8 Jun 2014

Russell, Ch.21 of Principles of Mathematics, ‘Numbers as Expressing Magnitude’, summary notes

Corry Shores
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[The following is summary and quotation. All boldface, underlining, and bracketed commentary are mine. Please consult the original text, as I did not follow it closely. Proofreading is incomplete, so mistakes are still present.]

Bertrand Russell

Principles of Mathematics

Part 3: Quantity

Ch.21: Numbers as Expressing Magnitude

Brief Summary:

Magnitudes are always measurable. Some are measurable numerically but others only by comparisons of more, less, and equal.  Extensive magnitudes are ones that are numerically measurable, and intensive magnitudes are ones that are not numerically measurable. There are two kinds of extensive magnitudes, divisibilities and distances. Divisibilities are numerically measurable by dividing them into equally sized parts and counting those parts. Distances [according to Russell] do not admit of parts and cannot combine to form larger wholes. This is because [according to Russell’s conception of distance] a distance is understood as a set of relations between terms such that these relations may determine the relative values for combinations of those terms. Thus although its parts (the terms) cannot combine to form the whole distance, we can still measure the distance on the basis of the parts but totaling the number of relations between them. In a distance, there are a series of terms: the end terms on both extremes and the intermediate terms between them. The set of intermediate terms is called the ‘stretch.’ We can measure the whole by counting the number of terms plus one of the end terms. [Consider something though would be measured as 3 units long, and composed of the terms 0,1,2, and 3. The end terms are 0 and 3, and there are 2 intermediate terms. To get the total value of three, we need to add an end term.]



“The purpose of the present chapter is to explain what is meant by measurement, what are the classes of magnitudes to which it applies and how it is applied to those classes.” [176]

When we measure magnitudes, we bring into metrical correspondence one things metrical parts with some series of numbers.

Measurement of magnitudes is, in its most general sense, any method by which a unique and reciprocal correspondence is established between all or some of the magnitudes of a kind and all or some of the numbers, integral, rational, or real, as the case may be. […] In this general sense, measurement demands some one-one relation between the numbers and magnitudes in question


Russell now says that there are two metaphysical positions that say all magnitudes are capable of measurement in this one-one fashion. [Russell in the following will distinguish events and events in “the dynamical causal series.” I do not know what he means, but his examples seem understandable.] In one position we find correspondences between intensive quantities and spatial ones [for example, luminosities are given numerical values such that one can be double the other.] Also under this position, in psychophysics, mental qualities are in psychophysical parallelism [meaning that increases in a physical stimulus can correspond to increases in the intensity of the experience.] The other position leads to universal measurability. It is suggested in Kant’s “Anticipations of Perception.” [What Russell seems to be noting is that for Kant, intensive magnitudes can vary but can also be compared; for example, the sun is so-many times the brightness as the moon. Russell also notes that this view allows us to say that “a child derives as much pleasure from one chocolate as from two acid drops.” (177) So the universal measure seems to be that one intensity is quantified in relation to another, and any intensity is then quantifiable in relation to an extensive magnitude.] [Russell then goes on evoking Cantor’s notion of continuity and explains how we might then conceive of magnitudes that are incapable of being measured by real numbers. See pp.177-178]


Russell will now look at the “more usual and concrete sense of measurement.” [178] He will examine the sense of measurement that allows us to say that one magnitude is double another. He says that “measurement demands that […] there should be an intrinsic meaning to the proposition ‘this magnitude is double that.’” But later we will learn what is meant by intrinsic meaning. When a quantity is inherently divisible, then one magnitude is double another when the first is equal to two equal instances of the second. “a magnitude A is double of B when it is the magnitude of two quantities together, each of these having the magnitude B.” [178]


This notion of measuring magnitude on the basis of divisibility is fine for finite magnitudes, but breaks down for infinite ones. In finite cases, one magnitude is double another when it has twice as many of that other’s parts. [Consider for example one segment that is twice another in size. If we think of both as being infinitely divided, then both will have an equal number of parts but still have different magnitudes.]

But in the case of infinite wholes, the matter is by no means so simple. Here the number of simple parts (in the only senses of infinite number hitherto discovered) may be equal without equality in the magnitude of divisibility. We require here a method which does not go back to simple parts.

[Russell then discusses the issue of measuring in metrical geometry, but says we discuss it fully not until a future section.]

[In the next paragraph it seems Russell is saying something like the following. Assume a magnitude is divisible and is measurable by means of its divisions. To each division would correspond a number. This means that both by means of numbers and divisions we may measure the magnitude. I think his next point might be that in the case of non-Euclidean geometry, for example, this does not work, perhaps because the metrics of space are not constant. “there is strictly no ground for saying that the divisibility of a sum of two units is twice as great as that of one unit”. [179]


[It seems in the following Russell distinguishes two kinds of addition: addition of wholes that creates a new whole, and additions that create no more than a combination of two things.]

In the above case we still had addition in one of its two fundamental senses, i.e. the combination of wholes to form a new whole. But in other cases of magnitude we do not have any such addition. The sum of two pleasures is not a new pleasure, but is merely two pleasures. The sum of two distances is also not properly one distance. But in this case we have an extension of the idea of addition. Some such extension must always be possible where measurement is to be effected in the more natural and restricted sense which we are now discussing. I shall first explain this generalized addition in abstract terms, and then illustrate its application to distances.

[The next paragraph seems to be the explanation “in abstract terms.” Please consult the text (p.180) to figure out for yourself what Russell means. It seems that he is saying that even though distances cannot be added to make a new whole distance, we can still measure their totality using a third distance which will allow us to determine the value of the first two’s combination, even though they do not actually form a whole.]

By distance, Russell means something broader than just ‘space.’ [Russell’s broader idea of distance seems to be a matter of terms that have a common third measure that allows for a sort of standardized comparison of all the terms. But please consult his definition and interpret it for yourself.] Instead for him distance means

a set of quantitative asymmetrical relations of which one and only one holds between any pair of terms of a given class; which are such that, if there is a relation of the kind between a and b, and also between b and c, then there is one of the kind between a and c, the relation between a and c being the relative product of those between a and b, b and c; this product is to be commutative, i.e. independent of the order of its factors; and finally, if the distance ab be greater than the distance ac, then, d being any other member of the class, db is greater than dc.

Russell now will explain the convention which allows for such distances to be numerically measurable. Consider a series of intervals all of the same value. We can calculate larger sections by counting how many segments it has. But since distances are not really composed of parts, it is only a convention that allows us to assign numbers to measure the value of such divisions. He explains this more later in the text.


[The following is important because he defines the term stretch. It seems it can simply be understood as the series of terms between any two. What is important is that the stretch is not the same as a distance. It seems more like just a series of terms whose total number can measure the distance. So lets consider a distance that would go from 0 to 10, and it has the terms, in all, 0,1,2,3,4,5,6,7,8,9,10. The stretch would be 1,2,3,4,5,6,7,8,9, thus a total of 9 terms. But the distance is 10. We can measure the distance on the basis of the terms by adding one of the end terms. He also seems to have us consider a distance divided into an infinity of terms. But the whole distance is finite. He seems to say that we convert the infinite number of terms into finite stretches to measure the whole.]

The importance of the numerical measurement of distance, at least as applied to space and time, depends partly upon a further fact, by which it is brought into relation with the numerical measurement of divisibility. In all series there are terms intermediate between any two whose distance is not the minimum. These terms are determinate when the two distant terms are specified. The intermediate terms may be called the stretch from a0 to an. The whole composed of these terms is a quantity, and has a divisibility measured by the number of terms, provided their number is finite. If the series is such that the distances of consecutive terms are all equal, then, if there are n − 1 terms between a0 and an, the measure of the distance is proportional to n. Thus, if we include in the stretch one of the end terms, but not the other, the measures of the stretch and the distance are proportional, and equal stretches correspond to equal distances. Thus the number of terms in the stretch measures both the distance of the end terms and the amount of divisibility of the whole stretch. When the stretch contains an infinite number of terms, we estimate equal stretches as explained above. It then becomes an axiom, which may or may not hold in a given case, that equal stretches correspond to equal distances. In this case, coordinates measure two entirely distinct magnitudes, which, owing to their common measure, are perpetually confounded.


Now Russell will discuss some problems this might have in geometry. [See p.182 for details.]


So we examined two classes of magnitudes, divisibilities and distances, and we have seen how both can be measured. These two classes cover extensive magnitudes. [Russell then clarifies that although we normally think that all extensive magnitudes are divisible, for him not all are divisible, as for example distances. However, he explained the way the can be numerically measurable as though having parts. See p.182.]

All other quantities are intensive, and they are not numerically measurable. However, even though they are not numerically measurable, this does not mean we cannot for example judge one to be equal to another. In fact,

Quantities not susceptible to numerical measurement can thus be arranged in a scale of greater and smaller magnitudes, and this is the only strictly quantitative achievement of even numerical measurement. We can know that one magnitude is greater than another, and that a third is intermediate between them; also, since the differences of magnitudes are always magnitudes, there is always (theoretically, at least) an answer to the question whether the difference of one pair of magnitudes is greater than, less than, or the same as the difference of another pair of the same kind. And such propositions, though to the mathematician they may appear approximate, are just as precise and definite as the propositions of Arithmetic. Without numerical measurement, therefore, the quantitative relations of magnitudes have all the definiteness of which they are capable—nothing is added, from the theoretical standpoint, by the assignment of correlated numbers.

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

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