De l’Infini, Mathematique. Par Louis Couturat, Ancien Élève de l’École normale supérieure, Agrégé de Philosophie, Liciencié ès Sciences Mathématiques, Docteur ès Lettres. Paris: Félix Alcan, 1896. Pp. xxiv., 659.
THE relation of number to quantity forms perhaps the most difficult, as well as the most fundamental, of the problems of mathematical philosophy. On this problem at least three radically different views may be taken. We may, with Mill and most thorough-going empiricists, regard number as empirically derived from quantity, and quantity itself as a datum in experience. Or we may regard number as wholly a priori, and quantity as the result of applying to experienced data the a priori category of number. This view has been much advocated in France of late years, especially in M. Hannequin’s important work on Atomism.1 Lastly, we may hold that number and quantity are wholly independent categories, and that the application of number to quantity, as it occurs in measurement, has no deeper motive than one of convenience.
The last of these three is the view of M. Couturat, who is forced, in the course of an able apology for mathematical infinity, to devote most of his space to the relations of quantity and number. Infinite quantity, he urges, is given a priori, and does not stand or fall with infinite number. To maintain this thesis it is necessary to establish the independence of quantity, and this independence is, in fact if not in form, the chief theme of his work.
The work is divided into two parts, of which the first, on the generalisation of number, is content to exhibit and analyse those results of mathematical science which bear on the question at issue; while the second part, on number and quantity, adopts a critical and philosophical attitude, and endeavours to establish, by philosophical considerations, the legitimacy of the notions of infinity and the continuum. Whichever may be the most valuable of these two parts, the second is the most interesting to the student of philosophy, and will be the more fully discussed here.
The first part is divided into four books: the first three deal with the generalisation of number as it appears in arithmetic, algebra and geometry respectively, while the fourth deals with mathematical infinity. The aim of the first three books is to exhibit the growing necessity and the diminishing conventionality of this generalisation in the three sciences in question. In arithmetic, whose primary subject-matter is positive integers alone, fractions, negative numbers and imaginary numbers can only be introduced in an arbitrary and conventional manner. We take two integers in a given order, regarded as forming a couple with certain arithmetical properties, and establish arbitrary definitions of the equality, addition, and multiplication of two such couples. According to the definitions chosen, one of the three kinds of generalised number results.2 We cannot arithmetically introduce these generalisations in the ordinary way, for unity, in pure number, is indivisible, and such an expression as 1/3 has no meaning except what we choose to assign to it. The arithmetical generalisation is, therefore, of necessity an arbitrary and apparently motiveless process, giving rise to symbols which are without arithmetical meaning, and are only subject by convention to arithmetical operations. Irrational numbers are still more difficult to introduce arithmetically, and they first involve infinity. They express, in arithmetic, the mere absence of a number of any of the former kinds at certain points in the scale, and arise from the possibility of dividing all rational numbers, in an infinite number of ways, into two classes, such that any member of the first class is smaller than any member of the second, but no assignable member is the largest or smallest of the respective classes. There is thus a gap, which can only be filled in by a symbol expressing the absence of a rational number at the point in question.
The algebraical generalisation, as M. Couturat calls it, is less formal, and proceeds from the desire to be always able to assign roots to equations of the first or second degree. Thus equations of the first degree, to be always soluble, require negative numbers and fractions; equations of the second degree require irrational and imaginary numbers. But the algebraical generalisation is unable to obtain the so-called transcendental numbers (e and p for instance), which are not roots of any algebraical equation of finite degree. These mark the distinction between quantity and number, and point to the former to justify the generalisation of number.
Proceeding to the geometrical generalisation, M. Couturat points out that, even in algebra, there would be no need to demand that equations should always have a root, if these equations were not the statement of real problems arising outside algebra. Such problems occur in geometry, where first we find the true interpretation, as quantities, of the symbols which, from a numerical point of view, appeared to indicate impossible problems. All varieties of number, including transcendental and imaginary numbers, find here their justification and their motive.
In all this there is, from one standpoint, nothing to criticise: it is a clear and lucid account of the motives which have led mathematicians to abandon the restriction to positive integers which the pure idea of number would seem to impose. When the unit, divisible and possessed of qualities, is substituted for the abstract unity of arithmetic, all the apparatus of mathematical analysis inevitably arises. But the unit itself is not thereby freed from difficulties, and the contradictions latent in the quantitative unit appear to be inadequately realised by our author. Of this we have evidence in his fourth book, on mathematical infinity, where he argues that infinite quantities are actually given, and enforces his contention by considering the intersection of two lines which gradually become parallel. These have no intersection at a finite distance, but they cannot, he says, suddenly cease to have an intersection, for such a breach of continuity, though not logically contradictory, would be irrational (p. 216). This is the first application of a somewhat dubious principle, borrowed from Cournot, and emphatically stated in the preface (p. x), according to which philosophy has to choose, between several equally logical alternatives, the most rational one. The use of this principle, which is frequent, seems to me not very happy, and indeed may be employed to cloak what is really intellectual capitulation. M. Couturat appears to be an idealist, and on critical occasions appeals to the reason as against the understanding (see e.g. pp. 537, 565). But the function of reason, in the opinion of most idealists, is not to pronounce between alternatives which logic leaves undecided, but rather to find a logically possible alternative where the understanding finds none. This view of reason seems never to occur to our author, and the possibility of contradictions in results to which the understanding appears forced, though peculiarly evident in the case of mathematical infinity, is strenuously denied throughout the work. His rational principles, of which the principle of continuity is the chief, are thus introduced more or less arbitrarily, and appear as dogmas rather than genuinely necessary axioms. The scoffer might even be tempted to identify his “reason” with common sense.
Of this general criticism, we shall have abundant illustration in the second part of the work, which deals philosophically with the relation between number and quantity, and with the objections to infinite quantity. The first book, on number, begins with an excellent criticism of the empirical theory of number, by which M. Couturat means, not the much-refuted theory of Mill, but the formalist theory of Helmholtz and Kronecker. This theory defines numbers as mere signs with a fixed order of sequence, and endeavours to deduce the properties of cardinal numbers from this definition of ordinals. Our author insists – I think with complete success – that we must begin with cardinal numbers, and that the attempts of Helmholtz and Kronecker involve either a vicious circle or a petitio.
He then proceeds (bk. 1, ch. 3) to the rationalist theory of whole numbers – one of the very few subjects in epistemology, I suppose, upon which there is at present a consensus among experts. Number requires only a logical and formal unit, created by thought: all the objects of a numbered collection must be regarded as units in this sense, and in so far identical, while the whole collection must, in turn, be regarded as a complex unity. The idea of unity is an a priori idea, and is never given by the data of sense. It is doubly involved in number, which is a “unity of a plurality of unities” (p. 361).3 In the next chapter, after urging that number requires no schematism, as Kant had maintained – for it is not successive enumeration, but simultaneous apprehension, which is needed (p. 354) – our author begins to tread on more dangerous ground, by the consideration of infinite number. One would have supposed that the condition of being a completed whole, which he has urged as necessary to number, would have precluded the possibility of infinite number. But M. Couturat boldly contends that a collection is given as a whole, as soon as we have a law by which any required number of its members can be constructed, and from which no member is exempt. The conditions for a number of a collection may, therefore, be satisfied, even if the collection is infinite, and successive enumeration of all its terms is impossible (p. 351). This is certainly the only hope of saving infinite number from contradiction, and M. Couturat has made the most of it. As he deals with the same topic more fully at a later stage (book 3), I shall postpone the criticism of this view till we have discussed book 2, which deals with quantity.
The main contention of book 2 is, that quantity is as a priori as number, but is an entirely independent category. The idea of quantity, he says, is indefinable and irreducible: to define it as what can be increased or diminished involves an obvious circle (pp. 367-9). If quantity be indeed an independent category, we must of course agree that it cannot be defined in terms of other categories: but one could wish that M. Couturat had made more attempt to show that it is a category free from contradictions, and that there is some way of knowing about it without the help of number. Measurement, he admits, consists in the numerical expression of quantity, and thus introduces axioms due to the nature of number, and producing a conflict between number and quantity (pp. 404-5). But he thinks that the equality and addition of quantities of the same kind can be effected without reference to division into units, and therefore without dependence on number (p. 404). Chapters 1 and 2 of this book give the axioms of equality and addition, which are declared a priori, and are stated in a form apparently free from reference to number. But they seem very insufficient for the foundation of a science of quantity. Thus he says, for example, that equality in general cannot be defined, but as soon as we have the idea of any kind of quantity, we have the idea of equal quantities in this kind, or rather of the same quantity in different objects (pp. 372-3). This seems vague, and it might be objected that we cannot have the idea of a kind of quantity until we know what we mean by equal quantities of this kind, since it is equality and inequality which constitute quantitative relations. Again he says (p. 389) that the sum of two quantities is a quantity of the same kind. If this axiom were really independent of division into units, it would hold equally of intensive quantities, but this is not the case. The sum of two temperatures, for example, is meaningless.
He has also what he calls the axiom of the modulus (p. 399), according to which there exists a certain real quantity, called zero, such that, when added to any other quantity, it leaves that quantity unchanged. Having been told previously (p. 232) that a zero distance is not nothing, but just as real as any other distance, we are not surprised at this axiom; but it is a pity that no attempt is made to explain what a zero quantity is. Zero would seem to be about as non-existent as anything could be; for it is defined as nothing but quantity, and further as containing no quantity. We are not told how to make something of this apparent nonentity; and it is even supposed that objections to infinity will be silenced by the argument that the same objections apply to zero (p. 436). Zero is said to have a rational, not a logical, necessity (p. 402); without zero, it is said, measurement would be impossible. It seems strange to assign rational necessity to anything so grossly contradictory as mathematical zero, and its necessity for measurement seems at best doubtful. Temperature, for example, has a purely fictitious zero, and yet is measured by the thermometer. The fact is, I think, that quantitative zero is a limit necessarily arising out of the infinite divisibility of extensive quantities, and that M. Couturat’s axioms are only applicable to extensive quantities. In this way division into units is surreptitiously introduced; but by immediately declaring every axiom in turn a priori, without analysis of its grounds, the necessity of reference to number is concealed.4
There are several other points in this book which call for criticism. On p. 416, the axiom of continuity is given in the form: “If all the quantities of a kind can be divided into two classes, such that all the quantities of the one are smaller (or larger) than all the quantities of the other, there exists a quantity of this kind which represents this mode of division, and which is at once larger than all the quantities of the inferior class, and smaller than all the quantities of the superior class.” Against this definition a dilemma may be presented: either the word all is not to be taken as including the quantity which lies between the two classes, in which case the definition applies equally to discreta, e.g., the series of natural numbers; or the word all is to be taken rigidly, in which case there is a palpable contradiction in supposing another quantity of the same kind, which does not belong to either of the classes into which all the quantities of the kind have been divided. That such a contradiction must necessarily hold of continua, I have no wish to deny; but it seems bold to say, of an axiom involving this contradiction, that though indemonstrable, “it possesses an evidence almost equal to that of analytical judgments” (p. 416), and that its justification is not logical but rational (p. 554). Again, in discussing the definition of number as the ratio of quantities of the same kind, he defends the definition from the vicious circle which it appears to involve, by declaring that ratio is a fundamental idea not susceptible of definition (p. 426). In view of the obvious method of defining ratio by means of number, this almost reminds one of Mill’s “final inexplicability.” Finally he says that incommensurable quantities must have a ratio, for otherwise two continuous variables would have a ratio one moment and none the next (p. 428), and that to identify unity with the quantitative unit is impossible, since it either supposes unity divisible or the unit indivisible. He does not seem to realise that the possibility of continuous variation, and the impossibility of indivisible units, are the very points which a champion of quantity, as against number, has to establish.
The third book consists of an answer to the usual objections to infinity, which is given in the form of a dialogue between a finitist and an infinitist. The proofs that infinite number is impossible, he says, all rest on one of two fallacies: (1) that infinity is the largest of all numbers, and (2) that all infinities are equal. Both these are false, he says, and the difficulties to which they have given rise are all resolved by Cantor’s transfinite numbers (p. 455). The actual infinity of the series of natural numbers is not contradictory, he says, for it results from their law of formation, by which they are given as a totality.5 That the resulting infinity may reveal a latent contradiction in the law of formation itself, appears not to occur to him. Thus when the finitist objects that an infinite collection can never be really given or really a whole, the infinitist replies that, in that case, number itself must be contradictory (p. 471) – a conclusion from which, one would think, a bold disputant would scarcely shrink. But infinite quantity, he says, is even simpler than infinite number, for it does not necessarily consist of a collection of units. It is difficult to see in what sense a quantity is infinite, unless it is compared with some unit, of which an infinite number would be required to reproduce it. But M. Couturat holds that an infinite quantity can be given to begin with as a totality, and that measurement can be effected otherwise than by the addition of elements (p. 483). How the latter is possible, he does not explain; integration, which he gives as an example, is essentially an addition of elements. The possibility of the former, one would think, is sufficiently disproved by the stock argument, that mathematical infinity consists essentially in the absence of totality, the absence of completed synthesis. But infinite quantity, as M. Couturat says, depends on continua, and these, like most of the fundamentals in his work, he regards as given by reason, not by logic (pp. 497-8). For one equipped only with logic, it is impossible to follow into this fortress of reason, where the shafts of logic cannot penetrate.
In the last book, conclusions are drawn from the previous discussions. Beginning with number, it is pointed out that number, like the concept, depends on generalisation and abstraction; it is, indeed, the other face of the concept, the extension corresponding to a given intension. But since it requires that its constituents should each be united, and all be similar, its application to reality is never wholly legitimate, for similarity and unity seem to exist in inverse ratio. The latter is best found in organisms, the former in homogeneous continua. All this is excellent, like what was said on number in book 2; but it is a pity that a similar process of criticism is not applied to continuous quantity, which stands at least as much in need of it. As regards this conception, M. Couturat points out that it is not given by sensible experience, since sensation can never reach an accuracy sufficient to exclude discreteness. Thus irrationals suffice to prove that quantity is not empirical; it is in fact, he says, a priori and due to reason. With the justification of quantity comes the justification of infinity, which reason, he says, does not obtain by successive synthesis, but sees from the start (p. 565). From this conclusion, he goes on to Kant’s antinomies, whose theses, since the realised infinite is not impossible, he declares to be all false (p. 567). The antitheses, therefore, he regards as true, and Kant’s objection to them, he says, is due to the schematism of the categories, which led him to regard quantitative synthesis as necessarily successive, while every quantity, in fact, is given first as a whole, not as a synthesis of parts.
It seems to me that this argument, as well as the whole argument against the “finitists” throughout the work, rests on a misapprehension of their position. That infinity follows necessarily from certain premisses – e.g., the reality of space and time as something more than relations – must be admitted; that infinity is useful and unobjectionable in mathematics, is by this time almost self-evident; but that mathematical infinity is philosophically valid might, I imagine, be met by two converging lines of argument. The first and more usual argument would urge the contradictions of infinity, which M. Couturat, I think, has not succeeded in disproving; the second might urge that, in all the cases where infinity is unavoidable, there has been some undue hypostatising of relations, which makes the attainment of a completed substantive whole impossible. These lines of argument, however, can be only suggested within the space of a review.
Finally, I wish to urge the very solid merits of the work, to which it is difficult, in the course of detailed criticism, to do justice. The position taken up is the only one from which infinite quantity can be philosophically defended, and the main thesis is carefully and consistently worked out. To take up an unpopular cause is always praiseworthy, and almost always useful; in the case of M. Couturat it is certainly both, and his book is likely long to remain the classic advocate of mathematical infinity.
B. RUSSELL
1 Essai Critique sur l’Hypothèse des Atomes. Paris, 1894. M. Hannequin’s book and M. Couturat’s should be compared, as they deal ably with almost the same theme from different standpoints.
2 For example, fractions are defined by the following conventions: (a, b) = (c, d) if ad = bc ; (a, b) + (c, d) = (ad + bc, bd) ; (a, b) × (c, d) = (ac, bd).
3 “Units” would perhaps be a better word, but as there is only one word in French for unit and unity, the distinction has to be made by translation.
4 On one of these axioms, the axiom that of two magnitudes of the same kind one must be the larger, there is a rather serious inconsistency; it is denied as regards imaginaries on p. 46, and affirmed to be a priori on p. 384.
5 There is some inconsistency on this point. Sometimes it is argued that all the natural numbers are given by successive addition of unity, sometimes, with Cantor, that another principle of formation is also required. Compare, e.g., pp. 364, 465.
*
Bertrand Russell, Review of Louis Couturat, De l’Infini mathématique, Mind, n.s. 6, no. 21 (Jan 1897), 112-19