LECTURE 11 (206 b 33-207 a 31)

THE DEFINITION OF THE INFINITE

             382. Having shown how the infinite exists, the Philosopher explains here what the infinite is.

             Concerning this he makes three points. First he explains what the infinite is. Secondly, where he says, 'It is reasonable . . .' (207 a 32), he explains the meaning [ratio] of those things which are said of the infinite. Thirdly, where he says, 'It remains to dispose . . .' (208 a 5), he answers the arguments given above.

             Concerning the first part he makes two points. First he explains what the infinite is by refuting a false definition. Secondly, where he says, '. . . for it is from this . . .' (207 a 19), he refutes a false opinion which follows from this false definition.

             Concerning the first part he makes three points. First he states his intention. Secondly, where he says, 'This is indicated . . .' (207 a 2), he explains his position. Thirdly, where he says, 'Hence, Parmenides . . .' (207 a 15), he draws a conclusion from what he has said.

             383. He says, therefore, first that the infinite must be defined contrary to the way in which some have defined it. For some have said that the infinite is 'that beyond which there is nothing'. But on the contrary, it must be said that the infinite is 'that beyond which there is always something'.

             384. Next where he says, 'This is indicated . . .' (207 a 2), he explains his position.

             First he shows that his definition is good. Secondly, where he says, 'On the other hand . . .' (207 a 9), he shows that the definition of the ancients is worthless.

             First he shows with an example that the infinite is 'that beyond which there is always something'.

             For some say that rings are infinite because of the fact that they are circular and because it is always possible to take a part in addition to a part already taken.

             But this is not said properly but according to a certain similitude. For in order for a thing to be infinite, it is required that beyond any part taken there be some other part, in such a way that the part which was previously taken is never taken again. But this is not so in a circle. For the part which is taken after another part is different only from the part which has just been taken, but not from all the parts previously taken. For one part can be taken many times, as is clear in circular motion.

             If, therefore, rings are said to be infinite because of this similitude, it follows that that which is truly infinite is that beyond which something else can always be taken, if one wishes to take its quantity. For the quantity of the infinite cannot be comprehended. Rather if one wishes to take it, he will take part after part to infinity, as was said above.

             385. Next where he says, 'On the other hand . . .' (207 a 9), he proves with the following argument that the definition of the ancients is worthless. 'That beyond which there is nothing' is the definition of the perfect and the whole. He proves this as follows.

             A whole is defined as that which lacks nothing. Thus we speak of a whole man or a whole arc, in which none of the things which they should possess is missing. And as we say this with respect to a singular whole, such as this or that particular thing, so also this intelligibility [ratio] belongs to that which is truly and properly whole, i.e., the universe, outside of which there simply is nothing. When, however, something is deficient because of the absence of something intrinsic, then it is not a whole.

             Therefore, it is clear that this is the definition of a whole: a whole is that outside of which there is nothing. But a whole and the perfect are either altogether the same or else are close according to nature. He says this because a 'whole' is not found in simple things which do not have parts. However we do call such things 'perfect'. From this, then, it is clear that the perfect is that which has nothing outside of itself. But nothing which lacks an end is perfect, for the end is the perfection of each thing. Moreover, the end is the terminus of that which has an end. Hence nothing which is infinite and unterminated is perfect. Therefore, the definition of the perfect, i.e., 'that beyond which there is nothing', does not belong to the infinite.

             386. Next where he says, 'Hence Parmenides . . .' (207 a 15), he draws a certain conclusion from what he has said.

             Since the definition of the whole does not belong to the infinite, it is clear that Parmenides spoke better than Melissus. For Melissus said that the whole universe is infinite, whereas Parmenides said that the whole is 'bounded by exerting itself equally from the middle'. In this way he makes the body of the universe to be spherical. For in a spherical figure lines are brought from the centre to the boundary, i.e., the circumference, according to equality, fighting as it were, equally among themselves.

             And it is rightly said that the whole universe is finite. For the whole and the infinite are not consequent upon each other as if continuous with each other, as thread is joined to thread in weaving. There was a proverb in which things which are consequent upon each other were said to be continuous as thread with thread.

             387. Next where he says, '. . . for it is from this . . .' (207 a 19), he refutes a certain false opinion which has arisen from the false definition mentioned above.

             He treats this first as it commonly applies to all, and secondly, as it especially applies to Plato, where he says, 'If it contains . . .' (207 a 28).

             He says, therefore, first that since they thought that the infinite is joined to the whole, they took it as a 'dignity' of the infinite, i.e., as something per se known, that it contains all and has all in itself. They did this because the infinite has a certain likeness to the whole, as that which is in potency has a likeness to act. For the infinite insofar as it is in potency is like matter with respect to the perfection of magnitude and is like a whole in potency, but not in act. This is clear from the fact that a thing is called infinite insofar as it can be divided into something smaller and insofar as addition as the opposite of division can occur, as was said above. Therefore, the infinite in itself, i.e., according to its proper nature [ratio], is a whole in potency and is imperfect, such as matter which does not have perfection.

             However, it is not a whole and finite in itself, i.e., according to the proper nature [ratio] by which it is infinite. Rather it is such with respect to another, i.e., with respect to an end and a whole, to which it is in potency. For division which can go on to infinity, insofar as it is terminated in something, is said to be perfect, and insofar as it moves toward the infinite, it is imperfect. Now since a whole contains and matter is contained, it is clear that the infinite as such does not contain but is contained. That is, that which pertains to the infinite in act is always contained by something greater, insofar as it is possible to take something beyond it.

             388. From the fact that the infinite is, as it were, being in potency, not only does it follow that the infinite is contained and does not contain, but also two other conclusions follow. One of these is that the infinite as such is unknown, because it is, as it were, matter having no species, i.e., form, as was said. And matter is known only through form.

             The other conclusion which follows from the same thing is that the infinite has the nature [ratio] of a part rather than of a whole. For matter is related to the whole as a part. And the infinite is rightly related as a part, insofar as only some part of it can be taken in act.

             389. Next where he says, 'If it contains . . .' (207 a 28), he refutes the opinion of Plato, who placed the infinite in both sensible and intelligible things.

             He says that from this it is also clear that if the great and the small, to which Plato attributed the infinite, are in both sensible things and intelligible things as containing (because to contain is attributed to the infinite), then it follows that the infinite would contain intelligible things.

             But it seems to be absurd and impossible for the infinite, since it is unknown and indeterminate, to contain and determine intelligible things. For the known is not determined by the unknown, but vice versa.