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*Principles of Mathematics*

Part 5: Infinity and Continuity

Ch.40: The Infinitesimal and the Improper Infinite

Brief Summary:

Until recently (ca. 1900), continuity was understood by means of the concept of the infinitesimal. But now that concept has been abandoned. [336]

The infinitesimal has been given certain senses, but none have been mathematically precise. It is for example the distance between a point and its immediate neighbor. But we now know that there is no such thing.

The infinitesimal has, in general, been very vaguely defined. It has been regarded as a number or magnitude which, though not zero, is less than any finite number or magnitude. It has been the dx or dy of the Calculus, the time during which a ball thrown vertically upwards is at rest at the highest point of its course, the distance between a point on a line and the next point, etc., etc. But none of these notions are at all precise. The dx and dy, as we saw in the last chapter, are nothing at all: dy/dx is the limit of a fraction whose numerator and denominator are finite, but is not itself a fraction at all. The time during which a ball is at rest at its highest point is a very complex notion, involving the whole philosophic theory of motion; in Part VII we shall find, when this theory has been developed, that there is no such time. The distance between consecutive points presupposes that there are consecutive points—a view which there is every reason to deny. And so with most instances—they afford no precise definition of what is meant by the infinitesimal.

[336]

§310

[We should first examine the axiom of Archimedes. We want to know if two values are finite in relation to one another or infinite in relation to one another. Consider values 4 and 6. We can multiply 4 by 2 and get 8, which is larger than 6. This means they are finite in relation to one another, or their difference in value is finite. Now consider 4 and the cardinal value for the natural numbers. Or let’s just say, consider 4 and infinity. There is no finite number that we can multiply 4 by in order to obtain a number greater than infinity. That means they are infinite in relation to one another, or their difference is infinite. So the first example illustrates relative finitude. Absolute finitude would require some anchoring points you say, 0 and 1, and as well a principle of composing finite numbers, namely mathematical induction, the successor function. As we can see, the notion of relative finitude applies to any kind of magnitude, but absolute infinity has more limited application to numbers, classes and divisibilities. And also note that an inch and a foot both are magnitudes consisting of an infinity of terms (leading up to their total value, all the sizes smaller than an inch that are implicitly contained within it). So both an inch and a foot are absolute infinities. However, they are finite in relation to one another and are thus relative finitudes. So “any two numbers, classes, or divisibilities, which are both absolutely finite are also relatively finite; but the converse does not hold”.]

There is, so far as I know, only one precise definition, which renders the infinitesimal a purely relative notion, correlative to something arbitrarily assumed to be finite. When, instead, we regard what had been taken to be infinitesimal as finite, the correlative notion is what Cantor calls the improper infinite (Uneigentlich-Unendliches). The definition of the relation in question is obtained by denying the axiom of Archimedes, just as the transfinite was obtained by denying mathematical induction. If P, Q be any two numbers, or any two measurable magnitudes, they are said to be finite with respect to each other when, if P be the lesser, there exists a finite integer n such that nP is greater than Q. The existence of such an integer constitutes the axiom of Archimedes and the definition of relative finitude. It will be observed that it presupposes the definition of absolute finitude among numbers—a definition which, as we have seen, depends upon two points, (1) the connection of 1 with the logical notion of simplicity, or of 0 with the logical notion of the null-class; (2) the principle of mathematical induction. The notion of relative finitude is plainly distinct from that of absolute finitude. The latter applies only to numbers, classes and divisibilities, whereas the former applies to any kind of measurable magnitude. Any two numbers, classes, or divisibilities, which are both absolutely finite are also relatively finite; but the converse does not hold. For example, ω and ω.2, an inch and a foot, a day and a year, are relatively finite pairs, though all three consist of terms which are absolutely infinite.

[337]

[Russell will now definite the infinitesimal and improper infinite. Consider 2 values. If no matter what finite value we multiply one by that it can in no case be greater than the other, then this term is infinitesimal or improperly infinite. This can only apply to numbers and not magnitudes.]

The definition of the infinitesimal and the improper infinite is then as follows. If P, Q be two numbers, or two measurable magnitudes of the same kind, and if, n being any finite integer whatever, nP is always less than Q, then P is infinitesimal with respect to Q, and Q is infinite with respect to P. With regard to numbers, these relative terms are not required; for if, in the case supposed, P is absolutely finite, then Q is absolutely infinite; while if it were possible for Q to be absolutely finite, P would be absolutely infinitesimal—a case, however, which we shall see reason to regard as impossible. Hence I shall assume in future that P and Q are not numbers, but are magnitudes of a kind of which some, at least, are numerically measurable. It should be observed that, as regards magnitudes, the axiom of Archimedes is the only way of defining, not only the infinitesimal, but the infinite also. Of a magnitude not numerically measurable, there is nothing to be said except that it is greater than some of its kind, and less than others; but from such propositions infinity cannot be obtained. Even if there be a magnitude greater than all others of its kind, there is no reason for regarding it as infinite. Finitude and infinity are essentially numerical notions, and it is only by relation to numbers that these terms can be applied to other entities.

[337]

§311

[Russell will now consider instances of infinitesimal values. We first consider divisible magnitudes. If we compare something with a finite number of parts to one with an infinite number, than the first is infinitesimal in relation to it. But we cannot compare such magnitudes on the basis of placing into a ratio the cardinal numbers of their parts. Russell gives two reasons. The first is that we cannot place transfinite values into ratios (his explanation begins with saying we cannot place two transfinite cardinals into ratios. His example is of a finite and a transfinite. So for some reason it still applies in this other case). He second reason is equally unclear, but it seems he is saying that in order to make our original comparison, the divisibilities of each magnitude must be equal, but that is not the case for the transfinite value for some reason. Here is the text:]

The next question to be discussed is: What instances of infinitesimals are to be found? Although there are far fewer instances than was formerly | supposed, there are yet some that are important. To begin with, if we have been right in regarding divisibility as a magnitude, it is plain that the divisibility of any whole containing a finite number of simple parts is infinitesimal as compared with one containing an infinite number. The number of parts being taken as the measure, every infinite whole will be greater than n times every finite whole, whatever finite number n may be. This is therefore a perfectly clear instance. But it must not be supposed that the ratio of the divisibilities of two wholes, of which one at least is transfinite, can be measured by the ratio of the cardinal numbers of their simple parts. There are two reasons why this cannot be done. The first is, that two transfinite cardinals do not have any relation strictly analogous to ratio; indeed, the definition of ratio is effected by means of mathematical induction. The relation of two transfinite cardinals α, γ expressed by the equation αβ = γ bears a certain resemblance to integral ratios, and αβ =γδ may be used to define other ratios. But ratios so defined are not very similar to finite ratios. The other reason why infinite divisibilities must not be measured by transfinite numbers is, that the whole must always have more divisibility than the part (provided the remaining part is not relatively infinitesimal), though it may have the same transfinite number. In short, divisibilities, like ordinals, are equal, so long as the wholes are finite, when and only when the cardinal numbers of the wholes are the same; but the notion of magnitude of divisibility is distinct from that of cardinal number, and separates itself visibly as soon as we come to infinite wholes.

[337-338]

We can even have examples where one thing is infinitely less divisible than another, as for example a line compared to a square. [This is an example of an infinitesimal. But it seems Russell is saying that they are just relative infinitesimals and not the kind we are more concerned with, like in the infinitesimal calculus.]

Two infinite wholes may be such that one is infinitely less divisible than the other. Consider, for example, the length of a finite straight line and the area of the square upon that straight line; or the length of a finite straight line and the length of the whole straight line of which it forms part (except in finite spaces); or an area and a volume; or the rational numbers and the real numbers; or the collection of points on a finite part of a line obtainable by von Staudt’s quadrilateral construction, and the total collection of points on the said finite part.* All these are magnitudes of one and the same kind, namely divisibilities, and all are infinite divisibilities; but they are of many different orders. The points on a limited portion of a line obtainable by the quadrilateral construction form a collection which is infinitesimal with respect to the said portion; this portion is ordinally infinitesimal† with respect to any bounded area; any bounded area is ordinally infinitesimal with respect to any bounded volume; and any bounded volume (except in finite spaces) is ordinally infinitesimal with respect to all space. In all these cases, the word infinitesimal is used strictly according to the above definition, obtained from the axiom of Archimedes. What makes these various | infinitesimals somewhat unimportant, from a mathematical standpoint, is, that measurement essentially depends upon the axiom of Archimedes, and cannot, in general, be extended by means of transfinite numbers, for the reasons which have just been explained. Hence two divisibilities, of which one is infinitesimal with respect to the other, are regarded usually as different kinds of magnitude; and to regard them as of the same kind gives no advantage save philosophic correctness. All of them, however, are strictly instances of infinitesimals, and the series of them well illustrates the relativity of the term infinitesimal.

[338-339]

[Russell examines another example of comparing magnitudes divided infinitely. It is not clear to me, but it seems to be saying that if the divisions of a magnitude get smaller than the finite, then if we add up all their values, it will be 0. But please read it for yourself to decide what it means.]

An interesting method of comparing certain magnitudes, analogous to the divisibilities of any infinite collections of points, with those of continuous stretches is given by Stolz,* and a very similar but more general method is given by Cantor.† These methods are too mathematical to be fully explained here, but the gist of Stolz’s method may be briefly explained. Let a collection of points x' be contained in some finite interval a to b. Divide the interval into any number n of parts, and divide each of these parts again into any number of parts, and so on; and let the successive divisions be so effected that all parts become in time less than any assigned number δ. At each stage, add together all the parts that contain points of x' . At the mth stage, let the resulting sum be Sm. Then subsequent divisions may diminish this sum, but cannot increase it. Hence as the number of divisions increases, Sm must approach a limit L. If x' is compact throughout the interval, we shall have L = b − a; if any finite derivative of x' vanishes, L = 0. L obviously bears an analogy to a definite integral; but no conditions are required for the existence of L. But L cannot be identified with the divisibility; for some compact series, e.g. that of rationals, are less divisible than others, e.g. the continuum, but give the same value of L.

[339]

§312

[Normally we think of the infinitesimal as composing a dense or ‘compact’ series. For, if all its parts were finite and there are infinitely many, than the whole segment would be infinite. If the parts were 0, then it would have 0 value. But if there were infinitely many infinitely small part, then those infinitive values would cancel one another generating a finite value. Russell will show that either it is impossible for the parts to be infinitesimal or at least that if they were, they would be indefinable. First he establishes that any segment is infinitely divisible, because between any two values is another. Next, he explains that segments can be added by placing one at the end of the other, which increases the total magnitude. If the added segments are equal, the new total will be double. Segments without terminal endings included in them (where they tend toward limits without attaining them), we can add them by adding such terminal segments. So we can define any finite multiple of segments (by adding them). For some reason, it seems we will draw these conclusions: if a smaller segment obeys the axiom of Archimedes with respect to the larger (if no matter how many times we multiply it, it will not be greater than the larger), then the larger will contain all the terms coming after the smaller. However, if the smaller is infinitesimal with respect to larger ones, then the larger one will not contain points of the first segment. (This is too unclear for me to understand). (It seems now we are working with the idea that an infinite segment cannot be increased by doubling it. Only terminating segments can.) Thus the larger segment is not terminating. On account of this, for some reason Peano concludes that the larger segment cannot be an element in finite magnitudes. Russell draws a stronger conclusion. an infinitesimal cannot have determinate bounds. So it cannot be added so to produce larger segments. Consult the original text:]

The case in which infinitesimals were formerly supposed to be peculiarly evident is that of compact series. In this case, however, it is possible to prove that there can be no infinitesimal segments,‡ provided numerical measurement be possible at all—and if it be not possible, the infinitesimal, as we have seen, is not definable. In the first place, it is evident that the segment contained between two different terms is always infinitely divisible; for since there is a term c between any two a and b, there is another d between a and c, and so on. Thus no terminated segment can contain a finite number of terms. But segments defined by a class of terms may (as we saw in Chapter 34) have no limiting term. In this case, however, provided the segment does not consist of a single term a, it will contain some other term b, and therefore an infinite number of terms. Thus all segments are infinitely divisible. The next | point is to define multiples of segments. Two terminated segments can be added by placing a segment equal to the one at the end of the other to form a new segment; and if the two were equal, the new one is said to be double of each of them. But if the two segments are not terminated, this process cannot be employed. Their sum, in this case, is defined by Professor Peano as the logical sum of all the segments obtained by adding two terminated segments contained respectively in the two segments to be added.* Having defined this sum, we can define any finite multiple of a segment. Hence we can define the class of terms contained in some finite multiple of our segment, i.e. the logical sum of all its finite multiples. If, with respect to all greater segments, our segment obeys the axiom of Archimedes, then this new class will contain all terms that come after the origin of our segment. But if our segment be infinitesimal with respect to any other segment, then the class in question will fail to contain some points of this other segment. In this case, it is shown that all transfinite multiples of our segment are equal to each other. Hence it follows that the class formed by the logical sum of all finite multiples of our segment, which may be called the infinite multiple of our segment, must be a non-terminated segment, for a terminated segment is always increased by being doubled. “Each of these results”, so Professor Peano concludes, “is in contradiction with the usual notion of a segment. And from the fact that the infinitesimal segment cannot be rendered finite by means of any actually infinite multiplication, I conclude, with Cantor, that it cannot be an element in finite magnitudes” (p. 62). But I think an even stronger conclusion is warranted. For we have seen that, in compact series, there is, corresponding to every segment, a segment of segments, and that this is always terminated by its defining segment; further that the numerical measurement of segments of segments is exactly the same as that of simple segments; whence, by applying the above result to segments of segments, we obtain a definite contradiction, since none of them can be unterminated, and an infinitesimal one cannot be terminated.

[339-340]

[Next Russell will argue that rational and real numbers cannot be made of infinitesimals. He seems to be saying that the real numbers are made of rational numbers. The real numbers are a class of real numbers. So any member of them will as well contain rational numbers, no matter how small. But an infinitesimally small term does not contain with in any rational numbers, because it is too small. Hence the real numbers cannot be made of infinitesimals. He might very well be saying something else, so please consider the original:]

In the case of the rational or the real numbers, the complete knowledge which we possess concerning them renders the non-existence of infinitesimals demonstrable. A rational number is the ratio of two finite integers, and any such ratio is finite. A real number other than zero is a segment of the series of rationals; hence if x be a real number other than zero, there is a class u, not null, of rationals such that, if y is a u, and z is less than y, z is an x, i.e. belongs to the segment which is x. Hence every real number other than zero is a class containing rationals, and all rationals are finite; consequently every real number is finite. Consequently if it were possible, in any sense, to speak of infinitesimal numbers, it would have to be in some radically new sense.

[340]

§313

[Russell now examines an interesting question regarding orders of infinity and infinitesimality of functions. Russell does not draw any conclusions (although he seems to want at the end to say that this material supports the notion that infinitesimals are mathematical fictions), so I will just place the very technical material below:]

I come now to a very difficult question, on which I would gladly say nothing—I mean, the question of the orders of infinity and infinitesimality of functions. On this question the greatest authorities are divided: Du Bois Reymond, Stolz, and many others, maintaining that these form a special class of magnitudes, in which actual infinitesimals occur, while Cantor holds strongly that the whole theory is erroneous. To put the matter as simply as possible, consider a function f(x) whose limit, as x approaches zero, is zero. It may happen that, for some finite real number α, the ratio f(x)/x^{α}has a finite limit as x approaches zero. There can be only one such number, but there may be none. Then α, if there is such a number, may be called the order to which f(x) becomes infinitesimal, or the order of smallness of f(x) as x approaches zero. But for some functions, e.g. 1/log x, there is no such number α. If α be any finite real number, the limit of 1/x^{α}logx, as x approaches zero, is infinite. That is, when x is sufficiently small, 1/x^{α}log x is very large, and may be made larger than any assigned number by making x sufficiently small—and this whatever finite number α may be. Hence, to express the order of smallness of 1/log x, it is necessary to invent a new infinitesimal number, which may be denoted by 1/g. Similarly we shall need infinitely great numbers to express the order of smallness of (say) e^{−1/x}as x approaches zero. And there is no end to the succession of these orders of smallness: that of 1/log (log x), for example, is infinitely smaller than that of 1/log x, and so on. Thus we have a whole hierarchy of magnitudes, of which all in any one class are infinitesimal with respect to all in any higher class, and of which one class only is formed of all the finite real numbers.In this development, Cantor finds a vicious circle; and though the question is difficult, it would seem that Cantor is in the right. He objects (loc. cit.) that such magnitudes cannot be introduced unless we have reason to think that there are such magnitudes. The point is similar to that concerning limits; and Cantor maintains that, in the present case, definite contradictions may be proved concerning the supposed infinitesimals. If there were infinitesimal numbers j, then even for them we should haveLim_{x = 0}1/ (log x. x^{j}) = 0since x^{j}must ultimately exceed ½. And he shows that even continuous, differentiable and uniformly growing functions may have an entirely ambiguous order of smallness or infinity: that, in fact, for some such functions, this order oscillates between infinite and infinitesimal values, according to the manner in which the limit is approached.Hence we may, I think, conclude that these | infinitesimals are mathematical fictions. And this may be reinforced by the consideration that, if there were infinitesimal numbers, there would be infinitesimal segments of the number-continuum, which we have just seen to be impossible.[341-342]

§314

[Russell now summarizes. He has shown that the infinitesimal can never be anything but a relative term. When it does have an absolute meaning, it is indistinguishable from finitude (perhaps this is from the idea that an infinitely small segment cannot be increased by doubling it, so were a segment made of infinitesimals, they would have to have the properties of finite magnitudes.) There are cases of infinitesimals, for example the side of a square is infinitesimal compared with its area. But mathematicians consider each magnitude as different in kind and thus incomparable. We also saw that compact (dense) series cannot be made of infinitesimals. Thus the infinitesimal has not many important manifestations and it is not important mathematically.]

Thus to sum up what has been said concerning the infinitesimal, we see, to begin with, that it is a relative term, and that, as regards magnitudes other than divisibilities, or divisibilities of wholes which are infinite in the absolute sense, it is not capable of being other than a relative term. But where it has an absolute meaning, there this meaning is indistinguishable from finitude. We saw that the infinitesimal, though completely useless in mathematics, does occur in certain instances—for example, lengths of bounded straight lines are infinitesimal as compared to areas of polygons, and these again as compared to volumes of polyhedra. But such genuine cases of infinitesimals, as we saw, are always regarded by mathematics as magnitudes of another kind, because no numerical comparison is possible, even by means of transfinite numbers, between an area and a length, or a volume and an area. Numerical measurement, in fact, is wholly dependent upon the axiom of Archimedes, and cannot be extended as Cantor has extended numbers. And finally we saw that there are no infinitesimal segments in compact series, and—what is closely connected—that orders of smallness of functions are not to be regarded as genuine infinitesimals.[342]The infinitesimal, therefore—so we may conclude—is a very restricted and mathematically very unimportant conception, of which infinity and continuity are alike independent.

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

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