order to show the relevance of the problem of infinity to the problem of
the reality of objects of sense. In the remainder of the present
lecture, I wish to state and explain the problem of infinity, to show
how it arose, and to show the irrelevance of all the solutions proposed
by philosophers. In the following lecture, I shall try to explain the
true solution, which has been discovered by the mathematicians, but
nevertheless belongs essentially to philosophy. The solution is
definitive, in the sense that it entirely satisfies and convinces all
who study it carefully. For over two thousand years the human intellect
was baffled by the problem; its many failures and its ultimate success
make this problem peculiarly apt for the illustration of method.
The problem appears to have first arisen in some such way as the
following.[22] Pythagoras and his followers, who were interested, like
Descartes, in the application of number to geometry, adopted in that
science more arithmetical methods than those with which Euclid has made
us familiar. They, or their contemporaries the atomists, believed,
apparently, that space is composed of indivisible points, while time is
composed of indivisible instants.[23] This belief would not, by itself,
have raised the difficulties which they encountered, but it was
presumably accompanied by another belief, that the number of points in
any finite area or of instants in any finite period must be finite. I do
not suppose that this latter belief was a conscious one, because
probably no other possibility had occurred to them. But the belief
nevertheless operated, and very soon brought them into conflict with
facts which they themselves discovered. Before explaining how this
occurred, however, it is necessary to say one word in explanation of the
phrase "finite number." The _exact_ explanation is a matter for our next
lecture; for the present, it must suffice to say that I mean 0 and 1 and
2 and 3 and so on, for ever--in other words, any number that can be
obtained by successively adding ones. This includes all the numbers that
can be expressed by means of our ordinary numerals, and since such
numbers can be made greater and greater, without ever reaching an
unsurpassable maximum, it is easy to suppose that there are no other
numbers. But this supposition, natural as it is, is mistaken.
[22] In what concerns the early Greek philosophers, my knowledge is
largely derived from Burnet's valuable work, _Early Greek Philosophy_
(2nd ed., London, 1908). I have also been greatly assisted by Mr D. S.
Robertson of Trinity College, who has supplied the deficiencies of my
knowledge of Greek, and brought important references to my notice.
[23] _Cf._ Aristotle, _Metaphysics_, M. 6, 1080b, 18 _sqq._, and
1083b, 8 _sqq._
Whether the Pythagoreans themselves believed space and time to be
composed of indivisible points and instants is a debatable question.[24]
It would seem that the distinction between space and matter had not yet
been clearly made, and that therefore, when an atomistic view is
expressed, it is difficult to decide whether particles of matter or
points of space are intended. There is an interesting passage[25] in
Aristotle's _Physics_,[26] where he says:
"The Pythagoreans all maintained the existence of the void, and said
that it enters into the heaven itself from the boundless breath,
inasmuch as the heaven breathes in the void also; and the void
differentiates natures, as if it were a sort of separation of
consecutives, and as if it were their differentiation; and that this
also is what is first in numbers, for it is the void which
differentiates them."
[24] There is some reason to think that the Pythagoreans distinguished
between discrete and continuous quantity. G. J. Allman, in his _Greek
Geometry from Thales to Euclid_, says (p. 23): "The Pythagoreans made
a fourfold division of mathematical science, attributing one of its
parts to the how many, ~to poson~, and the other to the how much, ~to
pelikon~; and they assigned to each of these parts a twofold division.
For they said that discrete quantity, or the _how many_, either
subsists by itself or must be considered with relation to some other;
but that continued quantity, or the _how much_, is either stable or in
motion. Hence they affirmed that arithmetic contemplates that discrete
quantity which subsists by itself, but music that which is related to
another; and that geometry considers continued quantity so far as it
is immovable; but astronomy (~ten sphairiken~) contemplates continued
quantity so far as it is of a self-motive nature. (Proclus, ed.
Friedlein, p. 35. As to the distinction between ~to pelikon~,
continuous, and ~to poson~, discrete quantity, see Iambl., _in
Nicomachi Geraseni Arithmeticam introductionem_, ed. Tennulius,
p. 148.)" _Cf._ p. 48.
[25] Referred to by Burnet, _op. cit._, p. 120.
[26] iv., 6. 213b, 22; H. Ritter and L. Preller, _Historia Philosophiae
Graecae_, 8th ed., Gotha, 1898, p. 75 (this work will be referred to in
future as "R. P.").
This seems to imply that they regarded matter as consisting of atoms
with empty space in between. But if so, they must have thought space
could be studied by only paying attention to the atoms, for otherwise it
would be hard to account for their arithmetical methods in geometry, or
for their statement that "things are numbers."
The difficulty which beset the Pythagoreans in their attempts to apply
numbers arose through their discovery of incommensurables, and this, in
turn, arose as follows. Pythagoras, as we all learnt in youth,
discovered the proposition that the sum of the squares on the sides of a
right-angled triangle is equal to the square on the hypotenuse. It is
said that he sacrificed an ox when he discovered this theorem; if so,
the ox was the first martyr to science. But the theorem, though it has
remained his chief claim to immortality, was soon found to have a
consequence fatal to his whole philosophy. Consider the case of a
right-angled triangle whose two sides are equal, such a triangle as is
formed by two sides of a square and a diagonal. Here, in virtue of the
theorem, the square on the diagonal is double of the square on either of
the sides. But Pythagoras or his early followers easily proved that the
square of one whole number cannot be double of the square of
another.[27] Thus the length of the side and the length of the diagonal
are incommensurable; that is to say, however small a unit of length you
take, if it is contained an exact number of times in the side, it is not
contained any exact number of times in the diagonal, and _vice versa_.
[27] The Pythagorean proof is roughly as follows. If possible, let the
ratio of the diagonal to the side of a square be _m_/_n_, where _m_
and _n_ are whole numbers having no common factor. Then we must have
_m_2 = 2_n_2. Now the square of an odd number is odd, but _m_2, being
equal to 2_n_2, is even. Hence _m_ must be even. But the square of an
even number divides by 4, therefore _n_2, which is half of _m_2, must
be even. Therefore _n_ must be even. But, since _m_ is even, and _m_
and _n_ have no common factor, _n_ must be odd. Thus _n_ must be both
odd and even, which is impossible; and therefore the diagonal and the
side cannot have a rational ratio.
Now this fact might have been assimilated by some philosophies without
any great difficulty, but to the philosophy of Pythagoras it was
absolutely fatal. Pythagoras held that number is the constitutive
essence of all things, yet no two numbers could express the ratio of the
side of a square to the diagonal. It would seem probable that we may
expand his difficulty, without departing from his thought, by assuming
that he regarded the length of a line as determined by the number of
atoms contained in it--a line two inches long would contain twice as
many atoms as a line one inch long, and so on. But if this were the
truth, then there must be a definite numerical ratio between any two
finite lengths, because it was supposed that the number of atoms in
each, however large, must be finite. Here there was an insoluble
contradiction. The Pythagoreans, it is said, resolved to keep the
existence of incommensurables a profound secret, revealed only to a few
of the supreme heads of the sect; and one of their number, Hippasos of
Metapontion, is even said to have been shipwrecked at sea for impiously
disclosing the terrible discovery to their enemies. It must be
remembered that Pythagoras was the founder of a new religion as well as
the teacher of a new science: if the science came to be doubted, the
disciples might fall into sin, and perhaps even eat beans, which
according to Pythagoras is as bad as eating parents' bones.
The problem first raised by the discovery of incommensurables proved, as
time went on, to be one of the most severe and at the same time most
far-reaching problems that have confronted the human intellect in its
endeavour to understand the world. It showed at once that numerical
measurement of lengths, if it was to be made accurate, must require an
arithmetic more advanced and more difficult than any that the ancients
possessed. They therefore set to work to reconstruct geometry on a basis
which did not assume the universal possibility of numerical
measurement--a reconstruction which, as may be seen in Euclid, they
effected with extraordinary skill and with great logical acumen. The
moderns, under the influence of Cartesian geometry, have reasserted the
universal possibility of numerical measurement, extending arithmetic,
partly for that purpose, so as to include what are called "irrational"
numbers, which give the ratios of incommensurable lengths. But although
irrational numbers have long been used without a qualm, it is only in
quite recent years that logically satisfactory definitions of them have
been given. With these definitions, the first and most obvious form of
the difficulty which confronted the Pythagoreans has been solved; but
other forms of the difficulty remain to be considered, and it is these
that introduce us to the problem of infinity in its pure form.
We saw that, accepting the view that a length is composed of points, the
existence of incommensurables proves that every finite length must
contain an infinite number of points. In other words, if we were to take
away points one by one, we should never have taken away all the points,
however long we continued the process. The number of points, therefore,
cannot be _counted_, for counting is a process which enumerates things
one by one. The property of being unable to be counted is characteristic
of infinite collections, and is a source of many of their paradoxical
qualities. So paradoxical are these qualities that until our own day
they were thought to constitute logical contradictions. A long line of
philosophers, from Zeno[28] to M. Bergson, have based much of their
metaphysics upon the supposed impossibility of infinite collections.
Broadly speaking, the difficulties were stated by Zeno, and nothing
material was added until we reach Bolzano's _Paradoxien des
Unendlichen_, a little work written in 1847-8, and published
posthumously in 1851. Intervening attempts to deal with the problem are
futile and negligible. The definitive solution of the difficulties is
due, not to Bolzano, but to Georg Cantor, whose work on this subject
first appeared in 1882.
[28] In regard to Zeno and the Pythagoreans, I have derived much
valuable information and criticism from Mr P. E. B. Jourdain.
In order to understand Zeno, and to realise how little modern orthodox
metaphysics has added to the achievements of the Greeks, we must
consider for a moment his master Parmenides, in whose interest the
paradoxes were invented.[29] Parmenides expounded his views in a poem
divided into two parts, called "the way of truth" and "the way of
opinion"--like Mr Bradley's "Appearance" and "Reality," except that
Parmenides tells us first about reality and then about appearance. "The
way of opinion," in his philosophy, is, broadly speaking,
Pythagoreanism; it begins with a warning: "Here I shall close my
trustworthy speech and thought about the truth. Henceforward learn the
opinions of mortals, giving ear to the deceptive ordering of my words."
What has gone before has been revealed by a goddess, who tells him what
really _is_. Reality, she says, is uncreated, indestructible,
unchanging, indivisible; it is "immovable in the bonds of mighty chains,
without beginning and without end; since coming into being and passing
away have been driven afar, and true belief has cast them away." The
fundamental principle of his inquiry is stated in a sentence which would
not be out of place in Hegel:[30] "Thou canst not know what is not--that
is impossible--nor utter it; for it is the same thing that can be
thought and that can be." And again: "It needs must be that what can be
thought and spoken of is; for it is possible for it to be, and it is not
possible for what is nothing to be." The impossibility of change follows
from this principle; for what is past can be spoken of, and therefore,
by the principle, still is.
[29] So Plato makes Zeno say in the _Parmenides_, apropos of his
philosophy as a whole; and all internal and external evidence supports
this view.
[30] "With Parmenides," Hegel says, "philosophising proper began."
_Werke_ (edition of 1840), vol. xiii. p. 274.
The great conception of a reality behind the passing illusions of sense,
a reality one, indivisible, and unchanging, was thus introduced into
Western philosophy by Parmenides, not, it would seem, for mystical or
religious reasons, but on the basis of a logical argument as to the
impossibility of not-being. All the great metaphysical systems--notably
those of Plato, Spinoza, and Hegel--are the outcome of this fundamental
idea. It is difficult to disentangle the truth and the error in this
view. The contention that time is unreal and that the world of sense is
illusory must, I think, be regarded as based upon fallacious reasoning.
Nevertheless, there is some sense--easier to feel than to state--in
which time is an unimportant and superficial characteristic of reality.
Past and future must be acknowledged to be as real as the present, and a
certain emancipation from slavery to time is essential to philosophic
thought. The importance of time is rather practical than theoretical,
rather in relation to our desires than in relation to truth. A truer
image of the world, I think, is obtained by picturing things as entering
into the stream of time from an eternal world outside, than from a view
which regards time as the devouring tyrant of all that is. Both in
thought and in feeling, to realise the unimportance of time is the gate
of wisdom. But unimportance is not unreality; and therefore what we
shall have to say about Zeno's arguments in support of Parmenides must
be mainly critical.
The relation of Zeno to Parmenides is explained by Plato[31] in the
dialogue in which Socrates, as a young man, learns logical acumen and
philosophic disinterestedness from their dialectic. I quote from
Jowett's translation:
"I see, Parmenides, said Socrates, that Zeno is your second self in his
writings too; he puts what you say in another way, and would fain
deceive us into believing that he is telling us what is new. For you, in
your poems, say All is one, and of this you adduce excellent proofs; and
he on the other hand says There is no Many; and on behalf of this he
offers overwhelming evidence. To deceive the world, as you have done, by
saying the same thing in different ways, one of you affirming the one,
and the other denying the many, is a strain of art beyond the reach of
most of us.
"Yes, Socrates, said Zeno. But although you are as keen as a Spartan
hound in pursuing the track, you do not quite apprehend the true motive
of the composition, which is not really such an ambitious work as you
imagine; for what you speak of was an accident; I had no serious
intention of deceiving the world. The truth is, that these writings of
mine were meant to protect the arguments of Parmenides against those who
scoff at him and show the many ridiculous and contradictory results
which they suppose to follow from the affirmation of the one. My answer
is an address to the partisans of the many, whose attack I return with
interest by retorting upon them that their hypothesis of the being of
the many if carried out appears in a still more ridiculous light than
the hypothesis of the being of the one."
[31] _Parmenides_, 128 A-D.
Zeno's four arguments against motion were intended to exhibit the
contradictions that result from supposing that there is such a thing as
change, and thus to support the Parmenidean doctrine that reality is
unchanging.[32] Unfortunately, we only know his arguments through
Aristotle,[33] who stated them in order to refute them. Those
philosophers in the present day who have had their doctrines stated by
opponents will realise that a just or adequate presentation of Zeno's
position is hardly to be expected from Aristotle; but by some care in
interpretation it seems possible to reconstruct the so-called "sophisms"
which have been "refuted" by every tyro from that day to this.
[32] This interpretation is combated by Milhaud, _Les
philosophes-geometres de la Grece_, p. 140 n., but his reasons do not
seem to me convincing. All the interpretations in what follows are
open to question, but all have the support of reputable authorities.
[33] _Physics_, vi. 9. 2396 (R.P. 136-139).
Zeno's arguments would seem to be "ad hominem"; that is to say, they
seem to assume premisses granted by his opponents, and to show that,
granting these premisses, it is possible to deduce consequences which
his opponents must deny. In order to decide whether they are valid
arguments or "sophisms," it is necessary to guess at the tacit
premisses, and to decide who was the "homo" at whom they were aimed.
Some maintain that they were aimed at the Pythagoreans,[34] while others
have held that they were intended to refute the atomists.[35] M.
Evellin, on the contrary, holds that they constitute a refutation of
infinite divisibility,[36] while M. G. Noel, in the interests of Hegel,
maintains that the first two arguments refute infinite divisibility,
while the next two refute indivisibles.[37] Amid such a bewildering
variety of interpretations, we can at least not complain of any
restrictions on our liberty of choice.
[34] _Cf._ Gaston Milhaud, _Les philosophes-geometres de la Grece_,
p. 140 n.; Paul Tannery, _Pour l'histoire de la science hellene_,
p. 249; Burnet, _op. cit._, p. 362.
[35] _Cf._ R. K. Gaye, "On Aristotle, _Physics_, Z ix." _Journal of
Philology_, vol. xxxi., esp. p. 111. Also Moritz Cantor, _Vorlesungen
ueber Geschichte der Mathematik_, 1st ed., vol. i., 1880, p. 168, who,
however, subsequently adopted Paul Tannery's opinion, _Vorlesungen_,
3rd ed. (vol. i. p. 200).
[36] "Le mouvement et les partisans des indivisibles," _Revue de
Metaphysique et de Morale_, vol. i. pp. 382-395.
[37] "Le mouvement et les arguments de Zenon d'Elee," _Revue de
Metaphysique et de Morale_, vol. i. pp. 107-125.
The historical questions raised by the above-mentioned discussions are
no doubt largely insoluble, owing to the very scanty material from which
our evidence is derived. The points which seem fairly clear are the
following: (1) That, in spite of MM. Milhaud and Paul Tannery, Zeno is
anxious to prove that motion is really impossible, and that he desires
to prove this because he follows Parmenides in denying plurality;[38]
(2) that the third and fourth arguments proceed on the hypothesis of
indivisibles, a hypothesis which, whether adopted by the Pythagoreans or
not, was certainly much advocated, as may be seen from the treatise _On
Indivisible Lines_ attributed to Aristotle. As regards the first two
arguments, they would seem to be valid on the hypothesis of
indivisibles, and also, without this hypothesis, to be such as would be
valid if the traditional contradictions in infinite numbers were
insoluble, which they are not.
[38] _Cf._ M. Brochard, "Les pretendus sophismes de Zenon d'Elee,"
_Revue de Metaphysique et de Morale_, vol. i. pp. 209-215.
We may conclude, therefore, that Zeno's polemic is directed against the
view that space and time consist of points and instants; and that as
against the view that a finite stretch of space or time consists of a
finite number of points and instants, his arguments are not sophisms,
but perfectly valid.
The conclusion which Zeno wishes us to draw is that plurality is a
delusion, and spaces and times are really indivisible. The other
conclusion which is possible, namely, that the number of points and
instants is infinite, was not tenable so long as the infinite was
infected with contradictions. In a fragment which is not one of the four
famous arguments against motion, Zeno says:
"If things are a many, they must be just as many as they are, and
neither more nor less. Now, if they are as many as they are, they will
be finite in number.
"If things are a many, they will be infinite in number; for there will
always be other things between them, and others again between these. And
so things are infinite in number."[39]
[39] Simplicius, _Phys._, 140, 28 D (R.P. 133); Burnet, _op. cit._,
pp. 364-365.
This argument attempts to prove that, if there are many things, the
number of them must be both finite and infinite, which is impossible;
hence we are to conclude that there is only one thing. But the weak
point in the argument is the phrase: "If they are just as many as they
are, they will be finite in number." This phrase is not very clear, but
it is plain that it assumes the impossibility of definite infinite
numbers. Without this assumption, which is now known to be false, the
arguments of Zeno, though they suffice (on certain very reasonable
assumptions) to dispel the hypothesis of finite indivisibles, do not
suffice to prove that motion and change and plurality are impossible.
They are not, however, on any view, mere foolish quibbles: they are
serious arguments, raising difficulties which it has taken two thousand
years to answer, and which even now are fatal to the teachings of most
philosophers.
The first of Zeno's arguments is the argument of the race-course, which
is paraphrased by Burnet as follows:[40]
"You cannot get to the end of a race-course. You cannot traverse an
infinite number of points in a finite time. You must traverse the half
of any given distance before you traverse the whole, and the half of
that again before you can traverse it. This goes on _ad infinitum_, so
that there are an infinite number of points in any given space, and you
cannot touch an infinite number one by one in a finite time."[41]
[40] _Op. cit._, p. 367.