Infinity

1. Infinity, in its original Intention, attributed to Space, Duration, and Number.

1. Infinity, in its original sense, applies to space, 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.

The best way to understand the idea called INFINITY is to consider what the mind most directly regards as infinite. Then consider how the mind forms the idea of infinity.

FINITE and INFINITE seem to me to be looked upon by the mind as the MODES OF QUANTITY, 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 space, duration, and number, which we have considered in the foregoing chapters. It is true, that we cannot but be assured, that the great God, 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 God’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 God, 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.

The mind seems to me to regard FINITE and INFINITE as MODES OF QUANTITY. These terms apply primarily, in their original meaning, only to things with parts. Such things can increase or decrease by adding or removing even the smallest part. These include the ideas of space, duration, and number, considered in the preceding chapters. We cannot doubt that the great God, to whom all things belong and from whom all things come, is incomprehensibly infinite. Yet our weak and limited minds apply our idea of infinity to this first and supreme Being primarily in two respects. These are his duration and his presence everywhere. I think we apply infinity more figuratively to his power, wisdom, goodness, and other attributes. These attributes are more properly described as inexhaustible and incomprehensible, and so on. For when we call THESE ATTRIBUTES infinite, our only idea of their infinity reflects and resembles endless number or extent. We think of the number or extent of the acts or objects of God’s power, wisdom, and goodness. We can never imagine these acts or objects as so great or numerous that God’s attributes would not exceed them. This remains true however far we multiply these acts or objects in thought, using the full infinity of endless number. I do not claim to say how these attributes exist in God, who is infinitely beyond our limited understanding. These attributes undoubtedly contain every possible perfection. I say only that this is how we conceive of these attributes, and these are our ideas of their infinity.

2. The Idea of Finite easily got.

2. The idea of the finite is easy to grasp.

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.

The mind, then, sees finite and infinite as MODIFICATIONS of space and duration. The next question is HOW THE MIND COMES BY THESE IDEAS. The idea of the finite presents no great difficulty. The portions of space we perceive through our senses bring the idea of the finite into our minds. Likewise, the usual intervals we use to measure time and duration, such as hours, days, and years, have limits. The difficulty is how we acquire the BOUNDLESS IDEAS of eternity and immensity. For the objects we encounter fall so far short of approaching such vastness or bearing any proportion to it.

3. How we come by the Idea of Infinity.

3. How we get the idea of infinity.

Every one that has any idea of any stated lengths of space, 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 space by further additions remaining still the same, he hence takes the idea of infinite space.

Anyone with an idea of a fixed length of space, such as a foot, can repeat that idea. By adding another foot to the first, they form the idea of two feet. Adding a third gives them three feet, and they can continue adding without ever reaching an end. They can keep adding a foot, or, if they prefer, double the length they have imagined. Or they can use any other length, such as a mile, the earth’s diameter, or that of the orbis magnus. Whatever length they choose, they can double or otherwise multiply that length as often as they please. After doubling the length in their thoughts and enlarging their idea as much as they please, nothing changes. They have no more reason to stop than when they began. Nor are they any closer to the end of such additions. Their ability to enlarge their idea of space by further additions remains the same. From this ability, they form the idea of infinite space.

4. Our Idea of Space boundless.

4. Our idea of space has no limits.

This, I think, is the way whereby the mind gets the IDEA of infinite space. It is a quite different consideration, to examine whether the mind has the idea of such a boundless space 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 space in itself is actually boundless, to which imagination the idea of space 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 space 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 space, 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 space 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 space, 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 space, though ever so little, here amongst bodies; and if it be possible for body to move in or through that empty space;—nay, it is impossible for any particle of matter to move but into an empty space; the same possibility of a body’s moving into a void space, beyond the utmost bounds of body, as well as into a void space interspersed amongst bodies, will always remain clear and evident: the idea of empty pure space, 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 space, 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.

This, I think, is how the mind gets the IDEA of infinite space. Whether the mind has an idea of such boundless space ACTUALLY EXISTING is a quite different question. Our ideas do not always prove that things exist. But since the question arises here, I suppose I may say that we are APT TO THINK space itself is actually boundless. The idea of space or extension naturally leads us to think so. We consider space either as the extension of a body or as existing independently, with no solid matter occupying it. We not only have an idea of such empty space. I have also proved, I think, from the motion of bodies, that empty space must exist. However we consider space, the mind cannot find or imagine any end to it. Nor can the mind reach a stopping point, however far its thoughts extend through space. Boundaries made of solid bodies, even walls of unbreakable stone, cannot stop the mind from thinking of space extending farther. Instead, such boundaries make that further reach of thought easier and greater. For no one can doubt that extension exists as far as those bodies reach. And when we reach the outermost edge of a body, what can stop the mind there? What can convince the mind that space ends there, when the mind sees that space does not end there? Indeed, the mind is convinced that the body itself can move into the space beyond. Suppose that the motion of bodies requires empty space, however small, here among bodies. Suppose also that a body can move into or through that empty space. Indeed, no particle of matter can move except into empty space. Then a body can just as clearly move into empty space beyond the outermost limits of all bodies. That possibility remains as clear and evident as movement into empty space among bodies. For the idea of pure, empty space is exactly the same within and beyond the limits of all bodies. Such spaces differ in size, but not in nature. And nothing prevents a body from moving into the space beyond. So wherever the mind places itself in thought, whether among bodies or far from them all, space remains the same. Within this uniform idea of space, the mind can find no boundary or end. The very nature and idea of each part of space must therefore lead the mind to conclude that space is actually infinite.

5. And so of Duration.

5. The same applies to duration.

As, by the power we find in ourselves of repeating, as often as we will, any idea of space, 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.

We can repeat any idea of space as often as we wish, and so we get the idea of IMMENSITY. In the same way, we can repeat our idea of any length of time, adding endlessly as we count. By doing so, we arrive at the idea of ETERNITY. For we find that we can no more reach an end to these repeated ideas than to numbers. Everyone recognizes that counting has no end. But again, having an IDEA of eternity is quite different from knowing whether ANY REAL BEING has existed eternally. On this question, I say that anyone who considers something now existing must necessarily conclude that Something has existed eternally. But I have discussed this question elsewhere, so I will say no more about it here. Instead, I will turn to some other considerations about our idea of infinity.

6. Why other Ideas are not capable of Infinity.

6. Why other ideas cannot be infinite.

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 space and duration; since they may be as easily, and as often, repeated in our minds as the other: and yet nobody 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 repetition, the idea of infinity; because, with this endless repetition, 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 space, duration, and number, being capable of increase by repetition, 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.

Suppose our idea of infinity comes from our ability to repeat our own ideas endlessly in our minds. Someone may then ask why we do not attribute infinity to other ideas besides space and duration. We can repeat other ideas in our minds just as easily and just as often. Yet nobody ever thinks of infinite sweetness or infinite whiteness. We can still repeat the idea of sweetness or whiteness as often as the idea of a yard or a day. My answer is this: all ideas considered as having parts can give us the idea of infinity through repetition. These ideas must also be capable of growing through the addition of equal or smaller parts. Because the repetition is endless, the increase continues, and there CAN be no end to it. But other ideas do not work this way. Adding even the smallest part increases the largest idea of space or duration I currently have. But consider my most complete idea of the whitest whiteness. Adding another idea of equal or lesser whiteness does not increase or enlarge my idea at all. Nor can I add an idea of greater whiteness than the whitest I already have. For this reason, different ideas of whiteness and similar qualities are called degrees. Ideas made up of parts can grow whenever even the smallest part is added. But consider the idea of white that one patch of snow gave us yesterday. Now take another idea of white from another patch of snow we see today. If we combine these ideas in our minds, they merge, as it were, into one. The idea of whiteness does not increase at all. And if we add a lesser degree of whiteness to a greater one, we diminish it rather than increase it. Ideas not made up of parts cannot be increased to whatever extent people please. Nor can these ideas extend beyond what people have received through their senses. But space, duration, and number can increase through repetition. So these ideas leave the mind with an idea of endless room for more. We cannot imagine any point where further addition or progress must stop. And so these ideas ALONE lead our minds toward the thought of infinity.

7. Difference between infinity of Space, and Space infinite.

7. The difference between the infinity of space and infinite space.

Though our idea of infinity arise from the contemplation of quantity, and the endless increase the mind is able to make in quantity, 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 quantity the mind can be thought to have, and so discourse or reason about an infinite quantity, as an infinite space, or an infinite duration. For, as our idea of infinity being, as I think, AN ENDLESS GROWING IDEA, but the idea of any quantity 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 space, and the idea of a space infinite. The first is nothing but a supposed endless progression of the mind, over what repeated ideas of space it pleases; but to have actually in the mind the idea of a space infinite, is to suppose the mind already passed over, and actually to have a view of ALL those repeated ideas of space which an ENDLESS repetition can never totally represent to it; which carries in it a plain contradiction.

Our idea of infinity comes from considering quantity and the mind’s ability to increase quantity endlessly. The mind does so by repeatedly adding whatever amounts it chooses. Yet I suspect we greatly confuse our thoughts when we attach infinity to any idea of quantity the mind might have. We then discuss or reason about an infinite quantity, such as infinite space or infinite duration. For our idea of infinity is, I think, AN ENDLESSLY GROWING IDEA. But any idea of quantity in the mind is, at that moment, LIMITED to the quantity represented. (However great that quantity may be, it cannot be greater than it is.) So attaching infinity to that idea is like fitting a fixed measure to something that keeps growing. I therefore think the following distinction is more than a pointless subtlety. We must carefully distinguish between the idea of the infinity of space and the idea of an infinite space. The first means only imagining the mind progressing endlessly through whatever repeated ideas of space it chooses. But actually having an idea of infinite space means supposing the mind has already completed that progression. The mind would already have ALL those repeated ideas of space in view. Yet an ENDLESS repetition can never present all those ideas to the mind. This supposition contains a clear contradiction.

8. We have no Idea of infinite Space.

8. We have no idea of infinite space.

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 space, 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 space 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 space 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 space 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 passing 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 space, or (which is the same thing) a number infinite, i. e. of a space or number which the mind actually has, and so views and terminates in; and of a space 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 space 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.

Perhaps this will be a little clearer if we consider numbers. Anyone who reflects on numbers can easily see that they are infinite. Everyone recognizes that adding more numbers brings us no closer to an end. But however clear this idea of endless numbers may be, something else is even clearer. It is absurd to suppose we have a complete idea of an infinite number. Whatever POSITIVE ideas we have of space, time, or number, however great, they remain finite. But suppose we imagine an inexhaustible amount remaining, with all limits removed. We allow the mind to go on thinking endlessly, without ever completing the idea. This is our idea of infinity. The idea seems fairly clear when we consider only the absence of an end. Yet when we try to picture infinite space or time, the idea becomes very obscure and confused. It combines two very different parts, which may even be inconsistent. For suppose someone forms an idea of a space or number as great as they please. Clearly, the mind RESTS AND STOPS at that idea. But this conflicts with the idea of infinity, which CONSISTS IN AN ASSUMED ENDLESS PROGRESSION. I think this explains why we become so easily confused when reasoning about infinite space, time, and so on. We fail to recognize that the parts of such an idea are inconsistent. So whichever part we draw conclusions from, the other always causes confusion. An idea of motion without movement would similarly confuse anyone who tried to reason from it. Such an idea is no better than an idea of motion at rest. To me, the idea of infinite space or, equivalently, an infinite number seems much the same. It means a space or number that the mind actually grasps, takes in, and stops at. Yet it also means a space or number that thought can never reach through constant, endless expansion and progression. For however large my idea of space may be, it is no larger than it is when I have it. I may be able to double it the next moment, and continue doing so forever. But only what has no limits is infinite. And our idea of infinity is an idea in which our thoughts can find no limits.

9. Number affords us the clearest Idea of Infinity.

9. Numbers give 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.

But of all ideas, number gives us, I think, the clearest and most distinct idea of infinity we can have. As I have said, even when considering infinity in space and time, the mind uses numbers and their repetition. We think, for example, of millions and millions of miles or years. These are distinct ideas, and numbers are the best way to keep them from becoming a confused jumble. In such a jumble, the mind loses its way. The mind can add together as many millions of known lengths of space or time as it pleases. Even then, the clearest idea of infinity the mind can reach is a confused, incomprehensible sense of what remains. There are always more numbers to add, with no end or boundary in sight.

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

10. How our ideas of infinity in numbers differ from our ideas of infinity in time and space.

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.

Perhaps the following observation will shed a little more light on our idea of infinity. It may show us that infinity is NOTHING BUT THE INFINITY OF NUMBER APPLIED TO DEFINITE PARTS OF WHICH WE HAVE CLEAR IDEAS. Consider that we do not generally think of number as infinite, though we tend to think of duration and extension that way. The reason is that, with number, we stand at one end, so to speak. Since nothing in number is LESS than one unit, we stop there and reach an end. But we can set no limit to adding numbers or increasing them. Number is therefore like a line with one end at our position. The other end keeps extending forward, beyond anything we can imagine. But space and duration are different. In thinking about duration, we imagine this line of numbers extending BOTH ways to an inconceivable, indefinite, and infinite length. Anyone who reflects on their idea of Eternity will see this. I suppose they will find that Eternity is simply this infinity of number extended in both directions. These directions are called a parte ante (into the past) and a parte post (into the future). For when we consider eternity into the past, what do we do? We begin with ourselves and the present time. Then, in our minds, we repeatedly add years, ages, or any other definite period of past duration. We look ahead to continuing this addition with all the infinity of number. And when we consider eternity into the future, we begin with ourselves in just the same way. We count repeated periods yet to come, continuing to extend the line of numbers as before. Together, these two directions make up the infinite duration we call ETERNITY. Whether we look forward or backward, that duration appears infinite. This is because we always turn the infinite end of number in the direction we are looking. That infinite end is our power to keep adding more.

11. How we conceive the Infinity of Space.

11. How we understand 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.

The same happens with space. We imagine ourselves, as it were, at the center, and extend lines of numbers endlessly in all directions. Starting from ourselves, we count a yard, a mile, the earth’s diameter, or the orbis magnus (the earth’s orbit). Because numbers are infinite, we can add more of these distances as often as we wish. We have no more reason to limit these repeated ideas than we have to limit numbers. In this way, we form an idea of immensity with no fixed limit.

12. Infinite Divisibility.

12. Dividing Without End.

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.

And since our thoughts can never reach a final limit to dividing any amount of matter, division also seems infinite. This infinity, too, involves the infinity of numbers. But there is a difference. In considering the infinity of space and duration, we only add numbers. Dividing matter, however, is like dividing a unit into fractions. The mind can continue this division endlessly, just as it can continue adding. Indeed, division is still only the addition of new numbers. Yet adding gives us no more of a POSITIVE idea of infinitely large space than division gives us of an infinitely small body. Our idea of infinity is, so to speak, an idea that keeps growing or slipping away. It continues in a limitless progression that can stop nowhere.

13. No positive Idea of Infinity.

13. No definite 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.

I think it is hard to find anyone absurd enough to claim a POSITIVE idea of an actual infinite number. A number’s infinity lies only in our ability to keep adding any group of units to any previous number. We can continue adding for as long as we like, and by as much as we like. The same is true of infinity in space and duration. This ability always leaves the mind room for endless additions. Yet some people imagine they have positive ideas of infinite duration and space. I think one question would be enough to overturn any such positive idea of infinity. Ask someone claiming to have that idea whether they could add to it. The question would easily expose the mistake in claiming such a positive idea. I think we can have no positive idea of space or duration except through repeated units of measurement. Any such space or duration must consist of, and correspond in size to, a number of feet, yards, days, or years. These are the common measures we have ideas of in our minds. We use these measures to judge the size of quantities of this kind. An infinite idea of space or duration must therefore consist of infinitely many parts. So its infinity can only be the infinity of a number CAPABLE of further addition. It cannot be an actual positive idea of an infinite number. For I think it is clear that adding finite things can produce an idea of infinity only as numbers do. All lengths of which we have positive ideas are finite things of this kind. Numbers consist of finite units added to one another. They suggest infinity only through our ability to keep increasing the sum by adding more units of the same kind. Yet we never come even slightly closer to the end of that progression.

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

14. Why we cannot form a concrete idea of an infinite 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, A PARTE ANTE, or of a duration without a beginning, is but a negative idea.

Those who try to prove their idea of infinity is positive seem to me to use an amusing argument. Their argument rests on the denial of an end. An end, they say, is negative, so denying an end gives us something positive. But consider that the end of a physical body is simply its outer edge or surface. Anyone who considers this may not be so quick to agree that an end is merely negative. Someone who sees that the end of their pen is black or white will probably reach a similar conclusion. They will think that the end is something more than a pure negation. Nor, when applied to duration, is an end simply the absence of existence. More properly, it is the last moment of existence. But these people insist that an end is nothing but the mere absence of existence. I am sure they cannot deny, however, that the beginning is the first instant of existence. No one conceives of that beginning as a mere negation. So, by their own argument, the idea of eternity A PARTE ANTE, or duration without a beginning, is merely negative.

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

15. What is positive and what is negative in our idea of infinity.

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.

I admit that the idea of infinity contains something positive in everything we apply it to. When we think of infinite space or time, we usually begin by imagining something very large. We might imagine millions of ages or miles, then perhaps double and multiply that amount several times. Everything we gather in our thoughts this way is positive. It combines many positive ideas of space or time. But we have no clear, positive idea of what remains beyond that amount. We are like a sailor trying to measure the sea’s depth. He lowers a long stretch of his sounding line but reaches no bottom. He therefore knows the sea is that many fathoms deep, and more. But he has no clear idea how much deeper it is. Suppose he could always add more line, and the weight kept sinking without ever stopping. He would be in much the same position as a mind seeking a complete, positive idea of infinity. Whether his line is ten or ten thousand fathoms long, it reveals equally little about what lies beyond. It gives only this vague, comparative idea: this is not all, and one can still go farther. The mind has a positive idea of whatever amount of space it grasps. But when the mind tries to make that space infinite, its idea remains imperfect and incomplete. The mind is always enlarging the space, always going farther. Whatever space the mind surveys while considering vastness forms a clear picture and a positive idea in the understanding. But infinity is still greater. 1. The idea of SO MUCH, then, is positive and clear. 2. The idea of GREATER is also clear, but it is only comparative. It is the idea of SO MUCH GREATER THAT IT CANNOT BE FULLY GRASPED. 3. And this is plainly negative, not positive. No one has a clear, positive idea of the size of any expanse without fully grasping its dimensions. Yet that size is what we seek in the idea of infinity. And no one, I think, claims to grasp the dimensions of anything infinite. Consider the claim that someone has a clear, positive idea of a quantity without knowing how large it is. That is as reasonable as claiming to know clearly and positively the number of grains of sand on the seashore. Yet the person does not know how many grains there are, only that there are more than twenty. Someone has just such a complete, positive idea of infinite space or time when they say it is LARGER THAN some definite amount. That amount might be ten, one hundred, one thousand, or any other number of miles or years. They have, or can have, a positive idea of that definite amount. And this comparison is, I think, our whole idea of infinity. So whatever lies beyond our positive idea TOWARDS infinity remains obscure. It has the vague uncertainty of a negative idea. I know that I neither do nor can grasp everything I want to grasp. It is too large for my limited, finite capacity. Such an idea must be very far from positive and complete. Most of what I want to grasp is left out, replaced by the vague suggestion that it is still greater. Suppose you measure a certain amount of any quantity, or go a certain distance, without reaching the end. Saying you have not reached the end means only that the quantity is greater. So denying an end to any quantity is simply saying that it is larger. Denying any end at all means carrying this “still larger” along through every increase in quantity your thoughts can make. It means adding this IDEA OF STILL GREATER to ALL the ideas of quantity you have, or could possibly have. Now, whether such an idea is positive, I leave everyone to consider.

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

16. We have no concrete idea of time lasting forever.

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.

I ask those who claim to have a positive idea of eternity: does their idea of duration include succession? If not, they should explain how their idea of duration differs when applied to an eternal Being and a finite being. Perhaps others, like me, will admit to these people that our understanding is weak on this point. We acknowledge that our idea of duration forces us to think in a particular way. Whatever has duration has existed longer today than it had yesterday. To avoid succession in external existence, these people may return to the schools’ punctum stans, or standing point. But I suppose this will do little to improve matters or clarify our positive idea of infinite duration. Nothing is more inconceivable to me than duration without succession. Besides, if punctum stans means anything, it is not a quantity. So it cannot be either finite or infinite. But suppose our limited understanding cannot separate succession from any duration at all. Then our idea of eternity can only be an INFINITE SUCCESSION OF MOMENTS OF DURATION IN WHICH ANYTHING EXISTS. Whether anyone has, or can have, a positive idea of an actual infinite number, I leave that person to consider. Let him consider until his infinite number becomes so large that he himself can add nothing more to it. As long as he can increase the number, I suspect he himself will find his idea a little too small to represent positive infinity.

17. No complete Idea of Eternal Being.

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.

I think every thoughtful, rational being must arrive at the idea of an eternal, wise Being who had no beginning. They need only examine their own existence or the existence of anything else. I am sure I have such an idea of infinite duration. But denying a beginning is only denying something positive. Such a denial hardly gives me a positive idea of infinity. Whenever I try to stretch my thoughts to infinity, I admit I am at a loss. I find I cannot clearly understand infinity.

18. No positive Idea of infinite Space.

18. No definite 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.

Anyone who thinks they have a definite idea of infinite space will find, on reflection, that they do not. They can no more form a definite idea of the greatest space than of the smallest. The smallest space seems easier to grasp and more within our understanding. Yet even here, we can only imagine something smaller than any space we can picture clearly. All our DEFINITE ideas of quantities, whether large or small, have limits. But our COMPARATIVE idea has no limits, since we can always add to the larger quantity or subtract from the smaller. Whatever lies beyond our definite idea, whether greater or smaller, remains obscure. Our only idea of what lies beyond is our ability to enlarge one quantity and reduce the other WITHOUT END. A pestle and mortar could make a particle of matter indivisible as soon as a mathematician’s sharpest thinking could. A surveyor with a measuring chain could measure infinite space as soon as a philosopher’s fastest thoughts could reach it. Neither could the philosopher fully grasp infinite space in thought, which is what having a definite idea of infinity requires. Anyone thinking of a cube one inch across has a clear and definite idea of the cube. They can also picture cubes half, a quarter, or an eighth of an inch across, and so on. Eventually, they have an idea of something very small. But they still cannot grasp the incomprehensible smallness that division can produce. The smallness beyond what they can picture remains as far beyond their thoughts as when they began. So they never reach a clear and definite idea of the smallness that follows from infinite divisibility.

19. What is positive, what negative, in our Idea of Infinite.

19. What is positive and what is negative in our idea of infinity.

Every one that looks towards infinity does, as I have said, at first glance make some very large idea of that which he applies it to, let it be space or duration; and possibly he wearies his thoughts, by multiplying in his mind that first large idea: but yet by that he comes no nearer to the having a positive clear idea of what remains to make up a positive infinite, than the country fellow had of the water which was yet to come, and pass the channel of the river where he stood:

As I have said, everyone who considers infinity first imagines a very large amount of space or time. Perhaps they tire their minds by repeatedly multiplying that first large amount in their thoughts. But this brings them no closer to a clear, definite idea of what remains to make up an actual infinity. They are like the countryman standing beside a river. He had no clearer idea of all the water still to come and flow through the river channel:

20. Some think they have a positive Idea of Eternity, and not of infinite Space.

20. Some think they have a definite idea of eternity, but not of infinite space.

There are some I have met that put so much difference between infinite duration and infinite space, that they persuade themselves that they have a positive idea of eternity, but that they have not, nor can have any idea of infinite space. The reason of which mistake I suppose to be this—that finding, by a due contemplation of causes and effects, that it is necessary to admit some Eternal Being, and so to consider the real existence of that Being as taken up and commensurate to their idea of eternity; but, on the other side, not finding it necessary, but, on the contrary, apparently absurd, that body should be infinite, they forwardly conclude that they can have no idea of infinite space, because they can have no idea of infinite matter. Which consequence, I conceive, is very ill collected, because the existence of matter is no ways necessary to the existence of space, no more than the existence of motion, or the sun, is necessary to duration, though duration uses to be measured by it. And I doubt not but that a man may have the idea of ten thousand miles square, without any body so big, as well as the idea of ten thousand years, without any body so old. It seems as easy to me to have the idea of space empty of body, as to think of the capacity of a bushel without corn, or the hollow of a nut-shell without a kernel in it: it being no more necessary that there should be existing a solid body, infinitely extended, because we have an idea of the infinity of space, than it is necessary that the world should be eternal, because we have an idea of infinite duration. And why should we think our idea of infinite space requires the real existence of matter to support it, when we find that we have as clear an idea of an infinite duration to come, as we have of infinite duration past? Though I suppose nobody thinks it conceivable that anything does or has existed in that future duration. Nor is it possible to join our idea of future duration with present or past existence, any more than it is possible to make the ideas of yesterday, to-day, and to-morrow to be the same; or bring ages past and future together, and make them contemporary. But if these men are of the mind, that they have clearer ideas of infinite duration than of infinite space, because it is past doubt that God has existed from all eternity, but there is no real matter co-extended with infinite space; yet those philosophers who are of opinion that infinite space is possessed by God’s infinite omnipresence, as well as infinite duration by his eternal existence, must be allowed to have as clear an idea of infinite space as of infinite duration; though neither of them, I think, has any positive idea of infinity in either case. For whatsoever positive ideas a man has in his mind of any quantity, he can repeat it, and add it to the former, as easy as he can add together the ideas of two days, or two paces, which are positive ideas of lengths he has in his mind, and so on as long as he pleases: whereby, if a man had a positive idea of infinite, either duration or space, he could add two infinites together; nay, make one infinite infinitely bigger than another—absurdities too gross to be confuted.

I have met some people who draw a sharp distinction between infinite duration and infinite space. They convince themselves that they have a positive idea of eternity, a definite conception of what it is. Yet they believe they have no idea of infinite space, and cannot have one. I suppose their mistake arises in the following way. By carefully considering causes and effects, they find that they must accept the existence of some Eternal Being. They therefore regard that Being’s actual existence as filling and matching their idea of eternity. But they find no need to believe that matter is infinite. On the contrary, infinite matter seems absurd to them. So they hastily conclude that they cannot conceive of infinite space because they cannot conceive of infinite matter. I think this conclusion is badly drawn. Matter is no more necessary for space to exist than motion or the sun is necessary for duration to exist. Yet duration is usually measured by motion or the sun. I have no doubt that someone can imagine a space ten thousand miles square without any object that large. Equally, someone can imagine ten thousand years without any object that old. Imagining space without matter seems just as easy to me as imagining a bushel measure without grain. Or we can imagine the hollow inside a nutshell without a kernel. Our idea of infinite space does not require an infinitely extended solid body to exist. Likewise, our idea of infinite duration does not require the world to be eternal. And why should we think our idea of infinite space needs actual matter to support it? We have as clear an idea of infinite future duration as of infinite past duration. Yet I suppose nobody thinks anything can conceivably exist now, or have existed, in that future duration. Nor can we combine our idea of future duration with present or past existence. That is no more possible than making our ideas of yesterday, today, and tomorrow identical. Nor can we bring past and future ages together and make them occur at the same time. But suppose these people think their ideas of infinite duration are clearer than their ideas of infinite space. Their reason is that God has undoubtedly existed from all eternity, while no actual matter extends throughout infinite space. Some philosophers, however, believe that God’s infinite presence fills infinite space, just as his eternal existence fills infinite duration. Those philosophers must be allowed to have as clear an idea of infinite space as of infinite duration. Yet I think neither group has any positive idea of infinity in either case. Whatever positive idea of a quantity someone has, that person can repeat the quantity and add it to the original. This is as easy as adding together the ideas of two days or two paces. These are positive ideas of lengths of time or space held in the mind. The person can continue adding for as long as they please. So, if someone had a positive idea of infinite duration or space, that person could add two infinities together. Indeed, one infinity could be made infinitely greater than another. These absurdities are too obvious to need refuting.

21. Supposed positive Ideas of Infinity, cause of Mistakes.

21. Assuming we have positive ideas of infinity leads to mistakes.

But yet if after all this, there be men who persuade themselves that they have clear positive comprehensive ideas of infinity, it is fit they enjoy their privilege: and I should be very glad (with some others that I know, who acknowledge they have none such) to be better informed by their communication. For I have been hitherto apt to think that the great and inextricable difficulties which perpetually involve all discourses concerning infinity,—whether of space, duration, or divisibility, have been the certain marks of a defect in our ideas of infinity, and the disproportion the nature thereof has to the comprehension of our narrow capacities. For, whilst men talk and dispute of infinite space or duration, as if they had as complete and positive ideas of them as they have of the names they use for them, or as they have of a yard, or an hour, or any other determinate quantity; it is no wonder if the incomprehensible nature of the thing they discourse of, or reason about, leads them into perplexities and contradictions, and their minds be overlaid by an object too large and mighty to be surveyed and managed by them.

But if, after all this, some people convince themselves they have clear, definite, comprehensive ideas of infinity, let them enjoy that privilege. I would be very glad to learn more from what they have to share. So would some others I know, who admit, as I do, that they have no such ideas. Until now, I have tended to see the endless, unresolvable difficulties surrounding all discussions of infinity as clear signs of a problem. Whether we discuss infinite space, duration, or divisibility, these difficulties reveal shortcomings in our ideas of infinity. They also show how far the nature of infinity exceeds what our limited minds can grasp. People discuss and debate infinite space or duration as if their ideas of these things were complete and definite. They speak as if these ideas were as complete and definite as their ideas of the names they use. Or as if they understood infinity as clearly as a yard, an hour, or any other fixed quantity. So we should not be surprised if the incomprehensible nature of their subject leads them into confusion and contradictions. Nor should we be surprised if their minds are overwhelmed by something too vast and powerful to examine and handle.

22. All these are modes of Ideas got from Sensation and Reflection.

22. All these are forms of ideas gained through sensation and reflection.

If I have dwelt pretty long on the consideration of duration, space, and number, and what arises from the contemplation of them,—Infinity, it is possibly no more than the matter requires; there being few simple ideas whose MODES give more exercise to the thoughts of men than those do. I pretend not to treat of them in their full latitude. It suffices to my design to show how the mind receives them, such as they are, from sensation and reflection; and how even the idea we have of infinity, how remote soever it may seem to be from any object of sense, or operation of our mind, has, nevertheless, as all our other ideas, its original there. Some mathematicians perhaps, of advanced speculations, may have other ways to introduce into their minds ideas of infinity. But this hinders not but that they themselves, as well as all other men, got the first ideas which they had of infinity from sensation and reflection, in the method we have here set down.

If I have spent considerable time discussing duration, space, number, and the infinity we conceive through considering them, perhaps the subject requires it. Few simple ideas have forms that occupy human thought more than these do. I do not claim to discuss these ideas in their full scope. For my purpose, showing how the mind receives these ideas, such as they are, from sensation and reflection is enough. I also aim to show that our idea of infinity comes from the same sources, like all our other ideas. Infinity may seem far removed from anything we sense or any operation of our minds. Nevertheless, our idea of infinity originates in sensation and reflection. Some mathematicians engaged in advanced thinking may perhaps have other ways of bringing ideas of infinity into their minds. But even those mathematicians, like everyone else, first gained their ideas of infinity from sensation and reflection, as described here.