PURE IDEAL TIME AND EMPYRICAL TIME.
112. Time is not only conceived as a general order of change, or
as a relation of being and not-being; but also as something fixed,
which can be measured with exactness. Thus, before the creation of
the world, we conceive not only an abstract order, or time, but a
time composed of years, of centuries, or some other terms. But this,
if we closely examine it, is only an idea in which we conceive the
phenomena of experience under a general view, taking them out of
actuality and contemplating them in the sphere of possibility. Neither
the years nor the centuries existed when there was nothing by which
they could be measured. If we imagine a sort of vague line of duration
prolonged to infinity, abstracting it from the measure and the object
measured, we become the sport of our imagination, and are entangled in
contradictions from which it is difficult to extricate ourselves.
113. The pure and abstract idea of time admits no measure; it is a mere
relation of being and not-being. The measure is possible only when the
idea of time is combined with the phenomena of experience.
Subject as we are to change, and situated amid beings as changeable
as ourselves, we should certainly fall into the greatest confusion of
our ideas, if in this ebb and flow of external as well as internal
existences which appear to us, we had not the greatest facility in
referring them to fixed measures, which are the thread that guides us
in this labyrinth of continual variations.
114. Two things are required for this measure: first, a suitable
phenomenon, and secondly, the idea of number. The common idea of time
which serves for the ordinary purposes of life of these three elements:
the pure idea of time, or the relation of being and not-being;
secondly, a suitable phenomenon to which we apply this pure idea; and
thirdly, the numeration of the changes of this phenomenon. Apply this
observation to all the measures of time, and you will find these three
elements always sufficient, but always indispensable also.
115. From this we deduce the necessity of time, even considered
empirically; for it involves two ideas, the one metaphysical, and the
other mathematical, applied to a fact. The metaphysical idea is the
relation of being and not-being; the mathematical idea is number; and
the fact is the sensible phenomenon, as, for example, the solar, or
human motion. Metaphysics and arithmetic take charge of the absolute
certainty; the fact observed answers for the experimental certainty;
and as, on the other hand, this phenomenon is supposed to be certain,
because, in case it were necessary we could abstract it from the
reality, and attend only to the possibility; it follows that time, even
considered empirically, may become the object of the exact sciences.
116. This theory does not make time a purely subjective condition,
nor grant it a nature independent of things; it reconciles the pure
intellectual order with the order of experience; and places man in
communication with the real world, without creating a contradiction in
his ideas.
CHAPTER XVII.
RELATIONS OF THE IDEA OF TIME AND THE PRINCIPLE OF CONTRADICTION.
117. Let us explain the true meaning of the principle of contradiction.
"It is impossible for any thing to be and not be at the same time."
The connection of the ideas contained in this principle seems at
first sight to be explained without any difficulty; so that, to raise
questions as to its true sense is to place ourselves in contradiction
with one of the fundamental truths on which rests the edifice of our
knowledge. For, if there be any doubt as to the true meaning of the
principle, it may be understood in several ways, and then there will be
another doubt as to whether the generality of men understand it as they
ought to, and whether, consequently, it is for them a solid foundation
of knowledge.
This difficulty ceases to be one when we reflect that the most
evident axioms may be considered in two manners: empirically,
or scientifically; or in other words, inasmuch as they are the
application, or the object of analytical examination. In the first
manner, they are equally certain and equally clear to all men; in
the second, they are subject to difficulties. The principle,--things
equal to a third are equal to each other,--considered empirically, is
absolutely certain and evident to all men: all men, from the wisest
to the most ignorant, compare things with a third, when they wish to
ascertain their equality or inequality; this is only an application of
the principle. If you ask them the reason of this proceeding, although
they may not enunciate the axiom in its precise terms, they refer to it
in different ways: "These two tables are equal, because I have measured
them, and they are each four feet square." Probably the generality of
men, not accustomed to reflect on their knowledge, would not express
the principle in universal and precise terms; as, "These two tables
are equal, because they have a common measure, and things equal to a
third are equal to each other." Yet they are just as clearly certain
of the principle, and apply it, without any danger of error, in all
real and possible cases. This is what I call the empirical knowledge of
principles,--a knowledge which is perfect in the direct order, and is
defective only in the reflex order.[32]
[32] See Bk. I., Ch. III.
It is very easy to reconcile the difficulty in the analysis of the
principle, with its clearness when applied to ordinary purposes, or
to those of science. Thus, in the example given, the analysis of the
term _equal_ leads to the analysis of the term _quantity_: reflection
can discover in this difficulties which, although they do not disturb
mankind in the possession of truth, are difficulties notwithstanding.
Geometry is undoubtedly a science perfectly evident and certain; but
who can deny that the idea of extension presents serious difficulties,
when examined before the tribunal of metaphysics? Universal arithmetic
is, beyond all doubt, a science; yet the ideas of quantity and number,
which are indispensable to it, give rise to the most abstruse questions
of metaphysics and ideology. In general, it may be said that there
is no branch of our knowledge which is exempt from difficulties,
considered in its root; but these difficulties, arising from
reflection, do not in any way lessen the certainty of direct knowledge.
Hence it is no objection that the analysis of the principle of
contradiction presents difficulties; nor are we therefore to fear for
the firmness of the edifice of our knowledge. It would be of no service
to us not to attend to these difficulties, if they really existed; a
difficulty does not vanish because we shut our eyes so as not to see
it. Let us not, therefore, vainly fear to examine the true sense of the
principle of contradiction.
118. It seems that this principle either does not exist, or has no
meaning, unless we presuppose the idea of time; and, on the other
hand, we cannot conceive time, unless we presuppose the principle of
contradiction. Do we thus fall into a vicious circle, and this too in
the fundamental principle of all our knowledge? This is a difficulty
which I shall first develop and present more clearly.
The principle of contradiction presupposes the idea of time,
because there would be no contradiction if being and not-being were
not referred to the same time. This last condition is altogether
indispensable; for, suppressing the simultaneousness, there is no
contradiction in a thing both being and not-being. Not only is there
no contradiction in this, but it is a thing which we constantly meet
with, in every thing around us. We see being and not-being in things
which pass from existence to non-existence, or from non-existence to
existence.
Although the simultaneousness may not be expressed in the principle of
contradiction, it is always understood, so that we should gain nothing
by adopting Kant's formula.[33] In whatever terms the principle may be
enunciated, it is always true that the same thing cannot both be and
not be at the same time, but may very well be at one time and not be at
another time.
[33] See Bk. I., Ch. XX.
The idea of time is therefore necessary in order that the contradiction
may follow in some cases, and disappear in others. If the time implies
simultaneousness, it generates the contradiction; if it implies
succession, it destroys the contradiction; because being and not-being
are impossible, unless we presuppose a successive duration, among the
different parts of which, things that would otherwise be contradictory
are distributed.
119. The idea of time also presupposes the principle of contradiction;
for, if time, in things, is only being and not-being, and in the
intellect, the perception of this being and not-being; we cannot
perceive time without having perceived being and not-being; and as
these ideas, without succession, involve a contradiction, we must
perceive the principle of contradiction when we perceive time. I have
said that succession implies the mutual exclusion of the things which
succeed; now, the first exclusion is the principle of contradiction:
in perceiving time, we perceive succession; therefore we have already
perceived the contradiction.
120. These remarks might incline us to believe it necessary to choose
between a vicious circle, which is inadmissible in the foundation of
all our knowledge, and an explanation of time, independently of being
and not-being. If we conceived time as existing by itself, as a sort of
line prolonged to infinity; as a form of things, but distinct from them
all; as a vague capacity in which successive beings might be placed,
just as we situate co-existences in space,--then the idea of time would
not be explained by the principle of contradiction, and we could only
say that it was completed by it. "When we say that it is impossible
for the same thing to be and not be at the same time, but that it is
possible for the same thing to be and not be at different times, the
contradiction is affirmed or taken away, accordingly as the being and
not-being are referred to the same point or to different points in this
vague extension, this infinite line, which we call successive duration,
and in which we conceive changeable things to be distributed." This
explanation is convenient; but it has a defect, that it cannot stand a
philosophical examination, as we have seen in the preceding chapters.
We must therefore have recourse to another class of considerations.
121. To solve this difficulty, it is necessary to determine precisely
the meaning of our ideas. The expression of _vicious circle_ is
improperly applied to this case. If we understand this, the whole
difficulty is solved at once. In explaining things, which are not
identical, a circle is a defect, and is called vicious; but when
two things are identical at bottom, although they appear distinct,
because presented under various aspects, it is impossible to explain
one without stumbling, so to speak, on the other, or to approach one
without meeting the other. Because they are presented under different
aspects we are led to believe them distinct; but examining them
analytically, we abstract the difference of aspect, and penetrate to
the reality, and discover the point where they are united, or, rather,
where they are absolutely identified.
122. We may draw from these observations a criterion which we may
use in a great many cases. When, in explaining two objects, we find
ourselves led alternately from the one to the other, without any
possibility of avoiding a circle, we may suspect that objects, which
appear distinct, are not so in reality, and that the objects presented
to the eyes of our understanding are not two objects, but only one
object perceived in different ways.
123. This is true in the present instance. In explaining the principle
of contradiction we encounter the idea of time, and in defining time
we encounter the principle of contradiction, or the ideas of being and
not-being. This is a circle, but an inevitable one; and therefore it
ceases to be vicious.
124. What is the meaning of the principle of contradiction? Its true
meaning is, that being excludes not-being, and not-being excludes
being; that the nature of these conceptions is such that the
affirmation of one implies the negation of the other, not only in the
order of our ideas, but in reality. Let us call A any being whatever:
the principle of contradiction means that A excludes not-A, and not-A
excludes A. If we think A, the conception of not-A disappears; and
if we think not-A, the conception of A disappears. If we affirm A in
reality, we deny not-A, and if we affirm not-A in reality, we deny
A. This is the true meaning of the principle of contradiction. If we
reflect, we shall find that, as far as possible, we have abstracted the
idea of time; for we have only considered the mutual exclusion of A and
not-A, in reference to a _simul_, an indivisible point of duration,
which, involving no succession, does not give us the idea of time. I
said, _as far as possible_; because as soon as we think A and not-A,
the idea of succession, and consequently of time, arises in our mind.
125. A and not-A imply contradiction; but not so that they absolutely
cannot be realized. The exclusion is conditional; that is, it exists as
long as the contradictory extremes are simultaneous, or referred to an
indivisible _now_; but we discover no intrinsic necessity of existence
in the idea of A: consequently, although we know that while A is,
not-A cannot be, we can very well conceive that A may cease to be, and
not-A may begin to be. There is, in, that case, no contradiction, and
we can easily reconcile in our mind the two ideas of A and not-A, by
referring them to different instants.
126. Hence the perception of time implies the perception of beings that
are not necessary,--of beings which, when they exist, may cease to
exist, and when they do not exist, may begin to exist. The difference
between necessary and contingent being is, that the existence of the
former absolutely excludes its non-existence, while the existence of
the latter excludes its non-existence only conditionally, or on the
supposition of simultaneousness.
127. This is why the principle of contradiction requires the condition
of time. The objects which we perceive are changeable; there is nothing
either in their nature or in their modifications which involves
existence. If they are, they may cease to be; and if this change does
not constantly occur in their substance, it does in their accidents.
Therefore we cannot affirm the absolute, but only the conditional
contradiction of their being and not-being; it exists only on the
supposition of simultaneousness.
128. If we conceived only necessary being, we could have no idea
of time: its existence absolutely excludes its non-existence, and
therefore the contradiction would be always absolute, never conditional.
129. A most important consequence results from this analysis. The
perception of time with us implies the perception of the non-necessity
of things. When we perceive a being which is not necessary, we perceive
a being which may cease to be, in which case we have the idea of
succession, of real or possible time. Here another reflection arises
which is also important: the idea of time is the idea of contingency:
the consciousness of time is the consciousness of our weakness.
130. The idea of time is so deep in our mind, that without it we could
not form the idea of the _me_. The consciousness of the identity of
the _me_ supposes a link[34] which it is impossible to find without
memory. Memory necessarily involves the relation of _past_, and,
consequently, the idea of time.
[34] See Bk. I., Ch. XXV.
CHAPTER XVIII.
SUMMING UP.
Let us collect together the doctrines of the preceding chapters.
131. Time is a question difficult to explain. Whoever denies this
difficulty shows that he has meditated but superficially on the matter.
132. Motion is measured by time; but it is not a sufficient definition
of time to call it the measure of motion.
133. It is impossible to find a primitive measure of motion; we must,
at last, take some measure or another, and although arbitrarily chosen,
we must refer motion to it. It should be the most uniform measure
possible.
134. The resemblance between the ideas of time and space creates a
suspicion that they ought to be explained in a similar manner.
135. There is no duration without something which endures; therefore
there is no duration separate from things. If nothing existed, there
could be no duration.
136. There is no succession without things which succeed: therefore
succession cannot be realized as a form independent of things, although
it may be conceived in the abstract by itself.
137. Time implies _before_ and _after_, and, consequently, succession.
It is succession itself, because in conceiving succession, we conceive
time.
138. Succession involves the exclusion of some things by others. This
exclusion may either be founded on the essence of things, or be derived
from an external cause.
139. Time, therefore, involves exclusion: it is the general idea of the
order of changes, or of the mutual relation of being and not-being.
140. If there were no change there would be no time.
141. No time had passed before the existence of the world. There was no
other duration than eternity.
142. Eternity is the existence of the infinite being, without any
alteration either actual or possible.
143. Time is not any thing absolute and independent of things, but is
really in them. It is the order between being and not-being.
144. Co-existence is merely the existence of various beings. To
conceive many beings without the idea of the negation of being, is to
have the perception of co-existence.
145. Time may be considered under three aspects; the present, the past,
and the future. All other relations of time, differently expressed in
different idioms, are only combinations of these.
146. The present is the only absolute time: it is conceived without
relation to the past or the future; but the past and the future are not
conceived without relation to the present.
147. The idea of present accompanies the very idea of being; or rather,
it is confounded with the idea of existence; that which has no present
existence is not being.
148. The idea of past time is the perception of not-being, or of a
being that has been destroyed, in relation to a present being: the idea
of future time is the perception of a possible being proceeding from a
cause already determined, and in relation to a present being.
149. The idea of time is excited by experience; but it cannot be called
a fact of mere observation; for this would be opposed to its intrinsic
necessity, by virtue of which it is the object of the exact sciences.
150. Still less can we say, that this idea is confined to the sensible
order, since it includes every manner of change in general, whether
sensible or supersensible.
151. The idea of time being the perception of the order between being
and not-being, this relation, considered in general, belongs to the
pure intellectual order. The transition to experience is realized in
the same manner as in other general and indeterminate conceptions.
152. It is necessary to make a distinction between pure ideal time
and empirical time: pure ideal time is the relation between being and
not-being, considered in the greatest generality and the most complete
indeterminateness; empirical time is the same relation subjected to a
sensible measure.
153. To measure this succession, three things are necessary, and their
union forms the idea of empirical time. They are, first, the pure idea
of being and not-being, or of change; secondly, the application of this
idea to a sensible phenomenon, as, for example, the solar motion; and
thirdly, the idea of number applied to the determining of the changes
of this phenomenon.
154. We thus conceive why empirical time implies a true necessity, and
is the object of science. Of the three elements which compose it, the
first is a metaphysical idea, the second, a mathematical idea, and the
third, a fact of observation, to which these ideas are applied. If this
fact be not real, it must, at least, be possible, in order to save the
necessity of the calculation which is based upon it.
155. There is a close relation between the idea of time and the
principle of contradiction. Each is explained by the other, yet this
is not a vicious circle. The principle of contradiction consists in
the mutual exclusion of being and not-being, and the idea of time is
the perception of the order between being and not-being. Analysis must
therefore lead to a part which is identical in both, to the comparison
of the ideas of being and not-being.
156. Without the idea of time, memory would be impossible; consequently
also, the unity of consciousness.
CHAPTER XIX.
A GLANCE AT THE IDEAS OF SPACE, NUMBER, AND TIME.
157. We may now mark out and determine with perfect exactness the
necessary elements which form the object of the natural and exact
sciences. This is not only curious, but highly important; for it
presents under the simplest aspect, an immense field of knowledge, the
limits of which expand, as we advance; so that, it is impossible to
assign a limit to progress.
158. Space, number, and time, are the three elements of all the
natural and exact sciences. All else contained in them pertains to
mere experience, to the order of contingent facts, which involve no
necessity, and cannot strictly be the objects of science.
159. Universal arithmetic is founded on the idea of numbers, geometry
on that of space, and the idea of time places us in communication with
the sensible world, so as to determine the relations of its phenomena.
These phenomena are isolated contingent facts, and cannot become the
object of science, until subjected to the general ideas of space,
number, and time.
160. Hence, there are two parts in every natural science; the
theoretic, and the experimental. The former is founded on necessary
ideas, the latter on contingent facts; the first without the second,
would not come down to the real world; the second without the first,
would not rise to the regions of science.
161. The natural sciences merit the name of science, in proportion
to the quantity of necessary elements which they contain, and the
closeness of the connection by which they unite with them contingent
facts. But as no natural science can be conceived, without contingent
facts, so there is none entirely free from the contingency which they
communicate.
162. These observations reveal a great simplicity in the elements of
science, and we may push this simplicity much farther, if we recollect
what has been said when analyzing the ideas of number and time.
163. The idea of number arises from the idea of being and not-being:
the same is also true of the idea of time; therefore, at bottom, these
ideas are but one, though presented under different aspects.
164. Hence, all the natural and exact sciences may be reduced to two
elements: the intuition of extension, and the general conception of
being. Extension is the basis of all sensible intuitions: externally,
it is a necessary condition of the relations which we conceive in the
corporeal world; internally, it is a perception, without which the
sensibility could not represent external objects. The conception of
being, is the basis of all conceptions; developed in different ways,
it generates the ideas of number and time; and these, combined with
extension, constitute the necessary part of all the natural and exact
sciences.
165. The ideas of space, number, and time are common to all men; the
proof that they are identical to all is, that, in their application,
all are led to the same results, and in speaking of them they all
use the same expressions. All men measure space, and its various
dimensions; they all count, they all conceive time: why, then, is there
so great difficulty in explaining these ideas? why such difference of
opinion among philosophers? Here we have a confirmation of what we have
said[35] of the strength of direct perception, and the weakness of
reflex. When we content ourselves with the direct perception of space,
of number, and of time, our ideas are clear, and the understanding
feels its strength and energy, it extends the sphere of its knowledge
beyond all limits, and raises the edifice of the mathematical and
exact sciences. But as soon as it turns upon itself, and, leaving the
direct perception, passes to the reflex, endeavoring to perceive its
perception, its strength fails, and it falls into a confusion which
gives rise to interminable disputes. We scarce perceive that idea,
which, a moment ago, we applied to every thing, which penetrated
all our cognitions, and circulated, like our life, through all our
perceptions; but in its isolation, and its purity, it continually
escapes from us; mingled with all things, we see that it is something
distinct from them; we separate it from one, and it unites with
another; we make an effort to cut it off from all that is not itself,
and the mind feels a kind of dizziness come over it, every thing
vanishes from before it, and, unable to reach the reality, it is forced
to be contented with names, which it pronounces and repeats a thousand
times, turning over in them the little reality which they contain.
[35] See Bk. I., Chap. III.
167. One of the causes of this weakness and of the errors which are
its ordinary consequence, is, as I have before said, our mad desire of
representing every idea as an internal form, or image, whereas we ought
to consider that in many cases there is only a perception, a simple
act in the lowest depth of our mind,--an act, which can be represented
by nothing, which resembles nothing, and which cannot be explained
in words, because it cannot be decomposed, and it is only present as
a simple fact of consciousness. But this fact of consciousness is an
active fact; by it we penetrate into things, and see what they have
in common, and separate it from what is particular, establishing in
our mind, as it were, a central, culminating point, from which we
contemplate the internal and the external world, and roam through the
boundless regions of possibility.
BOOK EIGHTH.
THE INFINITE.
CHAPTER I.
TRANSITORY VIEW OF THE ACTUAL STATE OF PHILOSOPHY.
1. In the works on transcendental philosophy which have been published
of late years, we find the words infinite, absolute, indeterminate,
unconditioned, frequently repeated, and made to play a very prominent
part in the explanation of the most recondite secrets which can be
presented to the consideration of man. The words finite, relative,
determinate, conditional, are easily combined with these; and from this
combination they pretend that a ray of light will arise to dissipate
the darkness of philosophical questions.
2. In spite of the bad use many make of such words, we must confess
that the fact indicated is consoling by reason of the great desire
there is to use them. This desire marks an effort in the human mind
to raise itself from the mire in which the impious school of the last
century has sunk it.
3. What was the world in the eyes of the false philosophers who
preceded the French revolution? A mass of matter, subject to simple
mechanical laws of motion, the whole explanation of which was given
in two words, blind necessity. What was the human mind? Nothing but
matter. What was thought? A modification of matter. In what did the
difference between thinking and non-thinking matter consist? In a
little greater or less subtilty, in a more or less happy disposition
of atoms. What was morality? An illusion. What were sentiments? A
material phenomenon. What was the origin of man? That of matter,--a
phenomenon offered by a quantity of molecules, which at one moment
happen to be disposed one way, and a moment after in a very different
way. If you inquired if there were a destiny beyond the grave, We argue
that question! they would answer with a scornful smile. Have you such
a word as religion? The scorn increased and changed into contempt. Do
you recognize the dignity of the human race? O, yes! we admit this
dignity, and we are of opinion that it is of the same nature as that
of the brutes, only it has reached a higher degree of perfection. We
do not deny that your form may be more noble and elegant than that of
the monkey, nor do we dispute the superiority of your intelligence;
but we would have you take good care not to make pretensions to a
nobler origin or a loftier destiny. The course of ages may develop and
perfect the monkey form, and render it equal with yours; it may develop
and perfect his cerebral organs, so that from this very monkey, whose
extravagant motions and ridiculous attitudes now amuse, men will be
born such as were Plato, Saint Augustine, Leibnitz, or Bossuet.
4. With such a system, it was useless to deal in ideas: they retained
only sensations. Whatever could occupy the mind of man, whether
the most imbecile or endowed with the loftiest genius, was nothing
more than a sensation transformed. The very brutes possessed all
the elements of human intelligence; to think was only to feel more
perfectly. Such was the last term of their analysis; such the result of
their most accurate observation; such the solution their profoundest
philosophy gave to the problems of man's understanding. Plato,
Aristotle, Saint Augustine, Saint Thomas, Descartes, Malebranche and
Leibnitz were nothing but sublime dreamers, whose genius strongly
contrasted with their ignorance of the true nature of things. None of
them knew any thing about ideology or metaphysics; these sciences were
an unknown world until Locke and Condillac came and discovered them.
5. This school, as fatal as frivolous, has involved and stifled mind in
matter. The butterfly could not unfold his wings of fair and various
colors; he was forced to lay them off and to change into a stupid
and filthy worm, entangled in a covering as loathsome and unclean as
itself. In this consisted progress. The limit of ideological perfection
was to deny ideas; that of metaphysical studies, to deny spirits; that
of morals, to deny morality; that of society, to deny authority; that
of politics, to establish license; that of religion, to deny God. Thus,
human reason, thinking to advance, marched in a retrograde direction;
and proposed to raise the edifice of its knowledge, when there was
nothing left to demolish: thus they imagined to attain a scientific
result by denying every thing, and by finally denying themselves.
6. At present, there is a reaction against so degrading a philosophy.
We have only to open the writings of the philosophers of this age to
convince ourselves of this consoling truth. We everywhere meet the word
idea in contraposition to that of sensation; that of mind to that of
matter; that of activity of thought to that of bodily motion; those
of cause, order, liberty, of free will, morality, infinity. The ideas
which accompany them are sometimes inexact, sometimes extravagant; but
at the bottom of all this we distinguish an anxious desire to rise from
the abyss down to which an atheistical and material philosophy had
dragged the human mind. Some who have contributed to the reaction do
not admit a free and intelligent God, distinct from the universe. What
we have said above is therefore true, that pantheism is atheism in
disguise; nevertheless, the atheism of the pantheist now-a-days is an
atheism which is ashamed to confess itself such, and which sometimes,
perhaps, deceives itself, being persuaded that it is not.
7. The atheism of modern philosophers unites itself with the infinite:
it does not reject those great ideas which as relics of a primitive
tradition were common in the old world, and were afterwards fixed,
cleared up and elevated by the superior teaching of Christianity. The
philosophy of the last century sat down in the darkness and shadow
of death and declared itself alone in possession of light and life.
Philosophy now still remains in obscurity, but it is not satisfied with
it, and gropes about in the dark, seeking some outlet to the regions
of light. Hence those desperate efforts to resolve itself, not into
matter, but into the focus of intelligence, into the _me_, that is, the
mind: hence the continual use of the words absolute, unconditional,
infinite, words which, notwithstanding they ordinarily lead to
absurdities, do yet indicate a sublime aspiration.
8. These observations show that we do not confound the philosophy
of to-day with that of the past century; that we do not regard the
pantheism of to-day as a pure materialism, and that, notwithstanding
the atheism of which we accuse the doctrine of certain philosophers, we
do not deny that they have, even in the midst of their extravagance,
preserved a kind of horror of it, and that, lost as they are in the
labyrinth of their speculations, they seek the thread which shall
conduct them to the gates of truth.
9. This act of justice we willingly render to modern philosophers, but
it will not prevent us from combating their pretension to a merit they
do not possess. They style themselves restorers of the spirituality of
the soul, and of human liberty; and when they speak of God they almost
exact a tribute of gratitude from him for having replaced him upon
his throne. Before making such proud pretensions they ought to have
considered that they are even yet far from the truth with respect both
to God and to man, not only as Christianity has at all times taught it,
but also as the most illustrious modern philosophers have professed it.
They are ambitious to be called restorers, but their restoration with
its licentious frequency is a new revolution, at times as terrible as
the evil it attempts to combat.
10. Another consideration ought to have moderated their zeal to be
thought inventors, which is that they have said nothing concerning God,
the human mind, thought, ideas, the liberty of freewill, which may not
be read in all the works of the philosophers who flourished before, or
even in the beginning of, the eighteenth century. Open the text-books
of the schools, and you will find many things which they would have us
believe to be important discoveries. The great philosophers gloried in
knowing what they had before learned when children. The philosophical
tradition of sound ideas was not interrupted during the past century.
In many parts of Europe schools existed which taught them with
scrupulous fidelity. And besides human schools, there was that of the
God-Man, the Church of Jesus Christ, which, among its supernatural
dogmas, preserved even natural truths, notwithstanding the senseless
efforts which have been made to obliterate them.
11. To what, then, are the invention and restoration reduced? Invention
there is not, either with respect to God, to the human mind, or to
morality, for nothing true has been said of them which had not already
been said. Restoration, properly so called, there is not; for what does
not perish cannot be restored. The truth exists; and has been known and
revered during the whole six thousand years it has refused to bow the
knee to Baal. Let not deserters say, when they turn and come back to
the truth that they have restored it, but that they have recovered it;
not that they give, but that they receive it; not that they enlighten
the world, but that they are blind, and that it is the goodness of
Providence which opens their eyes to the light.
CHAPTER II.
IMPORTANCE AND ANOMALY OF THE QUESTIONS ON THE IDEA OF THE INFINITE.
12. The examination of the idea of the infinite is of the highest
importance, not only because we meet it in various sciences, the
exact sciences among others, but because it is one of the principal
characteristics by which we distinguish God from creatures. A finite
God would be no God; an infinite creature would not be a creature.
In the scale of finite beings we discover a gradation, by which
they are interlinked; the less perfect, as they are perfected, go
on approaching the perfect; and there are, preserving the limits of
each one's nature, points of comparison by which we may measure their
respective distances. Between the finite and the infinite there is no
comparison; all measures are inadequate and as nothing. We pass from
an imperceptible drop to an immense ocean; from the atom which escapes
observation to the abundance of matter diffused through all space;
and much as these transitions express, they are as nothing to the
transition from the finite to the infinite; these oceans, compared with
the infinite truth, become in their turn imperceptible drops, and thus
an interminable scale baffles the efforts of the mind in search of
something to correspond to its idea. The examination of the idea of the
infinite ought to occupy an important place in the study of philosophy,
although it served for no other purpose than the contemplation of
infinite greatness.
13. The disputes on the idea of the infinite, not only in relation
to its nature, but also to its existence, present a strange anomaly.
If it exists in our mind it ought to fill it entirely, so that it
must be impossible to cease to perceive it. Yet it is well known that
philosophers dispute even on the existence of this idea; although it is
an infinite treasure, those who possess it doubt its reality--just as
the heroes in romance, when they find themselves in a castle richly and
splendidly adorned, imagine it the effect of enchantment.
14. The mere dispute as to whether the idea of the infinite be positive
or negative, is equivalent to the question of its existence. If it is
negative, it expresses an absence of being; if positive, the plenitude
of being. What question can be more vital to an idea than the dispute
whether it represents the absence or the plenitude of being?
15. Here again we meet the fact which we have observed in the preceding
discussions. Reason, after digging at its own foundations, is
threatened with death under the ruins of its loftiest edifices.
CHAPTER III.
HAVE WE THE IDEA OF THE INFINITE?
16. If we had no idea of the infinite, the word would have no meaning
to us, and when used it would not be understood.
17. Whatever may be the nature and perfection of our idea of the
infinite, it is certain that it involves something fixed, and common
to all intelligences. We apply the idea to things of very different
orders, and it is always understood in the same sense by all men. Even
the difficulty we find in attempting to explain it, in itself or in its
applications, proceeds from the idea itself; it is a difficulty which
we all meet with, because we all conceive in the same manner what is
understood by the infinite, taken in general.
18. Infinite and indefinite express very different meanings. The
infinite implies the absence of limits; the indefinite implies that
these limits retire continually from us; it abstracts their existence,
and only says that they cannot be assigned.
19. Whatever exists is finite or infinite; for it either has limits or
it has not: in the first case, it is finite; in the second, infinite:
there is no medium between yes and no.
20. Hence, properly speaking, there is in reality nothing indefinite;
this word only expresses a mode of conceiving things, or rather a
vagueness in the conception, or indecision in the judgment. When we do
not know the limits of any thing, and, on the other hand, do not dare
to affirm its infinity, we call it indefinite. Thus, space is called
indefinite by those who see no way of assigning a limit to it, and yet
are unwilling to say that it is infinite. Even in ordinary language we
call a thing indefinite which has no limits assigned to it; thus, we
say "a concession has been made for an indefinite time," although it is
limited to some time which has not been determined.
21. The idea of the infinite does not consist in conceiving that
another quantity may always be added to a given quantity, or that
a perfection may be made more intense; this expresses only the
possibility of a series of conceptions by which we endeavor to approach
the absolute idea of the infinite. It is easy to see that the absolute
idea is something distinct from those conceptions, because we regard it
as a type to which the series of connections is referred, but which it
can never equal, no matter how greatly prolonged.
22. Let us consider the words in which we naturally express what passes
within us when we think of the infinite.
What is an infinite line? A line which has no limits. Is it a million,
or a billion miles in length? There is no number to express its length;
it will always be greater than the number. But do we not approach the
infinite in proportion as we prolong a finite line? Certainly, in so
far as _approaching_ means only placing quantities which are found in
what we approach; but not in so far as it means that this difference
can be assigned. There is no comparison between the finite and the
infinite; and therefore it is not possible to assign the difference
between them. Would an infinite line be formed by the addition of all
finite lines? No; for we can conceive the multiplication of each of the
terms of the addition, and therefore an increase in the infinite, which
would be absurd. Would the infinity of the line consist in our not
knowing its limits, or not thinking of them? No; but in its not having
them.
23. Thus, we see, that the idea of the infinite, is in the reach of the
most common intellects, and expresses only what any person of ordinary
understanding would say, even though he had never occupied himself
with philosophical studies; that the idea of the infinite is in our
understanding, as a constant type, to which all finite representations
are unable to arrive. We know the conditions which must be fulfilled,
but at the same time, we see the impossibility of fulfilling them. When
any one tries to persuade us of the contrary, we reflect on the idea
of the infinite, and say: "No; it is a contradiction of infinity; it
is not infinite, but finite." We distinguish perfectly well between
the absence of the perception of the limit and its non-existence. If
any one tries to make us confound these two ideas, we answer, "No;
they must not be confounded; there is a great difference between our
not perceiving an object and the non-existence of that object, and we
are not now examining whether we conceive the limit, but whether it
exists." Though the limit retire and hide itself, so to speak, from our
eyes, we are not deceived: it exists, or does not exist. If it exists,
the condition involved in the conception of infinity is not fulfilled,
and the object is not infinite, but finite; if it does not exist, there
is true infinity,--the condition is complied with.
24. When the idea of the infinite is considered in general, it can
never be confounded with the idea of the finite. There is a line which
divides them, and which prevents all error; for it is the principle of
contradiction itself; it is the distinction between _yes_ and _no_.
When we say _finite_, we affirm the limit; when we say _infinite_, we
deny it. No ideas can be clearer or more exact.
CHAPTER IV.
THE LIMIT.
25. The word _infinite_ is equivalent to _not finite_, and seems to
express a negation. But negations are not always truly such, although
the terms imply it; for if that which is denied, be a negation, the
denial of it is an affirmation. This is the reason why two negatives
are said to be equivalent to an affirmative. If I say, it has not
varied, and you deny it, you deny my negation; for it is the same thing
to deny that it has not varied, as to affirm that it has varied. In
order, therefore, to determine whether the word _infinite_ expresses a
true negative, we must know what is meant by the word _finite_.
26. The finite is that which has a limit. A limit is the term beyond
which there is nothing of the object limited. The limits of a line,
are the points beyond which the line does not extend; the limit of
a number, is the extreme where the number stops; the limit of human
knowledge, is the point to which we may arrive, but which we cannot
go beyond. A limit being a negation, to deny a limit, is to deny a
negation, and is consequently an affirmation.
27. It is easy to see from these examples, that a limit in the ordinary
sense, expresses an idea distinct from what mathematicians define
it. They call a limit every expression, whether finite, infinite,
or a nullity, which a quantity may continually approach without
ever reaching. Thus, the value 0/a is the limit of the decrement of
a fraction, the numerator of which is variable x/a; because, if we
suppose X to be constantly diminishing, the fraction will approach
the expression 0/a, without ever being confounded with it, so long as
X does not entirely disappear. If we suppose (b + x)/a an expression
in which X is decreasing, the expression will continually approach (b
+ 0)/a = b/a, which will be the limit of the fraction. If we suppose
the expression a/x, in which X is decreasing, we shall continually
approach the expression a/0 = ∞, an infinite value which
the fraction can never attain, until X becomes 0, which cannot happen,
because X is a true quantity. These examples show that mathematicians
admit limits which are finite, infinite, or a nullity, and prove that
mathematicians employ the word limit in a different sense from its
ordinary as well as philosophical meaning.
28. A limit, therefore, expresses a true negation, and the word
finite, or limited, necessarily involves a negative idea. That which is
not, is not limited; therefore the finite is not an absolute negation.
An absolute negation is nothing, and we do not call the finite nothing.
Therefore, in the idea of finite are contained being, and a negation
of another being. A line one foot in length, involves the positive
value of one foot, and the negation of all value of more than a foot.
Therefore, the finite, in so far as finite, involves a negation
relatively to a being. If we could express this idea in the abstract,
using the word finity, as we have the word infinity, we should say that
finity in itself expresses only the negation of being relatively to a
being.
29. Hence, the word infinite is not negative; for it is the negation
of a negation. The infinite is the not-finite; it is that which has no
negation of being, consequently that which possesses all being.
30. We have, therefore, an idea of the infinite, and this idea is not a
pure negation. But it must not be supposed that we have arrived at the
last term of the analysis of the infinite. We are still far from it,
and it is even doubtful whether we shall obtain any satisfactory result
after long investigations.
CHAPTER V.
CONSIDERATIONS ON THE APPLICATION OF THE IDEA OF THE INFINITE TO
CONTINUOUS QUANTITIES, AND TO DISCRETE QUANTITIES, IN SO FAR AS THESE
LAST ARE EXPRESSED IN SERIES.
31. One of the characteristic properties of the idea of the infinite is
application to different orders. This gives occasion to some important
considerations which greatly assist to make this idea clear in our
mind.
32. From the point where I am situated I draw a line in the direction
of the north; it is evident that I may prolong this line infinitely.
This line is greater than any finite line can be; for the finite line
must have a determinate value, and therefore, if placed on the infinite
line, will reach only to a certain point. This line, therefore, seems
to be strictly infinite in all the force of the word, because there is
no medium between the finite and the infinite, and we have shown that
it is not finite, since it is greater than any finite line; therefore
it must be infinite.
This demonstration seems to leave nothing to be desired; yet there is
a conclusive argument against the infinity of this line. The infinite
has no limits, and this line has a limit, because, starting from the
point from which it is drawn in the direction of the north, it does not
extend in the direction of the south.
33. This line is greater than any finite line; but we may find another
line greater still. If we suppose it produced in the direction of the
south, it will be greater by how much it is produced towards the south;
and if it be infinitely produced in this direction, its length will be
twice that of the first line.
34. By the infinite prolongation of a line in two opposite directions
we seem to obtain an absolutely infinite line; for we cannot conceive
a lineal value greater than that of a right line infinitely prolonged
in opposite directions. But it is not so: by the side of this right
line another may be drawn, either finite or infinite, and the sum of
the two will form a lineal value greater than that of the first line;
therefore that line is not infinite, because it is possible to find
another still greater. And as, on the other hand, we may draw infinite
lines and prolong them infinitely, it follows that none of them can
form an infinite lineal value, because it is only a part of the lineal
sum resulting from the addition of all the lines.
35. Reflecting on this apparent contradiction in our ideas, we discover
that the idea of the infinite is indeterminate, and consequently
susceptible of different applications. Thus, in the present instance,
it cannot be doubted that the right line, prolonged to infinity,
has some infinity, since it is certain that it has no limit in its
respective directions.
36. This example would lead us to believe that the idea of the infinite
represents nothing absolute to us; because even among those objects
which are presented the most clearly to our mind, such as the objects
of sensible intuition, we find infinity under one aspect which is
contradicted one by another.
37. What we have observed of lineal values is also true of numerical
values expressed in series. Mathematics speak of infinite series, but
there can be no such series. Let the series be _a_, _b_, _c_, _d_,
_e_, ....: it is called infinite if its terms continue _ad infinitum_.
It cannot be denied that the series is infinite under one aspect; for
there is no limit which puts an end to it in one sense; but it is
evident that the number of its terms will never be infinite, because
there are others greater; such, for instance, is the series continued
from left to right, if continued from right to left at the same time,
in this manner:
.......... _e_, _d_, _c_, _b_, | _a_, _b_, _c_, _d_, _e_, ..........
In this case the number of terms is evidently twice as great as in the
first series.
Therefore the series which are called infinite are not infinite, and
cannot be so, in the strict sense of the term.
38. But what is still more strange is, that the series is not infinite,
even though we suppose it continued in opposite directions; for by its
side we may imagine another, and the sum of the terms of both will be
greater than the terms of either; therefore neither will be infinite.
As it is evident that whatever be the series, we can always imagine
others, it follows that there can be no infinite series in the sense in
which mathematicians use the word series to express a continuation of
terms, not excluding the possibility of other continuations besides the
supposed infinite continuation.
39. The objections against lineal infinity apply equally to surfaces.
If we suppose an infinite plane, it is evident that we can describe an
infinity of planes distinct from the first plain and intersecting it in
a variety of angles; the sum of all these surfaces will be greater than
any one of them. Therefore the infinite extension of a plain in all
directions does not constitute a truly infinite surface.
40. A solid expanding in all directions seems to be infinite; but if
we consider that the mathematical idea of a solid does not involve
impenetrability, we shall see that inside of the first solid a second
may be placed, which, added to the first, will give a value double
that of the first alone. Let S be the empty space which we imagine
to be infinite; and let W be a world of equal extension placed in it
and filling it; it is evident that S + W are greater than S alone.
Therefore, although we suppose S to be infinite, = ∞, W also = ∞;
therefore S + W = ∞ + ∞ = 2 ∞. And as this value expresses the size,
the first is not infinite because it can be doubled. If we take the
impenetrability, the operation may proceed _ad infinitum_.
Therefore the first infinite, far from being infinite, seems to be a
quantity susceptible of infinite increase.
CHAPTER VI.
ORIGIN OF THE VAGUENESS AND APPARENT CONTRADICTIONS IN THE APPLICATION
OF THE IDEA OF THE INFINITE.
41. The difficulties in the application of the idea of infinity,
seem on the one hand, to prove that either this idea does not exist
in us, or is very confused; and on the other hand, that we possess
it, and in a very perfect degree. Why do we discover that numbers
are not infinite, although at first they seem to be? Why do we deny
the infinity of certain dimensions, notwithstanding their infinite
prolongation in one sense? Because, on examining these objects, we find
that they do not correspond to the type of infinity. If this type did
not exist in our mind, how could it be possible for us to make use of
it? How could we compare beings with it, if we did not know it? Is it
possible to know when any thing arrives at a turn, if we have no idea
of that turn? It is comparing without a point of comparison; that is,
it is exercising a contradictory act.
42. Although these arguments in favor of the existence of the idea of
the infinite, if we examine our own mind, we cannot deny that we find
there a certain vagueness and confusion which inspire strong doubts as
to the reality of this idea. What is presented to our mind, when we
think of the infinite? The imagination abandoned to itself, extends
space, expands dimensions, multiplies numbers indefinitely, but it
offers nothing to the intellect which has the marks of infinity. If we
leave the imagination, and regard the understanding only, it gives a
type by which to judge of the infinity or not-infinity of the objects
presented to it, but if we reflect on the type itself, it loses the
clearness it possessed before, and we even ask if the type really
exists.
43. Do we, therefore, deny the existence of this idea? are we going to
renounce our intention of explaining it? We do neither. I believe that
it is necessary to admit the idea, that it is not impossible to explain
it, and that we may even point out the reason of its obscurity.
44. Before passing further, I wish to observe, that one of the causes
of the difficulties in the explanation of the idea of the infinite,
arises from our not distinguishing the intuitive from the abstract
cognition.[36] Many difficulties would be avoided by attending to
this distinction. When we say that the idea of the infinite is not
intuitive, but abstract, we give the key to the solution of the
principal objections brought against it.
[36] See Book V., Ch. XI.
45. We have no intuitive idea of infinity; that is to say, this idea
does not present to our mind an infinite object; we can have this
intuition only when we see the essence of God, which will happen in a
future life.
46. If we had now the intuition of an infinite object, we should see
its perfections as they are, with their true marks; or rather, we
should see how all the perfections dispersed among limited beings, are
united in one infinite perfection. We could not refer the idea of the
infinite to determinate objects, as, for example, to extension, because
these objects contradict the idea. It would be impossible for us to
modify the idea in different ways, and apply it first in one sense,
and then in another very different sense. The idea is one, and simple;
it would, therefore, always relate to an object which is also one and
simple, not vague and indeterminate, as now, but with the determination
of a necessary existence and an infinite perfection. We should have
intuition of infinite being, as we have intuition of the facts of
our consciousness: our cognition of it would be that of an object
eminently incommunicable, as predicate to any order of finite beings;
and it would be as manifest a contradiction, to apply the idea of this
infinity to any number or extension, as it would be to identify an act
of our consciousness with external objects.
47. The indeterminate character in which the idea of the infinite is
presented us, and the ease with which we modify it in various ways, and
apply it to different objects, in different senses, proves that this
idea is not intuitive, but abstract and indeterminate, that it is one
of those general conceptions, by the aid of which the mind obtains a
certain knowledge not afforded by intuition.
This will explain the origin of the vagueness of our idea of infinity.
Indeterminate conceptions, and because they are indeterminate, relate
to no particular object, or quality, which may be conceived by itself
alone, as something which may be realized; they do not contain those
determinations which fix our cognition in an absolute manner. The
indeterminate manner in which they present any property of beings,
causes a difference in the application, accordingly as the particular
properties, which are combined with the general, are different. If we
take a right-angle triangle, in which we know the measure of all the
sides and angles, the determinateness of the idea avoids the vagueness
of the intellect, and prevents the application of this idea to cases
different from that which is determinate and fixed. But if we take a
right-angle, in general, without determining the value of its sides
and angles, its applications may be infinite. The more general and
indeterminate the idea of a triangle becomes, the greater is the
variety of its applications.
48. Indeterminate ideas, in order to represent any thing, must be
applied to some property which is the condition of their actual or
possible realization. Until this application is made, they are pure
intellectual forms, which represent nothing determinate. I do not
mean by this, that these ideas are empty conceptions, which cannot be
applied outside of the sensible order, as was maintained by Kant;[37]
but only that granting them an universal value, I deny that they have
by themselves alone a value representative of any thing that can be
realized, beyond the property which they express. The idea of a _pure_
triangle can not be realized, for every _real_ triangle would contain
something more than is in the idea: it would be a right-angled or
oblique-angled, etc., all which, the pure idea abstracts. The object
will be indeterminate, in proportion to the indeterminateness of the
properties contained in the conception; consequently, that which
is presented to the understanding will also be more vague, and the
applications which may be made of the idea, will be more varied and
numerous, as is the case in the ideas of being, not-being, limit, and
the like.
[37] See Book V., Chapters XIV., XV., and XVI.
CHAPTER VII.
FUNDAMENTAL EXPLANATION OF THE ABSTRACT IDEA OF THE INFINITE.
49. Supposing that our idea of the infinite is not intuitive but
abstract, let us see how its true nature may be explained.
We have the ideas of being and of its opposite, not-being; these ideas
considered in themselves are general, indeterminate, and may be
applied to every thing which is subjected to our experience.
We may affirm and deny something of every limited being: we may affirm
what it is: we may deny what it is not: the limit is only conceived as
such when something is denied of it.
50. The activity of our being is unceasing, but it is limited by
the absence or the resistance of objects; the external world is an
assemblage of beings presenting a great variety of limitations.
Therefore both internal and external experience give us the idea of the
finite, that is, of a being which involves some not-being. The brute
has sensible perception, but no understanding: it _is_ sensitive, and
herein it has being; it _is not_ intelligent, and herein it is limited.
Man is sensitive and intelligent; the limit of the brute is not the
limit of man. Among intelligent beings some understand more than
others; therefore the limit of all is not the same.
51. Since we find a limit in both internal and external experience, it
is evident that we can form the general idea of limit, that is, of a
negation applied to an object.
52. The same experience teaches that what is the limit of some
things is not the limit of others, and that the limit applied to one
object must be denied of another. When we compare different beings
together, we frequently find ourselves _denying certain_ limits. As
our understanding has the faculty of generalizing, it is evident that
we may conceive in general the negation of _certain_ limits, and form
an indeterminate conception, including the two ideas of _negation_ and
_limit_.
53. I do not see what objection can be made either to the possibility
or to the existence of this conception; but as this fact is necessary
for the explanation of the idea of infinity, I shall make some further
observations for the purpose of confirming it.
We have an idea of negation in general; this is a primitive fact of our
mind: without it no negative judgments would be possible, nor could
we even know the principle of contradiction. It is impossible for any
thing to be _and not be_ at the same time; when we say _not be_ we
express a negation, we therefore have the conception of negation. This
conception is general, because it involves no determination; we speak
of not-being without applying it to any particular object, nor even to
any determinate species or genus. Therefore the conception of negation
is general and absolutely undetermined.
54. We have the idea of limit; for, as we have seen, it is a negation
applied to a being. We have also the idea of the negation of limit; for
just as we conceive the limit as applied or applicable, we may and do
conceive it as not applied or not applicable. At every moment we deny
certain limits; this idea generalized becomes the negation in general
of limit in general.
55. After these remarks we may establish what is contained in the
idea of the infinite. This idea is a general conception involving the
conception of being in general, and the negation of limit in general.
The union of these two conceptions constitutes the abstract idea of the
infinite.
56. The general conception of the negation of limit gives us an idea
of infinity in the abstract, but not any infinite thing. Without
the intuitive cognition of an infinite object, and with only a very
imperfect idea of it, we may speak of infinity without falling into
contradiction, and determine the cases in which it may be applied to a
being or to an order of beings, whether real or possible. Man has many
ideas of this vague kind, which nevertheless answer his necessities. We
shall make this palpable by examples.
57. Suppose we take an uneducated person and point out to him a number
of learned men, telling him that one of them knows more than all the
rest. The uneducated person has no idea of what the man knows who
knows the most, nor the man who knows the least; he has no idea of the
degrees of science, nor of what science itself is; but he possesses the
general ideas of degree, of more and less, and also of knowledge, and
this enables him to speak, without contradiction or confusion, of the
greater science of the one and the less science of the others, and even
to solve with certainty the questions concerning the science of those
individuals, in so far as these questions are contained in the general
idea that the science of one is greater than that of all the others.
A servant in an establishment where the most beautiful products of
art are collected, may speak of them all without contradiction or
confusion, although he may be incapable of knowing their merit, and
entirely ignorant of the circumstances which constitute the beauty of
the objects. It is sufficient for him to have the idea of perfection
or beauty in general, and to arrange by certain arbitrary signs the
degrees of perfection or beauty of the objects, in order to be able
to point them out to visitors, and talk of the greater skill of one
artist, the poorer success of another; the greater effect and value of
the works of the former, and the inferiority of those of the second,
and to make other remarks of a similar nature, which at first might
make us suppose him a consummate artist, or, at the least, an amateur
of a great intellect and exquisite taste.
58. It would be easy to show by other examples, how fruitful some
general ideas are, and how they may undergo innumerable combinations,
without presenting any thing determinate to the intellect. This is
precisely what happens with the idea of the infinite: in vain we ask
what there is within us which corresponds to it: the conception of
being in general and of the negation of limit present nothing fixed,
except certain abstract conditions to which we continually reduce the
objects which come under our intuition, or are presented to us with
certain characteristic properties which permit us to form a less vague
idea of the negation of limit.
CHAPTER VIII.
THE DEFINITION OF INFINITY CONFIRMED BY APPLICATION TO EXTENSION.
59. We have explained the idea of infinity in general, by the
indeterminate conceptions of being and the negation of limit. In order
to assure ourselves that the explanation is well grounded, and that we
have pointed out the essential marks of the conception, let us examine
whether their application to determinate objects corresponds to what we
have established in general.
If the idea of infinity is what we have defined it to be, we may apply
it to all objects of sensible intuition or of the pure understanding,
and we shall obtain the results which we ought to obtain, including the
anomalies already referred to.[38]
[38] See Chap. V.
60. The anomalies, or, rather, the contradictions which we seem to find
in the applications of the idea of the infinite, when any thing is
presented to us as infinite which we afterwards discover not to be so,
originate in the application of this idea under different conditions.
This variety would not be possible if the idea represented any thing
determinate; but as it only contains the negation of limit in general
joined to being in general, it follows that we subject this negation to
particular conditions in each case, and therefore when we pass to other
conditions, the general idea cannot give us the same result.
61. A line drawn from the point where we are situated in the direction
of the north, and produced infinitely, gives us an infinite and a
not-infinite. This contradiction is only apparent; there is really
only the difference of result caused by the condition under which the
general idea is applied.
When we consider a line infinitely produced towards the north, we do
not apply the idea of the infinite to a lineal value in the abstract,
but to a right line starting from a point and produced only in one
direction. The result is what it should be. The negation of limit is
affirmed under a condition; the infinite which results is subject to
that condition. It may be said that there is no medium between the
infinite and the not-infinite; but it is easy to solve this difficulty,
if we observe that yes and no, to be contradictory, must be referred to
the same thing, which is not the case when the conditions of the object
are changed.
62. If instead of a line produced in one direction only, we had wished
to apply the negation of limit to a right line in general, it is
evident that we should have been obliged to produce the line in the two
opposite directions: which would have given us another infinite under a
new condition.
We have before seen that not even in this case can we have a lineal
value strictly infinite; because this right line only forms a part
of the sum of lines which we can imagine. Is it then infinite, or is
it not? It is both, if we make the proper distinction. It will be
infinite, or we shall have the idea of infinity or negation of limit,
applied to a right line _alone_; but if instead of one _right line
alone_, we take a lineal value, without any condition, the supposed
line will not be infinite; the negation of the limit is not applied
under that condition; the result must therefore be different.
63. We find the same anomaly, if we take two lines alone. Let us
suppose a right line infinitely produced in both directions, and by
its side let us describe a curve with continual undulations extending
infinitely in a direction parallel to the right line. Both lines will
be infinite if we consider only their direction, abstracting their
lineal value; but if we regard this value the curve is greater than
the straight line; for it is evident if we take a part of the curve
corresponding to a part of the straight line, and extend or straighten
this part of the curve, it will be greater than the corresponding part
of the straight line; as this may be done throughout the whole length
of the lines, the lineal value of the curve must be greater than that
of the straight line in proportion to the law of its undulations.
64. This may suffice to show how the idea of infinity may be applied
under different conditions and produce different results, without
any contradiction. What is infinite under one aspect is not so under
another aspect; hence we have the _orders of infinities_ which figure
so largely in mathematics; but I say again that these contradictions
are not susceptible of any explanation if we attribute an absolute
value to the idea of the infinite, instead of considering it as the
abstract representation of the negation of limit.
65. Is it possible to conceive in a right line or curve an absolutely
infinite length or lineal value, to which we may apply the negation
of limit absolutely? I think not: for whatever be the line under
consideration we can always draw others, which, added to the first,
will give a value greater than that of the first above. This is a
case in which there is a contradiction between the negation of limit
and the condition to which it is subjected. You demand a lineal value
to which the negation of limit may be applied absolutely; and on the
other hand you require that this lineal value should be found in a
determinate line, which by the fact of its being determinate, excludes
the absolute negation of limit. The problem supposes contradictory
_data_; therefore the result must be a contradiction.
66. What must we suppose in order to conceive an absolutely infinite
lineal value? We need only suppose no condition which excludes the
absolute negation of limit. We must here distinguish between the pure
conception and the sensible intuition in which it is expressed. The
conception of infinite lineal value exists from the moment that we
unite the two general conceptions of lineal value and negation of
limit. But the sensible intuition, which may represent this conception,
is not so easy to imagine, even in general. To arrive at it we must
imagine a space without any limit; and then considering in general all
the lines whether right lines or curves, which may be drawn in it, in
all directions, and under all possible conditions, we must take the
sum of all these lineal values; and the result will be an absolutely
infinite lineal value; for we shall have applied the negation of limit
without any restriction.
67. We may obtain in the same way an infinite superficial value; for
it is evident that we may apply to it all that we have said of lineal
values.
68. In all these cases we apply the negation of limit to extension
considered only in some of its dimensions. If we wish to obtain
an absolutely infinite extension, we must abstract no dimension;
consequently the absolutely infinite of this order, is extension in
all its dimensions with the absolute negation of limit. But it is also
to be observed that we must presuppose an absolutely infinite value of
extension in order to obtain an absolutely infinite value of lines or
surfaces; because it is equivalent to presupposing an infinite space in
which the lines and surfaces may be drawn in all directions and under
all possible conditions.
CHAPTER IX.
CONCEPTION OF AN INFINITE NUMBER.
69. Can we conceive an infinite number? On one side, it seems not;
because we doubt its possibility, and if we possessed this idea we
should have no doubt of its existence. On the other side, it seems
that we can conceive an infinite number; for we know immediately when
a number is not infinite, and we could not know this if we had not the
idea of infinite number.
Our observations on infinite series would seem to prove that the idea
of infinite number is an illusion; for we find those numbers which we
believed infinite, not to be so.
I think this question may be solved on the same principles as those
of the last chapter. I see no difficulty in admitting the idea of an
infinite number, nor how any contradiction can proceed from it.
70. Number is a collection of units; it is a general idea, because
to conceive the number, we do not need to know of what class, or
how many the units may be. The idea of number in general abstracts
absolutely all such determinations. It is evident that, whatever number
we imagine, we can always conceive another still greater, and if we
assign a limit to a number, we can always remove it indefinitely, so
that the limit of one is not the limit of the other. To the idea of
number, we unite the idea of a limit and of the negation of another
limit. Therefore, if we unite to the idea of number in general, the
idea of the negation of limit in general, we shall obtain the idea of
an infinite number.
71. What does this idea represent? It represents nothing determinate:
it is an entirely abstract conception, formed of two other abstract
conceptions, those of number and the negation of limit. No determinate
object corresponds to it; it is a work of our understanding referred
to objects in general, without a determination of any sort. We may now
solve the difficulties previously intimated.
72. Why is a series of terms presented to us as infinite, which, when
we examine it closely, we find wants some of the marks of infinity?
Because, in the first instance, we apply the negation of limit under a
condition which we take no notice of in the second instance.
Set us the series _a_, _b_, _c_, _d_, _e_, ..........
It is evident that we may continue it infinitely, and conceive the
negation of all limit of this continuation: in this sense, the number
of terms is infinite; for the idea of the negation of limit is really
applied to the series. When we ask if the number of terms is absolutely
infinite, we abstract the condition under which we had united the
negation of limit. That, therefore, which is infinite in one instance
is not so in another. Still there is not any contradiction because the
yes and the no refer to different suppositions.
73. Let us take a line and measure it by feet. Producing this line
we multiply the number of feet; and we may conceive the negation of
all limit of this multiplication. The number of feet will then be
infinite. If instead of a foot we take an inch as the unit of measure,
we shall have a number twelve times as great. This number would also
be infinite, and thus we should have two infinite numbers, one of them
greater than the other. Is there any contradiction in this? Certainly
not: there is only a different combination of ideas. In the first case,
the idea of the negation of limit was subordinated to the condition of
the division of the line into feet: whereas, in the second case, we
introduce a different condition; the division of the line into inches.
74. But, it may be said, these numbers, considered in themselves,
abstracted from their relation to feet or inches, are equal or they
are not equal; consequently they are infinite or not infinite. The
objection vanishes as soon as we correct the error which supports it.
When we abstract all relation to determinate divisions, we consider
number in general; on this supposition there are not two cases, but
only one; there cannot then be a relation of greater or less. We have
only the conception of number in general combined with the idea of the
negation of limit in general; therefore the result must be an infinite
number in the abstract.
The difficulty consists in a contradiction which escapes our sight
at first. We abstract particular conditions in order to know if the
numbers are in themselves infinite or not; and at the same time we do
not abstract them, because it is only in reference to them that the
objection has any meaning, since it supposes the division into various
kinds of units. When, therefore, we speak of particular numbers, and
at the same time pretend to consider them in themselves, we fall into
a contradiction, because we take the numbers both with and without
particular conditions at the same time.
75. From all that has been said, we may conclude that the conception
of infinite number, abstracted from the nature and relations of the
things numbered, involves no contradiction, since it contains only the
two ideas of number, as a collection of beings, and of the absolute
negation of limit; but we cannot affirm from this alone, that an
infinite number can be realized. Infinite number cannot become actual
without an infinite collection of beings; and these beings, when
realized, cannot be abstract beings, which contain nothing else but
being; they must have characteristic qualities, and must be subject to
the conditions imposed by these qualities. As we absolutely abstract
these conditions in the general conception, it is not possible to
discover, from the conception alone, the contradiction which they
may imply. Hence, although there is no contradiction contained in
the conception, there may still be in the reality. In the same
manner, certain mechanical theories are perfectly conceivable, but
they cannot be reduced to practice on account of the opposition of
the matter to which they should be applied. Finite beings are the
matter on which indeterminate and metaphysical conceptions are to be
realized; the possibility of the conceptions does not absolutely prove
the possibility of the beings. The reality may draw with it certain
determinations involving a contradiction which was latent in the
general conception, and is made manifest by the reality.
CHAPTER X.
CONCEPTION OF INFINITE EXTENSION.
76. Is infinite extension conceivable? This conception includes
two ideas: the idea of extension, and the idea of the negation of
limit. The idea of extension is a general conception, referring to
the intuition which, whatever may be in itself and in its object,
represents extension and the union of the three dimensions, the pure
form of which is space. It is evident that we can unite, in one
conception, the two ideas of extension in general and the negation of
limit; and if this is what is called the idea of infinite extension, it
is clear that we have this idea. This conception of infinite extension,
abstracts all conditions of the reality; we do not know whether there
be, in the nature of extended things, any thing which prevents the
absolute infinity of their extension; consequently, we are ignorant
whether there is or is not any latent contradiction, which the general
conception does not reveal to us.
77. It must be remembered that I am speaking of the idea and not of
the sensible representation of extension; for although I hold that it
is possible for us to have the conception of an infinite extension, I
do not think the same with respect to its sensible representation. The
latter may be indefinitely expanded, but it cannot become infinite.
Reason demonstrates this impossibility which consciousness makes known
to us. Internal sensible representations are only the repetition of
the external, or at least are formed from the elements which these
latter furnish. Sight and touch are the two senses which produced
the representation of extension, and they both imply a limit. Touch
only reaches that which is immediate to it, and sight cannot see
with a limit which sends the rays of light to it. Internal sensible
representations must always retain this limitation; their object may be
expanded, or the limit removed to a greater distance, but to destroy
this limit would be to destroy themselves. Therefore, the imagination
of an infinite extension is impossible to every sensitive being.
78. I have proposed above (§ 40) an objection against the infinity of
extension, in so far as we may represent it as a size without limits.
The objection was, that as the idea of impenetrability is not
contained in the conception of a solid, we may imagine an infinite
series of infinites placed one inside of another. This difficulty
is only conclusive when speaking of the conception of a solid which
contains something more than the pure idea of extension. The idea of
extension necessarily implies that some parts are _outside_ of others,
and it is not possible to conceive extension otherwise. It is certain
that a body may be situated in a part of space; taking from this body
its impenetrability, we may put another body in the same place, and so
on to infinity; but in that case we conceive something besides pure
extension, we unite something, although in a general and indeterminate
manner, to the idea of things situated in space; otherwise we should
not distinguish the space, representing pure extension, from the solids
placed in it, nor should we distinguish these solids from one another,
if we did not recognize in them some difference, although general and
undetermined.
79. It seems most probable that the pure idea of an infinite extension
is contained in the idea of an infinite size, which is nothing more
than the idea of space. Whatever else is introduced into the idea is
a foreign element, adding to pure extension something which does not
belong to it, such is the difference between extended beings, although
conceived in an indeterminate manner.
CHAPTER XI.
POSSIBILITY OF INFINITE EXTENSION.
80. What are we to think as to the possibility of the infinities which
we conceive? Let us examine the question.
Is an infinite extension possible? There is no incompatibility between
the idea of extension and the negation of limit, at least, according
to our way of conceiving them. It is more difficult for us to conceive
extension absolutely limited, than to conceive it unlimited: beyond all
limit, we imagine space without end.
81. Neither do we discover any impossibility in the existence of
an unlimited extension, if we consider the question in relation to
the divine omnipotence. Beyond all extension God can create another
extension; if we suppose that he has applied his creative power to all
the extension possible, he must have created an infinite extension.
82. Here a difficulty arises. If God had created an infinite extension
he could not create another extension; his power would be exhausted,
and consequently it would not be infinite.
This difficulty proceeds from understanding infinite power in a false
sense. When we say that God can do all things, we do not mean that he
can do things that are contradictory: omnipotence is not an absurd
attribute, as it would be if applied to things that are absurd. An
absolutely infinite extension is contradictory in relation to another
distinct extension; for, being absolutely infinite, it contains all
possible extensions. If we suppose it to exist, no other is possible:
to affirm that God could not produce another, is not to limit his
omnipotence, but only to say that he cannot do a thing which is absurd.
83. We will make this solution clearer. The intelligence of God is
infinite; and he cannot understand more than he now understands; all
progress would suppose imperfection, because it would involve a change
from a less to a greater intelligence. If, then, we say that God will
never understand more than he does now, do we limit his intelligence?
Certainly not. He cannot understand more, because he understands
all that is real and all that is possible, and we cannot, without
contradiction, conceive that he can understand more than he now does:
this is not to limit his intelligence, but to affirm its infinity: it
is not susceptible of perfection, because it is infinite. This will
enable us to understand the expression _cannot_, as applied to God.
What is denied is not a perfection, but an absurdity: wherefore St.
Thomas very opportunely observes, that we should much better say that
the thing cannot be done, than that God cannot do it.
CHAPTER XII.
SOLUTION OF VARIOUS OBJECTIONS AGAINST THE POSSIBILITY OF AN INFINITE
EXTENSION.
84. The discussions on the possibility of an infinite extension are of
a very ancient date. How could it be otherwise? Must not the glorious
spectacle of the universe, and the space which we imagine beyond the
boundaries of all worlds, naturally have given rise to questions as to
the existence or possibility of a limit to this immensity?
Some philosophers think an infinite extension impossible. Let us see on
what they found their opinion.
85. Extension is a property of a finite substance, and that which
belongs to a finite thing cannot be infinite; therefore it is
impossible to conceive infinity of any kind in a finite being. This
argument is not conclusive. It is true that an extended substance is
finite, in the sense that it does not possess absolute infinity such
as is conceived in the Supreme Being; but it does not follow from
this that it cannot be infinite under certain aspects. Neither is it
correct to say that no finite substance can have an infinite property,
because the properties flow from the substance, and the infinite cannot
proceed from the finite. In order that this argument may be valid, it
is necessary to prove that all the properties of a being emanate from
its substance: figures are accidental properties of bodies, and yet
many of them have no relation to the substance, and are mere accidents
which appear or disappear, not by the internal force of the substance,
but by the action of an external cause. We see extension in bodies; but
as we know not the essence of corporeal substance, we cannot say how
far this property is connected with the substance, whether it is an
emanation from it, or only something which has been given to it and may
be taken from it without any essential alteration.[39]
[39] See Bk. III., Chaps. XIX., XXI., XXIV., XXV., XXVI., XXVII. and
XXVIII.
Moreover, when we say that the infinite cannot proceed from the finite,
we do not deny that an infinite property may proceed from a substance
finite in its essence.
When we admit the infinite property, we admit at the same time all that
is necessary in the substance in order that this property may have its
root in it, so long as we do not deny the character of finite which
essentially belongs to every creature. When we deny that creatures are
or can be infinite, we speak of essential infinity, of that infinity
which implies necessity of being and absolute independence under every
aspect; but we do not deny them a relative infinity, such as that of
extension.
To undertake to prove that infinite extension is impossible, because
every property of a finite substance must be finite, is equivalent
to supposing the very thing in dispute; for the precise question is,
whether one of these properties, namely, extension, can be infinite. In
order to establish the negative proposition, "No property of a finite
substance can be infinite," it is necessary to prove this of extension.
Hence the argument which we are imposing implies, in some manner, a
begging of the question, when they found it on a general proposition
which can only be certain when the present question is solved.
86. Infinite extension ought to be the greatest of all extensions, but
there is no such extension. From any given extension God can take away
a certain quantity; for example, a yard: in that case the infinite
extension would become finite, for it would be less than the first;
and as the difference between the two extensions is only a yard, it is
clear that not even the first could be infinite; for it is impossible
that there should be only the difference of one yard between the finite
and the infinite.
This difficulty merits a serious consideration: at first sight it
seems so conclusive that no possibility of a satisfactory solution is
conceivable.
The proposition that the difference between the finite and the infinite
cannot be finite, is not wholly correct. We must first of all take
notice that the difference between two quantities, whether finite or
infinite, cannot be absolutely infinite, in the sense of diminution.
Difference is the excess of one quantity over another, and necessity
implies a limit; for as the excess only is considered, the quantity
exceeded is not contained in the difference. Calling the difference
D, the greater quantity A, and the smaller _a_, I say that D can in
no hypothesis be infinite. By the supposition D = A - _a_; therefore
D + _a_ = A; in order that D may equal A it is necessary to add to it
_a_; therefore D cannot be infinite. If we suppose A = ∞,
we shall have D = A - _a_ = ∞ - _a_, or D + _a_ = ∞.
Therefore to make D infinite we must add to it _a_, and we
can never have D = ∞ unless _a_ = 0; but in that case there
would be no true difference, since the equation, D = A - _a_, would be
converted into D = A - 0 = A, and the difference would not be real but
imaginary.
It follows from this that no difference between two positive quantities
can be absolutely infinite; if it is so in some sense, it is not so in
the sense of diminution; and the union of these two ideas of difference
and infinity results in a contradiction.[40]
[40] I am speaking of the difference between _positive_ quantities; for
with regard to other quantities we may express an infinite difference
algebraically. Let the two quantities be (∞ - _a_) and (-_a_). The
difference between them will be expressed in this equation, D=(∞ - _a_)
- (-_a_) = ∞ - a + a = ∞.
The difference between an infinite quantity and a given finite quantity
cannot be another given finite quantity, but it must be infinite in
some sense. Let us suppose an infinite line and a given finite line,
the difference between them cannot be expressed by a given finite
lineal value. For supposing the second line to be a finite and a given
line, we may place it upon the infinite line in any of its directions,
and from any point in it it will reach a certain point of the infinite
line. If we suppose a second given finite line, representing the
difference between the other two lines, we ought to place it upon
the infinite line at the point where the other terminates; and it is
evident that it will terminate at another point determined by its
length; therefore it will not measure the whole of the difference
between the infinite and the finite lines.
We obtain the same result in algebraic expressions. If A be a given
finite value, the difference between A and ∞ cannot be another given
finite value. For, expressing the difference by D, we shall have ∞ - D
± A D. Therefore, D + A = ∞; consequently, if both were given finite
values, an infinite would result from two given finite values, which
is absurd.
Hence, a difference may be in some sense infinite, according to the
meaning we attach to the term infinity. If from the point where we are
situated, we draw a line towards the north and produce it infinitely,
and then produce it, also, infinitely towards the south, the difference
between either of these lines and the sum of them both, will be
infinite only in a certain sense. This is also verified by algebraic
expressions. If we have the infinite value equal 2∞, and compare it
with ∞, the result is 2∞ - ∞ = ∞.
In general, from any infinite value we may subtract any finite
difference in relation to it, so long as the subtrahend is not a given
finite value. Let ∞ be the infinite value,--I say that we can find
in it any finite value; for, ∞ being an infinite value, A contains
all finite values of the same order; therefore it contains the finite
value, A; consequently we may form the equation, ∞ - A = B. Whatever be
the value of B, the relation of B to ∞ is A; for by only adding A to B
we obtain ∞. The equation, ∞ - A = B, gives B + A = ∞, and also ∞ - B =
A; and as A is a given value according to the supposition, and A is the
given finite difference between ∞ and B, it follows that we may find a
finite difference to every infinite value.
We may infer from this that the possibility of assigning a finite
difference to an infinite extension, does not prove any thing against
its true infinity. The infinite, and because it is infinite, contains
all that belongs to the order in which it is infinite. We may take any
sure value, and considering it as a difference, and we shall obtain a
finite difference. But far from proving the absence of infinity, this
confirms its existence; for it shows that all the finite is contained
in the infinite.
In this case, the subtrahend would be infinite under a certain aspect;
but not in the order of diminution, because it wants the quantity which
is taken from it.
87. There is another argument against the absolute infinity of
extension, which seems to have more weight than any of those which
precede, and I cannot see why it has never occurred to those who argue
against this possibility. It is this,--we suppose an infinite extension
to exist. God can annihilate it, and then create another equally
infinite. The sum of both is greater than either alone; therefore
neither of them alone is infinite. This annihilation we may suppose as
often as we wish; hence we may have a series of infinite extensions.
The terms of this series cannot exist at the same time, since one
actual infinite extension excludes all others. Therefore, as the sum
of the extensions is greater than any number of particular extensions,
the absolute infinite extension must be found, not in the particular
extensions, but in the sum, and hence an actual infinite extension is
intrinsically impossible.
To solve this difficulty we must distinguish between extension and the
thing extended: the whole question turns on the intrinsic possibility
of the infinity of extension, considered in itself, abstracting
absolutely the subject in which it is found. The difficulty places
before our sight a series of successive infinite extensions; but in
reality this succession is in the beings which are extended, and the
number of which goes on increasing; but not in the extension itself.
The pure idea of infinite extension in the one case, is not increased
by the new extensions which are produced; the extension appears,
disappears, reappears, and again disappears, but is not increased.
The succession shows the intrinsic possibility of its appearance
and its disappearance, its essential contingency, because it is not
repugnant for it to cease to exist when it exists, or to pass again
from non-existence to existence. If we examine our ideas, we shall
find that we cannot increase the infinite extension which we conceive,
by any imaginable supposition; and that whatever we may do, is reduced
to a succession of productions and annihilations. The idea of infinite
extension seems to be a primitive part of our mind; the infinity
which we imagine in space, is only the attempt which our mind makes
to express its idea in reality. Created with sensible intuition, we
have received the power of expanding this intuition on an infinite
scale,--to do this we require the idea of an infinite extension.
CHAPTER XIII.
EXISTENCE OF INFINITE EXTENSION.
88. The question of the possibility of an infinite extension is very
different from that of its existence. The first we answer in the
affirmative, the second in the negative.
Descartes maintained that the extension of the world is indefinite; but
this is a term which, although it has a very rational meaning when it
refers to the compass of our understanding, has no meaning when applied
to things. There is no objection to saying that the extension of the
world is indefinite, if it only means that we cannot assign its limits;
but in the reality, the limits exist or do not exist, indifferently
of our power of assigning them; there is no medium between yes and
no; therefore there is no medium between the existence and the
non-existence of these limits. If they exist, the extension of the
world is finite; if they do not exist, it is infinite;--in either case,
the word indefinite expresses nothing.
The argument of Descartes proves nothing, or it proves the true
infinity of the world. For, if we must remove its limits indefinitely
because we always conceive indefinitely an extension beyond every
other extension, as, on the other hand, we know that this series of
conceptions has no limit, we may at once transfer the unlimitedness
to the object which corresponds to those conceptions, and affirm that
the extension of the world is absolutely infinite. Unfortunately,
the argument of Descartes is without any basis; for it consists in a
transition from the ideal, or, rather, imaginary order, to the real
order, which is contrary to good logic.[41]
[41] See Book III., Chapter VIII.
89. Leibnitz maintained, that although God could have made the material
universe finite in its extension, it is more in conformity with his
wisdom not to have done so. "Thus I do not say," he writes,[42] "as is
here imputed to me, that God cannot give limits to the extension of
matter; but the appearance is that he does not wish it, but preferred
to give it more." The opinion of Leibnitz is founded on his system of
optimism, which is open to a multitude of objections, but it is not the
place here to examine them.
[42] _Lettres entre Leibnitz et Clarke_, Vième Écrit. de Leibnitz, § 73.
90. To speak frankly my own opinion, I say that this is a question
which cannot be solved on purely philosophic principles; for, as
the ideas contain no intrinsic necessity, either for or against the
existence of an infinite extension, we must look for its solution to
what experience teaches us. All the time occupied in attempting to
solve this question is lost. What we can assert is, that the extension
of the world exceeds all appreciation; and as the science of astronomy
advances, greater depths are discovered in the ocean of space. Where is
the shore? or is there any? Reason cannot answer such questions. What
do we, poor insects, know, whose life is but a momentary dwelling on
this little ball of dust, which we call the globe of the earth?
CHAPTER XIV.
POSSIBILITY OF AN ACTUAL INFINITE NUMBER.
91. Is an infinite number possible? Does the union of the idea of
number with the idea of the absolute negation of limit, involve any
contradiction which prevents the realization of the conception?
Whatever number we may conceive, we can always conceive one still
greater: this seems to show that no existing number can be absolutely
infinite. If we suppose this number to be realized, an intelligence
may know it, and may multiply it by two, three, or any other number;
therefore the number may be increased, and consequently it is not
infinite.
This difficulty is far from being conclusive, if we examine it
carefully. The intellectual act of which it speaks, would be impossible
on the supposition of the existence of an infinite number. If the
intelligence should not know the infinity of the number, it might
make the multiplication, but it would fall into a contradiction
through its ignorance; for the number being absolutely infinite,
could not be increased; its multiplication would be an absurdity,
and the intelligence making it, would combine two ideas which would
still be repugnant, although not known to be so by the intelligence.
If the absolute infinity of the existing number were known to the
intelligence, the idea of multiplication could never be associated
with it; for the intelligence would know that all possible products
already exist.
92. An absolutely infinite number cannot be expressed in the algebraic
or geometrical values; the attempt so to express it limits it in a
certain sense, and therefore destroys its absolute infinity. If the
expression ∞, represented an absolutely infinite number, it
would not be susceptible of any combination which would increase it: to
suppose that it may be multiplied by other numbers, finite or infinite,
is to take its infinity in another than an absolute sense.
The fraction a/0 does not express an infinite value in all the
strictness of the word; for it is evident that whatever be the value of
a/0 it will always be less than 2a/0 or, in general, less than na/0 n
representing a value greater than unity.
93. Neither can an infinite number be represented in geometrical values.
Let us take a line one foot long. It is evident that if we produce
this line infinitely in opposite directions, the number of feet will
be in some sense infinite, since the foot is supposed to be repeated
infinite times: the expression of the number of the feet will be the
expression of an infinite value. Now, I say that this number is not
infinite, because there are other numbers still greater. In each foot
there are twelve inches; therefore, the number of inches contained
in the line will be twelve times as great as the number of feet;
consequently the number of feet is not infinite. Neither is the number
of inches infinite; for they in their turn may be divided into lines,
the lines into points; and it is evident that the number of the smaller
quantities will be proportionally greater than the number of the
greater quantities. There will be twelve times as many inches as feet,
twelve times as many lines as inches, and twelve times as many points
as lines; and this progression can never end, because the value of a
line is infinitely divisible.
94. Pushing to infinity the divisibility of an infinite line, we seem
to have an infinite number in the elements which constitute it; but a
slight reflection will dissipate this illusion. For it is evident that
we can draw other infinite lines by the side of the supposed infinite
line; and since according to the supposition, each of them may be
infinitely divided, it follows that the sum of the elements of all the
lines will give a greater number than the sum of the elements of any
one of them.
95. If we wish to find an infinite number of parts in values of
extension, we must suppose a solid infinite in all its dimensions, with
all its parts infinitely divided. But not even then should we have an
absolutely infinite number, although we should have the greatest which
can be represented in values of extension.
Conceding that an infinite extension existed which is infinitely
divisible, the number of its parts would not be absolutely infinite;
for we can conceive other beings besides extended beings, and
considering both under the general idea of being, we might unite them
in a number which would be greater than that of extended beings alone.
96. No imaginable species of beings infinitely multiplied, can give an
absolutely infinite number. The reason is the same as that given in the
last paragraph: the existence of beings of one species does not render
the existence of beings of another species impossible. Therefore,
besides the supposed infinity of the number of beings of a determinate
species, there are other numbers which, united with this, produce a
number greater than the pretended infinity.
97. The existence of an absolutely infinite number requires: first, the
existence of infinite species of beings; and secondly, the existence of
infinite individuals of each species. Let us see if these conditions
can be realized.
98. There seems to be no doubt of the intrinsic possibility of
infinite species. The scale of beings is between two extremes, nothing
and infinite perfection: the space between these extremes is infinite;
and beings may be distributed on it in an infinite gradation.
99. Admitting the intrinsic possibility of an infinite gradation in
the scale of beings, the question occurs, whether their possibility is
only ideal, or also real, that is, may be realized. God is infinitely
powerful; if the infinite gradation is intrinsically possible, God can
produce it; for whatever is intrinsically possible falls within the
reach of divine omnipotence. On the other hand, supposing, as we must,
the liberty of God, there is no doubt but God is free to create all
that he can create. If then there is nothing repugnant in an infinity
of the species of beings distributed in an infinite gradation, these
beings may exist if God will it. Therefore denying all limit to the
number of species and of individuals of each species, it seems that
the infinite number would exist, since it is impossible to imagine any
increase or limitation in the collection of all beings.
On this supposition the most perfect created beings possible would
exist, and no more perfect being in the sphere of creatures could be
conceived. All that can be imagined would already exist, from nothing
to infinite perfection.
100. Still it must be observed that the collection of created beings,
whatever be their perfection, are necessarily subject to the condition
of dependence on another being; a condition from which the infinite
being above is essentially exempt. This condition involves limitation;
therefore, all created beings must be finite.
101. Does the character of finite, which is met with in all created
beings, involve a determinate limit beyond which they cannot pass? If
this limit exists, is not the number of possible species also limited?
And if these species are not infinite, is not an infinite number an
illusion?