Commentary Aristotle Physics
Lecture 8. An infinite in act in sensible things cannot be granted
We may begin with a rational argument and show as follows that there is no such thing. If “determined by a surface” is the definition of body, there cannot be an infinite body either intelligible or sensible. Nor can number taken in abstraction be infinite, for number or that which has number, is numerable. If, then, the numerable can be numbered, it would also be possible to go through the infinite.
If, on the other hand, we investigate the question more in accordance with principles appropriate to physics, we are led as follows to the same result. The infinite body must be either compound or simple; yet neither alternative is possible.
Compound the infinite body will not be, if the elements are finite in number. For they must be more than one, and the contraries must always balance, and no one of them can be infinite.
If one of the bodies falls in any degree short of the other in potency—suppose fire is finite in amount while air is infinite and a given quantity of fire exceeds in power the same amount of air in any ratio provided it is numerically definite—the infinite body will obviously prevail over and annihilate the finite body.
On the other hand, it is impossible that each should be infinite. Body is what has extension in all directions, and the infinite is indeterminate with regard to distance, so that the infinite body would be extended in all directions to infinity.
Nor can the infinite body be one and simple, whether it is, as some hold, a thing over and above the elements (from which they generate the elements) or is not thus qualified.
We must consider the former alternative; for there are some people who make this the infinite, and not air or water, in order that the other elements may not be annihilated by the element that is infinite.
They have contrariety with each other—air is cold, water moist, fire hot. If one were infinite, the others would have ceased to be by now. As it is, they say, the infinite is different from them and is their source.
It is impossible, however, that there should be such a body, not because it is infinite—on that point, a general proof can be given that applies equally to all, air, water, or anything else—but simply because there is, as a matter of fact, no such sensible body alongside the so-called elements.
Everything can be resolved into the elements of which it is composed. Hence, the body in question would have been present in our world here, alongside air and fire and earth and water; but nothing of the kind is observed. Nor can fire or any other of the elements be infinite.
For generally, and apart from the question of how any of them could be infinite, the all, even if it were unlimited, cannot either be or become one of them, as Heraclitus says that, at some time, all things become fire. (The same argument applies also to the one that the physicists suppose to exist alongside the elements, for everything changes from contrary to contrary, such as from hot to cold.)
349. After rejecting the opinion of the earlier philosophers who did not speak of the infinite as natural philosophers, separating it from sensible things, the Philosopher now shows there is no infinite even in the sense in which the natural philosophers posited it.
First, he shows this by logical reasons;
second by natural reasons, at “if, on the other hand” (204b10; [353]).
The first set of reasons are called “logical” not because they proceed logically from logical terms, but because they proceed in a logical manner, meaning from common and probable propositions, which is the characteristic of the dialectical syllogism.
350. He therefore gives (204b4) two logical reasons. In the first of these, it is shown that there is no infinite body. For the definition of body is that it is determined by a surface, just as the definition of a line is that its terms are points. But no body determined by a surface is infinite. Therefore, no body is infinite, whether it be sensible—that is, a natural body—or intelligible, that is, a mathematical body. The word rational should be here expounded as “logical”: indeed, logic is called rational philosophy.
351. The second reason shows that there is no infinite multitude. For everything countable can be numbered, and, consequently, passed through by counting. But every number and whatever has a number is countable. Therefore, every such thing can be passed through. Therefore, if any number, whether separated or existing in sensible things, be infinite, it follows that the infinite can be passed through, which is impossible.
352. Notice that these reasons are tenable and share the same premises. For they do not conclude of necessity, for whoever posits an infinite body would not concede that it would in its very account be terminated by a surface, except perhaps potentially, although this is probable and well-known. Similarly, whoever would posit an infinite multitude would not admit it to be a number or to have a number. For number adds to multitude the account of measure, because a number is multitude measured by unity, as is said in Metaphysics 10. For this reason, number is considered to be a species of discrete quantity, but multitude is not; it is, rather, a transcendental.
353. Then, at “if, on the other hand” (204b10), he gives reasons from natural philosophy to show that there is no infinite body in act. In connection with these reasons, one must consider that, since Aristotle had not yet proved that the heavenly body was of another essence from that of the four elements, and the common opinion of his time was that it was of the same nature as the four elements, he therefore proceeds in these reasonings as though there were no other sensible body outside of the four elements. This is in keeping with his custom, since he always, before proving that which is his own belief, proceeds from what is supposed by the common opinion of others. Hence, after he proved in De caelo 1.2 that the heavens are of another nature from the elements, he repeats, for the sake of the certitude of the truth, the consideration of the infinite, showing unqualifiedly that no sensible body is infinite.
However, here he first shows that there is no sensible infinite body on the supposition that the elements are finite in number;
second, he shows the same thing in a universal way, at “the preceding consideration” (205a7; [358]).
He therefore says first that, when one proceeds naturally, according to the principles of natural science, one is able to better consider, with more certitude, that there is no sensible infinite body from what will be said. For every sensible body is either simple or composite.
354. First, therefore, he shows that there is no composite sensible body that is infinite, supposing that the elements are finite according to multitude. For it cannot be that one of the elements is infinite and the others finite, because the composition of any compound body requires that there be a number of elements and that the contraries in it be somehow in equilibrium. If this were not so, the composition could not endure, for the strongest would destroy all the others, since the elements are contrary. But, if one of the elements were infinite, no equilibrium would ensue as long as the other elements were finite, because there is no proportion between infinite and finite. Therefore, it cannot be that only one of the elements in the composite be infinite.
But someone could claim that the infinite element might have such weak power in acting that it would not destroy the finite elements which are stronger; for example, if the infinite were air and the finite fire. And, therefore, to remove this objection, he says that, no matter how much less the power of that one infinite body is than that of the finite body (for example, if fire be infinite and air finite) nevertheless an infinite accumulation of air would be equal in power to the fire. For if the power of the fire is one hundred times greater than an equal quantity of air, then if the air be multiplied by a hundred, it will equal the fire in power; and yet air multiplied a hundred times is multiplied according to a finite number and is exceeded by the power of the whole infinite amount of air. Hence, it is clear that even the power of the fire will be overcome by the power of infinite air; thus, the infinite will excel and corrupt the finite, no matter how powerful its nature.
355. Similarly, it cannot be that any of the elements out of which a compound body is composed be infinite, because it is a property of a body that it have dimensions in every direction, and not in length only, as in a line, or in length and width only, as on a surface. But the account of the infinite is to have infinite distance or dimensions. Therefore, the infinite body should have infinite dimensions in every direction. Thus, it cannot be that one body result from a number of infinite bodies, because each occupies the whole world, unless you posit that two bodies interpenetrate, which is impossible.
356. Therefore, having shown that a composite body cannot be infinite, he now proves that neither can a simple body be infinite, nor one of the elements, nor any mean among the elements (for instance, as vapor is a mean between air and water). For some posited this last as a principle, stating other things to be generated from it. And they said that this was something infinite, but not air or water or any of the other elements, because the other elements would be corrupted by whichever one was supposed as infinite. For the elements have contrariety one to the other, since air is humid, water cold, fire hot, and earth dry. Hence, if one of them were infinite, it would destroy the others, since one contrary is disposed to be corrupted by another. And that is why they said that something other than the elements was infinite, from which, as from a principle, the elements arose.
Now, he states this position to be impossible not only as to its maintaining such a mediate body to be infinite, since there will be the same common argument about fire, air, and water as there is about the mediate body, but also as to its positing some elemental principle in addition to the elements. For there is found no sensible body outside of those things called the elements, namely, air, water, and the like. But this would have to be the case if anything besides the elements should enter into the composition of such bodies. If, therefore, anything else should enter into the composition of those bodies in addition to the four elements, it would follow that we should find here some simple body besides the elements by the resolution of the above bodies into their elements. It follows, therefore, that the stated position is false as to its positing of some simple body besides the known elements.
357. He further shows by a general argument that none of the elements can be infinite. For if any of the elements were infinite, it would be impossible for the whole universe to be anything but that element. It would likewise be necessary that all the other elements be changed into it, or have already been changed into it, due to the excess of power of the infinite over other things, as Heraclitus says that, at some future time, all things will be converted into fire because of the excelling power of fire.
And the same reason holds good for one of the elements and for some other body that some natural philosophers create besides the elements.
For it is necessary that this other body have contrariety toward the elements, since other things are asserted to be generated from it, and change does not take place except from one contrary to another, as in the case of going from hot to cold, as shown above.
This middle body would, therefore, in this way, destroy the other elements by reason of its contrariety.