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 Mr. Haldane on Infinity  (1908)*

By Bertrand Russell

THIS address falls into two parts, of which the first may be described as an endeavour to restate shortly and intelligibly the parts of Hegel’s greater Logic which deal with mathematics, while the second, having dismissed the “false” infinite, proceeds to some meditations on the “true” infinite, i.e., the Absolute. These two parts have, so far as I can see, no connexion except that the word “infinite” happens to be used both for a mathematical concept and for a (quite different) philosophical concept. What is rather curious is that Mr. Haldane himself seems to recognise this absence of connexion, for he says (p. 1): “An infinite series suggests, or ought to suggest, nothing analogous to an infinite God.” Apparently he conceives, however, that there is a dialectical connexion between them – that the infinite series, when its contradictions are brought to light and synthesised, is found to be merely an inadequate expression of an infinite God. To one who holds, as I do, that there are no contradictions about an infinite series, this connexion of course fails. I shall, therefore, say no more about the “true” infinite, confining myself to what Mr. Haldane says about the “false” infinite. I may remark, however, that if I had the bestowal of the adjectives “true” and “false,” I should interchange them, since I think the “false” infinite logically faultless, and the “true” infinite a mere chimæra.

When Mr. Haldane speaks of “modern logic,” he must be understood as meaning the logic of Hegel (and his disciples); and when he speaks of “the conception of infinity,” he must be understood as meaning the conception of infinity as it existed in Hegel’s day. For this reason, what he says about the Infinitesimal Calculus is in the main not relevant to the present state of that science. It is true that he quotes (pp. 4, 5) two American authors, Dr. Bledsoe and Prof. Buckingham, in support of his contentions. These authors are, unfortunately, unknown to me except through Mr. Haldane's quotations. In these quotations they describe, very accurately, the state of mind of an intelligent pupil who is being taught the Calculus by an ignorant teacher. They do not say that the teacher they have in mind is ignorant, but if he is not, the things they say are no longer true. It seems a pity that Mr. Haldane has not (apparently) had the good fortune to come across any of the recognised modern authorities on the Calculus, such (for example) as the Encyklopädie der mathematischen Wissenschaften.1 If Mr. Haldane had read any such authority, he would have found that, while his criticisms of Leibniz (p. 5) are fully justified, no such criticisms are applicable to the present theory of the Calculus. He would there have learnt that the whole subject proceeds without ever introducing the infinitesimal, that the fundamental conception is that of a limit, and that a limit is something quite different from what non-mathematicians suppose it to be. I will endeavour to explain these points, though it should be said that only careful study makes it possible to grasp them thoroughly or to see their bearings.

Mr. Haldane sums up “the broad working conceptions of the Differential Calculus” in two propositions (p. 7):

(1) “A differential Coefficient expresses the rate of change of a function with respect to its independent variable.”

(2) “If there be a fixed magnitude to which a variable magnitude can be made as nearly equal as we please, and if it be impossible that the variable magnitude can ever be exactly equal to the fixed magnitude, the fixed magnitude is called the limit of the variable magnitude.”

He proceeds: “Now, this second proposition bears the mark of the cloven hoof. The idea of negligible difference is present in it.”

As to the first of these propositions, it should be observed that, if the modern “arithmetization” of mathematics is not all a mistake, it puts the cart before the horse. “Rate of change” is a phrase which can only be interpreted by means of a differential coefficient, and therefore we must first define the differential coefficient before we can speak of the rate of change. As regards the second proposition, it does not constitute a definition of “limit,” but a property which belongs to limits of a certain particular sort, namely limits of magnitudes in compact series of magnitudes in which, given a magnitude a, and a difference of magnitude c, there is always a magnitude b, other than a, such that the difference of a and b is less than c. This is an extremely special case, not realised in most of the series in which limits exist. As for the “idea of negligible difference,” this is not present even in this special case. The reason why Mr. Haldane supposes it present is, presumably, that he attributes to mathematicians the belief that a variable “ultimately” becomes equal to its limit. No such belief is involved in the notion of a limit, and in many cases it would be palpably inapplicable.

In order to be clear on the subject, it is necessary to realise, in the first place, that a variable does not vary. A variable is a symbol which is to have one of a certain set of values, without its being decided which one. It does not have first one value of the set, and then another; it has at all times some value of the set, where, so long as we do not replace the variable by a constant, the “some” remains unspecified. Another point on which it is important to be clear is the following. A limit belongs, properly speaking, not to a variable, but to a class contained in a series. When we speak of the limit of a variable, we mean the limit of the class of values of the variable. A third point on which confusion is common is, that a limit does not belong to a class intrinsically, but only in relation to some series in which the class is contained; a class may have one limit in one series, and another in another, or it may have a limit in one series, and no limit in another. Finally, a limit must not be conceived as something to which the successive terms of the class approach indefinitely near; they may all be at an infinite distance from the limit, or at a distance which remains permanently greater than some given finite distance; or the series concerned may be one in which there is no such thing as distance or difference.

The definition of a limit is as follows. Given any series, and a class a of terms belonging to the series, a term x belonging to the series is called the upper limit of a if every member of a precedes x, ,and every term of the series which precedes x precedes some member of a; x is called the lower limit of a if every member of a follows x, and every term of the series which follows x follows some member of a.

A few illustrations will make this definition clearer.

(1) Let our series consist of all rational numbers of the form 2m − 1/n, where m and n are finite integers (not zero), and the terms are arranged in order of magnitude. Then all terms of the form 2 − 1/n belong to this series; that is, the class whose members are

2 − 1, 2 − 1/2, 2 − 1/3, 2 − 1/4 …

is contained in the series. These terms, in the whole series of rational numbers, would have 2 for their upper limit, but 2 is not a member of our series. In our series, they still have an upper limit, but the limit is 3. For every term of the above class is less than 3, and any member of the series which is less than 3 is less than some member of the above class. But it cannot be said that the idea of “negligible difference” is present here. The difference between 3 and 2 − 1/n does not diminish without limit as n increases; on the contrary, the difference is always greater than 1, and has 1 for its lower limit in the whole series of rationals.

(2) Let our series consist of the rational numbers in order of magnitude, and let the class a consist of all those rationals whose square is less than 2. Then this class has no upper limit. For any rational x which comes after every member of a cannot have its square less than 2, by definition, or equal to 2, because no rational has its square equal to 2; therefore its square is greater than 2. But the rational numbers whose square is greater than 2 have no lower limit, therefore there are rationals less than x and yet greater than every member of a. Therefore no rational x is the upper limit of a. But if instead of taking the rational numbers only, we admit, irrational numbers also, and choose a out of the series of rational and irrational numbers combined, then a has a limit, namely, √2. Or again, if our series consists of all rationals whose square is less than 2, followed by all rationals whose square is not less than 4, then a has an upper limit, namely 2.

(3) If we consider the whole series of integers, finite and infinite, arranged in order of magnitude, then the class of finite integers, considered as part of this series, has an upper limit, namely the smallest of the infinite integers (which is the number of finite integers). Here, so far from any question of “negligible difference” coming in, the difference between the finite integers and their limit remains. constant and infinite.

The notion of a limit is, in fact, a purely ordinal notion, involving no reference to quantity whatever, and applicable to cases in which the “difference” between different terms of a series has no definable. meaning, as, for example, to series of points in space or moments in time. In the particular case in which our whole series is that of rational numbers or that of real numbers, it happens that, if x is the limit of a class a, then the limit of the class of differences xy, where y is any member of a, is zero. But obviously we cannot define limits by this property, first because it involves a circle in definition, and secondly because, even if this objection were circumvented, the definition is only applicable to certain very special cases, not to series in general.

We thus see that limits involve no such notion as “negligible difference” or infinitesimals. When, therefore, a differential is defined as a limit, it must not be inferred that infinitesimals are necessary to its definition. And, in fact, the so-called “infinitesimal calculus” nowhere assumes infinitesimals. The notation “dy/dx,” introduced by Leibniz, survives in practice owing to its technical convenience; but in fundamental definitions the notation “f ' (x),” for the differential or derivative of f (x) is more commonly employed, in order to avoid the fallacious suggestion of a fraction, with infinitesimals dy, and dx for its numerator and denominator. I will not here attempt to give the full definition of the derivative of a function, as it is very complicated, and depends upon the definition of the limit of a function for a given value of its argument. The definition of the derivative will be found in any standard work,2 and should be studied by those who wish to criticise the Calculus. Neglecting' niceties, however, a few remarks may serve to avoid misunderstandings.

The derivative f ' (x) of a function f (x) is the limit (in case there is one) of

[ f (x + δ) − f (x) ] / δ

as δ approaches zero either from above or below. This means that a number z is called the derivative of f (x) in the point x, if, given any number ε, there is a number η such that, if δ is any number (other than 0) between η and -η, [ f (x + δ) − f (x) ] / δ differs from z by a number which lies between ε and -ε. It is to be observed that [ f (x + δ) − f (x) ] / δ is a function of δ, and that f ' (x) is the limit of this function for δ = 0, not the value for δ = 0.

The value of a function for a given argument is a different conception from its limit for that argument; sometimes the two are equal, sometimes both exist and are unequal, sometimes one exists and the other does not. In the present case, though the function has a limit for δ = 0 wherever f (x) can be differentiated, the function never has a value for δ = 0. In finding its limit for δ = 0, only finite values of δ are involved; it is a sheer misunderstanding to suppose that δ becomes infinitesimal in the limit.

Another point which is important to observe is that functions, even when they are continuous, do not in general have derivatives; it is a stroke of luck when a continuous function has a derivative. When, therefore, philosophers argue as if differentiability were a consequence of continuity, they are guilty of a mathematical error, though one which, until about forty years ago, the mathematicians shared with them.

The theory of limits, and the modern treatment of the Calculus as based on this theory, are by no means easy, and require careful study; but philosophic criticism which is unaware of them can hardly hope to be fruitful. In Hegel’s day, the procedure of mathematicians was full of errors, which Hegel did not condemn as errors, but welcomed as antinomies; the mathematicians, more patient than the philosophers, have removed the errors by careful detailed work on every doubtful point. A criticism of mathematics based on Hegel can, therefore, no longer be regarded as applicable to the existing state of the subject.

In spite of the title of Mr. Haldane's Address, he does not deal with the modern mathematical conception of the infinite, or attempt to show that it is in any way self-contradictory. I have therefore not undertaken to defend it against the epithet “false,” but have attempted rather to bring to the notice of philosophers certain facts which, so far as I can see, invalidate practically everything that has been written by philosophers, from Leibniz downwards, on the logical basis of the Calculus. These facts are not very new, and not it all in dispute. It seems time, therefore, that philosophers who write on the subjects in which they are relevant should begin to become aware of them, as doubtless they would have done sooner but for the technical difficulties with which the study of mathematics is beset.

B. RUSSELL


*  Bertrand Russell, “Mr. Haldane on Infinity,” Mind, n.s. 17, no. 66 (Apr 1908), 238-42  This essay is a repsonse to R. B. Haldane’s presidential speech to the Aristotelian Society, “The Methods of Modern Logic and the Conception of Infinity”, delivered at the opening meeting of the 29th session, Nov 4, 1907

1 I mention this as being the most official, but exactly the same account of the matter will be found in any other well-informed modern book. See for example Dini’s Theorie der Functionen einer veränderlichen reellen Grösse (Leipzig, 1892)

2 E.g., Encyklopädie der mathematischen Wissenschaften, ii., A 1, pp. 20-22; ii., A 2, p. 60