Introduction to Mathematical Philosophy

to be defined, namely, the property which Cantor calls that of being "closed" (abgeschlossen). This, as we saw, was first defined as consisting in the fact that all the limiting-points of a series belong to it. But this only has any effective significance if our series is given as contained in some other larger series (as is the case, e.g., with a selection of real numbers), and limiting-points are taken in relation to the larger series. Otherwise, if a series is considered simply on its own account, it cannot fail to contain its limiting-points. What Cantor means is not exactly what he says; indeed, on other occasions he says something rather different, which is what he means. What he really means is that every subordinate series which is of the sort that might be expected to have a limit does have a limit within the given series; i.e. every subordinate series which has no maximum has a limit, i.e. every subordinate series has a boundary. But Cantor does not state this for every subordinate series, but only for progressions and regressions. (It is not clear how far he recognises that this is a limitation.) Thus, finally, we find that the definition we want is the following:— A series is said to be "closed" (abgeschlossen) when every progression or regression contained in the series has a limit in the series. We then have the further definition:— A series is "perfect" when it is condensed in itself and closed, i.e. when every term is the limit of a progression or regression, and every progression or regression contained in the series has a limit in the series. In seeking a definition of continuity, what Cantor has in mind is the search for a definition which shall apply to the series of real numbers and to any series similar to that, but to no others. For this purpose we have to add a further property. Among the real numbers some are rational, some are irrational; although the number of irrationals is greater than the number of rationals, yet there are rationals between any two real numbers, however [Pg 103] little the two may differ. The number of rationals, as we saw, is . This gives a further property which suffices to characterise continuity completely, namely, the property of containing a class of members in such a way that some of this class occur between any two terms of our series, however near together. This property, added to perfection, suffices to define a class of series which are all similar and are in fact a serial number. This class Cantor defines as that of continuous series. We may slightly simplify his definition. To begin with, we say: A "median class" of a series is a sub-class of the field such that members of it are to be found between any two terms of the series. Thus the rationals are a median class in the series of real numbers. It is obvious that there cannot be median classes except in compact series. We then find that Cantor's definition is equivalent to the following:— A series is "continuous" when (1) it is Dedekindian, (2) it contains a median class having terms. To avoid confusion, we shall speak of this kind as "Cantorian continuity." It will be seen that it implies Dedekindian continuity, but the converse is not the case. All series having Cantorian continuity are similar, but not all series having Dedekindian continuity. The notions of limit and continuity which we have been defining must not be confounded with the notions of the limit of a function for approaches to a given argument, or the continuity of a function in the neighbourhood of a given argument. These are different notions, very important, but derivative from the above and more complicated. The continuity of motion (if motion is continuous) is an instance of the continuity of a function; on the other hand, the continuity of space and time (if they are continuous) is an instance of the continuity of series, or (to speak more cautiously) of a kind of continuity which can, by sufficient mathematical [Pg 104] manipulation, be reduced to the continuity of series. In view of the fundamental importance of motion in applied mathematics, as well as for other reasons, it will be well to deal briefly with the notions of limits and continuity as applied to functions; but this subject will be best reserved for a separate chapter. The definitions of continuity which we have been considering, namely, those of Dedekind and Cantor, do not correspond very closely to the vague idea which is associated with the word in the mind of the man in the street or the philosopher. They conceive continuity rather as absence of separateness, the sort of general obliteration of distinctions which characterises a thick fog. A fog gives an impression of vastness without definite multiplicity or division. It is this sort of thing that a metaphysician means by "continuity," declaring it, very truly, to be characteristic of his mental life and of that of children and animals. The general idea vaguely indicated by the word "continuity" when so employed, or by the word "flux," is one which is certainly quite different from that which we have been defining. Take, for example, the series of real numbers. Each is what it is, quite definitely and uncompromisingly; it does not pass over by imperceptible degrees into another; it is a hard, separate unit, and its distance from every other unit is finite, though it can be made less than any given finite amount assigned in advance. The question of the relation between the kind of continuity existing among the real numbers and the kind exhibited, e.g. by what we see at a given time, is a difficult and intricate one. It is not to be maintained that the two kinds are simply identical, but it may, I think, be very well maintained that the mathematical conception which we have been considering in this chapter gives the abstract logical scheme to which it must be possible to bring empirical material by suitable manipulation, if that material is to be called "continuous" in any precisely definable sense. It would be quite impossible [Pg 105] to justify this thesis within the limits of the present volume. The reader who is interested may read an attempt to justify it as regards time in particular by the present author in the Monist for 1914-5, as well as in parts of Our Knowledge of the External World. With these indications, we must leave this problem, interesting as it is, in order to return to topics more closely connected with mathematics. [Pg 106]