by Corry Shores
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Bertrand Russell
Principles of Mathematics
Part 5: Infinity and Continuity
Ch.41: Philosophical Arguments Concerning the Infinitesimal
Brief Summary:
The concept of the infinitesimal as it is used in calculus involves the idea that there is a sequence of consecutive infinitely small values. But were this to be true, then there would be a one-one correspondence between the terms of one series and those of the other to which the first is being differentially related, resulting always in the ratio 1/1. However, calculus finds these differential ratios to have many values other than 1/1. So the infinitesimal leads to contradictions and must not be used in mathematics and presumably for that reason not in philosophy either.
Summary
§315
Previously Russell had argued against using the concept of the infinitesimal to account for continuity. Now he will address philosophical arguments which want to defend the infinitesimal. For this he will examine Cohen’s Princip der Infinitesimalmethode und seine Geschichte.
§316
The differential in calculus no longer needs the concept of the infinitesimal.
In the above exposition, the differential appeared as a philosophically unimportant application of the doctrine of limits. Indeed, but for its traditional importance, it would scarcely have deserved even mention. And we saw that its definition nowhere involves the infinitesimal. The dx and dy of a differential are nothing in themselves, and dy/dx is not a fraction. Hence, in | modern works on the Calculus, the notation f' (x) has replaced dy/dx, since the latter form suggests erroneous notions
[344]
But Cohen treats “the dx and the dy treated as separate entities, as real infinitesimals, as the intensively real elements of which the continuum is composed (pp. 14, 28, 144, 147).” [344] Because Cohen does not defend this concept of the infinitesimal, it seems to not be in question. “This view is certainly assumed as self-evident by most philosophers who discuss the Calculus. Let us see for ourselves what kind of grounds can be urged in its favour.” [344]
§317
Although Cohen may have understood the infinitesimals in terms of space and time, Russell will concern himself only with “such arguments as can be derived from purely numerical instances.” [344]
§318
Cohen rejects “the view that the infinitesimal calculus can be independently derived by mathematics from the method of limits.” It seems his reasoning is that (a) the method of limits presupposes a conception of equality [which is problematic for some reason, perhaps because it presupposes the idea of magnitude, but that is the second problem], and (b) the method of limits presupposes the concept of magnitude, but the concept of magnitude presupposes the concept of limit. [It may not be necessary now to fully understand how Cohen arrive at these conclusions. Russell will just show why they do not hold in mathematics.]
This method, he says (p. 1), “consists in the notion that the elementary conception of equality must be completed by the exact notion of the limit. Thus in the first place the conception of equality is presupposed. . . . Again, in the second place, the method of | limits presupposes the conception of magnitude. . . . But in the presupposed conception of magnitude the limiting magnitude is at the same time presupposed. The equality which is defined in the elementary doctrine of magnitude pays no attention to these limiting magnitudes. For it, magnitudes count as equal if and although their difference consists in a limiting magnitude. Hence the elementary conception of equality must be—this is the notion of the method of limits—not so much completed as corrected by the exact conception of the limit. Equality is to be regarded as an earlier stage of the limiting relation.”
[Russell 344-345, quoting Cohen p.1]
§319
But, Russell notes, “equality has no relevance to limits”. [Russell’s explanation for this can be found pp.345-346. He notes that the simplest concept of limit is ω, the limit of the ordinal numbers, but it does not involve the concept of equality (perhaps because it does not equal the largest of the ordinals; it is a figure above it). He gives the example of a diminishing series tending toward a value, which might seem like the sum equals the limit value, but in fact we do not need to think of it that way. See the noted pages for details.]
And Russell has already explained how magnitude is not involved in the concept of limits [limits are understood in terms of numerical series and not as series of diminishing magnitudes. See p.346 for details].
§320
[Russell then addresses the argument that magnitude presupposes the concept of limits, but limits presuppose the concept of magnitude. Yet, limits do not require the concept of magnitude. See p.346 for details.]
§321
The biggest mistake Cohen makes is that he thinks limits introduce a new meaning of equality. For magnitudes, there is only one meaning, and it does not involve the notion of approximation [no matter how close]. Cohen thinks that numbers do not have equality but only identity, which is (misleadingly) expressed using the equals sign. [Perhaps magnitudes would be two things that have identical values, and thus can be equal like two weights on a scale. But if two numbers are equal, that means they are the same number. There is not two cases of 2. There is only 2, but 2 can be expressed many ways, like 8/4. What this has to do with calculus is not so clear. But Russell goes on to say it seems that for Cohen, the infinitesimal magnitude added to a value y is equal to y because it is so close an approximation. Russell then reminds us that there is no such thing as dx and dy in calculus. Please read the text to get a more detailed understanding of Russell’s argument, pp.346-347.]
I imagine that what Cohen means may be expressed as follows. In forming a differential coefficient, we consider two numbers x and x + dx, and two others y and y + dy. In elementary Arithmetic, x and x + dx would count as equal, but not in the Calculus. There are, in fact, two ways of defining equality. Two terms may be said to be equal when their ratio is unity, or when their difference is zero. But when we allow real infinitesimals dx, x and x + dx will have the ratio unity, but will not have zero for their difference, since dx is different from absolute zero. This view, which I suggest as equivalent to Cohen’s, depends upon a misunderstanding of limits and the Calculus. There are in the Calculus no such magnitudes as dx and dy. There are finite differences Δx and Δy, but no view, however elementary, will make x equal to x + Δx. There are ratios of finite differences, Δy/Δx, and in cases where the derivative of y exists, there is one real number to which Δy/Δx can be made to approach as near as we like by diminishing Δx and Δy. This single real number we choose to denote by dy/dx; but it is not a fraction, and dx and dy are nothing but typographical parts of one symbol. There is no correction whatever of the notion of equality by the doctrine of limits; the only new element introduced is the consideration of infinite classes of terms chosen out of a series.
[347]
§322
[In the following Russell notes another of Cohen’s claims, but that claim is not explained. Perhaps it is saying that because dx and dy are infinitesimal, they do not extend in space, and thus they are inextensive. The extensive can be contrasted with the intensive. In Kant, the extensive magnitude is divisible into metrical parts, but the intensive is not. In ch.21, Russell discusses magnitudes and their measure, and he says that extensive magnitudes are numerically measurable but intensive magnitudes are not; they only admit of more or less. Also in that chapter Russell notes how for Kant, intensive magnitudes are realities that can be more or less in magnitude, like more or less bright. Let’s first look at Russell’s passage in this chapter.]
As regards the nature of the infinitesimal, we are told (p. 15) that the differential, or the inextensive, is to be identified with the intensive, and the differential is regarded as the embodiment of Kant’s category of reality.
[347]
[The differential ratio of dx to dy is inextensive for Cohen, because dx and dy are infinitesimal and thus do not extend in space. Being inextensive does not necessarily mean intensive yet. dx and dy are inextensive, as is their ratio. However, their ratio forms a quantity that is intensive, meaning that (using Russell’s definition) they are measurable in terms of being more or less but not numerically by counting parts. Kant’s concept of reality regards it as something that admits of degrees, and thus somehow the differential is the embodiment of Kant’s category of reality. I have not read Cohen yet, so I do not know his reasoning. But to give our own, we might say that Kant (especially according to a Deleuzean interpretation) regards our experiences as being experiences of reality, but these experiences are the experiences of the variations from moment to moment. We only experience degrees of difference and thus only intensities. It is only by means of recollection that we experience by means of synthesis stretches of time and the extensity of spatial objects. Thus reality in its most basic form are variations, which are understood as correlated infinitesimal differences, tiny changes over tiny moments.] [So Cohen understands dx and dy as being terms in a series, or as differences between consecutive terms. Russell will explain why they are neither, and instead they only represent stretches (series of intermediate terms) containing an infinity of terms, or “distances corresponding to such stretches”. So for Russell, dx and dy are not infinitesimals but rather merely tiny finite values that are infinitely divisible like any other finite value. Then Russell distinguishes series of numbers from series of measurable stretches or distances. Space and time, for example, are this second kind that are made of stretches or distances. But, dx and dy are not consecutive terms, because our series is compact (between any two there is another, and as we saw, there cannot be consecutive terms in a compact series, because there is no ‘next’ term; for there always is a ‘more next’ term, then another, and another, without end.) After considering some complications, Russell finds a possible way to tentatively conceive of dx and dy as being the distances of consecutive points. Russell will show why this is still absurd. He thinks it leads to the conclusion that all differential relations dx/dy would have to have the same value, either positive or negative 1. He could perhaps be saying the following. Suppose like Cohen we think that although a distance is infinitely divisible, it ultimately divides into smallest parts. These parts do not have a finite value. However, there are infinitely many such parts in a finite distance. dx and dy are thought of as such infinitesimal distances. Thus any finite distance along the x axis (or x series) is made up of an infinity of dx’s, which measure the distances between the consecutive points, and likewise for y. But the points between which dx and dy stand correlate in a one-one fashion, since these points are real values of the number line. This would seem to imply that dx and dy are always constant values, and thus dx/dy is always positive or negative one. This is because no other points intervene between them. Thus regardless of the supposed relative values of any dx/dy pairing, each themselves cannot have a value any different than the equally spaced points on which they are found, and thus must always be equal. Most likely Russell is making a different argument, which I cannot discern, so it is important to read the ‘mathematical arguments’ on page 348 for a more certain interpretation. Russell then puts these mathematical arguments aside, and says that since dx/dy have a numerical ratio, they must be numerically measurable, even though they are intensive magnitudes. (And recall that for Russell intensive magnitudes are not numerically measurable.) But Russell does not see how we might numerically measure them. So first we suppose that x and y are numbers. Then we suppose that x and x + dx are consecutive. Now, how are we to regard y + dy? We have four options. They either (a) are consecutive, (b) are identical, (c) have a finite number of terms between them, or (d) have an infinite number of terms between them. Cases c and d I think would be cases where dy is a stretch, that is, a series of terms between two end terms. Russell says that if it is a stretch, then dy/dx will always be either zero, integral, or infinite. I do not know why. Let’s suppose that these results follow from b, c, and d, as a possible way to start our explanation. If y and y + dy are identical, that means dy is 0, and making dy / dx be 0 over some other figure and thus 0. If there is a finite number of terms between y and y + dy, then that means dy/dx would have some integer value, and maybe that is what Russell means by ‘integral’. However, I do not know what he means here; if it has something to do with integral calculus I cannot discern it; and that dy/dx would have an integer value does not to me seem problematic, so I cannot interpret that. If there are an infinity of terms between y and y + dy, that means we have infinity over one (or some finite value) and thus dy/dx is infinity. In all three cases this is absurd (although the absurdity of the second case I cannot understand. It is also possible that the results “zero, or integral, or infinite” are not results of cases b, c, and d respectively. But if that were so, I understand the situation even less.) Russell then goes on to say that even if y is not constant, dy/dx must be positive or negative 1. It seems he proves this by considering the two ways we can conceive of dy and dx, that is, as being either stretches or distances. If they were stretches, that means no matter the size of y, it will have the same size of infinity of dy-components as x has of dx-components, and they will correlate always in a one-one fashion. Since for stretches the number of terms determines the magnitude, and because the number of terms is equal and correspondent in a one-one fashion in both x and y, then dy/dx will always be 1/1. He then says that if y is not constant (sticking still with dy and dx being stretches), dy/dx will still have to be positive or negative one. He has us consider the function y = x2. And here x and y are positive real numbers. But again, if they are stretches, the same one-one correspondence will apply and thus the same problem results. Now he has us consider if we measure by distances and not stretches. He seems to be using the same reasoning. He says dy and dx are always the distance from one number to the next along the distances y and x. He then has us consider a function for which dy/dx = 2 for x = 1 and y = 1. On the one hand, the function tells us dy/dx should be two. But since there is this one-one correspondence and since x and y are equal, it would also have to be 1/1, which is absurd. This means that no matter how we conceive of consecutive values, it will lead to absurdities when applied in calculus. I would like to point out possible reasons we might not have to come to Russell’s conclusion. Russell’s argument begins by conceptualizing the infinitesimals as being the consecutive intervals between the real number values taken to be infinitely close. So under this conception, we would think that there are an infinity of infinitely small increments making up the value x and the value y, and each such increment corresponds to a real number value, which is like the total of all the infinitely small increments leading up to it. Let’s think about a geometrical interpretation, for example the curve described by y = x2. When x is 2, y is 4, and the rise/run of the tangent at that place along the curve is 4/1. Russell’s problem is that this implies that as we move to the next real number value, x + dx, we would skip over 3 points of y on the y axis, since we are jumping by 4. (In fact, since the variations are exponential, when we get to the next one, we might have jumped over even more than 4.) But that does not mean the next four real number values do not have a corresponding y value. Consider: the function says for example we are going from (2,4) to (3,9) and so on [for (x,y) coordination, with the difference between values assumed to be infinitely small]. Thus we see that we are always skipping y values when we are keeping the x values constant. However, what happens when we go up the scale of y values? What if we went from (2,4) to (x,5)? What is the value of x? Would it not also have a value, which would be between the afore-determined x-values? And as we go up the scale of y, the x values will grow relatively slower. There seems then to be an impossible contradiction if we assume consecutive infinitesimal values making up the spaces between points on the x and y axes. But perhaps there is a flaw in how Russell sets up the problem. He equates real numbers along x and y with infinitesimal increments along x and y (or along x/y). When two successive numbers are real, then there is always another between them. So there are not consecutive real numbers whose values can be assigned. The infinitesimal interval would be smaller than the interval between any givable pair of reals. So maybe we cannot, like Russell does, equate the infinitesimal increments with the real’s increments. So if we go up an infinitesimal increment along the x axis, that does not mean that it must correspond to an increment along the y which is equal in magnitude. How we are to better conceptualize such successions of infinitesimals I am not sure, but it does seem to be fairly certain that they are not equal to the succession of real numbers and thus Russell’s criticism might not hold. I have placed the entirety of this paragraph below, because it deserves a better interpretation than I can give it.]
As regards the nature of the infinitesimal, we are told (p. 15) that the differential, or the inextensive, is to be identified with the intensive, and the differential is regarded as the embodiment of Kant’s category of reality. This view (in so far as it is independent of Kant) is quoted with approval from Leibniz; but to me, I must confess, it seems destitute of all justification. It is to be observed that dx and dy, if we allow that they are entities at all, are not to be identified with single terms of our series, nor yet with differences between consecutive terms, but must be always stretches containing an infinite number of terms, or distances corresponding to such stretches. Here a distinction must be made between series of numbers and series in which we have only measurable distances or stretches. The latter is the case of space and time. Here dx and dy are not points or instants, which alone would be truly inextensive; they are primarily numbers, and hence must correspond to infinitesimal stretches or distances—for it would be preposterous to assign a numerical ratio to two points, or—as in the case of | velocity—to a point and an instant. But dx and dy cannot represent the distances of consecutive points, nor yet the stretch formed by two consecutive points. Against this we have, in the first place, the general ground that our series must be regarded as compact, which precludes the idea of consecutive terms. To evade this, if we are dealing with a series in which there are only stretches, not distances, would be impossible: for to say that there are always an infinite number of intermediate points except when the stretch consists of a finite number of terms would be a mere tautology. But when there is distance, it might be said that the distance of two terms may be finite or infinitesimal, and that, as regards infinitesimal distances, the stretch is not compact, but consists of a finite number of terms. This being allowed for the moment, our dx and dy may be made to be the distances of consecutive points, or else the stretches composed of consecutive points. But now the distance of consecutive points, supposing for example that both are on one straight line, would seem to be a constant, which would give dy/dx = ±1. We cannot suppose, in cases where x and y are both continuous, and the function y is one-valued, as the Calculus requires, that x and x + dx are consecutive, but not y and y + dy; for every value of y will be correlated with one and only one value of x, and vice versâ; thus y cannot skip any supposed intermediate values between y and y + dy. Hence, given the values of x and y, even supposing the distances of consecutive terms to differ from place to place, the value of dy/dx will be determinate; and any other function y' which, for some value of x, is equal to y, will, for that value, have an equal derivative, which is an absurd conclusion. And leaving these mathematical arguments, it is evident, from the fact that dy and dx are to have a numerical ratio, that if they be intensive magnitudes, as is suggested, they must be numerically measurable ones: but how this measurement is effected, it is certainly not easy to see. This point may be made clearer by confining ourselves to the fundamental case in which both x and y are numbers. If we regard x and x + dx as consecutive, we must suppose either that y and y + dy are consecutive, or that they are identical, or that there are a finite number of terms between them, or that there are an infinite number. If we take stretches to measure dx and dy, it will follow that dy/dx must be always zero, or integral, or infinite, which is absurd. It will even follow that, if y is not constant, dy/dx must be ±1. Take for example y = x2, where x and y are positive real numbers. As x passes from one number to the next, y must do so likewise; for to every value of y corresponds one of x, and y grows as x grows. Hence if y skipped the number next to any one of its values, it could never come back to pick it up; but we know that every real number is among the values of y. Hence y and y + dy must be consecutive, and dy/dx = 1. If we measure by distances, not stretches, the distance dy must be fixed when y is given, and the distance dx when x is given. Now if x = 1, y = 1, dy/dx = 2; but, since x and y are the same number, dx and dy must be equal, since | each is the distance to the next number: therefore dy/dx = 1, which is absurd. Similarly, if we take for y a decreasing function, we shall find dy/dx = − 1. Hence the admission of consecutive numbers is fatal to the Calculus; and since the Calculus must be maintained, the Calculus is fatal to consecutive numbers.
[347-349]
§323
[First Russell notes that perhaps some of the problems that have arisen result from the conceptualization of going from one term to the next being a matter of physical motion (like a point moving along the x-axis) when it is really more of a numerical progress without real temporal and spatial properties. He then goes on to challenge Cohen’s idea that inextensive infinitesimals are equatable with intensive magnitudes. To explain his reasoning for this, let’s consider first an example of an intensive magnitude, let’s say the brightness of a light. We would never say that it is smaller than some extensive magnitude. It is just a different kind of magnitude. But the infinitesimal is smaller than any extensive magnitude, and for that reason should not be considered intensive.]
§324
[In this last paragraph, Russell sums up his argument so far against infinitesimals: they are (1) unnecessary (because they are not needed for calculus), (2) erroneous (because he showed in a prior chapter that they are obtained through an “illegitimate use of mathematical inductions) and (3) self-contradictory (because they lead to such contradictions as the one mentioned above regarding their consecutivity).]
We cannot, then, agree with the following summary of Cohen’s theory (p. 28): “That I may be able to posit an element in and for itself, is the desideratum, to which corresponds the instrument of thought reality. This instrument of thought must first be set up, in order to be able to enter into that combination with intuition, with the consciousness of being given, which is completed in the principle of intensive magnitude. This presupposition of intensive reality is latent in all principles, and must therefore be made independent. This presupposition is the meaning of reality and the secret of the concept of the differential.” What we can agree to, and what, I believe, confusedly underlies the above statement, is, that every continuum must consist of elements or terms; but these, as we have just seen, will not fulfil the function of the dx and dy which occur in old-fashioned accounts of the Calculus. Nor can we agree that “this finite” (i.e. that which is the object of physical science) “can be thought as a sum of those infinitesimal intensive realities, as a definite integral” (p. 144). The | definite integral is not a sum of elements of a continuum, although there are such elements: for example, the length of a curve, as obtained by integration, is not the sum of its points, but strictly and only the limit of the lengths of inscribed polygons. The only sense which can be given to the sum of the points of the curve is the logical class to which they all belong, i.e. the curve itself, not its length. All lengths are magnitudes of divisibility of stretches, and all stretches consist of an infinite number of points; and any two terminated stretches have a finite ratio to each other. 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.
[350]
Source:
Bertrand Russell. Principles of Mathematics. London/New York: Routledge, 2010 [1st published 1903].
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