As Far As I Know, There Are Infinitely Many Stars -- Grice on "∞" and "ℵ"

Speranza

--- is an example by Grice in "Way in the Way of Words" -- his argument against Malcolm.

Let us reconsider what we mean, 'infinity', or

"∞"


or

"ℵ" (aleph).

We can start with the excellent job done by R. B. Jones at

http://www.rbjones.com/rbjpub/philos/classics/locke/ctb2c17.htm

 

where Locke's Essay is converted to HTML for RBJones.com by first edition 1994/10/29 last
modified 2009/3/30.

Chapter XVII

 

"Of Infinity"

 Locke writes:

 

"[Let us speak of] "infinity", in its original intention [i.e. signification], 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 we cannot do it better than by considering 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,  &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 idea of "finite" is 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."

"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."

"For let us consider our idea of space as 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."


"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."

"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 as follows.

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, &c. 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."

"There is a 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.

"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."

"Number affords us the clearest idea of infinity.


But of all other ideas, 
it is number, as I have said, which I think furnishes us with the clearest
and  most distinct idea of infinity we are capable of.

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

"Our different conceptions of the infinity of number can be 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.

"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."

"Let us speak of 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."

"There is No positive idea of infinity."

"Though it be hard, I think, to find anyone so absurd as to say he has the 
positive idea of an actual infinite number;- the infinity whereof lies only
in a  power still of adding any combination of units to any former number,
and that as  long and as much as one will; the like also being in the
infinity of space and  duration, which power leaves always to the mind room for
endless additions;- yet  there be those who imagine they have positive ideas
of infinite duration and  space.

It would, I think, be enough to destroy any
such positive idea of  infinite, to ask him that has it,- whether he could
add to it or no; which would  easily show the mistake of such a positive
idea.

We can, I think, have no  positive idea of any space or duration which is
not made up of, and commensurate  to, repeated numbers of feet or yards, or
days and years; which are the common  measures, whereof we have the ideas in
our minds, and whereby we judge of the  greatness of this sort of
quantities.

And therefore, since an infinite idea of  space or duration must needs be
made up of infinite parts, it can have no other  infinity than that of
number capable still of further addition; but not an  actual positive idea of a
number infinite.

For, I think it is evident, that the  addition of finite
things together (as are all lengths whereof we have the  positive ideas) can
never otherwise produce the idea of infinite than as number  does; which,
consisting of additions of finite units one to another, suggests  the idea of
infinite, only by a power we find we have of still increasing the  sum, and
adding more of the same kind; without coming one jot nearer the end of  such
progression."

"Let us show 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."

"Let us consider 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."

"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."

"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."

"There is No positive idea of infinite space."

"He that thinks he has a positive idea of infinite space, will, when he 
considers it, find that he can no more have a positive idea of the greatest, 
than he has of the least space.

For in this latter, which seems the easier
of  the two, and more within our comprehension, we are capable only of a
comparative  idea of smallness, which will always be less than any one whereof
we have the  positive idea.

All our positive ideas of any quantity, whether
great or little,  have always bounds, though our comparative idea, whereby we
can always add to  the one, and take from the other, hath no bounds.

For
that which remains, either  great or little, not being comprehended in that
positive idea which we have,  lies in obscurity; and we have no other idea of
it, but of the power of  enlarging the one and diminishing the other,
without ceasing.

A pestle and  mortar will as soon bring any particle of matter to
indivisibility, as the  acutest thought of a mathematician; and a surveyor
may as soon with his chain  measure out infinite space, as a philosopher by
the quickest flight of mind  reach it, or by thinking comprehend it; which
is to have a positive idea of it. 

He that thinks on a cube of an inch
diameter, has a clear and positive idea of  it in his mind, and so can frame one
of 1/2, 1/4, 1/8, and so on, till he has  the idea in his thoughts of
something very little; but yet reaches not the idea  of that incomprehensible
littleness which division can produce.

What remains of  smallness is as far from
his thoughts as when he first began; and therefore he  never comes at all to
have a clear and positive idea of that smallness which is  consequent to
infinite divisibility."

"What is positive, what negative, in our idea of infinite.

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."

Rusticus expectat dum defluat amnis, at ille
Labitur, et labetur in  omne volubilis oevum.

"Some think they have a positive idea of eternity, and 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
used 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 infinities together;  nay, make one infinite infinitely
bigger than another- absurdities too gross to  be confuted."

"The Supposed positive ideas of infinity is the cause of 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."

"All these are modes of ideas got from 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."