12. Real numbers.

Every measure of size implies a diffuse notion of real numbers. From the point of view of mathematics, the origins of the theory of real numbers must go back to the progressive formation, in Babylonian science, of a system of numbering capable (in principle) of denoting values as near as required to every real number [232]. The possession of such a system, and the confidence in numerical calculations which can not but follow from this, terminated inevitably, in fact, with a “naive” notion of real numbers, which is hardly different from that which one finds again today (linked to the decimal system of numbering) in elementary teaching or with physicists and engineers; this notion does not allow itself to be defined with exactness, but it can be expressed by saying that a number is considered as defined by the possibility of obtaining approximate values and introducing these into the calculations: which, besides, implies necessarily a certain degree of confusion between the measures of size given by experience, which are naturally not susceptible to indefinite approximation, and such “numbers” as $\sqrt{2}$ (supposing that there exists an algorithm for the indefinite approximation of this number).

A similar “pragmatic” point of view reappears therefore in all the schools of mathematics where calculating skills take over from the desire for rigour and theoretical preoccupations. It is these latter, on the contrary, which dominate Greek mathematics: so we owe to it the first rigorous and coherent theory of ratios of quantities, that is to say, essentially the real numbers; it is the end result of a series of discoveries on ratios and in particular on incommensurable ratios, of which it is difficult to exaggerate the importance in the history of Greek thought, but of which, in the absence of precise texts, we can only just discern the major lines. Greek mathematics in its beginnings is inseparably linked to speculations, partly scientific, partly philosophical and mystical, on proportions, similitudes and ratios, in particular the “simple ratios” (expressible by fractions with small numerator and denominator); and it was one of the characteristic tendencies of the Pythagorean school to claim to explain all by integers and the ratio of integers. But it was the Pythagorean school, precisely, that discovered the incommensurability of the side of a square with its diagonal (the irrationality of $\sqrt{2}$): the first example, no doubt, of a proof of impossibility in mathematics; just the fact of stating this question implies a clear distinction between a ratio and its approximate values, and suffices to indicate the immense gap which separates Greek mathematicians from their predecessors.$^1$

We are badly informed about the movement of ideas that accompanied and followed this important discovery.$^2$ We will limit ourselves to indicating summarily the principal ideas that are at the basis of the theory of ratios of quantities, a theory which, codified by the great mathematician Eudoxus (contemporary and friend of Plato), was definitively adopted by classical Greek mathematics, and is known to us through the Elements of Euclid [107], where it is given a masterly exposition (in Book V of these Elements):

1) The word and the idea of number are strictly reserved for integers > 1 (1 is the monad and not a number properly speaking), to the exclusion, not only of our irrational numbers, but even of those that we call rational numbers, these being, for the Greek mathematicians of the classical era, ratios of numbers. There is there much more than a simple question of terminology, the word number being linked for the Greeks (and for the moderns until recent times) to the idea of a system with a double law of composition (addition and multiplication): the ratios of integers are conceived by the classical Greek mathematicians as operators, defined on the set of integers or on a part of this set (the ratio of p to q is the operator which, to N, makes correspond, if N is a multiple of q, the integer p.(N/q)), and making up a multiplicative group, but not a system with a double law of composition. In this, the Greek mathematicians were separating themselves voluntarily from the "logisticians" or professional calculators, who had, like their Egyptian and Babylonian predecessors, no scruples at treating as numbers fractions or the sum of an integer and a fraction. It seems besides that they imposed on themselves this restriction on the idea of number for motives more philosophical than mathematical, and following on from the thoughts of the first Greek thinkers on the unit and the multiple, the unit not being capable of subdivision (in this system of thought) without losing by that selfsame action its character of unit.$^3$

$^1$ The discovery of the irrationality of $\sqrt{2}$ is attributed by some to Pythagoras himself, without sufficient authority it would seem; by others to some Pythagorean of the Vth century; there is agreement, on the testimony of Plato in the Theetete in attributing to Theodorus of Cyrene the proof of the irrationality of $\sqrt{3}, \sqrt{5}$, "an so on therefore until $\sqrt{17}$", following which Theetete would either have obtained a general proof for $\sqrt{N}$ ($N =$ an integer not a perfect square), or in any case (if, as may be, the proof of Theodorus was general in its principle) have proceeded to a classification of certain types of irrationals. It is not known if these first proofs of irrationality proceeded by an arithmetic or geometric path: see on that [148], chap. IV; cf. also [153 a], [321] and [151]. $^2$ On this consult in particular [153 a], [309 b and c], and further the works cited in the preceding note, as well as [317 c]. $^3$ Plato ([250], book VII, 525th) mocks the calculators "who change the unit for small change" and tells us that, where these divide, the scholarly multiply: which means that, for example, for the mathematician, the equality of two ratios $a/b$ and $c/d$ is realised, not by dividing $a$ by $b$ and $c$ by $d$, which in general leads to a

2) The theory of quantities is axiomatically based, and at the same time for all kinds of quantities (one finds allusions to previous theories which, from what appears, treated separately lengths, areas, volumes, time, etc.). The quantities of one kind are characterised by the fact that they are capable of being compared (that is to say that a definition of equality, which is properly speaking an equivalence, and the relations $>$ and $<$ are assumed), of being added and subtracted ($A + B$ is defined, and $A - B$ if $A > B$), and satisfy the axiom of "Archimedes" so-called; this is clearly conceived, from the beginning, as a keystone of the edifice (it is in fact indispensable for all axiomatic characterisations of real numbers); it is a pure accident that it has been given the name of Archimedes, and this latter insists, in the introduction to his "Quadrature of the Parabola" ([5 b], v. II, p. 265), on the fact that this axiom has been used by his predecessors, that it plays an essential role in the work of Eudoxus, and that its consequences are no less sure than the determinations of areas and volumes made without its help.$^4$

It is easy to see that, from this axiomatic foundation, follows necessarily the theory of real numbers. It must be noted that, for Eudoxus, the quantities of a given kind form a system with one internal law of composition (addition), but that this system has an external law of composition with as operators the ratios of quantities, these being conceived of as forming an abelian multiplicative group. $A$ and $A'$ being quantities of the same kind, and similarly $B$ and $B'$, the ratios of $A$ and $A'$ and of $B$ and $B'$ are defined as equal if, whatever the integers $m$ and $m'$, $mA < m'A'$ forces $mB < m'B'$ and $mA > m'A'$ forces $mB > m'B'$; inequalities between ratios are defined in an analogous manner. That these ratios form a domain of operators for every kind of quantity is equivalent to the axiom (not made explicit but often used in the drafting of Euclid) of the existence of a fourth proportional: a ratio $A/A'$ being given and $B'$ being given, there exists a $B$, of the same kind as $B'$, such that $B/B' = A/A'$. Thus the highly original idea of Eudoxus allowed identification between the domains of operators defined for all kinds of quantities;$^5$ in an analogous manner, the set of integers (see above) can be identified with a subset of the set of ratios of quantities, namely with the set of rational ratios (ratios of commensurable quantities); however, because of

calculus of fractions (it is thus that the Egyptians and Babylonians would also have operated), but in checking that $a.d = b.c$; and other similar facts.

$^4$ A manifest allusion to polemics that have not been kept for us: it could be believed that it is a modern speaking of Zermelo's axiom.

$^5$ Thus it was possible to do in all rigour what was currently being done by the Greek mathematicians when they considered as proved a theorem on proportions as soon as it had been proved for all rational ratios. It seems that before Eudoxus, an attempt at constructing a theory which would have attained the same objectives had been made by defining the ratio $A/A'$ of two quantities by what we would call in modern language the terms of the continuous fraction which expresses it; about these attempts, to which the algorithm so-called of "Euclid" for the search for a common measure for $A$ and $A'$ if it exists (or for the determination of the G. C. D.) was naturally leading, cf. [17 a].

the fact that these ratios, in as much as they are operators on the integers, are (in general) defined only for a subset of the set of integers, it was still necessary to develop their theory separately (Book VII of Euclid).

The domain of universal operators thus constructed was therefore for Greek mathematicians the equivalent of what is for us the set of real numbers; it is clear besides that with the addition of quantities and the multiplication of ratios of quantities, they possessed the equivalent of what for us is the field of real numbers, even though in a much less handy form.$^6$ It is possible, on the other hand, to ask oneself if they had conceived of these sets (set of quantities of a given kind, or sets of ratios of quantities) as complete in our sense; it is not easy to see, otherwise, why they would have assumed (without even feeling the need to turn it into an axiom) the existence of a fourth proportional; furthermore, certain texts seem to refer to ideas of this kind; finally they certainly assume as obvious that a curve, capable of being described by a continuous movement, can not pass from one side to the other of a straight line without cutting it, a principle that they used for example in their research on the duplication of the cube (construction of $\sqrt{2}$ by the intersection of curves) and which is essentially equivalent to the property under consideration; however, the texts that we possess do not allow us to know with complete precision their ideas on this point.

Thus therefore is the state of the theory of real numbers in the classical era of Greek mathematics. As admirable as the construction of Eudoxus was, and not leaving anything to be desired from the point of view of coherence and rigour, it must be admitted that it lacked in suppleness, and was hardly favourable for the development of numerical calculations and especially of algebraic calculations. What is more, its logical necessity could only be appreciated by minds taken with rigour and exercised in abstraction; it is therefore natural that with the decline of Greek mathematics, one sees reappearing little by little the "naive" point of view which had been kept by means of the tradition of the logicians; it is that which dominates for example with Diophantus [91 a], a true follower of this tradition rather than official Greek science; this latter, while reproducing for form's sake the Euclidean definition of number, understands really by the word "number", the unknown of the algebraic problems whose solution is, either an integer, or a fractional number, or even an irrational.$^7$ Although this change in attitude, about the subject of number, was linked to one of the most important ad-

$^6$ So unhandy that the Greek mathematicians, to translate the algebraic science of the Babylonians into their language, had found it necessary to use systematically a means of quite another order, namely the correspondence between two lengths and the area of a rectangle constructed with these two lengths as sides: which is not properly speaking a law of composition, and does not allow the convenient writing of algebraic relations of a degree higher than the second.

$^7$ "The 'number' is found to be not rational", Diophantus, book IV, problem IX. On this return to the naive notion of number, cf. also Eutocius, in his Commentary on Archimedes ([5 b], v. III, pp. 120-126).

vances in the history of mathematics, namely the development of Algebra (see p. 48), it does not constitute, be it well understood, progress in itself, but rather regress.

It is not possible for us to follow here all the vicissitudes of the idea of number through Hindu, Arab and Occidental mathematics until the end of the Middle Ages: it is the "naive" notion of number that dominates; and, even though the Elements of Euclid were used as basis for the teaching of mathematics during this period, it is likely that the doctrine of Eudoxus remained generally not understood because the necessity for it did not arise any more. The "ratios" of Euclid were most often qualified as "numbers"; the rules of calculation of integers were applied to them, obtaining thus exact results, without seeking to analyse deeply the reasons for the success of these methods.

We see however already R. Bombelli, in the middle of the XVIth century, writing on this subject, in his Algebra [28 b],$^8$ a point of view that (on condition that we assume acquired the results of book V of Euclid) is essentially correct; having recognised that once the unit of length is chosen there is a bijective correspondence between the lengths and the ratios of quantities, he defines the various algebraic operations (assuming the unit fixed, of course) on the lengths, and, representing numbers by lengths, obtains the geometric definition of the field of real numbers (a point of view for which the credit is most often given to Descartes), and gives thus to his Algebra a solid geometrical basis.$^9$

But the Algebra of Bombelli, even though singularly advanced for his times, did not go beyond the extraction of radicals and the solution by radicals of equations of the 2nd, 3rd and 4th degrees; of course the possibility of extraction of radicals is assumed by him without discussion. Simon Stévin [295], also, adopts an analogous point of view on the subject of number, which is for him that which marks a measure of quantity, and which he considers as essentially "continuous" (without making precise the meaning he gives to this word); if he distinguishes between the "geometrical numbers" and the "arithmetical numbers", it is only from the accident of their mode of definition, without there being for him there a difference of kind; here besides is his last word on the subject: "We conclude therefore that there are no absurd, irrational, irregular, inexplicable or deaf numbers; but that there is in them such excellence and agreement that we have subject matter for meditation night and day in their admirable perfection" ([295], p. 10). On the other hand, having been the first to have turned the tool of decimal fractions into

$^8$ It is a matter here of book IV of this Algebra, which remained unpublished to our day; it matters little for the object of the statement below, whether the ideas of Bombelli on this subject were known or not by his contemporaries. $^9$ We will not enter here into the history of the use of negative numbers, which is the province of Algebra. Let us note however that Bombelli, in this same place, gives with perfect clarity the definition, purely formal (such as it could be found in a modern Algebra), not only of negative quantities, but also of complex numbers.

a method of calculation, and proposed for them a notation already close to ours, he conceived clearly that these fractions furnish an indefinite approximation algorithm for every real number, as he brings out in his Appendice algebraique of 1594 "containing a general rule for all Equations" (a booklet of which the only known example was burnt at Louvain in 1914; but see [295], p. 88). Such an equation having been put into the form $P(x) = Q(x)$ (where $P$ is a polynomial of degree greater than that of the polynomial $Q$, and $P(0) < Q(0)$), we substitute for $x$ the numbers 10, 100, 1000, ... until we find $P(x) > Q(x)$, which, he says, determines the number of digits of the root; then (if for example the root is found to have two digits) we substitute 10, 20, ... which determines the digit of tens; and then the same for the following digit, then for the successive decimal digits: "And proceeding thus infinitely", he says, "one reaches infinitely near to that required" ([295], p. 88). As can be seen, Stévin had (no doubt the first) a clear idea of Bolzano's theorem and had recognised in this theorem the essential tool for the systematic solution of numerical equations; an intuitive conception of the numerical continuum is recognised there, at the same time, so clear that there remained little to do to make it definitively precise.

However, in the two centuries that followed, the definitive establishment of correct methods was twice slowed down by the development of two theories of which we do not have to give the history here: the infinitesimal calculus, and the theory of series. Across the discussions that they brought up, we recognise, as at all periods of the history of mathematics, the perpetual balancing between the seekers occupied with going forward, at the cost of some insecurity, persuaded that there will always be time later to consolidate the conquered territory, and the critical spirits, who (without necessarily giving an inch to the first as far as intuitive faculties and inventive talents go) do not believe they are wasting their efforts in devoting some effort to precise expression and to rigorous justification of their concepts. In the XVIIth century, the principal object of debate is the notion of the infinitely small, which, justified a posteriori by the results which it allows us to reach, seemed in open opposition to the axiom of Archimedes; and we see the most open spirits of the time finish by adopting a point of view little different from that of Bombelli, and which is distinguished from it mainly by the greater attention paid to the rigorous methods of the ancients; Isaac Barrow (the master of Newton, and who himself played an important part in the creation of the infinitesimal calculus) gives a brilliant exposé of it in his Lessons in Mathematics delivered at Cambridge in 1664-65-66 [16 a and b]; recognising the necessity, in order to find again for the subject of numbers the proverbial "geometrical certainty", of returning to the theory of Eudoxus, he presents at length, and very judicially, the defence of this latter (who, as he witnessed, appeared intelligible to many of his contemporaries) against those who tackled him with obscurity or even absurdity. On the other hand, defining numbers as symbols which denote the ratio of quantities, and susceptible of being combined amongst themselves by the operations of arithmetic, he obtains the field of real numbers, in terms taken up again after him by Newton in his Arithmetic and of which his successors until Dedekind and Cantor would not change anything.

But it is at about this time that the method of expansion in series was introduced, which soon, in the hands of impenitent algebraists, takes on an exclusively formal character and turns away the attention of mathematicians from the questions of convergence that the sane use of series raises in the domain of real numbers. Newton, the principal creator of this method, was still conscious of the necessity of considering these questions: and, if he had not sufficiently elucidated them, he had recognised at least that the power series that he was introducing converged "most of the time" at least as well as a geometric series (whose convergence was already known to the ancients) for small values of the variable ([233 a], v. I, pp. 3-26); at about the same time, Leibniz had observed that an alternating series, with terms decreasing in absolute value and tending to 0, is convergent; in the following century, d'Alembert, in 1768, expresses doubts on the use of non-convergent series. But the authority of Bernoulli and especially Euler means that such doubts are exceptional at this time.

It is clear that the mathematicians who would have had the habit of using series in numerical calculation would never have neglected thus the notion of convergence; and it is not chance that the first who, in this domain as in many others, had brought about a return to correct methods, was a mathematician who, already from the prime of youth, had a love of numerical calculation: C. F. Gauss, who, almost a child, had used the algorithm of the arithmetico-geometric mean,$^{10}$ could not avoid forming for himself a clear idea of limits; and we see him, in a fragment that dates from 1800 (but was only published in our time) ([124 a], v X$_1$, p. 390), defining precisely, on the one hand the upper bound and the lower bound, on the other hand the upper limit and the lower limit of a sequence of real numbers; the existence of the first (for a bounded sequence) seeming to be assumed as obvious, and the latter having been correctly defined as limits for $n$ tending to $+\infty$, of $\sup_{p \geq 0} u_{n+p}, \inf_{p \geq 0} u_{n+p}$. Gauss, as well, gives also, in his memoir of 1812 on the hypergeometric series ([124 a], v. III, p. 139) the first model of a discussion about convergence undertaken, as he says, "in all rigour, and done to satisfy those whose preferences go to the rigorous methods of the ancient geometers": it is true that this discussion, forming a secondary point in the memoir, does not go back to first principles in the theory of series; it is Cauchy who establishes these first, in his Cours d'Analyse of 1821 ([56 a], (2), v. III), in a manner correct at all points, starting from the Cauchy criterion clearly stated, and assumed to be obvious; as, for the definition of number, he keeps to the point of view of Barrow and Newton, it can thus be said that for him

$^{10}$ $x_0, y_0$ being given and $> 0$, let $x_{n+1} = (x_n + y_n)/2, y_{n+1} = \sqrt{x_n y_n}$; for $n$ tending to $+\infty$, $x_n$ and $y_n$ tend (very fast) to a common limit, called the arithmetico-geometric mean of $x_0$ and $y_0$; this function is intimately linked to elliptic functions and formed the point of departure for important work of Gauss on this subject.

the real numbers are defined by the axioms of quantities and the criterion of Cauchy: which indeed is sufficient to define them.

It is at the same time that another important aspect of the theory of real numbers is definitively clarified. As we have said, it has always been assumed as geometrically obvious that two continuous curves can not cross without intersecting; a principle that (made suitably precise) would be equivalent, it also, to the property of the straight line being a complete space. This principle is still part of the basis of the "rigorous" proof given by Gauss in 1799 of d'Alembert's theorem, according to which every polynomial with real coefficients admits a real or complex root ([124 a], v. III, p. 1); the proof of the same theorem, given by Gauss in 1815 ([124 a], v. III, p. 31), relies, in the same way as a previous attempt by Lagrange, on the principle, analogous but simpler, according to which a polynomial can not change sign without becoming zero (a principle that we have already seen used by Stévin). In 1817, Bolzano gives, starting with the criterion of Cauchy, a complete proof of this latter principle, which he obtains as a particular case of an analogous theorem about continuous numerical functions of a numerical variable [27 c]. Stating clearly (before Cauchy) the "Cauchy criterion", he seeks to justify it by an argument that, in the absence of all arithmetical definitions of a real number, was only and could only be a vicious circle; but, once this point is admitted, his work is entirely correct and quite remarkable, as containing, not only the modern definition of a continuous function (given here for the first time), with a proof of the continuity of polynomials, but even the proof of the existence of the lower bound of an arbitrary bounded set of real numbers (he does not speak of sets, but, which amounts to the same thing, of properties of real numbers). On the other hand, Cauchy, in his Cours d'Analyse ([56 a], (2), v. III), defining, he also, continuous functions of one or more numerical variables, proves equally that a continuous function of one variable can not change sign without becoming zero, and this by the same argument as Simon Stévin, which naturally becomes correct, once continuity is defined, as soon as the Cauchy criterion is used (or as soon as the equivalent principle of "nested intervals" so-called is assumed, as Cauchy does at this point, of which the convergence of indefinite decimal fractions is of course only a particular case).

Once this point is reached, it only remained for mathematicians to make precise and develop the results acquired, by correcting a few errors and filling a few gaps. Cauchy, for example, had believed for a time that a convergent series, whose terms are continuous functions of a variable, has as sum a continuous function: the correction of this point by Abel, during the course of his important work on series ([1], v. I, p. 219; cf. also v. II, p. 257 and passim), ended finally by the elucidation by Weierstrass, in his lectures (unpublished, but which had a considerable influence), of the notion of uniform convergence (see p. 205). On the other hand, Cauchy had, without sufficient justification, assumed the existence of a minimum of a continuous function in one of the proofs given by him for the existence of roots of a polynomial; it is again

Weierstrass who brought clarity to questions of this kind by proving in his lectures the existence of a minimum for functions of numerical variables, defined on closed bounded intervals; it is following on from his critique of the unjustified application of this theorem to sets of functions (of which the "Dirichlet principle" is the best known example) that the movement of ideas begins which ends (see p. 143) with the general definition of compact spaces and the modern statement of the theorem.

At the same time, Weierstrass, in his lectures, had recognised the logical interest that there is in completely separating the idea of real numbers from the theory of quantities: to use this, in effect, reduces to defining axiomatically the set of points of the straight line (therefore definitively the set of real numbers) and assuming the existence of such a set; although this way of doing things is essentially correct, it is evidently preferable to start only with rational numbers, and to deduce the real numbers from there by completion.$^{11}$ This is what was done, by different methods, and independently the one from the other, by Weierstrass, Dedekind, Méray and Cantor; whereas the procedure of "cuts" so-called, proposed by Dedekind ([79], v. II, pp. 315-334) was much closer to the definitions of Eudoxus, the other proposed methods are nearer to the method used since Hausdorff to complete a metric space. It is at this moment also that Cantor starts to develop the theory of sets of real numbers, of which Dedekind had had the first idea [48], obtaining thus the principal elementary results on the topology of the straight line, the structure of open sets, of closed sets, the notions of derived sets, perfect totally discontinuous sets, etc..

With Cantor, the theory of real numbers took, with few exceptions, its definitive form; let us indicate briefly in what way it has been extended. Apart from the work in general topology (see p. 143), and applications to Integration (see p. 221), it is mainly a case of research on the structure and classification of sets of points on the straight line and of numerical functions of real variables; they have their origin in the works of Borel [32 a], aiming mainly at measure theory, but which ended up among other things at the definition of "Borel sets": these are sets belonging to the smallest family of subsets of $\mathbf{R}$, containing the intervals, and closed with respect to union and countable intersection and the operation C. Intimately linked to these sets are the so-called "Baire" functions, that is to say those that can be obtained starting with continuous functions by the operation of limits, repeated "trans-

$^{11}$ Indeed, the question of existence, that is to say, in modern language, the consistency of the theory of real numbers, is thus reduced to the analogous question for rational numbers, on condition however that the theory of abstract sets is assumed known (since completion presupposes the notion of generic subset of an infinite set); in other words, everything is reduced to this latter theory, since one can deduce from it the theory of rational numbers. On the contrary, if it is assumed that the Theory of sets is not available, it is impossible to reduce the consistency of the theory of real numbers to that of arithmetic, and it becomes again necessary to give it an independent axiomatic characterisation.

finitely"; they were defined by Baire in the course of important work where he abandons entirely the measure point of view in order to tackle systematically the qualitative and "topological" aspect of these questions [11 a]: it is on this occasion that he defines and studies for the first time semi-continuous functions, and with a view to characterise limit functions of continuous functions, he introduces the important notion of a thin set ("first category" set in the terminology of Baire) (see p. 165). As for the numerous works that have followed those of Baire, and which are due mainly to the Russian and especially Polish schools, we can only draw attention to their existence here (see p. 166).