Thus children, either for want of names to mark the several progressions of numbers, or not having yet the faculty to collect scattered ideas into complex ones, and range them in a regular order, and so retain them in their memories, as is necessary to reckoning, do not begin to number very early, nor proceed in it very far or steadily, till a good while after they are well furnished with good store of other ideas: and one may often observe them discourse and reason pretty well, and have very clear conceptions of several other things, before they can tell twenty.
And some, through the default of their memories, who cannot retain the several combinations of numbers, with their names, annexed in their distinct orders, and the dependence of so long a train of numeral progressions, and their relation one to another, are not able all their lifetime to reckon, or regularly go over any moderate series of numbers.
For he that will count twenty, or have any idea of that number, must know that nineteen went before, with the distinct name or sign of every one of them, as they stand marked in their order; for wherever this fails, a gap is made, the chain breaks, and the progress in numbering can go no further. So that to reckon right, it is required, (1) That the mind distinguish carefully two ideas, which are different one from another only by the addition or subtraction of ONE unit: (2) That it retain in memory the names or marks of the several combinations, from an unit to that number; and that not confusedly, and at random, but in that exact order that the numbers follow one another. In either of which, if it trips, the whole business of numbering will be disturbed, and there will remain only the confused idea of mult.i.tude, but the ideas necessary to distinct numeration will not be attained to.
8. Number measures all Measurables.
This further is observable in number, that it is that which the mind makes use of in measuring all things that by us are measurable, which princ.i.p.ally are EXPANSION and DURATION; and our idea of infinity, even when applied to those, seems to be nothing but the infinity of number.
For what else are our ideas of Eternity and Immensity, but the repeated additions of certain ideas of imagined parts of duration and expansion, with the infinity of number; in which we can come to no end of addition?
For such an inexhaustible stock, number (of all other our ideas) most clearly furnishes us with, as is obvious to every one. For let a man collect into one sum as great a number as he pleases, this mult.i.tude how great soever, lessens not one jot the power of adding to it, or brings him any nearer the end of the inexhaustible stock of number; where still there remains as much to be added, as if none were taken out. And this ENDLESS ADDITION or ADDIBILITY (if any one like the word better) of numbers, so apparent to the mind, is that, I think, which gives us the clearest and most distinct idea of infinity: of which more in the following chapter.
CHAPTER XVII.
OF INFINITY.
1. Infinity, in its original Intention, attributed to s.p.a.ce, Duration, and Number.
He that would know what kind of idea it is to which we give the name of INFINITY, cannot do it better than by considering to what infinity is by the mind more immediately attributed; and then how the mind comes to frame it.
FINITE and INFINITE seem to me to be looked upon by the mind as the MODES OF QUANt.i.tY, and to be attributed primarily in their first designation only to those things which have parts, and are capable of increase or diminution by the addition or subtraction of any the least part: and such are the ideas of s.p.a.ce, duration, and number, which we have considered in the foregoing chapters. It is true, that we cannot but be a.s.sured, that the great G.o.d, of whom and from whom are all things, is incomprehensibly infinite: but yet, when we apply to that first and supreme Being our idea of infinite, in our weak and narrow thoughts, we do it primarily in respect to his duration and ubiquity; and, I think, more figuratively to his power, wisdom, and goodness, and other attributes which are properly inexhaustible and incomprehensible, &c. For, when we call THEM infinite, we have no other idea of this infinity but what carries with it some reflection on, and imitation of, that number or extent of the acts or objects of G.o.d"s power, wisdom, and goodness, which can never be supposed so great, or so many, which these attributes will not always surmount and exceed, let us multiply them in our thoughts as far as we can, with all the infinity of endless number.
I do not pretend to say how these attributes are in G.o.d, who is infinitely beyond the reach of our narrow capacities: they do, without doubt, contain in them all possible perfection: but this, I say, is our way of conceiving them, and these our ideas of their infinity.
2. The Idea of Finite easily got.
Finite then, and infinite, being by the mind looked on as MODIFICATIONS of expansion and duration, the next thing to be considered, is,--HOW THE MIND COMES BY THEM. As for the idea of finite, there is no great difficulty. The obvious portions of extension that affect our senses, carry with them into the mind the idea of finite: and the ordinary periods of succession, whereby we measure time and duration, as hours, days, and years, are bounded lengths. The difficulty is, how we come by those BOUNDLESS IDEAS of eternity and immensity; since the objects we converse with come so much short of any approach or proportion to that largeness.
3. How we come by the Idea of Infinity.
Every one that has any idea of any stated lengths of s.p.a.ce, as a foot, finds that he can repeat that idea; and joining it to the former, make the idea of two feet; and by the addition of a third, three feet; and so on, without ever coming to an end of his additions, whether of the same idea of a foot, or, if he pleases, of doubling it, or any other idea he has of any length, as a mile, or diameter of the earth, or of the orbis magnus: for whichever of these he takes, and how often soever he doubles, or any otherwise multiplies it, he finds, that, after he has continued his doubling in his thoughts, and enlarged his idea as much as he pleases, he has no more reason to stop, nor is one jot nearer the end of such addition, than he was at first setting out: the power of enlarging his idea of s.p.a.ce by further additions remaining still the same, he hence takes the idea of infinite s.p.a.ce.
4. Our Idea of s.p.a.ce boundless.
This, I think, is the way whereby the mind gets the IDEA of infinite s.p.a.ce. It is a quite different consideration, to examine whether the mind has the idea of such a boundless s.p.a.ce ACTUALLY EXISTING; since our ideas are not always proofs of the existence of things: but yet, since this comes here in our way, I suppose I may say, that we are APT TO THINK that s.p.a.ce in itself is actually boundless, to which imagination the idea of s.p.a.ce or expansion of itself naturally leads us. For, it being considered by us, either as the extension of body, or as existing by itself, without any solid matter taking it up, (for of such a void s.p.a.ce we have not only the idea, but I have proved, as I think, from the motion of body, its necessary existence,) it is impossible the mind should be ever able to find or suppose any end of it, or be stopped anywhere in its progress in this s.p.a.ce, how far soever it extends its thoughts. Any bounds made with body, even adamantine walls, are so far from putting a stop to the mind in its further progress in s.p.a.ce and extension that it rather facilitates and enlarges it. For so far as that body reaches, so far no one can doubt of extension; and when we are come to the utmost extremity of body, what is there that can there put a stop, and satisfy the mind that it is at the end of s.p.a.ce, when it perceives that it is not; nay, when it is satisfied that body itself can move into it? For, if it be necessary for the motion of body, that there should be an empty s.p.a.ce, though ever so little, here amongst bodies; and if it be possible for body to move in or through that empty s.p.a.ce;--nay, it is impossible for any particle of matter to move but into an empty s.p.a.ce; the same possibility of a body"s moving into a void s.p.a.ce, beyond the utmost bounds of body, as well as into a void s.p.a.ce interspersed amongst bodies, will always remain clear and evident: the idea of empty pure s.p.a.ce, whether within or beyond the confines of all bodies, being exactly the same, differing not in nature, though in bulk; and there being nothing to hinder body from moving into it. So that wherever the mind places itself by any thought, either amongst, or remote from all bodies, it can, in this uniform idea of s.p.a.ce, nowhere find any bounds, any end; and so must necessarily conclude it, by the very nature and idea of each part of it, to be actually infinite.
5. And so of Duration.
As, by the power we find in ourselves of repeating, as often as we will, any idea of s.p.a.ce, we get the idea of IMMENSITY; so, by being able to repeat the idea of any length of duration we have in our minds, with all the endless addition of number, we come by the idea of ETERNITY. For we find in ourselves, we can no more come to an end of such repeated ideas than we can come to the end of number; which every one perceives he cannot. But here again it is another question, quite different from our having an IDEA of eternity, to know whether there were ANY REAL BEING, whose duration has been eternal. And as to this, I say, he that considers something now existing, must necessarily come to Something eternal. But having spoke of this in another place, I shall say here no more of it, but proceed on to some other considerations of our idea of infinity.
6. Why other Ideas are not capable of Infinity.
If it be so, that our idea of infinity be got from the power we observe in ourselves of repeating, without end, our own ideas, it may be demanded,--Why we do not attribute infinity to other ideas, as well as those of s.p.a.ce and duration; since they may be as easily, and as often, repeated in our minds as the other: and yet n.o.body ever thinks of infinite sweetness or infinite whiteness, though he can repeat the idea of sweet or white, as frequently as those of a yard or a day? To which I answer,--All the ideas that are considered as having parts, and are capable of increase by the addition of an equal or less parts, afford us, by their repet.i.tion, the idea of infinity; because, with this endless repet.i.tion, there is continued an enlargement of which there CAN be no end. But for other ideas it is not so. For to the largest idea of extension or duration that I at present have, the addition of any the least part makes an increase; but to the perfectest idea I have of the whitest whiteness, if I add another of a less equal whiteness, (and of a whiter than I have, I cannot add the idea,) it makes no increase, and enlarges not my idea at all; and therefore the different ideas of whiteness, &c. are called degrees. For those ideas that consist of part are capable of being augmented by every addition of the least part; but if you take the idea of white, which one parcel of snow yielded yesterday to our sight, and another idea of white from another parcel of snow you see to-day, and put them together in your mind, they embody, as it were, all run into one, and the idea of whiteness is not at all increased and if we add a less degree of whiteness to a greater, we are so far from increasing, that we diminish it. Those ideas that consist not of parts cannot be augmented to what proportion men please, or be stretched beyond what they have received by their senses; but s.p.a.ce, duration, and number, being capable of increase by repet.i.tion, leave in the mind an idea of endless room for more; nor can we conceive anywhere a stop to a further addition or progression: and so those ideas ALONE lead our minds towards the thought of infinity.
7. Difference between infinity of s.p.a.ce, and s.p.a.ce infinite.
Though our idea of infinity arise from the contemplation of quant.i.ty, and the endless increase the mind is able to make in quant.i.ty, by the repeated additions of what portions thereof it pleases; yet I guess we cause great confusion in our thoughts, when we join infinity to any supposed idea of quant.i.ty the mind can be thought to have, and so discourse or reason about an infinite quant.i.ty, as an infinite s.p.a.ce, or an infinite duration. For, as our idea of infinity being, as I think, AN ENDLESS GROWING IDEA, but the idea of any quant.i.ty the mind has, being at that time TERMINATED in that idea, (for be it as great as it will, it can be no greater than it is,)--to join infinity to it, is to adjust a standing measure to a growing bulk; and therefore I think it is not an insignificant subtilty, if I say, that we are carefully to distinguish between the idea of the infinity of s.p.a.ce, and the idea of a s.p.a.ce infinite. The first is nothing but a supposed endless progression of the mind, over what repeated ideas of s.p.a.ce it pleases; but to have actually in the mind the idea of a s.p.a.ce infinite, is to suppose the mind already pa.s.sed over, and actually to have a view of ALL those repeated ideas of s.p.a.ce which an ENDLESS repet.i.tion can never totally represent to it; which carries in it a plain contradiction.
8. We have no Idea of infinite s.p.a.ce.
This, perhaps, will be a little plainer, if we consider it in numbers.
The infinity of numbers, to the end of whose addition every one perceives there is no approach, easily appears to any one that reflects on it. But, how clear soever this idea of the infinity of number be, there is nothing yet more evident than the absurdity of the actual idea of an infinite number. Whatsoever POSITIVE ideas we have in our minds of any s.p.a.ce, duration, or number, let them be ever so great, they are still finite; but when we suppose an inexhaustible remainder, from which we remove all bounds, and wherein we allow the mind an endless progression of thought, without ever completing the idea, there we have our idea of infinity: which, though it seems to be pretty clear when we consider nothing else in it but the negation of an end, yet, when we would frame in our minds the idea of an infinite s.p.a.ce or duration, that idea is very obscure and confused, because it is made up of two parts, very different, if not inconsistent. For, let a man frame in his mind an idea of any s.p.a.ce or number, as great as he will; it is plain the mind RESTS AND TERMINATES in that idea, which is contrary to the idea of infinity, which CONSISTS IN A SUPPOSED ENDLESS PROGRESSION. And therefore I think it is that we are so easily confounded, when we come to argue and reason about infinite s.p.a.ce or duration, &c. Because the parts of such an idea not being perceived to be, as they are, inconsistent, the one side or other always perplexes, whatever consequences we draw from the other; as an idea of motion not pa.s.sing on would perplex any one who should argue from such an idea, which is not better than an idea of motion at rest. And such another seems to me to be the idea of a s.p.a.ce, or (which is the same thing) a number infinite, i. e. of a s.p.a.ce or number which the mind actually has, and so views and terminates in; and of a s.p.a.ce or number, which, in a constant and endless enlarging and progression, it can in thought never attain to.
For, how large soever an idea of s.p.a.ce I have in my mind, it is no larger than it is that instant that I have it, though I be capable the next instant to double it, and so on in infinitum; for that alone is infinite which has no bounds; and that the idea of infinity, in which our thoughts can find none.
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 s.p.a.ce and duration, when the mind pursues the idea of infinity, it there makes use of the ideas and repet.i.tions 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 s.p.a.ce 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 s.p.a.ce 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 a.s.signable 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 s.p.a.ce.
The same happens also in s.p.a.ce, 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 s.p.a.ce 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 s.p.a.ce 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 s.p.a.ce 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 s.p.a.ce. 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 s.p.a.ce 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 quant.i.ties. And therefore, since an infinite idea of s.p.a.ce 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 Quant.i.ty.
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, a PARTE ANTE, or of a duration without a beginning, is but a negative idea.
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 s.p.a.ce 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 ama.s.s together in our thoughts is positive, and the a.s.semblage of a great number of positive ideas of s.p.a.ce 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 s.p.a.ce, 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 s.p.a.ce 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, n.o.body, I think, pretends to in what is infinite. For to say a man has a positive clear idea of any quant.i.ty, 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-sh.o.r.e, 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 s.p.a.ce 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 quant.i.ty measured so much, or gone so far, you are not yet at the end, is only to say that that quant.i.ty is greater. So that the negation of an end in any quant.i.ty 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 quant.i.ty; and adding this IDEA OF STILL GREATER to ALL the ideas you have, or can be supposed to have, of quant.i.ty. Now, whether such an idea as that be positive, I leave any one to consider.