OF THE "PHILOSOPHY OF THE INFINITE."
In the last Chapter but one I stated that Schelling propounded a
Philosophy of the Absolute, the Absolute being the original basis
of truth in which the two opposite elements, Ideas and Facts, are
identified, and that Hegel also founded his philosophy on the Identity
of these two elements. These German philosophies appear to me, as I
have ventured to intimate, of small or no value in their bearing on
the history of actual science. I have in the history of the sciences
noted instances in which these writers seem to me to misconceive
altogether the nature and meaning of the facts of scientific history;
as where[298] Schelling condemns Newton's Opticks as a fabric of
fallacies: and where[299] Hegel says that the glory due to Kepler has
been unjustly transferred to Newton. As it appears to me important that
English philosophers should form a just estimate of Hegel's capacity of
judging and pronouncing on this subject, I will print in the Appendix a
special discussion of what he has said respecting Newton's discovery of
the law of gravitation.
Recently attempts have been made to explain to English readers
these systems of German philosophy, and in these attempts there are
some points which may deserve our notice as to their bearing on the
philosophy of science. I find some difficulty in discussing these
attempts, for they deal much with phrases which appear to me to
offer no grasp to man's power of reason. What, for instance, is the
_Absolute_, which occupies a prominent place in these expositions? It
is, as I have stated, in Schelling, the central basis of truth in which
things and thoughts are united and identified. To attempt to reason
about such an "Absolute" appears to me to be an entire misapprehension
of the power of reason. Again; one of the most eminent of the
expositors has spoken of each system of this kind as a _Philosophy of
the Unconditioned_[300]. But what, we must ask, is the _Unconditioned_?
That which is subject to no conditions, is subject to no conditions
which distinguish it from any thing else, and so, cannot be a matter of
thought. But again; this _Absolute_ or _Unconditioned_ is (if I rightly
understand) said to be described also by various other names; _unity_,
_identity_, _substance_, _absolute cause_, the _infinite_, _pure
thought_, &c. As each of these terms expresses some condition on which
the name fixes our thoughts, I cannot understand why they should any of
them be called the _Unconditioned_; and as they express very different
thoughts, I cannot understand why they should be called by the same
name. From speculations starting from such a point, I can expect
nothing but confusion and perplexity; nor can I find that anything else
has come of them. They appear to me more barren, and more certain to be
barren, of any results which have any place in our real knowledge, than
the most barren speculations of the schoolmen of the middle ages: which
indeed they much resemble in all their features--their acuteness, their
learning, their ambitious aim, and their actual failure.
2. But leaving the Absolute and the Unconditioned, as notions which
cannot be dealt with by our reason without being something entirely
different from their definitions, we may turn for a moment to another
notion which is combined with them by the expositors of whom I speak,
and which has some bearing upon our positive science, because it enters
into the reasonings of mathematics: I mean the notion of _Infinite_.
Some of those who hold that we can know nothing concerning the
Absolute and the Unconditioned, (which they pretend to prove, though
concerning such words I do not conceive that anything can be true or
false,) hold also that the Infinite is in the same condition;--that we
can know nothing concerning what is Infinite;--therefore, I presume,
nothing concerning infinite space, infinite time, infinite number, or
infinite degrees.
To disprove this doctrine, it might be sufficient to point out that
there is a vast mass of mathematical science which includes the notion
of infinites, and leads to a great body of propositions concerning
Infinites. The whole of the infinitesimal calculus depends upon
conceiving finite magnitudes divided into an infinite number of parts:
these parts are infinitely small, and of these parts there are other
infinitesimal parts infinitely smaller still, and so on, as far as we
please to go. And even those methods which shun the term _infinite_, as
Newton's method of Ultimate Ratios, the method of Indivisibles, and the
method of Exhaustions of the ancient geometers, do really involve the
notion of infinite; for they imply a process continued without limit.
3. But perhaps it will be more useful to point out the fallacies of
the pretended proofs that we can know nothing concerning Infinity and
infinite things.
The argument offered is, that of infinity we have no notion but the
negation of a limit, and that from this negative notion no positive
result can be deduced.
But to this I reply: It is not at all true that our notion of what is
infinite is merely that it is _that_ which has no limit. We must ask
further that _what_? that space? that time? that number?--And if that
space, that what kind of space? That line? that surface? that solid
space?--And if that line, that line bounded at one end, or not? If that
surface, that surface bounded on one, or on two, or on three sides?
or on none? However any of these questions are answered, we may still
have an infinite space. Till they are answered, we can assert nothing
about the space; not because we can assert nothing about infinites; but
because we are not told what _kind_ of infinite we are talking of.
In reality the definition of an Infinite Quantity is not negative
merely, but contains a positive part as well. We assume a quantity
of a certain kind which may be augmented by carrying onward its
limits in one or more directions: this is a finite quantity of a
given kind. We _then_--when we have thus positively determined the
kind of the quantity--suppose the limit in one or more directions to
be annihilated, and thus we have an infinite quantity. But in this
infinite quantity there remain the positive properties from which we
began, as well as the negative property, the negation of a limit; and
the positive properties joined with the negative property may and do
supply grounds of reasoning respecting the infinite quantity.
4. This is lore so elementary to mathematicians that it appears almost
puerile to dwell upon it; but this seems to have been overlooked, in
the proof that we can have no knowledge concerning infinites. In such
proof it is assumed as quite evident, that all infinites are equal.
Yet, as we have seen, infinites may differ infinitely among themselves,
both in quantity and in kind. A German writer is quoted[301] for an
"ingenious" proof of this kind. In his writings, the opponent is
supposed to urge that a line _BAC_ may be made infinite by carrying
the extremity _C_ infinitely to the right, and again infinite by
carrying the extremity _B_ infinitely to the left; and thus the line
infinitely extended both ways would be double of the line infinite on
one side only. The supposed reply to this is, that it cannot be so,
because one infinite is equal to another: and moreover that what is
bounded at one end _A_, cannot be infinite: both which assumptions are
without the smallest ground. That one infinite quantity may be double
of another, is just as clear and certain as that one finite quantity
may. For instance, if one leaf of the book which the reader has before
him were produced infinitely upwards it would be an infinite space,
though bounded at the bottom and at both sides. If the other leaf were
in like manner produced infinitely upwards it would in like manner be
infinite; and the two together, though each infinite, would be double
of either of them.
5. As I have said, infinite quantities are conceived by conceiving
finite quantities increased by the transfer of a certain limit, and
then by negativing this limit altogether. And thus an infinite number
is conceived by assuming the series 1, 2, 3, 4, and so on, up to a
limit, and then removing this limit altogether. And this shows the
baselessness of another argument quoted from Werenfels. The opponent
asks, Are there in the infinite line an infinite number of feet?
Then in the double line there must be twice as many; and thus the
former infinite number did not contain all the (possible) unities;
(numerus infinitus non omnes habet unitates, sed præter eum concipi
possunt totidem unitates, quibus ille careat, eique possunt addi).
To which I reply, that the definition of an infinite number is not
that it contains all possible unities: but this--that the progress of
numeration being begun according to a certain law, goes on without
limit. And accordingly it is easy to conceive how one infinite number
may be larger than another infinite number, in any proportion. If, for
instance, we take, instead of the progression of the natural numbers 1,
2, 3, 4, &c. and the progression of the square numbers 1, 4, 9, 16, &c.
any term of the latter series will be greater than the corresponding
term of the other series in a ratio constantly increasing, and the
infinite term of the one, infinitely greater than the corresponding
infinite term of the other.
6. In the same manner we form a conception of infinite time, by
supposing time to begin now, and to go on, after the nature of time,
without limit; or by going back in thought from the present to a past
time, and by continuing this retrogression without limit. And thus we
have time infinite _a parte ante_ and _a parte post_, as the phrase
used to run; and time infinite both ways includes both, and is the most
complete notion of eternity.
7. Perhaps those who thus maintain that we cannot conceive anything
infinite, mean that we cannot form to ourselves a definite image of
anything infinite. And this of course is true. We cannot form to
ourselves an image of anything of which one of the characteristics is
that it is, in a certain way, unlimited. But this impossibility does
not prevent our reasoning about infinite quantities; combining as
elements of our reasoning, the absence of a limit with other positive
characters.
8. One of the consequences which is drawn by the assertors of the
doctrine that we cannot know anything about Infinity, is that we cannot
obtain from science any knowledge concerning God: And I have been the
more desirous to show the absence of proof of this doctrine, because I
conceive that science _does_ give us some knowledge, though it be very
little, of the nature of God: as I shall endeavour to show hereafter.
For instance, I conceive that when we say that God is an _eternal_
Being, this phraseology is not empty and unmeaning. It has been used
by the wisest and most thoughtful men in all ages, and, as I conceive,
may be used with undiminished, or with increased propriety, after
all the light which science and philosophy have thrown upon such
declarations. The reader of Newton will recollect how emphatically he
uses this expression along with others of a cognate character[302]:
"God is eternal and infinite, ... that is, He endures from eternity
to eternity, and is present from infinity to infinity.... He is not
eternity and infinity, but eternal and infinite. He is not duration and
space, but He endures and is present. He endures always, and is present
everywhere, and by existing always and everywhere He constitutes
duration and space." We shall see shortly that the view to which we are
led may be very fitly expressed by this language.
But I will first notice some other aspects of this philosophy.
FOOTNOTES:
[Footnote 298: _Hist. Ind. Sc._ b. ix. c. iii.]
[Footnote 299: _Ibid._ b. vii. c. ii.]
[Footnote 300: Sir W. Hamilton's Note on the _Philosophy of the
Unconditioned_.]
[Footnote 301: Werenfels in Mr. Mansel's _Bampton Lectures_, lect. ii.
Note 15.]
[Footnote 302: _Scholium Generale_ at the end of the _Principia_.]