore, since in the text it is said that mathematical things do not subsist, it must not be interpreted to mean that mathematical universals taken universally do not subsist (for it would be ridiculous to bring forward this reason): but that mathematical things as such, taken particularly, do not subsist; or, which is the same thing, that mathematical things as such have no individual existing in the nature of things. And therefore they are neither in the universal nor in the particular: and on this account they cannot be beings. Which cannot be said of other things taken universally. And thus the nullity of the consequence made to the opposite is evident: and why it is said singularly of mathematical things that they do not have being.

V. To the second doubt, however³, it is answered that, since statements must always be interpreted according to the subject matter, therefore, although the inference is not valid, it abstracts from matter and motion, therefore from an end, as is clear in Book XII of the Metaphysics, text 37⁴, where an end is found in immobile things; yet this inference is valid: mathematical entities abstract from matter and motion, therefore from an end: because mathematical entities are not by nature apt to have an end except the end of matter and motion, since in reality they are natural beings (entia naturalia). For if they abstract from matter and motion, they must abstract from the end of matter and motion; and if they abstract from such an end, they abstract entirely from an end, because they are not by nature apt to have an end in any other way.

And by this the response to the second objection is also clear. For when it is said, the end has the character of a mover, ly mover can signify the pure causality of the end: and thus "mover" is said equivocally with respect to motion properly so called. But it is not taken in this way here. – It can also signify both the causality of the end and its effect: and then it is equivalent to what I say, causing motion. And thus it is taken in the matter at hand. Nor is there any equivocation: but motion is taken properly on both sides, and it is specified what sort of end that of the mathematicians is. As if he were saying that, because the end of the mathematicians has the character of a mover, that is, of causing motion properly so called, just as does the end of natural things, it follows that, if mathematics abstracts from matter and motion, it also abstracts from an end. – And thus both objections are resolved from the same root.

VI. Secondly, in the same reply⁵ he satisfies the argument. And the argument is indeed this: Mathematical things are not good; mathematical things are beings; therefore some beings are not good (in the second mode of the third figure). – The reply, however, consists in the denial of the argument: because it is a sophism of figure of speech, or from the qualified to the unqualified. For mathematical things seem indeed to signify beings: and yet they signify beings abstracted in this way. And therefore in the text both the inferred conclusion is denied, namely, that some beings are not good; and the conclusion that ought to be inferred is conceded, namely, that some beings, as they are subject to such abstraction, are not good. Nor is this unfitting, as is said in the text: because the notion of being also abstracts from the notion of the good. Just as, because the notion of man is prior to risibility, man can be taken in some prior aspect (because in the first mode of speaking per se), in which there belongs to him to be a rational animal, and there does not belong to him to be risible: nor nevertheless can it be inferred, therefore some man is not risible; but rather, therefore some man, in some prior aspect, abstracts from risibility, in which he does not abstract from the notion of man.

VII. Concerning this part a doubt arises, how these two stand together: mathematical entities, formally speaking, are neither in act nor in potency, as was said⁶; and, mathematical entities have the character of being: since being is said only of that which is in act or in potency.

To this it may be briefly said that these two propositions, rightly understood, stand together. For the first does not deny being universally: rather, it denies being thus, namely in such a mode of abstraction. For a triangle abstracted from sensible matter neither is, nor can be (namely according to such a mode of being), in the nature of things. And this is what is said at the beginning of this response. – The second, however, affirms confusedly some being. Now it is clear that, with the negation of one mode of being, there stands the affirmation of being in general of the same subject: because it can be verified according to another mode of being. Mathematical things, therefore, both are not beings in such a way, and absolutely are beings.

VIII. And if it is pressed against our position: Therefore the same must be said, that they are not good in such a way, but are absolutely good: and consequently it is badly said that mathematical things are not good — this consequence must be denied. For in order for something to have formally the ratio of being (ens), it suffices that it be of such a nature in itself that existence in the order of real things is not repugnant to it, however this may come about, whether through its own abstraction or not. But for something to have formally the ratio of the good, beyond this it is required that the thing itself be taken in relation to esse, or to an end, etc. Now, a triangle and other mathematical objects, according to their quiddities, possess this, namely that existing is not repugnant to them; and therefore they retain formally the ratio of being. But because they are not considered in relation to esse and to an end in the order of real things, therefore they abstract from the ratio of the good formally, though not fundamentally; and therefore esse is denied of them. – A manifest sign of this is that no conclusion there is demonstrated on account of the good or of better being; no demonstration there proceeds through a final or efficient cause (which look to esse in the order of real things), but only through the formal cause: for that belongs properly to the good, this to being (ens).