1. Foundations of mathematics; logic; set theory.

The study of what is usually called the "foundations of mathematics", which has been carried out ceaselessly since the beginning of the 19th century, was impossible to bring to fruition except with the help of a parallel effort to systematise Logic, at least those parts that govern the links between mathematical statements. So it is not possible to separate the history of set theory and the formalisation of mathematics from that of "Mathematical Logic". But traditional logic, like that of the modern philosophers, covers in principle, an area of applications far greater than Mathematics. Therefore the reader must not expect to find in what follows a history of Logic, even in a very summarised form; we have limited ourselves as far as possible to retracing the evolution of Logic only in so far as it impinged on that of Mathematics. It is because of this that we will say nothing about the non-classical logics (many-valued logics, modal logics); all the more so will we be unable to tackle the history of those controversies which, from the Sophists to the Vienna School, have never stopped dividing philosophers both as to the possibility and the manner of applying Logic to objects in the real world or to concepts of human thought.

That there was a well-developed prehellinic mathematics is not today in any doubt. Not only are the notions (already very abstract) of whole number and of the measurement of quantities commonly used in the most ancient documents which have reached us from Egypt or Chaldea, but Babylonian algebra, because of the elegance and sureness of its methods, should not be thought of as a simple collection of problems solved by empirical fumbling. And, if nothing is found in the texts which resembles a "proof" in the formal meaning of the word, it is reasonable to believe that the discovery of such methods of solution, whose generality appears through particular numerical applications, was not possible without a minimum of logical links (perhaps not entirely conscious, but rather like those on which a modern algebraist depends when he undertakes a calculation, before "setting down formally" all the details) ([232], pp. 203 ff.).

The essential originality of the Greeks consisted precisely of a conscious effort to order mathematical proofs in a sequence such that passing from one link to the next leaves no room for doubt and constrains universal assent.

That Greek mathematicians made use, in the course of their research, just like modern mathematicians, of "heuristic", rather than convincing, arguments, is what was proved for example (if there were any need), by the "treatise of method" of Archimedes [153 c]; note also in this, allusions to results "found, but not proven" by earlier mathematicians.$^1$ But from the first detailed texts that are known to us (and which date from the middle of the Vth century), the ideal "canon" of a mathematical text is properly settled. It will find its highest expression in the great classics, Euclid, Archimedes and Apollonius; the notion of proof, in these authors, differs in no way from ours.

We have no texts allowing us to follow the first steps of this "deductive method", which seems to us already near perfection at the exact moment when we become aware of its existence. One can only think that it fits fairly naturally into the perpetual search for "explanations" of the world which characterises Greek thought and which is so discernible already amongst the Ionian philosophers of the VIIth century; moreover tradition is unanimous in ascribing the development and refinement of the method to the Pythagorean School, in a period which fits between the end of the VIth century and the middle of the Vth century.

It is on this "deductive" mathematics, fully conscious of its goals and methods, that the philosophical and mathematical thought of subsequent times will be concentrated. We will see on the one hand the establishment little by little of "formal" Logic modelled on mathematics, to conclude with the creation of formalised languages; on the other hand, mainly starting at the beginning of the XIXth century, the basic concepts of mathematics will be queried more and more and a great effort will be made to clarify their nature, especially after the birth of the Theory of sets.

THE FORMALISATION OF LOGIC.

The general impression which seems to emerge from the (very patchy) texts that we have on Greek philosophical thought of the Vth century, is that it is dominated by an increasingly conscious effort to extend to the whole field of human thought the procedures for conducting discussion put in hand with so much success by contemporary rhetoric and mathematics - in other words, to create Logic in its most general form. The tone of philosophical writings

$^1$ Notably Democritus, to whom Archimedes attributes the discovery of the formula giving the volume of a pyramid ([153 c], p. 13). This allusion must be put alongside a famous fragment attributed to Democritus (but of contested authenticity) where he states: "No one has ever surpassed me in constructing figures by means of proofs, not even the Egyptian 'harpedonaptes', as they are called." ([89], v. I, p. 439 and v. II, 1, pp. 727-728). The comment by Archimedes and the fact that proofs (in the classical sense) have never been found in those Egyptian texts which have reached us, lead us to think that the "proofs" to which Democritus refers were no longer considered such in the classical period, and would not be so today either.

undergoes at this time a sudden change: while in the VIIth and VIth centuries the philosophers affirm or vaticinate (or at most outline vague arguments, based on equally vague analogies), starting with Parmenides and especially Zenon, they argue and try to draw out general principles which can serve as a basis for their dialectic: it is in Parmenides that one finds the first statement of the principle of the excluded middle, and the proofs "by contradiction" of Zenon of Elea remain famous. But Zenon writes in the middle of the Vth century; and, whatever the uncertainties of our documentation,$^2$ it is very likely that at this time, mathematicians, in their own sphere, were currently using these principles.

As we said above, it is not for us to retrace the innumerable difficulties which abound at every step in the gestation of this Logic, and the controversies which result, from the Eleatics to Plato and Aristotle, via the Sophists; let us pick up here only the role played in this evolution by the assiduous cultivation of oratorical art and the analysis of language which is one of its corollaries, developments that it is agreed should be attributed mainly to the Sophists of the Vth century. On the other hand, even if the influence of mathematics is not always explicitly recognised, it is none the less manifest, in particular in the writings of Plato and Aristotle. It can be said that Plato was almost obsessed by mathematics; without being himself an inventor in this area, he kept up, after a particular period in his life, with the discoveries by contemporary mathematicians (many of whom were his friends or his pupils), and never after ceased from a most direct kind of interest, even going so far as to suggest new directions of research; so it is that constantly, in his writings, mathematics serves as illustration or model (and sometimes even, as with the Pythagoreans, feeding his leanings towards mysticism). As for his pupil Aristotle, he could not have avoided receiving the minimum of mathematical foundations required of the pupils of the Academy, and a volume has been produced consisting of extracts of his work that are about mathematics or refer to it [153 d]; but he seems never to have made a great effort to keep in touch with the mathematical movements of his time, and he only quotes in this area results that had been popularised a long time before. This displacement will besides only become more marked with the majority of later philosophers, of whom many, due to the lack of technical preparation, believe themselves in all good faith to be speaking knowledgeably about mathematics, whereas they will only be referring to a stage long since overtaken in its evolution.

The end product of this period, as far as Logic is concerned, is the monumental work of Aristotle [6], whose great merit is to have succeeded in systematising and codifying for the first time procedures of reasoning which

$^2$ The best classical example of proof by contradiction in mathematics is the proof of the irrationality of $\sqrt{2}$, to which Aristotle often refers; but modern experts have not been able to date this discovery with any accuracy, some putting it at the beginning and others at the very end of the Vth century (see p. 148 and the references cited there concerning this topic).

had remained vague or unformulated in his predecessors.$^3$ We must above all retain here, as our aim, the general thesis of this work, to know that it is possible to reduce all correct reasoning to the systematic application of a small number of immutable rules, which are independent of the particular nature of the objects in question (an independence which is clearly demonstrated by the representation of the concepts or propositions by means of letters — probably borrowed by Aristotle from the mathematicians). But Aristotle concentrates his attention almost exclusively on a particular type of relations and logical chains, making up what he calls the “syllogism”: it is mainly a case of relations that we would translate nowadays in the form $A \subset B$ or $A \cap B \neq \emptyset$ in the language of the theory of sets,$^4$ and with the way of chaining these relations or their negations, by means of the schema

Aristotle was still sufficiently informed of the mathematics of his era in order to have seen that schemas of this type were not sufficient to take account of all the logical operations used by mathematicians, nor, all the more, of the other applications of Logic ([6], An. Pr.,I, 46).$^5$ At least the deep study of the different forms of “syllogisms” which he undertakes (and which is almost entirely consecrated to the elucidation of the perpetual difficulties raised by the ambiguity or the obscurity of the objects on which the reasoning bears) gives him among others the chance to formulate the rules for obtaining the negation of a proposition ([6], An. Pr., I, 46). It is also to Aristotle that is

$^3$ In spite of the simplicity and the “obviousness” that the rules of logic formulated by Aristotle appear to lay out for us, it is only necessary to put them back in their historical setting in order to appreciate the difficulties that presented themselves in trying to obtain a precise statement of these rules, and the effort that Aristotle must have put forth in order to succeed: Plato, in his dialogues, where he is addressing a cultured public, still lets his characters get embroiled with questions as elementary as the relationships between the negation of $A \subset B$ and the relation $A \cap B = \emptyset$ (in modern notation), however making the correct answer appear later [264].

$^4$ The corresponding statements of Aristotle are “All A is B” and “Some A is B”; in this notation A (the “subject”) and B (the “predicate”) take the place of concepts, and to say that “All A is a B” means that it is possible to assign the concept B to each object to which it is possible to assign the concept A (A is the concept “man” and B the concept “mortal” in the classic example). The interpretation that we give to it consists in considering the set of objects to which the concepts A and B respectively can be applied; it is the point of view “of extension” so called, already known to Aristotle. But he considers mainly the relation “All A is B” from another point of view, that “of comprehension” where B is seen as one of the concepts that constitute in some way the more complex concept A, or, as Aristotle says, “belong” to it. At first sight, the two points of view seem as natural one as the other, but the point of view “of comprehension” has been a constant source of problems in the development of Logic (it seems further removed from intuition than the first, and leads fairly easily to errors, notably in schemes involving negations; cf. [69a], pp.21-32).

$^5$ For a critical discussion of syllogisms and their deficiencies, see for example ([69a], pp. 432-441) or ([164], pp. 44-50).

due the credit for having distinguished with great precision the role of "universal" propositions from that of "particular" propositions, the first sketch of quantifiers.$^6$ But it is too well known how the influence of his writings (often interpreted in a narrow and unintelligent way), which remain still very influential until well into the XIXth century, was to encourage philosophers in their neglect of the study of mathematics, and to block the progress of formal Logic.$^7$

However this latter continued to make progress in Antiquity, in the surroundings of the Megaric and Stoic schools, rivals of the Peripatetics. Our information about these tenets are unfortunately all at second hand, often passed on by adversaries or mediocre commentators. The essential progress achieved by these logicians consists, it would appear, of the formation of a "propositional calculus" in the meaning which it has today: instead of being restricted, like Aristotle, to propositions of the particular form $A \subset B$, they state rules about completely indeterminate propositions. Further, they had analysed the logical links between these rules in such a deep way that they knew how to deduce them all from five of them, set down as "unprovable", by means of procedures very similar to modern methods [23]. Unfortunately their influence was fairly ephemeral, and their results were to fall into oblivion until the day when they were rediscovered by the logicians of the XIXth century. Aristotle remains as the uncontested master in Logic until the XVIIth century; it is known in particular that the scholastic philosophers are entirely under his sway, and if their contribution to formal logic is far from negligible [25], it does not contain any progress at the highest level compared with the achievements of the philosophers of Antiquity.

It is appropriate, however, to note here that it does not seem that the works of Aristotle or his successors had a notable influence on mathematics. The Greek mathematicians conducted their research along the path opened by the Pythagoreans and their successors of the IVth century (Theodorus, Theeta, Eudoxus) without apparently bothering with formal logic in the presentation of their results: a finding that should hardly astonish when one compares the flexibility and precision acquired already by that time by mathematical reasoning, to the very rudimentary state of Aristotelian logic. And when logic will overtake this stage, it is again the new acquisitions from mathematics that will guide it in its evolution.

With the development of algebra, it was indeed impossible to avoid being struck by the analogy between the rules of formal logic and the rules of algebra, the ones like the others having in common the property of being applicable to objects (propositions or numbers) which are not precisely determined. And when in the XVIIth century algebraic notation took its definitive

$^6$ The absence of real quantifiers (with the modern meaning) until the end of the XIXth century, has been one of the reasons for the stagnation of formal Logic. $^7$ The case of an eminent academic is quoted who, at a recent conference at Princeton in the presence of Gödel, would allegedly have said that nothing new had been done in Logic since Aristotle!

form in the hands of Viète and Descartes, almost immediately one can see appearing diverse attempts at a symbolic notation intended for the representation of logical operations; but before Leibniz, these tentative efforts, for example like that of Herigone (1644) at writing down the proofs of elementary Geometry, or that of Pell (1659) at writing down those of Arithmetic, remain very superficial and do not lead to any progress in the analysis of mathematical reasoning.

With Leibniz, we are in the presence of a philosopher who is also a mathematician of the first rank, and who will know how to draw out of his mathematical experience the germ of the ideas that will bring formal logic out of the scholastic dead-end.$^8$ A universal spirit if ever there were one, inexhaustible source of original and fruitful ideas, Leibniz would be interested even more in Logic as it found itself at the very heart of his grand projects for formalising language and thought, at which he never ceased working throughout his life. Trained in his childhood in scholastic logic, he was seduced by the idea (going back to Raymond Lulle) of a method which would reduce all human concepts to primitive concepts, making up an “Alphabet of human thought”, and would rebuild them in a quasi mechanical way to obtain all true propositions ([198 b], v. VII, p. 185; cf. [69a], chap. II). Still very young, he had also conceived another much more original idea, that of the usefulness of symbolic notations as an “Ariadne’s string” string of thought:$^9$ “The true method”, he said, “must provide us with a filum Ariadnes, that is to say a kind of sensitive and coarse means that guides the mind, in the same way as lines drawn in geometry and the type of operations that are prescribed to apprentices in Arithmetic. Without that our mind would not know how to go along a long path without straying.” ([198 b], v. VII, p. 22; cf. [69 a], p. 90). Knowing little of the mathematics of his time until about his 25th year, it is at first in the form of a “universal language” that he presents his projects ([69 a], chap. III); but as soon as he comes in contact with Algebra, he adopts it as model for his

$^8$ Although Descartes and (to a lesser degree) Pascal devoted part of their philosophical work to the foundations of mathematics, their contribution to the progress of formal Logic is negligible. No doubt this arises from the fundamental tendency of their thought, the effort to break free of the scholastic tutelage, which caused them to reject everything that could be linked to it, and most especially formal Logic. In fact in his Thoughts on the geometric spirit Pascal, as he himself recognises, restricts himself essentially to moulding in well cast formulae the known principles from Euclidean proofs (for example, the famous precept: “Always substitute mentally the definitions in place of the defined” ([244], v.IX, p.280) was essentially known to Aristotle ([6], Top., VI, 4; [153 d], p.187)). As for Descartes, the rules of reasoning that he states are above all psychological precepts (fairly vague) and not logical criteria; as the criticism of Leibniz shows ([69 a], p. 94 and 202-203), they only have a subjective span as a result.

$^9$ Of course, the interest in such a symbolism had not escaped the predecessors of Leibniz as far as mathematics was concerned, and Descartes, for example, recommends replacing entire figures “by very short symbols” (XVIth Rule for the direction of the mind; [85 a], v. X, p. 454). But nobody before Leibniz had insisted with such vigour on the universal range of this principle.

"universal Characteristic". By that he means a type of symbolic language, capable of expressing without ambiguity all human thought, of reinforcing our powers of reasoning, of avoiding errors by means of an effort of entirely mechanical concentration, finally constructed so that "the chimera that the person himself who presents them does not hear, would be impossible to write down in these characters" ([198 a], v. I, p. 187). In innumerable passages in his writings where Leibniz alludes to this grandiose project and to the progress to which its attainment would lead (cf. [69 a], chap. IV and VI), it is seen with what clarity he had conceived the notion of a formalised language, a pure combination of symbols of which only the chaining is important,$^{10}$ in such a way that a machine would be able to produce all the theorems,$^{11}$ and that all controversies would be resolved by a simple calculation ([198 b], v. VII, p. 198-203). If these hopes might appear excessive, it is none the less true that it is to this constant tendency of Leibniz's thought that must be tied a good part of his mathematical work, beginning with his work on the symbolism of the infinitesimal Calculus (see p. 190 ff.); he was himself perfectly aware of this, and linked explicitly also to his "Characteristic" his ideas on indicial notation and determinants ([198 a], v. II, p. 204; cf. [69 a], pp. 481-487) and his sketch of "geometrical Calculus" (see pp. 50 and 61 ff.; cf. [69 a], chap IX). But in his mind the essential part had to be symbolic Logic, or, as he puts it a "Calculus ratiocinator", and though he does not succeed in creating this calculus, at least we see him trying to at least three times. On his first attempt, he has the idea of associating with each "primitive" term a prime number, each term made up of several primitive terms being represented by the product of the corresponding prime numbers;$^{12}$ he seeks to translate into this system the usual rules of syllogism, but runs up against major complications caused by negation (that he tries, fairly naturally, to represent by a change of sign) and abandons rapidly this path ([198 c], pp. 42-96; cf. [69 a], pp.326-344). In later attempts, he seeks to give Aristotelian logic a more algebraic form; sometimes he keeps the notation $AB$ for the conjunction of two concepts; sometimes he uses the notation $A + B;$^{13} he notes (in multiplicative notation) the law of idempotence $AA = A$, remarks that one can substitute the proposition "all A is B" by the equality $A = AB$ and that one can recover starting from there most of the rules of Aristotle by a purely algebraic calculation ([198 c], pp. 229-237 and 356-399; cf. [69 a], pp.345-364); he also has the idea of the empty set ("non Ens"), and recognises

$^{10}$ It is striking to see him quoting here as examples of reasoning "of the right type", "a collector's account" or even a legal text ([198 b], v. IV, p. 295). $^{11}$ One knows that this conception of a "logical machine" is used these days in metamathematics, where it is of great usefullness ([181], chap. XIII). $^{12}$ The idea was taken up successfully by Gödel in his work on metamathematics, in a slightly different form (cf. [130 a] and [181], p. 254). $^{13}$ Leibniz only seeks to introduce disjunction into his calculus in a few fragments (where he denotes it by $A + B$) and does not seem to have succeeded in handling simultaneously this operation and conjunction in a satisfactory way ([69 a], p. 363).

for instance the equivalence of the propositions “all A is B” and “A.(not B) is not” (loc. cit.). Further, he remarks that his logical calculus is applicable not only to the logic of concepts, but also to that of propositions ([198 c], p. 377). He appears to be therefore very near to “Boolean calculus”. Unfortunately, it appears that he was not able to free himself completely from the scholastic influence; not only does he set himself as almost the only aim of his calculus the transcription, into his notation, of the rules of syllogism,$^{14}$ but he goes as far as sacrificing his most felicitous ideas to the desire to recover completely the rules of Aristotle, even those that were incompatible with the idea of an empty set.$^{15}$

The work of Leibniz remained for the most part unpublished until the beginning of the XXth century, and had only little direct influence. During the whole of the XVIIIth and the beginning of the XIXth centuries, different authors (de Segner, J. Lambert, Ploucquet, Holland, De Castillon, Gregonne) sketched attempts similar to those of Leibniz, without ever progressing substantially beyond the point at which he had stopped; their work only had a feeble effect, which means that most of them ignored all results due to their predecessors.$^{16}$ It is in any case under the same conditions that G. Boole, who must be considered to be the real creator of modern symbolic logic [29], writes. His key idea consists in putting himself systematically in the position of considering “extension”, so of calculating directly with sets, and writing $xy$ for the intersection of two sets, and $x + y$ for their union when $x$ and $y$ have no element in common. He introduces as well a “universe” denoted 1 (the set of all elements) and the empty set denoted 0, and he writes $1 - x$ for the complement of $x$. As Leibniz had done, he interprets the relation of inclusion by the relation $xy = x$ (from which he extracts without difficulty the justification of the rules of classical syllogism) and his notation for union and complements gives his system a flexibility that was missing in his forerunners.$^{17}$ Furthermore, by associating with each proposition the set of “cases” in which it holds, he interprets the relation of implication as an inclusion, and his calculus of sets gives him in this way the rules of the “propositional calculus”.

$^{14}$ Leibniz knew very well that Aristotelian logic was insufficient to translate formally mathematical texts, but, in spite of some attempts, he never succeeded in improving it in this respect ([69 a], pp. 435 and 560).

$^{15}$ These consist of the rules “of conversion” so-called based on the postulate that “All A is a B” entails “Some A is a B”, which assumes naturally that A is not empty.

$^{16}$ The influence of Kant, from the middle of the XVIIIth century, without doubt played some part in the lack of interest aroused by formal logic at this time; he feels that “we have no need of any new invention in logic”, the shape given to it by Aristotle being sufficient for all applications that can be made of it ([178], v. VIII, p. 340). Concerning the dogmatic concepts of Kant concerning mathematics and logic, one can consult [69b].

$^{17}$ Note especially that Boole uses the distributivity of intersection over union, which seems to have been noticed for the first time by J. Lambert.

In the second half of the XIXth century, Boole’s system formed the basis for the work of an active school of logicians who improved it and completed it at several points. It is thus that Jevons (1864) enlarged the meaning of the operation of union $x + y$ by extending it to the case in which $x$ and $y$ are arbitrary; A. de Morgan in 1858 and C. S. Peirce in 1867 proved the duality laws

De Morgan tackles also, in 1860, the study of relations, defining inversion and the composition of binary relations (that is to say the operations corresponding to the operations $G^{-1}$ and $G_1 \circ G_2$ on graphs).$^{19}$ All this work is systematically expounded and developed in the massive and prolific work of Schröder [277]. But it is fairly interesting to note that the logicians of whom we have been speaking do not seem very interested in the application of their results to mathematics and that, on the contrary, Boole and Schröder especially seem to have as their principal aim to develop “Boolean” algebra by imitating the methods and problems of classical algebra (often in a fairly artificial way). The reasons for this attitude must doubtless be seen in the fact that the Boolean calculus still lacked a facility for transcribing most of mathematical reasoning,$^{20}$ and only supplied in this way a very partial answer to the great dream of Leibniz. The construction of a formalism better adapted to mathematics — of which the introduction of variables and quantifiers, due independently to Frege [117 a, b, c] and C. S. Peirce [248 b], make up the major stage — was the work of logicians and mathematicians who, unlike the above, had above all as their aim applications to the foundations of mathematics.

Frege’s project [117 b and c] was to create a foundation for arithmetic based on a logic formalised by a “writing of concepts” (Begriffsschrift) and we will come back later (p. 29) to the way in which he defines the natural numbers. His work is characterised by extreme precision and attention to detail in the analysis of concepts; it is because of this tendency that he introduces many a distinction which turns out to be of great importance in modern logic: for instance it is he who first distinguishes between the statement of a proposition and the assertion that this proposition is true, between

18 It must be noted that statements equivalent to these rules are already to be found in several scholastic philosophers ([25], p.67 ff.). 19 However the notion of “Cartesian” product of two arbitrary sets does not seem to be explicitly introduced until G. Cantor ([47], p. 286); it is also Cantor who first defines exponentiation $A^B$ (loc. cit., p. 287); the general notion of an infinite product is due to A. N. Whitehead ([333], p.369). The use of graphs of relations is fairly recent; if exception is made, naturally, of the classical case of real valued functions of a real variable, it seems to appear for the first time amongst the Italian geometers, notably C. Segre, in their study of algebraic correspondences. 20 For each relation obtained from one or more given relations by use of our quantifiers, it would be necessary, in this calculus, to introduce an ad hoc notation of the type of the notation $G^{-1}$ and $G_1 \circ G_2$ (cf for example [248 b]).

the relation of belonging and that of inclusion, between an object $x$ and the set $\{x\}$ reduced to this single object, etc.. His formalised logic, which contains not only "variables" with the meaning used in mathematics, but also "propositional variables" representing indeterminate relations, susceptible to quantification, would later (through the work of Russell and Whitehead) supply the fundamental tool for metamathematics. Unfortunately, the symbols which he adopts are hardly inspiring, of a terrifying typographic complexity and far removed from the practice of mathematicians; which had the result of turning away these latter and reducing considerably the influence of Frege on his contemporaries.

Peano's goal was at the same time much vaster and much more down to earth; it consisted in publishing a "Formulary of mathematics", written entirely in formal language and containing, not only mathematical logic, but all the results of the most important branches of mathematics. The speed with which he managed to complete this ambitious project, helped by a host of enthusiastic collaborators (Vailati, Pieri, Padoa, Vacca, Vivanti, Fano, Burali-Forti) is witness to the excellence of the symbolism which he had adopted: following closely the current practice of mathematicians, and introducing numerous well-chosen abbreviating symbols, his language remains as well fairly easily legible, thanks notably to an ingenious system for replacing brackets by full stop separators [246 f]. Much notation due to Peano is today adopted by the majority of mathematicians: we quote $\in, \supset$ (but, contrary to the present use, with the meaning of "is contained in" or "implies"$^{21}$), $\cup, \cap, A - B$ (set of differences $a - b$ where $a \in A$ and $b \in B$). On the other hand, it is in the "Formulary" that is found for the first time a thorough analysis of the general notion of function, of those of direct image$^{22}$ and reciprocal image, and the remark that a sequence is only a function defined on $\mathbf{N}$. But quantification, with Peano, is subject to hampering restrictions (one can only, in principle, quantify, in his system, relations of the form $A \to B, A \Leftrightarrow B$ or $A = B$). Further the almost fanatical zeal of some of his disciples laid them wide open to ridicule; criticism, often unjustified, by H. Poincaré in particular, was a heavy blow to the Peano school and became an obstacle to the diffusion of his doctrines in the mathematical world.

With Frege and Peano the essential elements of the formal languages used today were acquired. The most widespread is doubtless that hammered out by Russell and Whitehead in their great work "Principia Mathematica", which happily links the precision of Frege and the convenience of Peano [266]. Most actual formal languages are differentiated from it only by changes of secondary importance, aimed at simplifying its use. Among the most ingenious, we quote the "functional" writing of relations (for instance $\in xy$ instead of

$^{21}$ This indicates well to what extent was rooted, even in him, the old habit of thinking "in comprehension" rather than "in extension".

$^{22}$ The introduction of this seems due to Dedekind, in his work "Was sind und was sollen die Zahlen", of which we will speak later ([79], v. III, p.348).

$x \in y$), thought up by Lukasiewicz, thanks to which brackets can be completely omitted; but the most interesting is without doubt the introduction by Hilbert of the symbol $\tau$, which allows the consideration of the quantifiers $\exists$ and $\forall$ as abbreviation signs, the avoidance of the introduction of the "universal" functional symbol $\iota$ of Peano and Russell (which is only applied to functional relations), and finally which avoids the need to formulate the axiom of choice in the theory of sets ([163 a], v.III, p. 183).

THE NOTION OF TRUTH IN MATHEMATICS.

Mathematicians have always been sure that they prove "truths" or "true propositions"; such a conviction can obviously only be sentimental or metaphysical, and it is not by getting on to mathematical ground that it can be justified, nor even given a meaning that does not make it a tautology. The history of the concept of truth in mathematics is the concern therefore of the history of philosophy and not of that of mathematics; but the evolution of this concept has had an undeniable influence on that of mathematics, and because of this we can not let it go by in silence.

Let us note first that it is as rare to see a mathematician in possession of a strong philosophical culture as to see a philosopher who has an extensive knowledge of mathematics; the opinions of mathematicians on topics in philosophy, even when these questions are concerned with their field, are most often opinions received at second or third hand, coming from doubtful sources. But, precisely because of this, it is these average opinions which interest the mathematical historian, at least as much as the original views of thinkers such as Descartes or Leibniz (to mention two who were also mathematicians of the first rank), Plato (who at least kept up with the mathematics of his time), Aristotle or Kant (of whom the same could not be said).

The traditional notion of mathematical truth is that which goes back to the Renaissance. In this concept, there is no great difference between those objects which are the concern of mathematicians and those which are the concern of natural science; both are knowable and man has obtained a grasp on them both through intuition and through reason; there are no grounds to doubt either intuition or reason, which are fallible only if used incorrectly. "One must have", says Pascal, "altogether a wrong spirit to reason badly about principles so great that it is almost impossible for them to escape" ([244], v. XII, p. 9). Descartes, by his fireside, convinced himself that "there have been only Mathematicians who were able to find some proofs, that is to say some sure and certain reasons" ([85 a], v.VI, p. 19) and that (if his tale is to be believed) well before he had built up a metaphysics in which "that very same thing", he says, "that I previously took as a rule, namely that the objects that we can visualise very clearly and very distinctly are all true, is only ensured because God is or exists, and that he is a perfect being" ([85 a], v. VI, p. 38). If Leibniz objects to Descartes that it cannot be seen how to recognise that an idea is "clear and distinct",$^{23}$ he also considers axioms as obvious and undisputed consequences of the definitions as soon as the expressions are understood.$^{24}$ It must not be forgotten either that, in the language of that time, mathematics includes many sciences that we do not recognise anymore as such, and sometimes even as far as the art of the engineer; and in the confidence that they inspire, the surprising success of their applications to "natural philosophy", to the "mechanical arts", to navigation, comes in for a large part.

In this way of looking at things, axioms are no more susceptible to being discussed or put into question than the rules of deduction; at most the choice can be left to each person, according to his preferences, to reason "in the way of the ancients" or to give free reign to his intuition. The choice of point of departure is also a matter of individual preference, and one sees appearing numerous "editions" of Euclid where the solid logical framework of the Elements becomes a strange travesty; some surveys are given of the infinitesimal calculus and rational mechanics, supposedly deduced from fundamentals which are remarkably badly established; and Spinoza was perhaps giving in good faith his Ethics as being proved in the manner of geometers "more geometrico demonstrata". If it is hard to find in the XVIIth century two mathematicians who agree on any matter whatsoever, if the polemics occur daily, endless and acrimonious, the notion of truth remains none the less not in question. "Only having one truth about each object", says Descartes, "whoever finds it knows as much as can be known about it" ([85 a], v. VI, p. 21).

Although no Greek mathematical text of the high period on these questions has survived, it is probable that the point of view of Greek mathematicians on this subject had many more nuances. It is by experience only that the rules of deduction were elaborated to the point of inspiring complete confidence; before they could be considered to be above all discussion, it was necessary to go through mainly fumblings and paralogisms. It would also be

$^{23}$ "Those that have given us methods" he says in this context "give, without doubt, some beautiful precepts, but not the means to obey them." ([198 b], v. VII, p. 21). And elsewhere, making fun of the Cartesian rules, he compares them to the recipes of the alchemists: "Take what you need, act as you must, and you will obtain that for which you wish!" ([198 b], v. IV, p. 329).

$^{24}$ On this point, Leibniz is still under scholastic influence; he still thinks that propositions establish a link from "subject" to "predicate" among concepts. As soon as concepts have been reduced to "primitive" concepts (which, as we have seen, is one of his fundamental ideas), everything reduces, for Leibniz, to checking relations of "inclusion" by means of what he calls "identical axioms" (essentially the propositions $A = A$ and $A \subset A$) and the principle of "substitution of equivalents" (if $A = B$, $A$ can be replaced everywhere by $B$ ([69 a], pp. 184-206)). It is interesting in this context to note that, in conformity with his desire to reduce everything to Logic and to "prove everything that is provable", Leibniz proves the symmetry and transitivity of the relation of equality, starting from the axiom $A = A$ and the principle of substitution of equivalents ([198 a], v. VII, pp. 77-78).

to misconstrue the critical spirit of the Greeks, their taste for discussion and sophistry, to imagine that the “axioms” themselves that Pascal judged to be the most obvious (and that, according to a legend spread by his sister, he had, with infallible instinct, discovered himself during his childhood) were not the subject of long discussions. In an area that was not that of geometry strictly speaking, the paradoxes of the Eleates have preserved for us some traces of these polemics; and Archimedes when he makes the observation ([5 b], v. II, p. 265) that his predecessors made use in many situations of the axiom to which we usually ascribe his name, adds that what is proved by means of this axiom “has been accepted no less than what has been proved without it”, and that it is sufficient that his own results be accepted on the same basis. Plato, in accordance with his metaphysical views, presents mathematics as a means of access to a “truth in itself” and the objects it deals with as having a real existence in the world of ideas; he characterises none the less the mathematical method precisely in a famous extract from the Republic: “Those that are involved with geometry and arithmetic ... assume even and odd, three kinds of angles; they treat them as known objects: once assumed, they esteem that they no longer need to give account of them either to themselves or to others, [considering it] as clear to each one; and starting there, they proceed in sequence, in order to reach by common agreement the goal that their research had suggested” ([250], Book VI, 510 c-e). That which makes up a proof is therefore firstly a point of departure supplying an arbitrary start (even though “clear to everyone”), and beyond which, he says a bit further on, one does not try to dig; and then, a discussion that follows in sequence a series of intermediate stages; finally, at each step, the consent of the interlocutor guaranteeing the correctness of the reasoning. It must be added that once the axioms are stated, no new appeal to intuition is allowed on principle: Proclus, quoting Géminus, recalls that “we have learnt from the pioneers of this science themselves, to take no account of conclusions which are only plausible when it is a case of reasoning which must be part of our geometric doctrine” ([153 e], v.I, p. 203).

Thus it is to experience and the cauldron of criticism that is due the elaboration of the rules of mathematical reasoning; and if it is true, as has been argued in a plausible way [317 d], that Book VIII of Euclid has kept for us part of the arithmetic of Archytas, it is not surprising to see there the rigidity of the rather pedantic reasoning that does not fail to appear in all mathematical schools where “rigour” is discovered or believed to have been discovered. But, once having entered into the practice of mathematicians, it does not seem that these rules of reasoning have ever been doubted until quite recently: if with Aristotle and the Stoics, some of the rules are deduced from others by schemes of reasoning, the primitive rules are always assumed to be evident. In the same way, having gone back to the “hypotheses”, “axioms” and “postulates” that appeared to them to supply a solid foundation for the science of their time (such as those for example that they put forward in the first "Elements" that tradition ascribes to Hippocrates of Chio, about 450 B.C.), the Greek mathematicians from the classical period seem to have bent all their efforts to the discovery of new results rather than to a critique of these foundations that, at that time, could not have failed to be sterile; and, putting aside all metaphysical preoccupations, it is from this general accord amongst mathematicians on the bases of their science that the text of Plato quoted above bears testimony.

On the other hand Greek mathematicians do not seem to have believed it possible to explain the "primary notions" that were their points of departure, straight line, surface, ratio of quantities; if they give "definitions" of them, it is obviously a crisis of conscience and without any illusions about their range. It goes without saying that on the other hand, about definitions other than those of "primary notions" (definitions often called "nominal"), the Greek mathematicians and philosophers had perfectly clear ideas. It is in this context that intervenes explicitly, for the first time no doubt, the question of "existence" in mathematics. Aristotle does not miss observing that a definition does not imply the existence of the object defined, and that there must be over and above that either a postulate or a proof. No doubt his observation was derived from the practice of mathematicians; in any case Euclid takes care to postulate the existence of a circle, and to prove that of an equilateral triangle, of parallels, of a square, etc. as he introduces them into his arguments ([153 e], Book I); these proofs are "constructions"; in other words, he exhibits, basing himself on the axioms, mathematical objects that he proves satisfy the definitions that he needs to justify.

Thus we see Greek mathematics in the classical era reaching a kind of empirical certainty (whatever the metaphysical bases for it from some philosopher or other might be); if it cannot be conceived that the rules of reasoning can be questioned, the success of Greek science, and the feeling that exists of the inopportunity of a critical revision, play a great part in the confidence that the axioms themselves inspire, a confidence that is rather of the order of that (almost unlimited, as well) that was put during the last century in the principles of theoretical physics. It is in any case what is suggested by the motto of the school "nihil est in intellectu quod non prius fuerit in sensu", which is precisely that against which Descartes stands out, as not giving a firm enough basis to that which Descartes hoped to extract from the use of reason.

One must come down to the beginning of the XIXth century to see mathematicians recover from the arrogance of a Descartes (not to say that of a Kant or that of a Hegel, this latter somewhat late, as was fashionable, about the science of his time$^{25}$), to a position as well balanced as that of the Greeks. The first blow to the classical concepts is the establishment of non-Euclidean

$^{25}$ In his inaugural dissertation, he "proves" that there were at most seven planets, the same year that the eighth was being discovered.

hyperbolic geometry by Gauss, Lobatschevsky and Bolyai at the beginning of the century. We will not undertake to retrace here in detail the origins of this discovery, the outcome of numerous unfruitful attempts to prove the parallel postulate (see [105 a and b]). At the time its effect on the principles of mathematics was perhaps not so deep as is sometimes said. It simply forces the abandonment of the pretensions of the previous century to the “absolute truth” of Euclidean geometry, and all the more, the Leibnizian point of view that definitions imply axioms; these latter no longer appear as at all “obvious”, but rather as hypotheses of which it must be determined whether they are adapted to the mathematical representation of the world of the senses. Gauss and Lobatschevsky believe that the debate between the different possible geometries can be determined by experience ([206], p. 76). It is also the point of view of Riemann, of whom the famous inaugural Lecture “On the hypotheses which form the foundation of geometry” had as aim to furnish a general mathematical framework for the various natural phenomena: “What remains to be resolved” he says, “is the question of knowing to what extent and up to what point these hypotheses are found to be confirmed by experience” ([259 a], p. 284). But that is a problem which visibly has nothing anymore to do with Mathematics; and none of the previous authors seem to put into question that, even if a “geometry” does not correspond to an experimental reality, its theorems remain no less “mathematical truths”.

All the same, if it so, it is certainly no longer to an unlimited confidence in classical “geometric intuition” that such a conviction is due; the description that Riemann seeks to give of “multiplicities n times extended”, the object of his work, only relies on “intuitive” considerations to reach a justification for the introduction of “local co-ordinates”; from that moment on he apparently seems to feel himself on solid ground, namely that of Analysis. But this latter is grounded in the end on the concept of real number, which remained until then of a very intuitive nature; and progress in the theory of functions was leading in this respect to some very worrying results: with the research of Riemann himself on integration, and more so with the examples of curves without tangents, constructed by Bolzano and Weierstrass, it is the whole of the pathology of mathematics that was beginning. For a century we have seen so many monsters of this species that we are a bit blasé, and the most weird teratological characters must be accumulated in order still to astound us. But the effect produced on the majority of mathematicians of the XIXth century went from disgust to consternation: “How” H. Poincaré asks himself, “can intuition deceive us at this point?” ([251 d], p. 19); and Hermite (not without a spark of humour which the commentators of this famous sentence

26 Cf. the arguments of Poincaré in favour of “simplicity” and of the “convenience” of Euclidean geometry ([251 c], p. 67), as well as the analysis whereby, a bit later, he reaches the conclusion that experience does not furnish an absolute criterion for the choice of one geometry rather than another as a framework for natural phenomena. 27 Again this word is only justified for $n \leq 3$; for larger values of $n$, it is a case of arguing by analogy.

do not all seem to have perceived) declares that he "turns away with fear and horror from this lamentable plague of continuous functions that do not have a derivative" ([160], v. II, p. 318). The worst was that these phenomena, so contrary to common sense, could no longer be laid at the door of ideas badly elucidated, as at the time of the "indivisibles" (see p. 173), since they survived after the reform of Bolzano, Abel and Cauchy, who had established the foundation of the notion of limit in a manner as rigorous as the theory of ratios (see p. 154). It is thus fully to the gross and incomplete character of our geometric intuition that the account must be laid, and it is reasonable that since then it has remained discredited quite justifiably as a means of proof.

This realisation would inevitably react on classical mathematics, starting with geometry. In whatever respect the axiomatic construction of Euclid had been held, more than one imperfection had been noticed, and that already in antiquity. It was the postulate on parallels that had been the object of the greatest number of criticisms and attempts at proof; but the followers and commentators of Euclid had also attempted to prove other postulates (notably that of the equality of right angles) or recognised the insufficiency of certain definitions, such as those of the straight line or plane. In the XVIth century, Clavius, an editor of the Elements, notes the absence of a postulate guaranteeing the existence of the fourth proportional; for his part, Leibniz remarks that Euclid uses geometric intuition without mentioning it explicitly, for example when he admits (Elements, Book I, prop. 1) that two circles, each of which goes through the centre of the other have a common point ([198 b], v. VII, p. 166). Gauss (who himself did not deny himself the use of such topological considerations) draws attention to the role played in Euclidean constructions by the notion of a point (or a straight line) being "between" two others, notion that is however not defined ([124 a], v. VIII, p. 222). Finally, the use of displacements — notably in the "case of equality of triangles" — long assumed to be obvious,$^{28}$ was soon to appear to the critics of the XIXth century as relying also on unstated axioms. One ends up thus, in the period from 1860 to 1885, with different partial revisions of the beginnings of geometry (Helmholtz, M\'eray, Houël) tending to remedy some of these gaps. But it is only with M. Pasch [245] that the abandonment of all appeals to intuition is a programme properly formulated and followed with full rigour. The success of his enterprise soon brought him numerous emulators who, principally between 1890 and 1910, gave quite varied statements of the axioms of Euclidean geometry. The most famous of these works were those of Peano, written in his symbolic language [246 d], and especially the "Grundlagen der Geometrie" of Hilbert [163 c], appearing in 1899, a book which, by its lucidity and the depth of its exposition, was to become immediately,

$^{28}$ It must be noted however that, already in the XVIth century, a commentator on Euclid, J. Peletier, protested against this means of proof, in terms close to those of modern critics ([153 e], v. I, p. 249).

with full justification, the charter for modern axiomatics, even to the extent of leading to forgetting its forerunners. It is indeed that, not content with giving there a complete system of axioms for Euclidean geometry, Hilbert classifies these axioms into different groups of different types, and sets himself to determine the exact area of influence of each of these groups of axioms, not only in developing the logical consequences of each of them in isolation, but also in discussing the different "geometries" obtained when one omits or modifies certain of these axioms (geometries amongst which those of Lobatschevsky and of Riemann appear only as particular cases$^{29}$); he thus puts clearly in the picture, in an area considered until then as one of those nearest the reality of the senses, the freedom of which the mathematician disposes in his choice of postulates. In spite of the disarray caused in more than one philosopher by these "metageometries" with strange properties, the thesis of the "Grundlagen" was rapidly adopted almost unanimously by mathematicians; H. Poincaré, although hardly guilty of favouring formalism, recognised in 1902 that the axioms of geometry are conventions, for which the notion of "truth", as it is normally understood, has no more meaning ([251 c], pp. 66-67). "Mathematical truth" resides thus uniquely in logical deduction starting from premises arbitrarily set by axioms. As will be seen later (pp. 35 ff.), the validity of the rules of reasoning used in making these deductions was itself soon to be put in question, bringing about thus a complete reshaping of the basic conceptions of mathematics.

OBJECTS, MODELS AND STRUCTURES.

A) Objects and structures of mathematics. — From Antiquity to the XIXth century, there is a common understanding as to what are the principal objects of the mathematicians; they are exactly those that are mentioned by Plato in the passage quoted earlier (p. 13); numbers, quantities and figures. If, at first, must be added the objects and phenomena which are the subject of Mechanics, Astronomy, Optics and Music, these "mathematical" disciplines are always clearly separated, amongst the Greeks, from Arithmetic and Geometry, and starting with the Renaissance they fairly quickly attain the rank of independent sciences.

Whatever the philosophical nuances by which the conception of mathematical objects are coloured by such and such a mathematician or philosopher, there is at least one point on which there is unanimity: it is that these objects are given to us and it is not in our power to assign to them arbitrary properties, in the same way as a physicist cannot change a natural phenomenon. Truth to say, psychological reactions doubtless form one part

$^{29}$ The one which appears to have struck his contemporaries most deeply is the "non-Archimedean" geometry, that is to say the geometry having as base field an ordered non-archimedean field (commutative or not) which (in the commutative case) had been introduced a few years earlier by Veronese [318].

of these views, reactions which are not for us to follow up, but which each mathematician knows when he wears himself out in vain efforts to seize upon a proof which seems to slip away endlessly. From there to assimilate this resistance to the obstacles that the world of senses sets in our path there is only a single step; and even today, more than one, who claims an intransigent formalism, would subscribe voluntarily, in his innermost being, to this admission of Hermite: "I believe that numbers and the functions of Analysis are not the arbitrary result of our minds; I think that they exist outside of us, with the same character of necessity as the things of objective reality, and we meet them or discover them, and study them, as do the physicists, the chemists and the zoologists" ([160], v. II, p. 398).

There is no question in the classical conception of mathematics, of straying away from the study of numbers and figures; but this official doctrine, to which every mathematician believes himself bound to bring his verbal adherence, becomes no less bit by bit an intolerable burden, as new ideas accumulate. The embarrassment of the algebraists up against negative numbers does not cease until analytical Geometry gives it a convenient "interpretation"; but, well into the XVIIIth century still, d'Alembert, discussing the question in the Encyclopédie ([75 a], article NÉGATIF), loses his nerve suddenly after a column of fairly confused explanations, and is content to conclude that "the rules of algebraic operations on negative quantities are generally assumed by everybody and perceived generally as exact, whatever idea is linked elsewhere to these quantities". As for imaginary numbers, the scandal is far bigger; for if they are "impossible" roots and if (until about 1800) no way of "interpreting" them is seen, how can one, without contradiction, talk of these indefinable beings, and above all why introduce them? D'Alembert keeps here a prudent silence and does not even state these questions, no doubt because he realises that he would not be able to answer other than was done naively by A. Girard a century earlier ([129], f. 22): "It could be said: what use are these impossible solutions? I answer: for three things, for the certainty of the general rule, and that there is no other solution, and for its usefulness."

In Analysis, the situation, in the XVIIth century, was no better. It is a happy circumstance that analytical Geometry was to appear, as if at a designated point, to give a "representation" in the shape of a geometric figure, of the great creation of the XVIIth century, the notion of a function, and so assist powerfully (with Fermat, Pascal or Barrow) at the birth of the infinitesimal Calculus (cf. p. 193). But it is known on the other hand, to what philosophico-mathematical controversies the notions of the infinitely small and indivisible were going to give rise. And if d'Alembert is happier here, and recognises that in the "metaphysics" of the infinitesimal Calculus there is nothing other than the notion of limit ([75 a], articles DIFFÉRENTIEL and LIMITE, and [75 b]), he is no more able than his contemporaries, to understand the real meaning of expansion in divergent series, and to explain the paradox of exact results obtained at the end of calculations with expressions deprived of any numerical interpretation. Finally, even in the domain of "geometric certainty", the Euclidean framework bursts: when Stirling, in 1717, does not shrink from saying that a certain curve has a "double imaginary point at infinity" ([299], p. 93 of the new edition), he would certainly be in trouble trying to link such an "object" to commonly understood notions; and Poncelet who, at the beginning of the XIXth century, gives considerable development to such ideas in founding projective geometry (see p. 131), is content still to invoke as justification a "principle of continuity" which is altogether metaphysical.

One can conceive that, under these conditions (and at the exact moment when, paradoxically, the "absolute truth" of mathematics is proclaimed with the greatest force), the notion of proof seems to become more and more blurred during the XVIIIth century, since it is out of the question to set down, as did the Greeks, the notions on which reasoning is conducted, and their fundamental properties. The return towards rigour, which starts at the beginning of the XIXth century, brings some improvement to this state of things, but does not stop for all that the stream of new notions: one sees thus appearing in Algebra the imaginaries of Galois ([123], p. 113-127), the ideal numbers of Kummer [188 b], which are followed by vectors and quaternions, n-dimensional spaces, multivectors and tensors (see pp. 61 ff.), not to speak of Boolean algebra. No doubt great progress (which precisely allows the return to rigour, without losing any of the conquests of previous times) occurs with the possibility of giving "models" for these new notions in more classical terms: the ideal numbers or the imaginaries of Galois are interpreted by means of the theory of congruences (see pp. 82 ff.), n-dimensional geometry only appears (if so desired) as a pure language for stating results in algebra "with n variables"; and for the classical imaginary numbers — of which the geometric representation by the points of a plane (see pp. 161 ff.) marks the beginning of this flowering of Algebra — there is soon the choice between the geometric "model" and an interpretation in terms of congruences (cf. p. 82). But mathematicians begin at last to feel sharply that it is to fight against a natural slope along which their work is dragging them, and that it must be allowable, in mathematics, to reason about objects that have no sensible "interpretation": "It is not of the essence of mathematics", says Boole in 1854, "to be occupied with the ideas of number and quantity" ([29], v. II, p. 13).30 The same preoccupation leads Grassmann, in his "Ausdehnungslehre"

30 Leibniz, in this respect, appears again as a precursor: "universal Mathematics" he says, "is, so to speak, the Logic of the imagination", and must deal with "everything in the domain of the imagination which is susceptible to exact determination" ([198 c], p. 348; cf. [69 a], pp.290-291); and for him, the masterpiece of Mathematics thus understood is what he calls "Combinatorics" or the "Art of formulas", by which he means essentially the science of abstract relations between mathematical objects. But just as until then the relations considered in mathematics were almost exclusively relations of quantities (equality, inequality, ratio), Leibniz conceives many other types of relations which, in his opinion, should have been studied of 1844, to present his calculus in a form from which the notions of number or of geometric object are, at first, excluded.$^{31}$ And, a bit later, Riemann, in his inaugural Lecture, takes care at the beginning not to speak of "points", but rather of "determinations" (Bestimmungsweise), in his description of "multiplicities $n$ times extended", and underlines that, in such a multiplicity, the "metric relations" (Massverhältnisse) "can only be studied for abstract quantities and can only be represented by formulae; under certain conditions, one can however decompose them into relations of which each taken in isolation is susceptible to a geometric representation, and in that way it is possible to express the results of the calculation in a geometric form" ([259 a], p. 276).

From that moment, the expansion of the axiomatic method is an accomplished fact. If, for a while longer, it is believed useful to control, when possible, the "abstract" results by geometric intuition, at least it is admitted that the "classical" objects are no longer the only ones that the mathematician can legitimately study. It is that — precisely because of the multiple "interpretations" or "models" that are possible — it has been recognised that the "nature" of mathematical objects is in the end secondary, and that it matters little, for example, that a result is presented as a theorem of "pure" geometry, or as a theorem of algebra by the means of analytical geometry. In other words, the essence of mathematics — this elusive notion that could until then only have been expressed by vague names such as "general rule" or "metaphysics" — appears as the study of relationships between objects that are only (voluntarily) known and described by some of their properties, precisely those that are put as axioms at the foundations of their theory. It is this that had already been clearly seen by Boole in 1847, when he wrote that mathematics deals with "operations considered in themselves, independently of the diverse objects to which they can be applied" ([29], v. I, p. 3). Hankel,

systematically by mathematicians, such as the relation of inclusion, or what he calls the relation of univalent or multivalent "determination" (that is to say the notions of mapping and of correspondence) ([69 a], pp. 307-310). Many other modern ideas appear in his writing on this subject: he remarks that the different relations of equivalence of classical geometry have in common the properties of symmetry and transitivity; he conceives also the notion of a relation which is compatible with the relation of equivalence, and expressly notes that an arbitrary relation does not necessarily have this property ([69 a], pp. 313-315). Of course, he presupposes there as elsewhere the use of a formalised language, and even introduces a symbol meant to denote an arbitrary relation ([69 a], p. 301).

$^{31}$ It must be recognised that his language, of a very philosophical bent, was hardly meant to seduce the majority of mathematicians, who would feel uneasy in front of a formula such as the following: "Pure Mathematics is the science of the individual object in as much as it is born in thought" (Die Wissenschaft des besonderen Seins als eines durch das Denken gewordenen). But the context shows that Grassmann meant by that in a fairly precise way axiomatic mathematics in the modern sense (except that he fairly curiously follows Leibniz in considering that the bases of this "formal science", as he calls it, are the definitions and not the axioms); in any case, he insists, like Boole, on the fact that "the name of science of quantities does not fit the whole of mathematics" ([134], v. I, pp.22-23) in 1867, inaugurating the axiomatisation of algebra, defends a mathematics that is "purely intellectual, a pure theory of forms, which has as its purpose, not the combination of quantities, or of their images, the numbers, but objects of thought ("Gedankendinge") to which may correspond effective objects or relations, even though such a correspondence is not necessary" ([146], p. 10). Cantor, in 1883, echoes this claim of a "free mathematics" by proclaiming that "mathematics is entirely free in its development, and its concepts are only linked by the necessity of being consistent, and are co-ordinated with concepts introduced previously by means of precise definitions" ([47], p.182). Finally, the revision of Euclidean geometry succeeds in spreading and popularising these ideas. Pasch himself, although still attached to a certain "reality" of geometric objects, recognises that geometry is in fact independent of their significance, and consists purely in the study of their relations ([245], p. 90); a concept that Hilbert pushes to its logical conclusion in underlining that even the names of the basic notions of a mathematical theory can be chosen arbitrarily,$^{32}$ and that Poincaré expresses by saying that the axioms are "disguised definitions", thus completely reversing the scholastic point of view.

It would thus be tempting to say that the modern notion of "structure" is attained in substance around 1900; in fact it will need still another thirty years of apprenticeship before it appears in all its glory. It is no doubt not difficult to recognise structures of the same kind when they have a fairly simple nature; for the group structure, for example, this stage was reached already by the middle of the XIXth century. But at the same time, Hankel can be seen fighting — without quite succeeding — to bring out the general ideas of field and extension, that he does not succeed in expressing except in the form of a "principle of permanence" which is semi metaphysical [146], and which will be only formulated in a definitive form by Steinitz [294 a] 40 years later. Above all it has been fairly difficult, in this matter, to free oneself from the impression that mathematical objects are "given" to us with their structure; only a fairly long use of functional Analysis has been able to familiarise modern mathematicians with the idea that, for example, there are several "natural" topologies on the rational numbers, and several measures on the number line. With this disassociation the transition to general definition of structures was finally realised.

B) Models and isomorphisms. — The intervention of the notion of "model" or "interpretation" of one mathematical theory by means of another will have been noticed on several occasions. That is not a recent idea, and no

$^{32}$ As in a famous anecdote, Hilbert was keen to express this idea by saying that one could replace the words "point", "straight line" and "plane" by "table", "chair" and "beer mug" without changing any of the geometry. It is curious that one can find already in d'Alembert an anticipation of this pun: "One can give words whatever meaning one wants" he writes in the Encyclopédie ([75 a], article DÉFINITION); "[one could] strictly speaking create parts of Geometry which are exact (but ridiculous) by giving the name triangle to what is normally called a circle".

doubt what can be seen here is a manifestation ceaselessly recurring of a deep feeling for the unity of the various "mathematical sciences". If one can take as authentic the traditional maxim "All things are numbers" of the first Pythagoreans, that can be considered as the trace of a first attempt to bring back the geometry and algebra of the times to arithmetic. Although the discovery of the irrationals seemed to close for ever that route, the reaction to which it gave birth in Greek mathematics was a second attempt at synthesis this time taking geometry as the basis, and absorbing among others the solution methods for algebraic equations inherited from the Babylonians.33 It is known that this conception was to remain until the fundamental reform of R. Bombelli and of Descartes, assimilating every measure of quantity to a measure of length (in other words, to a real number; cf. p. 151). But with the creation of analytical geometry by Descartes and Fermat, the tendency is again reversed, and a much tighter fusion of geometry and algebra is obtained, but this time to the benefit of algebra. Further, elsewhere, Descartes goes further and conceives of the essential unity of "all these sciences that are commonly called Mathematics ... Even though their subjects are different", he says "they still do not fail to converge, in that they consider nothing other than the various relations or ratios that are to be found there" ([85 a], v. VI, pp. 19-20).34 However, this point of view only tended to make Algebra the fundamental mathematical science; a conclusion against which Leibniz protests vigorously, who also himself, as has been seen, conceived of a "universal Mathematics", but on a much vaster scale and already quite close to modern ideas. Making precise the "accord" of which Descartes spoke, he glimpses, in fact, for the first time, the general notion of isomorphism (which he calls "similitude"), and the possibility of "identifying" relations or operations that are isomorphic; he gives as examples addition and multiplication ([69 a], pp. 301-303). But these audacious views remained without echo amongst his contemporaries, and one must await the expansion of Algebra which takes place around the middle of the XIXth century (see pp. 51 ff.) to see the beginnings of the realisation of the Leibnizian dreams. We have already underlined that it is at that moment that the "models" multiply and that it becomes usual to go from one theory to another by a simple change of language; the most striking example of this is perhaps duality in projective geometry (see p. 132), where the practice, frequent at that time, of printing face to face, in two columns, the theorems "dual" one to the other, doubtless

33 Arithmetic remains however outside this synthesis; and it is known that Euclid, after having developed the general theory of ratios between arbitrary quantities, develops independently the theory of rational numbers, instead of considering them as particular cases of ratios of quantities (see pp. 148). 34 It is fairly interesting, on this point, to see Descartes getting closer to arithmetic and to "combinations of numbers", the "arts ... where order reigns more fully, as are those of artisans who make cloth or carpets, or those of women who embroider or make lace" ([85 a], v. X, p. 403), as if in anticipation of modern studies on symmetry and its relations with the notion of group (cf. [331 c]).

plays a great part in the full realisation of the notion of isomorphism. From a more technical point of view, certainly the notion of isomorphic groups for abelian groups is known to Gauss, for groups of permutations to Galois (cf. pp. 51 ff.); it is acquired in a general way for any group whatsoever around the middle of the XIXth century.35 Following this, with each new axiomatic theory, it is a natural development to define a notion of isomorphism; but it is only with the modern notion of structure that it was finally recognised that every structure carries within itself a notion of isomorphism, and that it is not necessary to give a special definition of it for each type of structure.

C) The arithmetisation of classical mathematics. — The use, more and more widespread, of the notion of "model" was also going to allow the realisation in the XIXth century of the unification of mathematics dreamed of by the Pythagoreans. At the beginning of the century, whole numbers and continuous quantities seemed as irreconcilable as ever in antiquity; real numbers remain linked to the notion of geometric quantity (at least to that of length), and it is to this latter that appeal had been made for the "models" of negative numbers and imaginary numbers. Even rational numbers were traditionally attached to the idea of the splitting of a quantity into equal parts; only the whole numbers remained apart, as "exclusively the product of our minds" as is said by Gauss in 1832, putting them in opposition to the notion of space ([124 a], v. VIII, p. 201). The first efforts to bring together Arithmetic and Analysis were made at first with rational numbers (positive and negative) and are due to Martin Ohm (1822); they were taken up again around 1860 by several authors, notably Grassmann, Hankel and Weierstrass (in his unpublished lectures); it is to this latter that it appears is due the idea of obtaining a "model" of the positive rational numbers or of whole negative numbers by considering classes of pairs of whole numbers. But the most important step remained to be taken, namely to find a "model" of irrational numbers in the theory of rational numbers; around 1870, it had become an urgent problem, in view of the necessity, after the discovery of "pathological" phenomena in Analysis, to eliminate every trace of geometric intuition in the vague notion of "quantity" in the definition of real numbers. It is known that this problem was solved around this time, almost simultaneously by Cantor, Dedekind, Méray and Weierstrass, and using fairly different methods (see p. 155).

From this moment, whole numbers became the foundation of all classical mathematics. Further, "models" based on Arithmetic acquired still more importance with the extension of the axiomatic method and the conception of mathematical objects as free creations of the mind. There remained in fact one restriction to this freedom claimed by Cantor, the question of "existence" which had already preoccupied the Greeks, and which arose here in a much

35 The word "isomorphism" itself is introduced in group theory at about the same time; but at first, it is used to designate surjective homomorphisms, qualified as "merihedral isomorphisms", whereas isomorphisms proper are called "holohedral isomorphisms"; this terminology would remain in use until the work of E. Noether.

more immediate way, precisely since all appeals to an intuitive representation had now been abandoned. We will see later (pp. 36 ff.) of what a philosophico-mathematical maelstrom the notion of “existence” was going to be the centre in the first years of the XXth century. But in the XIXth century that had not been reached, and to prove the existence of a mathematical object having given properties, it was simply, as for Euclid, “constructing” an object with given properties. That was precisely what arithmetic “models” were for: once the real numbers were “interpreted” in terms of whole numbers, complex numbers and Euclidean geometry were also, thanks to analytical Geometry, and it was the same for all new algebraic objects introduced since the beginning of the century; finally — a discovery that had a great effect — Beltrami and Klein had even obtained Euclidean “models” of the non-Euclidean geometries of Lobatschevsky and of Riemann (see p. 134), and in consequence “arithmetised” (and thereby completely justified) these theories which had at first sight aroused such distrust.

D) The axiomatisation of arithmetic.—It was in line with this evolution that a subsequent turn towards the foundations of arithmetic itself was made, and indeed that is what can be seen around 1880. It appears that before the XIXth century no attempt had been made to define addition and multiplication of whole numbers except by a direct appeal to intuition; Leibniz is the only one who, faithful to his principles, warns expressly that “truths” as “obvious” as $2 + 2 = 4$ are no less susceptible to proof if one thinks about the definitions of the numbers involved ([198 b], v. IV, p. 403; cf. [69 a], p. 203); and he did not consider at all that commutativity of addition and multiplication were intrinsic.$^{36}$ But he does not take his thoughts on this subject any further, and about the middle of the XIXth century, still no progress in this direction had been made: Weierstrass himself, whose lectures contributed considerably to extend the “arithmeticising” point of view, does not seem to have felt the need for a logical clarification of the theory of whole numbers. The first steps in this direction seem to be due to Grassmann, who, in 1861 ([134], v. II$_2$, p. 295) gives a definition of addition and multiplication for whole numbers, and proves their fundamental properties (commutativity, associativity, distributativity) using only the operation $x \mapsto x + 1$ and the principle of induction. This latter had been clearly conceived and used for the first time in the XVIIth century by B. Pascal ([244], v. III, p. 456)$^{37}$ — even though one can find in Antiquity some more or less conscious applications of it — and was currently used by mathematicians since the second half of the XVIIth century. But it is only in 1888 that Dedekind ([79], v. III, pp.359-361) stated a complete system of axioms for arithmetic (a system repeated 3 years later by Peano and usually known by his name [246 c]), which contained in

$^{36}$ As examples of non-commutative operations, he points to subtraction, division and exponentiation ([198 b], v. VII, p. 31); he had at one time even tried to introduce such operations into his logical calculus ([69 a], p. 353). $^{37}$ See also [45] particular a precise formulation of the principle of induction (that Grassmann uses yet without stating it explicitly).

With this axiomatisation, it seemed that the definitive foundations of mathematics had been reached. In fact, at the very moment when the axioms of arithmetic were being clearly formulated, these same, for many mathematicians (starting with Dedekind and Peano themselves) were already deprived of this role of primordial science, in favour of the latest arrival among the theories of mathematics, the theory of sets; and the controversies that were going to unravel around the notion of whole number cannot be isolated from the great "crisis of the foundations" of the years 1900-1930.

THE THEORY OF SETS.

It can be said that at all times, mathematicians and philosophers have used reasoned arguments from the theory of sets in a more or less conscious way; but in the history of their conceptions of this subject, one must separate sharply all questions linked to the idea of cardinal number (and in particular to the notion of infinity) from those that only introduce the notions of belonging and inclusion. These latter are among the most intuitive and seem never to have raised controversy: it is on them that one can most easily base a theory of syllogism (as Leibniz and Euler were to show us), or axioms such as "the whole is greater than the sum of its parts", without talking about that which, in geometry, is concerned with intersections of curves and surfaces. Until the end of the XIXth century, no problem arises in talking about the set (or "class" with some authors) of objects possessing such and such a given property;38 and the famous "definition" given by Cantor ("By a set is meant a gathering into one whole of objects which are quite distinct in our intuition or our thought" ([47], p. 282)) will give rise, at the time of its publication, to hardly any objections.39 It is altogether different as soon as the notion of set is mixed with that of number or quantity. The question of the infinite divisibility of the line (doubtless posed already by the first Pythagoreans) was, as is known, to lead to considerable philosophical difficulties: from the Eleates to Bolzano and Cantor, mathematicians and philosophers will throw themselves without success against the paradox of the finite quantity made up of an infinity of points without size. It would be of no interest to us to retrace, even summarily, the interminable and impassioned polemics that are aroused by this problem, that constituted a ground particularly favourable to metaphysical or theological digressions; let us note only the point of view

38 We have seen earlier that Boole does not even hesitate to introduce in his logical calculus a "Universe" 1, the set of all objects; it does not appear that at the time this conception was criticised, even though it has been rejected by Aristotle, who gave a proof, fairly obscure, aiming to show its absurdity ([6], Met. B, 3, 998 b). 39 Frege seems to be one of the rare contemporary mathematicians who, not without cause, spoke out against the wave of similar "definitions" ([117 c], v. I, p. 2).

at which, from Antiquity, most mathematicians stop. It consists essentially in refusing to debate, short of being able to conclude in an irrefutable way—an attitude that we will find again among the modern formalists; in the same way as these latter arrange for the elimination of the intervention of "paradoxical" sets (see below pp. 31 ff.), the classical mathematicians carefully avoid introducing into their reasoning the "actual infinity" (that is to say sets made up of an infinity of objects conceived as existing simultaneously, at least in their thought), and content themselves with "potential infinity", that is to say the possibility of increasing every given quantity (or of decreasing it if it is a case of a "continuous" quantity).40 If this point of view contained a certain dose of hypocrisy,41 it allowed however the development of the larger part of classical mathematics (including the theory of ratios and later the infinitesimal Calculus);42 it even seemed an excellent safeguard, above all after the quarrels caused by the infinitely small, and had become a dogma almost universally subscribed to until well into the XIXth century.

A first glimmer of the general notion of equipotence appears in a remark of Galileo ([122 b], v. VIII, pp. 78-80): he observes that the map $n \mapsto n^2$ establishes a bijective correspondence between the natural numbers and their squares and that it follows that the axiom "the whole is greater than the part"

40 A typical example of this conception is the statement of Euclid: "For each given quantity of prime numbers, there is one that is bigger" that we state nowadays by saying that the set of prime numbers is infinite. 41 Classically, one had obviously the right to say that a point belonged to a straight line, but to extract from that that a straight line "consists of points" would be to violate the taboo of the actual infinity, and Aristotle devotes long developments to justify this interdiction. It is likely that, in order to escape all objections of this kind, in the XIXth century many mathematicians avoid talking about sets, and reason systematically "by comprehension"; for example, Galois does not talk of the field of numbers, but only of the properties common to all the elements of such a field. Even Pasch and Hilbert, in their axiomatic presentations of Euclidean geometry, still abstain from saying that straight lines and planes are sets of points; Peano is the only one who uses freely the language of the theory of sets in elementary geometry. 42 No doubt the reason for this fact must be found in the circumstance that the sets appearing in classical mathematics belonged to a small number of simple types, and could for the most part be completely described by a finite number of numerical "parameters", so much so that their study reduces in the end to that of a finite set of numbers (it is so thus for example for algebraic curves and surfaces, which for a long time, are about the only "figures" of classical geometry). Before the progress of Analysis imposes, in the XIXth century, the consideration of arbitrary subsets of the straight line or $\mathbf{R}^n$, one can only rarely find sets that stray from the preceding types; for example Leibniz, always original, considers as a "geometric locus" the closed disc deprived of its centre, or (in a curious presentiment of the theory of ideals) considers that in arithmetic a whole number is "the type" of the set of its multiples, and remarks that the set of multiples of 6 is the intersection of the set of multiples of 2 and the set of multiples of 3 ([198 b], v. VII, p. 292). From the beginning of the XIXth century, one becomes familiar, in algebra and in the theory of numbers, with sets of this last type, as the classes of quadratic forms, introduced by Gauss, or fields and ideals, defined by Dedekind before the Cantorian revolution.

could not be applied to an infinite set. But far from inaugurating a rational study of infinite sets, this remark seems to have had no other effect than to reinforce mistrust of actual infinity; it is already the conclusion of Galileo himself, and Cauchy, in 1833, only quotes him in order to approve of his attitude.

The necessities of Analysis — and particularly the deep study of functions of real variables, which was taking place during the whole of the XIXth century — are at the origins of what was to become the modern theory of sets. When Bolzano, in 1817, proves the existence of the lower limit of a set bounded below in $\mathbf{R}$ [27 c], he still reasons, like the greater part of his contemporaries, "by comprehension", speaking not of an arbitrary set of real numbers, but of an arbitrary property of these latter. But when, 30 years later, he writes his "Paradoxien des Unendlichen" [27 b] (published in 1851, three years after his death), he does not hesitate to claim the right to the existence of the "actual infinity" and to speak of arbitrary sets. He defines, in this work, the general notion of the equipotence of two sets, and proves that two compact intervals in $\mathbf{R}$ are equipotent; he notes also that the characteristic difference between finite sets and infinite sets consists in that an infinite set E is equipotent to a subset distinct from E, but he does not give any convincing proof of this assertion. The general tone of this work is in any case more philosophical than mathematical, and for lack of disassociating sufficiently clearly the notion of the power of a set from that of size or order of infinity, Bolzano fails in his attempts to construct infinite sets of greater and greater power, and lets himself be drawn on this occasion into mixing his reasoning with considerations on divergent series that are totally devoid of sense.

It is due to the genius of G. Cantor that the theory of sets as we know it today was created. He also started with Analysis, and his work on trigonometric series, inspired by those of Riemann, bring him naturally, in 1872, to a first attempt to classify "exceptional" sets which occur in this theory,$^{43}$ by means of the notion of successive "derived sets", which he introduces at this point. It is no doubt in connection with these researches, and also with his method for defining the real numbers ([47], pp. 92-97), that Cantor begins to be interested in the problems of equipotence, for in 1873, he remarks that the set of rational numbers (or the set of algebraic numbers) is countable; and in his correspondence with Dedekind, which starts about this time [48], he is seen posing the question of the equipotence between the set of whole numbers and the set of real numbers, which he succeeds in solving negatively a few weeks later. Then, from 1874 on, it is the problem of dimension that preoccupies him, and for three years he seeks in vain to establish the impossibility of a bijective correspondence between $\mathbf{R}$ and $\mathbf{R}^n(n > 1)$, before succeeding, to

$^{43}$ It is a case of sets $E \subset \mathbf{R}$ such that, if a trigonometric series $\sum_{-\infty}^{+\infty} c_n e^{nix}$ converges to 0 except at the points of $E$, it necessarily follows that $c_n = 0$ for all $n$ ([47], p. 99).

his great surprise,$^{44}$ in defining such a correspondence. Having such results as new as they were surprising, he devotes himself entirely to the theory of sets. In a series of 6 memoirs published in the Mathematische Annalen between 1878 and 1884, he tackles both the problems of equipotence, the theory of totally ordered sets, the topological properties of $\mathbf{R}$ and of $\mathbf{R}^n$ and the problem of measure; and it is admirable to see with what clarity these notions appear in his hands little by little, notions which appear so inextricably tied to the classical conception of the "continuum". From 1880, he has the idea of iterating "transfinitely" the construction of "derived sets"; but this idea takes on life only two years later, with the introduction of well ordered sets, one of the most original discoveries of Cantor, thanks to which he can tackle a detailed study of cardinal numbers and formulate the "problem of the continuum" [47].

It was impossible that such bold conceptions, reversing a twice millennial tradition, and leading to results so unexpected and of such a paradoxical appearance, would be accepted without resistance. In fact, amongst the mathematicians who were influential at that time in Germany, Weierstrass was the only one to follow with some favour the work of Cantor (his erstwhile pupil); this latter was to run up on the other hand against the irreducible opposition of Schwarz and especially of Kronecker.$^{45}$ It is, it would appear, as much the constant tension generated by the opposition to his ideas, as his fruitless efforts to prove the continuum hypothesis, which caused in Cantor the first symptoms of a nervous illness from which his mathematical productivity would suffer.$^{46}$ He only really took up again his interest in the theory of sets in 1887, and his last publications date from 1895-1897; he develops there mostly the theory of totally ordered sets and the calculus of ordinals. He had also proved in 1890 the inequality $m < 2^m$; however, not only did the problem of the continuum remain without answer, but there remained in the theory of cardinals a more serious gap, for Cantor had been unable to establish the existence of a well ordering between arbitrary cardinals. This gap was going to be filled, on the one hand by the theorem of F. Bernstein (1897) showing that the relations $a \leq b$ and $b \leq a$ imply $a = b$,$^{47}$ and above all by the theorem of Zermelo [342 a] proving the existence of a well order on every set — a theorem conjectured already in 1883 by Cantor ([47], p. 169).

$^{44}$ "I can see it, but I do not believe it" he writes to Dedekind ([48], p. 34); in French in the text).

$^{45}$ The contemporaries of Kronecker have made frequent allusions to his doctrinaire position on the foundations of mathematics; it must be presumed that he expressed himself more explicitly in his personal contacts than in his publications (where, as far as the role of natural numbers is concerned, he only takes up again some fairly banal remarks on "arithmetisation" around 1880) (cf. [327 b], in particular pp. 14-15).

$^{46}$ For this period of the life of Cantor, see [275 b].

$^{47}$ This theorem had already been obtained by Dedekind in 1887, but its proof was not published ([79], v. III, p. 447).

However Dedekind, from the beginning, had not ceased to follow with a sustained interest the researches of Cantor; but while this latter concentrated his attention on infinite sets and their classification, Dedekind followed his own thoughts on the notion of number (which had already led him to his definition of irrationals by “cuts”). In his opuscule Was sind und was sollen die Zahlen, published in 1888, but the essentials dating from 1872-78 ([79], v. III, p. 335), he shows how the notion of natural number (on which, as has been seen, the whole of classical Mathematics had ended up by relying) could itself be derived from fundamental notions in the theory of sets. Developing (no doubt the first explicitly) the elementary properties of arbitrary maps from one set to another (neglected until Cantor, who was only interested in bijective correspondences), he introduces, for every map $f$ of a set $E$ into itself, the notion of “chain” of an element $a \in E$ relative to $f$, namely the intersection of the sets $K \subset E$ such that $a \in K$ and $f(K) \subset K$.48 He then takes as his definition of an infinite set $E$ the fact that there exists a bijective map $\phi$ of $E$ into $E$ such that $\phi(E) \neq E$;49 if further there exists such a map $\phi$ and an element $a \notin \phi(E)$ for which $E$ is the chain of $a$, Dedekind says that $E$ is “simply infinite”, remarks that the “axioms of Peano” are then satisfied and shows (before Peano) how starting from there all the theorems of elementary arithmetic are obtained. The only missing part of his exposé is the axiom of infinity that Dedekind (following Bolzano) believes himself able to prove by considering the “world of thought” of humans (“Gedankenwelt”) as a set.50

On another tack, Dedekind had been led by his work on arithmetic (and notably the theory of ideals) to consider the notion of ordered set in a more

48 It is on a very analogous notion that the second proof given by Zermelo for his theorem relies [342 b]. 49 We have seen that Bolzano had already noted this characterisation of infinite sets, but his work (hardly disseminated, it would appear, in mathematical circles) was unknown to Dedekind at the time when he was writing “Was sind und was sollen die Zahlen”. 50 Another method for defining the notion of natural number and for establishing the fundamental properties was proposed by Frege in 1884 [117 b]. He seeks first to give to the notion of cardinal of a set a more precise meaning than Cantor; at that time this latter had only defined the notions of equipotent sets and of sets having a power at most equal to that of another, and the definition of “cardinal number” that he was to give later ([47], p. 282) was as obscure and unusable as the definition of a straight line in Euclid. Frege, always careful about precision, has the idea of taking as definition of the cardinal of a set $A$ the set of all sets equipotent to $A$ ([117 b], §68); then, having defined $\phi(a) = a + 1$ for all cardinals ($§ 76$), he puts himself in the set $C$ of all cardinals, and defines the relation “$b$ is a $\phi$-successor to $a$” as meaning that $b$ belongs to the intersection of all the sets $X \subset C$ such that $\phi(a) \in X$ and $\phi(X) \subset X$ ($§ 79$). Finally, he defines a natural number as a $\phi$-successor of 0 ($§ 83$; all these definitions are of course expressed by Frege in his language of the “logic of concepts”). Unfortunately this construction was to reveal itself to be defective, the set $C$ or set of sets equipotent with a set $A$ was “paradoxical” (see below).

general setting than Cantor. While this latter restricts himself exclusively to totally ordered sets,$^{51}$ Dedekind tackles the general case, and makes in particular a deep study of lattices ([79], v. II, pp. 236-271). This work was hardly noticed at the time; although these results, found again by several authors, have been the object of numerous publications since 1935, their historical importance arises much less from the possibilities for applications, fairly thin no doubt, of this theory, than from the fact that they constituted one of the first examples of careful axiomatic construction. On the other hand, the first results of Cantor on sets which were countable or had the power of the continuum were rapidly to have numerous and important applications right up to the most classical questions of Analysis$^{52}$ (without talking naturally of those parts of the Cantorian work that inaugurated General Topology and measure theory; on that see pp. 141 and 221). Moreover, from the last years of the XIXth century the first uses of the principle of transfinite induction were appearing, having become, especially after the proof of the theorem of Zermelo, an indispensable tool in all parts of modern mathematics. Kuratowski was to give, in 1922, an often more amenable version of this principle, avoiding the use of well ordered sets ([189 a], p. 89); it is in this form, found again later by Zorn [344], that it is mainly used at the present time.$^{53}$

Towards the end of the XIXth century, the essential concepts of Cantor had thus gained their cause.$^{54}$ We have seen that, at about this time, the formalisation of mathematics is achieved and the use of the axiomatic method is almost universally agreed. But, at the same time, a "crisis of foundations" of a rare violence was opening up, which was to shake the mathematical world for more than 30 years, and to seem at times to compromise, not only all the recent acquisitions, but even the most classical parts of Mathematics.

$^{51}$ It is interesting to note that, among these latter, Cantor never wanted to admit the existence of "non archimedean" ordered groups because they introduced the notion of the "actual infinitely small" ([47], pp. 156 and 172). Such order relations arose naturally in the researches of Du Bois-Reymond on orders of infinity (cf. p. 202) and were studied systematically by Veronese [318].

$^{52}$ From 1874, Weierstrass had given notice, in a letter to Du Bois-Reymond, of an application to functions of a real variable of the theorem of Cantor on the possibility of writing the rational numbers as a sequence ([329 b], p. 206).

$^{53}$ Because of this the interest that was shown in the ordinals of Cantor has greatly diminished; in general, in any case, many of the results of Cantor and his successors on the arithmetic of ordinals and non denumerable cardinals have remained until now fairly isolated.

$^{54}$ The official consecration of the theory of sets is manifested already at the first international Congress of mathematicians (Zurich, 1897), where Hadamard [141] and Hurwitz give notice of important applications to Analysis. The growing influence of Hilbert at this time contributed greatly to the spread of the ideas of Cantor, especially in Germany.

THE PARADOXES OF THE THEORY OF SETS AND THE CRISIS OF FOUNDATIONS.

The first "paradoxical" sets appeared in the theory of cardinals and ordinals. In 1897, Burali-Forti remarked that one cannot consider the existence of a set consisting of all the ordinals, because this set would be well ordered, and so isomorphic to one of its segments distinct from the whole, which is absurd [42].55 In 1899, Cantor observes (in a letter to Dedekind) that it is no longer possible to speak of the cardinals forming a set, nor to speak of the "set of all sets" without reaching a contradiction (the set of subsets of this latter "set" $\Omega$ would be equipotent to a subset of $\Omega$, which is contrary to the inequality $m < 2^m$) ([47], pp. 444-448). In 1905 finally, Russell analysing the proof of this inequality, shows that the reasoning that establishes it proves (without appealing to the theory of cardinals) that the notion of the "set of all sets which are not elements of themselves" is, it also, contradictory [266].56

It could be believed that such "antinomies" would only be manifested in the peripheral regions of mathematics, characterised by the consideration of sets of a "size" inaccessible to intuition. But other "paradoxes" were soon to threaten the most classical parts of mathematics. Berry and Russell ([266], v. I, pp.63-64), simplifying an argument of J. Richard [258], observe in effect that the set of whole numbers of which the definition can be expressed in less than sixteen words in English is finite, but that it is however contradictory to define a whole number as "the least whole number which is not definable by less than sixteen words in English", because this definition consists of fifteen words only.

Although such reasoning, so far from that usually used by mathematicians, may have appeared to many of them as a kind of riddle, they pointed the way none the less to the necessity for a revision of the bases of mathematics, aiming to eliminate "paradoxes" of this nature. But although there was unanimity concerning the urgency of this revision, radical divergences were soon to appear concerning the manner of realising it. For a first group of mathematicians, be they "idealists", be they "formalists",57 the situation created by the "paradoxes" of the theory of sets was very analogous to that which resulted, in geometry, from the discovery of non-Euclidean geometries

55 This remark had already been made by Cantor in 1896 (in a letter to Hilbert, unpublished). 56 The reasoning of Russell is getting close to the paradoxes of antiquity of which the type is the celebrated "Liar", the subject of innumerable commentaries in formal classical Logic: it consists in knowing whether the man who says "I lie" says or does not say the truth in pronouncing these words (cf. [267]). 57 The divergence between these two schools is mostly philosophical, and we cannot here go into more details on this topic; the essence is that they come together on properly mathematical ground. For example, Hadamard, a typical representative of the "idealists", adopts, where the validity of reasoning in the theory of sets is concerned, a point of view very close to the formalists, but without expressing it in an axiomatic form ([12], p.271).

or of "pathological" curves (such as curves without tangents); it must lead to a similar, but more general, conclusion namely that it is hopeless to seek to base an arbitrary mathematical theory on an appeal (explicit or not) to "intuition". The position can be summed up by the very words of the principal adversary of the formalist school : "The formalist" says Brouwer ([39 a], p. 83), "submits that human reason does not have at its disposal exact pictures of straight lines or numbers greater than ten, for example ... It is true that certain relations between mathematical entities, that we take as axioms, we can deduce from other relations according to fixed rules, with the conviction that in this way we derive truth from other truths by logical reasoning ... [But] for the formalist, mathematical exactitude resides only in the development of the sequence of relations, and is independent of the meaning that one could wish to give to the relations or entities that they link."

It consists, therefore, for the formalist, in giving to the theory of sets an axiomatic basis quite analogous to that of elementary geometry, where one does not concern oneself with what the "objects" are that are called "sets", nor what is meant by the relation $x \in y$, but where one enumerates the conditions to be imposed on this latter relation; of course, this must be done in such a way as to include, as far as possible, all the results of the theory of Cantor, all the while making it impossible for "paradoxical" sets to exist. The first example of such an axiomatisation was given by Zermelo in 1908 [342 c]; he avoids sets that are "too large" by the introduction of an "axiom of selection (Aussonderung)", a property $P(x)$ only determining a set made up of the elements which have it if $P(x)$ already entails a relation of the form $x \in A$.58 But the elimination of paradoxes analogous to "Richard's paradox" could only be done by restricting the meaning attached to the notion of "property"; on that, Zermelo contents himself with describing in very vague terms a type of property that he calls "defined", and to indicate that one must limit oneself to these latter in the application of the axiom of selection. This point was made precise by Skolem [286 a] and Fraenkel [113 b]; as they observed, its elucidation forces us to place ourselves in a completely formalised system, where the notions of "property" and of "relation" have lost all "significance" and have become simple designations for the collections formed following explicit rules. That necessitates of course that one must put into the system the rules of Logic that are used, which was not yet the case in the system of Zermelo-Fraenkel.

Other axiomatisations of the theory of sets were proposed following on from this. Let us mention principally that of von Neumann [342 a and c] which, more than the system of Zermelo-Fraenkel, approached the primitive conception of Cantor: this latter, to avoid paradoxical sets, had already proposed ([47], pp. 445-448), in his correspondence with Dedekind, to distinguish

58 For example, the paradox of Russell would only become valid in the system of Zermelo if one could prove there the relation $(\exists z)((x \notin x) \Rightarrow (x \in z))$; of course, such a proof, if it could be obtained, would have as immediate consequence the necessity of a substantial modification of the system in question.

between two kinds of sets, the “multiplicities” (“Vielheiten”) and the “sets” (“Mengen”) proper, the second ones being characterised in that they can be thought of as a single object. It is this idea that von Neumann makes precise in distinguishing two types of objects, “sets” and “classes”; in his system (almost completely formalised), classes are distinguished from sets in that they cannot be placed on the left of the sign $\in$. One of the advantages of such a system is that it rehabilitates the notion of “universal class” used by the logicians of the XIXth century (which naturally is not a set); let us point out also that the system of von Neumann avoids (for the theory of sets) the introduction of axiom schemes, replaced by suitable axioms (which makes its logical study easier). Some variants of the system of von Neumann have been given by Bernays and Gödel [130 b].

The elimination of paradoxes seems to have been well carried out by the systems above, but at the cost of restrictions which cannot but seem to be very arbitrary. In favour of the system of Zermelo-Fraenkel, one can say that it restricts itself to formulating restrictions that only sanction the current practice in the applications of the notion of set to the various mathematical theories. The systems of von Neumann and of Gödel are further removed from the usual conceptions; on the other hand, it is not excluded that it is easier to insert certain mathematical theories still in their beginnings in the framework provided by such systems rather than in the narrower framework of the system of Zermelo-Fraenkel.

Certainly one could not affirm that any of these solutions give the impression of being definitive. If they satisfy the formalists, it is that these latter refuse to take into consideration the individual psychological reactions of each mathematician; they assume that a formalised language has completed its task when it can transcribe mathematical reasoning into a form lacking ambiguity, and so useful as a vehicle for mathematical thought; everyone is free, they would say, to think what he likes about the “nature” of mathematical objects or the “truth” of the theorems that he uses, so long as his reasoning can be transcribed into the common language.59

In other words, from a philosophical point of view, the attitude of the formalists consists in being disinterested in the problem set by the “paradoxes”, in abandoning the platonic position which aimed to attribute to mathematical notions an intellectual “content” common to all mathematicians. Many mathematicians draw back faced with this break with tradition. Russell, for example, seeks to avoid paradoxes by analysing their structure in a deeper way. Taking up again an idea first brought forward by J. Richard (in the article [258] where he displayed his “paradox”) and developed later by H. Poincaré [251 c], Russell and Whitehead observe that the definitions of para-

59 Hilbert, however, seems always to have believed in an objective mathematical “truth” ([163 c], pp. 315 and 323). Even some formalists who, like H. Curry, have a stance very close to that which we have just outlined, reject with a kind of indignation the idea that mathematics could be considered a simple game, and want absolutely to see in it an “objective science” ([73], p. 57).

doxical sets all violate the following principle, called the “principle of the vicious circle”: “An element of which the definition involves the totality of the elements of a set can not belong to that set” ([266], v. I, p. 40). Also it is this statement that serves as basis for the Principia, and it is to satisfy it that the “theory of types” is developed in this work. Like that of Frege by which it is inspired, the logic of Russell and Whitehead contains “propositional variables”; the theory of types proceeds to a classification of these different variables, of which the main lines are the following. Starting from a “domain of individuals” which are not specified and which can be qualified as “objects of order 0”, the relationships where the variables (free or tied) are individuals, are said to be “objects of the first order”; and in general, the relations in which the variables are objects of order $\leq n$ (one at least being of order $n$) are said to be “objects of order $n + 1$”.$^{60}$ A set of objects of order $n$ can then only be defined by a relation of order $n + 1$, a condition that allows the elimination without trouble of the paradoxical sets.$^{61}$ But the principle of the “hierarchy of types” is so restrictive that by adhering to it strictly, one ends up with a mathematics of inextricable complexity.$^{62}$ To escape this consequence, Russell and Whitehead are constrained to introduce an “axiom of reducibility” affirming the existence, for every relation between “individuals”, of a relation of the first order to which it is equivalent; a condition quite as arbitrary as the axioms of the formalists, and which reduces considerably interest in the construction of the Principia. So the system of Russell and Whitehead has had more success with the logicians than with the mathematicians; it is in any case not entirely formalised,$^{63}$ and as a result there are numerous obscure details. Various efforts have been made to simplify and clarify this system (Ramsey, Chwistek, Quine, Rosser); tending to use languages that are more and more fully formalised, these authors replace the rules of the Principia (which still had a certain intuitive foundation) with restrictions only taking account of the wording of the aggregations being con-

$^{60}$ That is only in fact the beginning of the classification of “types”, of which a faithful account could not be given without going into some very long developments; the reader wishing for more detailed explanations could in particular take note of the introduction to v. II of Principia Mathematica [266]. $^{61}$ In the system of Russell and Whitehad, the relation $x \in x$ can not therefore be legally written down, as against the system of Zermelo-Fraenkel, for example. $^{62}$ For example, equality is not a primitive notion in the system of Principia: two objects $a, b$ are equal if for every property $P(x), P(a)$ and $P(b)$ are equivalent propositions. But this definition does not make sense in the theory of types: in order to make sense, one would have to specify at least the “order” of $P$, and so one would be led to distinguish amongst an infinity of relations of equality! Zermelo had in any case remarked, already in 1908 [342 b], that numerous classical mathematical definitions (for example that of a lower limit of a set in $\mathbf{R}$) do not obey the “principle of the vicious circle”, and that the adoption of this principle thus risked ruling out important parts of the most traditional mathematical theories. $^{63}$ Russell and Whitehead (as already Frege) keep to the classical position as regards mathematical formulae, which for them must always have a “meaning” related to an activity underlying the thought.

sidered; not only do these rules appear to be as gratuitous as the prohibitions formulated in the systems of Zermelo-Fraenkel or of von Neumann, but, being further removed from the practice of mathematicians, they have, in many cases, led to unacceptable consequences, which the author had not foreseen (such as the paradox of Burali-Forti, or the negation of the axiom of choice).

For mathematicians of the preceding schools, it was above all a case of not renouncing any part of the heritage of the past: "From the paradise that Cantor has created for us", says Hilbert ([163 c], p. 274), "no one should be able to chase us". To do this, they are willing to accept limitations on mathematical reasoning, hardly essential because consonant with use, but which do not seem to be imposed by our mental customs and the intuitive idea of the notion of set. Anything seems to them preferable to the intrusion of psychology into the criteria of validity in mathematics; and rather than "putting into account the characteristics of our brains", as Hadamard says ([12], p. 270), they are resigned to imposing on the mathematical domain limits which are for the most part arbitrary, so long as they include classical Mathematics and do not risk hampering future progress.

It is a very different matter when we see the attitude of mathematicians belonging to the tendency of which it remains for us to speak. If the formalists accept the renunciation of the control of the "eyes of the mind" so far as mathematical reasoning is concerned, the mathematicians that have been called "empiricists", "realists" or "intuitionists" refuse this abdication; they must have a kind of internal certainty guaranteeing the "existence" of the mathematical objects with which they are concerned. As long as it was only a case of renouncing spatial intuition, there was no serious objection, since the arithmetical "models" allowed retrenchment behind the intuitive notion of whole numbers. But irreducible opposition manifests itself when it is a matter of reducing the notion of whole number to that (much less intuitively precise) of set, then of imposing on the handling of sets barriers without intuitive foundation. The first in time of these opponents (and the one who, by the authority of his genius, was to exercise the most influence) was H. Poincaré; having admitted, not only the axiomatic point of view for geometry and the arithmetisation of Analysis, but also a good part of the theory of Cantor (which he was one of the first to apply fruitfully in his work), he refuses on the other hand to conceive that arithmetic may be able, it also, to be justified by an axiomatic treatment; the principle of complete induction seems to him in particular a fundamental intuition of our mind, where it is impossible to see a pure convention$^{64}$ [251 c]. Hostile in principle to formalised languages, of which he contested the usefulness, he confuses constantly the

$^{64}$ Poincaré goes as far as saying in substance that it is impossible to define a structure satisfying all the axioms of Peano, with the exception of the principle of complete induction ([251 c], p. 65); the example (due to Padoa) of the whole numbers with the mapping $x \mapsto x + 2$ replacing $x \mapsto x + 1$ shows that this assertion is inaccurate. It is a curious point that it is found already in Frege, almost in the same words ([117 b], p. 21 e).

notion of whole number in formalised mathematics, and the use of whole numbers in the theory of proof, which was starting up then, and of which we will speak later; no doubt it was difficult at that time to make as clearly as today — after 50 years of study and discussion — this distinction which however had been clearly felt by a Hilbert or a Russell.

The critics of this kind multiplied after the introduction of the axiom of choice by Zermelo in 1904 [342 a]. Its use in many previous proofs in Analysis or the theory of sets had until then gone by almost unnoticed;$^{65}$ it is in following an idea suggested by Erhard Schmidt that Zermelo, explicitly formulating this axiom, deduced from it in an ingenious way a satisfactory proof of the existence of a well ordering on every set. Coming at the same time as the "paradoxes", it appears that this new way of reasoning, by its odd allure, had thrown many a mathematician into confusion; it is enough to see the strange misunderstandings which surged up about this topic, in the following volume of Mathematische Annalen, under the signature of mathematicians who were as familiar with the Cantorian methods as Schoenflies and F. Bernstein. The criticisms of E. Borel, published in this same volume, are more substantial and attach themselves clearly to the point of view expressed by H. Poincaré about whole numbers; they are developed and discussed in an exchange of letters between E. Borel, Baire, Hadamard and Lebesgue, who had remained in the classical tradition in French mathematics [12]. Borel starts by denying the validity of the axiom of choice because it contains in general an uncountable infinity of choices, which is inconceivable intuitively. But Hadamard and Lebesgue observe that a countable infinity of arbitrary successive choices is no more intuitive, since it consists of an infinity of operations, that it is impossible to conceive as really taking place. For Lebesgue, who enlarged the debate, everything reduces to knowing what is meant by saying that a mathematical object "exists"; it is necessary, for him, that one "names" explicitly a property defining it in a unique way (a "functional" property, we would say); for a function such as the one used by Zermelo in his reasoning, that is what Lebesgue calls a "law" of choice; if, he continues, this requirement is not satisfied, and that ones limits oneself to "thinking" about this function instead of "naming" it, can one be sure that during the reasoning process one is always thinking of the same thing ([12], p. 267)? In any case, this brings Lebesgue to new doubts; already the question of the choice of a single element in a set seems to him to raise difficulties: one must be sure that such an element "exists", that is to say that one can "name" at least one of the elements

$^{65}$ In 1890, Peano, proving his theorem on the existence of integrals of differential equations, remarks that he would be led naturally to "apply an infinity of times an arbitrary law under which to a class an individual of that class is made to correspond"; but he adds also that such a reasoning is inadmissible in his eyes ([246 c], p. 210). In 1902, B. Levi had remarked that this same reasoning was implicitly used by F. Bernstein in a proof in the theory of cardinals [199].

of the set.$^{66}$ Can one then speak of the “existence” of a set of which one does not know how to “name” each element? Already Baire does not hesitate to deny the “existence” of the set of all subsets of a given infinite set (loc. cit., pp. 263-264); in vain does Hadamard observe that these requirements lead to renouncing even the possibility of speaking of the real numbers: it is to just that conclusion that E. Borel finally agrees. Putting aside the fact that countability seems to have acquired freehold, one has returned just about to the classical position of the adversaries of the “actual infinite”.

All these objections were not very systematic; it was down to Brouwer and his school to undertake a complete remould of mathematics guided by similar, but even more radical, principles. We would not dare to summarise here a doctrine as complex as intuitionism, which consists as much of psychology as mathematics, and we will limit ourselves to indicating some of the most striking aspects, referring for more details to the work of Brouwer himself [39 a and b] and to the exposition by Heyting [162]. For Brouwer, Mathematics is identical with the “exact” part of our thought, based on the first intuition of the sequence of natural numbers, and which it is impossible to translate without mutilation into a formal system. It is in any case only “exact” in the thought of mathematicians, and it is a chimera to hope to develop an instrument of communication between them which is not subject to all the imperfections and ambiguities of language; at most it is possible to hope to awaken in the interlocutor a favourable state of mind by descriptions that are more or less vague ([162], pp. 11-13). Intuitionist mathematics attaches hardly more importance to logic than to language: a proof is not conclusive by virtue of logical rules fixed once and for all, but because of the “immediate evidence” of each of its steps. This “evidence” must moreover be interpreted in a more restrictive way than by E. Borel and his followers: it is thus that in intuitionist mathematics, it cannot be said that a relation of the form “$R$ or (not $R$)” is true (principle of the excluded middle), unless for all system of values given to the variables appearing in $R$, one can prove that one of the two propositions $R$, “not $R$” is true; for example, from the equation $ab = 0$ between two real numbers, it cannot be concluded that “$a = 0$ or $b = 0$”, for it is easy to create explicit examples of real numbers $a, b$ for which one has $ab = 0$ without knowing, at the actual time, how to prove either of the two propositions $a = 0, b = 0$ ([162], p. 21).

$^{66}$ The supposed “choice” of an element in a set has in fact nothing to do with the axiom of choice; it consists merely of a figure of speech, and everywhere where one expresses oneself in this way, one is in reality only using the most elementary logical rules (where the sign $\tau$ does not occur). Of course, the application of these rules to a set $A$ requires that it has been proved that $A \neq \emptyset$; it is on this point that the argument of Lebesgue bears, for such a proof is only valid for him precisely when one has “named” an element of $A$. For example, Lebesgue does not consider as valid the reasoning of Cantor ([47], pp. 115-118) proving the existence of transcendental numbers; this existence is only proved for him because it is possible to “name” transcendental numbers, such as the Liouville numbers or the numbers $e$ or $\pi$.

It is not astonishing that, starting from such principles, the intuitionist mathematicians finish up with results that are very different from the classical theorems. A whole part of these disappear, for example most of the "existential" theorems in Analysis (such as the theorems of Bolzano and Weierstrass for real functions); if a function of a real variable "exists" in the intuitionist sense, it is ipso facto continuous; a bounded monotone sequence of real numbers does not necessarily have a limit. On the other hand, many classical notions become ramified, for the intuitionist, into several distinct fundamental notions: there are thus two notions of convergence (for a sequence of real numbers) and eight of countability. It goes without saying that transfinite induction and its applications to modern Analysis are (as for the greater part of the theory of Cantor) condemned without appeal.

It is only in this way that, according to Brouwer, mathematical propositions can acquire a "content"; the formalist reasonings that go beyond what the intuitionists allow are judged to be without value, since one can no longer give them a "meaning" to which the intuitive notion of "truth" could be applied. It is clear that such judgements can only rely on a pre-existing notion of "truth", of a psychological or metaphysical nature. It is to say in practice that they escape all discussion.

It is undoubted that the vigorous attacks that came from the intuitionist camp have sometimes forced not only the mathematical schools in the avant-garde, but even the followers of traditional mathematics to go on the defensive. A famous mathematician has admitted having been impressed by the attacks to the point of having voluntarily restricted his work to branches of mathematics judged to be "certain". But such cases can hardly have been frequent. The intuitionist school, of which the memory is no doubt destined to remain only as an historical curiosity, would at least have been of service by having forced its adversaries, that is to say definitely the immense majority of mathematicians, to make their position precise and to take more clearly notice of the reasons (the ones of a logical kind, the others of a sentimental kind) for their confidence in mathematics.

METAMATHEMATICS

The absence of contradiction has at all times been considered as a sine qua non condition for all mathematics, and from the time of Aristotle, logic was sufficiently developed so that it was perfectly well known that from an inconsistent theory anything could be deduced. The proofs of "existence", considered as indispensable since Antiquity, visibly had no other aim than to guarantee that the introduction of a new concept did not risk entailing a contradiction, particularly when this concept was too complicated to fall immediately under "intuition". We have seen how this requirement became more pressing with the arrival of the axiomatic point of view in the XIXth century and how the construction of arithmetical "models" responded to it. But arithmetic itself, could it be inconsistent? A question that without doubt would not have been thought of before the end of the XIXth century, so much did the whole numbers appear to belong to what was most certain in our intuition; but after the “paradoxes” everything seemed to have been questioned, and one can understand that the feeling of insecurity that they created had brought mathematicians, around 1900, to bear with more care on the problem of the non-contradiction of arithmetic, so as to save at least classical mathematics from shipwreck. So this problem is the second of those that Hilbert enumerated in his famous lecture at the international Congress of 1900 ([163 a], v. III, pp. 229-301). In doing this, he laid down a new principle that was to have great repercussions: whereas in traditional logic the non-contradiction of a concept only made it “possible”, it was equivalent for Hilbert (at least for mathematical concepts defined axiomatically) to the existence of the concept. This implied apparently the necessity of proving a priori the non-contradiction of a mathematical theory before even being able to develop it legitimately; it is just in this way that H. Poincaré understands it when, to make a breach in formalism, he takes to his account the idea of Hilbert, underlining with a mischievous pleasure the extent to which the formalists were far, at that stage, from being able to achieve it ([251 c], p. 163). We will see later how Hilbert was to pick up the gauntlet; but before that, it must be noted here that, under the influence of his authority and that of Poincaré, the requirements set down by this latter were to be for a long time accepted without reservation as well by the formalists as by their adversaries. One consequence was the belief, very widespread among the formalists, that the theory of proof of Hilbert was an integral part of mathematics, of which it constituted the indispensable prolegomena. This dogma does not seem justified to us,$^{67}$ and we consider that the intervention of metamathematics in the exposition of logic and mathematics can and should be reduced to the very elementary part that is concerned with the handling of abbreviating symbols and criteria of deduction. It is thus not a case, contrary to what Poincaré thinks, of “claiming the freedom from contradiction”, but rather to consider, with Hadamard, that the absence of contradiction, even when it is not proven, is noted ([12], p. 270).

It remains for us to give a brief historical sketch of the efforts of Hilbert and his school, and to retrace summarily, not only the evolution that was finally to lead to the negative result of Gödel and justify a posteriori the scepticism of Hadamard, but also all the progress that followed on from this, concerning the knowledge of the mechanism of mathematical reasoning, and which makes modern metamathematics an autonomous science of incontestable interest.

$^{67}$ In a pure formalist doctrine, the words “there exists” in a formalised text have no more “significance” than any others, and there is no need to consider other types of “existence” in formalised proofs.

Already in 1904, in a lecture at the international Congress ([163 c], pp. 247-261), Hilbert attacks the problem of consistency in arithmetic. He observes first that there is no question of proving it by making use of a model,$^{68}$ and outlines the principles of another method: he proposes considering the true propositions of formalised arithmetic as collections of signs without meaning, and to prove that in using the rules governing the formation and linkage of these collections, it is impossible to obtain a collection which is a true proposition and whose negation is also a true proposition. He sketches even a proof of this nature for a formalism less extensive than that of arithmetic; but, as was observed by H. Poincaré soon after ([251 c], p. 185), this proof uses essentially the principle of recurrence, and so appears to rely on a vicious circle. Hilbert did not answer this criticism immediately, and about fifteen years passed by without anyone attempting to develop his ideas; it is in 1917 only that (motivated by the wish to answer the attacks of the intuitionists) he harnesses himself again to the problem of the foundations of mathematics, which he will not cease from now on to consider until the end of his scientific career. In his work on this subject, which is spread out from 1920 to 1930 approximately, and in which a whole school of young mathematicians take part (Ackermann, Bernays, Herbrand, von Neumann), Hilbert brings out little by little the principles of his "theory of proof" in a more precise way: acknowledging implicitly the validity of the criticism of Poincaré, he admits that in metamathematics, the arithmetical reasoning that is used can only be based on our intuition of the whole numbers (and not on formalised arithmetic); to do this it appears to be essential to restrict this reasoning to "finite procedures" ("finite Prozesse") of a type allowed by the intuitionists: for example, a proof by contradiction cannot prove the metamathematical existence of a collection or sequence of collections, there must be an explicit law of construction.$^{69}$ On the other hand Hilbert enlarges in two directions his initial programme: not only does he tackle the consistency of arithmetic, but he aspires also to prove the consistency of the theory of real numbers and even that of the theory of sets;$^{70}$ further, to the problems of consistency come to be added those of independence of axioms, of completeness and of decision. We will pass quickly in review these various questions and point out the principal research to which they have given rise.

$^{68}$ The "models" supplied by the definitions of Dedekind or Frege will only shift the question, by reducing it to the consistency of the Theory of sets, a problem which is without doubt more difficult than the consistency of Arithmetic, and which was to appear even more so at a time when no serious attempt to avoid the "paradoxes" had been proposed.

$^{69}$ For a more detailed and precise description of finite procedures allowed in metamathematics, one can consult, for instance, the thesis of J. Herbrand [158].

$^{70}$ When one is speaking of the consistency of the theory of real numbers, one assumes that this theory is defined axiomatically, without using the theory of sets (or at least abstaining from using certain axioms of this latter, such as the axiom of choice or the axiom of the set of subsets).

1. FOUNDATIONS OF MATHEMATICS; LOGIC; SET THEORY.

The proof of the independence of a system of propositions $A_1, A_2, \ldots, A_n$ consists in showing that, for each index $i$, $A_i$ is not a theorem in the theory $T_i$ obtained by taking as axioms the $A_j$ with index $j \neq i$. This point will be established if one knows a consistent theory $T_i'$ in which the $A_j (j \neq i)$ are theorems, as well as "not $A_i$"; thus one can consider the problem in two ways depending on whether or not one allows that certain theories (such as arithmetic or the theory of sets) are consistent. In the second case, one has to deal with a problem of "absolute" consistency. On the contrary, the first type of problem reduces to a problem of "relative" consistency, by means of a construction of appropriate "models", and many proofs of this kind have been thought up, well before mathematics had taken up a completely formalised aspect: it is enough to remember the models of non-Euclidean geometry (see p. 133), the questions of the independence of the axioms of elementary geometry dealt with by Hilbert in the "Grundlagen der Geometrie" [163 c] as well as the work of Steinitz on the axiomatisation of Algebra and those of Hausdorff and his successors on that of Topology.

A theory $\mathscr{T}$ is said to be complete if, for every proposition $A$ of $\mathscr{T}$ (not containing any letter other that the constants of $\mathscr{T}$), one of the propositions $A,$ (not $A$), is a theorem of $\mathscr{T}$.$^{71}$ If one puts on one side certain very rudimentary formalisms, of which the completeness is easily proved ([164], p. 35), the results obtained in this area are essentially negative; the first in time, is due to K. Gödel [130 a] who showed that, if $\mathscr{T}$ is consistent and if the axioms of formalised arithmetic are theorems of $\mathscr{T}$, then $\mathscr{T}$ is not complete. The fundamental idea of his ingenious method consists in establishing a bijective correspondence (of course, by means of "finite processes") between the meta-mathematical statements and certain propositions in formalised arithmetic; we will restrict ourselves to a broad sketch.$^{72}$ For each collection $A$ which is a term or a relation of $\mathscr{T}$, one begins by associating (by a procedure which is an explicit construction, which can be applied quasi mechanically) a whole number $g(A)$, in a bijective way. In the same way, to each proof $D$ of $\mathscr{T}$ (considered as a succession of collections) one can associate a whole number $h(D)$ in a bijective way. Finally, one can give a procedure for constructing explicitly a relation $P(x, y, z)$ of $\mathscr{T}$,$^{73}$ such that, in $\mathscr{T}$, $P(x, y, z)$ requires that $x, y, z$ are whole numbers, and satisfies the following two conditions:

1° if $D$ is a proof of $A(\lambda)$, where $A(x)$ is a relation of $\mathscr{T}$, and $\lambda$ is an explicit whole number (that is to say a term of $\mathscr{T}$ which is a whole number), then $P(\lambda, g(A(x)), h(D))$ is a theorem of $\mathscr{T}$;

$^{71}$ This is often expressed by saying that if $A$ is not a theorem of $\mathscr{T}$, the theory $\mathscr{T}'$ obtained by adding $A$ to the axioms of $\mathscr{T}$ is inconsistent.

$^{72}$ For more details see [130 a] or ([181], pp. 181-258).

$^{73}$ The detailed description of $g(A), h(D)$ and $P(x, y, z)$ is very long and meticulous, and the writing out of $P(x, y, z)$ would require a number of symbols so great that it is impossible in practice, but no mathematician thinks that diminishes in any way the value of these constructions.

2° if the explicit whole number $\mu$ is not of the form $h(D)$, or if $\mu = h(D)$ and if $D$ is not a proof of $A(\lambda)$, then (not $P(\lambda, g(A(x)), \mu)$) is a theorem of $\mathscr{T}$.

Then let $S(x)$ be the relation (not $(\exists z)P(x, y, z)$), and let $\gamma = g(S(x))$, which is a term of $\mathscr{T}$. If $\mathscr{T}$ is consistent, there is no proof of the proposition $S(\gamma)$ in $\mathscr{T}$. In fact, if $D$ were such a proof, $P(\gamma, g(S(x)), h(D))$ would be a theorem of $\mathscr{T}$; but this relation is none other than $P(\gamma, \gamma, h(D))$, and it follows that $(\exists z)(P(\gamma, \gamma, z)$ would also be a theorem of $\mathscr{T}$; as this last relation is equivalent to (not $S(\gamma)$), $\mathscr{T}$ would be inconsistent. On the other hand what has just been said shows that for every explicit whole number $\mu$, (not $P(\gamma, \gamma, \mu)$) is a theorem of $\mathscr{T}$. The result is that there is no proof, in $\mathscr{T}$, of (not $S(\gamma)$), for this last relation is equivalent to $(\exists z)P(\gamma, \gamma, z)$ and the existence of a whole number $\mu$ such that $P(\gamma, \gamma, \mu)$ would require that $\mathscr{T}$ were inconsistent, by virtue of what comes before.$^{74}$ This metamathematical theorem of Gödel has been subsequently generalised in various directions ([181], chap. XI).$^{75}$

is intuitively true, since one has a proof in $\mathscr{T}$ of (not $P(\gamma, \gamma, \mu)$) for each explicit whole number $\mu$; and yet this proposition is not provable in $\mathscr{T}$. This situation is to be compared with a result obtained previously by Löwenheim and Skolem (see [286 a]): this latter defines metamathematically a relation between two natural whole numbers $x, y$ which, if one writes $x \in y$, satisfies the axioms of von Neumann for the theory of sets. From which, at first sight, a new "paradox", since in this "model" all the infinite sets would be countable, contrary to the inequality of Cantor $m < 2^m$. But in fact, the "relation" defined by Skolem can not be written in the formalised theory of sets, any more than the "theorem" stating that the set of subsets of an infinite set has only a countable infinity of "elements". At bottom, this "paradox" is only a more subtle form of the banal remark that one will never be able to write down more than a finite number of collections in a formalised theory, and that it is therefore absurd to conceive of an uncountable set of terms of the theory; remark close to that which had already led to "Richard's paradox". Similar reasoning proves that the formalisation of the theory of sets is indispensable if one wants to preserve the essentials of the Cantorian construction. Mathematicians seem to agree to conclude also that there is only just a superficial concordance between our "intuitive" conceptions of the notion of set or whole number, and the formalisms which are supposed to account for them; the disagreement begins when it is a question of choosing between the one and the other.

$^{74}$ In fact, this latter part of the reasoning supposes a bit more than the consistency of $\mathscr{T}$, namely what is called the "ω-consistency" of $\mathscr{T}$: this means that there is no relation $R(x)$ in $\mathscr{T}$ such that $R(x)$ requires $x \in \mathbf{N}$ and that, for each explicit whole number $\mu$, $R(\mu)$ be a theorem of $\mathscr{T}$ even though $(\exists x)(x \in \mathbf{N} \text{ and } \text{not } R(x)))$ is also a theorem of $\mathscr{T}$. Rosser has shown besides that one can modify the reasoning of Gödel in such a way as only to assume the consistency of $\mathscr{T}$ ([181], p. 208).

$^{75}$ The analogy of the reasoning of Gödel with the sophistry of the liar should be noted: the proposition $S(\gamma)$ affirms its own falsity, when one interprets it in metamathematical terms! One remarks also that the proposition of the cause and does not tie up in any natural way with any mathematical problem. Much more interesting is the fact that if $\mathscr{T}$ denotes the theory of sets (with the system of axioms of von Neumann-Bernays), neither the continuum hypothesis nor its negation are provable in $\mathscr{T}$. This remarkable result was established in two stages: in 1940, Gödel proved that the theory obtained by adjoining to $\mathscr{T}$ the continuum hypothesis $2^{N_0} = N_1$ was not inconsistent [130 b]; then, in 1963, P. Cohen proved the same when one adjoins to $\mathscr{T}$ the relation $2^{N_0} = N_2$ (or $2^{N_0} = N_n$ for an arbitrary whole number $n > 1$) [67].

The decision problem ("Entscheidungsproblem") is without doubt the most ambitious of all those which are stated in metamathematics: it consists in knowing whether, for a given formal language, one can imagine a "universal procedure" which is quasi mechanical which, when applied to any relation of the formalism in question, indicates in a finite number of operations whether that relation is true or not.$^{76}$ The solution of this problem was already part of the grand design of Leibniz, and it appears that at one time the school of Hilbert believed its realisation was quite close. It is a fact that one can describe such procedures for formalisms containing few primitive symbols and axioms ([181], pp. 136-141). But the efforts made to make precise the decision problem, by outlining exactly what is meant by "universal procedure" have until now resulted only in negative results ([181], pp.432-439). Besides, the solution of the decision problem for a theory $\mathscr{T}$ entrains immediately the knowledge of whether $\mathscr{T}$ is consistent or not, since it suffices to apply the "universal procedure" to a relation of $\mathscr{T}$ and its negation; and we will see that it is impossible to solve the question in this way for the usual mathematical theories.$^{77}$

It is indeed in the question of the consistency of mathematical theories — the origin and very heart of Metamathematics — that the results have revealed themselves to be the most deceiving. During the years 1920-1930, Hilbert and his school had developed new methods for attacking these problems; after having proved the consistency of partial formalisms, covering part of arithmetic (cf. [158], [324 d]), they believed they had reached the goal and proved, not only the consistency of arithmetic, but also that of the theory of sets, when Gödel, relying on the incompleteness of arithmetic, deduced from it the impossibility of proving, by the "finite procedures" of Hilbert, the consistency of every theory $\mathscr{T}$ containing this latter.$^{78}$

$^{76}$ A neighbouring question is that of the existence of "algorithms" for the solution of certain problems, a question which we are unable to examine here (Turing, Post, P. Novikov).

$^{77}$ One must distinguish carefully the decision problem from the belief, shared by numerous mathematicians, and often expressed vigorously by Hilbert in particular, that for every mathematical proposition, one will end up one day by knowing whether it is true, false or undecidable. That is a pure act of faith, the critique of which escapes our discussion.

$^{78}$ With the notation introduced above, the precise result of Gödel is the following. To say that $\mathscr{T}$ is consistent means that there is no proof, in $\mathscr{T}$, of the relation

The theorem of Gödel does not however shut the door completely on attempts to prove consistency, so long as one abandons (at least partially) the restrictions of Hilbert concerning "finite procedures". It is thus that Gentzen in 1936 [127], succeeds in proving the consistency of formalised arithmetic, by using "intuitively" transfinite induction up to the countable ordinal $\varepsilon_0$. The value of the "certainty" that one can attach to such reasoning is without doubt less convincing than for that which satisfies the initial requirements of Hilbert, and is essentially a matter of personal psychology for each mathematician; it remains no less true that similar "proofs" using "intuitive" transfinite induction up to a given ordinal, would be considered important progress if they could be applied, for example, to the theory of real numbers or to a substantial part of the theory of sets.

$0 \neq 0$; this requires as a consequence, that for every explicit whole number $\mu$, (not $P(0, g(x \neq x), \mu)$) is a theorem of $T$. Let us consider therefore the proposition $(\forall z)((z \in \mathbf{N}) \Rightarrow \text{not } P(0, g(x \neq x), z))$, which we will denote by $C$; in "translating" into formal arithmetic the reasoning (reproduced above) by which it is shown metamathematically that "if $T$ is consistent, there is no proof that $S(\gamma)$ is in $T$", one can establish that $C \Rightarrow (\text{not } (\exists z)(P(\gamma, \gamma, z)))$ is a theorem of $T$, in other words, that $C \Rightarrow S(\gamma)$ is a theorem of $T$. It follows that, if $T$ is consistent, $C$ is not a theorem of $T$, since under these conditions $S(\gamma)$ is not a theorem of $T$. That is the exact statement of the theorem of Gödel.

$^{79}$ Gentzen associates with each proof $D$ of formalised arithmetic an ordinal $\alpha(D) < \varepsilon_0$; on the other hand, he describes a procedure which, starting from each proof $D$ which ends with a contradiction, supplies a proof $D'$ also ending at a contradiction and such that $\alpha(D') < \alpha(D)$; the theory of well ordered sets allows us to conclude that such a proof $D$ cannot exist (a type of reasoning which extends the classical "infinite descent" of the theory of numbers).