of motion on the ground that what is moved must always attain the
middle point sooner than the end-point, on which we gave our opinion
in the earlier part of our discourse." _Phys._, vi. 9. 939B (R.P.
136). Aristotle seems to refer to _Phys._, vi. 2. 223AB [R.P. 136A]:
"All space is continuous, for time and space are divided into the same
and equal divisions.... Wherefore also Zeno's argument is fallacious,
that it is impossible to go through an infinite collection or to touch
an infinite collection one by one in a finite time. For there are two
senses in which the term 'infinite' is applied both to length and to
time, and in fact to all continuous things, either in regard to
divisibility, or in regard to the ends. Now it is not possible to
touch things infinite in regard to number in a finite time, but it is
possible to touch things infinite in regard to divisibility: for time
itself also is infinite in this sense. So that in fact we go through
an infinite, [space] in an infinite [time] and not in a finite [time],
and we touch infinite things with infinite things, not with finite
things." Philoponus, a sixth-century commentator (R.P. 136A, _Exc.
Paris Philop. in Arist. Phys._, 803, 2. Vit.), gives the following
illustration: "For if a thing were moved the space of a cubit in one
hour, since in every space there are an infinite number of points, the
thing moved must needs touch all the points of the space: it will then
go through an infinite collection in a finite time, which is
impossible."
Zeno appeals here, in the first place, to the fact that any distance,
however small, can be halved. From this it follows, of course, that
there must be an infinite number of points in a line. But, Aristotle
represents him as arguing, you cannot touch an infinite number of points
_one by one_ in a finite time. The words "one by one" are important. (1)
If _all_ the points touched are concerned, then, though you pass through
them continuously, you do not touch them "one by one." That is to say,
after touching one, there is not another which you touch next: no two
points are next each other, but between any two there are always an
infinite number of others, which cannot be enumerated one by one. (2)
If, on the other hand, only the successive middle points are concerned,
obtained by always halving what remains of the course, then the points
are reached one by one, and, though they are infinite in number, they
are in fact all reached in a finite time. His argument to the contrary
may be supposed to appeal to the view that a finite time must consist of
a finite number of instants, in which case what he says would be
perfectly true on the assumption that the possibility of continued
dichotomy is undeniable. If, on the other hand, we suppose the argument
directed against the partisans of infinite divisibility, we must suppose
it to proceed as follows:[42] "The points given by successive halving of
the distances still to be traversed are infinite in number, and are
reached in succession, each being reached a finite time later than its
predecessor; but the sum of an infinite number of finite times must be
infinite, and therefore the process will never be completed." It is very
possible that this is historically the right interpretation, but in this
form the argument is invalid. If half the course takes half a minute,
and the next quarter takes a quarter of a minute, and so on, the whole
course will take a minute. The apparent force of the argument, on this
interpretation, lies solely in the mistaken supposition that there
cannot be anything beyond the whole of an infinite series, which can be
seen to be false by observing that 1 is beyond the whole of the infinite
series 1/2, 3/4, 7/8, 15/16, ...
[42] _Cf._ Mr C. D. Broad, "Note on Achilles and the Tortoise,"
_Mind_, N.S., vol. xxii. pp. 318-9.
The second of Zeno's arguments is the one concerning Achilles and the
tortoise, which has achieved more notoriety than the others. It is
paraphrased by Burnet as follows:[43]
"Achilles will never overtake the tortoise. He must first reach the
place from which the tortoise started. By that time the tortoise will
have got some way ahead. Achilles must then make up that, and again the
tortoise will be ahead. He is always coming nearer, but he never makes
up to it."[44]
[43] _Op. cit._
[44] Aristotle's words are: "The second is the so-called Achilles. It
consists in this, that the slower will never be overtaken in its
course by the quickest, for the pursuer must always come first to the
point from which the pursued has just departed, so that the slower
must necessarily be always still more or less in advance." _Phys._,
vi. 9. 239B (R.P. 137).
This argument is essentially the same as the previous one. It shows
that, if Achilles ever overtakes the tortoise, it must be after an
infinite number of instants have elapsed since he started. This is in
fact true; but the view that an infinite number of instants make up an
infinitely long time is not true, and therefore the conclusion that
Achilles will never overtake the tortoise does not follow.
The third argument,[45] that of the arrow, is very interesting. The text
has been questioned. Burnet accepts the alterations of Zeller, and
paraphrases thus:
"The arrow in flight is at rest. For, if everything is at rest when it
occupies a space equal to itself, and what is in flight at any given
moment always occupies a space equal to itself, it cannot move."
[45] _Phys._, vi. 9. 239B (R.P. 138).
But according to Prantl, the literal translation of the unemended text
of Aristotle's statement of the argument is as follows: "If everything,
when it is behaving in a uniform manner, is continually either moving or
at rest, but what is moving is always in the _now_, then the moving
arrow is motionless." This form of the argument brings out its force
more clearly than Burnet's paraphrase.
Here, if not in the first two arguments, the view that a finite part of
time consists of a finite series of successive instants seems to be
assumed; at any rate the plausibility of the argument seems to depend
upon supposing that there are consecutive instants. Throughout an
instant, it is said, a moving body is where it is: it cannot move during
the instant, for that would require that the instant should have parts.
Thus, suppose we consider a period consisting of a thousand instants,
and suppose the arrow is in flight throughout this period. At each of
the thousand instants, the arrow is where it is, though at the next
instant it is somewhere else. It is never moving, but in some miraculous
way the change of position has to occur _between_ the instants, that is
to say, not at any time whatever. This is what M. Bergson calls the
cinematographic representation of reality. The more the difficulty is
meditated, the more real it becomes. The solution lies in the theory of
continuous series: we find it hard to avoid supposing that, when the
arrow is in flight, there is a _next_ position occupied at the _next_
moment; but in fact there is no next position and no next moment, and
when once this is imaginatively realised, the difficulty is seen to
disappear.
The fourth and last of Zeno's arguments is[46] the argument of the
stadium.
[46] _Phys._, vi. 9. 239B (R.P. 139).
The argument as stated by Burnet is as follows:
First Position. Second Position.
A .... A ....
B .... B ....
C .... C ....
"Half the time may be equal to double the time. Let us suppose three
rows of bodies, one of which (A) is at rest while the other two (B, C)
are moving with equal velocity in opposite directions. By the time they
are all in the same part of the course, B will have passed twice as many
of the bodies in C as in A. Therefore the time which it takes to pass C
is twice as long as the time it takes to pass A. But the time which B
and C take to reach the position of A is the same. Therefore double the
time is equal to the half."
Gaye[47] devoted an interesting article to the interpretation of this
argument. His translation of Aristotle's statement is as follows:
"The fourth argument is that concerning the two rows of bodies, each row
being composed of an equal number of bodies of equal size, passing each
other on a race-course as they proceed with equal velocity in opposite
directions, the one row originally occupying the space between the goal
and the middle point of the course, and the other that between the
middle point and the starting-post. This, he thinks, involves the
conclusion that half a given time is equal to double the time. The
fallacy of the reasoning lies in the assumption that a body occupies an
equal time in passing with equal velocity a body that is in motion and a
body of equal size that is at rest, an assumption which is false. For
instance (so runs the argument), let A A ... be the stationary bodies of
equal size, B B ... the bodies, equal in number and in size to A A ...,
originally occupying the half of the course from the starting-post to
the middle of the A's, and C C ... those originally occupying the other
half from the goal to the middle of the A's, equal in number, size, and
velocity, to B B ... Then three consequences follow. First, as the B's
and C's pass one another, the first B reaches the last C at the same
moment at which the first C reaches the last B. Secondly, at this moment
the first C has passed all the A's, whereas the first B has passed only
half the A's and has consequently occupied only half the time occupied
by the first C, since each of the two occupies an equal time in passing
each A. Thirdly, at the same moment all the B's have passed all the C's:
for the first C and the first B will simultaneously reach the opposite
ends of the course, since (so says Zeno) the time occupied by the first
C in passing each of the B's is equal to that occupied by it in passing
each of the A's, because an equal time is occupied by both the first B
and the first C in passing all the A's. This is the argument: but it
presupposes the aforesaid fallacious assumption."
[47] _Loc. cit._
First Position. Second Position.
B B' B" B B' B"
. . . . . .
A A' A" A A' A"
. . . . . .
C C' C" C C' C"
. . . . . .
This argument is not quite easy to follow, and it is only valid as
against the assumption that a finite time consists of a finite number of
instants. We may re-state it in different language. Let us suppose three
drill-sergeants, A, A', and A", standing in a row, while the two files
of soldiers march past them in opposite directions. At the first moment
which we consider, the three men B, B', B" in one row, and the three men
C, C', C" in the other row, are respectively opposite to A, A', and A".
At the very next moment, each row has moved on, and now B and C" are
opposite A'. Thus B and C" are opposite each other. When, then, did B
pass C'? It must have been somewhere between the two moments which we
supposed consecutive, and therefore the two moments cannot really have
been consecutive. It follows that there must be other moments between
any two given moments, and therefore that there must be an infinite
number of moments in any given interval of time.
The above difficulty, that B must have passed C' at some time between
two consecutive moments, is a genuine one, but is not precisely the
difficulty raised by Zeno. What Zeno professes to prove is that "half of
a given time is equal to double that time." The most intelligible
explanation of the argument known to me is that of Gaye.[48] Since,
however, his explanation is not easy to set forth shortly, I will
re-state what seems to me to be the logical essence of Zeno's
contention. If we suppose that time consists of a series of consecutive
instants, and that motion consists in passing through a series of
consecutive points, then the fastest possible motion is one which, at
each instant, is at a point consecutive to that at which it was at the
previous instant. Any slower motion must be one which has intervals of
rest interspersed, and any faster motion must wholly omit some points.
All this is evident from the fact that we cannot have more than one
event for each instant. But now, in the case of our A's and B's and C's,
B is opposite a fresh A every instant, and therefore the number of A's
passed gives the number of instants since the beginning of the motion.
But during the motion B has passed twice as many C's, and yet cannot
have passed more than one each instant. Hence the number of instants
since the motion began is twice the number of A's passed, though we
previously found it was equal to this number. From this result, Zeno's
conclusion follows.
[48] _Loc. cit._, p. 105.
Zeno's arguments, in some form, have afforded grounds for almost all the
theories of space and time and infinity which have been constructed from
his day to our own. We have seen that all his arguments are valid (with
certain reasonable hypotheses) on the assumption that finite spaces and
times consist of a finite number of points and instants, and that the
third and fourth almost certainly in fact proceeded on this assumption,
while the first and second, which were perhaps intended to refute the
opposite assumption, were in that case fallacious. We may therefore
escape from his paradoxes either by maintaining that, though space and
time do consist of points and instants, the number of them in any finite
interval is infinite; or by denying that space and time consist of
points and instants at all; or lastly, by denying the reality of space
and time altogether. It would seem that Zeno himself, as a supporter of
Parmenides, drew the last of these three possible deductions, at any
rate in regard to time. In this a very large number of philosophers have
followed him. Many others, like M. Bergson, have preferred to deny that
space and time consist of points and instants. Either of these solutions
will meet the difficulties in the form in which Zeno raised them. But,
as we saw, the difficulties can also be met if infinite numbers are
admissible. And on grounds which are independent of space and time,
infinite numbers, and series in which no two terms are consecutive, must
in any case be admitted. Consider, for example, all the fractions less
than 1, arranged in order of magnitude. Between any two of them, there
are others, for example, the arithmetical mean of the two. Thus no two
fractions are consecutive, and the total number of them is infinite. It
will be found that much of what Zeno says as regards the series of
points on a line can be equally well applied to the series of fractions.
And we cannot deny that there are fractions, so that two of the above
ways of escape are closed to us. It follows that, if we are to solve the
whole class of difficulties derivable from Zeno's by analogy, we must
discover some tenable theory of infinite numbers. What, then, are the
difficulties which, until the last thirty years, led philosophers to the
belief that infinite numbers are impossible?
The difficulties of infinity are of two kinds, of which the first may be
called sham, while the others involve, for their solution, a certain
amount of new and not altogether easy thinking. The sham difficulties
are those suggested by the etymology, and those suggested by confusion
of the mathematical infinite with what philosophers impertinently call
the "true" infinite. Etymologically, "infinite" should mean "having no
end." But in fact some infinite series have ends, some have not; while
some collections are infinite without being serial, and can therefore
not properly be regarded as either endless or having ends. The series of
instants from any earlier one to any later one (both included) is
infinite, but has two ends; the series of instants from the beginning of
time to the present moment has one end, but is infinite. Kant, in his
first antinomy, seems to hold that it is harder for the past to be
infinite than for the future to be so, on the ground that the past is
now completed, and that nothing infinite can be completed. It is very
difficult to see how he can have imagined that there was any sense in
this remark; but it seems most probable that he was thinking of the
infinite as the "unended." It is odd that he did not see that the future
too has one end at the present, and is precisely on a level with the
past. His regarding the two as different in this respect illustrates
just that kind of slavery to time which, as we agreed in speaking of
Parmenides, the true philosopher must learn to leave behind him.
The confusions introduced into the notions of philosophers by the
so-called "true" infinite are curious. They see that this notion is not
the same as the mathematical infinite, but they choose to believe that
it is the notion which the mathematicians are vainly trying to reach.
They therefore inform the mathematicians, kindly but firmly, that they
are mistaken in adhering to the "false" infinite, since plainly the
"true" infinite is something quite different. The reply to this is that
what they call the "true" infinite is a notion totally irrelevant to the
problem of the mathematical infinite, to which it has only a fanciful
and verbal analogy. So remote is it that I do not propose to confuse the
issue by even mentioning what the "true" infinite is. It is the "false"
infinite that concerns us, and we have to show that the epithet "false"
is undeserved.
There are, however, certain genuine difficulties in understanding the
infinite, certain habits of mind derived from the consideration of
finite numbers, and easily extended to infinite numbers under the
mistaken notion that they represent logical necessities. For example,
every number that we are accustomed to, except 0, has another number
immediately before it, from which it results by adding 1; but the first
infinite number does not have this property. The numbers before it form
an infinite series, containing all the ordinary finite numbers, having
no maximum, no last finite number, after which one little step would
plunge us into the infinite. If it is assumed that the first infinite
number is reached by a succession of small steps, it is easy to show
that it is self-contradictory. The first infinite number is, in fact,
beyond the whole unending series of finite numbers. "But," it will be
said, "there cannot be anything beyond the whole of an unending series."
This, we may point out, is the very principle upon which Zeno relies in
the arguments of the race-course and the Achilles. Take the race-course:
there is the moment when the runner still has half his distance to run,
then the moment when he still has a quarter, then when he still has an
eighth, and so on in a strictly unending series. Beyond the whole of
this series is the moment when he reaches the goal. Thus there certainly
can be something beyond the whole of an unending series. But it remains
to show that this fact is only what might have been expected.
The difficulty, like most of the vaguer difficulties besetting the
mathematical infinite, is derived, I think, from the more or less
unconscious operation of the idea of _counting_. If you set to work to
count the terms in an infinite collection, you will never have completed
your task. Thus, in the case of the runner, if half, three-quarters,
seven-eighths, and so on of the course were marked, and the runner was
not allowed to pass any of the marks until the umpire said "Now," then
Zeno's conclusion would be true in practice, and he would never reach
the goal.
But it is not essential to the existence of a collection, or even to
knowledge and reasoning concerning it, that we should be able to pass
its terms in review one by one. This may be seen in the case of finite
collections; we can speak of "mankind" or "the human race," though many
of the individuals in this collection are not personally known to us. We
can do this because we know of various characteristics which every
individual has if he belongs to the collection, and not if he does not.
And exactly the same happens in the case of infinite collections: they
may be known by their characteristics although their terms cannot be
enumerated. In this sense, an unending series may nevertheless form a
whole, and there may be new terms beyond the whole of it.
Some purely arithmetical peculiarities of infinite numbers have also
caused perplexity. For instance, an infinite number is not increased by
adding one to it, or by doubling it. Such peculiarities have seemed to
many to contradict logic, but in fact they only contradict confirmed
mental habits. The whole difficulty of the subject lies in the necessity
of thinking in an unfamiliar way, and in realising that many properties
which we have thought inherent in number are in fact peculiar to finite
numbers. If this is remembered, the positive theory of infinity, which
will occupy the next lecture, will not be found so difficult as it is to
those who cling obstinately to the prejudices instilled by the
arithmetic which is learnt in childhood.
LECTURE VII
THE POSITIVE THEORY OF INFINITY
The positive theory of infinity, and the general theory of number to
which it has given rise, are among the triumphs of scientific method in
philosophy, and are therefore specially suitable for illustrating the
logical-analytic character of that method. The work in this subject has
been done by mathematicians, and its results can be expressed in
mathematical symbolism. Why, then, it may be said, should the subject be
regarded as philosophy rather than as mathematics? This raises a
difficult question, partly concerned with the use of words, but partly
also of real importance in understanding the function of philosophy.
Every subject-matter, it would seem, can give rise to philosophical
investigations as well as to the appropriate science, the difference
between the two treatments being in the direction of movement and in the
kind of truths which it is sought to establish. In the special sciences,
when they have become fully developed, the movement is forward and
synthetic, from the simpler to the more complex. But in philosophy we
follow the inverse direction: from the complex and relatively concrete
we proceed towards the simple and abstract by means of analysis,
seeking, in the process, to eliminate the particularity of the original
subject-matter, and to confine our attention entirely to the logical
_form_ of the facts concerned.
Between philosophy and pure mathematics there is a certain affinity, in
the fact that both are general and _a priori_. Neither of them asserts
propositions which, like those of history and geography, depend upon the
actual concrete facts being just what they are. We may illustrate this
characteristic by means of Leibniz's conception of many _possible_
worlds, of which one only is _actual_. In all the many possible worlds,
philosophy and mathematics will be the same; the differences will only
be in respect of those particular facts which are chronicled by the
descriptive sciences. Any quality, therefore, by which our actual world
is distinguished from other abstractly possible worlds, must be ignored
by mathematics and philosophy alike. Mathematics and philosophy differ,
however, in their manner of treating the general properties in which all
possible worlds agree; for while mathematics, starting from
comparatively simple propositions, seeks to build up more and more
complex results by deductive synthesis, philosophy, starting from data
which are common knowledge, seeks to purify and generalise them into the
simplest statements of abstract form that can be obtained from them by
logical analysis.
The difference between philosophy and mathematics may be illustrated by
our present problem, namely, the nature of number. Both start from
certain facts about numbers which are evident to inspection. But
mathematics uses these facts to deduce more and more complicated
theorems, while philosophy seeks, by analysis, to go behind these facts
to others, simpler, more fundamental, and inherently more fitted to form
the premisses of the science of arithmetic. The question, "What is a
number?" is the pre-eminent philosophic question in this subject, but it
is one which the mathematician as such need not ask, provided he knows
enough of the properties of numbers to enable him to deduce his
theorems. We, since our object is philosophical, must grapple with the
philosopher's question. The answer to the question, "What is a number?"
which we shall reach in this lecture, will be found to give also, by
implication, the answer to the difficulties of infinity which we
considered in the previous lecture.