Disputationes metaphysicae

1. This question can be treated in a twofold sense according to the twofold acceptation of numerical unity, namely either for quantitative unity or for singular unity; for so Aristotle said, Metaphysics III, text 14: Whether you say one in number or singular makes no difference; for in this way we expound the singular as that which is one in number. According to these two acceptations, then, two questions or opinions are to be treated.

Various opinions on quantitative unity

2. The first opinion was that of Avicenna, which St. Thomas reports, I, q. 11, a. 1, and In I, dist. 14, q. 1, a. 3, and In Metaph. IV, lect. 2, and the Commentator in the same place, comm. 3, who asserted that the unity which comes to beings from quantity is the very transcendental unity of things; whence he concluded that one adds quantity over being. Which opinion, taken so universally, is plainly false; both because it will be necessary to say either that there are no beings except quantified ones, or that unity is not an adequate attribute of being, but proper to material beings; and also because in material and quantified beings themselves there exists a proper unity and integrity, which each being has in itself, which is not formally from quantity, although it is not without quantity, as was touched upon in sect. 3, and will be said again in the following disputation and in the disputation on quantity.

3. But if this opinion only asserted that the unity of quantity is one of the transcendental unities, it would have some apparent probability. For, just as being is common to quantity and to the other predicaments, so transcendental one is common to all of them; whence, just as being is determined to the individual predicaments through proper modes of entity, and transcendental being, as included in the individual predicaments, is nothing other than such or such being, so transcendental one is determined to the individual predicaments through such modes of unity, and transcendental one as included in the individual predicaments will be nothing other than such or such one. Therefore transcendental unity in quantity is nothing other than such unity as quantity by its nature postulates. Therefore in this way it seems it can be said that quantitative unity is not distinguished from transcendental, not because it is convertible with it, but because it is a certain subjective part of it, in the way in which it can also be said that quantity is not distinguished from being. And in this way Durandus taught that opinion, In I, dist. 24, q. 2, who consequently says that in the individual predicaments special modes of unity and multitude can be distinguished, yet that there are no peculiar names imposed to signify these except in quantity, because the notion of divisibility or of the divisible continuum, from whose division number results, as is said in Physics III, c. 7, is found in a peculiar mode in quantity.

4. And this opinion proceeds by distinguishing quantity from the quantified thing; for those who think that quantity adds nothing to substance consequently say that transcendental one and that which is the principle of number are in no way distinguished among themselves, which Gregory tries to defend by many reasons, In I, dist. 24, qq. 1 and 2. Not that he thinks every one is quantified, but conversely, because he thinks every one is transcendental, and quantitative one is nothing other than transcendental one in substance or in material accidents. But of the distinction of quantity from the quantified thing we must speak elsewhere; now we suppose that quantity has an entity distinct from the others. Which being supposed, the opinion of Durandus can be confirmed by this reason; for being and transcendental one are convertible; therefore of whatever the one is said, the other too is predicated; but quantity is being; therefore it is transcendental one; but it is not transcendental one except by its quantitative unity; therefore that is contained under transcendental unity. Otherwise it would be necessary to feign two unities in one quantity, one transcendental, the other quantitative, which is superfluous and can scarcely be conceived by the mind.

5. But Averroes, above, and St. Thomas disprove this opinion of Avicenna, who consequently teach that quantitative unity differs from transcendental in this, that it adds to being something positive and real which transcendental unity does not add. From which some elicit that quantitative unity adds to quantity itself something positive, and in this way differs from the transcendental unity of quantity itself, and by reason of that positive thing is the principle of number, which is posited as a species of quantity, whereas transcendental multitude is not judged to make up a proper and real species of being. So Javellus holds, Metaphysics IV, q. 8, with Capreolus, In I, dist. 24, sole question, where he adduces various places of St. Thomas, and he is founded chiefly on this, that if quantitative unity were not something positive besides the negative indivision of quantity, neither would quantitative number be some certain real species, nor would there be any reason for constituting that number among the species of things rather than the number of any beings whatever. But if you ask what it is which unity adds over quantity, Capreolus seems to answer, in the explanation of the first conclusion, that in quantity two things are to be distinguished, namely the form of continuity, and the being which such continuous quantity gives to the subject; and he indicates that unity adds over quantity this latter being: Because the form (he says) of continuity is divisible into many, and therefore does not deserve to be called unity, of whose notion is indivisibility; but the being which that form gives to the subject is called unity, because it is in no way divisible, because, if that form or the subject is divided, the first being does not remain, nor is it divided, but each part of the divided thing has a new being and does not have that which it had before. And he cites Avicenna, Metaphysics III, c. 2, saying that unity is substantially that very thing which is not divided. Whence Capreolus indicates that this positive thing which quantitative unity adds over quantity is really distinct from it. But Giles, in the same distinction, q. 2, dub. 2, holding the same opinion, thinks of the positive addition that that addition is distinguished by reason only in this way, that quantity, in so far as it extends the parts of substance, is called continuous quantity, but in so far as it perfects substance, no account being taken of the parts, is called unity. Others, however, say that quantitative unity adds over quantity the notion of measure, from St. Thomas, I, q. 11, a. 2, and In Metaph. IV, lect. 3.

6. But this opinion is without doubt false, as Hervaeus, Thomas of Strasbourg, Durandus, Gregory and others noted in the said distinction. For that distinction which Capreolus feigns is fictitious and plainly unintelligible. Both because the being which quantity gives (according to the truer opinion) is not a thing distinct from actual and real quantity itself, as will be said generally below; and also because, granted that it were distinct, it would nevertheless not be more distinguished in quantity than the form of whiteness is distinguished in whiteness from the being which it gives to the subject, or in the form of stone the form itself from the being which it gives to matter; therefore either this distinction is not enough for the unity of quantity to be said to add something over quantity, or the same must be said of the unity of whiteness and of any other created form whatever; and moreover because, that being distinct from quantity being admitted, it is feigned without foundation to be more indivisible than quantity itself; for who would believe that, when wood is divided, in the individual divided parts the same quantity remains which was before, and that the same being of quantity which quantity itself conferred does not remain? For there is no greater reason for the corruption of one than of the other; indeed all too who think that being is really distinguished from the form from which it comes think at least that naturally and by ordinary power being is inseparable from it while it remains. But if perhaps Capreolus says that he is not speaking of the being of existence of quantity, but of a certain other distinct from quantity and its existence, this indeed needs no refutation, because it is a mere fiction devised without foundation. And moreover, in whatever way that being is explained, it cannot be more indivisible than quantity itself, because, just as the whole quantity gives being to the whole subject, so a part of quantity to a part of the subject, nor from the partial quantity which is in one part of the subject does any being come which is in the other parts of the subject, nor conversely; that being, therefore, must necessarily be coextended with quantity itself, nor can it be more indivisible than it. Especially since it is material and corporeal, and in a divisible subject. Nor is it true, what Capreolus says, that the form of continuity cannot be called unity because indivisibility is of the notion of unity; for above it has been proved by us that not indivisibility but only indivision is of the notion of unity, and therefore nothing prevents the form of unity, that is, in which unity is founded, from being divisible, provided it is undivided. This declaration of Capreolus, therefore, stands by no reason.

7. The other distinction of Giles too is none at all, except only in words; both because quantity does not perfect substance otherwise than by extending it into parts, and also because it is a contradiction to say that quantity has the notion of numerical unity in so far as it abstracts from parts; for it is of the intrinsic notion of this unity that it be in a quantified subject having parts, and it adds almost this alone over transcendental unity. The other exposition, finally, about the notion of measure is easily refuted; both because the notion of measure does not pertain per se primarily to unity, but follows upon it, as St. Thomas rightly teaches, In Metaph. IV, lect. 8, and Book X, lect. 2. And also because this notion of measure, if it is considered as regards aptitude, which is in the thing itself, is found in the same proportion in transcendental one with respect to multitude as in quantitative one with respect to the number of quantity; but if it is considered as regards the use of men, this matters little to the notion of unity as it exists in itself. Finally, against this whole opinion there stand both the testimonies of Aristotle and all the reasons by which it was proved above that the notion of one is placed in negation; for all of these proceed in the same proportion in numerical or quantitative unity, as is evident from Aristotle, Metaphysics V, text 11, and Book X, from the beginning, and from the reason given above, because, after anyone has feigned an entity or positive notion which this unity adds over quantity, if he understands it in the mind or in reality to be separated, and quantity alone to remain undivided, he will understand that one quantity remains; therefore whatever is added is added superfluously and without foundation.

The true notion of quantitative unity is declared

8. It is to be noticed, then, in order that both the thing itself and the opinion of St. Thomas in this part may be understood, what we were saying above, at the end of sect. 3, that in material substance there can be considered intrinsic and entitative unity, and accidental unity belonging to it denominatively from the affection of quantity; whence it comes about that quantity itself can be considered either as it is one in itself, or as by reason of it that which is affected by it is rendered one. This being supposed, it must first of all be said as regards this part that transcendental unity is not the quantitative unity which belongs to material substance denominatively and accidentally by means of quantity. This is evident from what has been said above, because neither in other things does transcendental unity consist in this, as is self-evident, nor in substance itself, or any other material form whatever, as is evident from what was said in sect. 3. For transcendental unity is most intrinsic to each being and belongs by virtue of its entity alone with a negation adjoined, as has been shown; whence, quantity being removed, intrinsic transcendental entitative unity is understood even in material substance. And it is confirmed, because substance, as it is one through quantity, is simply speaking one per accidens in the genus of being, since it consists of things of diverse predicaments; but substance, as it is transcendentally one, is not one per accidens, but per se; otherwise material substance could not have unity per se in its proper genus, which is very absurd. And the arguments which Gregory adduces in the aforesaid place also rightly prove this.

9. Next it must be said that in quantity itself the unity by which it itself is one is not other than the transcendental unity applied¹ to such a being, namely to quantity; this too the arguments of Durandus rightly prove, and those which we adduced in reporting his opinion. And from these it follows, first, that numerical or quantitative unity with respect to quantity adds nothing besides indivision, which is only a certain negation, as was said above. So hold the other authors cited, and Soncinas, Metaphysics X, q. 2; Soto, on the Categories, chapter on Quantity, q. 1. And the reasons made against Capreolus and others prove this. And it is evident also, because this unity is transcendental with respect to quantity; therefore it is of the same notion in it as it is in other things; just as, therefore, in other beings unity is the undivided entity of each, so in quantity its unity is quantity itself or the continuity of quantity as undivided. You will say: when continuous quantity is divided, its quantitative unity is lost and number results, as is said in Physics III, and yet transcendental unity is not lost, because, as we were saying above, the same entity remains in the individual divided parts which they had in the continuum; therefore they are diverse unities. The answer is by denying the minor, because, although the divided parts retain the same entity which they had as partial in the continuum, nevertheless simply the entity of the continuum was not placed in those parts alone, but also in their union and the term by which they were joined, or rather in that whole composite which resulted from all these, whose unity, both quantitative and transcendental, consisted in that whole as undivided; whence that unity is destroyed by division because the same indivision does not remain which was before, and because that whole is dissolved in the notion of a certain whole.

10. Secondly it is inferred from what has been said that quantitative or numerical unity, in so far as it belongs denominatively to substance, adds over substance a positive thing under a negation, namely quantity itself with indivision. So I think St. Thomas is to be understood; and Fonseca rightly noted this, Metaphysics V, c. 13, q. 3, sect. 3, from the same St. Thomas, In Metaph. IV, lect. 3. And it is easily evident, because, just as substance is quantified through quantity adjoined to it, so also it is one quantitatively on account of the one quantity which it has in itself; likewise hence it often comes about that substances which in their genus do not have transcendental unity are in some way called one quantitatively by reason of quantity, in which way a rod is called one which is partly green, partly dry.

11. Thirdly, finally, it is evident from what has been said how transcendental unity is related to quantitative unity; for in no sense are they convertibly the same, as the things adduced above evidently prove. If, however, we speak of quantitative unity according to itself or with respect to quantity, then they are related as inferior and superior; for it has been shown that quantitative unity is a certain transcendental unity, although not every transcendental unity is also quantitative. Whence transcendental one is found in all the predicaments; but the unity of quantity in the one predicament alone. But if we speak of quantitative unity as it belongs accidentally and denominatively to substance, or to other material accidents through quantity, then it differs from transcendental unity; for the latter is intrinsic and per se accompanying the entity of the thing; but the former is as it were extrinsic, in so far as it is adjoined through another entity. But because this numerical unity too, with respect to substance, is real, it is therefore necessary that it be in some way contained under the transcendental; and it is contained in the way in which substance as quantified is contained under being; just as, therefore, substance by the adjoining of quantity is accidentally quantified, and from both results a certain being composed per accidens, so it is also accidentally one, and that unity which results from both is also per accidens and proportioned to such a composite, and with respect to it is transcendental, although with respect to substance it is accidental. And from these things it is also evident that quantitative number and multitude keep among themselves the same proportion as numerical and transcendental one keep. But what is to be said of the notion of number, namely whether it constitutes a proper genus and species of being, and whether in this it differs from multitude, is to be said below in treating of quantity.

Transcendental unity is compared with singular unity

12. The second opinion to be examined in this place is that of some who say that transcendental unity is the individual unity of each singular being, which is also wont to be called numerical unity, in a diverse signification from that which was treated in the preceding opinion; for there numerical unity was taken for that which is the principle of quantitative number. For, because our knowledge begins from sense, and through sense we can number those things which have magnitude, therefore the multitude of these things has specially received the name of number, and therefore in this special mode the unity founded in quantity is called numerical unity. But here numerical unity is taken more broadly as it is distinguished against generic or specific, and is found in any singular being whatever, and it is called numerical in so far as from it multitude or transcendental number can be made up; but it is called individual or singular in so far as, as such, it is incommunicable to many inferiors, in which it differs from formal unity, both specific and generic. This opinion says, then, that this singular unity is the proper and real attribute of being, or transcendental unity. The foundation can be that transcendental unity, which is an attribute of real being, must be real unity; but in things there is no real unity besides numerical and singular unity; therefore that alone is the adequate attribute of being; therefore that alone is transcendental unity. And it is confirmed, for this unity is convertible with real being; for everything which is one in this way is real being. For although a being of reason can be called one in number, and so genera, species and individuals are distinguished by the dialecticians in beings of reason themselves, nevertheless that is as it were equivocally, by mere extrinsic denomination; but properly and rigorously speaking, whatever is simply one in number is real being; and conversely every real being is singular, and consequently one in number, because nothing is called real being except what is in things, or what is capable of real being; but in things there is nothing except the singular, nor is anything capable of real existence except an individual thing; therefore real being and one in number or singular are convertible; therefore the aforesaid singular unity is transcendental unity or an attribute of being.

13. But this opinion, if we attend to the common mode of speaking and conceiving, can seem difficult and singular. For first of all Aristotle, Metaphysics V, c. 6, text 11, among the unities of being posits not only singular unity, but also specific, generic and analogous, and universally he says: Whatever things do not have division, in so far as they do not have it, are so called one. And in Book IV, text 6, he gathers hence that it pertains to metaphysics to treat of genus and species, because these are in being, namely as comprehended among its attributes; but they are not comprehended except in so far as they are contained under unity. Moreover, the adequate attribute of some common object must belong not only to the inferiors contained under such an object, but also to the common object itself taken according to itself and formally; therefore to be one must belong not only to individual beings, but also to being as such; but as such it is not one in number, but only analogically and by formal notion; therefore. A similar argument is that of whatever being is said, transcendental one is said; but being is said not only of individuals, namely Peter and Paul, but also of common notions precisely taken, as of man, horse, etc.; for, although perhaps in these universality is through reason, as we shall say below, nevertheless the universal natures themselves are real, and are called true real beings; and so Aristotle, Peri Hermeneias I, c. 5, distinguishes that of things some are universal, others particular; man as such, therefore, is real being; therefore one by transcendental unity; and yet he is not one by singular and numerical unity; therefore transcendental unity extends more widely than individual or singular unity.

Resolution of the question

14. This difficulty depends much on the understanding of that difficult question about transcendental² and numerical unity, how they differ, and whether each is a true unity existing in things, and consequently whether universal unity too is in things. But since these must be treated singly for the exact explanation of this matter, it is now briefly said that the transcendental unity of which we are now treating is not to be limited to singular or universal, material or formal unity, but comprehends every unity which can be found in some real being or in the formal notion of real being. Transcendental unity, therefore, will be any notion whatever of real being per se, in so far as it is undivided adequately and according to itself. I say of being per se, in the way in which it was explained above by us to be the object of metaphysics; for, if it is taken more broadly, it has no unity, even formal, as has been said. I say also as undivided, because neither the notion of being alone nor the negation alone makes unity, but the undivided notion of being, as was shown above. Finally I say adequately and according to itself, because unity in each thing or real objective concept must be attended to according to its proper notion, and not according to some superior, inferior or extraneous one; for these are either inadequate or in a certain way per accidens; but the unity of each is adequate to it and belongs to it per se. Whence man is called one essentially because according to his proper adequate notion he is not divided in essential notion, although in individuals he is entitatively divided; but man will not be called simply one in that way because he is undivided in the notion of animal; for that indivision would not suffice for the aforesaid unity, because it does not belong to man adequately and per se primarily, but by reason of the superior. From these things, then, the notion of unity taken in common has been sufficiently explained. But in order that it may be explained more distinctly and radically, it is necessary that we speak in particular of individual, formal and universal unity, which are the proper metaphysical unities.