Historical note

(Chapters I through IV)

“Aliquot selectos homines rem intra quinquennium absolvere posse puto” Leibniz ([12 b], vol. 7, p. 187).

(Numbers in brackets refer to the bibliography at the end of this Note.)

The study of what is usually called the “foundations of mathematics”, which has been unflaggingly pursued since the beginning of the nineteenth century, could not have achieved success without a parallel effort of systematization of logic, or at least of those parts of logic which govern the relationships of mathematical propositions. Thus the history of the theory of sets and the formalization of mathematics cannot be separated from that of “mathematical logic”. But traditional logic, like that of the modern philosophers, extends in principle over a field of applications far wider than mathematics. The reader will therefore not find a history of logic, even in a summary form, in this Note; we have limited ourselves, so far as possible, to retracing the evolution of logic only in so far as it has influenced the evolution of mathematics. Thus we shall say nothing about non-classical logics (many-valued logics, modal logics); a fortiori, we shall not go into the history of the controversies which, from the time of the Sophists to that of the Vienna school, have divided philosophers as to the possibility and manner of applying logic to the objects of the external world or the concepts of the human mind.

¶ Today it is well established that mathematics was already highly developed in pre-Hellenic times. Not only are the (extremely abstract) notions of integers and measurement of magnitudes freely used in the most ancient documents which have come down to us from Egypt and Chaldaea, but the algebra of the Babylonians, by the elegance and sureness of its methods, cannot be regarded as simply a collection of problems solved by empirical devices. And, although there is nothing in the texts resembling a “proof” in the formal sense of the word, we may justly consider that the discovery of such methods of solution, whose generality is apparent from the particular numerical applications treated, could not have been achieved without a minimum of logical reasoning (perhaps not entirely conscious, but rather of the type on which a modern algebraist relies when undertaking a calculation, before writing out all the details) ([1], pp. 203 ff.).

The originality of the Greeks consists precisely in a conscious effort of arranging mathematical proofs in a sequence so that the passage from one step to the next leaves nothing in doubt and compels universal assent. Of course, the Greek mathematicians, just like their present-day successors, made use of “heuristic” rather than rigorous arguments in the course of their researches; for example, Archimedes, in his Treatise on Method [4 bis], refers to results “found, but not proved” by earlier mathematicians [^1]. But, from the earliest detailed texts known to us (which date from the middle of the fifth century B.C.) the ideal “canon” of a mathematical text is fixed. It found its complete realization in the works of the great masters Euclid, Archimedes, and Apollonius. Their notion of proof differs in no respect from ours.

We do not possess any text which would allow us to trace the first steps of this “deductive method”. At its first known appearance, it is already nearly perfected. We can only think that it was a natural development from the perpetual search for “explanations” of the world, which characterized Greek thought from the time of the Ionian philosophers of the seventh century B.C. onward; furthermore, tradition is unanimous in ascribing the development and perfection of the method to the Pythagorean school, at a period somewhere between the end of the sixth century and the middle of the fifth.

This “deductive” mathematics, precise in its aims and rigorous in its methods, was to occupy the philosophical and mathematical reflections of the succeeding ages. We see, on the one hand, the edifice of “formal” logic slowly being built up, on the model of mathematics, and culminating in the creation of formalized languages; and on the other hand, principally from the beginning of the nineteenth century onward, especially after the advent of the theory of sets, the concepts which lie at the root of mathematics being more and more closely questioned and examined in an effort to clarify their nature.

The formalization of logic

The general impression that we get from the (extremely incomplete) texts in our possession is that Greek philosophic thought of the fifth century B.C. was dominated by an increasingly conscious effort to extend to the whole field of human thought the procedures of articulate discourse which were put to use with such great success in rhetoric and contemporary mathematics — in other words, to create Logic, in the most general sense of the word. The tone of philosophical writings underwent an abrupt change in this epoch. Whereas in the seventh and sixth centuries the philosophers asserted and prophesied (or, at best, sketched vague arguments, founded on equally vague analogies), from Parmenides and especially Zeno onward they reasoned and sought to extract general principles to serve as a basis for their dialectic. The first statement of the principle of the excluded middle is to be found in the work of Parmenides, and the proofs “by contradiction” of Zeno of Elea have remained famous. But Zeno flourished in the middle of the fifth century and, in spite of the uncertainty of documentation [^2], it is very likely that at this period the mathematicians also used these principles in their own sphere.

As we have already said, it is not within the scope of this book to relate the innumerable difficulties which arose at each stage in the development of this Logic, and the resulting polemics which occupied the Eleatic school, and the Sophists, as well as Plato and Aristotle. Let us merely note the part played in this evolution by the assiduous culture of the art of oratory and the analysis of language which naturally followed from it — developments which are attributed mainly to the Sophists of the fifth century. On the other hand, the influence of mathematics, if not always explicitly acknowledged, is no less clear, in particular in the writings of Plato and Aristotle. It could be said that Plato was almost obsessed by mathematics; although himself not an inventor in this domain, he kept himself informed of the discoveries of the contemporary mathematicians (many of whom were his friends or his pupils), and remained always most directly interested in the subject, to the extent of suggesting new directions for research. In his writings, it is mathematics which constantly serves as illustration or model (and on occasion, as with the Pythagoreans, feeds his penchant toward mysticism). Aristotle, as Plato’s pupil, could hardly have failed to receive the minimum of mathematical education demanded of the pupils at the Academy, and a volume of selections from his writings which relate or allude to mathematics has been compiled [2 bis]. However, he seems never to have made much effort to keep in contact with the mathematical movements of his times, and in this domain he quotes only results which had long since become common knowledge. The later philosophers were, for the most part, even less well acquainted with mathematics. Many of them lacked the necessary technical knowledge, and their references to mathematics relate to stages long since passed in the evolution of the subject.

The culmination of this period, in so far as logic is concerned, is the monumental work of Aristotle [2], whose great merit is to have succeeded for the first time in systematizing and codifying methods of reasoning which had hitherto remained vague or unformulated [^3]. For our purpose we need to keep in mind the general thesis of this work, namely that it is possible to reduce every correct argument to the systematic application of a small number of immutable rules which are independent of the particular nature of the objects under consideration (this independence is clearly brought out by the use of letters to denote concepts or propositions — a usage which Aristotle very probably borrowed from the mathematicians). But Aristotle concentrated his attention almost exclusively on a particular type of relations and chains of reasoning, namely what he called “syllogisms”; essentially, these are relations of the form $A \subset B$ or $A \cap B \neq \varnothing$, in the language of set theory [^4], and chains of such relations or their negations linked together by means of the scheme

Aristotle was too well acquainted with the mathematics of his time not to have realized that schemes of this nature were not sufficient to account for all the logical operations used by mathematicians, nor a fortiori for other applications of logic ([2], Analytica Priora, I, 35; [2 bis], pp. 25-26) [^5]. At any rate, he undertook a close study of the various forms of "syllogism" (almost entirely devoted to the elucidation of the perpetual difficulties which arise from the ambiguity or obscurity of the terms involved in reasoning), which led him to formulate, among other things, rules for taking the negation of a proposition ([2], Analytica Priora, I, 46). To Aristotle also belongs the merit of having distinguished with great clarity the role of "universal" propositions from that of "particular" propositions : a first step toward quantifiers [^6]. But it is only too well known how the influence of his writings — often narrowly and unintelligently interpreted — which remained considerable up to the nineteenth century, led philosophers to neglect the study of mathematics and impeded the progress of formal logic [^7]. Nevertheless, logic continued to make progress in antiquity, in the Megarian and Stoic schools, the rivals of the Peripatetics. Our knowledge of these doctrines is unfortunately all at second hand, often derived from their adversaries or mediocre commentators. It seems that the contribution of these logicians was the foundation of a "propositional calculus" in the modern sense of the word; instead of limiting themselves, like Aristotle, to propositions of the particular form $A \subset B$, they stated rules concerning entirely indeterminate propositions. Moreover, they analyzed so closely the logical relationships between these rules that they were able to deduce them all from five elementary rules, which were classed as "unprovable", by methods very similar to those we described in Chapter I, §§ 2 and 3 [5]. Unfortunately their influence was ephemeral, and their results lapsed into oblivion until rediscovered by the logicians of the nineteenth century. Aristotle remained the unchallenged master, in logic, until the seventeenth century. In particular, the scholastic philosophers were entirely under his influence, and although their contribution to formal logic is far from negligible, it contains no advance of the first order in comparison with that due to the philosophers of antiquity.

However, it does not appear that the works of Aristotle or his successors caused any notable repercussions in mathematics. The Greek mathematicians pursued their research along the paths opened by the Pythagoreans and their successors of the fourth century (Theodorus, Theaetetus, Eudoxus) without apparently concerning themselves with formal logic in the presentation of their results. This is hardly surprising when one compares the flexibility and precision of mathematical reasoning at this period with the extremely rudimentary state of Aristotelian logic. And, later, when logic passed beyond this stage, its evolution was guided by the new conquests of mathematics.

With the development of algebra, one could hardly fail to be struck by the similarity between the rules of formal logic and those of algebra, both being applicable to unspecified objects (propositions or numbers). And when, in the seventeenth century, algebraic notation took its definitive form in the hands of Vieta and Descartes, almost immediately there appeared various attempts to represent logical operations symbolically. But before Leibniz these attempts (as for example, that of Hérigone (1644) to write symbolically the proofs of elementary geometry, or that of Pell (1659) to do the same for arithmetic) were superficial and did not lead to any progress in the analysis of mathematical reasoning.

Leibniz was a philosopher who was also a mathematician of the first rank, and who was able to draw from his mathematical experience the germs of the ideas which were to release formal logic from the scholastic impasse [^8]. A universal genius if ever there was one, and an inexhaustible source of original and fertile ideas, Leibniz was all the more interested in logic because it lay at the heart of his grand projects of formalizing language and thought, at which he continued to work throughout his life. Having broken with scholastic logic already in his childhood, he was attracted by the idea (going back to Raymond Lulle) of a method of resolving all human concepts into primitive concepts, which thus constituted an “alphabet of human thoughts", and recombining them in a quasi-mechanical fashion so as to obtain all true propositions ([12 b], vol. VII, p. 185; cf. [12 bis], Chap. II). When still a very young man, he also conceived another, much more original idea, that of the utility of symbolic notation as a "thread of Ariadne" of thought [^9]. "The true method", he said "should provide us with a filum Ariadnes, that is to say, a certain sensible material means which will lead the intellect, like the lines drawn in geometry and the forms of operations which are prescribed to novices in arithmetic. Without this, our minds will not be able to travel a long path without going astray" ([12 b], vol. VII, p. 22; cf. [12 bis], p. 90). He had little knowledge of the mathematics of his time until the age of twenty-five, and first presented his projects in the form of a "universal language". But when he came into contact with algebra, he adopted it as the model for his "universal characteristic". By this phrase he meant a sort of symbolic language, capable of expressing all human thoughts unambiguously, of strengthening our powers of deduction, of avoiding errors by a purely mechanical effort of attention, and constructed in such a way that "les chimeres, que celuy meme qui les avance n'entend pas, ne pourrount pas estre ecrites en ces caracteres" ([12 a], vol. I, p. 187). In countless passages in his writings where Leibniz alludes to this grandiose project and the progress which its achievement would bring (cf. [12 bis], Chapters IV and VI), one sees how clearly he conceived the notion of a formalized language, as a pure combination of signs in which only the order of writing is important [^10], so that a machine would be capable of producing all the theorems [^11] and all controversies could be resolved by a simple calculation ([12 bis], vol. VII, pp. 198-203). Although these hopes may appear extravagant, it is nonetheless true that this constant theme in Leibniz' thought underlies a good part of his mathematical output, beginning with his work on the symbolism of infinitesimal calculus. He himself was perfectly aware of this fact and explicitly associated his ideas on indicial notation and determinants ([12 a], vol. II, p. 240; cf. [12 bis], pp. 481-487) and his sketch of "geometrical calculus" (cf. [12 bis], Chapter IX) with his "characteristic". In his mind, the essential part of his scheme was to be symbolic logic or, in his words, a "Calculus ratiocinator"; and if he did not succeed in creating this calculus, he made at least three attempts at it. In the first attempt he had the idea of associating with each "primitive" term a prime number, every term formed of several primitive terms being represented by the product of the corresponding prime numbers [^12]; he sought to translate into this system the usual rules of syllogism, but met considerable complications caused by negation (which he attempted, naturally enough, to represent by a change of sign) and soon abandoned this approach ([12 c], pp. 42-96; cf. [12 bis], pp. 326-344). In later attemps, he sought to give Aristotelian logic a more algebraic form. Sometimes he retained the notation $\mathrm{AB}$ for the conjunction of two concepts, sometimes he used the notation $\mathrm{A} + \mathrm{B}$ [^13]. He observed (in the multiplicative notation) the law of idempotence $\mathrm{AA} = \mathrm{A}$, noted that the proposition "every A is B" may be replaced by the equality $\mathrm{A} = \mathrm{AB}$, and that from this starting-point most of the rules of Aristotle may be obtained by purely algebraic calculation ([12 c], pp. 229-237 and 356-399; cf. [12 bis], pp. 345-364). Also he had the idea of the empty concept ("non Ens") and recognized, for example, the equivalence of the propositions "every A is B" and "A. (not B) is not true" (loc. cit.). Furthermore, he noted that his logical calculus applied not only to the logic of concepts but also to that of propositions ([12 c], p. 377). Thus he came very close to "Boolean calculus". Unfortunately, he seems not to have succeeded completely in casting off the scholastic influence; not only did he limit the aim of his calculus almost entirely to the transcription of the rules of syllogism in his notation [^14], but he went as far as sacrificing his most felicitous ideas to the desire of recovering the Aristotelian rules in their entirety, even those which are incompatible with the notion of the empty set [^15].

The work of Leibniz remained, for the most part, unpublished until the beginning of the twentieth century, and had little direct influence. Throughout the eighteenth century and the beginning of the nineteenth, various authors (de Segner, Lambert, Ploucquet, Holland, De Castillon, Gergonne) gave outlines of attempts which were similar to those of Leibniz but which never really passed beyond the point at which he had halted. Their works made very little impression, and most of them were unaware of the results of their predecessors [^16]. Another who wrote under the same conditions was G. Boole, who can be considered as the true creator of modern symbolic logic [16]. His master idea was to take the "extensional" point of view systematically and thus to calculate directly with sets, denoting by $xy$ the intersection of two sets, and by $x + y$ their union when $x$ and $y$ are disjoint. He also introduced a "universe", denoted by 1 (the set of all objects), and the empty set, denoted by 0, and he wrote $1 - x$ for the complement of $x$. As Leibniz had done, he interpreted the relation of inclusion by the relation $xy = x$ (from which he easily deduced the justifications of the rules of the classical syllogism), and his notation for unions and complements gave his system a flexibility which its predecessors had lacked [^17]. Moreover, by associating with each proposition the set of "cases" in which it is satisfied, he interpreted the relation of implication as an inclusion, and his calculus of sets gave him in this way the rules of "propositional calculus".

In the second half of the nineteenth century, Boole's system became the basis of the work of an active school of logicians, who improved and completed it in various respects. Thus Jevons (1864) widened the sense of the operation of union $x + y$ by extending it to the case where $x$ and $y$ are arbitrary. A. De Morgan in 1858, and C. S. Peirce in 1867, proved the duality relations

In 1860 De Morgan also began the study of relations, and defined inversion and composition of binary relations (i.e., the operations which correspond to the operations $\overset{-1}{G}$ and $G_1 \circ G_2$ on graphs) [^19]. All this work was systematically expounded and developed in the massive and prolix volumes of Schröder [23]. But it is rather curious to observe that the logicians we have just mentioned seemed scarcely interested in the application of their results to mathematics and that, on the contrary, the principal aim of Boole, and Schröder in particular, was apparently to develop “Boolean” algebra by imitating the methods and duplicating the problems of classical algebra (often in a rather artificial fashion). No doubt the reason for this attitude lies in the fact that Boolean calculus was still not suitable for transcribing the majority of mathematical arguments [^20] and therefore was only a very incomplete answer to the grand dream of Leibniz. The construction of formalisms better adapted to mathematics — the introduction of variables and quantifiers, due independently to Frege [24] and C. S. Peirce [22 b], the capital step forward in this direction — was the work of logicians and mathematicians who, in contrast to their predecessors, were primarily interested in applications to the foundations of mathematics.

Frege’s project [24 b and c] was to base arithmetic on logic formalized by means of a symbolism to which he gave the name Begriffsschrift (concept-script) ; we shall return later to the method by which he defined the natural integers. His publications were characterized by extreme precision and minute care in the analysis of concepts, which led him to introduce many distinctions that have since shown themselves to be of great importance in modern logic. For example, he was the first to distinguish between the statement of a proposition and the assertion that the proposition is true; between the relation of membership and that of inclusion; between an object $x$ and the set $\{x\}$ whose only member is $x$, etc. This formalized logic, which contains not only “variables” in the mathematical sense of the word but also “propositional variables” which represent undetermined relations and are susceptible of quantification, was later (through the work of Russell and Whitehead) to provide the fundamental tool of metamathematics. Unfortunately the notation adopted by Frege was unsuggestive, of appalling typographical complexity, and far removed from current mathematical usage; and the resulting unreadability had the effect of considerably diminishing his influence on his contemporaries.

Peano’s aim was both more ambitions and more down-to-earth. He set about publishing a Formulaire de mathematiques written entirely in formalized language, and containing not merely mathematical logic but all the results of the most important branches of mathematics. The speed with which he succeeded in completing this ambitious project, assisted by a constellation of enthusiastic collaborators (Vailati, Pieri, Padoa, Vacca, Vivanti, Fano, Burali-Forti) bears witness to the excellence of the system he had adopted. Closely following current mathematical usage, and introducing many well-chosen abbreviating symbols, his language succeeded moreover in being fairly readable, thanks mainly to an ingenious system of replacing brackets by separating points [29 c]. Many of the notations due to Peano are now in common use by the majority of mathematicians : for example, $\in$, $\supset$ (but, contrary to the present usage, in the sense of "is contained" or "implies" [^21]), $\cup$, $\cap$, $\mathrm{A} - \mathrm{B}$ (the set of differences $a - b$, where $a \in \mathrm{A}$ and $b \in \mathrm{B}$). Besides this, the Formulaire contains, for the first time, a close analysis of the general notion of a function, of direct [^22] and inverse images, and the remark that a sequence is just a function defined on $\mathbf{N}$. But in Peano's hands quantification is subjected to irksome restrictions (in his system, in principle, only relations of the form $\mathrm{A} \Rightarrow \mathrm{B}$, $\mathrm{A} \Leftrightarrow \mathrm{B}$, or $\mathrm{A} = \mathrm{B}$ can be quantified). Moreover, the almost fanatic zeal of some of his disciples laid him wide open to ridicule, and the often unjust criticism of H. Poincaré in particular was a serious blow to Peano's school and hindered the dissemination of his doctrines in the mathematical world.

With Frege and Peano we have the essential ingredients of the formalized languages in use today. The most famous of such languages is, without doubt, that created by Russell and Whitehead in their great work Principia Mathematica, which happily combines the precision of Frege with the convenience of Peano. Most of the formalized languages in current use differ from each other only in modifications of secondary importance whose object is to simplify their use. One of the most ingenious is the "functional" notation for relations (for example, $\in xy$ in place of $x \in y$), due to Lukasiewicz, which does away completely with the need for brackets. But the most interesting is certainly the introduction by Hilbert of the symbol $\tau$, which allows one to consider the quantifiers $\exists$ and $\forall$ as abbreviating symbols, to avoid the "universal" functional symbol $\iota$ of Peano and Russell (which applies only to functional relations) and to dispense with the axiom of choice in the theory of sets ([31], p. 183 [^23]).

The notion of truth in mathematics

Mathematicians have always been convinced that what they prove is "true". It is clear that such a conviction can be only of a sentimental or metaphysical order, and cannot be justified, or even ascribed a meaning which is not tautological, within the domain of mathematics. The history of the concept of truth in mathematics therefore belongs to the history of philosophy and not of mathematics; but the evolution of this concept has had an undeniable influence on the development of mathematics, and for this reason we cannot pass over it in silence.

First of all, a mathematician who is well schooled in philosophy is as rare as a philosopher with a wide knowledge of mathematics. The opinions of mathematicians on philosophical questions, even questions which touch on their own science, are most often second- or third-hand and from sources of doubtful value. But, precisely for this reason, these “average” opinions are of as much interest to the historian of mathematics as the original views of thinkers such as Descartes or Leibniz (to cite two who were also mathematicians of the highest order), Plato (who was at least familiar with the mathematics of this time), and Aristotle or Kant (for whom one cannot say as much).

The traditional notion of mathematical truth goes back to the Renaissance. In this conception, there is no great difference between the objects studied by mathematics and those studied by the natural sciences. Both are knowable, and can be apprehended by intuition and by reason; neither intuition nor reason is fallible if properly employed. “Only an entirely false intellect”, says Pascal, “could reason wrongly from principles so obvious that it is almost impossible for them to escape notice” ([11], vol. XII, p. 9). Descartes convinced himself that “it is only mathematicians who have been able to find proofs, that is to say, certain and evident reasons” ([10], vol. VI, p. 19), and this (if we accept his account) before he had constructed a metaphysic in which he says, “this, which a little while ago I took as a rule, namely that the things we conceive very clearly and very distinctly are all true, is assured only because God is or exists and is a perfect being” ([10], vol. VI, p. 38). Although Leibniz objects that it is not obvious how one is to recognize that an idea is “clear and distinct” [^24], yet he too considers the axioms to be self-evident and ineluctable consequences of the definitions as soon as the terms are understood [^25]. Moreover, it should not be forgotten that at this period mathematics embraced many sciences — sometimes even the art of the engineer which we would not nowadays regard as mathematical. The remarkable success of its applications to “natural philosophy”, the “mechanical arts”, and navigation was a principal cause of the confidence it inspired.

From this point of view the axioms were no more open to discussion or doubt than the rules of logical reasoning; at most, it was a matter of personal choice whether one reasoned “in the manner of the ancients” or gave free rein to one’s intuition. The choice of starting point was also a matter of individual preference. Thus many editions of Euclid’s work were published in which the solid logical framework of the Elements was strangely travestied, and “deductive” accounts of infinitesimal calculus or rational mechanics were given in which the foundations were singularly badly laid. Spinoza therefore perhaps acted in good faith when he claimed that his Ethics was “proved in the manner of the geometers” (more geometrico demonstrata). Although it is hard to find two mathematicians in the seventeenth century in agreement on any single question whatsoever, although the controversies were frequent, interminable, and acrimonious, yet the notion of truth was free from all imputation. “Since there is only one truth for each thing”, said Descartes, “whoever finds it knows as much as can be known” ([10], vol. VI, p. 21).

Although no Greek mathematical text of the great period on these questions has been preserved, it is probable that the point of view of the Greek mathematicians on this subject was much less rigid. It is only by experience that the rules of reasoning have been developed to the point of inspiring complete confidence; before they could be considered as above all discussion they must have passed through many hesitations and tentative forms. Furthermore, it would have been out of keeping with the critical spirit of the Greeks and their taste for discussion and sophistry, if those “axioms” which Pascal considered to be self-evident (and which, according to a legend originated by his sister, he himself discovered, with an infallible instinct, as a child) had not been the object of long discussions. In a domain which is not, strictly speaking, that of geometry, the paradoxes of the Eleatic philosophers have preserved for us some traces of such controversies; and Archimedes, when he observes ([4], p. 265) that his predecessors have on many occasions made use of the axiom which bears his name, adds that what is proved with the help of this axiom “has been admitted no less than what is proved without it”, and that it is enough for him that his own results should be admitted on the same footing. Plato, in conformity with his metaphysical views, presented mathematics as a means of access to “truth in itself”, and the objects with which it deals as existing in their own right in the world of ideas. He characterizes the mathematical method with accuracy in a famous passage of the Republic: “Those who study geometry and arithmetic... assume the existence of odd and even numbers, and three kinds of angles; these things they take as known, and consider that there is no need to justify them either to themselves or to others, because they are self-evident to everyone; and, starting from them, they proceed consistently step by step to the propositions which they set out to examine” (Book VI, 510c-e). Thus what constitutes a proof is first of all a starting point, which is to some extent arbitrary (although “self-evident to everyone”) and from which, he says a little later, one does not attempt to go further back; next, a path which passes in order through a sequence of intermediate stations; finally, at each step the consent of the interlocutor guaranteeing the correctness of the reasoning. It should be added that, once the axioms had been stated, no new appeal to intuition was in principle admitted; Proclus, quoting Geminus, recalls that “we have taught even the pioneers in this science to take no account of merely plausible conclusions when concerned with reasoning that is to form a part of our geometric doctrine” ([3], vol. I, p. 203).

Thus the rules of mathematical reasoning were developed by experience and the fire of criticism. If it is true, as has been plausibly argued [^26], that Book VIII of Euclid has preserved for us a part of the arithmetic of Archytas, then it is not surprising to find there the somewhat pedantic stiffness of reasoning which never fails to appear in any mathematical school which discovers or thinks it has discovered “rigour”. But once these rules of reasoning had entered into the common practice of mathematicians, it does not appear that they were ever subjected to doubt until a very recent epoch; although Aristotle and the Stoics deduced certain of these rules from certain others by schemes of reasoning, the primitive rules were always admitted as self-evident. Likewise, having gone back as far as the “hypotheses”, “axioms”, and “postulates” which seemed to them to provide a solid foundation for the science of their period (such, for example, as must have appeared in the first Elements, which tradition ascribes to Hippocrates of Chio, about 450 B.C.), the Greek mathematicians of the classical period seem to have devoted their efforts to the discovery of new results rather than to a critique of the foundations, which at this period could only have been sterile; and, metaphysical preoccupations apart, the text from Plato quoted earlier bears witness to this general agreement among mathematicians concerning the bases of their science.

Furthermore, the Greek mathematicians seem not to have believed that the “primary notions” which served for them as a starting point — straight line, surface, ratio of magnitudes — were capable of further elucidation. If they gave “definitions”, it was visibly to clear their consciences and without any illusions about the validity of these definitions. It goes without saying that, by contrast, the Greek mathematicians and philosophers had perfectly clear ideas on definitions other than those of the “primary notions” (often called “nominal” definitions, and playing the same role as our “abbreviating symbols”). In this connection there intervenes explicitly, without doubt for the first time, the question of “existence” in mathematics. Aristotle did not omit to remark that a definition does not imply the existence of the thing defined, and that either a postulate or a proof of existence is necessary. Undoubtedly his observation was derived from the practice of mathematicians. In every case Euclid is careful to postulate the existence of the circle, to prove that of the equilateral triangle, parallel lines, the square, etc. as he finds it necessary to introduce them into his arguments ([3], Book I); these proofs are “constructions” — in other words, he exhibits, with the aid of his axioms, mathematical objects which he then proves satisfy the definitions in question.

Thus we see Greek mathematics of the classical period come to a sort of empirical certainty (whatever may have been the metaphysical basis, according to this or that philosopher). If the rules of mathematical reasoning are considered to be above question, then the success of Greek science, and the feeling that critical revision would have been inopportune, are largely due to the confidence inspired by the axioms : a confidence of the same order as that accorded in the last century to the principles of theoretical physics. It is what is suggested by the proverb “nihil est in intellectu quod non prius fuerit in sensu” with which Descartes rightly took issue as not providing a firm enough basis for what he proposed to achieve by the use of reason.

It was not until the beginning of the nineteenth century that mathematicians retreated from the arrogant position of Descartes (not to speak of Kant or of Hegel; the latter was a little behind the time, as was to be expected, in relation to the science of his age [^27]) to a position as flexible as that of the Greeks. The first blow to the classical conceptions was the construction of hyperbolic non-Euclidean geometry by Gauss, Lobatschevsky, and Bolyai at the beginning of the century. We shall not undertake here a detailed account of the genesis of this discovery, the culmination of numerous fruitless attempts to prove the parallel postulate. At the time, its effect on the principles of mathematics was perhaps not so deep as is sometimes asserted. It simply demanded the abandonment of the claims of the preceding century to the “absolute truth” of Euclidean geometry and, a fortiori, of the Leibnizian view that the definitions imply the axioms; the latter no longer appear at all as “self-evident” but rather as hypotheses which are to be judged according to whether they are adapted to the mathematical representation of the physical world. Gauss and Lobatschevsky believed that the debate between the various possible geometries could be settled by experiment ([15], p. 76). This was also the point of view of Riemann, the aim of whose famous inaugural lecture “On the hypotheses which lie at the foundations of geometry” was to provide a general mathematical framework for various natural phenomena. “The question remains to be settled”, he said, “as to how far and to what extent these hypotheses will be confirmed by experiment” ([19], p. 284). But clearly this is a problem which no longer has anything to do with mathematics; and none of the authors just mentioned seems to have had any doubts that, even if a “geometry” did not correspond to experimental reality, its theorems were any less “mathematically true” [^28].

However, if this is so, such a conviction can no longer be attributed to an unlimited confidence in classical “geometrical intuition”. The description, which Riemann sought to give of “manifolds $n$ times extended”, the object of his work, relies on “intuitive” considerations [^29] only to the point of justifying the introduction of “local coordinates”; from then on, he seems to have felt that he was on firm ground, namely that of Analysis. But the latter is ultimately based on the concept of real number, which up to that time had remained on a very intuitive basis; and progress in the theory of functions was leading to most disturbing results in this respect. With the work of Riemann himself on integration, and even more with the examples of curves with no tangents, constructed by Bolzano and Weierstrass, the whole pathology of mathematics was beginning to emerge. A century later, we have seen so many monsters of this sort that we have become a little blasé. But the effect produced on the majority of nineteenth century mathematicians ranged from disgust to consternation. “How”, asked H. Poincaré, “can intuition deceive us at this point?” ([33 b], p. 19); and Hermite (not without a turn of humour, which not all the commentators on this celebrated phrase seem to have perceived) declared that he “turned away with fright and horror from this lamentable plague of continuous functions with no derivatives” ([27], vol. II, p. 318). The most serious feature of the malaise was that it was no longer possible to attribute these phenomena, so contrary to common sense, to badly elucidated notions, as in the days of the “indivisibles”, because they arose after the reforms of Bolzano, Abel, and Cauchy, which had succeeded in founding the notion of limit as rigorously as the theory of proportions. They were therefore to be ascribed to the gross and incomplete character of our geometrical intuition, which since that time has been discredited as a method of proof.

This conclusion inevitably affected classical mathematics, beginning with geometry. However much respect was paid to the axiomatic construction of Euclid, the imperfections in it had been noticed, from antiquity onward. The parallel postulate had been the object of the largest number of criticisms and attempts at proof; but Euclid’s successors and commentators had also sought to prove other postulates (notably that of the equality of right angles) or had recognized the inadequacy of certain definitions, such as those of a straight line or a plane. In the 16th century Clavius, an editor of the Elements, noticed the absence of a postulate guaranteeing the existence of the fourth proportional; Leibniz, for his part, remarked that Euclid makes use of geometrical intuition without mentioning it explicitly, for example when he assumes (Elements, Book I, Proposition 1) that two circles, each of which passes through the centre of the other, have a common point ([12 b], vol. VII, p. 166). Gauss (who did not deny himself the use of such topological considerations) drew attention to the part played in Euclid’s constructions by the notion of a point (or a line) situated “between” two others, a notion which nevertheless had not been defined ([14], vol. VIII, p. 222). Finally, the use of displacements — notably in the case of “congruent triangles” — long accepted as standing to reason ([^30]), soon appeared to the critics of the nineteenth century as a concept which rested on unformulated axioms. Thus, in the period from 1860 to 1885, many partial revisions of the foundations of geometry appeared (Helmholtz, Méray, Houël) with the purpose of filling some of these gaps. But it was not until M. Pasch [28] that the abandonment of all appeal to intuition was clearly formulated as a programme and carried out with perfect rigour. The success of his enterprise soon produced numerous emulators who, principally between 1890 and 1910, gave varied presentations of the axioms of Euclidean geometry. The most famous of these works were Peano’s, written in his symbolic language [29 b], and above all Hilbert’s Grundlagen der Geometrie, which appeared in 1899, and by the lucidity and profundity of its exposition immediately and justly became the model of modern axiomatics, to the extent that its predecessors were forgotten. Hilbert was not content with giving a complete system of axioms for Euclidean geometry, but classed them in various groups according to their nature, and set about determining the exact range of each group of axioms, not only by developing the logical consequences of each group in isolation, but also by discussing the various “geometries” obtained by suppressing or modifying certain axioms (geometries among which those of Lobatschevsky and Riemann appear as special cases [^31]). In this way he brought clearly into view, in a domain which until that time had been considered as one of the closest to external reality, the liberty which a mathematician has in the choice of his postulates. In spite of the disarray caused among some philosophers by these “metageometries” with their weird properties, the thesis of the Grundlagen was rapidly and almost unanimously adopted by mathematicians. Thus H. Poincaré, who could hardly be suspected of partiality towards formalism, asserted in 1902 that the axioms of geometry are conventions for which the notion of “truth”, in the everyday sense of the word, has no meaning ([33 a], pp. 66-67). “Mathematical truth” therefore consists entirely in logical deduction from premises posed arbitrarily as axioms. As we shall see later, the validity of the rules of reasoning according to which the deductions operate was soon to be called in question, leading thus to a complete overhauling of the conceptions at the base of mathematics.

Objects, models, structures

A. Mathematical objects and structures. From antiquity to the nineteenth century there was common agreement on the principal objects of study of a mathematician, namely those mentioned by Plato in the passage quoted earlier: numbers, magnitudes, and figures. Although, for the Greeks, this list should be enlarged to include the objects and phenomena proper to mechanics, astronomy, optics, and music, they nevertheless always made a clear distinction between these “mathematical” disciplines and arithmetic and geometry; and from the time of the Renaissance the former soon acquired the status of independent sciences.

Whatever the philosophical nuances which coloured the conception of mathematical objects in the mind of this or that mathematician or philosopher, there was at least one point of unanimous agreement : that these objects are given and that it does not lie in our power to ascribe arbitrary properties to them, any more than a physicist can alter a natural phenomenon. In truth, these views are partly the product of psychological reactions, which it is not for us to go into, but which every mathematician is well aware of when he exhausts himself in vain efforts to obtain a proof which seems perpetually to escape him. From there it is only one step to assimilating this resistance to external obstacles; and even today more than one mathematician, who parades an intransigent formalism, would voluntarily subscribe in his conscience to this avowal by Hermite : “I believe that the numbers and functions of Analysis are not the arbitrary product of our minds; that they exist outside of us, with the same character of necessity as the objects of the material world, and we meet or discover them, and study them, just like the physicists, chemists, and zoologists ([27], vol. II, p. 398).

In the classical conception, mathematics was the study of only numbers and figures. But this official doctrine, to which every mathematician felt bound to give his verbal adherence, gradually became more and more of an intolerable constraint under the pressure of new ideas. The embarrassment of algebraists in the presence of negative numbers vanished only when analytical geometry provided them with a convenient “interpretation”. But even in the eighteenth century d’Alembert (although a convinced “positivist”), when discussing the question in the Encyclopédie [13], suddenly lost courage after a column of somewhat confused explanations, and contented himself with concluding that “the rules of algebraic operations on negative quantities are generally admitted by everyone and are generally received as correct, whatever interpretation is to be attached to these quantities”. As regards imaginary numbers, the scandal was far worse still; for if they were “impossible” roots and if (until about 1800) there was no known way of “interpreting” them, how could these undefinable objects be talked about without contradiction, and above all how could they be introduced? On this topic d’Alembert prudently kept his silence, and did not even state the questions, no doubt because he realized that he could not answer in terms other than those naïvely used by A. Girard a century earlier [9] : “On pourroit dire : à quoy sert ces solutions qui sont impossibles? Je réponds : pour trois choses, pour la certitude de la reigle générale, et qu’il n’y a point d’autres solutions, et pour son utilité”.

In analysis, the situation in the eighteenth century was hardly better. It was a happy coincidence that analytical geometry appeared, as if on cue, to provide an “interpretation”, in the form of a geometrical figure, of the great creation of the 17th century, the notion of function, and thus to assist powerfully (in the hands of Fermat, Pascal, and Barrow) at the birth of infinitesimal calculus. But the philosophico-mathematical controversies to which the notions of infinitely small and indivisible quantities gave rise are only too well known. And although d’Alembert was on firmer ground here, and realized that in the “metaphysics” of the infinitesimal calculus there is nothing but the concept of limit, he was no more able than his contemporaries to comprehend the true meaning of expansions in divergent series, or to explain the paradox of correct results obtained by calculations with expressions which are devoid of all numerical interpretation. Finally, even in the domain of “geometrical certainty”, the Euclidean façade was cracking. When Stirling, in 1717, did not hesitate to say that a certain curve has an “imaginary double point at infinity” [^32], he would have been hard put to relate such an “object” to commonly received notions; and Poncelet, who, at the beginning of the nineteenth century, gave a considerable impetus to such ideas by founding projective geometry, was content to invoke as justification an entirely metaphysical “principle of continuity”.

In these circumstances (and, paradoxically, at the very time when the “absolute truth” of mathematics was being proclaimed with the greatest insistence), the notion of proof seemed to become more and more blurred as the eighteenth century ran its course, because the notions employed, and their fundamental properties, could not be definitively fixed in the manner of the Greeks. The return to rigour, which was set in motion at the beginning of the nineteenth century, brought some improvements to this state of affairs, but did not correspondingly stem the flood of new ideas. Thus in algebra appeared the imaginaries of Galois, the ideal numbers of Kummer, followed by vectors and quaternions, $n$-dimensional spaces, multivectors, and tensors, not to mention Boolean algebra. Undoubtedly one of the great advances (which made possible a return to rigour without abandonment of any part of the conquests of the past) was the possibility of constructing “models” of these new notions in more classical terms. Thus ideal numbers and the imaginary numbers of Galois were interpreted in terms of the theory of congruences, $n$-dimensional geometry could be regarded (if one so wished) purely as a language for expressing the results of algebra “of $n$ variables”; and, as regards the classical imaginary numbers — whose geometrical representation by the points of a plane marked the beginning of this efflorescence of algebra — there was soon the choice between this geometrical “model” and an interpretation in terms of congruences. But mathematicians at last began to feel that this makeshift approach was contrary to the natural development of their science and that it should be legitimate, in mathematics, to work with objects which have no concrete “interpretation”. “It is not of the essence of mathematics”, said Boole in 1854, “to be conversant with the ideas of number and quantity” ([^33]).

The same considerations led Grassmann, in his “Ausdehnungslehre” of 1844, to present his calculus in a form in which the notions of number and geometrical object were completely excluded from the start [^34]. And a little later, Riemann, in his inaugural lecture, was careful not to speak of “points” but of “determinations” (Bestimmungsweise) in his description of “manifolds $n$ times extended”, and emphasized that in such a manifold the “metrical relations” (Massverhältnisse) “can be studied only for abstract quantities and can be represented only by formulae; under certain assumptions they can however be decomposed into relations, each of which separately is capable of a geometrical representation, and thereby the results of calculations may be expressed in geometrical terms” ([19], p. 276).

From this moment onward, the enlargement of the axiomatic method was an accepted fact. Although for some time yet it was felt to be useful to check “abstract” results, whenever possible, against geometrical intuition, at least it was agreed that the “classical” objects were not the only legitimate objects of study for a mathematician. Precisely because of the multiplicity of possible “interpretations” or “models”, it came to be recognized that the “nature” of mathematical objects is ultimately of secondary importance, and that it matters little, for example, whether a result is presented as a theorem of “pure” geometry or as a theorem of algebra via analytical geometry. In other words, the essence of mathematics — that fleeting and insubstantial notion which hitherto could only be expressed by vague names such as “reigle generale” or “metaphysic” — appeared as the study of relations between objects which do not of themselves intrude on our consciousness, but are known to us by means of some of their properties, namely those which serve as the axioms at the basis of their theory. Boole had understood this clearly in 1847 when he wrote that mathematics is concerned with “operations considered in themselves, independently of the various ways in which they may be applied” ([16 a], p. 3). Hankel, in 1867, inaugurating the axiomatization of algebra, defended mathematics as being “purely intellectual, a pure theory of forms whose purpose is the study, not of the combination of magnitudes or of their images, namely numbers, but objects of thought (Gedankendinge), which may correspond to concrete objects or relations, although such a correspondence is not necessary” ([20], p. 10). Cantor, in 1883, echoes this claim for a “free mathematics”, proclaiming that “mathematics is entirely free in its development, and its concepts are bound only by the necessity that they should be non-contradictory and coordinated with concepts previously introduced by precise definitions” ([25], p. 182). Finally, the revision of Euclidean geometry helped to disseminate and popularize these ideas. Pasch himself, although still attached to a certain “reality” of geometrical objects, realized that geometry is a fact independent of their signification, and consists purely in the study of their relations ([28], p. 90): a conception which Hilbert pushed to its logical conclusion by emphasizing that even the names of the basic notions of a mathematical theory may be chosen at random [^35], and which Poincaré expressed by saying that axioms are “disguised definitions”, thereby completely reversing the point of view of the scholastic philosophers.

It is therefore tempting to assert that the modern notion of “structure” was substantially in existence by 1900, but in fact another thirty years of preparation were required before it made its full-fledged appearance. Certainly it is not difficult to recognize structures of the same species when they are of a sufficiently simple nature; with group-structures, for example, this point was attained in the middle of the nineteenth century. But at the same period Hankel was still struggling, without complete success, to extract the general ideas of a field and an extension field, which he managed to express only in the semi-metaphysical form of a “principle of permanence” [20], and which were definitively formulated only by Steinitz forty years later. It has been especially difficult to escape from the feeling that mathematical objects are “given” together with their structure, and it has taken many years of functional analysis to make modern mathematicians familiar with the idea that, for example, there are several “natural” topologies on the set of rational numbers, and several measures on the real line. With this dissociation the passage to the general definition of structures such as has been presented in this book, was finally achieved.

B. Models and isomorphisms. In several places we have had occasion to refer to the notion of a “model” or an “interpretation” of a mathematical theory constructed with the help of another theory. This is not a recent idea, and it should undoubtedly be seen as a continually recurring manifestation of a deep-lying feeling of the unity of the various “mathematical sciences”. If the traditional maxim “All things are numbers” of the early Pythagoreans is taken as authentic, it may be considered as the vestige of a first attempt to reduce the geometry and algebra of the time to arithmetic. Although the discovery of irrationals appeared to close this path for good, the reaction it provoked in Greek mathematics was a second attempt at synthesis, this time taking geometry as the basis and embracing, among other things, the methods of solution of algebraic equations inherited from the Babylonians [^36]. This conception reigned supreme until the fundamental reform of R. Bombelli and Descartes, which assimilated every measurement of magnitudes to a measurement of length (in other words, to a real number). But with the creation of analytical geometry by Descartes and Fermat, the trend was again reversed, and a far closer fusion of geometry and algebra was obtained, this time to the benefit of algebra. Descartes went further and at one stroke conceived of the essential unity of “all the sciences which are commonly called mathematical... Although their objects are different”, he wrote, “they are all concordant with one another in that they are concerned only with the various ratios or proportions which are present” ([10], vol. VI, pp. 19-20) [^37]. However, this point of view tended only to make algebra the fundamental mathematical science: a conclusion against which vigorous protests were uttered by Leibniz, who, as we have seen, had also conceived of a “universal mathematics”, but on a far vaster scale and already in close accord with modern ideas. Making quite explicit the “concordance” of which Descartes spoke, he perceived for the first time what is effectively the general notion of isomorphism (which he called “similitude”) and the possibility of “identifying” isomorphic relations or operations; as an example he addition and multiplication ([12 bis], pp. 301-303). But these bold conceptions found no echo among his contemporaries, and the realization of his dreams had to await the expansion of algebra in the middle of the nineteenth century. We have already observed that it was at this point that the “models” multiplied and the habit of passing from one theory to another by simple changes of language was formed; the most striking example of the latter is perhaps that of duality in projective geometry, where the practice of printing theorems “dual” to each other side by side in two columns undoubtedly had a large share in the acceptance of the notion of isomorphism. On a more technical level, it is certain that the notion of isomorphic groups was known to Gauss for commutative groups, and to Galois for groups of permutations; it was in defined for general arbitrary groups about the middle of the nineteenth century [^38]. Thereafter, with each new axiomatic theory it became natural to define a notion of isomorphism; but it is only with the modern notion of structure that it has been finally recognized that every structure contains in itself a notion of isomorphism, and that it is superfluous to give a special definition for each species of structures.

C. The arithmetization of classical mathematics. The increasingly widespread use of the notion of “model” was also to permit the nineteenth century mathematicians to achieve the unification of mathematics dreamed of by the Pythagoreans. At the beginning of the century, whole numbers and continuous quantities appeared to be as irreconcilable as they were in antiquity; the real numbers were related to the notion of geometric magnitude (especially that of length), and the “models” of negative numbers and imaginary numbers were constructed on this basis. Even the rational numbers were traditionally associated with the idea of “subdivision” of a magnitude into equal parts. Only the integers remained apart, as “exclusive products of our intellect”, as Gauss said in 1832, when contrasting them with the notion of space ([14], vol. VIII, p. 201). The first attempts, by Martin Ohm (1822), to bring together arithmetic and analysis were concerned with the rational numbers (positive and negative); they were taken up again around 1860 by several authors, notably Grassmann, Hankel, and Weierstrass (in unpublished lectures). To Weierstrass is apparently due the idea of obtaining a “model” of the positive or negative rational numbers by considering classes of ordered pairs of natural integers. But the most important step still remained to be taken, namely to construct a “model” of the irrational numbers from within the theory of rational numbers. By 1870 this had become an urgent problem because of the necessity, after the discovery of “pathological” phenomena in analysis, to purge every trace of geometrical intuition and the vague notion of “magnitude” from the definition of real numbers. The problem was in fact solved, at about this time, almost simultaneously by Cantor, Dedekind, Méray, and Weierstrass, using quite different methods from one another.

From then on the integers became the foundation of all classical mathematics. Furthermore, the “models” founded on arithmetic acquired still greater importance with the extension of the axiomatic method and the conception of mathematical objects as free creations of the human intellect. For there remained a limitation to the freedom claimed by Cantor, namely the limitation raised by the question of “existence”, which had already occupied the Greeks, and which arose now so much more urgently precisely because all appeal to intuitive representation was now abandoned. We shall see later what a philosophico-mathematical maelström was to be generated by the notion of “existence” in the early years of the twentieth century. But in the nineteenth century this stage had not yet been reached, and to prove the existence of a mathematical object having a given set of properties meant simply, just as for Euclid, to “construct” an object with the required properties. This was precisely the purpose of the arithmetical “models”; once the real numbers had been “interpreted” in terms of integers, then the same was done for complex numbers and Euclidean geometry, thanks to analytic geometry, and likewise for all the new algebraic objects introduced since the beginning of the century. Finally, in a discovery which achieved great fame, Beltrami and Klein obtained Euclidean “models” of the non-Euclidean geometries of Lobatschevsky and Riemann and thereby “arithmetized” (and completely justified) these theories which at first had aroused so much distrust.

D. The axiomatization of arithmetic. It was natural, in this line of development, that attention should next be directed toward the foundations of arithmetic itself, and in fact this is what happened around 1880. Apparently, before the nineteenth century, no one had attempted to define addition and multiplication in any other way than by a direct appeal to intuition. Leibniz alone, faithful to his principles, pointed out explicitly that “truths” as “obvious” as $2 + 2 = 4$ are no less capable of proof, if one reflects on the definitions of the numbers which appear therein ([12 b], vol. IV, p. 403; cf. [12 bis], p. 203); and he did not by any means regard the commutativity of addition and multiplication as self-evident [^39].[^39] He did not, however, push his reflections on this subject any further, and in the middle of the nineteenth century progress had still not been made in this direction. Even Weierstrass, whose lectures contributed greatly to the the spread of "arithmetizing" point of view, seems not to have realized the necessity of a logical clarification of the theory of integers. The first steps in this direction are apparently due to Grassmann who, in 1861 ([17], vol. $\mathrm{II}_2$, p. 295), gave a definition of addition and multiplication of integers and proved their fundamental properties (commutativity, associativity, distributivity), using nothing but the operation $x \to x + 1$ and the principle of induction. The latter had been clearly conceived and used for the first time in the seventeenth century by B. Pascal ([11], vol. III, p. 456) [^40][^40] — though more or less explicit applications of it are to be found in the mathematics of antiquity — and it was in current use by mathematicians from the second half of the seventeenth century. But it was not until 1888 that Dedekind ([26], vol. III, pp. 359-361) enunciated a complete system of axioms for arithmetic (reproduced three years later by Peano and usually known under his name [29 a]), which contained in particular a precise formulation of the principle of induction (which Grassmann had used without enunciating it explicitly).

With this axiomatization it seemed that the definitive foundations of mathematics had been attained. But, in fact, at the very moment when the axioms of arithmetic were being clearly formulated, arithmetic itself was being dethroned from this role of primordial science in the eyes of many mathematicians (beginning with Dedekind and Peano), in favour of the most recent of mathematical theories, namely the theory of sets; and the controversies which were to surround the notion of integers cannot be isolated from the great "crisis of foundations" of the years 1900-1930.

The theory of sets

It may be said that at every period of history mathematicians and philosophers have used arguments of the theory of sets more or less consciously. But in the history of their conceptions on this subject, it is necessary to separate clearly all questions relating to the idea of cardinal numbers (and, in particular, relating to the notion of infinity) from those which involve only the notions of membership and inclusion. The latter are intuitively clear and seem never to have given rise to controversies. They serve as the easiest foundation for a theory of the syllogisms (as Leibniz and Euler were to show), for axioms such as “the whole is greater than the part”, and for that part of geometry which is concerned with intersections of curves and surfaces. Up to the end of the nineteenth century, there was equally no objection raised to speaking of the set (or “class” according to some writers) of all objects having one or another given property [^41]; and the celebrated “definition” given by Cantor (“By a set we mean a grouping into one entity of distinct objects of our intuition or our thought”) ([25], p. 282) provoked hardly any objections at the time of its publication [^42]. But the situation was quite different where the notions of number or magnitude became entangled with the notion of set. The question of the indefinite divisibility of length (raised, certainly, by the early Pythagoreans) led, as is well known, to considerable philosophical difficulties; from the Eleatic school to Bolzano and Cantor, mathematicians and philosophers unsuccessfully came up against the paradox of the finite magnitude composed of an infinite number of points without magnitude. For us it is irrelevant to retrace, even in summary, the interminable and passionate polemics aroused by this problem, which constituted a particularly favourable terrain for metaphysical or theological aberrations. Let us only take note of the point of view held since antiquity by the majority of mathematicians. It consisted essentially in refusing to discuss the question since it could not be irrefutably decided, an attitude which we shall find again among the modern formalists: just as the latter contrive to eliminate all appearances of “paradoxical” sets (see later, page 328), so the classical mathematicians carefully avoided introducing into their arguments the “actual infinite” (that is to say, sets containing an infinity of objects conceived as existing simultaneously, at least in thought) and contented themselves with the “potential infinite”, i.e., the possibility of increasing any given magnitude (or of diminishing it, if a “continuous” magnitude was under consideration) [^43]. If this point of view contained a certain measure of hypocrisy [^44], it nevertheless allowed the development of the greater part of classical mathematics (including the theory of proportion and, later, the infinitesimal calculus) [^45]; it appeared to be an excellent lifeline, especially after the quarrels aroused by infinitesimals, and had become an almost universally accepted dogma until well into the nineteenth century.

A first glimmer of the general notion of equipotence appeared in a remark of Galileo ([8], vol. VIII, pp. 78-80). He observed that the mapping $n \to n^2$ established a one-to-one correspondence between the natural integers and their squares, and consequently that the axiom “the whole is greater than the part” could not be applied to infinite sets. But, far from inaugurating a rational study of infinite sets, this remark seems to have had no other effect than to reinforce distrust of the “actual infinite”; this was the moral that Galileo himself drew, and Cauchy quotes him in 1833 only to endorse his attitude.

The requirements of analysis — and especially the deeper study of functions of real variables, which was pursued throughout the nineteenth century — are at the origin of what was to become the modern theory of sets. When Bolzano in 1817 proved the existence of the greatest lower bound of a set bounded below in $\mathbf{R}$, he argued, as most of his contemporaries did, in terms of “comprehension” and considered, not an arbitrary set of real numbers, but an arbitrary property of them. But when, thirty years later, he wrote his Paradoxien des Unendlichen [18] (published in 1851, three years after his death), he did not hesitate to claim the possibility of existence for the “actual infinite” and to speak of arbitrary sets. In this work he defined the general notion of equipotence of two sets and proved that any two compact intervals in $\mathbf R$ are equipotent (provided each contains more than a single point). He observed also that the characteristic difference between finite and infinite sets lies in the fact that an infinite set $E$ is equipotent to some proper subset of $E$, but he gave no convincing proof of this assertion. The general tone of the work is more philosophical than mathematical; and since he did not distinguish precisely enough between the notion of the power of a set and that of magnitude or order of infinity, Bolzano failed in his attempts to construct infinite sets of greater and greater powers, and his discussion in this connection was entangled with considerations of divergent series which are entirely devoid of meaning.

The creation of the theory of sets, as it is understood today, is due to the genius of G. Cantor. He too began as an analyst, and his work on trigonometric series, inspired by that of Riemann, led him naturally in 1872 to a first essay in classifying the “exceptional” sets which arise in this theory [^46] through the notion of successive “derived sets”, which he introduced in this connection. Undoubtedly it was these researches, and also his method for defining real numbers, which led Cantor to interest himself in problems of equipotence; for in 1873 he noted that the set of rational numbers (or the set of algebraic numbers) is countable. In his correspondence with Dedekind, which began at about this time [25 bis], he raised the question whether the set of integers and the set of real numbers are equipotent, and succeeded a few weeks later in showing that they are not. Next, from 1874 onward, the problem of dimension preoccupied him, and for three years he sought in vain to establish the impossibility of a one-to-one correspondence between $\mathbf R$ and $\mathbf R^n$ ($n>1$), before he came to construct such a correspondence, to his own stupefaction [^47]. Having obtained these new and surprising results, he devoted himself entirely to the theory of sets. In a series of six memoirs published in the Mathematische Annalen between 1878 and 1884, he considered problems of equipotence, the theory of totally ordered sets, topological properties of $\mathbf R$ and $\mathbf R^n$, and the problem of measure; and it is admirable to see how in his hands the notions which appeared to be so inextricably rooted in the classical conception of the “continuum” were gradually brought out. In 1880 he had the idea of iterating “transfinitely” the formation of “derived sets”; but this idea did not take substance until two years later, with the introduction of well-ordered sets — one of the most original of Cantor’s discoveries, thanks to which he was able to initiate a detailed study of cardinal numbers and to formulate the “problem of the continuum” [25].

It was not to be expected that such bold conceptions, which ran counter to a tradition two thousand years old and led to such unlikely results of so paradoxical an appearance, should have been accepted without resistance. In fact, among the influential mathematicians in Germany, Weierstrass was the only one to follow with some favour the work of Cantor (his former pupil); but Cantor met with the irreconcilable opposition of Schwarz, and above all of Kronecker [^48]. It seems to have been as much the constant tension generated by opposition to his ideas as his fruitless efforts to prove the continuum hypothesis which produced in Cantor the first symptoms of a nervous illness which was to affect his mathematical productivity [^49]. In fact he did not resume his interest in the theory of sets until about 1887, and his last publications, which are concerned mainly with the theory of totally ordered sets and the calculus of ordinals, date from the period 1895-97. He had also proved in 1890 the inequality $m < 2^m$. However, not only did the problem of the continuum remain unanswered, but there was a more serious gap in the theory of cardinals, for Cantor had not been able to prove the existence of a well-ordering relation between arbitrary cardinals. This gap was to be filled partly by the theorem of F. Bernstein (1897) showing that the relations $\alpha \leq \beta$ and $\beta \leq \alpha$ imply $\alpha = \beta$ [^50], and above all by the theorem of Zermelo [36 a] which proved the existence of a well-ordering on any set : a theorem that Cantor had conjectured already in 1883 ([25], p. 169).

Dedekind, however, had followed the work of Cantor with sustained interest since the beginning. But whereas the latter concentrated his attention on infinite sets and their classification, Dedekind pursued his own thoughts on the notion of number (which had already led to his definition of irrational numbers by means of "cuts"). In his pamphlet Was sind und was sollen die Zahlen, which was published in 1888, but whose essential content dates from the period 1872-8 ([26], vol. III, p. 335), he showed how the notion of natural integer (which, as we have seen, was now the basis for the whole of classical mathematics) could itself be derived from the fundamental notions of the theory of sets. He was certainly the first to develop explicitly the elementary properties of arbitrary mappings of one set into another (hitherto neglected by Cantor, who was interested only in injective correspondences) and introduced, for any mapping $f$ of a set E into itself, the notion of the "chain" of an element $a \in E$ relative to $f$, namely the intersection of all sets $K \subset E$ such that $a \in K$ and $f(K) \subset K$ [^51]. Next, he took as the definition of an infinite set E the fact that there exists a one-to-one mapping $\varphi$ of E into E such that $\varphi(E) \neq E$ [^52]. If, furthermore, there exists such a mapping $\varphi$ and an element $a \notin \varphi(E)$ for which the whole set E is the chain of $a$, Dedekind said that E is "simply infinite"; moreover, he noted that "Peano's axioms" are then satisfied and showed (before Peano did) how, from that starting-point, all the elementary theorems of arithmetic may be obtained. The only thing lacking in his presentation is the axiom of infinity, which Dedekind (following Bolzano) believed he could prove by considering the "world of thoughts" (Gedankenwelt) as a set [^53].

On another front, Dedekind had been led by his arithmetical work (and especially the theory of ideals) to envisage the notion of ordered set in a more general aspect than Cantor had done. Whereas the latter restricted himself entirely to totally ordered sets [^54], Dedekind investigated the general case and in particular made a close study of lattices ([26], vol. II, pp. 236-271). This work attracted hardly any attention at the time; and although his results, rediscovered by various authors, have been the object of numerous publications since 1935, their historical importance lies far less in the possible applications of this theory (which indeed are rather scanty) than in the fact it was one of the first examples of careful axiomatic construction. On the other hand, Cantor's first results on countable sets soon produced many important applications, even to the most classical questions of analysis [^55] (quite apart from those parts of his work which inaugurated general topology and measure theory; for these the reader is referred to the Historical Notes in General Topology). Moreover, the last years of the nineteenth century saw the first applications of the principle of transfinite induction which, especially since the proof of Zermelo's theorem, has become an indispensable instrument in all parts of modern mathematics. Kuratowski in 1922 gave a version of this principle, subsequently rediscovered by Zorn [45] and customarily named after him, which is often more convenient to apply since it avoids the use of well-ordered sets ([40], p. 89); it is this form of the principle that is generally used nowadays [^56].

By the end of the nineteenth century, therefore, the essential conceptions of Cantor had carried the day [^57]. As we have seen, the formalization of mathematics was achieved at the same period, and the use of the axiomatic method had become almost universally accepted. In other words, the intense labours of the years 1875-1895 had acquired for mathematics the essential content of the material presented in this book. But the same period saw the beginning of a "crisis of the foundations" of rare violence, which was to shake the mathematical world for more than thirty years and seemed at times to put in danger not only all the recent acquisitions but even the most classical parts of mathematics.

The paradoxes of the theory of sets and the crisis of the foundations

The first "paradoxical" sets appeared in the theory of cardinals and ordinals. In 1897, Burali-Forti noted that there cannot be a set formed of all the ordinals, for such a set would be well-ordered and therefore isomorphic to a proper segment of itself, which is absurd [^58]. In 1899, Cantor observed (in a letter to Dedekind) that we can no more say that the cardinals form a set, nor speak of "the set of all sets" without contradiction (the set of all subsets of the latter "set" $\Omega$ would be equipotent to a subset of $\Omega$, which is contrary to the inequality $\mathfrak{m} < 2^{\mathfrak{m}}$) ([25], pp. 444-448). Finally, in 1905, by analyzing the proof of this inequality Russell showed that the reasoning which establishes it also proves (without any appeal to the theory of cardinals) that the notion of "the set of all sets which are not elements of themselves" is self-contradictory [37] [^59].

It might be thought that such "antinomies" would appear only in the peripheral regions of mathematics characterized by the consideration of sets so "large" as to be inaccessible to intuition. But other paradoxes were soon to threaten the most classical parts of mathematics. Berry and Russell [37], simplifying an argument due to J. Richard [34], observed that the set of integers which can be named in less than nineteen syllables is finite, but that it is nevertheless contradictory to define an integer as "the least integer not nameable in fewer than nineteen syllables", since this definition contains only eighteen syllables.

Although such arguments, so remote from the current usage of mathematicians, must have appeared to many as a sort of play upon words, yet they indicated the need for a revision of the basis of mathematics in order to eliminate "paradoxes" of this nature. But although there was unanimous agreement on the urgency of such a revision, radical divergences of opinion soon arose as to the manner of achieving it. For one group of mathematicians, the “idealists” and the “formalists” [^60], the situation created by the “paradoxes” of set theory was very similar to that caused in geometry by the discovery of non-Euclidean geometries or “pathological” curves (such as curves with no tangents); it should therefore lead to a similar, but more general conclusion, namely that it is futile to attempt to found any mathematical theory on an appeal (explicit or otherwise) to “intuition”. This position may be summarized in the words of the principal adversary of the formalist school : “... the view of formalism”, said Brouwer ([38 a], p. 83), “which maintains that human reason does not have at its disposal exact images either of straight lines or of numbers larger than ten, for example... It is true that from certain relations among mathematical entities, which we assume as axioms, we deduce other relations according to fixed laws, in the conviction that in this way we derive truths from truths by logical reasoning... For the formalist, therefore, mathematical exactness consists merely in the method of developing the series of relations, and is independent of the significance one might want to give to the relations or the entities which they relate”.

For the formalist, therefore, what is required is to provide an axiomatic basis for the theory of sets quite analogous to that for elementary geometry, in which it is not necessary to know what the “things” are that are called “sets”, nor what the relation $x\in y$ means, but in which the conditions imposed on this relation are enumerated; naturally, this should be done in such a way as to include, so far as possible, all the results of Cantor’s theory, while rendering impossible the existence of “paradoxical” sets. The first example of such an axiomatization was given by Zermelo in 1908 [36 c]; he avoided sets which are “too big” by introducing a “selection axiom” (Aussonderungsaxiom), according to which a property $P\{x\}$ determines a set formed by the elements which possess this property only if $P\{x\}$ already implies a relation of the form $x\in A$ [^61]. But the elimination of paradoxes analogous to “Richard’s paradox” could be achieved only by restricting the sense attached to the notion of “property”.

In this connection, Zermelo was content to describe in extremely vague terms a type of properties which he called “definite” and to indicate that the application of the selection axiom must be limited to properties of this type. This point was made precise by Skolem [39] and Fraenkel [41] [^62]; as they observed, its elucidation demands a completely formalized system (such as that described in Chapters I and II of this book), in which the notions of “property” and “relation” have lost all “meaning” and have become simply designations for assemblies formed in accordance with explicit laws. This, of course, necessitates that the rules of logic used be incorporated in the system, which was not yet the case in the Zermelo-Fraenkel system; apart from this, it is essentially their system which we have described in Chapters I and II.

Other axiomatizations of the theory of sets were subsequently proposed. We shall consider principally that due to von Neumann [43 a and b], which comes closer than the Zermelo-Fraenkel system to Cantor’s primitive conception; the latter, in order to avoid “paradoxical” sets, had already proposed (in his correspondence with Dedekind; [25], pp. 445-448) to distinguish two kinds of sets, “multiplicities” (Vielheiten) and “sets” (Mengen) in the strict sense, the latter being characterized by the fact that they could be thought of as single objects. It is this idea which von Neumann made precise by distinguishing between two types of objects, “sets” and “classes”. In his system (almost completely formalized) the classes are distinguished from sets by the fact 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” (which, of course, is not a set) used by the logicians of the nineteenth century. We should add that von Neumann’s system avoids (for set theory) the introduction of axiom schemes, which are replaced by suitable axioms, thereby making the logical study of the system easier. Variants of von Neumann’s system have been given by Bernays and Gödel [44 b].

These systems seem to have succeeded in eliminating the paradoxes, but at the cost of restrictions which cannot but appear highly arbitrary. In favour of the Zermelo-Fraenkel system it can be said that it limits itself to formulating prohibitions which do no more than sanction current practice in the applications of the notion of set to various mathematical theories. The systems of von Neumann and Gödel are more remote from usual conceptions. On the other hand, we cannot exclude the possibility that it may be easier to insert the basis of some mathematical theories into the framework of such systems than into the more rigid framework of the Zermelo-Fraenkel system.

In any case, it cannot be said that any of these solutions gives the impression of finality. They satisfy the formalists, because the formalists refuse to take into consideration the individual psychological reactions of every mathematician; they consider that a formalized language has done its duty if it is capable of transcribing mathematical arguments in a form which contains no ambiguities, and hence of serving as a vehicle for mathematical thought. It is everyone’s right, they would say, to think what they like about the “nature” of mathematical entities or the “truth” of the theorems they use, provided that their reasoning can be transcribed into the common language [^63].

In other words, from a philosophical point of view, the attitude of the formalists is characterized by an indifference to the problem posed by the “paradoxes”, resulting from the abandonment of the Platonic position which claimed to attribute to mathematical notions an intellectual “content” common to all mathematicians. Many mathematicians rebelled against this break with tradition. Russell, for example, sought to avoid the paradoxes by analyzing their structure more deeply. Taking up an idea first advanced by J. Richard (in the article [34] in which he presented his “paradox”) and later developed by H. Poincaré [33 c], Russell and Whitehead observed that the definitions of the paradoxical sets all violate the following principle, called the “vicious circle principle” : “whatever involves all of a collection must not be one of the collection” ([37], vol. I, p. 40). This statement served as a basis for the Principia, and the “theory of types” was developed in order to accommodate it. Like that of Frege which inspired it, the logic of Russell and Whitehead (much vaster than the mathematical logic used in Chapter I of this Book) contains “propositional variables”. The theory of types proceeds to a classification of these variables, in broad outline as follows.

Starting from an undefined “domain of individuals”, which may be qualified as “objects of order 0”, relations in which the variables (free or bound) are individuals are called “first-order objects”; and in general, relations in which the variables are objects of order $\leq n$ (and at least one of them is of order $n$) are called “objects of order $n+1$” [^64]. A set of objects of order $n$ can therefore be defined only by a relation of order $n+1$, and this condition allows the elimination of paradoxical sets without any difficulty [^65]. But the principle of the “hierarchy of types” is so restrictive that, if strictly adhered to, it leads to a mathematics of inextricable complexity [^66]. To escape this consequence, Russell and Whitehead were obliged to introduce an “axiom of reducibility” which asserts the existence, for every relation between “individuals”, of an equivalent relation of the first order. This condition is as arbitrary as the axioms of the formalists, and reduces considerably the interest of the construction of the Principia. The system of Russell and Whitehead has been more popular with logicians than with mathematicians. Moreover, it is not completely formalized [^67], so that there are numerous obscurities of detail. Various efforts have been made to simplify and clarify it (Ramsey, Chwistek, Quine, Rosser); these authors tend to use more and more completely formalized languages, and replace the rules of the Principia (which still contain a certain intuitive substratum) by restrictions which take account only of the writing down of the assemblies considered; not only do these rules thus appear as gratuitous as the prohibitions formulated in the systems of Zermelo-Fraenkel or von Neumann, but also, being more remote from mathematical usage, they have in many cases led to unacceptable consequences which the author had not foreseen (such as Burali-Forti’s paradox, or the negation of the axiom of choice).

For the mathematicians of these schools, the essential aim was to avoid renouncing any part of the heritage of the past. “No one shall expel us”, said Hilbert ([30], p. 274), “from the paradise Cantor has created for us”. To achieve this aim, they were prepared to accept limitations on mathematical reasoning which, though not of vital importance because they conformed with mathematical usage, did not appear to be imposed by our mental habits and the intuitive notion of a set. For them, anything was preferable to the intrusion of psychology in the criteria of validity in mathematics; and rather than “take into account the properties of our brains”, as Hadamard said ([35], p. 270), they resigned themselves to the imposition of largely arbitrary boundaries, provided that all of classical mathematics was contained in them and that they did not threaten to impede later progress.

Quite different was the attitude of the mathematicians who adhered to the following doctrine. Whereas the formalists agreed to abandon control by the “eyes of the mind” so far as mathematical reasoning was concerned, the mathematicians who have been labeled as “empiricists”, “realists”, or “intuitionists” refused to consent to this abdication; they insisted on a sort of inner certainty guaranteeing the “existence” of the mathematical objects they studied. They had no serious objection to the renunciation of spatial intuition, because arithmetical “models” allowed them to shelter behind the intuitive notion of integers. But irreconcilable opposition arose over the question of reducing the notion of integer to that of set (which is intuitively far less clear) and then of erecting barriers, devoid of intuitive foundations, to the manipulation of sets. The first person to voice this opposition (and the most influential, by reason of the authority of his genius) was H. Poincaré. Having accepted not only the axiomatic point of view toward geometry and the arithmetization of analysis but also a good part of Cantor’s theory, he refused to consider that arithmetic, too, might be amenable to axiomatic treatment. The principle of mathematical induction, in particular, was in his eyes a fundamental intuition of our intellects and could not be regarded purely as a convention [^68] [33 c]. He was in principle against formalized languages, whose usefulness he contested, and was prone to confuse the notion of integers in formalized mathematics with the use of integers in the theory of proof (which was then emerging and of which we shall speak later). No doubt it was not easy at that time to make this distinction as precisely as we do today — after more than fifty years of study and discussion — although the distinction had been grasped clearly by men such as Hilbert and Russell.

Criticisms of this nature multiplied after the introduction of the axiom of choice by Zermelo in 1904 [36 a]. Its use in many previous proofs in analysis and set theory had hitherto gone almost unnoticed [^69], and it was by following up an idea suggested by Erhard Schmidt that Zermelo, explicitly stating this axiom (and in the two forms given in the Summary of Results, § 4, no. 10), deduced by an ingenious method (which we reproduced in essence in Chapter III, § 2) a satisfactory proof of the existence of a well-ordering on any set. It appears that, coming as it did at the same time as the “paradoxes”, this new method of reasoning, with its unfamiliar appearance, caused confusion among many mathematicians; one need only read the strange misconceptions which arose in this connection, in the following volume of the Mathematische Annalen, by authors as familiar with Cantor’s methods as Schoenflies and F. Bernstein. The criticisms of E. Borel, published in the same volume, have more substance and clearly reflect the point of view of Poincaré on integers; they were developed and discussed in an exchange of letters between E. Borel, Baire, Hadamard, and Lebesgue, which has become a classic of the French mathematical tradition [35]. Borel began by denying the validity of the axiom of choice because in general it involves an uncountable infinity of choices, which cannot be intuitively imagined. But Hadamard and Lebesgue observed that a countable infinity of successive arbitrary choices is no more intuitive, because it involves an infinite number of operations, which it is impossible to conceive of being actually carried out. In the view of Lebesgue, who enlarged the debate, everything depended on what was meant by the assertion that a mathematical object “exists”; for him, it is necessary to “name” explicitly a property (a “functional” property, we should say) which defines the object uniquely. A function such as that used by Zermelo in his proof is what Lebesgue called a “law” of choice. If he continued, this requirement is not satisfied, and we merely “consider” this function instead of “naming” it, can we be sure that throughout our reasoning we are always considering the same function? ([35], p. 267) This consideration led Lebesgue to new doubts : even the choice of a single element in a set appears to raise difficulties; it is necessary to be certain that such an element “exists”, i.e., that at least one element of the set can be “named” [^70]. Can we therefore speak of the “existence” of a set if we cannot “name” every element of it? Baire did not hesitate to deny the “existence” of the set of subsets of a given infinite set (loc. cit., pp. 263-264); in vain Hadamard observed that these restrictions would lead to renouncing even the right to speak of the set of real numbers, and E. Borel in the end came to this conclusion. Apart from the fact that countable sets seem to have been granted the franchise, we are almost back to the classical position of the opponents of the “actual infinite”.

None of these objections was very systematic; it was reserved to Brouwer and his school to undertake a complete recasting of mathematics on similar but even more radical lines. We shall not here presume to summarize a doctrine so complex as intuitionism, which is as much of philosophy as mathematics, and we shall do no more than indicate some of its most striking features, referring for more details to the works of Brouwer himself [38] and Heyting’s exposition [46]. In Brouwer’s view, mathematics is identical with the “exact” part of our thought, founded on the primary intuition of the sequence of natural numbers, and cannot be translated into a formal system without mutilation. Moreover, it is “exact” only in the minds of mathematicians, and it is illusory to hope to construct an instrument of communication between mathematicians which is free of all the imperfections and ambiguities of languages; the most that can be hoped for is to arouse in the interlocutor a favourable state of mind for more or less vague descriptions ([46], pp. 11-13). Intuitionist mathematics attaches hardly more importance to logic than to language; a proof is conclusive not by virtue of the rules of logic, fixed once and for all, but by reason of the “immediate self-evidence” of each of its steps. This “self-evidence” is moreover to be interpreted in an even more restrictive sense than that of E. Borel and his supporters; thus in intuitionist mathematics we may not assert that a relation of the form “$R$ or (not $R$)” is true (the principle of the excluded middle) unless, for every system of values given to the variables appearing in $R$, we can prove that one of the two propositions $R$, “not $R$” is true. For example, from the equation $ab=0$ between two real numbers, we may not infer “$a=0$ or $b=0$”, because it is easy to construct explicit examples of real numbers $a,b$ such that we have $ab=0$ without at present being able to prove either one of the two propositions $a=0,b=0$ ([36], p. 21).

It is hardly surprising that the intuitionist mathematicians, starting from such principles, were led to results quite different from the classical theorems. A large number of the latter disappeared, for example most of the “existence” theorems of 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, and a monotone bounded sequence of real numbers does not necessarily have a limit. Furthermore, many of the classical notions ramify into several fundamentally distinct notions in intuitionist mathematics. Thus there are two notions of convergence (for a sequence of real numbers) and eight notions of countability. It goes without saying that transfinite induction and its applications to modern analysis are, like most of Cantor’s theory, condemned without appeal.

According to Brouwer, it is only in this way that mathematical propositions can acquire a “content”; formalist arguments which go beyond what is admitted by intuitionism are judged to be worthless, because it is impossible to give them a “meaning” to which the intuitive notion of “truth” could be applied. Clearly, judgments of this sort can only rest on a previous notion of “truth”, which is of a psychological or metaphysical nature; that is to say, in practice, that they are beyond discussion.

Certainly, the vigorous attacks from the intuitionist camp have from time to time obliged not only the avant-garde mathematical schools, but even the partisans of traditional mathematics, to be on the defensive. A well-known mathematician has admitted to being impressed by these attacks to the point that he voluntarily restricted his work to those branches of mathematics considered to be “certain”. But such cases must have been rather uncommon. The intuitionist school, whose memory will undoubtedly survive only as a historical curiosity, has at least rendered the service of having obliged its opponents, that is to say the vast majority of mathematicians, to clarify their own positions and to become more consciously aware of the reasons (whether logical or sentimental) for their confidence in mathematics.

Metamathematics

Absence from contradiction has always been considered to be a sine qua non of all mathematics, and from the time of Aristotle logic was sufficiently developed for it to be realized that anything could be deduced from a contradictory theory. Proofs of “existence”, which have been regarded as indispensable ever since antiquity, clearly served only to guarantee that the introduction of a new concept did not risk entailing a contradiction, particularly if the concept was too complicated to fall immediately under “intuition”. We have seen how this demand for a proof of non-contradiction became more imperative with the advent of the axiomatic point of view in the nineteenth century, and how the construction of arithmetic “models” answered it. But might arithmetic itself not be contradictory? Such a question could not have been asked before the end of the nineteenth century; so much did the integers appear to belong to what is surest in our intuition. But after the “paradoxes” everything seemed open to doubt, and the feeling of insecurity they created understandably led mathematicians, round about 1900, to look more closely at the problem of the consistency of arithmetic, in order to save at least classical mathematics from shipwreck. Thus this problem was the second of those listed by Hilbert in his famous address to the International Congress of 1900 ([31], pp. 229-301).

In doing so, he put forward a new principle which was to make a deep impression. Whereas, in traditional logic, the non-contradiction of a concept only made the concept “possible”, for Hilbert it was equivalent to the existence of the concept (at any rate where axiomatically defined mathematical concepts were concerned). This point of view apparently entailed the necessity of proving a priori the consistency of a mathematical theory before even being able to develop the theory legitimately; certainly this was how the principle was understood by H. Poincaré, who took up Hilbert’s idea as a convenient stick with which to beat the formalists, by pointing out with malicious satisfaction how far the formalists in this period were from being able to satisfy this condition ([33 c], p. 163). We shall see later how Hilbert took up this challenge. But first, it should be noted that, under his influence and that of Poincaré, the demands put forward by the latter were accepted without reserve for a long time, as much by the formalists as by their opponents. One consequence was the belief, which was very widespread even among the formalists, that Hilbert’s theory of proof was an integral part of mathematics, and constituted an indispensable prolegomenon to mathematics proper. We explained in the Introduction why this dogma appears unjustified to us [^71], and we hold that the role of metamathematics in an account of logic and mathematics can and should be confined to the very elementary part which deals with the manipulation of abbreviating symbols and deductive criteria. Contrary to what Poincaré asserted, it is thus not a question of “claiming freedom from contradiction”, but rather of considering, as Hadamard did, that consistency, even when it cannot be proved, can be ascertained ([35], p. 270).

It remains for us to give a brief historical sketch of the efforts of Hilbert and his school. Although the theory of proof is not touched on in this series, it is not without interest to retrace in outline not only the evolution of ideas which finally led to Gödel’s negative result and justified a posteriori Hadamard’s scepticism, but also all the progress which has resulted in an understanding of the mechanism of mathematical reasoning, and which has elevated modern metamathematics to the position of an autonomous science of undeniable importance.

In 1904, in an address to the International Congress ([30], pp. 247-261), Hilbert attacked the problem of the consistency of arithmetic. First, he established that a proof of consistency could not be obtained by recourse to a “model” [^72], and he indicated in broad outline the principle of another method. He proposed to consider the true propositions of formalized arithmetic as assemblies of signs without any meaning and to show that, by using the rules governing the formation and juxtaposition of these assemblies, an assembly could never be obtained which was a true proposition and whose negation was also a true proposition. He sketched a proof of this nature for a formalism less elaborate than that of arithmetic; but, as H. Poincaré observed soon afterward ([33 c], p. 185), this proof made essential use of the principle of induction, and therefore appeared to be founded on a vicious circle. Hilbert did not reply to this criticism immediately, and fifteen years went by before anyone tried to develop his ideas. It was only in 1917 that (moved by the desire to answer the attacks of the intuitionists) he devoted himself again to the problem of the foundations of mathematics, which from then on occupied him continuously until the end of his scientific career. In his work on this subject, which stretches from 1920 to about 1930, and in which a whole school of young mathematicians (Ackermann, Bernays, Herbrand, von Neumann) were active participants, Hilbert gradually extracted the principles of his “theory of proof” in a more precise fashion. Realizing implicitly the justice of Poincaré’s criticism, he admitted that in metamathematics the arithmetical arguments used could be based only on our intuition of the integers (and not on formalized arithmetic). Thus it appeared essential to restrict these arguments to “finite procedures” (finte Prozesse) of a type allowed by the intuitionists. For example, a proof by reductio ad absurdum cannot establish the metamathematical existence of an assembly or a sequence of assemblies; it is necessary to give an explicit law of construction [^73]. Furthermore, Hilbert enlarged his initial program in two directions; not only did he attack the problem of the consistency of arithmetic, but he also hoped to prove the consistency of the theory of real numbers and even of the theory of sets [^74]. To the problems of consistency were added those of independence of the axioms, of categoricity, and the decision problem. We shall give a brief review of these various questions and mention the principal researches which they have occasioned.

A proof of the independence of a system of propositions $A_1$, $A_2$, ..., $A_n$ consists in showing that, for each index $i$, $A_i$ is not a theorem in the theory $\mathscr{T}_i$ obtained by taking as axioms the $A_j$ where $j \neq i$. All we need is to find a non-contradictory theory $\mathscr{T}'_i$ in which the $A_j$ $(j \neq i)$ are theorems, together with "not $A_i$"; we may therefore consider the problem in two aspects according as we do or do not assume that certain theories (such as arithmetic or set theory) are consistent. In the latter case, we have a problem of "absolute" consistency. On the other hand, the first type of problem is to be solved, like problems of "relative" consistency, by the construction of appropriate "models", and many proofs of this nature were devised well before mathematics had taken on a completely formalized aspect. We may cite as examples the models of non-Euclidean geometry, the questions of the independence of the axioms of elementary geometry treated by Hilbert in the Grundlagen der Geometrie [30], the work of Steinitz on the axiomatization of algebra, and the work of Hausdorff and his successors on the axiomatization of topology.

A theory $\mathscr{T}$ is said to be categorical if, for every proposition A in $\mathscr{T}$ which contains no letters other than the constants of $\mathscr{T}$, one of the two propositions A, (not A) is a theorem in $\mathscr{T}$ [^75]. Apart from some very rudimentary formalism whose categoricity is easily proved ([32], p. 35; cf. Chapter I, Appendix, Exercise 7), the results obtained in this area are essentially negative. The most important is due to K. Gödel who showed that if $\mathscr{T}$ is non-contradictory and if the axioms of formalized arithmetic are theorems in $\mathscr{T}$, then $\mathscr{T}$ is not categorical. The fundamental idea of his ingenious method consists in establishing a one-to-one correspondence (of course, by means of "finite procedures") between metamathematical statements and certain propositions of formalized arithmetic. We give here an outline sketch of the argument [^76]. With each assembly A which is a term or a relation in $\mathscr{T}$, we associate (by an explicit constructive procedure which can be applied quasi-mechanically) an integer $g(A)$ in a one-to-one manner. Likewise, with every proof D in $\mathscr{T}$ (considered as a succession of assemblies; cf. Chapter I, § 2, no. 2) we may associate an integer $h(D)$ in a one-to-one manner. Finally, we can give an explicit procedure for constructing a relation $P\{x,y,z\}$ in $\mathscr{T}$ [^77] such that, in $\mathscr{T}$, $P\{x,y,z\}$ implies that $x,y,z$ are integers, and satisfies the following two conditions :

(1) If D is a proof of $A\{\lambda\}$, where $A\{x\}$ is a relation in $\mathscr{T}$, and $\lambda$ is a definite integer (that is, a term in $\mathscr{T}$ which is an integer), then

is a theorem in $\mathscr{T}$.

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

Now let $S\{x\}$ be the relation (not $(\exists z)P\{x,x,z\}$), and let $\gamma=g(S\{x\})$, which is a term in $\mathscr{T}$. If $\mathscr{T}$ is not contradictory, there is no proof of the proposition $S\{\gamma\}$ in $\mathscr{T}$. For if D were such a proof, then

would be a theorem in $\mathscr{T}$. This relation is just $P\{\gamma,\gamma,h(D)\}$, and consequently $(\exists z)P\{\gamma,\gamma,z\}$ would also be a theorem in $\mathscr{T}$; but since the last relation is equivalent to (not $S\{\gamma\}$), $\mathscr{T}$ would be contradictory. Furthermore, what has just been said shows that, for each definite integer $\mu$, (not $P\{\gamma,\gamma,\mu\}$) is a theorem in $\mathscr{T}$. It follows that there is no proof in $\mathscr{T}$ of the proposition (not $S\{\gamma\}$), for this relation is equivalent to $(\exists z)P\{\gamma,\gamma,z\}$, and the existence of an integer $\mu$ such that $P\{\gamma,\gamma,\mu\}$ would imply that $\mathscr{T}$ was contradictory, by virtue of the preceding argument [^78]. This metamathematical theorem of Gödel has been subsequently generalized in various directions ([48], Chapter XI) [^79].

¶ The relation $S \{ \gamma \}$ in $\mathscr{T}$, which is thus shown to have the property that there exists no proof in $\mathscr{T}$ either of it or of its negation, has obviously been manufactured to fit the requirements of the argument and is not related in any natural way to any mathematical problem. What is 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 has been established in two steps : in 1940, Gödel proved that the theory obtained by adjoining to $\mathscr{T}$ the hypothesis $2^{\aleph_0} = \aleph_1$ is not contradictory [44b]; and quite recently. P. Cohen has proved that the same is met when the relation $2^{\aleph_0} = \aleph_2$ (or $2^{\aleph_0} = \aleph_n$ for any integer $n > 1$) is adjoined to $\mathscr{T}$ [49].

The decision problem (Entscheidungsproblem) is certainly the most ambitious of all in metamathematics. The problem is whether, for a given formalized language, a quasi-mechanical "universal procedure" can be conceived which, when applied to any relation whatever in the formalism under consideration, will indicate in a finite number of operations whether the relation is true or not. The solution of this problem formed a substantial part of the grand designs of Leibniz, and it seems that at one point the Hilbert school believed they were very close to a solution. It is a fact that such procedures can be described for formalisms which contain few primitive signs and axioms ([48], pp. 136-141; cf. Chapter I, Appendix, Exercise 7). But the efforts to make precise the decision problem by defining exactly what is to be meant by "universal procedure" have led so far only to negative results ([48], pp. 432-439). Moreover, the solution of the decision problem for a theory $\mathscr{T}$ would tell immediately whether or not $\mathscr{T}$ is contradictory, because the "universal procedure" could be applied to a relation in $\mathscr{T}$ and its negation; and, as we shall see, the possibility of resolving the question in this way is excluded, so far as the usual mathematical theories are concerned [^80].[^80] It is in fact in the question of consistency of mathematical theories — the origin and the very heart of mathematics — that the results have turned out to be most deceptive. During the years 1920-1930, Hilbert and his school developed new methods for attacking these problems. Having proved the consistency of some partial formalisms, covering a part of arithmetic (cf. [42], [43 c]), they thought they were on the verge of victory and that they would succeed in proving the consistency not only of arithmetic, but also of set theory, when Gödel used the non-categoricity of arithmetic to deduce that it is impossible to prove, by means of Hilbert's "finite procedures", the consistency of any theory E which contains arithmetic [^81].[^81] Nevertheless, Gödel's theorem does not close the door altogether to attempts to prove consistency, provided that Hilbert's restrictions concerning "finite procedures" are (at least partially) abandoned. Thus, in 1936, Gentzen [47] succeeded in proving the consistency of formalized arithmetic by "intuitive" use of transfinite induction as far as the countable ordinal $\varepsilon_0$ (Chapter III, § 6, Exercise 14) [^82].[^82] The degree of "certainty" which can be attributed to such an argument is undoubtedly less than for those which satisfy Hilbert's original restrictions, and is essentially a matter depending on the personal psychology of the individual mathematician. But it is no less true that similar "proofs" using "intuitive" transfinite induction as far as some given ordinal would be considered as an important step forward if they could be applied, for example, to the theory of real numbers or to an important part of set theory.

Furthermore, within set theory itself there are many problems of "relative" consistency connected with the many "hypotheses" at large in the theory. The most remarkable result in this area is again due to Gödel. In 1940 he proved that, if the theory whose axioms are those of the von Neumann-Bernays system, except for the axiom of choice, is consistent, then the theory obtained by adjoining to these axioms a very strong form of the axiom of choice (of the type which the use of the symbol $\tau$ gives us) and the generalized continuum hypothesis is also consistent [44 b]. More recently, I. Novak-Gal has shown that the consistency of the Zermelo-Fraenkel system implies that of the von Neumann-Bernays-Gödel system [^83].

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[^1]: Notably Democritus, to whom Archimedes attributes the discovery of the formula for the volume of a pyramid ([4 bis], p. 13). This allusion should be compared with a celebrated fragment ascribed to Democritus (but of doubtful authenticity) in which he declares : “nobody has ever surpassed me in the construction of figures by means of proofs, not even the Egyptian so-called ‘harpedonaptes’” (H. Diels, Die Fragmente der Vorsokratiker, 2nd edition, vol. I, p. 439 and vol. II.1, pp. 727-728, Berlin (Weidmann), 1906-1907). Archimedes’ remark, and the fact the Egyptian texts which have come down to us contain no proofs (in the classical sense), lead us to think that the “proofs” to which Democritus alludes were no longer considered to be proofs in the classical period, and would not be so considered today. [^2]: The most beautiful classical example of reductio ad absurdum in mathematics is the proof that $\sqrt{2}$ is irrational, to which Aristotle alludes on several occasions. But the experts have not succeeded in assigning any precise date to this discovery; some put it at the beginning and others right at the end of the fifth century B.C. (see the Historical Note to General Topology, Chapter IV, and the references there). [^3]: In spite of the simplicity and “obviousness” to our minds of the logical rules formulated by Aristotle, it is only necessary to place them in their historical context for us to appreciate the difficulties in the way of a precise conception of these rules, and to understand the effort which Aristotle must have required to surmount these difficulties. Plato, in his dialogues, which were addressed to an educated public, allows his characters to become entangled in questions as elementary as the relationship between the negation of $A \subset B$, and $A \cap B = \varnothing$ (in modern language) before producing the correct answer (cf. R. Robinson, “Plato’s consciousness of fallacy”, Mind, vol. 51 (1942), pp. 97-114). [^4]: The corresponding statements in Aristotle are “every A is B” and “some A is B”. In this notation, A (the “subject”) and B (the “predicate”) stand for concepts, and to say that “every A is B” means that we can attribute the concept B to every entity to which the concept A can be attributed (A is the concept “man” and B the concept “mortal” in the classical examples). The interpretation which we give in this series consists in considering the sets of entities to which the concepts A and B respectively apply; this is the “extensional” point of view and was already known to Aristotle. But Aristotle considers the relation “every A is a B” from another point of view, the “comprehensional”, in which B is envisaged as one of the concepts which in some sense constitute the more complex concept A, or, as Aristotle says, “belong” to A. At first sight, these two points of view seem equally natural; but the “comprehensional” point of view has been a constant source of difficulties in the development of logic (it seems more remote from intuition than the former, and leads rather easily to errors, especially in schemes which involve negations; cf. [12 bis], pp. 21-32). [^5]: For a critical discussion of the syllogism and its inadequacies, see, for example, [12 bis], pp. 432-441, or [32], pp. 44-50. [^6]: The absence of genuine quantifiers (in the modern sense) up to the end of the nineteenth century was one of the causes of the stagnation of formal logic. [^7]: We may mention that during a recent lecture at Princeton an eminent academic is reported to have said in the presence of Gödel that nothing new had been done in logic since Aristotle! [^8]: Although Descartes and (to a lesser extent) Pascal had devoted part of their philosophical works to the foundations of mathematics, their contribution to the progress of formal logic was negligible. Undoubtedly the reason for this limitation is to be found in the fundamental trend of their thoughts, which was to emancipate themselves from the scholastic yoke and which led them to reject everything connected with scholastic philosophy, and first of all formal logic. In his Réflexions sur l’esprit géométrique, Pascal essentially limited himself — as he himself realized — to fashioning into striking formulae the known principles of Euclid’s proofs (for example, the famous precept “Substituer toujours mentalement les définitions à la place des définis” ([11], vol. IX, p. 280) had already been known to Aristotle ([2], Topics, VI, 4; [2 bis], p. 87)). As to Descartes, the rules of reasoning which he put forward were, above all, psychological precepts (and rather vague ones) rather than logical criteria; and as Leibniz says ([12 bis], pp. 94 and 202-203), their validity is therefore only subjective. [^9]: Of course, the usefulness of such a symbolism had not escaped the predecessors of Leibniz so far as mathematics was concerned. Descartes, for example, recommended the replacement of whole figures "by very short signs" (XVI^e Règle pour la direction de l'esprit; [10], vol. X, p. 454). But no one before Leibniz had insisted so strongly on the universal range of application of this principle. [^10]: It is striking that Leibniz cites, as examples of "formal" reasoning, "un compte de receveur" or even a legal text ([12 b], vol. IV, p. 295). [^11]: This conception of a "logical machine" is used nowadays in metamathematics with great effect ([48], Chapter VIII). [^12]: This idea has been taken up successfully, in a slightly different form, by Gödel in his metamathematical works (cf. [44a] and [48], p. 254). [^13]: Leibniz does not attempt to introduce disjunction into his calculus, except in one or two fragments (where it is denoted by $\mathrm{A} + \mathrm{B}$), and seems not to have succeeded in simultaneously handling this operation and conjunction in a satisfactory manner ([12 bis], p. 363). [^14]: Leibniz knew perfectly well that Aristotelian logic was insufficient to translate mathematical texts into formal language, but in spite of various attempts he never succeeded in improving it in this respect ([12 bis], pp. 435 and 560). [^15]: These are the so-called "rules of conversion", based on the postulate that "every A is a B" implies "some A is a B", which of course presupposes that A is not empty. [^16]: The influence of Kant, from the middle of the eighteenth century onward, is without doubt partly responsible for the lack of interest in formal logic at this period. Kant thought that "we have no need of any new invention in logic", the form given to it by Aristotle being sufficient for all possible applications (Werke, vol. VIII, Berlin (B. Cassirer), 1923, p. 340). For the dogmatic conceptions of Kant on mathematics and logic, consult the article of L. Couturat, Revue de Métaph. et de Morale, vol. 12 (1904) pp. 321-383. [^17]: Note in particular that Boole uses the distributivity of intersection with respect to union, which seems to have been noticed for the first time by J. Lambert. [^18]: It should be noted that statements equivalent to these rules are to be found in the writings of certain scholastic philosophers ([6], pp. 67-ff). [^19]: However, the notion of the "cartesian" product of two arbitrary sets seems to have been first introduced by G. Cantor ([25], p. 286), who was also the first to define exponentiation $A^B$ (loc. cit., p. 287). The general notion of infinite product is due to A. N. Whitehead (Amer. Jour. of Math. 24 (1902), p. 369). The use of graphs of relations is fairly recent; apart, of course, from the classical case of real-valued functions of real variables, they appeared first in the work of the Italian geometers, especially C. Segre, in the study of algebraic correspondences. [^20]: For each relation obtained from one or more given relations by applying our quantifiers it would be necessary, in this calculus, to introduce an ad hoc notation of the type $G^{-1}$ and $G_1 \circ G_2$ (cf. for example [22 b]). [^21]: This indicates how deeply rooted, even in Peano, was the old habit of thinking in terms of "comprehension" rather than "extension". [^22]: The introduction of this appears to be due to Dedekind, in his work Was sind und was sollen die Zahlen, which we shall come to later ([26], vol. III, p. 348). [^23]: It should be observed that what Hilbert denotes by $\tau_x(\mathrm{A})$ in this context is denoted by $\tau_x(\text{not A})$ in Chapter I. [^24]: “Those who have given us methods”, he says in this context, “have given us without doubt good precepts, but not the means of observing them” ([12 b], vol. VII, p. 21); Elsewhere, deriding the rules of Descartes, he compares them to the alchemists’ formulae : “Take what is necessary, do as you ought, and you will obtain what you desire!” ([12 b], vol. IV, p. 329). [^25]: On this point, Leibniz was still under the influence of the schoolmen; he thought always of propositions as establishing a relationship of “subject” to “predicate” between concepts. As soon as the concepts had been resolved into “primitive” concepts (which, as we have seen, was one of his fundamental ideas), everything was reduced, for Leibniz, to a problem of verifying relations of “inclusion” by means of what he called the “identical axioms” (which are essentially the propositions $A = A$ and $A \subset A$) and the principle of “substitution of equivalents” (if $A = B$, we may replace $A$ everywhere by $B$, which is our scheme S6 of Chapter I, § 5) ([12 bis], pp. 184-206). It is interesting in this connection to note that, in accordance with his desire to reduce everything to logic and to “prove all that can be proved”, Leibniz proved the symmetry and transitivity of the equality relation, starting from the axiom $A = A$ and the principle of substitution of equivalents; the proofs he obtained are essentially those we gave in Chapter I, § 5 ([12 a], vol. VII, pp. 77-78). [^26]: Cf. B. L. van der Waerden, “Die Arithmetik der Pythagoreer”, Math. Ann. 120 (1947) pp. 127-153. [^27]: In his inaugural dissertation, he “proved” that there could exist only seven planets, in the same year that an eighth was discovered. [^28]: Cf. the arguments of Poincaré in favour of the “simplicity” and “convenience” of Euclidean geometry ([33 a], p. 67), and the analysis by which, a little later, he arrives at the conclusion that experience provides no absolute criterion for the choice of one geometry over another as the framework for natural phenomena. [^29]: Again, this word is justified only for $n \leq 3$; for larger values of $n$ it is in fact argument by analogy which is involved. [^30]: It should however be noted that, already in the sixteenth century, a commentator on Euclid, J. Peletier, protested against this method of proof, in terms close to these of modern critics ([3], vol. I, p. 249). [^31]: What seems to have struck Hilbert’s contemporaries most is “non-Archimedean” geometry, i.e., geometry over a non-Archimedean ordered base field (or division ring), the commutative case of which had been introduced some years before by Veronese (Fondamenti di geometria, Padova, 1891). [^32]: J. STIRLING, Lineae tertii ordinis Newtonianae... (1717). [^33]: In this respect again, Leibniz appeared as a forerunner. “The universal Mathematics”, he says, “is, so to speak, the logic of imagination”, and should concern itself with “everything in the domain of imagination which is capable of exact determination” ([12 c], p. 348; cf. [12 bis], pp. 290-291). In his view the keystone of mathematics so conceived is what he calls the “art of formulae”, i.e., essentially the science of abstract relations between mathematical objects. But whereas at that time practically the only relations considered in mathematics were relations of magnitude (equality, inequality, proportion), Leibniz conceived of many other types of relations which, in his opinion, ought to be systematically investigated by mathematicians, such as inclusion, or what he called the relation of (single-valued or many-valued)“ determination” (i.e., the notions of mapping and correspondence) ([12 bis], pp. 307-310). Many other modern ideas made their appearance under his pen in this context; he observed that the various equivalence relations in classical geometry have in common the properties of symmetry and transitivity; he conceived the notion of a relation which is compatible with an equivalence relation, and expressly observed that an arbitrary relation does not necessarily have this property ([12 bis], pp. 313-315). Of course, here as everywhere, he advocated the use of a formalized language and even introduced a sign designed to denote an undetermined relation ([12 bis], p. 301). [^34]: It should be noted that his quasi-philosophical language was hardly likely to attract the majority of mathematicians, who felt ill at ease in the presence of a statement such as the following : “Pure mathematics is the science of the particular entity in so far as it is created by thought” (die Wissenschaft des besonderen Seins als eines durch das Denken gewordenen). But it is apparent from the context that what Grassmann had quite clearly in mind was axiomatic mathematics in the modern sense (except that, rather curiously, he followed Leibniz in considering that the bases of this “formal science”, as he called it, were the definitions and not the axioms). In any case, like Boole, he emphasized that “the name of science of magnitudes is not a suitable one for the whole of mathematics” ([17], vol. I₁, pp. 22-23). [^35]: According to a well-known anecdote, Hilbert was wont to express this idea by saying that the words “point”, “line”, and “plane” could be replaced by “table”, “chair”, and “beer-mug” without changing geometry in the least. It is curious to find an anticipation of this conceit (in the older sense of the word) in d’Alembert : “We may assign to words any meanings we like”, he writes in the Encyclopédie ([13], s.v. DEFINITION); “(we could) build up the elements of Geometry in full rigour by calling a triangle what is usually called a circle”. [^36]: Arithmetic, however, remained outside this synthesis. Thus Euclid, having developed the general theory of proportions between arbitrary magnitudes, then developed the theory of rational numbers independently of the notion of ratios of magnitudes. [^37]: It is rather curious, in this connection, to see Descartes compare arithmetic with “combinations of numbers”, and the “arts... in which order is dominant, such as those of the weavers and the tapestry-makers, the women who embroider and make lace” ([10], vol. X, p. 403), as if in anticipation of modern studies of symmetry and its relationship with the notion of a group (cf. H. WEYL, Symmetry, Princeton University Press, 1952). [^38]: The word “isomorphism” was introduced into group theory at the same period. But at first it was used also to denote surjective homomorphisms, which were called “meriedric isomorphisms”, in contrast to “holoedric isomorphisms”, which were isomorphisms in the modern sense of the word. This terminology remained in use up to the time of E. Noether. [^39]: As examples of non-commutative operations, he cited subtraction, division, and exponentiation ([12 b], vol. VII, p. 31). At one time he even attempted to introduce such operations into his logical calculus ([12 bis], p. 353). [^40]: See H. FREUDENTHAL, Zur Geschichte der vollständigen Induktion, Arch. Int. Hist. Sci., 33 (1953), p. 17. [^41]: We said earlier that Boole did not hesitate to include in his logical calculus the “Universe” (denoted by 1), the set of all objects. It does not appear that this conception was criticized at the time, although it had been rejected by Aristotle, who gave a rather obscure proof which purported to demonstrate its absurdity ([21], Met. B, 3, 998 b). [^42]: Frege seems to have been one of the few contemporary mathematicians who, not without reason, protested at the spate of similar “definitions” ([24 c], vol. I, p. 2). [^43]: A typical example of this conception is Euclid’s statement : “For every given quantity of prime numbers, there is one which is greater than all of them”, which we would express nowadays by saying that the set of prime numbers is infinite. [^44]: Classically it would, of course, be correct to say that a point belongs to a line, but to draw the conclusion that a line is “composed of points” would violate the taboo of the “actual infinite”, and Aristotle devoted long passages to justifying this interdict. Probably in order to escape any objection of this sort many mathematicians in the nineteenth century avoided speaking of sets, and reasoned systematically in terms of “comprehension”; for example, Galois did not speak of a number field, 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 abstained from saying that lines and planes were sets of points; Peano was the only one who freely used the language of set theory in elementary geometry. [^45]: The reason for this fact is undoubtedly the circumstance that the sets envisaged in classical mathematics belong to a small number of single types, and can for the most part be completely described by a finite number of numerical “parameters”, so that consideration of them reduces to that of a finite set of numbers (this is the case, for example, with algebraic curves and surfaces, which for a long time constituted almost exclusively the “figures” of classical geometry). Before the progress of analysis led, in the nineteenth century, to the consideration of arbitrary subsets of the line and $\mathbf{R}^n$, it was rare to encounter sets which did not conform to the above types. For example, Leibniz, original as always, considered the closed disc minus its centre as a “geometrical locus”, or (by a curious presentiment of the theory of ideals) considered that in arithmetic an integer is the “type” of the set of its multiples, and remarked that the set of multiples of 6 is the intersection of the set of multiples of 2 and the set of multiples of 3 ([12 b], vol. VII, p. 292). From the beginning of the nineteenth century sets of this latter type became familiar, in algebra and in number theory: for example, the classes of quadratic forms, introduced by Gauss, and fields and ideals, defined by Dedekind before the Cantorian revolution. [^46]: These are sets $E \subset \mathbf R$ with the property that if a trigonometric series