1. Infinity, in its original intention, attributed 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.
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, etc. 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.
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 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.
4. Our idea of space boundless. 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.
5. And so of 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.
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 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 any 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 in 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 or 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, etc. are called degrees. For those ideas that consist of parts 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, and 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.
7. Difference between infinity of space, and space infinite. 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.
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, etc. 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.
9. Number affords us the clearest idea of infinity. But of all other ideas, it is number, as I have said,
which I think furnishes us with the clearest and most distinct idea of infinity we are capable of. For,
even in space and duration, when the mind pursues the idea of infinity, it there makes use of the
ideas and repetitions of numbers, as of millions and millions of miles, or years, which are so many
distinct ideas,--kept best by number from running into a confused heap, wherein the mind loses
itself; and when it has added together as many millions, etc., as it pleases, of known lengths of
space or duration, the clearest idea it can get of infinity, is the confused incomprehensible
remainder of endless addible numbers, which affords no prospect of stop or boundary.
10. Our different conceptions of the infinity of number contrasted with those of duration and
expansion. It will, perhaps, give us a little further light into the idea we have of infinity, and discover
to us, that it is nothing but the infinity of number applied to determinate parts, of which we have in
our minds the distinct ideas, if we consider that number is not generally thought by us infinite,
whereas duration and extension are apt to be so; which arises from hence,--that in number we are
at one end, as it were: for there being in number nothing less than an unit, we there stop, and are at
an end; but in addition, or increase of number, we can set no bounds: and so it is like a line, whereof
one end terminating with us, the other is extended still forwards, beyond all that we can conceive.
But in space and duration it is otherwise. For in duration we consider it as if this line of number were
extended both ways--to an unconceivable, undeterminate, and infinite length; which is evident to
any one 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 the 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 backwards, appears infinite, because we still turn
that way the infinite end of number, i.e., the power still of adding more.
11. How we conceive the infinity of space. The same happens also in space, wherein, conceiving
ourselves to be, as it were, in the centre, we do on all sides pursue those indeterminable lines of
number; and reckoning any way from ourselves, a yard, mile, diameter of the earth, or orbis
magnus,--by the infinity of number, we add others to them, as often as we will. And having no more
reason to set bounds to those repeated ideas than we have to set bounds to number, we have that
indeterminable idea of immensity.
12. Infinite divisibility. And since in any bulk of matter our thoughts can never arrive at the utmost
divisibility, therefore there is an apparent infinity to us also in that, which has the infinity also of
number; but with this difference,--that, in the former considerations of the infinity of space and
duration, we only use addition of numbers; whereas this is like the division of an unit into its
fractions, wherein the mind also can proceed in infinitum, as well as in the former additions; it being
indeed but the addition still of new numbers: though in the addition of the one, we can have no more
the positive idea of a space infinitely great, than, in the division of the other, we can have the
[positive] idea of a body infinitely little;--our idea of infinity being, as I may say, a growing or fugitive
idea, still in a boundless progression, that can stop nowhere.
13. No positive idea of infinity. Though it be hard, I think, to find anyone so absurd as to say he has
the positive idea of an actual infinite number;--the infinity whereof lies only in a power stil of adding
any combination of units to any former number, and that as long and as much as one will; the like
also being in the infinity of space and duration, which power leaves always to the mind room for
endless additions;--yet there be those who imagine they have positive ideas of infinite duration and
space. It would, I think, be enough to destroy any such positive idea of infinite, to ask him that has
it,--whether he could add to it or no; which would easily show the mistake of such a positive idea.
We can, I think, have no positive idea of any space or duration which is not made up of, and
commensurate to, repeated numbers of feet or yards, or days and years; which are the common
measures, whereof we have the ideas in our minds, and whereby we judge of the greatness of this
sort of quantities. And therefore, since an infinite idea of space or duration must needs be made up
of infinite parts, it can have no other infinity than that of number capable still of further addition; but
not an actual positive idea of a number infinite. For, I think it is evident, that the addition of finite
things together (as are all lengths whereof we have the positive ideas) can never otherwise produce
the idea of infinite than as number does; which, consisting of additions of finite units one to another,
suggests the idea of infinite, only by a power we find we have of still increasing the sum, and adding
more of the same kind; without coming one jot nearer the end of such progression.
14. How we cannot have a positive idea of infinity in quantity. They who would prove their idea of
infinite to be positive, seem to me to do it by a pleasant argument, taken from the negation of an
end; which being negative, the negation of 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 if they will have the end to be nothing but
the bare negation of existence, I am sure they cannot deny but the beginning is 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 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 of 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 ide