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Cet article tente de donner une vue d’ensemble du travail de Beth à une large communauté de philosophes des sciences. L’accent sera moins mis sur ses contributions importantes et largement discutées à la logique mathématique – où résident ses mérites les plus créatifs – que sur le contexte philosophique et culturel. J’adopte le motif que Beth a lui-même poursuivi pour la composition de son ouvrage majeur The Foundations of Mathematics, à savoir renforcer les liens entre la recherche fondamentale et la philosophie générale, en rendant les problèmes qui l’intéressaient accessibles à partir de leurs racines théoriques et biographiques. This article attempts to provide an overview of Beth’s work for a broad community of philosophers of science. Emphasis will be placed more on the philosophical and cultural context than on his important and widely discussed contributions to mathematical logic—where his most creative merits lie. I adopt the motive Beth himself pursued in writing his major work The Foundations of Mathematics, namely, to strengthen the links between fundamental research and general philosophy, making the problems he was interested in accessible from their theoretical and biographical roots.

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Texte intégral

I would like to thank Jamie van der Klaauw and Henri Krop (Erasmus School of Philosophy at the Erasmus University Rotterdam) for extensive language reviewing, a major overhaul in the original structure and the addition of material to a first version of this article. I am very grateful to an anonymous reviewer for several helpful suggestions.

1Here I take over Beth’s main motive for writing his “Foundations,” i.e., the wish “to strengthen the ties between foundational research and general philosophy” through this biography [1959, XIII], and to make available to philosophers in particular, but not exclusively, a broad survey of the problems that interested him.

1 Biography

2Evert Willem Beth was born on 7 July 1908 in Almelo (Overijssel). His father, H. J. E. Beth (1880-1952), was a mathematician who obtained his doctorate in 1910 at the University of Amsterdam under the supervision of Diederik Johannes Korteweg. In 1906, his father became a teacher at the HBS in Almelo and he became the principal in 1912 (“HBS” is the Dutch abbreviation of Higher Civic School, a secondary school without Latin and Greek, but with an emphasis on sciences and modern languages). In 1922, Beth’s father became the principal of the state’s HBS in Deventer and in Amersfoort in 1935. Beth’s mother was Hilligje de Groot (1882-1960). In 1947, Beth married Cornelia Pastoor (1896-1978), a singer and interior designer. Their marriage remained childless. Beth attended the HBS of his father and graduated in 1925 in Deventer. He then began studying pharmacy at Utrecht but soon switched to mathematics and physics. He obtained a master’s degree [doctoraal] in mathematics with physics and mechanics as subsidiaries in 1932 and a diploma in life insurance mathematics in 1933. He continued his studies in Utrecht, Leiden and Brussels. In 1935, he received another master’s degree in psychology with philosophy and mathematics as subsidiaries and also a doctorate in philosophy from the University of Utrecht with a thesis on “Reason and Intuition in Mathematics” [1935] under the supervision of the neo-Kantian philosopher Johannes Christiaan Franken (1891-1941). In his thesis, according to Beth himself, he “made the attempt to reconcile Kantianism with the results of the investigations into the foundations of mathematics” (Science a Road to Wisdom [1962a, 96]). The following year, Beth attended lectures by the philosopher, mathematician, and central figure in the significs circle, Gerit Mannoury, and by the logician Arend Heyting, a student and collaborator of the intuitionist mathematician Luitzen Egbertus Jan Brouwer. In 1940, Beth returned to college and matriculated as a law student in Amsterdam. In 1941, he received a bachelor’s degree [kandidaats] in law at the University of Amsterdam.

3In 1933, he joined the Genootschap voor Critische Philosophie, founded at Utrecht in 1923 (since 1938, due to the abandonment of the exclusively neo-Kantian orientation: Genootschap voor Wetenschappelijke Philosophie), where he represented, with the mathematician Pieter G. J. Vredenduin, the philosophy of mathematics. According to Beth, one of the merits of this Association “was to have pointed out the distinction between philosophy and ideology (in the broad sense of ‘Weltanschauung’).” Their neo-Kantian plea for the autonomy of philosophy implied interest in the attempts, stemming from research on foundations of exact sciences propagated by the Wiener Kreis, “to establish a very close association between philosophy and the sciences” [1962a, 95–96]. As early as 1933, he attended a series of lectures on the foundations of mathematics by Adolf Fraenkel at the University of Utrecht. His studies in Brussels in 1934-1935 brought him into contact with Marcel Barzin, Alfred Errera, Robert Feys, and Herman Jean de Vleeschauwer, and at the Paris Congress for Philosophy (1937) he made the acquaintance of Paul Bernays, Alfred Tarski, and Heinrich Scholz of Münster. Between 1935 and 1945 he was, with intervals, a mathematics teacher in secondary schools in several cities and in 1942-1943 he was a teaching assistant in mathematics at the Delft University of Technology. Apparently, Beth had difficulty teaching in secondary schools. During the war Beth participated in the secret meetings of the International Society for Significs where he met his future wife Cornelia.

4In the early 1940s his future colleague H. J. Pos stimulated Beth to broaden his philosophical outlook and he became interested in the history of philosophy.

5In 1946, he was appointed Extraordinary Professor of logic and its history and philosophy of the exact sciences in the Faculty of Mathematics and Physics and the Faculty of Humanities and Philosophy at the University of Amsterdam. This appointment fulfilled the wish of the Faculty of Physics and Mathematics in Amsterdam for a more science-orientated philosophy (since 1815, philosophy has been part of the Faculty of Humanities at Dutch universities). He became a full professor in 1949. His immediate colleagues in the Faculty of Physics and Mathematics were Heyting and, until 1951, Brouwer. In philosophy, they were Hendrik Josephus Pos until his death in 1955 and Hendrik Marie Johan Oldewelt. The precarious position of Beth’s chair came to an end in 1960 with a new law on higher education, which created a new faculty called the Centrale Interfaculteit. Philosophy was formally separated from humanities and offered courses open to students of all faculties. The ideal of such a renewal of academic philosophy was formulated as early as 1933, but it was not until after the war that a committee for a new law on academic education was established. Several times Beth had to intervene in public to overcome the existing opposition to a less literary concept of philosophy in Dutch universities.

6After the war Beth became active as an organizer and member of learned societies. As early as 1946, he supported the Société internationale de logique et de philosophie des sciences (SILPS) founded by Karl Dürr, Ferdinand Gonseth, Karl Popper and Paul Bernays [van Ulsen 2000]. In 1947, together with Heyting and Andreas Gerardus Maria van Melsen, he was the founder and first president of the Dutch Society of Logic and Philosophy of the Exact Sciences. He represented this local society in the International Union for Philosophy of Science (IUPS), founded in 1949 by Gonseth and Stanislas Docks, the latter having already founded in 1947 the International Academy of Philosophy of Science in Brussels, of which Beth was a member from the beginning. In the early 1950s, Gonseth courted the Association for Symbolic Logic (ASL), a member of the International Mathematical Union (IUM) and strongly influenced by Tarski, to join their forces. However, Tarski and Beth discouraged the proposed alliance, because they considered Gonseth’s way of handling things too autocratic, and besides, Gonseth’s and Docks’s aversion to formal logic caused them constant difficulties [van Ulsen 2000]. In fact, Beth wanted to have a special international organization for the philosophy of science and logic that was neither dependent on philosophers nor on mathematicians.

7In 1948, he became president of the ANvW General Dutch Society of Philosophy (founded in 1933) and he was a member of the editorial board of its journal. In the same year, he was the secretary of the 10th International Congress of Philosophy in Amsterdam, where the series “Studies in Logic and the Foundations of Mathematics” was launched, with Beth, Brouwer, and Heyting as editors. In 1952, he founded the Institute for the Foundations and the Philosophy of the Exact Sciences at the University of Amsterdam, which had a specialized library and worked closely with the Mathematical Institute and its director A. Heyting. In 1961, the University of Amsterdam signed a contract with Euratom, to prepare the development of a “thought machine,” designed to facilitate the solution of mathematical problems by computational means, and to carry out research on mechanical translation and mathematical linguistics. This project was housed at Beth’s institute. Psychological insights would not only aid in the construction of such machines, but would also serve to illuminate the psychology of thought. This temporarily provided the institute with an expanded staff of about ten scholars. The successor to Beth’s institute is now the Institute for Logic, Language and Computation (ILLC).

8In 1953, Beth became a member of the Koninklijke Akademie van Wetenschappen (Royal Dutch Academy of Science) as a philosopher in the “humanities” section although he was a trained scientist.

9Beth gave guest lectures at the Universities of Cambridge (1951). In 1951-1952 he stayed at UC Berkeley (invited by Tarski) and at Johns Hopkins in 1957. At the end of his life he obtained an honorary doctorate at the University of Ghent. Beth was in poor health, suffered from severe coughing fits, smoked constantly and had a hoarse voice. When he died on April 12, 1964, he was fifty-five years old.

2 “Scientific Philosophy” and the history of metaphysics

10Beth contrasted “philosophy” with a mere personal levensbeschouwing (“persuasion,” or “Weltanschauung”), which is “the whole of convictions, valuations, attitudes and aspirations.” This personal philosophy, influenced by milieu and life experiences, is subjective and helps the individual to face the problems of life. Philosophy, on the other hand, like science, is systematic, logically consistent, and general. However, as a philosophy of life, or philosophy “in accordance with its cosmological conception” (Kant), it deals with the same human problems as “levensbeschouwingen,” but as a form of science it tries to offer objectively tenable solutions. It discusses the possible points of view and critically reflects on the questions we have to face, distinguishing between “pseudo-questions” and real questions.

11For Beth, the development of science in recent centuries has led to the disappearance of the “ancient metaphysics” that both founded science and initiated man into the mysteries of life. Metaphysics originated in the Aristotelian theory of science. “According to this doctrine, the principles of science need not be detected by way of intentional systematic research [...]. Knowledge of principles [...] is proper to any man” (“Critical epochs in the development of the theory of science” [1950a, 30]). The principles are considered as self-evident and not subject to revision. “All that philosophers have to do is to test and sift current opinions regarding the principles” (ibid.). This premise became implausible with the progress of science. Its demise implied that research into the foundations of sciences would no longer be done by speculation, but by the methods of science. The Enlightenment flourished because it criticized traditional views that were dominant in Western Europe and unacceptable to the inhabitants of Eastern Europe and North America. Beth concluded his inaugural address by expressing the hope that the awakening peoples of the Orient will be able to adopt philosophy and other spiritual goods purified of all bias by the criticism of the “actual more radical Enlightenment.”

12In his little book entitled La Crise de la raison et la logique [1957b], Beth suggests as the final goal of his philosophical thought “to arrive at guidelines for the construction of a new doctrinal synthesis that can replace the great traditional philosophical system.” In fact, according to Beth, the philosophical climate of the postwar period is characterized by the tension between scientific rationalism, dominated by mathematical logic, and irrationalism. This creates a vacuum that the systems of traditional metaphysics—Beth mentions the Thomistic, phenomenological, and materialist movements (Les Fondements logiques des mathématiques [1950b, 184])—are not able to fill. The solution that Beth sees in his so-called “scientific philosophy,” which is “a kind of doctrinal synthesis of the various contemporary tendencies that will probably emerge as these tendencies are pushed to their extreme” (Épistémologie mathématique et psychologie [Beth & Piaget 1961, 4, my emphasis]).

13What exactly motivates this synthesis? Is it more than an expression of eclecticism, and what are the consequences for the philosophy of science? Five points seem to be important in answering these questions:

  • First, concerned with the divergences of different schools, he was inspired by the Dutch neo-Kantians (Ovink, Franken), by Mannoury’s significs, by Brouwer’s or, better, by Heyting’s less idealistic intuitionism and by the Vienna Circle: scientific philosophy applied logical analysis and significs, socio-psycho-linguistic or semiotic procedures [1962a, 31].
  • Second, in order to understand the differences of opinion regarding both philosophical thought and the foundations of science, Beth recommended studying their historical and philosophical background, which originated in the Aristotelian theory of science, inspired by Plato’s philosophy. But today we are faced with the problem that the Aristotelian postulates, characterized as “evidence,” “reality,” and “truth,” “turn out to be incapable of dealing in a satisfactory manner with the problems originating from contemporary foundational research” [1950a, 27]. Although Beth is convinced that the modern discussion requires historical reconstruction—this shows for example, through Aristotle’s illustrations, why it is not logic but mathematics which is so important to traditional speculative philosophy—his interest lies in a systematic reconstruction in which Plato and Aristotle serve as predecessors of Cantor, Frege, and Russell [1959, 25, 31, 37].
  • Third, research into the foundations of a particular science can only be done on the basis of a thorough knowledge of the field in question. With respect to logic and mathematics, it then becomes clear that mathematical logic, arithmetic, and set theory allow for a more precise discussion of the three philosophically relevant constants of mathematical thought: deductive method, algorithm, and infinity (Konstanten van het Wiskundig Denken [1963b]).
  • Fourth, refusing to link rationality and absolute evidence, Beth’s scientific philosophy offers a method between relativism and scientism, closely related to certain insights of Bernays and Gonseth. As an anti-metaphysician and anti-psychologist, Beth uses symbolic logic for logical analysis without rejecting an intuitive foundation, unlike logical empiricists.
  • Fifth, and finally, although Beth started his work with a traditional psychological perspective on logic, in the late thirties, under the influence of Scholz, he switched to the “modern” way of doing logic. So the problem he faces in his correspondence with Jean Piaget is this: is there a moderately naturalistic way to bring logic and psychology together retaining the normative element of logic and the descriptive element of psychology? Neither Piaget nor Beth give a clear answer or even manage to maintain a precise and stable formulation of the problem.

14How does this all fit together? Instead of seeing the synthesis envisaged by Beth as a consequence of the destruction of the extremes, we should rather consider the latter also as a particularly insistent cognoscendi rationes of the complementarity of solutions, that is, of a multidimensional solution that shows itself in the form of horizons of realities (Gonseth). In other words, different historical systems should not be judged by the same criterion, since they may pursue different goals. Admittedly, we are not yet talking about different ways of constructing worlds (Nelson Goodman), but about different ways of constructing the world. In the famous last paragraph of his main work The Foundations of Mathematics Beth writes:

With respect to the other spheres or zones, each sphere of reality has a considerable degree of autonomy. [...] On the other hand, it must of course be conceded that the different spheres of reality are not entirely disconnected. [...] However, it is not superfluous to utter once more a caution against any premature attempt at a precise statement concerning the connections between the different spheres of reality. It is the task of science to penetrate into the different spheres of reality and to discover their mutual connections. It seems reasonable however tentatively to consider the various spheres of reality as complementary aspects of one and the same substratum. [Beth 1959, 644–645]

15Such spheres of reality immediately imply the question of intuitive evidence in relation to Euclidean geometry or in relation to absolute simultaneity, which must be explained by a transformation of the concept of intuition and not by the adoption of an irrational relativism [1957b, 6]. What kind of transformation is this exactly?

16In a discussion following a talk by Gonseth in Brussels, Beth transforms intuition into a preliminary position of common sense, noting in 1947:

I completely agree with MM. Gonseth and Weyl that we need, as a point of reference, so to speak, the vulgar world, the world of common sense. [...] I would like to add that this vulgar world may well serve us as a point of reference but cannot be taken as a criterion of truth for more developed theories. [Beth & Gonseth 1948, 56, transl. G. H.]

17With respect to the philosophy of mathematics, which is at the center of his philosophical interests, Beth opposes in his Mathematical Thought [1965] the path proceeding from an Aristotelian-Thomistic tradition according to which the structure of mathematical language is fixed independently of experience or intuition, and results from a reduction either to a transcendental form or to logic, to the intuitionist doctrine. The latter understands mathematics and logic either as a process of abstraction and construction from perceptible data, or as a process of construction from a priori given forms [Beth 1959, 180–186]. Beth favors the positions of the intuitionist tradition, and he carefully analyzes intuitionistic mathematics in chapter 15. Nevertheless, he rejects the consciousness-theoretical position of Brouwer and seems to prefer a constructivist approach which insists on the efficacy of the construction itself, i.e., where Brouwer’s concept of “free creation” is transformed into “free expression” [1965, 184]. For Beth, mathematics does not refer to an intelligible mode, nor is mathematics a pure human creation: rather, it “reflects the structure of reality” [1965, 180] and, as a “free expression,” concerns our ability to consider different horizons of realities [1965, 125], which remain horizons without becoming autonomous spheres. This also applies to the relationship between natural language and the formal languages of logic and mathematics. As in Gonseth’s case, there is no gap between both, but rathera continuous connection.

18For example, he agrees with the Carnap-Bernays thesis that mathematical existence cannot simply consist in the absence of contradiction. Indeed, in mathematical practice, existence is most often conditioned by the linguistic framework chosen for a given purpose; it is therefore “internal” (Carnap) or “bezogen” (Bernays) [see 1965, 171]. But, for Carnap, the choice of a framework is subject to the principle of tolerance applicable to linguistic forms and is determined exclusively by practical considerations, according to which propositions of external existence are devoid of any cognitive content. Beth, on the other hand, believes in a free choice limited by an ontological commitment or by mathematical intuition (“Carnaps Views” [1963a, 499 sq.]). For Beth, this attitude confirms an interpretation he shares with Quine and Gonseth: the three thinkers agree that, in principle, there is no precise boundary between the acceptance of a language structure and the acceptance of an assertion formulated in that language.

19Although mathematics and logic are not unrelated to and dependent on other areas, they cannot be reduced to any other field. Their “individual character [derives] from the circumstance that on the one hand one obtains knowledge here to which a very high degree of certainty should be ascribed, while on the other hand it turns out to be very difficult to localize a definite cognitive object” (Aspects of Modern Logic [1967, 99]).

  • 2 Concerning Tarski semantics, Beth regrets Gonseth’s technical misunderstandings [1962a, 33].

20In his work on the philosophy of physics at the end of the 1940s, Beth sketched the shift from the “received” Aristotelian view of science as anticipatio naturae, in which phenomena are explained from evident or absolute first principles, to a semantic approach [interpretatio naturae], in which the rules of deduction are replaced by consequence relations in the sense of Tarski:2 the question, which statements will follow from the premises, is answered by constructions of the form “all theoretically possible states that satisfy the premises also satisfy the conclusion”, as Van Fraassen said [van Benthem & van Ulsen 2008].

3 Philosophy of culture

21With the new method that he had in mind, Beth wanted to emphasize the different aspects of reality through the filter constituted by the intellectual climate of the time, i.e., the tension between scientism and subjective relativism, which results in a crisis of rationality, or rather, of irrationalism. For the latter, he saw a long tradition going from Pascal, Rousseau, Maine de Biran, Kiergegaard, Schopenhauer, Hartmann, Nietzsche, James, Bergson, Maritain, Keyserling, Klages to Heidegger and in which Peirce stands for a less radical version. This irrationalism “in our Time” coincided with a cultural pessimism, which is partly stimulated by the “cultural-historical pessimism” propagated by Spengler. However, it is the history of mathematics that shows that science does not develop along organic lines and is culturally independent. More problematic is the inherent conservatism of social and political thinking, which even in progressive circles tends to be used as an ideological device to justify the power of dominant social groups. Such conservatism slows down the adaptation of social institutions to the changing needs of society. However, Beth sees no reason for despair and argues for the establishment of an ideologically and financially independent Institute forsocial and political research.

22Beth’s development of a scientific rationalism in order to escape both the positivism and the dogmatism of a metaphysical irrationalism comes down to a clarification of the exact and natural sciences in relation to the human and social sciences. He stressed that it would indeed be dangerous and reductive to believe that the latter can be reduced to their role of guiding individuals in their moral and political commitments in the broadest sense. The human and social sciences define the reflexive, sociological, historical and even literary framework of scientific activities. They provide a conceptual framework that the scientist uses in their deductions. In this way, the traditional claim of the fundamental objectivity of science is challenged. First, because some modern theorizations, such as the theory of relativity or quantum physics, make the relationship between theory and experience more problematic. Second, because the role of individual personalities and styles in the practice of science is evident. On an ontological level, therefore, one cannot continue to reason about the supposedly eternal objects of science. Since the boundaries that traditionally separate the exact sciences from the social sciences are permeable, philosophy should be rethought by logic, which as a rational critic is a culturally relevant discipline with empirical and pragmatic components. His theory of Semantic tableaus—a variant of Gerhard Gentzen’s predicate calculus of sequents—serves as an example for solving the Berkeley-Locke problem about the relation between the general and theparticular in mathematics.

23On the other hand, the autonomy of the scientific field becomes relative, without harming its identity and development. In The Foundations of Mathematics, Beth notes:

We cannot separate the spheres of physical and of subjective reality, so long as we remain inside the circle of our personal experience. A separation of these spheres urges itself upon us as soon as we consider the experiences of other people; at the same time it involves the necessity of penetrating into the social sphere of reality. “Pure physics” is for this reason impossible, even in abstracto. [...] The reader [...] will not be inclined, I hope, to interpret these words as a plea for “impure” science. I meant only to state the conviction that the frontiers between the various domains of human knowledge, important and even indispensable though their maintenance may be for the development of science, cannot be vindicated indefinitely. [Beth 1959, 645–646] The new frontier is not between different kinds of knowledge—there is a methodological monism as a mode of argument among humanities and other types of science—but between knowledge and action based on individual (value) awareness, the latter depends on character and education, the former on a plurality of techniques. In a softening of the position of the logical empiricists, Beth, while holding a traditional metaphysics of life to be impossible, considers a scientific philosophy of life to be merely incomplete:
The essential difference between a modern scientific philosophy of life and traditional metaphysics is that modern scientific philosophy would never fail to realize that its explanation of human experience will always and inevitably be incomplete and tentative. [1962a, 28]

24Finally, the problems of classifying the new technologies require us to consider a frontier space between our traditional criteria and constitute—as Beth has foreseen [1965, 166]—a third crisis of foundations.

4 Mathematical logic

25We owe Beth three main achievements in mathematical logic, whose systematic connection is described by Guillaume [1998-1999].

  1. In “On Padoa’s Method in the Theory of Definition” [1953], he introduced the notions of implicit and explicit definability and provided a proof for first-order logic of what is called Beth’s theorem: the implicit definability of a predicate as being fixed by giving other predicates—a primarily model-theoretic, i.e., semantic notion—coincides with its explicit definability in terms of other predicates in a formal language—a primarily proof-theoretic, i.e., syntactic notion. As van Benthem noted, “both concepts occur in science as well as in philosophy—the former under the poetic name supervenience. [...] Beth’s implicit definability is actually an invariance criterion: it says that transformations that preserve the other predicates preserve ‘automatically’ also the new predicate” [van Benthem 2008b, 8–9, transl. G. H.]. The Definition theorem, on the one hand, was used in a paper published with Tarski in [1956] on “Equilaterality as the only primitive notion of Euclidean geometry” and, on the other hand, was somewhat generalized by Craig using his interpolation theorem.
  2. Beth’s “Semantic tableaus” constitute a system of deduction similar to that introduced by Gentzen in 1934, and are more elegant and natural than the Frege-Hilbert systems. Very close to Hintikka’s game-theoretical semantics—Quine informed Beth about it in June 1955—and Lorenzen’s dialogical logic—in the Beth archive is a correspondence with Lorenzen on the interpretation of tableau in terms of play—the crucial idea of the tableau system is that the systematic search for all possible counterexamples to an argument leads, in case of failure, to a proof of the logical truth of the argument. Beth gave a completeness proof for his system and used it for a proof of the Löwenheim-Skolem-Gödel theorem, the Herbrand theorem and Gentzen’s subformula theorem. Beth gave a (classical) completeness proof with respect to a series of tableaus and mainly applied the tableau method to construct a semantics for intuitionistic logic predicate logic as formulated by Heyting. One must emphasize with Ulsen that,
    “during his entire professional career Beth was interested in intuitionistic logic, but he was himself not an intuitionist and disliked intuitionistic philosophy. Beth combined his semantic tableaus with trees and choice sequence, thus creating the Beth models (1956). With the application (and in his case duplication) of Brouwer’s fundamental theorem he avoided non-intuitionistic mathematics in his intuitionistic semantics and completeness proof. ” [van Ulsen 2000, 330]
    Beth also used the tableau to solve problems in philosophy. The most often discussed and well-known example is the Locke-Berkeley problem, already mentionned above (Über Lockes “Allgemeines Dreieck” [1956d]): it comes from the fact that in geometry one constructs a special case for a universally valid proof. Locke takes as an example the proof of the theorem that the sum of the angles of a triangle is equal to two right angles. We perform the proof for the particular case and draw a general conclusion from that particular case. How is that possible? According to Locke, the proof refers from the outset to a general triangle, the existence of which Berkeley denied, but could not give a conclusive proof by means of a triangle abstracted from all particularities. With his tableau method, Beth shows that the deduction obtained is a formal-logical one, which is not related to the necessity of an appeal to intuitive vision: the introduction of the special points A, B, Cin the scope of the formula
    XYZ[Tr(X,Y,Z) → U(X,Y,Z)]
    meaning that if X, Y, Z define a triangle then the sum of its angles is equal to two right angles, is motivated by the way this formula is to be treated in the construction of the semantic tableau, which can be proved to be closed. Nevertheless, this result does not prove that in other cases no appeal to intuition is always possible. Beth invalidated this objection by the “use of all the logical insights acquired up to now” and especially by the completeness theorem for logic [Beth 1967, 48 sq.]. Another application of the “semantic tableaus” is their use in mechanical theorem proving and more generally in computer science, a subject that did not even exist in Beth’s time.
  3. As emphasized above, Beth was not an intuitionist in Brouwer’s sense and yet worked significantly in intuitionistic logic. In “Semantic construction of intuitionistic logic” [1956c] he uses semantic tableaus to give a completeness proof of Heyting’s formalization of intuitionistic predicate logic. In doing so, he gives a classical proof by what is today known as the “Beth Model,” but he also tries to conduct an intuitionistic proof by appealing to Brouwer’s fan theorem, but this leads to criticism, especially from Kreisel [van Ulsen 2000, 289 sqq.]. Nevertheless, the paper was the beginning of research by “Dyson & Kreisel [1961] and later W. Veldman, H. L. Friedman and A. S. Troelstra on the completeness of intuitionistic logic from an intuitionistic point of view” [de Jongh & van Ulsen 1998-1999, 289–290]. It also seems obvious that Kripke’s famous papers on possible word semantics were not designed independently from Beth’s semantic tableaus for classical and intuitionistic logic.

5 Influence

26Beth founded the institutional tradition of logic in the Netherlands. Thanks to its open-minded, rational, and international orientation, his Amsterdam institute became one of the world’s leading centers for logic, computation, and philosophy. His contribution to philosophy and science reveals an unusual blend of fields of interest, extending across areas of cultural criticism, education in science, history of the exact sciences, history of philosophy, ancient philosophy, philosophy and psychology of logic, foundations of mathematics, physics, and schools of modern philosophy, examined with particular emphasis on the logical point of view. His successor as director of the Amsterdam Institute characterizes Beth’s impact in a conclusive formula: papers like Beth’s not only inform, they educate [van Benthem 2008a, 45]. Concerning Beth’s early pioneering interest in computation and automated reasoning, based on his tableaus, there is still a thriving community in computer science around these themes (https://tableaux2021.org/​). The very current problem of automatic processing of large data sets in science was already studied in the Euratom project carried out by Beth’s institute.

27It should be mentioned here that Beth’s Foundation of Mathematics [1959] is, with regard to its philosophical and historical comments, an indispensable source on the problems, methods and results of the foundations of mathematics around 1960. Besides axiomatics, existence, and the logico-set-theoretical paradoxes, it also presents the basic results of recursion theory, metalogics, and model theory. Outdated by its recurrent vision of history and by modern studies on the history of set theory and proof theory among others, it remains in its ambition a hitherto unsurpassed model for the combination of philosophy, history, and logic.

28The Evert Willem Beth Foundation was established in 1978 in accordance with the will of Ms. Cornelia Beth-Pastoor, Beth’s widow. Its aim is to promote logic and scientific philosophy in the broadest sense. To this end, the Foundation organizes symposia, invites prominent scholars and scientists to give annual lectures, grants awards and funds the E. W. Beth “Outstanding Disseration Prize.”

29The “Beth Archive,” which keeps a large correspondence (among others with Bernays, Bochenski, Brouwer, Carnap, Craig, Curry, Destouches, Feys, Fraenkel, Henkin, Heyting, Kleene, Kreisel, Lorenzen, Mannoury, Piaget, Popper, Quine, Scholz, Tarski, Vaught, and van der Waerden), is in the care of the Noord-Holland Archief in Haarlem. His library of several thousand of books was donated by his widow to the Institute of Foundational Research and the Philosophy of the Exact Sciences in 1974.

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Bibliographie

Primary Works

Beth, Evert Willem [1935], Rede en aanschouwing in de wiskunde (thesis), Groningen: Noordhoff.

Beth, Evert Willem [1937], L’évidence intuitive dans les mathématiques modernes, in : Travaux du IXe Congrès international de philosophie, Paris, 1–6 août 1937, vol. VI, 161–165.

Beth, Evert Willem [1940], Inleiding tot de wijsbegeerte der wiskunde, Antwerpen: Standaard-Boekh, reprints: Antwerpen-Brussels; Nijmegen-Utrecht: Standaard Boekh Dekker & Van de Vegt, 1942; Antwerpen: Standard-Boekkandel, 1953.

Beth, Evert Willem [1943], Verleden en toekomst der wetenschappelijke wijsbegeerte, De Gids, 107, 46–67.

Beth, Evert Willem [1944a], De Wijsbegeerte der Wiskunde van Parmenides tot Bolzano, Antwerpen; Nijmegen: Standaard-Boekhandel; Dekker & Van de Vegt.

Beth, Evert Willem [1944b], Geschiedenis der Logica, The Hague: Servire.

Beth, Evert Willem [1946a], De strekking en het bestaansrecht der metaphysica in verband met de toekomst der wijsbegeerte, Groningen; Batavia: Noordhoff.

Beth, Evert Willem [1946b], Metaphysica en wetenschap, De Gids, 109, 83–91, also in Door wetenschap tot wijsheid.

Beth, Evert Willem [1947], Hoe is wijsbegeerte mogelijk, Algemeen Nederlands Tijdschrift voor Wijsbegeerte, 40, 170–176.

Beth, Evert Willem [1948a], De wetenschap als cultuurfactor, in: De functie der wetenschap: drie voordrachten met discussie, edited by E. W. Beth, D. van Dantzig, & C. F. P. Stutterheim, The Hague: Leopolds, ‘s-Gravenhage, 7–40.

Beth, Evert Willem [1948b], Natuurphilosophie, Gorinchem: Noorduijn.

Beth, Evert Willem [1948c], Symbolische Logik und Grundlegung der exakten Wissenschaften, Bern: A. Francke.

Beth, Evert Willem [1948-1949], Analyse sémantique des théories physiques, Synthese, 7(3), 206–207, https://www.jstor.org/stable/20114040.

Beth, Evert Willem [1949], Towards an up-to-year philosophy of the natural sciences, Methodos, 1, 178–185.

Beth, Evert Willem [1950a], Critical epochs in the development of the theory of science, The British Journal of the Philosophy of Science, 1(1), 27–42, doi: 10.1093/bjps/i.1.27.

Beth, Evert Willem [1950b], Les Fondements logiques des mathématiques, Paris ; Louvain : Gauthier-Villars ; E. Nauwelaerts.

Beth, Evert Willem [1950c], Wijsgerige Ruimteleer, Antwerpen; Nijmegen: Standaard-boekhandel; Dekker & van de Vegt.

Beth, Evert Willem [1951], A topological proof of the theorem of Löwenheim-Skolem-Gödel, Indagationes mathematicae, 13, 436–444, doi: 10.1016/s1385-7258(51)50062-8.

Beth, Evert Willem [1953], On Padoa’s method in the theory of definition, Indagationes Mathematicae (Proceedings), 56, 330–339, doi: 10.1016/S1385-7258(53)50042-3.

Beth, Evert Willem [1955], Semantic Entailment and Formal Derivability, Mededelingen van de Koninklijke Nederlandse Akademie van Wetenschappen, vol. 18(13), Amsterdam: Noord-Hollandsche Uitgevers Maatschappij, repr. in Hintikka, J. (ed.) The Philosophy of Mathematics, Oxford: Oxford University Press, 1969, 9–41.

Beth, Evert Willem [1956a], L’Existence en mathématiques, Paris; Louvain: Gauthier-Villars; E. Nauwelaerts, Collection de logique mathématique, série A.

Beth, Evert Willem [1956b], Poincaré et la philosophie, in : H. Poincaré : Œuvres, edited by G. Darboux, Paris : Gauthier-Villars, vol. 11, 232–238.

Beth, Evert Willem [1956c], Semantic Construction of Intuitionistic Logic, Mededelingen van de Koninklijke Nederlandse Akademie van Wetenschappen, vol. 19(11), Amsterdam: Noord-Hollandsche Uitgevers Maatschappij.

Beth, Evert Willem [1956d], Über Lockes “Allgemeines Dreieck”, Kantstudien, 48, 361–380, doi: 10.1515/kant.1957.48.1-4.361.

Beth, Evert Willem [1957a], De weg der wetenschap: inleiding tot de methodeleer der empirische wetenschappen, Haarlem: Bohn.

Beth, Evert Willem [1957b], La Crise de la raison et la logique, Paris : Gauthier-Villars.

Beth, Evert Willem [1958], On machines which prove theorems, Simon Stevin, 32, 49–60.

Beth, Evert Willem [1959], The Foundations of Mathematics. A Study in the Philosophy of Sciences, Amsterdam: North-Holland.

Beth, Evert Willem [1962a], Door wetenschap tot wijsheid: verzameling wijsgerige studiën, Amsterdam: Amsterdam University Press, English version: Science, a Road to Wisdom, Dordrecht: Reidel, 1968.

Beth, Evert Willem [1962b], Formal Methods. An Introduction to Symbolic Logic and the Study of Effective Operations in Arithmetic and Logic, Dordrecht: Reidel.

Beth, Evert Willem [1963a], Carnaps views on the advantages of constructed systems over natural languages in the philosophy of science, in: The Philosophy of Rudolf Carnap, edited by P. A Schilpp, La Salle: Open Court, 469–502.

Beth, Evert Willem [1963b], Konstanten van het Wiskundig Denken, Mededelingen Kononklijke Nederlandse Akademie Wetenschappen, vol. 26, Amsterdam: Noord-Hollandsche Uitgevers Maatschappij.

Beth, Evert Willem [1965], Mathematical Thought. An Introduction to the Philosophy of Mathematics, Dordrecht: Reidel.

Beth, Evert Willem [1967], Moderne Logica, Assen: Van Gorcum, Engl. transl. Aspects of Modern Logic, Dordrecht : Reidel, 1970.

Beth, Evert Willem & Gonseth, Ferdinand [1948], Intervention dans la discussion de F. Gonseth “Les conceptions mathématiques et le réel”, in : Les Sciences et le Réel, Paris : Hermann, 56.

Beth, Evert Willem, Mays, Wolfe, et al. [1957], Epistémologie génétique et recherche psychologique, Paris : Presses universitaires de France, Nendeln : Kraus reprint, 1974.

Beth, Evert Willem & Piaget, Jean [1961], Épistémologie mathématique et psychologie, Paris : Presses universitaires de France, Engl. transl. Mathematical Epistemology and Psychology Dordrecht : Reidel, 1966.

Beth, Evert Willem & Piaget, Jean [1962], Implication, formalisation et logique naturelle, Paris : Presses universitaires de France.

Beth, Evert Willem & Tarski, A. [1956], Equilaterality as the only primitive notion of Euclidean geometry, Indagationes mathematicae, 18, 462–469, doi: 10.1016/s1385-7258(56)50062-5.

Further Readings

Barth, Else M. [1990], In the service of human society: Formal, informal or anti-logical? The philosophy of the logician Evert Willem Beth (1908-1964), Informal Logic, 12(1), 1–10, doi: 10.22329/il.v12i1.2596.

Barth, Else M., Heinzman, Gerhard, et al. (eds.) [1998-1999], Un logician consciencieux. La philosophie de Evert Willem Beth. Actes du Colloque Everth Willem Beth, Archives Henri-Poincaré et Fondation E. W. Beth, Nancy, 22–24 avril 1998., Philosophia Scientiæ, vol. 3, Paris : Kimé, http://www.numdam.org/issues/PHSC_1998-1999__3_4/.

van Benthem, Johan van [2008a], Beth’s influence today, in: Logic and Scientific Philosophy. An E. W. Beth Centenary Celebration, Amsterdam: Evert Willem Beth Foundation, 44–46.

van Benthem, Johan van [2008b], Constanten, of Variabelen, van het Logisch Denken, Technical notes (x) series, https://eprints.illc.uva.nl/id/eprint/673/.

van Benthem, Johan van & van Ulsen, Paul (eds.) [2008], Logic and Scientific Philosophy. An E. W. Beth Centenary Celebration, Amsterdam: Evert Willem Beth Foundation.

Bernays, Paul [1976], Mathematische Existenz und Widerspruchsfreiheit (1950), in: Paul Bernay: Abhandlungen zur Philosophie der Mathematik, Darmstadt: Wissenschaftliche Buchgesellschaft, 92–106.

Burdman Feferman, Anita & Feferman, Solomon [2010], Logic and methodology, center stage, Philosophiae Scientiæ, 14(1), 159–168, doi: 10.4000/philosophiascientiae.162.

Carnap, Rudolf [1958], Empiricism, semantics, and ontology (1950), in: R. Carnap: Meaning and Necessity, Chicago: University of Chicago Press, 205–221.

Craig, William [1957], Three uses of the Herbrand-Gentzen Theorem in relating model theory and proof theory, Journal of Symbolic Logic, 22(3), 269–285, doi: 10.2307/2963594.

Destouches, Jean-Louis (ed.) [1967], E. W. Beth Memorial Colloquium. Logic and Foundations of Science Paris, Institut Henri Poincaré, 19–21 May 1964, Paris: Reidel, doi: 10.1007/978-94-010-3520-0, this volume contains a complete bibliography.

Dyson, Verena Huber & Kreisel, Georg [1961], Analysis of Beth’s semantic construction of intuitionistic logic, Applied mathematics and statistics laboratories, technical report 3, Stanford University.

Guillaume, Marcel [1998-1999], Essai sur la genèse de la méthode des tableaux de Beth, Philosophia Scientiæ, 3(4), 235–277, http://www.numdam.org/item/PHSC_1998-1999__3_4_235_0/.

Heinzmann, Gerhard [1982], Schematisierte Strukturen: Eine Untersuchung über den “Idoneismus” Ferdinand Gonseths auf dem Hintergrund eines konstruktivistischen Ansatz, Bern; Stuttgart: Haupt.

Heinzmann, Gerhard, Trognon, Alain, et al. [2014], Correspondence Between Evert Willem Beth and Jean Piaget (1951–1955), in: Interdisciplinary Works in Logic, Epistemology, Psychology and Linguistics: Dialogue, Rationality, and Formalism, edited by M. Rebuschi, M. Batt, G. Heinzmann, F. Lihoreau, M. Musiol, & A. Trognon, Cham: Springer International Publishing, 45–93, doi: 10.1007/978-3-319-03044-9_4.

Hintikka, Jaakko K. [1955], Form and content in quantification theory, Acta Philosophica Fennica, 8, 7–55.

de Jongh, Dick & van Ulsen, Paul [1998-1999], Beth’s nonclassical valuations, Philosophia Scientiæ, 3(4), 279–302, www.numdam.org/item/PHSC_1998-1999__3_4_279_0/.

Kreisel, Georg [1958], Review of “E. W. Beth. La Crise de la raison et la logique, Paris : Gauthier-Villars, The Journal of Symbolic Logic, 23(1), 35–-37, doi : 10.2307/2964464.

Lorenzen, Paul [1959], Ein dialogisches Konstrutivitätskriterium, in: Infinitistic Methods, Proceedings of the Symposium on Foundations of Mathematics Warsaw 1959, Oxford; London; Warsaw: Pergamon Press, 193–200.

Peijnenburg, Jeanne [1994], Formal proof or linguistic process? Beth and Hintikka on Kant’s use of “Analytic”, Kantstudien, 85, 160–178, doi: 10.1515/kant.1994.85.2.160.

Staal, J. F. [1965], E. W. Beth, 1908-1964, Dialectica, 19(1–2), 158–184, doi: 10.1111/j.1746-8361.1965.tb01455.x.

van Ulsen, Paul [2000], E. W. Beth als logicus, Ph.D. thesis, University of Amsterdam, Amsterdam.

Visser, Henk [1998-1999], E. W. Beth and the logical empiricists, Philosophia Scientiae, 3(4), 49–76, http://www.numdam.org/item/PHSC_1998-1999__3_4_49_0/.

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Notes

1 Quotations from Beth’s work are are in general indicated with title and pages, quotations of other authors with their name and the year of publication. References to these authors are in the “Further Readings” section.

2 Concerning Tarski semantics, Beth regrets Gonseth’s technical misunderstandings [1962a, 33].

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Gerhard Heinzmann

Université de Lorraine, Université de Strasbourg, CNRS, AHP-PReST, Nancy (France)

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