Subchapter325. The world
continuity has borne among philosophers, especially since the time of Hegel, a
meaning toally unlike that given to it by Cantor. Thus
Hegel says, [Smaller Logic, Wallace’s translation.] “Quantity, as we saw, has
two sources: the exclusive unit, and the identification or equalisation of
these units. When we look, therefore, at its immediate relation to self, or at
the characteristic of selfsameness made explicit by abstraction, quantity is
Continuous magnitude; but when we look at the other characteristic, the One
implied in it, it is Discrete magnitude.” When we remember that quantity and
magnitude, in Hegel, both mean “cardinal number,” we may conjecture that this
assertion amounts to the following: “Many terms, considered as having a
cardinal number, must all be members of one class: in so far as they are each
merely an instance of the classconcept, they are indistinguishable one from
another, and in this aspect the whole which they compose is called continuous;
but in order to their maniness, they must be different instances of the
classconcept, and in this aspect the whole which they compose is called
discrete.” Now I am far from denying—indeed I strongly hold – that this
opposition of identity and diversity in a collection constitutes a fundamental
problem ofLogic, perhaps even the fundamental problem ofPhilo. And being
fundamental, it is certainly relevant that to the study of the mathematical
continuuum as to everything else. But beyond this general connexion, it has no
special relation to the mathematical meaning of continuity, as may be seen at
once from the fact that it has no reference whatever to order. In this chapter,
it is the mathematical meaning that is to be discussed. I have quoted the
philosophic meaning only in order to state definitely that this is not here in
question; and since disputes about words are futile, I must ask philosophers to
divest themselves, for the time, of their habitual associations with the word,
and allow it no signification but that obtained from Cantor’s definition.
Paragraph2. In confining
ourselves to the arithmetical continuum, we conflict in another way with common
preconceptions. Of the arithmetical continuum, M.Poincaré justly remarks, “The
continuum thus conceived is nothing but a collection of individuals arranged in
a certain order, infinite in number, it is true, but external to each other.
This is not the ordinary conception, in which there is supposed to be, between
the elements of the continuum, a sort of intimate bond which makes a whole of
them, in which the point is not prior to the line, but the line to the point.
Of the famous formula, the continuum is unity in multiplicity, the multiplicity
alone subsists, the unity has disappeared.” [Revue de Métaphysique de Morale,
Vol.1, p.26.]
It has always been held to be
an open question whether the continuum is composed of elements; and even when
it has been allowed to contain elements, it has been often alledged to be not
composed of these. This latter view was maintained even by so stout a supporter
of elements in everything as Leibniz. [See thePhilo.OfLeibiniz, by the present
author, Chap.IX.] But all these views are only possible in regard to such
continua as those of space and time. The arithmetical continuum is an object
selected by definition, consisting of elements in virtue of the definition, and
known to be embodied in at least one instance, namely the segments of the
rational numbers. I shall maintain in PartVI that spaces afford other instances
of the arithmetical continuum. The chief reason for the elaborate and
paradoxical theoreis of space and time and their continuity, which have been
constructed by philosophers, has been the supposed contradictions in a
continuum composed of elements. The thesis of the present chapter is, that
Cantor’s continuum is free from contradictions. This thesis, as is evident,
must be firmly established, before we can allow the possibility that spatiotemporal
continuity may be of Cantor’s kind. In this argument, I shall assume as proved
the thesis of the preceding chapter, that the continuity to be discussed does
not involve the admission of actual infinitesimals.
Subchapter327. In this capricious world, nothing is more capricious than
posthumous fame. One of the most notable victims of posterity’s lack of
judgement is the Eleatic Zeno. Having invented four arguments all immeasurably
subtle and profound, the grossness of subsequent philosophers pronounced him to
be a mere ingenious juggler, and his arguments to be one and all sophisms. After
two thousand years of continual refutation, these sophisms were reinstated, and
made the foundation of a mathematical renaissance, by a german professor who
probably never dreamed of any connexion between himself and Zeno. Weierstrass, by strictly banishing all infinitesimals, has at
least shown that we live in an unchanging world, and that the arrow, at every
moment of its flight, is truly at rest. The only point where Zeno
probably erred was in inferring (if he did infer) that, because there is no
change, the world must be in the same state at one time as at another. This
consequence by no means follows, and in this point the german professor is more
constructive than the ingenious greek. Weierstrass, being able to embody his
opinions inMath., where familiarity with truth eliminates the vulgar prejudices
of common sense, has been able to give to his propositions the respectable air
of platitutdes; and if the result is less delightful to the lover of reason
than Zeno’s bold defiance, it is at any rate more calculated to appease the
mass of academic mankind.
Zeno’s argumentys are specially
concerned with motion, and are not therefore, as they stand, relevant to our
present purpose. But it is instructive to translate them, so far as possible,
into arithmetical language. [Not being a greek scholar, I pretend to no
firsthand authority as to what Zeno really did say or mean. The form of his
four arguments which I shall employ is derived from the interesting article of
M.Noël, “Le Mouvement et les arguments de Zénon d’Elée,” Revue de Métaphysique
et de Morale, Vol.1, pp.107-125. These arguments are in any case well worthy of
consideration, and as they are, to me, merely a text for discussion, their
historical correctness is of little importance.]
Subchapter328. The first
argument, that of dichotomy, asserts: “There is no
motion, for what moves must reach the middle of its course before it reaches
the end.” That is to say, whatever motion we assume to have taken place,
this presupposes another motion, and this in turn another, and so on ad
infinitum. Hence there is an endless regress in the mere idea of any assigned
motion. This argument can be put into an arithmetical form, but it appears then
far less plausible. Consider a variable x which is capable of all real (or
rational) values between two assigned limits, say 0 and 1. The class of its
values is an infinite whole, whose parts are logically prior to it: for it has parts,
and it cannot subsist if any of the parts are lacking. Thus the numbers frmo 0
to 1 presuppose those from 0 to ½, these presuppose the numbers from 0 to ½,
and so on. Hence, it would seem, there is an infinite regress in the notion of
any infinite whole; but without such infinite wholes, real numbers cannot be
defined, and arithmetical continuity, which applies to an infinite series,
breaks down.
This argument may be met in two
ways, either of which, at first sight, might seem sufficient, but both of which
are really necessary. First, we may distinguish two kinds of infinite
regresses, of which one is harmless. Secondly, we may distinguish two kinds of
whole, the collective and the distributive, and asser that, in the latter kind,
parts of equal complexity with the whole are not logically prior to it. These
two points must be separately explained.
Subchapter329. An infinite
regress may be of two kinds. In the objectionable kind, two or more
propositions join to constitue the meaning of some proposition; of these
constituents, there is one at least whose meaning is similarly compounded; and
so on ad infinitum. This form of regress commonly results from circular
definitions. Such definitions may be explained in a manner analogous to that in
which continued fractions are developed from quadratic equations. Bit at every
stage the term to be defined will reappear, and no definition will result. Take
for example the following: “Two people are said to have the same idea when they
have ideas which are similar; and ideas are similar when they contain an
identical part.” If an idea may have a part which is not an idea, such a
definition is not logically objectionable; but if part of an idea is an idea,
then, in the second place where identity of idea occurs, the definition must be
substituted; and so on. Thus wherever the meaning of a proposition is in
question, an infinite regress is objectionable, since we never reach a
proposition which has a definite fmeaning. But many infinite regresses are not
of this form. If A be a proposition whose meaning is perfectly definite, and A
implies B, B implies C, and so on, we have an infinite regress of a quite
unobjectionable kind. This depends upon the fact that implication is a
synthetic relation, and that, although, if A be an aggregate of propositions, A
implies any proposition which is part of A, it by no means follows that any
proposition which A implies is part of A. Thus there is no logical necessity,
as there was in the previous case, to complete the infinite regress before A
acquires a meaning. If, then, it can be shown that the implication of the parts
in the whole, when the whole is an infinite class of numbers, is of this latter
kind, the regress suggested by Zeno’s argument of dichotomy will have lost its
sting.
Subchapter330. In order to show that this is
the case, we must distinguish wholes which are defined extensionally, i.e., by
enumerating their terms, from such as are defined intensionally, i.e. as the
class of terms having some given relation to some given term, or, more simply,
as a class of terms. (For a class of terms, when it forms a whole, is merely
all terms having the classrelation to a classconcept.) [For precise statements,
v. supra, PartI. Chaps. VI and X.] Now an extensional whole, at least so far as
human powers extend, is necessarily finite: we cannot enumerate more than a
finite number of parts belonging to a whole, and if the number of parts be
infinite, this must be known otherwise than by enumeration. But this is
precisely what a classconcept effects: a whole whose parts are the terms of a
class is completely defined when the classconcept is specified; and any
definite individual either belongs, or does not belong, to the class in
question. An individual of the class is part of the whole extension of the
class, and is logically prior to this extension taken collectively; but the
extension itself it definable without any reference to any specified
individual, and subsists as a genuine entity even when the class contains no
terms. And to say, of such a class, that it is infinite, is to say that, though
it has terms, the number of these terms is not any finite number, a proposition
which, again, may be established without the impossible process of enumerating
all finite numbers. And this is precisely the case of the real numbers between
0 and 1. They form a definite class, whose meaning is known as soon as we know
what is meant by real number, 0, 1, and between. The particular members of the
class, and the smaller classes contained in it, are not logically prior to the
class. Thus the infinite regress consists merely in the fact that every segment
of real or rational numbers has parts which are again segments; but these parts
are not logically prior to it, and the infinite regress is perfectly harmless.
Thus the solution of the difficulty lies in the theory of denoting and the
intensional definition of a class. With this an answer is made to Zeno’s first
argument as it appears in Arithmetic.
Subchapter331. The second of
Zeno’s arguments is the most famous: it is the one which concerns Achilles and
the tortoise. “The slower,” it says, “will never be
overtaken by the swifter, for the pursuer must first reach the point whence the
fugitive is departed, so that the slower must always necessarily remain ahead.”
When this argument is translated into arithmetical language, it is seen to be
concerned with the oneonecorrelation of two infinite classes. If Achilles were
to overtake the tortoise, then the course of the tortoise would be part of that
of Achilles; but, since each is at each omoment at some point of his course,
simultaneity establishes a oneoncorrelation between the positions of Achilles
and those of the tortoise. Now it folllows from this that the tortoise, in any
given time, visits just as many places as Achilles does; hence, so it is hoped
we shall conclude, it is impossible that the tortoise’s path should be part of
that of Achilles. This point is purely ordinal, and may be illustrated by
Arithmetic. Consider, for example, 1+2x and 2+x, and let x lie between 0 and 1,
both inclusive. For each value of 1+2x there is one and only one value of 2+x,
and vice versâ. Hence as x grows from 0 to 1, the number of values assumed by
1+2x will be the same as the number assumed by 2+x. But 1+2x started from 1 and
ends at 3, while 2+x started from 2 and ends at 3. Thus there should be half as
many values of 2+x as of 1+2x. This very serious difficulty has been resolved,
as we have seen, by Cantor; but as it belongs rather to the Philosophy of the
infinite than to that of the continuum, I leave its further discussino to the
next chapter.
Subchatper332. The third
argument is concerned with the arrow. “If everything is
in rest or in motion in a space equal to itself, and if what moves is always in
the instant, the arrow in its flight is immovable.” This has usually
been thought so monstrous a paradox as scarcely to deserve serious discussion. To my mind, I must confess, it seems a very plain statement
of a veryelementaryfact, and its neglect has, I think, caused the quagmire in
which the Philsoophy of change has long been immersed. In PartVII, I
shall set forth a theory of change which may be called static, since it allows
the justice of Zeno’s remarks. For the present, I wish to divest the remark of
all reference to change. We shall then find that it is a very important and
very widely applicable platitude, namely: “Every possible value of a variable
is a constant.” If x be a variable which can take all values from 0 to 1, all
the values it can take are definite numbers, such as ½ or 1/3, which are all
absolute constants. And here a few words may be inserted concerning variables.
A variable is a fundamental concept of Logic, as of daily life. Though it is
always connected with some class, it is not the class, nor a particular member
of the class, nor yet the whole class, but any member of the class. On the
other hand, it is not the concept “any member of the class,” but it is that (or
those) which this concept denotes. On the logical difficulties of this
conception, I need not now enlarge; enough has been said on this subject in
PartI. The usual x in Algebra, for example, does not stand for a particular
number, nor for all numbers, nor yet for the class number. This may be easily
seen by considering some identity, say (x+1)square = x-square + 2x + 1. This
certainly does not mean what it would become if, say, 391 were substitued for
x, though it implies that the result of such a substitution would be a true
proposition. Nor does it mean what results from substituting for x the
classconcept number, for we cannot ad 1 to this concept. For the same reason, x
does not denote the concept any number: to this, too, 1 cannot be added. It
denotes the disjunction formed by the various numbers; or at least this view
may be taken as roughly correct. The values of x are then the terms of the
disjunction; and each of these is a constant. This simple logical fact seems to
constitue the essence of Zeno’s contention that the arrow is always at rest.
Subchapter333. But Zeno’s
argument contains an element which is specially applicable to continua. In the
case of motion, it denies that there is such a thing as a state of motion. In
the general case of a continuous variable, it may be taken as denying actual
infinitesimals. For infinitesimals are an attempt to extend to the values of a
variable the variability which belongs to it alone. When once it is firmly
realised that all the values of a variable are constants, it becomes easy to
see, by taking any two such values, that their difference is always finite, and
hence that there are no infinitesimal differences. If x be a variable which may
take all real values from 0 to 1, then, taking any two of these values, we see
that their difference is finite, although x is a continuous variable. It is true
the difference might have been less than the one we chose; but if it had been,
it would still have been finite. The lower limit to possible difference is
zero, but all possible differences are finite; and in this there is no shadow
of contradiction. This static theory of the variable is due to the
mathematicians, and its absence in Zeno’s day led him to suppose that
continuous change was impossible without a state of change, which involves
infinitesimals and the contradiction of a body’s being where it is not.
Subchapter334. The last of Zeno’s arguments is that of the measure. This is
closely analogous to one which I employed in the preceding chapter, against
those who regard dx and dy as distances of consecutive terms. It is only
applicable, as M.Noël, points out [loc. cit. p.116], against
those who hold to indivisibles among stretches, the previous arguments being
held to have sufficiently refuted the partisans of infinite divisibility. We
are now to suppose a set of discrete moments and discrete places, motion
consisting in the fact that at one moment a body is in one of these discrete
places, in another at another.
Imagine three parallel lines composed of the points a, b, c, d; a’, b’,
c’, d’; a’’, b’’, c’’, d’’ respectively. Suppose the second line, in one
instant, to move all its points to the left by one place, while the third moves
them all one place to the right. Then although the instant is indivisible, c’,
which was over c’’, and is now over a’’, must have passed b’’ during the
instant; hence the instant is divisible, contra hyp. This argument is virtually
that by which I proved, in the preceding chapter, that, if there are
consecutive terms, then dy/dx=plusminus1 always; or rather, it is this argument
together with an instance in which dy/dx = 2 It may be put thus: Let y, z be
two functions of x, and let dy/dx = 1, dz/dx = -1. Then (d/dx)(y-z) = 2, which
contradicts the principle that the value of every derivative must be
plusminus1. To the argument in Zeno’s form, M.Evellin, who is an advocate of
indivisible stretches, replies that a’’ and b’ do not cross each other at all.
For if instants are indivisible, and this is they hypothesis, all we can say
is, that at one instant a’ is over a’, in the next, c’ is over a’’. Nothing has
happened between the instants, and to suppose that a’’ and b’ have crossed is
to beg the question by a covert appeal to the continuity of motion. This reply
is valid, I think, in the case of motion; both time and space may, without
positive contradiction, be held to be discrete, by adhering strictly to
distances in addition to stretches. Geometry, Kinematics, and Dynamics become
false; but there is no very good reason to think them true. In the case of
Arithmetic, the matter is otherwise, since no empirical question of exsitence
is involved. And in this case, as we see from the above argument concerning
derivatives, Zeno’s argument is absolutely sound. Numbers are entities whose
nature can be established beyond question; and among numbers, the various forms
of continuity which occur cannot be denied without positive contradiction. For
this reason the problem of continuity is better discussed in connexion with
numbers than in connection with space, time, or motion.
Subchapter335. We have now seen that Zeno’s arguments, though they prove
a very great deal, do not prove that the continuuum, as we have become
acquainted with it, contains any contradictions whatever. Since his day the
attacks on the continuum have not, so far as I know, been conducted with any
new or more powerful weapons. It only remains, therefore, to make a few general
remarks.
The notion to which Cantor gives the name of continuum may, of course,
be called by any other name in or out of the dictionary, and it is open to
every one to assert that he himself means something quite different by the
continuum. But these verbal questions are purely frivolous. Cantor’s merit
lies, not in meaning what other people mean, but in telling us what he means
himself, an almost unique merit, where continuity is concerned. He has defined,
accurately and generally, a purely ordinal notion, free, as we now see, from
contradictions, and sufficient for all Analysis, Geometry, and Dynamics. This
notion was presupposed in existing Mathematics, though it was not known exactly
what it was that was presupposed. And Cantor, by his almost unexampled
lucidity, has succesfully analysed the extremely complex nature of spatial
series, by which, as we shall see in PartVI, he has rendered possible a
revolution in the Philosophy of space and motion. The salient points in the
definition of the continuum are (1) the connexion with the doctrine of limits,
(2) the denial of infinitesimal segments. These two points being borne in mind,
the whole Philosophy of the subject becomes illuminated.
Subchapter336. The denial of infinitesimal segments resolves an antinomy
which had long been an open scandal, I mean the antinomy that the continuum
both does and does not consist of elements. We see now that both may be said,
though in different senses. Every continuum is a series consisting of terms,
and the terms, if not indivisible, at any rate are not divisible into new terms
of the continuum. In this sense there are elements. But if we take consecutive
terms together with their asymmetrical relation as constituting what may be
called (though not in the sense of Part IV) an ordinal element, then, in this
sense, our continuum has no elements. If we take a stretch to be essentially
serial, so that it must consist of at least two terms, then there are no elementary
sketches; and if our continuum be one in which there is distance, then likewise
there are no elementary distances. But in neither of these cases is there the
slightest logical ground for elements. The demand for consecutive terms
springs, as we saw in PartIII, from an illegitimate use of mathematical
induction. And as regards distance, small distances are no simpler than large
ones, but all, as we saw in PartIII, are alike simple. And large distances do
not presuppose small ones: being intensive magnitudes, they may exist where
there are no smaller ones at all. Thus the infinite regress from greater to
smaller distances or stretches is of the harmless kind, and the lack of
elements need not cause any logical inconvenience. Hence the antinomy is resolved,
and the continuum, so far at least as I am able to discover, is wholly free
from contradictions.
It
only remains to inquire whether the same conclusion holds concerning the
infinite, an inquiry with which this FifthPart will come to a close.
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