Paul Bernays Research Articles

NOMINALISTIC ORDINALS, RECURSION ON HIGHER TYPES, AND FINITISM

AbstractIn 1936, Gerhard Gentzen published a proof of consistency for Peano Arithmetic using transfinite induction up to ε0, which was considered a finitistically acceptable procedure by both Gentzen and Paul Bernays. Gentzen’s method of arithmetising ordinals and thus avoiding the Platonistic metaphysics of set theory traces back to the 1920s, when Bernays and David Hilbert used the method for an attempted proof of the Continuum Hypothesis. The idea that recursion on higher types could be used to simulate the limit-building in transfinite recursion seems to originate from Bernays. The main difficulty, which was already discovered in Gabriel Sudan’s nearly forgotten paper of 1927, was that measuring transfinite ordinals requires stronger methods than representing them. This paper presents a historical account of the idea of nominalistic ordinals in the context of the Hilbert Programme as well as Gentzen and Bernays’ finitary interpretation of transfinite induction.

FROM STENIUS’ CONSISTENCY PROOF TO SCHÜTTE’S CUT ELIMINATION FOR ω-ARITHMETIC

AbstractThe book Das Interpretationsproblem der Formalisierten Zahlentheorie und ihre Formale Widerspruchsfreiheit by Erik Stenius published in 1952 contains a consistency proof for infinite ω-arithmetic based on a semantical interpretation. Despite the proof’s reference to semantics the truth definition is in fact equivalent to a syntactical derivability or reduction condition. Based on this reduction condition Stenius proves that the complexity of formulas in a derivation can be limited by the complexity of the conclusion. This independent result can also be proved by cut elimination for ω-arithmetic which was done by Schütte in 1951.In this paper we interpret the syntactic reduction in Stenius’ work as a method for cut elimination based on invertibility of the logical rules. Through this interpretation the constructivity of Stenius’ proof becomes apparent. This improvement was explicitly requested from Stenius by Paul Bernays in private correspondence (In a letter from Bernays begun on the 19th of September 1952 (Stenius & Bernays, 1951–75)). Bernays, who took a deep interest in Stenius’ manuscript, applied the described method in a proof Herbrand’s theorem. In this paper we prove Herbrand’s theorem, as an application of Stenius’ work, based on lecture notes of Bernays (Bernays, 1961). The main result completely resolves Bernays’ suggestions for improvement by eliminating references to Stenius’ semantics and by showing the constructive nature of the proof. A comparison with Schütte’s cut elimination proof shows how Stenius’ simplification of the reduction of universal cut formulas, which in Schütte’s proof requires duplication and repositioning of the cuts, shifts the problematic case of reduction to implications.

Bernays and Set Theory

AbstractWe discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higher-order reflection principles.

Lieber Herr Bernays!, Lieber Herr Gödel! Gödel on finitism, constructivity and Hilbert's program*

1. Godel, Bernays, and Hilbert. The correspondence between Paul Bernays and Kurt Godel is one of the most extensive in the two volumes of Godel's collected works devoted to his letters of (primarily) scientific, philosophical and historical interest. It ranges from 1930 to 1975 and deals with a rich body of logical and philosophical issues, including the incompleteness theorems, finitism, constructivity, set theory, the philosophy of mathematics, and post- Kantian philosophy, and contains Godel's thoughts on many topics that are not expressed elsewhere. In addition, it testifies to their life-long warm personal relationship. I have given a detailed synopsis of the Bernays Godel correspondence, with explanatory background, in my introductory note to it in Vol. IV of Godel's Collected Works, pp. 41-

Gentzen's Proof of Normalization for Natural Deduction

AbstractGentzen writes in the published version of his doctoral thesis Untersuchungen über das logische Schliessen (Investigations into logical reasoning) that he was able to prove the normalization theorem only for intuitionistic natural deduction, but not for classical. To cover the latter, he developed classical sequent calculus and proved a corresponding theorem, the famous cut elimination result. Its proof was organized so that a cut elimination result for an intuitionistic sequent calculus came out as a special case, namely the one in which the sequents have at most one formula in the right, succedent part. Thus, there was no need for a direct proof of normalization for intuitionistic natural deduction. The only traces of such a proof in the published thesis are some convertibilities, such as when an implication introduction is followed by an implication elimination [1934–35, II.5.13]. It remained to Dag Prawitz in 1965 to work out a proof of normalization. Another, less known proof was given also in 1965 by Andres Raggio.We found in February 2005 an early handwritten version of Gentzen's thesis, with exactly the above title, but with rather different contents: Most remarkably, it contains a detailed proof of normalization for what became the standard system of natural deduction. The manuscript is located in the Paul Bernays collection at the ETH-Zurichwith the signum Hs. 974: 271. Bernays must have gotten it well before the time of his being expelled from Göttingen on the basis of the racial laws in April 1933.

Burgess on Plural Logic and Set Theory

John Burgess in a 2004 paper combined plural logic and a new version of the idea of limitation of size to give an elegant motivation of the axioms of ZFC set theory. His proposal is meant to improve on earlier work by Paul Bernays in two ways. I argue that both attempted improvements fail.

Paul Bernays' Philosophical Way

Paul Bernays was both a philosopher and a mathematician. He is famous for his logical-mathematical production (also in collaboration with David Hilbert), while his philosophical works have been given less consideration. The present article is an attempt to reconstruct the way that led Bernays from his early writings on ethics to his final epistemological thought.

Saunders Mac Lane: a mathematical autobiography

Saunders Mac Lane was an extraordinary mathematician, a dedicated teacher, and a good citizen who cared deeply about the values of science and education. In his autobiography, he gives us a glimpse of his and times, mixing the highly personal with professional observations. His recollections bring to life a century of extraordinary accomplishments and tragedies that inspire and educate. Saunders Mac Lane's life covers nearly a century of mathematical developments. During the earlier part of the twentieth century, he participated in the exciting happenings in Gottingen---the Mecca of mathematics. He studied under David Hilbert, Hermann Weyl, and Paul Bernays and witnessed the collapse of a great tradition under the political pressure of a brutal dictatorship. Later, he contributed to the more abstract and general mathematical viewpoints developed in the twentieth century. Perhaps the most outstanding accomplishment during his long and extraordinary career was the development of the concept of categories, together with Samuel Eilenberg, and the creation of a theory that has broad applications in different areas of mathematics, in particular topology and foundations. He was also a keen observer and active participant in the social and political events. As a member and vice president of the National Academy of Science and an advisor to the Administration, he exerted considerable influence on science and education policies in the post-war period. Mac Lane's autobiography takes the reader on a journey through the most important milestones of the mathematical world in the twentieth century.

III *-ON THE MEANING OF THE WORD ‘PLATONISM’ IN THE EXPRESSION ‘MATHEMATICAL PLATONISM’

The expression ‘platonism in mathematics’ or ‘mathematical platonism’ is familiar in the philosophy of mathematics at least since the use Paul Bernays made of it in his paper of 1934, ‘Sur le Platonisme dans les Mathematiques’. But he was not the first to point out the similarities between the conception of the defenders of mathematical realism and the ideas of Plato. Poincare had already stressed the ‘platonistic’ orientation of the mathematicians he called‘Cantorian’, as opposed to those who (like himself) were ‘pragmatist’ ones. I examine in this paper some very perplexing aspects of the use which is made at that time of a number of concepts, particularly ‘idealism’ (which generally designates what we would call ‘mathematical realism’) and ‘empiricism’ (which can designate almost any form of antirealism, even if, like for example intuitionism, it is not empiricist at all). There are, of course, historical reasons that may explain why it was for a time so easy and natural to use the words and the concepts in a way that may seem now very strange and to treat as if they were equivalent the two oppositions: realism/antirealism and idealism/empiricism.

The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program

After a brief flirtation with logicism around 1917, David Hilbertproposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays andWilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the development of axiomatic systems for everstronger and more comprehensive areas of mathematics, and finitisticproofs of consistency of these systems. Early advances in these areaswere made by Hilbert (and Bernays) in a series of lecture courses atthe University of Gottingen between 1917 and 1923, and notably in Ackermann's dissertation of 1924. The main innovation was theinvention of the ∈-calculus, on which Hilbert's axiom systemswere based, and the development of the ∈-substitution methodas a basis for consistency proofs. The paper traces the developmentof the ``simultaneous development of logic and mathematics'' throughthe ∈-notation and provides an analysis of Ackermann'sconsistency proofs for primitive recursive arithmetic and for thefirst comprehensive mathematical system, the latter using thesubstitution method. It is striking that these proofs use transfiniteinduction not dissimilar to that used in Gentzen's later consistencyproof as well as non-primitive recursive definitions, and that thesemethods were accepted as finitistic at the time.

How “rational” is “rationality”?

By taking serious a remark once made by Paul Bernays, namely that an account of the nature of rationality should begin with concept-formation, this article sets out to uncover both the restrictive and the expansive boundaries of rationality. In order to do this some implications of the perennial philosophical problem of the “coherence of irreducibles” will be related to the acknowledgement of primitive terms and of their indefinability. Some critical remarks will be articulated in connection with an over-estimation of rationality - concerning the influence of Kant’s view of human understanding as the formal law-giver of nature (the supposedly “rational structure of the world”), and the apparently innocent (subjectivist) habit to refer to experiential entities as ‘objects’. The other side of the coin will be highlighted with reference to those kinds of knowledge transcending the limits of concept-formation - culminating in formulating the four most basic idea-statements philosophy can articulate about the universe. What is found “in-between” these (restrictive) and (expansive) boundaries of rationality will then briefly be placed within the contours of a threefold perspective on the self-insufficiency of logicality - as merely one amongst many more dimensions conditioning human life. Although the meaning of the most basic logical principles - such as the logical principles of identity, non-contradiction and sufficient reason - will surface in our analysis, exploring some of the complex issues in this respect, such as the relationship between thought and language, will not be analysed. The important role of solidarity - as the basis of critique - will be explained and related both to the role of immanent criticism in rational conversation and the importance of acknowledging what is designated as the principle of the excluded antinomy (which in an ontic sense underlies the logical principle of non-contradiction). The last section of our discussion will succinctly illuminate the proper place of the inevitable trust we ought to have in rationality - while implicitly warning against the rationalistic over-estimation of it (its degeneration into a rationalist “faith in reason”). Our intention is to enhance an awareness of the reality that rationality is embedded in and borders on givens which are not open to further “rational” exploration - givens that both condition (in a constitutive sense) and transcend the limits of conceptual knowledge. Some of the distinctions and insights operative in our analysis are explained in Strauss 2000 and 2003. Yet, most of the systematic perspectives found in this analysis of rationality are only developed in this article for the first time. Since a different study is required to discuss related problems and results found within cognitive science, it cannot be discussed within one article.

信としての数学

In this paper, I consider a philosophical interpretation of Hilbert's program put forward by Paul Bernays. The paper consists, roughly, of two parts. In the first part, I briefly describe Bernays's polemic against "immanence philosophy" over the question concerning the epistemological status of theoretical sciences and its relation to the foundational controversy in mathematics in the 1920s. In the second part, I briefly discuss the theory of cognition contained in the transcendental philosophy of Jakob Friedrich Fries and his twentieth century evangelist, LeonardNelson, and consider Bernays's view against the background of this intellectual tradition. I conclude the paper by arguing for the importance of attending to Friesian influence in order to achieve a deeper understanding of Bernays's (and Hilbert's) philosophy of mathematics.

Step by Recursive Step: Church's Analysis of Effective Calculability

AbstractAlonzo Church's mathematical work on computability and undecidability is well-known indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was “Church's Thesis” put forward publicly only in April 1935, when it had been formulated already in February/March 1934? Why did Church choose to formulate it then in terms of Gödel's general recursiveness, not his own λ-definability as he had done in 1934? A number of letters were exchanged between Church and Paul Bernays during the period from December 1934 to August 1937; they throw light on critical developments in Princeton during that period and reveal novel aspects of Church's distinctive contribution to the analysis of the informal notion ofeffective calculability. In particular, they allow me to give informed, though still tentative answers to the questions I raised; the character of my answers is reflected by an alternative title for this paper,Why Church needed Gödel's recursiveness for his Thesis. In Section 5, I contrast Church's analysis with that of Alan Turing and explore, in the very last section, an analogy with Dedekind's investigation of continuity.

Ferdinand Gonseth 1890-1975

In 1946 Ferdinand Gonseth founded, together with Paul Bernays, Karl Dürr (the only Swiss representative of logical positivism), and Sir Karl Popper, the International Society for Logic and Philosophy of Science. In the next year the first number of Dialectica, a philosophical review devoted mainly to epistemological subjects, was issued. Gonseth's bibliography presents more than 190 items — roughly 25 books, several prefaces, and, for the rest, papers published in numerous reviews. A good deal of his publications concern mathematics — he was appointed professor of mathematics first in Berne and then in Zurich at the well-known Swiss Federal Institute of Technology — but he has never completely separated his mathematical activities from his philosophical work.

Response to Bernays

In his response to my article, “Historicism and Science: Thoughts on Quine” (Dialectica, Vol. 29, Nos. 2-3), Paul Bernays distinguishes between two aspects of the article: my criticism of Quine's theory of knowledge and my contention that the epistemological foundations of modern science lead to historicism. Bernays substantially accepts my criticism of Quine, particularly insofar as I reject Quine's behaviorism and “physicalism”. He opposes, however, my claim that historicism is implicit in the presuppositions of modern science. I argued that the historicism inherent in modern science is due to the essentially changeable character of modern science. Quine's theory of knowledge tacitly takes account of this changeable character a) through his view of sense experience as being constituted by posits, b) through his recognition that individual scientific statements are dependent upon a broad linguistic-conceptual framework that is itself subject to change, and c) through his pragmatic conception of reason and the concomitant emphasis upon the criterion of simplicity.

Zu der Abhandlung von Raymond L. Weiss «Historicism and Science: Thoughts on Quine»

DialecticaVolume 29, Issue 2-3 p. 167-172 Zu der Abhandlung von Raymond L. Weiss «Historicism and Science: Thoughts on Quine» Paul Bernays, Paul BernaysSearch for more papers by this author Paul Bernays, Paul BernaysSearch for more papers by this author First published: September 1975 https://doi.org/10.1111/j.1746-8361.1975.tb00656.xAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Volume29, Issue2-3September 1975Pages 167-172 RelatedInformation

Paul Bernays. On the original Gentzen consistency proof for number theory. Intuitionism and proof theory, Proceedings of the summer conference at Buffalo N.Y. 1968, edited by A. Kino, J. Myhill, and R. E. Vesley, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam and London1970, pp. 409–417.

Paul Bernays. On the original Gentzen consistency proof for number theory. Intuitionism and proof theory, Proceedings of the summer conference at Buffalo N.Y. 1968, edited by A. Kino, J. Myhill, and R. E. Vesley, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam and London1970, pp. 409–417.

Gerhard Gentzen. Investigations into logical deduction. English translation of 4422 by M. E. Szabo. American philosophical quarterly, vol. 1 (1964), pp. 288–306, and vol. 2 (1965), pp. 204–218. - Paul Bernays. Introduction.Therein, vol. 1, p. 288.

Gerhard Gentzen. Investigations into logical deduction. English translation of 4422 by M. E. Szabo. American philosophical quarterly, vol. 1 (1964), pp. 288–306, and vol. 2 (1965), pp. 204–218. - Paul Bernays. Introduction.Therein, vol. 1, p. 288. - Volume 35 Issue 1

Gerhard Gentzen. Investigations into logical deduction. English translation of 4422 by M. E. Szabo. American philosophical quarterly, vol. 1 (1964), pp. 288–306, and vol. 2 (1965), pp. 204–218. - Paul Bernays. Introduction. Therein, vol. 1, p. 288.

Gerhard Gentzen. Investigations into logical deduction. English translation of 4422 by M. E. Szabo. American philosophical quarterly, vol. 1 (1964), pp. 288–306, and vol. 2 (1965), pp. 204–218. - Paul Bernays. Introduction. Therein, vol. 1, p. 288.

REMARKS TO "THE END OF A PHASE"

DialecticaVolume 17, Issue 1 p. 49-50 REMARKS TO “THE END OF A PHASE” Paul Bernays, Paul Bernays ZürichSearch for more papers by this author Paul Bernays, Paul Bernays ZürichSearch for more papers by this author First published: March 1963 https://doi.org/10.1111/j.1746-8361.1963.tb01442.xCitations: 1AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat No abstract is available for this article.Citing Literature Volume17, Issue1March 1963Pages 49-50 RelatedInformation