8 Jun 2014

Russell, Ch.42 of Principles of Mathematics, ‘The Philosophy of the Continuum’, summary notes


by Corry Shores
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Bertrand Russell

Principles of Mathematics

Part 5: Infinity and Continuity

Ch.42: The Philosophy of the Continuum

Brief Summary:

Russell rejects the notion of the infinitesimal for defining continuous variation, and he uses instead Cantor’s notion of continuum. The infinitesimal account runs into problems with Zeno’s paradoxes, but Cantor’s version of continuum does not.





[Recall from ch.36 Cantor’s later definition of continuum, which regards it as a dense series of terms whose values are all definable by means of limits contained with in it. He does not see it as a magnitude with parts.] Philosophy has understood continua in a manner different from Cantor. For contrast Russell looks at Hegel’s distinction of continuous and discrete magnitudes, in which Hegel seems to be saying that when we see a magnitude as a continuum, we think of it being one thing that continuously varies; but when we regard a magnitude as discrete, we think of it as a plurality of units. Russell notes that while this distinction between identity and diversity might be a fundamental problem in logic, it is not relevant for the mathematical understanding continuity, because “it has no reference whatever to order” [and Cantor’s continuity is ordinal.] Russell will focus in this chapter on this mathematical meaning of continuum.


This sort of mathematical continuity is not like the other conception which thinks of a continuum as being a whole which is divided into constituent parts. Rather, it is an in infinite ordered series which together makes a continuum, rather than being a series which is obtained by dividing a continuum.

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.”
[Russell p.352, citing Poincaré Revue de Métaphysique et de Morale, Vol. I, p. 26.]

This non-mathematical view of continua might apply for time and space, but not for an arithmetical continuum, which 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.” (352) Russell thinks that the paradoxical theories of time and space result from regarding it as composed of elements. Cantor’s continuum, however, is free from such contradictions.  [Russell refers to the thesis proved in the preceding chapter. Here is that chapter’s conclusion:

There is no such thing as an infinitesimal stretch; if there were, it would not be an element of the continuum; the Calculus does not require it, and to suppose its existence leads to contradictions. And as for the notion that in every series there must be consecutive terms, that was shown, in the last chapter of Part III, to involve an illegitimate use of mathematical induction. Hence infinitesimals as explaining continuity must be regarded as unnecessary, erroneous and self-contradictory.


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 spatio-temporal 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.


Zeno’s paradoxes have a controversial history in philosophy. However, for Weierstrass, they “made the foundation of a mathematical renaissance”. [353] Like Zeno, he showed that the object in motion is truly at rest. But we need not follow to Zeno’s conclusion that this means the world never changes.

Weierstrass, by strictly banishing all infinitesimals, has at last 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, therefore 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 in mathematics, where familiarity with truth eliminates the vulgar prejudices of common sense, has been able to give to his propositions the respectable air of platitudes; 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.

Russell will translate them into arithmetical language. [353]


The first of Zeno’s arguments says that for something in motion to reach a destination, it needs first to reach a middle point, then the new middle point and so on. [But since there are an infinity of possible middle points, the object never gets to a first middle point.] Russell then translates this argument into arithmetical language. [He seems to be saying that between two numbers are always values which subdivide it. This might mean that we can never say that the sequence is fully constituted of its values. But we would need such a claim in order to define real numbers and arithmetical continuity.]

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 from 0 to 1 presuppose those from 0 to 1/2, these presuppose the numbers from 0 to 1/4, 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.

There are two ways of solving this. First we must distinguish two kinds of infinite regress. One of them is not problematic in this way. Secondly, we must distinguish two kinds of wholes: collective wholes and distributive wholes. In distributive wholes, “parts of equal complexity with the whole are not logically prior to it.” [354]


So first we look at the two kinds of infinite regress. Both kinds involve propositions, but the first kind involves the meaning of a proposition, and the second kind involves the implications. [It seems in the first case, a proposition can only be understood on the basis of another interpretation, which requires another, and so on. This happens when the meaning of a proposition is circular, in which case one proposition requires us understanding the meaning of another, while this second one requires us understanding the meaning of the first. Or, the second requires the understanding of the meaning of a third and so on, thus we never are able to establish the meaning of the first. In the second case, there is just an infinite change of inferences that come from the first proposition. What is important here is that we do not need to complete the chain of inferences in order to know the meaning of the first term. If Zeno’s argument is regressive just because the first term implies others, then it is not a problem, since the first term is definable as are all the rest.]

An infinite regress may be of two kinds. In the objectionable kind, two or more propositions join to constitute 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 expanded in a manner analogous to that in which continued fractions are developed from quadratic equations. But 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 ideas 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 meaning. 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.


To show that this paragraph of Zeno has the unproblematic form of regress, we first need to distinguish extentional and intensional wholes. [It seems the difference is that we understand the whole extensionally when we enumerate the terms, but we understand it intensionally when we understand all of its terms as referring to some given term. Perhaps one example of intensional would be defining the natural numbers in terms of the successor function applied to 1; I am just guessing, but the extensional whole would require each natural number to be listed.]

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 class-relation to a class-concept.)

[It is unclear to me, but in the following it seems Russell is saying that if we think that the infinite set of values needs to be defined extensionally, then we can never define it, and thus the paradox would have a problematic regress. However, if we define it intensionally by first defining the terms real number, 0, 1, and between, then we can say the set is complete because all the numbers are implied in these simple concepts. Explicating them might be an endless process, but that does not mean the whole is that is intensionally implied is somehow incomplete or undefinable.]

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 class-concept effects: a whole whose parts are the terms of a class is completely defined when the class-concept 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 is 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.


The second of Zeno’s arguments is of 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.”

[Before proceeding with Russell’s explanation, just consider: Achilles starts later. For him to overtake the tortoise, he needs first to reach the point where the tortoise is. But while getting there, the tortoise has created a new point to be crossed. So long as the tortoise is motion, he keeps creating more points and Achilles will never cross past the tortoise.] [Russell then translates this into arithmetical language. Recall that a set has an infinite (or ‘transfinite’) cardinal value when one of it parts has as many members as the whole it is a part of (see ch.37 and “Mathematics and the Metaphysicians”). He seems to be saying that if Achilles overtakes the tortoise, then the set of points that the tortoise crossed is a part of the set that Achilles crossed. However, since the tortoise’s set has the same number as Achilles, that means Achilles can never overtake the tortoise. Russell then gives a mathematical example that seems to work like this: So consider that there is an infinity of values between 0 and 1. Now also consider a set beginning at one and another set beginning at two, but both end with three. Now return again to all the values between 0 and 1. Have the set beginning at 2 increase by those values until it gets to three (this would be like adding 1 to 2). But in the set beginning with one, we would need to double each increment in order to increase two units to three. So make the first set increase by doubles of the values from 0 to 1. Now, both sets increased by the same number of increments. But the first increases two units total and the second increases one unit total. We would think then that that the first set would have twice as many parts, but they do not. Translated into the Achilles example, Achilles cannot pass the tortoise, because to do so he would need to cross more points than the tortoise, but there is an equal number of points to both the tortoise’s shorter path Achilles’ longer one.]

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 one-one correlation 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 moment at some point of his course, simultaneity establishes a one-one correlation between the positions of Achilles and those of the tortoise. Now it follows 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 discussion to the next chapter.


The third paradox is the arrow in flight.

“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.”

[Before looking at Russell’s explanation, consider the following. The moving body would occupy some number of points in space. But in order to move, it needs to advance to the next point. But time is made up of instants. And each instant it can only occupy as many points as it is long. Thus it can never move anywhere, since it can never occupy more points than the space it occupies.] [Please read Russell’s arithmetical explanation on page 356. I will offer a possible interpretation. He seems to be saying that the different positions of the arrow, or the figures in the arithmetical series, can be understood as values substituted for a variable, like in the above formulation which has us substitute all the values from 0 to 1 into the two equations 1 + 2x and 2 + x. But all such substitutions are constants, meaning they have a determinate value. How does this relate to this paradox? I will guess. It might be that there are no substitutions of x which would give one value and its immediate successor. But that is what is needed for the object to be in motion, or for the series to advance continuously.]

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 1/2 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 Part I. 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)2 = x2 + 2x + 1.
This certainly does not mean what it would become if, say, 391 were substituted 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 class-concept number, for we cannot add 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 constitute the essence of Zeno’s contention that the arrow is always at rest.





But if all the values we can substitute in for x are constants, that means their difference is always finite and thus not infinitesimal. Nonetheless, since such a formula could describe every possible position the object moves through, it is sufficient to account for its motion, Russell thinks. But Zeno was not aware of this way of understanding change. At his time a state of change was needed to explain change. [Note, the following below should be set as block quote, but for some reason I am unable to set it that way here.]

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 realized 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 differences 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.


The fourth of Zeno’s arguments regards measure. Russell says that this is like what he dealt with in the prior chapter. There he argued that dx and dy cannot be consecutive terms.  [It seems the problem now is not with consecutivity but with a continuum being made of discrete indivisibles, but see 357, since I understood Russell’s continuum to be made of discrete terms. Perhaps he means their intervals are always divisible and thus the continuum is not made of discrete indivisibles.] In his explanation, he seems to be giving an instance where we have one singular instant of motion, but it involves crossing two spatial locations. He gives this diagram:


In one instant, the second row moves one place to the left and the third row moves one place to the right. But c’ now aligns with a’’, which is two places apart from it. So in one supposedly indivisible instant, one point crosses two points, which suggests the instant has two smaller parts, when c’ aligned with b’’ and when it aligned with a’’. [357] He says this is virtually the argument he made in the prior chapter, which says that if a continuous series is made of consecutive terms, then when differentially related to the terms in another  continuous series, the ratio of dy/dx will always be positive or negative one. [This case is similar, because it likewise has for one infinitesimal increment in one series (one moment of time, one dx) it has moved two increments in another series (relative position to parallel points in another moving body, 2 dy’s).] M. Evellin is a proponent of indivisible stretches, and he says that a’’ and b’’ do not cross: “one instant a’ is over a’’, in the next, c’ is over a’’. [We might say then that c’ skips over b’’.] [Russell says that for physical motion, this might be true, since it is not really self-contradictory. However, it does not work in arithmetic “since no empirical question of existence is involved.” Please see p.358, as I cannot explain what he means. Then he seems to be saying that since we solved Zeno’s problem arithmetically, we should use this method when discussing the problem of continuity:] [the following should be block quoted]

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 the 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 existence 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 connection with numbers than in connection with space, time or motion.


Thus the arithmetical conception of continuum that Russell advocates does not suffer from the problems of Zeno’s paradox. Russell will not make some remarks.

The first is that someone might say that what Cantor calls a continuum is not a continuum in the sense involved in the paradoxes [and thus does not relate to them enough to fall victim to their problems]. Russell then goes on to praise the merits of Cantor’s notion of continuity. [The following should be block quoted.]

The salient points in the definition of the continuum are (1) the connection 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.


In this final section Russell notes that the notion of infinitesimal segments brought about the antinomy that the continuum both does and does not consist of segments [there are infinitesimal parts but they do not have an extensive value are thus not parts of the same kind as the extensive whole]. Russell’s denial of infinitesimal segments does not have the problem of this antinomy, but this is because we can say both that a line is made of segments but also that it is not, however, we do not in both cases mean this with the same sense. [For details as to why this is, see p.359.]


Bertrand Russell. Principles of Mathematics. London/New York: Routledge, 2010 [1st published 1903].

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