Harsanyi claimed that his Aggregation and Impartial Observer Theorems provide a justification for utilitarianism. This claim has been strongly resisted, notably by Sen and Weymark, who argue that while Harsanyi has perhaps shown that overall good is a linear sum of individuals’ von Neumann-Morgenstern utilities, he has done nothing to establish any con- nection between the notion of von Neumann-Morgenstern utility and that of well-being, and hence that utilitarianism does not follow. The present article defends Harsanyi (...) against the Sen-Weymark cri- tique. I argue that, far from being a term with precise and independent quantitative content whose relationship to von Neumann-Morgenstern utility is then a substantive question, terms such as ‘well-being’ suffer (or suffered) from indeterminacy regarding precisely which quantity they refer to. If so, then (on the issue that this article focuses on) Harsanyi has gone as far towards defending ‘utilitarianism in the original sense’ as could coherently be asked. (shrink)

Two scenarios illustrate uses of von Neumann–Morgenstern stable sets in the construction of choice functions. A comparison is made to the construction of choice functions by the selection of maximal elements. A characterization is given of choice functions that are von Neumann–Morgenstern stable set rationalizable by acyclic, asymmetric binary relations. Two examples illustrate the use of the characterization.

Sequential von Neumann–Morgernstern (VM) games are a very general formalism for representing multi-agent interactions and planning problems in a variety of types of environments. We show that sequential VM games with countably many actions and continuous utility functions have a sound and complete axiomatization in the situation calculus. This axiomatization allows us to represent game-theoretic reasoning and solution concepts such as Nash equilibrium. We discuss the application of various concepts from VM game theory to the theory of planning and (...) multi-agent interactions, such as representing concurrent actions and using the Baire topology to define continuous payoff functions. (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)

This up-to-date introduction to decision theory offers comprehensive and accessible discussions of decision-making under ignorance and risk, the foundations of utility theory, the debate over subjective and objective probability, Bayesianism, causal decision theory, game theory, and social choice theory. No mathematical skills are assumed, and all concepts and results are explained in non-technical and intuitive as well as more formal ways. There are over 100 exercises with solutions, and a glossary of key terms and concepts. An emphasis on foundational aspects (...) of normative decision theory (rather than descriptive decision theory) makes the book particularly useful for philosophy students, but it will appeal to readers in a range of disciplines including economics, psychology, political science and computer science. • Has over 100 end of chapter review questions and exercises with solutions • Includes a chapter on how to draw a decision matrix • Explains the link between individual decision making, game theory and social choice theory Contents Preface; 1. Introduction; 2. The decision matrix; 3. Decisions under ignorance; 4. Decisions under risk; 5. Utility; 6. The mathematics of probability; 7. The philosophy of probability; 8. Why should we accept the preference axioms; 9. Causal vs. evidential decision theory; 10. Bayesian vs. non-Bayesian decision theory; 11. Game theory I: basic concepts and zero sum games; 12. Game theory II: nonzero sum and co-operative games; 13. Social choice theory; 14. Overview of descriptive decision theory; Appendix A. Glossary; Appendix B. Proof of the von Neumann-Morgenstern theorem; Further reading; Index. (shrink)

1.1 In 1955, John Harsanyi proved a remarkable theorem:1 Suppose n agents satisfy the assumptions of von Neumann/Morgenstern (1947) expected utility theory, and so does the group as a whole (or an observer). Suppose that, if each member of the group prefers option a to b, then so does the group, or the observer (Pareto condition). Then the group’s utility function is a weighted sum of the individual utility functions. Despite Harsanyi’s insistence that what he calls the (...) Utilitarian Theorem embeds utilitarianism into a theory of rationality, the theorem has fallen short of having the kind of impact on the discussion of utilitarianism for which Harsanyi hoped. Yet how could the theorem influence this discussion? Utilitarianism is as attractive to some as it is appalling to others. The prospects for this dispute to be affected by a theorem seem dim. Yet a closer look shows how the theorem could make a contribution. To fix ideas, I understand by utilitarianism the following claims: (1) Consequentialism: Actions are evaluated in terms of their consequences only. (2) Bayesianism: An agent's beliefs about possible outcomes are captured probabilistically. (3) Welfarism: The judgement of the relative goodness of states of affairs is based.. (shrink)

Book Reviews Paul Erickson, Economics and Philosophy, FirstView Article.

Although expected utility theory has proven a fruitful and elegant theory in the finite realm, attempts to generalize it to infinite values have resulted in many paradoxes. In this paper, we argue that the use of John Conway's surreal numbers shall provide a firm mathematical foundation for transfinite decision theory. To that end, we prove a surreal representation theorem and show that our surreal decision theory respects dominance reasoning even in the case of infinite values. We then bring our (...) theory to bear on one of the more venerable decision problems in the literature: Pascal's Wager. Analyzing the wager showcases our theory's virtues and advantages. To that end, we analyze two objections against the wager: Mixed Strategies and Many Gods. After formulating the two objections in the framework of surreal utilities and probabilities, our theory correctly predicts that (1) the pure Pascalian strategy beats all mixed strategies, and (2) what one should do in a Pascalian decision problem depends on what one's credence function is like. Our analysis therefore suggests that although Pascal's Wager is mathematically coherent, it does not deliver what it purports to, a rationally compelling argument that people should lead a religious life regardless of how confident they are in theism and its alternatives. (shrink)

How ought you to evaluate your options if you’re uncertain about which axiology is true? One prominent response is Expected Moral Value Maximisation, the view that under axiological uncertainty, an option is better than another if and only if it has the greater expected moral value across axiologies. EMVM raises two fundamental questions. First, there’s a question about what it should even mean. In particular, it presupposes that we can compare moral value across axiologies. So to even understand EMVM, we (...) need to explain what it is for such comparisons to hold. Second, assuming that we understand it, there’s a question about whether EMVM is true. Since there are many plausible rivals, we need an argument to defend it. In this paper, I’ll introduce a representation theorem for axiological uncertainty to answer these two questions. Roughly, the theorem shows that if all our axiologies satisfy the von Neumann–Morgenstern axioms, and if the facts about which options are better than which in light of your uncertainty also satisfy these axioms as well as a Pareto condition, then these facts have a relevantly unique expected utility representation. If I’m right, this theorem at once affords us a compelling way to understand EMVM—and specifically intertheoretic comparisons—and a systematic argument for its truth. (shrink)

In his classic book “the Foundations of Statistics” Savage developed a formal system of rational decision making. The system is based on (i) a set of possible states of the world, (ii) a set of consequences, (iii) a set of acts, which are functions from states to consequences, and (iv) a preference relation over the acts, which represents the preferences of an idealized rational agent. The goal and the culmination of the enterprise is a representation theorem: Any preference relation (...) that satisfies certain arguably acceptable postulates determines a (finitely additive) probability distribution over the states and a utility assignment to the consequences, such that the preferences among acts are determined by their expected utilities. Additional problematic assumptions are however required in Savage's proofs. First, there is a Boolean algebra of events (sets of states) which determines the richness of the set of acts. The probabilities are assigned to members of this algebra. Savage's proof requires that this be a σ-algebra (i.e., closed under infinite countable unions and intersections), which makes for an extremely rich preference relation. On Savage's view we should not require subjective probabilities to be σ-additive. He therefore finds the insistence on a σ-algebra peculiar and is unhappy with it. But he sees no way of avoiding it. Second, the assignment of utilities requires the constant act assumption: for every consequence there is a constant act, which produces that consequence in every state. This assumption is known to be highly counterintuitive. The present work contains two mathematical results. The first, and the more difficult one, shows that the σ-algebra assumption can be dropped. The second states that, as long as utilities are assigned to finite gambles only, the constant act assumption can be replaced by the more plausible and much weaker assumption that there are at least two non-equivalent constant acts. The second result also employs a novel way of deriving utilities in Savage-style systems -- without appealing to von Neumann-Morgenstern lotteries. The paper discusses the notion of “idealized agent" that underlies Savage's approach, and argues that the simplified system, which is adequate for all the actual purposes for which the system is designed, involves a more realistic notion of an idealized agent. (shrink)

This paper offers a modified version of the certainty equivalence (CE) theory of utility for uncertain prospects and a new set of axioms as its basis. It shows that the CE and the von Neumann-Morgenstern (NM) approaches to uncertainty are opposite in spirit: The CE approach represents a flight from the world of uncertainty to the rules of certainty while the NM approach represents a flight from the world of certainty to one of uncertainty. The two approaches differ (...) even in their treatment of compound prospects and their actuarially identical simple counterparts. (shrink)

Ever since the Harsanyi-Sen debate, it is controversial whether someone's welfare should be measured by her von-Neumann-Morgenstern (VNM) utility, for instance when analysing welfare intensity, social welfare, interpersonal welfare comparisons, or welfare inequality. We prove that natural working hypotheses lead to a different welfare measure. It addresses familiar concerns about VNM utility, by faithfully capturing non-ordinal welfare features such as welfare intensity, despite resting on purely ordinal evidence such as revealed preferences or self-reported welfare comparisons. Using this welfare (...) measure instead of VNM utility alters social welfare analysis -- for instance, Harsanyi's `utilitarian theorem' now effectively supports prioritarianism. VNM utility is shown to be a hybrid object, determined by an interplay of two factors: welfare and attitude to intrinsic risk, i.e., to risk in welfare rather than outcomes. (shrink)

We revisit classical Utilitarianism by connecting and generalizing two ideas. The first is that there is a representation theorem possible for hedonic value similar to, but also importantly different from, the one provided by von Neumann and Morgenstern to measure decision utility. The idea is to use objective time, in place of objective chance, to measure hedonic value. This representation for hedonic value delivers a stronger kind of scale than von Neumann–Morgenstern utility, a ratio scale (...) rather than merely an interval scale. The second idea is that measurement on a ratio scale allows the meaningful aggregation of utilities over a group. This is aggregation by product rather than sum. Aggregation by product is known to have interesting Prioritarian consequences. Aggregation becomes complicated when the two approaches are mixed, when hedonic value is mixed with uncertainly. It becomes problematic when pain as well as pleasure is taken into account. (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)

My dissertation focuses on problems that arise when a group makes decisions that are in reasonable ways connected to the beliefs and values of the group members. These situations are represented by models of decision-theoretic aggregation: Suppose a model of individual rationality in decision-making applies to each of a group of agents. Suppose this model also applies to the group as a whole, and that this group model is aggregated from the individual models. Two questions arise. First, what sets of (...) reasonable conditions can we consistently impose on the aggregation? Second, what can we learn from insights gained in such models, in particular for ethics and political philosophy? The dissertation has three parts, each using a different model of individual rationality. Each part contains essays addressing questions of either the first or the second kind with regard to its underlying model. Part I uses a social-choice-theoretic model, in which agents are only assumed to rank alternatives. Arrow's "Impossibility Theorem" shows that certain conditions that it would be natural to impose on all aggregation cannot be so imposed, on pain of contradiction. I discuss the impact of this result on the evaluation of majority rule and on democratic theory. I also discuss Sen's "Liberal Paradox", which addresses the possibility of incorporating rights into a social-choice theoretic model. Part II uses the von Neumann/Morgenstern theory of expected utility. A result under this model has been regarded as providing an argument for utilitarianism. I locate a place within utilitarianism where the theorem can do some work. To prepare the ground, I interpret the von Neumann/Morgenstern theory as part of a theory of deliberation. Thereby, this version of expected utility theory can be embedded into the philosophy of practical reasoning. The underlying model in Part III is Bayesian decision theory. Whereas the other two inquiries focus on the philosophical assessment of formal results, the third makes a formal contribution. One essay surveys the area, and a second explores a curious phenomenon that occurs in determining outcomes under one type of Bayesian aggregation. (shrink)

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 issues, raised in Żukowski (arXiv:0809.0115v1, 2008), concerning the relevance of the von Neumann theorem for the single-system’s quantumness test proposed in Alicki and Van Ryn (J. Phys. A: Math. Theor. 41:062001, 2008) and performed for the case of single photon polarization in Brida et al. (Opt. Express 16:11750, 2008; arXiv:0811.3376, 2008) and the usefulness of Bell’s inequality for testing the idea of macroscopic quantum systems are discussed in some details. Finally, the proper quantum mechanical description of the (...) experiment with polarized photon beams is presented. (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)

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)

The present paper investigates the philosophical relationship between John von Neumann’s Nohidden-variable theorem and Bell’s inequalities. Bell erroneously takes the axiomatic method as implying a finality claim and thus ignores von Neumann’s strongly pragmatist stance towards mathematical physics. If one considers, however, Hilbert’s axiomatic method as a critical enterprise, Bell’s theorem improves von Neumann’s by defining a more appropriate notion of ‘ hidden variable’ that permits one to include Bohm’s interpretation which recovers the predictive content (...) of quantum mechanics. Contrary to Bell’s belief, accepting this model does not require adopting the metaphysically realist Böhm picture. If one takes the latter as a physical research programme one sees that it only partly disputes a common domain of facts with the mathematically oriented research programme of von Neumann. (shrink)

We show that the respective oversights in the von Neumann's general theorem against all hidden variable theories and Bell's theorem against their local-realistic counterparts are homologous. When latter oversight is rectified, the bounds on the CHSH correlator work out to be ±2√2 instead of ±2.

The literature of economic theory, like that of philosophy, abounds in prefaces and prolegomena. Methodology and analysis of concepts take an important place in a science which has not found the sure path of development. But there is no sure path for methodology either. The selfconscious methodology of social science has been largely a borrowing from that of physical science, where procedures have developed to a stage of considerable maturity. But the analogy falls down where guidance is most needed, at (...) the points where social science is most likely to develop new concepts and new types of structure. Philosophers have not been lacking, indeed, to belittle the entire enterprise, and to deny the possibility of anything that could strictly be called social science. (shrink)

People with the kind of preferences that give rise to the St. Petersburg paradox are problematic---but not because there is anything wrong with infinite utilities. Rather, such people cannot assign the St. Petersburg gamble any value that any kind of outcome could possibly have. Their preferences also violate an infinitary generalization of Savage's Sure Thing Principle, which we call the *Countable Sure Thing Principle*, as well as an infinitary generalization of von Neumann and Morgenstern's Independence axiom, which we (...) call *Countable Independence*. In violating these principles, they display foibles like those of people who deviate from standard expected utility theory in more mundane cases: they choose dominated strategies, pay to avoid information, and reject expert advice. We precisely characterize the preference relations that satisfy Countable Independence in several equivalent ways: a structural constraint on preferences, a representation theorem, and the principle we began with, that every prospect has a value that some outcome could have. (shrink)

The paper discusses some of the poorly explored links between the conceptual systems of logic in Kurt Gödel, the theory of automata in Alan Turing, and the theory of self-reproducing automata in John von Neumann. Traditional controversies are left aside (especially the opposition of Gödel and Turing in the view of mind) and attention is focused on the similarities between all three authors. In individual chapters, the text deals with: the form of differentiation of syntax and semantics in formal (...) system in Gödel, Turing and von Neumann; von Neumann’s variant of Gödel’s theorem and von Neumann’s and Gödel’s conception of Turing machine; and finally the same basis of the view of the relation between mind and automaton in all three authors. (shrink)

The renewed interest in the foundations of quantum statistical mechanics in recent years has led us to study John von Neumann’s 1929 article on the quantum ergodic theorem. We have found this almost forgotten article, which until now has been available only in German, to be a treasure chest, and to be much misunderstood. In it, von Neumann studied the long-time behavior of macroscopic quantum systems. While one of the two theorems announced in his title, the one (...) he calls the “quantum H-theorem,” is actually a much weaker statement than Boltzmann’s classical H-theorem, the other theorem, which he calls the “quantum ergodic theorem,” is a beautiful and very non-trivial result. It expresses a fact we call “normal typicality” and can be summarized as follows: For a “typical” finite family of commuting macroscopic observables, every initial wave function ψ0 from a micro-canonical energy shell so evolves that for most times t in the long run, the joint probability distribution of these observables obtained from ψ is close to.. (shrink)

We extend the topos-theoretic treatment given in previous papers of assigning values to quantities in quantum theory, and of related issues such as the Kochen-Specker theorem. This extension has two main parts: the use of von Neumann algebras as a base category (Section 2); and the relation of our generalized valuations to (i) the assignment to quantities of intervals of real numbers, and (ii) the idea of a subobject of the coarse-graining presheaf (Section 3).

The debate over the question whether quantum mechanics should be considered as a complete account of microphenomena has a long and deeply involved history, a turning point in which has been certainly the Einstein-Bohr debate, with the ensuing charge of incompleteness raised by the Einstein-Podolsky-Rosen argument. In quantum mechanics, physical systems can be prepared in pure states that nevertheless have in general positive dispersion for most physical quantities; hence in the EPR argument, the attention is focused on the question whether (...) the account of the microphysical phenomena provided by quantum mechanics is to be regarded as an exhaustive description of the physical reality to which those phenomena are supposed to refer, a question to which Einstein himself answered in the negative. However, there is a mathematical side of the completeness issue in quantum mechanics, namely the question whether the kind of states with positive dispersion can be represented as a different, dispersion-free kind of states in a way consistent with the mathematical constraints of the quantum mechanical formalism. From this point of view, the other source of the completeness issue in quantum mechanics is the no hidden variables theorem formulated by John von Neumann in his celebrated book on the mathematical foundations of quantum mechanics, the preface of which already anticipates the program and the conclusion concerning the possibility of ‘neutralizing’ the statistical character of quantum mechanics. (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.

I teoremi di Gödel suscitano un interesse sempre crescente nella riflessione filosofica contemporanea. Rimane però in discussione fra gli studiosi come si sia arrivati alla loro scoperta e quale sia il loro significato per il dibattito sui fondamenti delle scienze. Nel volume si ripercorrono le vicende che portarono alla formulazione dei teoremi di incompletezza, a partire dall’incontro tra von Neumann e Gödel al Congresso di Königsberg nel 1930, e si indaga, riferendosi in particolare al lavoro di von Neumann, (...) sull’impatto che questi teoremi hanno avuto per il progetto hilbertiano di fondazione della matematica e delle scienze più in generale. Ne risulta un’immagine delle indagini di Hilbert meno rigida di quella finora corrente, in linea con le più recenti acquisizioni della storiografia, e soprattutto più rispettosa della stessa prassi assiomatica, che oscillava per Hilbert tra ambizioni fondazionaliste e istanze pragmatiche. È stato von Neumann a rendere esplicita, dopo la scoperta dei teoremi di Gödel, quella che si può definire come la “svolta pragmatica” nell’assiomatica. Il libro propone inoltre un chiarimento riguardo alla scoperta di von Neumann del secondo teorema di incompletezza. (shrink)

Papers advocating a hidden-variable interpretation of quantum mechanics typically begin by emphasizing that John von Neumann’s no-go theorem does not apply to them. If authors are ontologically minded, their criticism also takes aim at his theory of measurement as expressed in his seminal 1932 book Mathematical Foundations of Quantum Mechanics Additionally, David Bohm and Basil Hiley have recently argued that “in so far as von Neumann effectively gave the quantum state a certain ontological significance, the net result (...) was to produce a confused and unsatisfactory ontology. This ontology is such that the collapse of the wave function must also have an ontological significance.”. (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)

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 (...) class='Hi'>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)

This article investigates the weak distributivity of Boolean σ-algebras satisfying the countable chain condition. It addresses primarily the question when such algebras carry a σ-additive measure. We use as a starting point the problem of John von Neumann stated in 1937 in the Scottish Book. He asked if the countable chain condition and weak distributivity are sufficient for the existence of such a measure.Subsequent research has shown that the problem has two aspects: one set theoretic and one combinatorial. Recent (...) results provide a complete solution of both the set theoretic and the combinatorial problems. We shall survey the history of von Neumann's Problem and outline the solution of the set theoretic problem. The technique that we describe owes much to the early work of Dorothy Maharam to whom we dedicate this article.§1. Complete Boolean algebras and weak distributivity. ABoolean algebrais a setBwith Boolean operationsa˅b,a˄b and −a, partial orderinga≤bdefined bya˄b=aand the smallest and greatest element,0and1. By Stone's Representation Theorem, every Boolean algebra is isomorphic to an algebra of subsets of some nonempty setS, under operationsa∪b,a∩b,S−a, ordered by inclusion, with0= ∅ and1=S.Complete Boolean algebras and weak distributivity.A Boolean algebrais a setBwith Boolean operationsa˅b,a˄b and -a, partial orderinga≤bdefined bya˄b=aand the smallest and greatest element.0and1. By Stone's Representation Theorem, every Boolean algebra is isomorphic to an algebra of subsets of some nonempty setS, under operationsa∪b,a∩b,S-a, ordered by inclusion, with0= ϕ and1=S. (shrink)

The paper summarizes expected utility theory, both in its original von Neumann-Morgenstern version and its later developments, and discusses the normative claims to rationality made by this theory.

Whereas many others have scrutinized the Allais paradox from a theoretical angle, we study the paradox from an historical perspective and link our findings to a suggestion as to how decision theory could make use of it today. We emphasize that Allais proposed the paradox asa normative argument, concerned with ‘the rational man’ and not the ‘real man’, to use his words. Moreover, and more subtly, we argue that Allais had an unusual sense of the normative, being concerned not so (...) much with therationality of choicesas with therationality of the agent as a person. These two claims are buttressed by a detailed investigation – the first of its kind – of the 1952 Paris conference on risk, which set the context for the invention of the paradox, and a detailed reconstruction – also the first of its kind – of Allais’s specific normative argument from his numerous but allusive writings. The paper contrasts these interpretations of what the paradox historically represented, with how it generally came to function within decision theory from the late 1970s onwards: that is, as an empirical refutation of the expected utility hypothesis, and more specifically of the condition of von Neumann–Morgenstern independence that underlies that hypothesis. While not denying that this use of the paradox was fruitful in many ways, we propose another use that turns out also to be compatible with an experimental perspective. Following Allais’s hints on ‘the experimental definition of rationality’, this new use consists inletting the experiment itself speak of the rationality or otherwise of the subjects. In the 1970s, a short sequence of papers inspired by Allais implemented original ways of eliciting the reasons guiding the subjects’ choices, and claimed to be able to draw relevant normative consequences from this information. We end by reviewing this forgotten experimental avenue not simply historically, but with a view to recommending it for possible use by decision theorists today. (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.

I demonstrate that Bell's theorem is based on circular reasoning and thus a fundamentally flawed argument. It unjustifiably assumes the additivity of expectation values for dispersion-free states of contextual hidden variable theories for non-commuting observables involved in Bell-test experiments, which is tautologous to assuming the bounds of ±2 on the Bell-CHSH sum of expectation values. Its premises thus assume in a different guise the bounds of ±2 it sets out to prove. Once this oversight is ameliorated from Bell's argument (...) by identifying the impediment that leads to it and local realism is implemented correctly, the bounds on the Bell-CHSH sum of expectation values work out to be ±2√2 instead of ±2, thereby mitigating the conclusion of Bell's theorem. Consequently, what is ruled out by any of the Bell-test experiments is not local realism but the linear additivity of expectation values, which does not hold for non-commuting observables in any hidden variable theories to begin with. I also identify similar oversight in the GHZ variant of Bell's theorem, invalidating its claim of having found an inconsistency in the premisses of the argument by EPR for completing quantum mechanics. Conceptually, the oversight in both Bell's theorem and its GHZ variant traces back to the oversight in von Neumann's theorem against hidden variable theories identified by Grete Hermann in the 1930s. (shrink)

Non-commuting quantities and hidden parameters – Wave-corpuscular dualism and hidden parameters – Local or nonlocal hidden parameters – Phase space in quantum mechanics – Weyl, Wigner, and Moyal – Von Neumann’s theorem about the absence of hidden parameters in quantum mechanics and Hermann – Bell’s objection – Quantum-mechanical and mathematical incommeasurability – Kochen – Specker’s idea about their equivalence – The notion of partial algebra – Embeddability of a qubit into a bit – Quantum computer is not Turing (...) machine – Is continuality universal? – Diffeomorphism and velocity – Einstein’s general principle of relativity – „Mach’s principle“ – The Skolemian relativity of the discrete and the continuous – The counterexample in § 6 of their paper – About the classical tautology which is untrue being replaced by the statements about commeasurable quantum-mechanical quantities – Logical hidden parameters – The undecidability of the hypothesis about hidden parameters – Wigner’s work and и Weyl’s previous one – Lie groups, representations, and psi-function – From a qualitative to a quantitative expression of relativity − psi-function, or the discrete by the random – Bartlett’s approach − psi-function as the characteristic function of random quantity – Discrete and/ or continual description – Quantity and its “digitalized projection“ – The idea of „velocity−probability“ – The notion of probability and the light speed postulate – Generalized probability and its physical interpretation – A quantum description of macro-world – The period of the as-sociated de Broglie wave and the length of now – Causality equivalently replaced by chance – The philosophy of quantum information and religion – Einstein’s thesis about “the consubstantiality of inertia ant weight“ – Again about the interpretation of complex velocity – The speed of time – Newton’s law of inertia and Lagrange’s formulation of mechanics – Force and effect – The theory of tachyons and general relativity – Riesz’s representation theorem – The notion of covariant world line – Encoding a world line by psi-function – Spacetime and qubit − psi-function by qubits – About the physical interpretation of both the complex axes of a qubit – The interpretation of the self-adjoint operators components – The world line of an arbitrary quantity – The invariance of the physical laws towards quantum object and apparatus – Hilbert space and that of Minkowski – The relationship between the coefficients of -function and the qubits – World line = psi-function + self-adjoint operator – Reality and description – Does a „curved“ Hilbert space exist? – The axiom of choice, or when is possible a flattening of Hilbert space? – But why not to flatten also pseudo-Riemannian space? – The commutator of conjugate quantities – Relative mass – The strokes of self-movement and its philosophical interpretation – The self-perfection of the universe – The generalization of quantity in quantum physics – An analogy of the Feynman formalism – Feynman and many-world interpretation – The psi-function of various objects – Countable and uncountable basis – Generalized continuum and arithmetization – Field and entanglement – Function as coding – The idea of „curved“ Descartes product – The environment of a function – Another view to the notion of velocity-probability – Reality and description – Hilbert space as a model both of object and description – The notion of holistic logic – Physical quantity as the information about it – Cross-temporal correlations – The forecasting of future – Description in separable and inseparable Hilbert space – „Forces“ or „miracles“ – Velocity or time – The notion of non-finite set – Dasein or Dazeit – The trajectory of the whole – Ontological and onto-theological difference – An analogy of the Feynman and many-world interpretation − psi-function as physical quantity – Things in the world and instances in time – The generation of the physi-cal by mathematical – The generalized notion of observer – Subjective or objective probability – Energy as the change of probability per the unite of time – The generalized principle of least action from a new view-point – The exception of two dimensions and Fermat’s last theorem. (shrink)

I argue that prioritarianism cannot be assessed in abstraction from an account of the measure of utility. Rather, the soundness of this view crucially depends on what counts as a greater, lesser, or equal increase in a person’s utility. In particular, prioritarianism cannot accommodate a normatively compelling measure of utility that is captured by the axioms of John von Neumann and Oskar Morgenstern’s expected utility theory. Nor can it accommodate a plausible and elegant generalization of this theory that (...) has been offered in response to challenges to von Neumann and Morgenstern. This is, I think, a theoretically interesting and unexpected source of difficulty for prioritarianism, which I explore in this article. (shrink)

Acts are functions from states of nature into finite-support distributions over a set of ‘deterministic outcomes’. We characterize preference relations over acts which have a numerical representation by the functional J(f)=min>∫u∘ f dP¦PϵC where f is an act, u is a von Neumann-Morgenstern utility over outcomes, and C is a closed and convex set of finitely additive probability measures on the states of nature. In addition to the usual assumptions on the preference relation as transitivity, completeness, continuity and (...) monotonicity, we assume uncertainty aversion and certainty-independence. The last condition is a new one and is a weakening of the classical independence axiom: It requires that an act f is preferred to an act g if and only if the mixture of f and any constant act h is preferred to the same mixture of g and h. If non-degeneracy of the preference relation is also assumed, the convex set of priors C is uniquely determined. Finally, a concept of independence in case of a non-unique prior is introduced. (shrink)

We prove a uniqueness theorem showing that, subject to certain natural constraints, all 'no collapse' interpretations of quantum mechanics can be uniquely characterized and reduced to the choice of a particular preferred observable as determine (definite, sharp). We show how certain versions of the modal interpretation, Bohm's 'causal' interpretation, Bohr's complementarity interpretation, and the orthodox (Dirac-von Neumann) interpretation without the projection postulate can be recovered from the theorem. Bohr's complementarity and Einstein's realism appear as two quite different (...) proposals for selecting the preferred determinate observable--either settled pragmatically by what we choose to observe, or fixed once and for all, as the Einsteinian realist would require, in which case the preferred observable is a 'beable' in Bell's sense, as in Bohm's interpretation (where the preferred observable is position in configuration space). (shrink)

This note is a generalization and improved interpretation of the main result of Karni and Schmeidler. A decision-maker is supposed to possess a preference relation on acts and another preference relation on state-prize lotteries, both of which are assumed to satisfy the von Neumann–Morgenstern axioms. In addition, the two preference relations restricted to a state of nature are assumed to agree. We show that these axioms are necessary and sufficient for the existence of subjective expected utility over acts (...) with state-dependent utility functions and a subjective probability measure. This subjective probability measure is unique when conditioned on the set of states of nature in which not all the prizes are equally desirable. (shrink)

Standard theories of expected utility require that preferences are complete, and/or Archimedean. We present in this paper a theory of decision under uncertainty for both incomplete and non-Archimedean preferences. Without continuity assumptions, incomplete preferences on a lottery space reduce to an order-extension problem. It is well known that incomplete preferences can be extended to complete preferences in the full generality, but this result does not necessarily hold for incomplete preferences which satisfy the independence axiom, since it may obviously happen that (...) the extension does not satisfy the independence axiom. We show, for incomplete preferences on a mixture space, that an extension which satisfies the independence axiom exists. We find necessary and sufficient conditions for a preorder on a finite lottery space to be representable by a family of lexicographic von Neumann–Morgenstern Expected Utility functions. (shrink)