An Essay Concerning Human Understanding By John Locke Book II: Of Ideas, Chapters 12-33

9. Number affords us the clearest Idea of Infinity.

But of all other ideas, it is number, as I have said, which I think furnishes us with the clearest and most distinct idea of infinity we are capable of. For, even in space and duration, when the mind pursues the idea of infinity, it there makes use of the ideas and repetitions of numbers, as of millions and millions of miles, or years, which are so many distinct ideas, — kept best by number from running into a confused heap, wherein the mind loses itself; and when it has added together as many millions, &c., as it pleases, of known lengths of space or duration, the clearest idea it can get of infinity, is the confused incomprehensible remainder of endless addible numbers, which affords no prospect of stop or boundary.

10. Our different Conceptions of the Infinity of Number contrasted with those of Duration and Expansion.

It will, perhaps, give us a little further light into the idea we have of infinity, and discover to us, that it is NOTHING BUT THE INFINITY OF NUMBER APPLIED TO DETERMINATE PARTS, OF WHICH WE HAVE IN OUR MINDS THE DISTINCT IDEAS, if we consider that number is not generally thought by us infinite, whereas duration and extension are apt to be so; which arises from hence, — that in number we are at one end, as it were: for there being in number nothing LESS than an unit, we there stop, and are at an end; but in addition, or increase of number, we can set no bounds: and so it is like a line, whereof one end terminating with us, the other is extended still forwards, beyond all that we can conceive. But in space and duration it is otherwise. For in duration we consider it as if this line of number were extended BOTH ways — to an unconceivable, undeterminate, and infinite length; which is evident to anyone that will but reflect on what consideration he hath of Eternity; which, I suppose, will find to be nothing else but the turning this infinity of number both ways, a parte ante and a parte post, as they speak. For, when we would consider eternity, a parte ante, what do we but, beginning from ourselves and the present time we are in, repeat in our minds ideas of years, or ages, or any other assignable portion of duration past, with a prospect of proceeding in such addition with all the infinity of number: and when we would consider eternity, a parte post, we just after the same rate begin from ourselves, and reckon by multiplied periods yet to come, still extending that line of number as before. And these two being put together, are that infinite duration we call ETERNITY which, as we turn our view either way, forwards or backward appears infinite, because we still turn that way the infinite end of number, i.e. the power still of adding more.

11. How we conceive the Infinity of Space.

The same happens also in space, wherein, conceiving ourselves to be, as it were, in the centre, we do on all sides pursue those indeterminable lines of number; and reckoning any way from ourselves, a yard, mile, diameter of the earth or orbis magnus, — by the infinity of number, we add others to them, as often as we will. And having no more reason to set bounds to those repeated ideas than we have to set bounds to number, we have that indeterminable idea of immensity.

12. Infinite Divisibility.

And since in any bulk of matter our thoughts can never arrive at the utmost divisibility, therefore there is an apparent infinity to us also in that, which has the infinity also of number; but with this difference, — that, in the former considerations of the infinity of space and duration, we only use addition of numbers; whereas this is like the division of an unit into its fractions, wherein the mind also can proceed in infinitum, as well as in the former additions; it being indeed but the addition still of new numbers: though in the addition of the one, we can have no more the POSITIVE idea of a space infinitely great, than, in the division of the other, we can have the positive idea of a body infinitely little; — our idea of infinity being, as I may say, a growing or fugitive idea, still in a boundless progression, that can stop nowhere.

13. No positive Idea of Infinity.

Though it be hard, I think, to find anyone so absurd as to say he has the POSITIVE idea of an actual infinite number; — the infinity whereof lies only in a power still of adding any combination of units to any former number, and that as long and as much as one will; the like also being in the infinity of space and duration, which power leaves always to the mind room for endless additions; — yet there be those who imagine they have positive ideas of infinite duration and space. It would, I think, be enough to destroy any such positive idea of infinite, to ask him that has it, — whether he could add to it or no; which would easily show the mistake of such a positive idea. We can, I think, have no positive idea of any space or duration which is not made up of, and commensurate to, repeated numbers of feet or yards, or days and years; which are the common measures, whereof we have the ideas in our minds, and whereby we judge of the greatness of this sort of quantities. And therefore, since an infinite idea of space or duration must needs be made up of infinite parts, it can have no other infinity than that of number CAPABLE still of further addition; but not an actual positive idea of a number infinite. For, I think it is evident, that the addition of finite things together (as are all lengths whereof we have the positive ideas) can never otherwise produce the idea of infinite than as number does; which consisting of additions of finite units one to another, suggests the idea of infinite, only by a power we find we have of still increasing the sum, and adding more of the same kind; without coming one jot nearer the end of such progression.

14. How we cannot have a positive idea of infinity in Quantity.

They who would prove their idea of infinite to be positive, seem to me to do it by a pleasant argument, taken from the negation of an end; which being negative, the negation on it is positive. He that considers that the end is, in body, but the extremity or superficies of that body, will not perhaps be forward to grant that the end is a bare negative: and he that perceives the end of his pen is black or white, will be apt to think that the end is something more than a pure negation. Nor is it, when applied to duration, the bare negation of existence, but more properly the last moment of it. But as they will have the end to be nothing but the bare negation of existence, I am sure they cannot deny but the beginning of the first instant of being, and is not by any body conceived to be a bare negation; and therefore, by their own argument, the idea of eternal, À PARTE ANTE, or of a duration without a beginning, is but a negative idea.

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