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

15. What is positive, what negative, in our Idea of infinite.

The idea of infinite has, I confess, something of positive in all those things we apply to it. When we would think of infinite space or duration, we at first step usually make some very large idea, as perhaps of millions of ages, or miles, which possibly we double and multiply several times. All that we thus amass together in our thoughts is positive, and the assemblage of a great number of positive ideas of space or duration. But what still remains beyond this we have no more a positive distinct notion of than a mariner has of the depth of the sea; where, having let down a large portion of his sounding-line, he reaches no bottom. Whereby he knows the depth to be so many fathoms, and more; but how much the more is, he hath no distinct notion at all: and could he always supply new line, and find the plummet always sink, without ever stopping, he would be something in the posture of the mind reaching after a complete and positive idea of infinity. In which case, let this line be ten, or ten thousand fathoms long, it equally discovers what is beyond it, and gives only this confused and comparative idea, that this is not all, but one may yet go farther. So much as the mind comprehends of any space, it has a positive idea of: but in endeavouring to make it infinite, — it being always enlarging, always advancing, — the idea is still imperfect and incomplete. So much space as the mind takes a view of in its contemplation of greatness, is a clear picture, and positive in the understanding: but infinite is still greater. 1. Then the idea of SO MUCH is positive and clear. 2. The idea of GREATER is also clear; but it is but a comparative idea, the idea of SO MUCH GREATER AS CANNOT BE COMPREHENDED. 3. And this is plainly negative: not positive. For he has no positive clear idea of the largeness of any extension, (which is that sought for in the idea of infinite), that has not a comprehensive idea of the dimensions of it: and such, nobody, I think, pretends to in what is infinite. For to say a man has a positive clear idea of any quantity, without knowing how great it is, is as reasonable as to say, he has the positive clear idea of the number of the sands on the sea-shore, who knows not how many there be, but only that they are more than twenty. For just such a perfect and positive idea has he of an infinite space or duration, who says it is LARGER THAN the extent or duration of ten, one hundred, one thousand, or any other number of miles, or years, whereof he has or can have a positive idea; which is all the idea, I think, we have of infinite. So that what lies beyond our positive idea TOWARDS infinity, lies in obscurity, and has the indeterminate confusion of a negative idea, wherein I know I neither do nor can comprehend all I would, it being too large for a finite and narrow capacity. And that cannot but be very far from a positive complete idea, wherein the greatest part of what I would comprehend is left out, under the undeterminate intimation of being still greater. For to say, that, having in any quantity measured so much, or gone so far, you are not yet at the end, is only to say that that quantity is greater. So that the negation of an end in any quantity is, in other words, only to say that it is bigger; and a total negation of an end is but carrying this bigger still with you, in all the progressions your thoughts shall make in quantity; and adding this IDEA OF STILL GREATER to ALL the ideas you have, or can be supposed to have, of quantity. Now, whether such an idea as that be positive, I leave any one to consider.

16. We have no positive Idea of an infinite Duration.

I ask those who say they have a positive idea of eternity, whether their idea of duration includes in it succession, or not? If it does not, they ought to show the difference of their notion of duration, when applied to an eternal Being, and to a finite; since, perhaps, there may be others as well as I, who will own to them their weakness of understanding in this point, and acknowledge that the notion they have of duration forces them to conceive, that whatever has duration, is of a longer continuance to-day than it was yesterday. If, to avoid succession in external existence, they return to the punctum stans of the schools, I suppose they will thereby very little mend the matter, or help us to a more clear and positive idea of infinite duration; there being nothing more inconceivable to me than duration without succession. Besides, that punctum stans, if it signify anything, being not quantum, finite or infinite cannot belong to it. But, if our weak apprehensions cannot separate succession from any duration whatsoever, our idea of eternity can be nothing but of INFINITE SUCCESSION OF MOMENTS OF DURATION WHEREIN ANYTHING DOES EXIST; and whether any one has, or can have, a positive idea of an actual infinite number, I leave him to consider, till his infinite number be so great that he himself can add no more to it; and as long as he can increase it, I doubt he himself will think the idea he hath of it a little too scanty for positive infinity.

17. No complete Idea of Eternal Being.

I think it unavoidable for every considering, rational creature, that will but examine his own or any other existence, to have the notion of an eternal, wise Being, who had no beginning: and such an idea of infinite duration I am sure I have. But this negation of a beginning, being but the negation of a positive thing, scarce gives me a positive idea of infinity; which, whenever I endeavour to extend my thoughts to, I confess myself at a loss, and I find I cannot attain any clear comprehension of it.

18. No positive Idea of infinite Space.

He that thinks he has a positive idea of infinite space, will, when he considers it, find that he can no more have a positive idea of the greatest, than he has of the least space. For in this latter, which seems the easier of the two, and more within our comprehension, we are capable only of a comparative idea of smallness, which will always be less than any one whereof we have the positive idea. All our POSITIVE ideas of any quantity, whether great or little, have always bounds, though our COMPARATIVE idea, whereby we can always add to the one, and take from the other, hath no bounds. For that which remains, either great or little, not being comprehended in that positive idea which we have, lies in obscurity; and we have no other idea of it, but of the power of enlarging the one and diminishing the other, WITHOUT CEASING. A pestle and mortar will as soon bring any particle of matter to indivisibility, as the acutest thought of a mathematician; and a surveyor may as soon with his chain measure out infinite space, as a philosopher by the quickest flight of mind reach it or by thinking comprehend it; which is to have a positive idea of it. He that thinks on a cube of an inch diameter, has a clear and positive idea of it in his mind, and so can frame one of 1/2, 1/4, 1/8, and so on, till he has the idea in his thoughts of something very little; but yet reaches not the idea of that incomprehensible littleness which division can produce. What remains of smallness is as far from his thoughts as when he first began; and therefore he never comes at all to have a clear and positive idea of that smallness which is consequent to infinite divisibility.

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