Infinity

SOCRATES. I want to know whether such things as appear to us to admit of more or less, or are denoted by the words ‘exceedingly,’ ‘gently,’ ‘extremely,’ and the like, may not be referred to the class of the infinite, which is their unity, for, as was asserted in the previous argument, all things that were divided and dispersed should be brought together, and have the mark or seal of some one nature, if possible, set upon them—do you remember?

PROTARCHUS. Yes.

SOCRATES. And all things which do not admit of more or less, but admit their opposites, that is to say, first of all, equality, and the equal, or again, the double, or any other ratio of number and measure—all these may, I think, be rightly reckoned by us in the class of the limited or finite; what do you say?

PROTARCHUS. Excellent, Socrates.

The infinite turns out to be the contrary of what it is said to be. It is not what has nothing outside it that is infinite, but what always has something outside it. This is indicated by the fact that rings also that have no bezel are described as ‘endless’, because it is always possible to take a part which is outside a given part. The description depends on a certain similarity, but it is not true in the full sense of the word. This condition alone is not sufficient: it is necessary also that the next part which is taken should never be the same. In the circle, the latter condition is not satisfied: it is only the adjacent part from which the new part is different.

Our definition then is as follows: A quantity is infinite if it is such that we can always take a part outside what has been already taken. On the other hand, what has nothing outside it is complete and whole. For thus we define the whole-that from which nothing is wanting, as a whole man or a whole box. What is true of each particular is true of the whole as such-the whole is that of which nothing is outside.

The infinite has rightly been represented as a circle. The straight line goes out farther and farther, and symbolizes the merely negative and bad infinite, which, unlike the true, does not return into itself. The free will is truly infinite, for it is not a mere possibility or disposition. Its external reality is its own inner nature, itself.