John von Neumann's proof that quantum mechanics is logically incompatible with hidden varibales has been the object of extensive study both by physicists and by historians. The latter have concentrated mainly on the way the proof was interpreted, accepted and rejected between 1932, when it was published, and 1966, when J.S. Bell published the first explicit identification of the mistake it involved. What is proposed in this paper is an investigation into the origins of the proof rather (...) than the aftermath. In the first section, a brief overview of the his personal life and his proof is given to set the scene. There follows a discussion on the merits of using here the historical method employed elsewhere by Andrew Warwick. It will be argued that a study of the origins of von Neumann's proof shows how there is an interaction between the following factors: the broad issues within a specific culture, the learning process of the theoretical physicist concerned, and the conceptual techniques available. In our case, the ‘conceptual technology’ employed by von Neumann is identified as the method of axiomatisation. (shrink)

According to what has become a standard history of quantum mechanics, von Neumann in 1932 succeeded in convincing the physics community that he had proved that hidden variables were impossible as a matter of principle. Subsequently, leading proponents of the Copenhagen interpretation emphatically confirmed that von Neumann's proof showed the completeness of quantum mechanics. Then, the story continues, Bell in 1966 finally exposed the proof as seriously and obviously wrong; this rehabilitated hidden variables and made serious foundational research (...) possible. It is often added in recent accounts that von Neumann's error had been spotted almost immediately by Grete Hermann, but that her discovery was of no effect due to the dominant Copenhagen Zeitgeist. We shall attempt to tell a more balanced story. Most importantly, von Neumann did not claim to have shown the impossibility of hidden variables tout court, but argued that hidden-variable theories must possess a structure that deviates fundamentally from that of quantum mechanics. Both Hermann and Bell appear to have missed this point; moreover, both raised unjustified technical objections to the proof. Von Neumann's conclusion was basically that hidden-variables schemes must violate the "quantum principle" that all physical quantities are to be represented by operators in a Hilbert space. According to this conclusion, hidden-variables schemes are possible in principle but necessarily exhibit a certain kind of contextuality. As we shall illustrate, early reactions to Bohm's theory are in agreement with this account. Leading physicists pointed out that Bohm's theory has the strange feature that particle properties do not generally reveal themselves in measurements, in accordance with von Neumann's result. They did not conclude that the "impossible was done" and that von Neumann had been shown wrong. (shrink)

Since the analysis by John Bell in 1965, the consensus in the literature is that von Neumann’s ‘no hidden variables’ proof fails to exclude any significant class of hidden variables. Bell raised the question whether it could be shown that any hidden variable theory would have to be nonlocal, and in this sense ‘like Bohm’s theory.’ His seminal result provides a positive answer to the question. I argue that Bell’s analysis misconstrues von Neumann’s argument. What von Neumann (...) proved was the impossibility of recovering the quantum probabilities from a hidden variable theory of dispersion free (deterministic) states in which the quantum observables are represented as the ‘beables’ of the theory, to use Bell’s term. That is, the quantum probabilities could not reflect the distribution of pre-measurement values of beables, but would have to be derived in some other way, e.g., as in Bohm’s theory, where the probabilities are an artefact of a dynamical process that is not in fact a measurement of any beable of the system. (shrink)

According to a popular narrative, in 1932 von Neumann introduced a theorem that intended to be a proof of the impossibility of hidden variables in quantum mechanics. However, the narrative goes, Bell later spotted a flaw that allegedly shows its irrelevance. Bell’s widely accepted criticism has been challenged by Bub and Dieks: they claim that the proof shows that viable hidden variables theories cannot be theories in Hilbert space. Bub’s and Dieks’ reassessment has been in turn challenged by (...) Mermin and Schack. Hereby I critically assess their reply, with the aim of bringing further clarification concerning the meaning, scope and relevance of von Neumann’s theorem. I show that despite Mermin and Schack’s response, Bub’s and Dieks’ reassessment is quite correct, and that this reading gets strongly reinforced when we carefully consider the connection between von Neumann’s proof and Gleason’s theorem. (shrink)

Shortly after Kurt Gödel had announced an early version of the 1st incompleteness theorem, John von Neumann wrote a letter to inform him of a remarkable discovery, i.e. that the consistency of a formal system containing arithmetic is unprovable, now known as the 2nd incompleteness theorem. Although today von Neumann’s proof of the theorem is considered lost, recent literature has explored many of the issues surrounding his discovery. Yet, one question still awaits a satisfactory answer: how did von (...) Neumann achieve his result, knowing as little as he seemingly did about the 1st incompleteness theorem? In this article, I shall advance a conjectural argument to answer this question, after having rejected the argument widely shared in the literature and having analyzed the relevant documents surrounding his discovery. The argument I shall advance strictly links two of the three letters written by von Neumann to Gödel in the late 1930 and early 1931 (i.e. respectively that of November 20, 1930 and that of January 12, 1931) and finds the key for von Neumann’s discovery in his prompt understanding of the Gödel sentence A – as the documents refer to it – as expressing consistency for a formal system that contains arithmetic. (shrink)

We discuss the content and significance of John von Neumann’s quantum ergodic theorem (QET) of 1929, a strong result arising from the mere mathematical structure of quantum mechanics. The QET is a precise formulation of what we call normal typicality, i.e., the statement that, for typical large systems, every initial wave function ψ0 from an energy shell is “normal”: it evolves in such a way that |ψt ψt| is, for most t, macroscopically equivalent to the micro-canonical density matrix. The (...) QET has been mostly forgotten after it was criticized as a dynamically vacuous statement in several papers in the 1950s. However, we point out that this criticism does not apply to the actual QET, a correct statement of which does not appear in these papers, but to a different (indeed weaker) statement. Furthermore, we formulate a stronger statement of normal typicality, based on the observation that the bound on the deviations from the average specified by von Neumann is unnecessarily coarse and a much tighter (and more relevant) bound actually follows from his proof. (shrink)

This paper reviews some major episodes in the history of the spatial isomorphism problem of dynamical systems theory. In particular, by analysing, both systematically and in historical context, a hitherto unpublished letter written in 1941 by John von Neumann to Stanislaw Ulam, this paper clarifies von Neumann's contribution to discovering the relationship between spatial isomorphism and spectral isomorphism. The main message of the paper is that von Neumann's argument described in his letter to Ulam is the very first proof (...) that spatial isomorphism and spectral isomorphism are not equivalent because spectral isomorphism is weaker than spatial isomorphism: von Neumann shows that spectrally isomorphic ergodic dynamical systems with mixed spectra need not be spatially isomorphic. (shrink)

1. The theory of classes presented in this paper is a simplification of that presented by J. von Neumann in his paper Die Axiomatisierung der Mengenlehre. However, this paper is written so that it can be read independently of von Neumann's. The principal modifications of his system are the following.(1) The idea of ordered pair is defined in terms of the other primitive concepts of the system. (See Axiom 4.3 below.)(2) A much simpler proof of the well-ordering theorem, based (...) on von Neumann's equivalence axiom (Axiom 2.2 below) is given. (More exactly, the theory of ordinal numbers, on which the proof of the well-ordering theorem is based, is simplified—see §8.)(3) Functions are not assumed to be defined for all arguments. In place of “ = A” we have “is undefined.” This makes possible a constructive interpretation of the system. (See §2.)2. We wish first to give a rough picture of the system which we are trying to construct. Suppose that we have the idea “class,” but no material from which to construct classes. Nevertheless, we can construct the class 0 having no elements. And then the class 1 having 0 as its only element. And then the classes {1}, and {0, 1}, etc., using always the previously constructed classes as elements. To extend this method to infinite classes, we must give rules telling what elements are to be included. Since the nature of the rules which we can use is not altogether clear, we try to formalize the whole system; that is, to set up a system of axioms which seems to characterize this system of classes. Now in this set of axioms we find it necessary to use the idea of function (in the “Axiom of Replacement”). (shrink)

The strange story of the von Neumann impossibility proof is recalled, and the even stranger story of later impossibility proofs, and how the impossible was done by de Broglie and Bohm. Morals are drawn.

Concerns a problem posed, but not solved, by Frege in part III of his Grundgesetze. As a preliminary to defining ‘real number’, Frege attempts to analyse the notion of a quantitative domain. He was unaware of the previous attempt of Otto Holder to do this; it is remarked how much weaker Frege's assumptions were in deriving theorems than Holder's. Frege deals with groups on which there is a right‐invariant semilinear ordering, although he does not use this terminology. He is uncertain (...) whether it can be deduced that the ordering is also left‐invariant, and proves as much as possible without invoking the assumption that it is. The independence proof, due to Dr Peter Neumann, establishes that it could not be deduced. (shrink)

Arrow’s axiomatic foundation of social choice theory can be understood as an application of Tarski’s methodology of the deductive sciences—which is closely related to the latter’s foundational contribution to model theory. In this note we show in a model-theoretic framework how Arrow’s use of von Neumann and Morgenstern’s concept of winning coalitions allows to exploit the algebraic structures involved in preference aggregation; this approach entails an alternative indirect ultrafilter proof for Arrow’s dictatorship result. This link also connects Arrow’s seminal (...) result to key developments and concepts in the history of model theory, notably ultraproducts and preservation results. (shrink)

John von Neumann was among the first to learn about Kurt Gödel’s results on the incompleteness of formal systems. Did this shape his views on the completeness of quantum mechanics? I will investigate this question from two viewpoints: von Neumann’s no-hidden-variables proof and his treatment of the quantum measurement problem.

Hilbert's axiomatization program of physical theories met an interesting challenge when it confronted the rise of quantum mechanics in the mid-twenties. The novelty of the mathematical apparatus of the then newly born theory was to be matched only by its substantial lack of any definite physical interpretation. The early attempts at axiomatization, which are described here, reflect all the difficulty of the task faced by Jordan, Hilbert, von Neumann and others. The role of von Neumann is examined in considerable detail (...) as he can be viewed here as the most outstanding of Hilbert's heirs. Von Neumann, especially in his work devoted to the proof of the impossibility of hidden variables, not only continued Hilbert's program but pushed it to its very limits, blending axiomatic rigor and interpretative commitment. (shrink)

It is a widespread belief that the Kochen-Specker theorem imposes a contextuality constraint on the ontology of beables in quantum hidden variables theories. On the other hand, after Bell’s influential critique, the importance of von Neumann’s wrongly called ‘impossibility proof’ has been severely questioned. However, Max Jammer, Jeffrey Bub and Dennis Dieks have proposed insightful reassessments of von Neumann’s theorem: what it really shows is that hidden variables theories cannot represent their beables by means of Hermitian operators (...) in Hilbert space. Hereby I show that i) the very same constraint can be derived from Gleason’s theorem, and that ii) if we consider the import of von Neumann’s and Gleason’s theorems, the relevance of the Kochen-Specker theorem for hidden variables theories gets substantially weakened: it does not force them to be contextual in any interesting sense of the term. (shrink)

In 1930, Johann von Neumann, together with Rudolf Carnap and Arend Heyting, participated in a conference held in Königsberg, called “Second Seminar on the Epistemology of Exact Sciences”. The idea behind the reunion of these three researchers was to compose a fairly faithful picture of the three main foundational programs of mathematics at the time: formalism, logicism, and intuitionism. The main objective of this paper is to propose an analysis of the text “The Formalist Foundation of Mathematics” presented by von (...) Neumann in this conference. We show how, from von Neumann's perspective, the problem of the consistency of mathematical theories results from the transformation of a genuinely philosophical question into a genuinely logical-mathematical question and how a proof of consistency could be seen as the solution to the foundational problem. In the final part of the text, we present some considerations about Gerhard Gentzen’s consistency proofs and their importance to contemporary proof theory. (shrink)

Gentzen writes in the published version of his doctoral thesis Untersuchungen über das logische Schliessen 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. (shrink)

Gentzen's systems of natural deduction and sequent calculus were byproducts in his program of proving the consistency of arithmetic and analysis. It is suggested that the central component in his results on logical calculi was the use of a tree form for derivations. It allows the composition of derivations and the permutation of the order of application of rules, with a full control over the structure of derivations as a result. Recently found documents shed new light on the discovery of (...) these calculi. In particular, Gentzen set up five different forms of natural calculi and gave a detailed proof of normalization for intuitionistic natural deduction. An early handwritten manuscript of his thesis shows that a direct translation from natural deduction to the axiomatic logic of Hilbert and Ackermann was, in addition to the influence of Paul Hertz, the second component in the discovery of sequent calculus. A system intermediate between the sequent calculus LI and axiomatic logic, denoted LIG in unpublished sources, is implicit in Gentzen's published thesis of 1934—35. The calculus has half rules, half “groundsequents,” and does not allow full cut elimination. Nevertheless, a translation from LI to LIG in the published thesis gives a subformula property for a complete class of derivations in LIG. After the thesis, Gentzen continued to work on variants of sequent calculi for ten more years, in the hope to find a consistency proof for arithmetic within an intuitionistic calculus. (shrink)

Gentzen writes in the published version of his doctoral thesis Untersuchungen über das logische Schliessen 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. (shrink)

Conventional wisdom holds that the von Neumann entropy corresponds to thermodynamic entropy, but Hemmo and Shenker (2006) have recently argued against this view by attacking von Neumann's (1955) argument. I argue that Hemmo and Shenker's arguments fail due to several misunderstandings: about statistical-mechanical and thermodynamic domains of applicability, about the nature of mixed states, and about the role of approximations in physics. As a result, their arguments fail in all cases: in the single-particle case, the finite particles case, and the (...) infinite particles case. (shrink)

As interpreted by Pattee, von Neumann’s Theory of Self-Reproducing Automata has proved to be a useful tool for understanding some of the difficulties and paradoxes of molecular biosemiotics. But is its utility limited to molecular systems or is it more generally applicable within biosemiotics? One way of answering that question is to look at the Theory as a model for one particular high-level biosemiotic activity, human language. If the model is not useful for language, then it certainly cannot be (...) generally useful to biosemiotics. Beginning with the Universal Turing Machine and continuing with von Neumann’s Theory and Pattee’s interpretation, the properties of universality, programmability, underspecification, complementarity of description/construction, and open-ended evolutionary potential are shown to be usefully applicable to language, thus opening a new line of inquiry in biosemiotics. (shrink)

Der Band enthält Beiträge von Juristen, Medizinern und Philosophen aus Australien, Estland, Polen, Rußland, Tschechien, den U.S.A. und Deutschland zu Themen der Ethik und Wissenschaftstheorie der Medizin. Die Mehrzahl der Beiträge sind von Nachwuchswissenschaftlern des College for Advanced Central European Studies an der Europa-Universität Viadrina erarbeitet worden. Sie wurden im Rahmen der Jahrestagung des Arbeitskreises für Ethik und Wissenschaftstheorie der Medizin in Ostmitteleuropa neben weiteren Beiträgen, die hier zum Abdruck kommen, zur Diskussion gestellt. Der Arbeitskreis beruht auf einer Kooperationsvereinbarung des (...) Interdisziplinären Zentrums für Ethik der Europa-Universität Viadrina Frankfurt (Oder) und des Instituts für Geschichte und Ethik der Medizin der Universität Halle-Wittenberg. Die Beiträge befassen sich u. a. mit der Gesundheitsreform in Polen, mit ethischen und rechtlichen Problemen bei der künstlichen Befruchtung, der Abtreibung, medizinischen Experimenten an Menschen, der Gentechnik, der Patientenautonomie, Außenseitermethoden in der Medizin, dem Patiententestament, dem Todesbegriff, Drogenkonsum, der Sterbehilfe und der Gentechnik. (shrink)

This book continues from where the authors' previous book, Structural Proof Theory, ended. It presents an extension of the methods of analysis of proofs in pure logic to elementary axiomatic systems and to what is known as philosophical logic. A self-contained brief introduction to the proof theory of pure logic is included that serves both the mathematically and philosophically oriented reader. The method is built up gradually, with examples drawn from theories of order, lattice theory and elementary geometry. (...) The aim is, in each of the examples, to help the reader grasp the combinatorial behaviour of an axiom system, which typically leads to decidability results. The last part presents, as an application and extension of all that precedes it, a proof-theoretical approach to the Kripke semantics of modal and related logics, with a great number of new results, providing essential reading for mathematical and philosophical logicians. (shrink)

What seem to be Kurt Gödel’s first notes on logic, an exercise notebook of 84 pages, contains formal proofs in higher-order arithmetic and set theory. The choice of these topics is clearly suggested by their inclusion in Hilbert and Ackermann’s logic book of 1928, the Grundzüge der theoretischen Logik. Such proofs are notoriously hard to construct within axiomatic logic. Gödel takes without further ado into use a linear system of natural deduction for the full language of higher-order logic, with formal (...) derivations closer to one hundred steps in length and up to four nested temporary assumptions with their scope indicated by vertical intermittent lines. (shrink)

This is the classic work upon which modern-day game theory is based. What began as a modest proposal that a mathematician and an economist write a short paper together blossomed, when Princeton University Press published Theory of Games and Economic Behavior. In it, John von Neumann and Oskar Morgenstern conceived a groundbreaking mathematical theory of economic and social organization, based on a theory of games of strategy. Not only would this revolutionize economics, but the entirely new field of scientific inquiry (...) it yielded--game theory--has since been widely used to analyze a host of real-world phenomena from arms races to optimal policy choices of presidential candidates, from vaccination policy to major league baseball salary negotiations. And it is today established throughout both the social sciences and a wide range of other sciences. (shrink)

Gentzen's original proof of the Hauptsatz used a rule of multicut in the case that the right premiss of cut was derived by contraction. Cut elimination is here proved without multicut, by transforming suitably the derivation of the premiss of the contraction.

Von Neumann argued by means of a thought experiment involving measurements of spin observables that the quantum mechanical quantity is conceptually equivalent to thermodynamic entropy. We analyze Von Neumann's thought experiment and show that his argument fails. Over the past few years there has been a dispute in the literature regarding the Von Neumann entropy. It turns out that each contribution to this dispute addressed a different special case. In this paper we generalize the discussion and examine the full matrix (...) of possibilities that are relevant for the evaluation and understanding of Von Neumann’s argument. (shrink)

This book represents the views of one of the greatest mathematicians of the twentieth century on the analogies between computing machines and the living human brain.

In spite of the many efforts made to clarify von Neumann’s methodology of science, one crucial point seems to have been disregarded in recent literature: his closeness to Hilbert’s spirit. In this paper I shall claim that the scientific methodology adopted by von Neumann in his later foundational reflections originates in the attempt to revaluate Hilbert’s axiomatics in the light of Gödel’s incompleteness theorems. Indeed, axiomatics continues to be pursued by the Hungarian mathematician in the spirit of Hilbert’s school. (...) I shall argue this point by examining four basic ideas embraced by von Neumann in his foundational considerations: a) the conservative attitude to assume in mathematics; b) the role that mathematics and the axiomatic approach have to play in all that is science; c) the notion of success as an alternative methodological criterion to follow in scientific research; d) the empirical and, at the same time, abstract nature of mathematical thought. Once these four basic ideas have been accepted, Hilbert’s spirit in von Neumann’s methodology of science will become clear. (shrink)

The Löwenheim-Skolem theorem was published in Skolem's long paper of 1920, with the first section dedicated to the theorem. The second section of the paper contains a proof-theoretical analysis of derivations in lattice theory. The main result, otherwise believed to have been established in the late 1980s, was a polynomial-time decision algorithm for these derivations. Skolem did not develop any notation for the representation of derivations, which makes the proofs of his results hard to follow. Such a formal notation (...) is given here by which these proofs become transparent. A third section of Skolem's paper gives an analysis for derivations in plane projective geometry. To clear a gap in Skolem's result, a new conservativity property is shown for projective geometry, to the effect that a proper use of the axiom that gives the uniqueness of connecting lines and intersection points requires a conclusion with proper cases (logically, a disjunction in a positive part) to be proved. The forgotten parts of Skolem's first paper on the Löwenheim-Skolem theorem are the perhaps earliest combinatorial analyses of formal mathematical proofs, and at least the earliest analyses with profound results. (shrink)

Much of the recent discussion of problematic aspects of quantum-mechanical measurement centers around that feature of quantum theory which is called "the projection postulate." This is roughly the claim that a change of a certain sort occurs in the state of a physical system when a measurement is made on the system. In this paper an argument for the projection postulate due to von Neumann is considered. Attention is focused on trying to provide an understanding of the notion of "the (...) state of a physical system" which is compatible with the argument von Neumann offers. An attempt is made to formulate the argument in terms of an objectivistic interpretation of probability concepts. It is seen that such an interpretation does not provide a suitable way of understanding the argument. An attempt is made to illustrate the source of this failure in terms of a non-quantum-mechanical example. (shrink)

The idea of quantum logic first appears explicitly in the short Section 5 of Chapter III. in von Neumann’s 1932 book on the mathematical foundations of quantum mechanics [31]; however, the real birthplace of quantum logic is commonly identified with the 1936 seminal paper co-authored by G. Birkhoff and J. von Neumann [5]. The aim of this review is to recall the main idea of the Birkhoff-von Neumann concept1 of quantum logic as this was put forward in the 1936 (...) paper. The review is motivated partly by two facts related to quantum logic: one, peculiar, is that the 1936 von Neumann concept is an almost totally neglected2 topic in the enormous quantum logic literature [17]; the other, not very well-known, is that von Neumann was never completely satisfied with how he had worked out quantum logic. (shrink)

A proof-theoretical analysis of elementary theories of order relations is effected through the formulation of order axioms as mathematical rules added to contraction-free sequent calculus. Among the results obtained are proof-theoretical formulations of conservativity theorems corresponding to Szpilrajn’s theorem on the extension of a partial order into a linear one. Decidability of the theories of partial and linear order for quantifier-free sequents is shown by giving terminating methods of proof-search.

In a series of lectures written around 1952, Schrödinger refers to von Neumann’s account of measurement in quantum mechanics as follows:I said quantum physicists bother very little about accounting, according to the accepted law, for the supposed change of the wave-function by measurement. I know of only one attempt in this direction, to which Dr. Balazs recently directed my attention. You find it in John von Neumann’s well-known book. With great acuity he constructs one analytical example. It does (...) not refer to any actual experiment, it is purely analytical. He indicates in a simple case a supplementary operator which, when added to the internal wave operator, would with any desired approximation turn the wave function as time goes on into an eigenfunction of the observable that is measured. He found it necessary to show that such a mechanism is analytically possible. The idea has not been taken up and worked out since — in about twenty years or more. Indeed I do not think it would pay. I do not believe any real measuring device is of this kind. (shrink)

In preparation for his article on Russell’s mathematical logic (1944), Gödel read carefully Peano’s Arithmetices Principia. His six pages of summary in the Gabelsberger shorthand contain a remarkable analysis of the formal structure of Peano’s proofs which is diametrically opposed to the common view that Peano’s treatise contained no formal deductive machinery.

In preparation for his article on Russell’s mathematical logic, Gödel read carefully Peano’s Arithmetices Principia. His six pages of summary in the Gabelsberger shorthand contain a remarkable analysis of the formal structure of Peano’s proofs which is diametrically opposed to the common view that Peano’s treatise contained no formal deductive machinery.

Gentzen writes in the published version of his doctoral thesis Untersuchungen über das logische Schliessen 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. (shrink)

A way is found to add axioms to sequent calculi that maintains the eliminability of cut, through the representation of axioms as rules of inference of a suitable form. By this method, the structural analysis of proofs is extended from pure logic to free-variable theories, covering all classical theories, and a wide class of constructive theories. All results are proved for systems in which also the rules of weakening and contraction can be eliminated. Applications include a system of predicate logic (...) with equality in which also cuts on the equality axioms are eliminated. (shrink)

We review the famous no-hidden-variables theorem in von Neumann’s 1932 book on the mathematical foundations of quantum mechanics. We describe the notorious gap in von Neumann’s argument, pointed out by Hermann and, more famously, by Bell. We disagree with recent papers claiming that Hermann and Bell failed to understand what von Neumann was actually doing.

This paper surveys John von Neumann's work on the mathematical foundations of quantum theories in the light of Hilbert's Sixth Problem concerning the geometrical axiomatization of physics. We argue that in von Neumann's view geometry was so tied to logic that he ultimately developed a logical interpretation of quantum probabilities. That motivated his abandonment of Hilbert space in favor of von Neumann algebras, specifically the type II1II1 factors, as the proper limit of quantum mechanics in infinite dimensions. Finally, we present (...) the reasons why his axiomatic program remained an “unsolved problem” in mathematical physics. A recent unpublished result by Huzimiro Araki, proving that no algebra with a tracial state defined on it, such as the type II1II1 factors, can support any (regular) representation of the canonical commutation relations, is also reviewed and its consequences for von Neumann's projects are discussed. (shrink)