of those minute objects, which appear to the senses, to be equal or
nearly equal to the objects, and finding by reason, that there are other
objects vastly more minute, we too hastily conclude, that these are
inferior to any idea of our imagination or impression of our senses.
This however is certain, that we can form ideas, which shall be no
greater than the smallest atom of the animal spirits of an insect a
thousand times less than a mite: And we ought rather to conclude, that
the difficulty lies in enlarging our conceptions so much as to form a
just notion of a mite, or even of an insect a thousand times less than a
mite. For in order to form a just notion of these animals, we must have
a distinct idea representing every part of them, which, according to the
system of infinite divisibility, is utterly impossible, and, recording
to that of indivisible parts or atoms, is extremely difficult, by reason
of the vast number and multiplicity of these parts.
SECT. II. OF THE INFINITE DIVISIBILITY OF SPACE AND
TIME.
Wherever ideas are adequate representations of objects, the relations,
contradictions and agreements of the ideas are all applicable to the
objects; and this we may in general observe to be the foundation of all
human knowledge. But our ideas are adequate representations of the
most minute parts of extension; and through whatever divisions and
subdivisions we may suppose these parts to be arrived at, they can never
become inferior to some ideas, which we form. The plain consequence is,
that whatever appears impossible and contradictory upon the comparison
of these ideas, must be really impossible and contradictory, without any
farther excuse or evasion.
Every thing capable of being infinitely divided contains an infinite
number of parts; otherwise the division would be stopt short by the
indivisible parts, which we should immediately arrive at. If therefore
any finite extension be infinitely divisible, it can be no contradiction
to suppose, that a finite extension contains an infinite number of
parts: And vice versa, if it be a contradiction to suppose, that
a finite extension contains an infinite number of parts, no finite
extension can be infinitely divisible. But that this latter supposition
is absurd, I easily convince myself by the consideration of my clear
ideas. I first take the least idea I can form of a part of extension,
and being certain that there is nothing more minute than this idea, I
conclude, that whatever I discover by its means must be a real quality
of extension. I then repeat this idea once, twice, thrice, &c., and find
the compound idea of extension, arising from its repetition, always
to augment, and become double, triple, quadruple, &c., till at last it
swells up to a considerable bulk, greater or smaller, in proportion as I
repeat more or less the same idea. When I stop in the addition of parts,
the idea of extension ceases to augment; and were I to carry on the
addition in infinitum, I clearly perceive, that the idea of extension
must also become infinite. Upon the whole, I conclude, that the idea of
all infinite number of parts is individually the same idea with that of
an infinite extension; that no finite extension is capable of containing
an infinite number of parts; and consequently that no finite extension
is infinitely divisible [Footnote 3.].
[Footnote 3. It has been objected to me, that infinite
divisibility supposes only an infinite number of PROPORTIONAL not of ALIQIOT parts, and that an infinite
number of proportional parts does not form an infinite
extension. But this distinction is entirely frivolous.
Whether these parts be calld ALIQUOT or PROPORTIONAL, they
cannot be inferior to those minute parts we conceive; and
therefore cannot form a less extension by their conjunction.]
I may subjoin another argument proposed by a noted author [Mons.
MALEZIEU], which seems to me very strong and beautiful.
It is evident,
that existence in itself belongs only to unity, and is never applicable
to number, but on account of the unites, of which the number is
composed. Twenty men may be said to exist; but it is only because one,
two, three, four, &c. are existent, and if you deny the existence of
the latter, that of the former falls of course. It is therefore utterly
absurd to suppose any number to exist, and yet deny the existence of
unites; and as extension is always a number, according to the common
sentiment of metaphysicians, and never resolves itself into any unite or
indivisible quantity, it follows, that extension can never at all exist.
It is in vain to reply, that any determinate quantity of extension is an
unite; but such-a-one as admits of an infinite number of fractions, and
is inexhaustible in its sub-divisions. For by the same rule these twenty
men may be considered as a unit. The whole globe of the earth, nay
the whole universe, may be considered as a unit. That term of unity
is merely a fictitious denomination, which the mind may apply to any
quantity of objects it collects together; nor can such an unity any more
exist alone than number can, as being in reality a true number. But the
unity, which can exist alone, and whose existence is necessary to that
of all number, is of another kind, and must be perfectly indivisible,
and incapable of being resolved into any lesser unity.
All this reasoning takes place with regard to time; along with an
additional argument, which it may be proper to take notice of. It is a
property inseparable from time, and which in a manner constitutes its
essence, that each of its parts succeeds another, and that none of them,
however contiguous, can ever be co-existent. For the same reason, that
the year 1737 cannot concur with the present year 1738
every moment must
be distinct from, and posterior or antecedent to another. It is certain
then, that time, as it exists, must be composed of indivisible moments.
For if in time we could never arrive at an end of division, and if
each moment, as it succeeds another, were not perfectly single and
indivisible, there would be an infinite number of coexistent moments,
or parts of time; which I believe will be allowed to be an arrant
contradiction.
The infinite divisibility of space implies that of time, as is evident
from the nature of motion. If the latter, therefore, be impossible, the
former must be equally so.
I doubt not but, it will readily be allowed by the most obstinate
defender of the doctrine of infinite divisibility, that these arguments
are difficulties, and that it is impossible to give any answer to them
which will be perfectly clear and satisfactory. But here we may
observe, that nothing can be more absurd, than this custom of calling a
difficulty what pretends to be a demonstration, and endeavouring by that
means to elude its force and evidence. It is not in demonstrations as
in probabilities, that difficulties can take place, and one argument
counter-ballance another, and diminish its authority. A demonstration,
if just, admits of no opposite difficulty; and if not just, it is a
mere sophism, and consequently can never be a difficulty. It is either
irresistible, or has no manner of force. To talk therefore of objections
and replies, and ballancing of arguments in such a question as this, is
to confess, either that human reason is nothing but a play of words, or
that the person himself, who talks so, has not a Capacity equal to such
subjects. Demonstrations may be difficult to be comprehended, because of
abstractedness of the subject; but can never have such difficulties as
will weaken their authority, when once they are comprehended.
It is true, mathematicians are wont to say, that there are here equally
strong arguments on the other side of the question, and that the
doctrine of indivisible points is also liable to unanswerable
objections. Before I examine these arguments and objections in detail,
I will here take them in a body, and endeavour by a short and decisive
reason to prove at once, that it is utterly impossible they can have any
just foundation.
It is an established maxim in metaphysics, That whatever the mind
clearly conceives, includes the idea of possible existence, or in other
words, that nothing we imagine is absolutely impossible.
We can form the
idea of a golden mountain, and from thence conclude that such a mountain
may actually exist. We can form no idea of a mountain without a valley,
and therefore regard it as impossible.
Now it is certain we have an idea of extension; for otherwise why do we
talk and reason concerning it? It is likewise certain that this idea,
as conceived by the imagination, though divisible into parts or inferior
ideas, is not infinitely divisible, nor consists of an infinite number
of parts: For that exceeds the comprehension of our limited capacities.
Here then is an idea of extension, which consists of parts or inferior
ideas, that are perfectly, indivisible: consequently this idea implies
no contradiction: consequently it is possible for extension really to
exist conformable to it: and consequently all the arguments employed
against the possibility of mathematical points are mere scholastick
quibbles, and unworthy of our attention.
These consequences we may carry one step farther, and conclude that all
the pretended demonstrations for the infinite divisibility of extension
are equally sophistical; since it is certain these demonstrations cannot
be just without proving the impossibility of mathematical points; which
it is an evident absurdity to pretend to.
SECT. III. OF THE OTHER QUALITIES OF OUR IDEA OF SPACE
AND TIME.
No discovery coued have been made more happily for deciding all
controversies concerning ideas, than that abovementioned, that
impressions always take the precedency of them, and that every idea,
with which the imagination is furnished, first makes its appearance in a
correspondent impression. These latter perceptions are all so clear and
evident, that they admit of no controversy; though many of our ideas are
so obscure, that it is almost impossible even for the mind, which forms
them, to tell exactly their nature and composition. Let us apply this
principle, in order to discover farther the nature of our ideas of space
and time.
Upon opening my eyes, and turning them to the surrounding objects,
I perceive many visible bodies; and upon shutting them again, and
considering the distance betwixt these bodies, I acquire the idea of
extension. As every idea is derived from some impression, which
is exactly similar to it, the impressions similar to this idea of
extension, must either be some sensations derived from the sight, or
some internal impressions arising from these sensations.
Our internal impressions are our passions, emotions, desires and
aversions; none of which, I believe, will ever be asserted to be the
model, from which the idea of space is derived. There remains therefore
nothing but the senses, which can convey to us this original impression.
Now what impression do oar senses here convey to us?
This is the
principal question, and decides without appeal concerning the nature of
the idea.
The table before me is alone sufficient by its view to give me the idea
of extension. This idea, then, is borrowed from, and represents some
impression, which this moment appears to the senses. But my senses
convey to me only the impressions of coloured points, disposed in a
certain manner. If the eye is sensible of any thing farther, I desire
it may be pointed out to me. But if it be impossible to shew any thing
farther, we may conclude with certainty, that the idea of extension is
nothing but a copy of these coloured points, and of the manner of their
appearance.
Suppose that in the extended object, or composition of coloured points,
from which we first received the idea of extension, the points were of
a purple colour; it follows, that in every repetition of that idea we
would not only place the points in the same order with respect to each
other, but also bestow on them that precise colour, with which alone we
are acquainted. But afterwards having experience of the other colours of
violet, green, red, white, black, and of all the different compositions
of these, and finding a resemblance in the disposition of coloured
points, of which they are composed, we omit the peculiarities of
colour, as far as possible, and found an abstract idea merely on that
disposition of points, or manner of appearance, in which they agree. Nay
even when the resemblance is carryed beyond the objects of one sense,
and the impressions of touch are found to be Similar to those of sight
in the disposition of their parts; this does not hinder the abstract
idea from representing both, upon account of their resemblance. All
abstract ideas are really nothing but particular ones, considered in
a certain light; but being annexed to general terms, they are able to
represent a vast variety, and to comprehend objects, which, as they are
alike in some particulars, are in others vastly wide of each other.
The idea of time, being derived from the succession of our perceptions
of every kind, ideas as well as impressions, and impressions of
reflection as well as of sensations will afford us an instance of an
abstract idea, which comprehends a still greater variety than that of
space, and yet is represented in the fancy by some particular individual
idea of a determinate quantity and quality.
As it is from the disposition of visible and tangible objects we receive
the idea of space, so from the succession of ideas and impressions we
form the idea of time, nor is it possible for time alone ever to make
its appearance, or be taken notice of by the mind. A man in a sound
sleep, or strongly occupyed with one thought, is insensible of time;
and according as his perceptions succeed each other with greater or
less rapidity, the same duration appears longer or shorter to his
imagination. It has been remarked by a great philosopher, that our
perceptions have certain bounds in this particular, which are fixed by
the original nature and constitution of the mind, and beyond which no
influence of external objects on the senses is ever able to hasten or
retard our thought. If you wheel about a burning coal with rapidity, it
will present to the senses an image of a circle of fire; nor will there
seem to be any interval of time betwixt its revolutions; meerly because
it is impossible for our perceptions to succeed each other with the same
rapidity, that motion may be communicated to external objects. Wherever
we have no successive perceptions, we have no notion of time, even
though there be a real succession in the objects. From these phenomena,
as well as from many others, we may conclude, that time cannot make
its appearance to the mind, either alone, or attended with a steady
unchangeable object, but is always discovered some PERCEIVABLE
succession of changeable objects.
To confirm this we may add the following argument, which to me seems
perfectly decisive and convincing. It is evident, that time or duration
consists of different parts: For otherwise we coued not conceive a
longer or shorter duration. It is also evident, that these parts are not
co-existent: For that quality of the co-existence of parts belongs to
extension, and is what distinguishes it from duration.
Now as time is
composed of parts, that are not coexistent: an unchangeable object,
since it produces none but coexistent impressions, produces none that
can give us the idea of time; and consequently that idea must be
derived from a succession of changeable objects, and time in its first
appearance can never be severed from such a succession.
Having therefore found, that time in its first appearance to the mind
is always conjoined with a succession of changeable objects, and that
otherwise it can never fall under our notice, we must now examine
whether it can be conceived without our conceiving any succession
of objects, and whether it can alone form a distinct idea in the
imagination.
In order to know whether any objects, which are joined in impression,
be inseparable in idea, we need only consider, if they be different
from each other; in which case, it is plain they may be conceived apart.
Every thing, that is different is distinguishable: and everything,
that is distinguishable, may be separated, according to the maxims
above-explained. If on the contrary they be not different, they are
not distinguishable: and if they be not distinguishable, they cannot be
separated. But this is precisely the case with respect to time, compared
with our successive perceptions. The idea of time is not derived from a
particular impression mixed up with others, and plainly distinguishable
from them; but arises altogether from the manner, in which impressions
appear to the mind, without making one of the number.
Five notes played
on a flute give us the impression and idea of time; though time be not
a sixth impression, which presents itself to the hearing or any other of
the senses. Nor is it a sixth impression, which the mind by reflection
finds in itself. These five sounds making their appearance in this
particular manner, excite no emotion in the mind, nor produce an
affection of any kind, which being observed by it can give rise to a new
idea. For that is necessary to produce a new idea of reflection, nor can
the mind, by revolving over a thousand times all its ideas of sensation,
ever extract from them any new original idea, unless nature has so
framed its faculties, that it feels some new original impression arise
from such a contemplation. But here it only takes notice of the manner,
in which the different sounds make their appearance; and that it may
afterwards consider without considering these particular sounds, but
may conjoin it with any other objects. The ideas of some objects it
certainly must have, nor is it possible for it without these ideas ever
to arrive at any conception of time; which since it, appears not as any
primary distinct impression, can plainly be nothing but different
ideas, or impressions, or objects disposed in a certain manner, that is,
succeeding each other.
I know there are some who pretend, that the idea of duration
is applicable in a proper sense to objects, which are perfectly
unchangeable; and this I take to be the common opinion of philosophers
as well as of the vulgar. But to be convinced of its falsehood we need
but reflect on the foregoing conclusion, that the idea of duration is
always derived from a succession of changeable objects, and can never
be conveyed to the mind by any thing stedfast and unchangeable. For it
inevitably follows from thence, that since the idea of duration cannot
be derived from such an object, it can never-in any propriety or
exactness be applied to it, nor can any thing unchangeable be ever said
to have duration. Ideas always represent the Objects or impressions,
from which they are derived, and can never without a fiction represent
or be applied to any other. By what fiction we apply the idea of time,
even to what is unchangeable, and suppose, as is common, that duration
is a measure of rest as well as of motion, we shall consider [Sect 5.]
afterwards.
There is another very decisive argument, which establishes the present
doctrine concerning our ideas of space and time, and is founded only on
that simple principle, that our ideas of them are compounded of parts,
which are indivisible. This argument may be worth the examining.
Every idea, that is distinguishable, being also separable, let us take
one of those simple indivisible ideas, of which the compound one of
extension is formed, and separating it from all others, and considering
it apart, let us form a judgment of its nature and qualities.
It is plain it is not the idea of extension. For the idea of extension
consists of parts; and this idea, according to t-he supposition, is
perfectly simple and indivisible. Is it therefore nothing? That is
absolutely impossible. For as the compound idea of extension, which is
real, is composed of such ideas; were these so many non-entities, there
would be a real existence composed of non-entities; which is absurd.
Here therefore I must ask, What is our idea of a simple and indivisible
point? No wonder if my answer appear somewhat new, since the question
itself has scarce ever yet been thought of. We are wont to dispute
concerning the nature of mathematical points, but seldom concerning the
nature of their ideas.
The idea of space is conveyed to the mind by two senses, the sight
and touch; nor does anything ever appear extended, that is not either
visible or tangible. That compound impression, which represents
extension, consists of several lesser impressions, that are indivisible
to the eye or feeling, and may be called impressions of atoms or
corpuscles endowed with colour and solidity. But this is not all. It is
not only requisite, that these atoms should be coloured or tangible,
in order to discover themselves to our senses; it is also necessary
we should preserve the idea of their colour or tangibility in order to
comprehend them by our imagination. There is nothing but the idea of
their colour or tangibility, which can render them conceivable by the
mind. Upon the removal of the ideas of these sensible qualities, they
are utterly annihilated to the thought or imagination.
Now such as the parts are, such is the whole. If a point be not
considered as coloured or tangible, it can convey to us no idea; and
consequently the idea of extension, which is composed of the ideas of
these points, can never possibly exist. But if the idea of extension
really can exist, as we are conscious it does, its parts must also
exist; and in order to that, must be considered as coloured or tangible.
We have therefore no idea of space or extension, but when we regard it
as an object either of our sight or feeling.
The same reasoning will prove, that the indivisible moments of time must
be filled with some real object or existence, whose succession forms the
duration, and makes it be conceivable by the mind.
SECT. IV. OBJECTIONS ANSWERED.
Our system concerning space and time consists of two parts, which
are intimately connected together. The first depends on this chain of
reasoning. The capacity of the mind is not infinite; consequently no
idea of extension or duration consists of an infinite number of parts
or inferior ideas, but of a finite number, and these simple and
indivisible: It is therefore possible for space and time to exist
conformable to this idea: And if it be possible, it is certain they
actually do exist conformable to it; since their infinite divisibility
is utterly impossible and contradictory.