Cursus Philosophicus Thomisticus, Secunda Pars Artis Logicae

There is no less difficulty in understanding the formal character of quantity which, by its extension, unites parts and is continuous quantity than in understanding that which, by its extension, separates them and is discrete quantity, or number. The familiar distinction between numbering number and numbered number must be ASSUMED. Numbering number is that which is the principle of numbering in the intellect, such as two, three, four, and so on: these are the principles by which we number every sort of matter. Numbered number, by contrast, is the things themselves, or the matter, that is subjected to this numbering. Numbered number may be taken generally for every multitude in any way numerable by the intellect, including one found among spiritual things; or, more particularly, for quantitative multitude, which, by reason of quantity, has a special character of measure in numbering. For this reason, the very discreteness of quantity, relative to the substance subjected to it, is sometimes called numbering number—that is, it supplies the principle by which something is numerically measurable as quantitative. This is taken from St. Thomas, Opusculum 48, treatise on Quantity, ch. 1.¹

THE DIFFICULTY, THEN, lies in explaining two things. First, whether this character of number in discrete quantity is something one per se, sufficient to constitute a true species of quantity. Second, from what form number derives its unity, given that it has parts really separate and divided.

THE OPINION OF MANY, THEN, is that, considering the multitude of which number consists, they have said that number is not a being per se but per accidens, and consequently must be excluded from the per se species of quantity. This view may be found in Fonseca, Metaphysics V, ch. 7, q. 5, sections 1 and 2; Suárez, Disputationes Metaphysicae 41, section 1; Vázquez, Prima Pars 26, ch. 3; and the Coimbra commentators, here, q. 2, art. 1.

Others, on the contrary, affirm that number is one per se by the unity of discrete quantity, although it is not one in the genus of continuous quantity. Both the Thomist and Scotist schools commonly follow this view, but they do not all explain its unity in the same way. Some say that number is one by unity of order, a view grounded in St. Thomas, Metaphysics VII, final lecture, on text 60², where he says that a syllable and a number receive their species from order. But, if taken precisely, this unity seems more relational than quantitative. Others say that the last unit gives number its form and unity, as St. Thomas explicitly says in Metaphysics VIII, lect. 3³; this must be understood of the last unit in order to the others. There is, however, a difficulty: in a multitude the last unit is not determined to be this unit rather than another, so it will never remain determinately one in reality. Others say that discreteness itself is the formal character of number, while the units making up the multitude are the subject and matter of this discreteness. This is commonly the view of Scotus’s disciples, as may be seen in Father Merinero’s commentary here on the chapter on Quantity; St. Thomas, Opusculum 48, treatise on the Predicament of Quantity, ch. 1, seems to support them.⁴ Yet it is unclear how discreteness could be one form simply in the genus of quantity rather than relation, while having as its subject, in which it really inheres, things so diverse and really separate. Indeed, discreteness seems to be nothing other than division. Division is not a form that constitutes number as one species, but only as a multitude and a separation of continuous quantity; and multitude as such is not a species of quantity.

Finally, others say that every number is a plurality or multitude not ordered to compose one principle of operations, yet is a species of quantity insofar as it has the character of measure; otherwise it has unity only from our intellect, and in reality has only measurability, which is the numerical plurality of its parts. This is Torrejon’s view here, disputation 3, q. 5. But this simply excludes number from the predicament of quantity: its unity is not real, and the character of measure common to every spiritual and corporeal quantitative multitude is not real either. Number’s unity is not taken from its order to some operation, just as the unity of continuous quantity is not, since quantity is not operative.

Therefore, to explain what seems more probable in this matter and more in accord with the mind of the Holy Doctor, we answer with two conclusions.

FIRST CONCLUSION: Number composed of quantitative units is a true and proper species of quantity.

This conclusion cannot be denied in Aristotle’s doctrine. He places number among per se quantities not only here in the Categories, but also in Metaphysics V, where he enumerates only the per se species of quantity and excludes the species of quantity per accidens.⁵ Nor can Aristotle be explained, as some say, to have placed number among the species of quantity not because it has true unity, but because it has its own mode of extension or multiplication and is commonly conceived as one thing with a definition and properties of its own—enough, they say, for it to be placed in a predicament, since not everything placed in a predicament has its own unity per se. This solution swallows the entire difficulty and destroys the rationale of predicamental order. For if number lacks true unity and is a being per accidens, it is excluded from the predicament: it has no per se definition, since it has no one per se quiddity; consequently, it has neither one genus nor only one difference, and therefore is not placed in one predicament, since a predicament is an ordering of genera and species through their differences. If Aristotle excluded a being per accidens from definition, as is clear in Metaphysics VII⁶, how could he place number, a being per accidens, per se in a predicament as constituted by genus and difference—that is, by a definition? Hence there can be no doubt about the Philosopher’s intention.

There can be even less difficulty about St. Thomas’s intention. He plainly affirms in several places that quantitative number is a being per se and is placed in the predicament of quantity: see Metaphysics V, lect. 15⁷; Metaphysics IV, lect. 2⁸; Sentences I, dist. 24, q. 1, art. 3⁹; De potentia, q. 9, art. 7¹⁰; Summa theologiae I, q. 11, art. 1, ad 1; and Quodlibet 10, art. 1.¹¹ There is no need to cite his words one by one, since everywhere he says that number is a species of quantity and is one per se.

Thus those who explain St. Thomas’s statement that the last unit constitutes the species of number by saying he speaks metaphorically have either not read him or have refused to explain him in the many other places where he teaches absolutely that number is something one per se and belongs in the predicament of quantity. We understand, then, that when he says the last unit gives number its species, he means a true species, one per se, not a metaphorical one. Thus, near the end of Metaphysics VII, lect. 13,¹² he says: “Duality is not two units, but something composed of two units; otherwise number would not be a being per se and truly, but per accidens, like things heaped together.” Here St. Thomas plainly excludes metaphorical composition and unity per accidens, which are precisely the reasons for excluding number from the predicament of quantity. The same may be seen in Metaphysics VIII, lect. 3¹³ and Metaphysics VII, final lecture, on text 60.¹⁴

THE BASIS for this conclusion is given by St. Thomas in the cited Quodlibet 10 in two ways, one a priori, the other a posteriori. The a priori basis is that number expresses true extension and a character of measure arising from division of the continuous, which is not expressed by multitude alone, taken confusedly in the manner of a heap or pile. The reasoning is as follows. Number founded on divided quantity has true and proper measurement by way of extension and multiplication, by which it can exhaust and equal the very parts and extension of the continuous. It does not have this character of measurement and extension as a quantity per accidens, as do time and motion, which are extended by reason of the continuous. Number’s own discrete measurement does not arise from continuous quantity, since it rather divides and dissolves its continuity. Therefore, the character of measure and extension that number has is not a quantity per accidens, that is, through another, but is distinct from the mode of measuring the continuous. That number’s measure is a measure in the genus of quantity and extension is clear from its measuring according to the properties of quantity, such as equality and inequality, evenness and oddness, increase and augmentation. Therefore this measure is quantitative. Accordingly, St. Thomas says in Sentences I, to Hannibald, dist. 24, q. 1, art. 1, ad 3,¹⁵ that “number simply is that whose numbered things make some aggregation.” This is clear because by counting and adding units we make an aggregation and an increase in an ordered way, that is, with a measure of before and after. Thus there is extension—that is, multiplication joined to an ordering of parts—and the sole difference from continuous quantity is that in the latter the parts are united, whereas here they are separate. This accords with St. Thomas’s common teaching that the unit, as the principle of number, adds a character of measurement to the character of unity that is a property of being, as is clear in Metaphysics IV, lect. 2¹⁶; Metaphysics V, lect. 8¹⁷; and Summa theologiae I, q. 11, art. 1. Since number is constituted from units, if, as they say, the principle of measurement consists in those units, number thereby introduces a special measurement and extension.

YOU WILL OBJECT: All this is found in spiritual multitude: one unit added to another makes more, which is to make an aggregation. Even and odd are found there as well, since three angels are odd and four are even. Therefore spiritual number has the same character as quantitative number.

We answer that what is lacking there is extension and aggregation arising from division of the continuous; there is only affirmation and negation of being, as St. Thomas rightly notes in Sentences I, dist. 24, q. 1, art. 3.¹⁸ Accordingly, no aggregation or division commensurable with continuous quantity, and thus of the same genus as it, is found there. This is what belongs to the character of quantitative number: because it is caused by division of the continuous, by its very character as number it measures and exhausts the parts of continuous quantity. Perhaps this is why the Commentator said in Metaphysics X, text 2, that the character of quantity or measure belongs primarily to discrete quantity: we measure the continuous itself by counting its parts, and no measure is better known than counting. St. Thomas explicitly teaches this in Opusculum 36, ch. 1,¹⁹ saying that “the character of measure is found first in discrete quantity.” Therefore number in the genus of quantity adds this to multitude taken in general: it exhausts the measurement of continuous quantity and makes an aggregation and addition of one to another. Spiritual multitude does not do this, although we number angels just as we do quantities from the standpoint of numbering number.

St. Thomas’s second argument is that quantitative number is the proper object of arithmetic; therefore it is a being per se. The antecedent is evident, since arithmetic has no object other than number, or discrete quantity. The consequence is also evident: an object per se knowable must be a being per se, with its own definition, which is the principle of a science. Therefore number is a being per se, and so there is no impediment to placing it among the species of quantity.

BUT THE ANSWER IS GIVEN first, that number is a being per accidens as a thing, but per se as something knowable; thus number, because it has a numerable unity, is the object of arithmetic. Second, number is formally one by a unity of reason, which apprehends number as though it were something one; this unity suffices for it to be an object of science. Third, arithmetic treats number materially—that is, as continuous quantities, not formally as a plurality of units.

But the first solution fails completely. If number is per accidens in the genus of things but one per se in the genus of knowables, then there can be a science of a being per accidens, since a being per accidens can be per se knowable. This is wholly impossible, since a being per accidens cannot be defined per se: it has no per se quiddity, as has often been proved from the Philosopher. What cannot be defined cannot be an object of science, since definition is the principle of demonstration and scientific procedure, as is clear from Posterior Analytics I.²⁰

The second solution is no better. Either that unity of reason is formal and per se in number, or it is only accidental, as every nature considered universally has a unity of reason insofar as it is universal. The latter is irrelevant: just as universality is a condition of the object of knowledge, not the very character that is the knowable object, so that unity of reason would be only a condition. If the former is said, then number is formally a being of reason, since it is formally one by reason; arithmetic would thus treat a being of reason, which it would be remarkable for these authors to concede, since they deny that logic treats of a being of reason. Then numbered number, which arithmetic treats, would not be distinguished from numbering number, since that unity of reason arises in number only from the act of numbering reason. But the arithmetician does not consider that rational denomination per se.

The third solution is even less satisfactory. When arithmetic is said to treat number materially, taking it for continuous quantities, this means either that it takes continuous quantity formally insofar as it is continuous, or materially under the character of being divided and discrete. It plainly does not treat the things themselves that are subject to quantity. If number is taken for continuous quantity formally, that belongs not to arithmetic but to geometry. If it is taken insofar as divided and discrete, then arithmetic treats number per se, not continuous quantity; it is only per accidens to arithmetic to consider ratios of numbers as they occur in lines, surfaces, or some other continuous quantity.

I SAY, SECOND: That by which number is made one per se is not the multitude of units itself, nor their aggregation or discreteness; all this is material in number and is a being per accidens. Rather, the last unit is that from which number derives its unity per se, insofar as several units are ordered under it.

Therefore, the form of discrete quantity does not inform all the parts as continuous quantity does, but orders them; these discrete parts can be extended only by being ordered under a last unit. The unity of number is like the unity of place. Several surfaces succeed one another at one distance, and with the surfaces several relations to the poles of the world; nevertheless the same place remains formally with respect to the fixed terminus it faces, although materially there are several surfaces, as is said in Physics IV²¹. So in number there is the character of a last unit terminating the others. That this or that designated unit performs the office of last unit is material and per accidens.

This conclusion follows from St. Thomas. First, near the end of Metaphysics VII, lect. 13,²² he says: “Duality is not two units, but something composed of units; otherwise number would not be one per se and truly, but per accidens, like things heaped together.” Here he plainly distinguishes the multitude of units, which belongs to heaping, from the character of number, which is one per se. This also accords with the well-known authority from the Philosopher, Metaphysics V, text 19,²³, lect. 16 in St. Thomas²⁴, that “six is not twice three, but once six.” Number, then, does not consist in the mere division of the continuous; it requires a unity per se that results from such division and multitude.

Moreover, we should note St. Thomas’s resolution in Metaphysics VIII, lect. 3,²⁵ where he speaks of the form that constitutes the specific unity of number: “Number is one per se insofar as the last unit gives number its species and unity, just as in things composed of matter and form something is one through the form and receives its unity and species from it. Those who speak of the unity of number as though number were not one per se cannot say by what it is one, if it is one. For since it is composed of many units, either it is not one simply, but the units are aggregated in it by way of heaping, which does not make one simply and consequently does not constitute anything in a species, so that number is not any species of being; or, if number is one simply and not through itself, one must say what makes it one from many units, and this cannot be assigned.” See how plainly St. Thomas teaches that the last unit gives number its species per se. If it does not, no formed or resulting thing made from many units can be found to give number its species. Therefore, on the side of multitude, discreteness, or division of quantity, nothing can be given that constitutes per se the species of a determinate number except the last unit, insofar as it presupposes the others and subordinates them to itself.

That this suffices for number to be one simply and not a being per accidens is drawn from St. Thomas’s teaching in Metaphysics VII, final lecture on text 60,²⁶ where he explains the difference between composition from many things that makes one simply and that which makes one in a qualified sense. “The reason for this difference,” he says, “is that sometimes a composite receives its species from some one thing, which is either form, as in a mixed body, or composition, as in a house, or order, as in a syllable and a number; then the whole composite must be one simply. Sometimes, however, a composite receives its species from the multitude of collected parts themselves, as in a heap, a people, and other such things; in these cases the whole composite is not one simply, but only in a qualified sense.” This teaching must be carefully observed in distinguishing the various modes of unity simply and per accidens, for all these things are not called one in the same way; they are different degrees. An angel and the heavens are one in a more perfect way, admitting no variety, than animals or living things, which undergo continual increase and decrease; and these are more one than a river, which is not the same in matter or form as the river flowing now and the one that flowed twenty years ago, but is the same only by reason of the place through which it flows. Artifacts, similarly, are one only on the side of their artificial form. Yet all these fall under the character of unity simply by St. Thomas’s rule: things are one simply when they receive their species from something one, whether that one is form, composition, or order; only things that receive their species not from one thing but from a multitude are one in a qualified sense. So we say that number, insofar as it expresses multitude, is a being per accidens; both multitude and discreteness—which discontinues or divides quantity—are material and cannot give unity per se. For if number received its species from them, it would receive it from multitude itself and thus be one only in a qualified sense. Number, then, receives its unity per se and discrete extension from all these units being ordered under one last unit, which terminates them, insofar as they are quantitative units that add one divisible unit to another, can be made equal to some continuum, and are terminated by the last unit.

It also follows that discreteness itself, or any other form intrinsic to the whole number, cannot constitute number. Either that form is really in all the units or only in one. If in several, it is either divided among them or undivided. If divided, it is multiplied and consequently constitutes unity per se no more than multitude itself. If undivided, it cannot be identified with so many separate and divided units. But if that form exists in only one of the units, then our point is established: something one, namely the last unit, gives number its species. Therefore this unit cannot give number its species as an intrinsic form. Hence St. Thomas does not reduce the unity of number to unity from form, but to unity from order, as is clear from the cited words of Metaphysics VII, lect. 13,²⁷ insofar as the other units are ordered to it as that which finally terminates and closes them.

Thus the form of continuous quantity, by informing, extends and orders the parts; discrete quantity, however, cannot extend and order by informing, but coordinates one unit with another, so that from the last unit the determinate quantity of the number is taken. For example, that there are just three, five, or any other determinate number depends on the last unit by which the number is terminated and thereby determined.

BUT A DIFFICULTY ARISES HERE, one that commonly troubles the defenders of this opinion: how can that last unit be certain and designated, if, in a number—for example, four—any unit can be the last, and this depends only on the designation made by the intellect that numbers, which would make it something of reason?

We answer that in number there is in reality a last unit according to formal determination and designation, not material designation: that is, insofar as it terminates and closes the units. Four does not go beyond four units, for otherwise it would not be four; this effect is real. That the effect proceeds determinately from this unit or that is material and accidental, since each unit is suited and able to produce it. Thus it is one thing to be last with respect to designation, another with respect to the effect of termination. The latter exists in reality; the former may depend on the intellect’s act of designation, but is material and per accidens to the constitution of number itself. Many things in reality labor under this kind of difficulty: some formality and its effect are found in reality, while the material designation of the subject or part in which that formality exists depends on the intellect. For example, a circle, being of finite and determinate quantity, must have a beginning, middle, and end; otherwise its quantity would not be finite. Yet the character of beginning, middle, or end can be designated at any part, since no such part is designated by the nature of the thing. Similarly, a point in the middle of a line is the end of one part and the beginning of another; yet whether it is the beginning or end with respect to this determinate part or another depends on the intellect’s designation and is material. The same applies to the unity of a place or river: the designated parts or surfaces are material to the unity of the place or river, provided they remain under some relation to the same terminus.

The reason for these points is that the form giving unity to these things does not have an altogether absolute designation, but a relative one. That this part is a beginning or end, last or first, is said relatively; nevertheless, since quantity is finite, there is in reality a last and a first, an end and a beginning, even though their relative designation is not fixed. Thus it is materially variable but formally fixed, because the formal character of what is last always has the character of form, not matter, while what is not last always has the character of matter and of what can be determined. What is called last denominatively and materially can vary in receiving that denomination. Therefore it is false, speaking formally, that some units are matter and form with respect to one another in number: only the character of being last is form. What is last denominatively can shed that formal denomination, so the thing that is now denominatively form can become denominatively matter, or conversely; the very formality of being last cannot become matter.

BUT YOU WILL ASK whether this formality of the last unit is a physical or metaphysical form of number, and, if it does not compose number intrinsically, what the intrinsic form of number is. Some think that the last unit is the difference of number and its metaphysical form. They are entirely mistaken, since a difference is predicated of the whole, whereas the last unit is not predicated of the whole number: five is not the last unit. We must therefore say that the last unit is a partial and extrinsic form with respect to the other units, although it composes number with them—not in the sense that an accident is composed of matter and form, but in the way that, among parts of one extension, one part determines another, as a line determines a surface and a surface a body, or one degree of intensity another, although all together belong to an integral quality or quantity.

When, however, one asks what the intrinsic form of number is, we answer that there is no intrinsic form on the side of the units, because the unity of number is not unity of form but of order, as we said from St. Thomas, Metaphysics VII, final lecture.²⁸ This order is not a pure predicamental relation, for then it would be a unity of relation; it is an order of discrete extension, formally and principally in the last unit, and participated in and presupposed in the others subordinated to it, as will be explained more fully below in the answer to the second argument. This is like the fact that the whole power of acting is in the final disposition and presupposed in the preceding ones, or that the habit of virtue is principally in one power and diffused or disposed, as it were, presuppositionally in another, as St. Thomas teaches in Summa theologiae I–II, q. 56, art. 2. Therefore, as to its formal character, number is principally and inherently not in several subjects but in one, namely the last unit; it is present in the others by order and participation, and the last unit is determined formally, not materially.

THE ARGUMENTS ARE ANSWERED.

Objections are usually made on two grounds: first, to prove that number is not a being per se but per accidens; second, to prove that it is not specified by the last unit.

As to the first point, the common argument is made to prove that number is not one per se but per accidens; therefore it is not a species of quantity.

The antecedent is proved first as follows: by the division of quantity alone, every other form being removed, quantitative multitude is constituted, and this is nothing other than number. But a character drawn from multitude does not express unity per se, as we showed above from St. Thomas. Second, the antecedent is proved because one per se is not made from several beings in act, as the common axiom has it. White is called a being per accidens because it consists of several beings in act, even though they are united as subject and form—much more closely than the units in number. All the more, then, number is a being per accidens, since it consists of divided units in act. Nor can one say it is one per se in the genus of discrete quantity, though not in the genus of continuous quantity. For if the discrete genus consists of several beings in act, it cannot be a genus of unity per se, but only per accidens. Third, several things cannot coalesce into one except by unity or union. Among the units of number there is no union, since they are separate and discontinuous, much less any unity. Therefore number is not one per se. Fourth: Unity is opposed to multitude and is destroyed by it, whereas number is a kind of multitude. How, then, is number one per se? Finally: Number is one per se either insofar as it is a multitude or insofar as it is discrete. It is not one per se insofar as it is a multitude, for then it would not belong to the genus of quantity or express unity. Nor is it one per se insofar as it is discrete, for then it would consist only in the privation of continuity and connection among parts. Hence the Philosopher says in Physics III, text 68,²⁹ that number is several units, and in Metaphysics VIII, ch. 3, that number is not one, but is like a heap.³⁰

THE ANSWER is to deny the antecedent. To the first proof we say that number is not constituted formally by active division, as is evident, but efficiently. Nor is it constituted by passive division alone, since quantity’s being divided or multiplied is material to the character of number: division precisely dissolves continuity, but does not order parts under a determinate measure of number. Division can of itself proceed infinitely or without limit. That it occurs determinately and under a certain measure belongs to the character of number; thus division is opposed only to unity or the union of the parts of a continuum. Just as mere union is not enough for a continuum, but the parts must be ordered and terminated, so conversely, mere separation of the union of parts and division are not enough for discrete quantity: they must be ordered to one last unit, and thus made determinate and finite. In this way discrete quantity takes on the character of a special quantitative measure. Hence St. Thomas said that duality is not two units but something composed—that is, resulting—from them, in Metaphysics VII, lect. 13,³¹ and that twice three is not formally six, but once six³², and that number has unity not of form or composition but of order.³³

To the second proof we answer that from several beings in act there is not one per se by unity of form, but there can be one by unity of order and measure, as from several beings in act one artifact, one sacrament, or one banquet is made. These are one per se insofar as this is opposed to unity per accidens, which is taken from multitude and not from something one—whether form, composition, or order—as we saw from St. Thomas, Metaphysics VII, final lecture. To the objection about white, we answer that white, taken formally, is one per se and definable by a unity drawn from its form, although connotatively and denominatively it includes a subject as the supposit for whiteness. If it is taken not formally but as the third thing constituted from whiteness and a subject, we say that it is a being per accidens, because these do not coalesce into some third unity. Likewise, number formally expresses the last unit under one order; if, however, it is taken as an aggregate of units and order, it is a being per accidens, like other concrete things taken as the constituted thing or aggregate.

To the third proof we answer that there are several modes of unity and union: unity of form, composition, or order, as St. Thomas enumerates. Number has no union or unity of form, continuity, or composition, but has quantitative unity of order under one last unit. The precise nature of this unity of order will be explained in the following argument.

To the fourth proof we say that unity of continuity and form is opposed to multitude and division, but unity of order and discreteness is not—such as is found in number under one last unit terminating the order.

To the final proof we answer that number is one per se insofar as it is discrete, understanding discreteness not privatively, as opposed to continuity—in that sense it is nothing other than division—but in an ordering and formal sense, insofar as it expresses discrete extension under one last unit. The Philosopher in Physics III³⁴ does not say that number is several units; he says that number is “more than one and a certain amount,” because, as the Holy Doctor explains there, lect. 12,³⁵ “every number is denominated from one; hence Metaphysics V says³⁶ that the substance of six consists in its being six times one, not in its being twice three or three times two, for it would follow that one thing had several definitions and several substances.” Here Aristotle and St. Thomas teach our resolution quite openly. In Metaphysics VIII Aristotle does not say that number is like a heap; he says that if number is one, it is either like a heap, or else one must explain by what it is one. He explains this there by saying that one part is like matter and another like form, namely the last unit, as St. Thomas also teaches in lect. 3. When St. Thomas says in Summa theologiae I, q. 30, art. 1, ad 4, that abstract and absolute number exists only in the intellect, he is speaking of numbering number, which is taken apart from every subject.

Second, an argument is made from the other side to prove that the last unit cannot give number its species, nor can any order of units to it do so.

For this order cannot exist in several subjects except by a relation through which they regard and order themselves to one another. Therefore this unity would be a unity of relation, not quantity, and would belong to the predicament of relation. If one says that it is not a predicamental order but an order secundum dici, and so can belong to quantity, the contrary follows: either this quantitative order is the very plurality of the divided quantities, or it is some different quantity supervening on them. If the first, number will be nothing other than multitude or a heap, and not one per se. If the second, the difficulty returns: how can that quantity be in diverse subjects, or, if it is in one, how does it affect them all?

Next, it is specifically proved that the last unit cannot be the form of number. Either that last unit gives number its species because it is a unit, or because it is last. Not because it is a unit, since then the first unit would also give number its species. Not because it is last, since it owes its being last to the intellect that designates it. By the nature of the thing, this unit is no more determined than that one; without a determinate form, however, there is no determinate species. Second, it is unclear how the last unit is united to the remaining units so as to make one per se with them. As we said, it is not an intrinsic form but is extrinsic to the remaining units. Who would believe that the unit in India is united per se with the unit in Spain to constitute a pair or some other number? Third: Either the unit that becomes last is the form of number through itself or through something added to it. If through itself, number adds nothing to the unit, because any unit by itself suffices to constitute number, provided other units are posited extrinsically. If through something added, one must say what that is: a mode or a thing; whether it is found in all the units as such, or only in one when it becomes last. Yet since this designation comes from the intellect, nothing real can result in the unit from it. Finally, if this unit is taken apart from the other units, either it has that mode of number, or that by which it is the form of number, or it does not. If it does, then number is preserved in a single unit, which is absurd. If it does not, then its being last and the form of number come from the intellect alone, and so are not something real.

THE ANSWER to the first part of the argument is that the form of number, which is the last unit, is not a form inhering in all the units; nevertheless, an order to such a unit is present in them. This is not merely a relative unity of order, but also a quantitative one—not because some quantity inheres in all the units, which would make it continuous quantity and give number unity of form rather than order, something St. Thomas denies, but because a new character of measurement and discrete extension results. The order in number is called quantitative because materially it depends on continuous quantity, as a body depends on a surface and a line, although formally it expresses another measurement of extension and aggregation. Thus number, insofar as it takes on the special character of a measure that discretely exhausts continuous quantity, is found only in the genus of quantity. Hence St. Thomas says in Metaphysics V, lect. 8,³⁷ that the character of the one convertible with being consists solely in indivision, while that of the one that is the principle of number consists in measurement. In this way the unity of order in number is distinguished from the unity of an army or a city, which are beings per accidens, because in them there is only an order of relation, insufficient for unity per se. Relation formally concerns several terms, and so unity of order is taken from multitude when it is merely relational. But the unity of quantitative order or extension in number is not merely unity of relation: it is the distinction of divided quantities under one last unit, under which the others are ordered and enclosed, as taking on a new mode of measuring in extension itself and in discrete quantity.

To the second part of the argument: as to the first proof, we answer that the last unit gives number its species not merely because it is a unit, but because it is the last unit. As to material designation, it has the character of lastness from the intellect, and so does not formally constitute number. But as to the effect and formality of terminating—that is, of closing all the units under a determinate terminus so that the number does not remain indeterminate and undecided—this is real.

When it is objected that without a determinate form there is no determinate species, we answer, from what was said above³⁸, that without a determinate form, at least formally, no determinate effect is produced. But if the formality is determinate while the material designation is indeterminate, a determinate species simply exists in reality. This is like the unity of a river, which exists in reality even though materially the water is not the same but is now this water, now that; nevertheless, because it flows under the same relation and succession with respect to its bed, the river remains the same. The same holds for the unity of a place, as different surfaces succeed one another at the same distance. Thus different units can succeed one another by the intellect’s designation under the same character of being last, and so have unity formally in reality, though not materially.

To the second proof we say that units distant from one another are united for the character of number in the same way as units that are not distant. Since they are assumed to be divided and separate, it makes no difference whether they are far apart or near. They are not united by continuity or connection, but by unity of order and discreteness, in which one stands to another by quantitative addition. There is no need here to inquire into another intrinsic form, as has often been said: number has not unity of form, according to what we said from St. Thomas, but quantitative unity of order in discrete quantity, insofar as it introduces a new character of measure that exists formally in the last unit and presuppositionally in the others.

To the third we answer that a unit considered in itself is inadequate and partial matter of number, but insofar as it is last, it is form. Again, its being last with respect to the effect and formality of terminating and closing all the units under the terminus of a determinate number exists in reality, although it receives its material designation from the intellect. This termination, or character of being last, is truly something in every determinate number, because this function of closing and terminating the units under the character of a number—say, five or six—is truly found in reality. This character of terminus seems to be the mode by which the remaining units are terminated, coordinated and coextended with it under the character of discrete extension, which is a new termination. Such a character or mode cannot be separated from the divided units, since they are divided determinately in act: division is not determinate unless a terminus is posited. Thus the last unit, under the formality of being last, cannot be separated from the others with respect to which it is last without the number being destroyed and changing, for example, from five to four.

To the final proof we say that a unit separated from the other units does not have the character of being last, not through any defect in itself, but through the absence of the others, which must necessarily be presupposed and required for it to be last. When it is inferred that being last is nothing real but comes from the intellect, we distinguish: as to the designation of that material unit taken as last, we grant it; as to the effects and formality of terminating and closing the units under a determinate measure and character of discrete extension, we deny it. When it is said that nothing is required for number except that several units be posited, we say that materially, on the side of matter, nothing but plurality of units is required; formally, however—that is, for number to take on a new character of extension and discrete measurement—something further is required: the ordering of the units under one last unit that terminates them, an order that is quantitative, as has been explained.