In the third place, it follows that this multitude of distinct substances is not merely potential, but actual.
This conclusion is very clear. For every multitude of actual parts is an actual multitude, or, as they say, a multitude in act. But in continuous matter all the parts are actual, although they are not actually separated. Therefore the multitude of such parts is an actual multitude.
The upholders of continuous matter do not admit that this multitude is actual; they contend that it is only potential. For were they to concede that it is actual, they would be compelled to admit either that it is actually finite, or that it is actually infinite. Now, they cannot say that it is actually finite, because this would be against the well-known nature of continuum, which admits of an endless division, and therefore contains a multitude of parts which has no end. On the other hand, they cannot say that it is actually infinite; because, even admitting the absolute possibility of a multitude actually infinite, it would still be absurd to assert that such is the case with a piece of matter having finite dimensions. Indeed, Leibnitz and Descartes did not hesitate to teach this latter absurdity; but they could not make it fashionable, and were soon abandoned even by their own disciples. Thus the difficulty remained; and philosophers, being unable to solve it, tried to decline it by denying that there can be in the continuum an actual multitude of parts. This was, in fact, the view of the old advocates of continuous matter, who uniformly admitted that the parts of an unbroken continuum are merely potential, and form a potential multitude. For, they say, the actual multitude results from actual division, and therefore has no existence in the undivided continuum.
This last view would be very good, if the continuum in question were successive—as is the case with movement and time, which are always in fieri, and exist only by infinitesimals in an infinitesimal present, or if the continuum in question were virtual, as is the case with any mensurable interval of space; for evidently in these continuums no actual multitude is to be found. But the case is quite different with continuous matter. For he who asserts the existence of continuous matter asserts the existence of a thing having parts formally distinct and simultaneous. He therefore affirms the actual existence of a formal multitude of distinct parts, or, in other terms, an actual multitude. To deny the actual multitude of the parts, on the plea that there is no actual division, is to take refuge in a miserable sophism, which consists in denying the substantial distinction of the parts on the ground that [pg 284] they are not divided, and in ignoring their actual being solely because they have not a certain special mode of being.
As to the axiom that “Number results from division,” two things are to be noticed. The first is that the term “division” here means mensuration, not separation. Thus we divide the day into twenty-four hours, without discontinuing time for all that; and in like manner we divide the length of a journey into miles without discontinuing space. This shows that the numbers obtained by the division of the continuum are only artificially or virtually discrete, and that the continuum remains unbroken. The second is that a number is not merely a multitude, but a multitude measured by a certain unit, as S. Thomas aptly defines it: Numerus est multitudo mensurata per unum. Hence, if the unit of measure is arbitrary (as is the case with all continuous quantities), the same quantity can be expressed by different numbers, according as a different unit is employed in measuring it. But so long as the unit is not determined, the quantity cannot be expressed by any definite number. And if the unit employed be less than any given finite quantity, the thing which is measured will contain a multitude of such units greater than any given number. All such units exist in the thing measured prior to its mensuration; and as such units are actual and distinct, there can be no doubt that they constitute an actual multitude.
Some modern advocates of continuous matter have imagined another means of evading the difficulty. Tongiorgi admits extended atoms of continuous matter, but denies that their parts are actually distinct. As, however, he confesses that extension requires parts outside of parts (Cosmol., n. 143), we may ask him: Are not such parts actually distinct? Distinction is a negation of identity; and surely parts existing actually outside of one another are not actually identical. They are therefore actually distinct. Now, to use the very words of the author, “where there are distinct parts there is a plurality of units, that is, a multitude, although the parts which are distinct be united in a common term, as is the case with the parts of continuum”;[77] and therefore it is manifest that the continuous atom involves actual multitude.
Liberatore does not entirely deny the actual distinction of the parts in continuous matter, but maintains that the distinction is incomplete, and accordingly cannot give rise to an actual multitude. The parts of a continuum, says he, are united in a common term; hence they are incompletely distinct, and make no number, but are all one. They are outside of one another, yet in such a manner as to be also inside of one another. They do not subsist in themselves, but in the whole. The whole displays many parts, but it is one, and its parts are so indeterminate that they cannot be measured except by an arbitrary measure.[78]
This view scarcely deserves to be discussed, as the author himself owns that it makes continuous matter seem somewhat contradictory—Contradictoriis quodammodo notis subditur—though he attributes this kind of contradiction to the opposition which exists between the matter [pg 285] and the form—an explanation which we do not admit for reasons which we shall give in our next article. But as to the assertion that the parts of a continuum, on account of their having a common term, are only incompletely distinct, we can show at once that the author is much mistaken. Incomplete distinction is a distinction which does not completely exclude identity. Hence where there is incomplete distinction there is also incomplete identity. Now, not a shadow of identity is to be found between any two parts of continuum. Therefore any two parts of continuum are completely distinct. Thus each of the twenty-four hours into which we divide the day is completely distinct from every other, although the one is united with the other in a common term; for it is evident that the common term, having no extension, is no part of extension, and therefore cannot originate identity between any two parts of extension. To say that there is some identity, and therefore an incomplete distinction, between two extensions, because they have a common term which has no extension, is to pretend that the unextended has some identity with the extended; and this pretension is absurd. We conclude that, in spite of all the efforts of our opponents, it is manifest that continuous matter would be an actual multitude of distinct, though not separated, substances.
Lastly, it follows that actual continuous matter would be an actual infinite multitude of substances.
This conclusion is fully warranted by the infinite divisibility of the continuum. But here again the advocates of material continuity contend that this divisibility is potential, and can never be reduced to act; whence they infer that the multitude of the parts is not actual, but potential. We, however, repeat that if the division is potential, the divisible matter is certainly actual; and therefore the potency of an infinite division presupposes an infinite multitude of distinct terms actually existing in the divisible matter. And as we have already shown that each distinct term must have a distinct substantial act, we must conclude that the least piece of continuous matter would consist of an infinite actual multitude of substances—a consequence whose monstrosity needs no demonstration.