XX. ON THE DIVISIBILITY OF THE CONTINUOUS EXTENDED.

FIRST of all we must know the difference between an entitative or real distinction and a mere distinction of reason. When two entities can really be divided from each other, or so separated that one or both continue to exist, we certainly must affirm that there is a real distinction between them, and not a mere distinction of reason. There cannot be any question about that; for if it were only a distinction of reason, there would be a real identity, the one would really be the other. Both would be the idem. But the idem cannot really be divided into parts that continue to exist independently of each other; just as a mathematical point cannot be divided into two points. That division therefore is a sure sign of real distinction.

But though this real divisibility shows a previously real distinction between the divisibles, it is not the only sign. We can deduce the real distinction otherwise. Now because what is here indicated is a true criterion, we say firstly, that the parts of an extended substance are really distinct, because by a finite or infinite virtue they are really divisible; secondly, that the potentia is really distinct from its actus, because it may be separated therefrom. The matter, for instance, may be separated from the form, the intellect from the determinate act of understanding, the will from the volition, the soul from virtue and grace, a substance from its accidents. But we must bear in mind that, although the divisibility of two things essentially supposes a real distinction between them, nevertheless the conception of the real distinction is essentially different from the conception of the divisibility, and a fortiori different from the conception of the division. To confound them together and identify them would be absurd.

Certainly then, in the continuous, the parts are really but indeterminately distinct, and so long as the parts are not divided, either really or in the mind, it is not a discrete quantity. Whence it follows that the parts of the continuous have in reality no number; therefore it is absurd to speak of the continuous as having an infinite number of parts, though we can conceive it as infinitely divisible, and mentally imagine it as divided into numerable and numbered parts.

To make our meaning clear, we must distinguish the mathematical continuity from physical continuity. In mathematics the continuous is the continuous quantity mentally abstracted from the real substance or body which is the subject of the same. In physics the continuous is this or that substance or body with the quantity whose subject it is. Have both or has either infinite parts?

To this we reply that in the mathematical continuous, just because it is not a discrete quantity, there are no parts, finite or infinite, in actu, but there are infinite parts in potentia. In the physical continuous there are no parts, finite or infinite, in actu, but there are finite, not infinite, parts in potentia.

That in the mathematical continuous there are no parts in actu, either finite or infinite, but only finite parts in potentia, is evident. A small quantity, being essentially extended, can never be reduced to such a state by division that at one part it should be an extended thing, and at the other a mathematical point, or that an extended thing should be divisible into two mathematical points. The continuous quantity therefore will be essentially divisible into extended parts ever divisible: so that we never can consider as impossible a further division. Thus the mathematical continuous is in potentia essentially divisible ad infinitum. The infinite is the goal or term that we never can reach. In that term each part would be unextended: 1/infinity = 0.

The physical continuous is not mere quantity. It is the solid substance in which, as in a subject, the quantity is; and therefore the quantity cannot be divided without dividing the substance. If we consider the mere quantity, we find the continuous divisible ad infinitum potentially and not actually; but having to consider the substance also, we must say that the physical continuous is neither actually nor potentially divisible ad infinitum. Some people tell us that a division ad infinitum is impossible because a substance, unless it were infinite, could not be capable of such a division. Others think that this is a paradox; for they cannot understand how in the continuous the division can stop, seeing that the continuous is quantum, and that quantity is per se divisible ad infinitum. But we have to see whether this impossibility of dividing the physical continuous ad infinitum, which cannot be derived from the ratio of quantity, can be derived from the ratio of substance, or of the concrete nature of divisible being. The Angelic Doctor tells us that it is so derived.

For every corporeal nature must be considered, firstly, in its intrinsic essence, and secondly in what we may call its extrinsication, by which it stretches out to occupy a place and impede occupation of the same place by any other body. This latter property originates from the substance itself as a force or virtue of it. We can easily conceive a minimum of extension, at which a corporeal substance is no longer able to extend itself, occupy a place and resist the occupation of the same place by other substances. "It must be understood," says St. Thomas, "that a body, which is complete in size, is to be considered in two ways, i.e. mathematically, or according to the quantity alone, and naturally, by considering in it the matter and the form. It is evident that a natural body cannot be infinite in actu. For every natural body has some determinate substantial form: and therefore, since accidents follow the substantial form, determinate accidents, one of which is quantity, must follow a determinate substantial form. Therefore every natural body has a determinate quantity for the greater and for the less."^{1} Hence he quotes elsewhere (Comm. in II. Sent., Distinct. xxx. Q. ii. a. 2) those words of Aristotle: Ideo est invenire minimam aquam et minimam carnem, quae si dividatur, non erit ulterius aqua et caro. Therefore, whether the body be elementary, e.g., gold, oxygen and the rest, or a compound of elementary substances, e.g., water, wood, marble, we shall come, in thought at least, to a limit at which it could not be divided again without ceasing to have force or virtue sufficient for co-extending and for resisting the occupation of its place by any other body. In other words, it would cease to have quantity, and remota quantitate, substantia omnis indivisibilis est.^{2}

According to this doctrine we have, both in elementary and composite bodies, true atoms, i.e. the smallest that can be; and we may fairly suppose that not mere mixtures, but true chemical combinations, in which the nature of the substance is changed, are made with the smallest corporeal substances. Now these minima, which, if again divided, would cease to have quantity and no longer be divisible, must have volume and weight, and be in a certain number, and occupy certain relative positions. Surely then it cannot be said that we Thomists are far from agreeing to those laws of chemistry which lay down that chemical combinations require the elementary substances to be in a certain number, in a certain volume, in a certain weight, and that, in combining, they must have a previously determined relative position according to the nature of each. But these laws must not be arbitrarily presupposed. They must be founded on facts or on well proved reasoning, or not be presupposed.

{1} Summa, P. i. Q. vii. a. 3.

{2} Contra Gent., iv. 65.