cificam. Thence comes the horror it has sometimes inspired in certain minds, for instance in Hermite, whose favorite idea was to compare the mathematical to the natural sciences. With most of us these prejudices have been dissipated, but it has come to pass that we have encountered certain paradoxes, certain apparent contradictions that would have delighted Zeno the Eleatic and the school of Megara. And then each must seek the remedy. For my part, I think, and I am not the only one, that the important thing is never to introduce entities not completely definable in a finite number of words. Whatever be the cure adopted, we may promise ourselves the joy of the doctor called in to follow a beautiful pathologic case. THE INVESTIGATION OF THE POSTULATES. On the other hand, efforts have been made to enumerate the axioms and postulates, more or less hidden, which serve as foundation to the different theories of mathematics. Professor Hilbert has obtained the most brilliant results. It seems at first that this domain would be very restricted and there would be nothing more to do when the inventory should be ended, which could not take long. But when we shall have enumerated all, there will be many ways of classifying all; a good librarian always finds something to do, and each new classification will be instructive for the philosopher. Here I end this review which I could not dream of making complete. I think these examples will suffice to show by what mechanism the mathematical sciences have made their progress in the past and in what direction they must advance in the future. PARIS, FRANCE. H. POINCARÉ. TRANSFINITE NUMBERS AND THE PRIN CIPLES OF MATHEMATICS. PART I. One result of Georg Cantor's discovery of the transfinite cardinal and ordinal numbers has been the development of more satisfactory views on the principles of mathematics. To this end, also, the symbolic logic of Peano, Frege, and Russell1 contributed by enabling one, for the first time, to reach precision in such subjects as the relation of logic to mathematics, and the meaning of "definition" and "existence." In this first part, I give an account of these things, and, in the second part, I will review the modifications in logic and in our views of the principles of mathematics which progress in the theory of aggregates has necessitated. I hope to show that, just as we have been forced, especially during the nineteenth century, to a more rigorous foundation of the methods and results of mathematical analysis, so we are forced to logical investigations by that development of mathematics to which I have just referred. In this article I wish to emphasize an aspect in the development of views on the principles of mathematics other than that of the gradual rapprochement of mathematics and logic and their final reconciliation owing to the good offices of the logic of relations as promulgated by De Morgan, C. S. Peirce, Schröder, Dedekind, Frege, Peano, and Russell. I wish to point out the service which the theory of transfinite numbers has done first, in drawing atten This symbolic logic is a great advance on the older symbolic logic, of which Schröder has given an excellent account (Vorlesungen über die Algebra der Logik, 3 volumes, Leipsic, 1890 and subsequent years; part of the third volume is not yet published). tion to what are known as "the contradictions of the theory of aggregates," and hence to the necessity for a remoulding of logic; and, secondly, in clearly separating cardinal and ordinal numbers in our minds, and so making us, by analogy, more precise in our distinctions of signless integers and positive integers, of the integer n and the ratio n: I, and so on. These distinctions have made us give up the "principle of permanence," which formerly played such a great part in mathematics, for we are compelled to admit that it consists in identifying things whose difference is clearly discernible. Thus, the advance of mathematics has brought it nearer and nearer to logic; the extent of the validity of mathematical conceptions and methods has been examined ever more closely; and it is not difficult to see that, by this, we have attained to a more thorough knowledge, and even, by the capacity which we have gained of avoiding those pseudo-problems to which methods extended beyond their domain of validity give rise, to a practical advance. I. Cantor was led to see the necessity for introducing certain definitely infinite numbers by his mathematical researches on infinite aggregates of points situated on a finite line (using a geometrical terminology for conceptions which are, in reality, purely arithmetical); but, logically, the theory is independent of this origin, and here I will give the independent grounds on which, in the Grundlagen, Cantor made the introduction of these numbers rest. Among the finite integers I, 2, ..., v, there is no greatest, but, although it would be contradictory to speak of a greatest finite integer μ (for there is always a greater one μ+1), there is no contradiction involved in introducing a new, non-finite number (), which is defined as the first number that follows all the numbers 1, 2, ..., V, (in their order of magnitude). The A very full historical account by me appeared in the Archiv der Math. und Phys. for 1906 and 1909, and the rest will appear shortly. ... This is the point which will be found to require for its adequate discussion, all the resources of logic (see below). interest that attaches to the introduction of a series of such "transfinite" numbers, the first ones of which Cantor has denoted: ... ... @, w+I, w+2, ..., w+v, w.2, w.2+I, w.v, ...w2,... being any finite integer, is, of course, to be seen from the history of those mathematical questions which necessitated the introduction of these numbers; but here we are only concerned with the question whether the conception of such numbers is logically possible, that is to say, leads to no contradiction." That Cantor, to most intents and purposes, showed this by his above introduction and subsequent definition of w, is true, and, further, he successfully classified and answered the objections made by philosophers and mathematicians, from the time of Aristotle, against the actual (or completed, as distinguished from the "potential" or "becoming") infinite. A characteristic and illuminating example of this criticism was given à propos of Dühring's arguments against the actual infinite (Eigentlich-Unendlich). These arguments can, said Cantor, be reduced, either to the statement that a definite finite number, however large, can never be infinite (a statement which is a truism) or that a variable unlimitedly great finite number can not be thought of with the predicate of definiteness, and hence also not with the predicate of being (which again immediately results from the essence of variability). To conclude, as Dühring does, the non-thinkability of definitely infinite numbers is like arguing that, because there are innumerable intensities of green, there can be no red. The use of transfinite numbers in important questions of mathematics has been shown, for example, by G. H. Hardy (Proc. Lond. Math. Soc. (2), vol. I, 1904, pp. 285-290) and myself (Mess. of Math., April, 1904, pp. 166-171, and Crelle's Journ. für Math., Bd. CXXVIII, 1905, pp. 169-210). Cantor (Grundlagen einer allgemeinen Mannichfaltigkeitslehre, Leipsic, 1883, pp. 18-20), maintained the thesis that the formation of concepts in mathematics is completely free, and has only to satisfy the condition of the logical consistency of these concepts with one another. Such concepts then have "existence" (in mathematics). Cf. below on the question whether "freedom from contradiction" is necessary or sufficient for the "existence" of a concept. Grundlagen, pp. 9-18, 43-46; Zur Lehre vom Transfiniten, Halle a. S., 1890 (reprint of Cantor's articles in the Zeitschr. f. Phil. u. philos. Kritik, Bde. LXXXVIII, XCI, and XCII, 1885-1887). 'See Grundlagen, pp. 44-45. The arguments against the infinite in mathematics have also been discussed exhaustively by Couturat (De l'infini mathématique, Paris, 1896, pp. The logically exact investigation as to the existence of numbers defined by an infinite process (as w is by the finite numbers, or an irrational number by the rationals) was begun by Russell, and I return to the question in the next section. The series of the transfinite numbers was, now, shown by Cantor to fall into certain divisions, which he called "numberclasses"; which are characterized by the property that, if a and B are any numbers of the same class, all the numbers (from I on) preceding a can be brought (in a different order, of course) into a correspondence, which is one-one, with all those preceding ß; and inversely Cantor expressed this by saying that the first class of numbers had the same "power" as the second, or that one, and only one, "power" belonged to each "number-class." Thus, in addition to the series of finite and transfinite (ordinal) numbers, there is a series of finite and transfinite powers; for finite aggregates the conceptions of power and (ordinal) number appear to coincide,10 and such an aggregate has always the same number, however it may be arranged; but a given infinite aggregate, though no re-arrangement can alter its power, since this attribute is, by the definition, independent of order, can have various (ordinal) numbers, in fact, any number of a certain class,-according to the way in which it is arranged. But, even when an aggregate is "simply ordered" (that is to say, when an "order" is given to the terms of an aggregate such that, if a and b be any two terms, a either precedes or follows b in virtue of some relation, not necessarily in order of space or time), it need not have an ordinal number. In fact, Cantor's ordinal numbers only apply to certain kinds of ordered aggregates, which he called "well-ordered," and which are characterized by the property that any selection of terms has, in the order of the original series, an element of lowest rank. Thus, the series 441-503) and by Russell (The Principles of Mathematics, vol. I, Cambridge, 1903, pp. 355-362). 'See below. 10 However, strictly speaking they do not coincide. The point is the same as the one about signless integers (classes) and positive integers (relations) referred to below. |