This paper deals mainly with generalizations of results in finitary combinatorics to infinite ordinals. It is well-known that for finite ordinals ∑bT<αβ is the number of 2-element subsets of an α-element set. It is shown here that for any well-ordered set of arbitrary infinite order type α, ∑bT<αβ is the ordinal of the set M of 2-element subsets, where M is ordered in some natural way. The result is then extended to evaluating the ordinal of the set of all n-element (...) subsets for each natural number n ≥ 2. Moreover, series ∑β<αf are investigated and evaluated, where α is a limit ordinal and the function f belongs to a certain class of functions containing polynomials with natural number coefficients. The tools developed for this result can be extended to cover all infinite α, but the case of finite α appears to be quite problematic. (shrink)

A sensible quality is a perceptible property, a property that physical objects (or events) perceptually appear to have. Thus smells, tastes, colors and shapes are sensible qualities. An egg, for example, may smell rotten, taste sour, and look cream and round.1,2 The sensible qualities are not a miscellanous jumble—they form complex structures. Crimson, magenta, and chartreuse are not merely three different shades of color: the first two are more similar than either is to the third. Familiar color spaces or color (...) solids capture, to a greater or lesser extent, these relations between the colors. The same goes for sensible qualities perceived in other modalities: middle C, high C, and D are not merely three different notes, and the taste of lemons, oranges, and sugar cubes are not merely three different tastes. How can this structure of appearance be explained? One idea is that sensible qualities are of two sorts: basic and derived. The basic sensible qualities are the building blocks from which the derived sensible qualities can be.. (shrink)

An appropriate kind of curved Hilbert space is developed in such a manner that it admits operators of $\mathcal{C}$ - and $\mathfrak{D}$ -differentiation, which are the analogues of the familiar covariant and D-differentiation available in a manifold. These tools are then employed to shed light on the space-time structure of Quantum Mechanics, from the points of view of the Feynman ‘path integral’ and of canonical quantisation. (The latter contains, as a special case, quantisation in arbitrary curvilinear coordinates when space is (...) flat.) The influence of curvature is emphasised throughout, with an illustration provided by the Aharonov-Bohm effect. (shrink)

One of the traditional ways mathematical ideas and even new areas of mathematics are created is from experiments. One of the best-known examples is that of the Fermat hypothesis, which was conjectured by Fermat in his attempts to find integer solutions for the famous Fermat equation. This hypothesis led to the creation of a whole field of knowledge, but it was proved only after several hundred years. This book, based on the author's lectures, presents several new directions of mathematical research. (...) All of these directions are based on numerical experiments conducted by the author, which led to new hypotheses that currently remain open, i.e., are neither proved nor disproved. The hypotheses range from geometry and topology (statistics of plane curves and smooth functions) to combinatorics (combinatorial complexity and random permutations) to algebra and number theory (continuous fractions and Galois groups). For each subject, the author describes the problem and presents numerical results that led him to a particular conjecture. In the majority of cases there is an indication of how the readers can approach the formulated conjectures (at least by conducting more numerical experiments). Written in Arnold's unique style, the book is intended for a wide range of mathematicians, from high school students interested in exploring unusual areas of mathematics on their own, to college and graduate students, to researchers interested in gaining a new, somewhat nontraditional perspective on doing mathematics. In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSRI and the AMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession. Titles in this series are co-published with the Mathematical Sciences Research Institute (MSRI). (shrink)

Gauss’s quadratic reciprocity theorem is among the most important results in the history of number theory. It’s also among the most mysterious: since its discovery in the late 18th century, mathematicians have regarded reciprocity as a deeply surprising fact in need of explanation. Intriguingly, though, there’s little agreement on how the theorem is best explained. Two quite different kinds of proof are most often praised as explanatory: an elementary argument that gives the theorem an intuitive geometric interpretation, due to Gauss (...) and Eisenstein, and a sophisticated proof using algebraic number theory, due to Hilbert. Philosophers have yet to look carefully at such explanatory disagreements in mathematics. I do so here. According to the view I defend, there are two important explanatory virtues—depth and transparency—which different proofs (and other potential explanations) possess to different degrees. Although not mutually exclusive in principle, the packages of features associated with the two stand in some tension with one another, so that very deep explanations are rarely transparent, and vice versa. After developing the theory of depth and transparency and applying it to the case of quadratic reciprocity, I draw some morals about the nature of mathematical explanation. (shrink)

It is almost universally believed that in quantum theory the two following statements hold: all transformations are achieved by a unitary interaction followed by a von-Neumann measurement; all mixed states are marginals of pure entangled states. I name this doctrine the dogma of purification ontology. The source of the dogma is the original von Neumann axiomatisation of the theory, which largely relies on the Schrődinger equation as a postulate, which holds in a nonrelativistic context, and whose operator version holds only (...) in free quantum field theory, but no longer in the interacting theory. In the present paper I prove that both ontologies of unitarity and state-purity are unfalsifiable, even in principle, and therefore axiomatically spurious. I propose instead a minimal four-postulate axiomatisation: associate a Hilbert space \ to each system \; compose two systems by the tensor product rule \; associate a transformation from system \ to \ to a quantum operation, i.e. to a completely positive trace-non-increasing map between the trace-class operators of \ and \; evaluate all joint probabilities through that of a special type of quantum operation: the state preparation. I then conclude that quantum paradoxes—such as the Schroedinger-cat’s, and, most relevantly, the information paradox—are originated only by the dogma of purification ontology, and they are no longer paradoxes of the theory in the minimal formulation. For the same reason, most interpretations of the theory interpret the same dogma, not the strict theory stripped of the spurious postulates. (shrink)

Recent years have been blessed with an abundance of logical systems, arising from a multitude of applications. A logic can be characterised in many different ways. Traditionally, a logic is presented via the following three components: 1. an intuitive non-formal motivation, perhaps tie it in to some application area 2. a semantical interpretation 3. a proof theoretical formulation. There are several types of proof theoretical methodologies, Hilbert style, Gentzen style, goal directed style, labelled deductive system style, and so on. The (...) tableau methodology, invented in the 1950s by Beth and Hintikka and later per fected by Smullyan and Fitting, is today one of the most popular, since it appears to bring together the proof-theoretical and the semantical approaches to the pre of a logical system and is also very intuitive. In many universities it is sentation the style first taught to students. Recently interest in tableaux has become more widespread and a community crystallised around the subject. An annual tableaux conference is being held and proceedings are published. The present volume is a Handbook a/Tableaux pre senting to the community a wide coverage of tableaux systems for a variety of logics. It is written by active members of the community and brings the reader up to frontline research. It will be of interest to any formal logician from any area. (shrink)

We investigate a possible form of Schrödinger’s equation as it appears to moving observers. It is shown that, in this framework, accelerated motion requires fictitious potentials to be added to the original equation. The gauge invariance of the formulation is established. The example of accelerated Euclidean transformations is treated explicitly, which contain Galilean transformations as special cases. The relationship between an acceleration and a gravitational field is found to be compatible with the picture of the ‘Einstein elevator’. The physical effects (...) of an acceleration are illustrated by the problem of the uniformly-accelerated harmonic oscillator. (shrink)

Zu seinen Lebzeiten war Kurt Gödel außerhalb der Fachwelt der Mathematiker, Philosophen und theoretischen Physiker kaum bekannt. Zu Beginn seiner Karriere schuf er beeindruckende Arbeiten zur Vollständigkeit und Beweisbarkeit formaler logischer Systeme, die zu seiner Dissertation und seiner Habilitations-schrift wurden und ihn unter Fachleuten weltberühmt machten. Seine Unvoll-ständigkeitssätze läuteten das Ende der formal-logischen Programme der Logizisten (Russell et al.) und der Formalisten (Hilbert et al.) ein. Später erzielte er auch signifikante Ergebnisse in der Mengenlehre. Nach seiner Emigration in die USA (...) (Princeton), widmete er sich mehr der Philosophie, dem Leitmotiv seines Lebens, und er fand auch eine einzigartige Lösung zu Einsteins Feld-gleichungen der Gravitation, sein "Gödel-Universum". Dieses Buch beschreibt sowohl den Gödel, der ein genialer Wissenschaftler war, und der gewagte und neuartige Hypothesen zu den Fundamenten der Mathe-matik und Physik hervorbrachte, ‒ als auch den Gödel, der ein perfekter Rationalist war, aber sein Alltagsleben nur mit Mühe meistern konnte und zeitlebens unter Depressionen, Angstneurosen und Hypochondrie litt. Ein Leben voller Paradoxen, in dem er trotz all seiner psychischen Probleme Beachtliches leistete und zu einem Vorbild für viele jüngere Wissenschaftler wurde. Das Buch liefert den Kontext zu seinen Errungenschaften, die ein verblüffend breites Spektrum intellektueller Unternehmungen darstellen, und zu seiner zunehmenden Geisteskrankheit; und es zeigt, wie er eine lange und erfolgreiche Karriere mit Hilfe seiner loyalen Ehefrau Adele und einigen seiner Freunde durchlaufen konnte. Dies ist eine faszinierende Geschichte der wissen-schaftlichen Genialität und der menschlichen Natur. (shrink)

We present a novel variant of decision making based on the mathematical theory of separable Hilbert spaces. This mathematical structure captures the effect of superposition of composite prospects, including many incorporated intentions, which allows us to describe a variety of interesting fallacies and anomalies that have been reported to particularize the decision making of real human beings. The theory characterizes entangled decision making, non-commutativity of subsequent decisions, and intention interference. We demonstrate how the violation of the Savage’s sure-thing principle, known (...) as the disjunction effect, can be explained quantitatively as a result of the interference of intentions, when making decisions under uncertainty. The disjunction effects, observed in experiments, are accurately predicted using a theorem on interference alternation that we derive, which connects aversion-to-uncertainty to the appearance of negative interference terms suppressing the probability of actions. The conjunction fallacy is also explained by the presence of the interference terms. A series of experiments are analyzed and shown to be in excellent agreement with a priori evaluation of interference effects. The conjunction fallacy is also shown to be a sufficient condition for the disjunction effect, and novel experiments testing the combined interplay between the two effects are suggested. (shrink)

A deductive system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{S}$$ \end{document} (in the sense of Tarski) is Fregean if the relation of interderivability, relative to any given theory T, i.e., the binary relation between formulas\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\{ \left\langle {\alpha,\beta } \right\rangle :T,\alpha \vdash s \beta and T,\beta \vdash s \alpha \},$$ \end{document}is a congruence relation on the formula algebra. The multiterm deduction-detachment theorem is a natural generalization of the (...) deduction theorem of the classical and intuitionistic propositional calculi (IPC) in which a finite system of possibly compound formulas collectively plays the role of the implication connective of IPC. We investigate the deductive structure of Fregean deductive systems with the multiterm deduction-detachment theorem within the framework of abstract algebraic logic. It is shown that each deductive system of this kind has a deductive structure very close to that of the implicational fragment of IPC. Moreover, it is algebraizable and the algebraic structure of its equivalent quasivariety is very close to that of the variety of Hilbert algebras. The equivalent quasivariety is however not in general a variety. This gives an example of a relatively point-regular, congruence-orderable, and congruence-distributive quasivariety that fails to be a variety, and provides what apparently is the first evidence of a significant difference between the multiterm deduction-detachment theorem and the more familiar form of the theorem where there is a single implication connective. (shrink)

Deformation quantisation is applied to ordinary Quantum Mechanics by introducing the star product in a configuration space combining a Riemannian structure with a Poisson one. A Hilbert space compatible with such a configuration space is designed. The dynamics is expressed by a Hermitian Hamiltonian containing a scalar potential and a one-form potential. As a simple illustration, it is shown how a particular type of non-commutativity of the star product is interpretable as generating the Zeeman effect of ordinary Quantum Mechanics.

A classical and quantum relativistic interacting particle formalism is revisited. A Hilbert space is achieved through the use of variable individual particle rest masses, but no c-number mass parameter is required for the relativistic free particle. Boosted center of momentum states feature in both the free and interacting model. The implications of a failure to impose simultaneity conditions at the classical level are explored. The implementation of these conditions at the quantum level leads to a finite uncertainty in interaction times, (...) perhaps more closely modeling the exchange of virtual particles in quantum field theory. This work is compared and contrasted with other variable mass models in the literature. (shrink)

This article presents modal versions of resource-conscious logics. We concentrate on extensions of variants of linear logic with one minimal non-normal modality. In earlier work, where we investigated agency in multi-agent systems, we have shown that the results scale up to logics with multiple non-minimal modalities. Here, we start with the language of propositional intuitionistic linear logic without the additive disjunction, to which we add a modality. We provide an interpretation of this language on a class of Kripke resource models (...) extended with a neighbourhood function: modal Kripke resource models. We propose a Hilbert-style axiomatisation and a Gentzen-style sequent calculus. We show that the proof theories are sound and complete with respect to the class of modal Kripke resource models. We show that the sequent calculus admits cut elimination and that proof-search is in PSPACE. We then show how to extend the results when non-commutative connectives are added to the language. Finally, we put the l.. (shrink)

A classical and quantum relativistic interacting particle formalism is revisited. A Hilbert space is achieved through the use of variable individual particle rest masses, but no c-number mass parameter is required for the relativistic free particle. Boosted center of momentum states feature in both the free and interacting model. The implications of a failure to impose simultaneity conditions at the classical level are explored. The implementation of these conditions at the quantum level leads to a finite uncertainty in interaction times, (...) perhaps more closely modeling the exchange of virtual particles in quantum field theory. This work is compared and contrasted with other variable mass models in the literature. (shrink)

We have seen that despite Feferman's results Gödel's second theorem vitiates the use of Hilbert-type epistemological programs and consistency proofs as a response to mathematical skepticism. Thus consistency proofs fail to have the philosophical significance often attributed to them.This does not mean that consistency proofs are of no interest to philosophers. We know that a ‘non-pathological’ consistency proof for a system S will use methods which are not available in S. When S is as strong a system as we are (...) willing to entertain seriously then a consistency proof for it will yield no epistemological gain. But in other cases philosophers might argue that the proof uses methods which are merely different rather than stronger than those available in the system in question. This claim has been made, for example, in the case of the constructive consistency proofs for elementary number theory. Similar philosophical investigations can be made on relative consistency proofs, since these differ from each other in the principles they employ. For example, most relative consistency proofs can be carried out within elementary number theory, but without using the theory of real numbers, no one has been able to prove the consistency of Quine's ML relative to that of his NF.What about the consistency of all mathematics or of some strong system for set theory? How do we answer the skeptic? Since here a convincing proof is not possible, we have established that the skeptic demands too much. We cannot be certain that our axioms are free from contradiction and must treat them as hypotheses which may be abandoned or modified in the face of further mathematical experience. This attitude is taken by many foundational workers who also go on to voice opinions about thelikelihood that various systems are consistent. Since these opinions are variously supported by appeals to the clarity of the mathematical concept formalized, the existence or non-existence of ‘weird’ models for the system and actual empirical experience with the system, this is surely a fruitful area for philosophical research. (shrink)

We explore the possibility of using quantum mechanical principles for hypercomputation through the consideration of a quantum algorithm for computing the Turing halting problem. The mathematical noncomputability is compensated by the measurability of the values of quantum observables and of the probability distributions for these values. Some previous no-go claims against quantum hypercomputation are then reviewed in the light of this new positive proposal.

The use of the symbol ∨for disjunction in formal logic is ubiquitous. Where did it come from? The paper details the evolution of the symbol ∨ in its historical and logical context. Some sources say that disjunction in its use as connecting propositions or formulas was introduced by Peano; others suggest that it originated as an abbreviation of the Latin word for “or,” vel. We show that the origin of the symbol ∨ for disjunction can be traced to Whitehead and (...) Russell’s pre-Principia work in formal logic. Because of Principia’s influence, its notation was widely adopted by philosophers working in logic (the logical empiricists in the 1920s and 1930s, especially Carnap and early Quine). Hilbert’s adoption of ∨ in his Grundzüge der theoretischen Logik guaranteed its widespread use by mathematical logicians. The origins of other logical symbols are also discussed. (shrink)

The brain is composed of electrically excitable neuronal networks regulated by the activity of voltage-gated ion channels. Further portraying the molecular composition of the brain, however, will not reveal anything remotely reminiscent of a feeling, a sensation or a conscious experience. In classical physics, addressing the mind–brain problem is a formidable task because no physical mechanism is able to explain how the brain generates the unobservable, inner psychological world of conscious experiences and how in turn those conscious experiences steer the (...) underlying brain processes toward desired behavior. Yet, this setback does not establish that consciousness is non-physical. Modern quantum physics affirms the interplay between two types of physical entities in Hilbert space: unobservable quantum states, which are vectors describing what exists in the physical world, and quantum observables, which are operators describing what can be observed in quantum measurements. Quantum no-go theorems further provide a framework for studying quantum brain dynamics, which has to be governed by a physically admissible Hamiltonian. Comprising consciousness of unobservable quantum information integrated in quantum brain states explains the origin of the inner privacy of conscious experiences and revisits the dynamic timescale of conscious processes to picosecond conformational transitions of neural biomolecules. The observable brain is then an objective construction created from classical bits of information, which are bound by Holevo’s theorem, and obtained through the measurement of quantum brain observables. Thus, quantum information theory clarifies the distinction between the unobservable mind and the observable brain, and supports a solid physical foundation for consciousness research. (shrink)

In the foundations of quantum mechanics Gleason’s theorem dictates the uniqueness of the state transition probability via the inner product of the corresponding state vectors in Hilbert space, independent of which measurement context induces this transition. We argue that the state transition probability should not be regarded as a secondary concept which can be derived from the structure on the set of states and properties, but instead should be regarded as a primitive concept for which measurement context is crucial. Accordingly, (...) we adopt an operational approach to quantum mechanics in which a physical entity is defined by the structure of its set of states, set of properties and the possible (measurement) contexts which can be applied to this entity. We put forward some elementary definitions to derive an operational theory from this State–COntext–Property (SCOP) formalism. We show that if the SCOP satisfies a Gleason-like condition, namely that the state transition probability is independent of which measurement context induces the change of state, then the lattice of properties is orthocomplemented, which is one of the ‘quantum axioms’ used in the Piron–Solèr representation theorem for quantum systems. In this sense we obtain a possible physical meaning for the orthocomplementation widely used in quantum structures. (shrink)

The paper argues against an intuitive reading of the Stone-von Neumann theorem as a categoricity result, thereby pointing out that this theorem does not entail any model-theoretical difference between the theories that validate it and those that don't.

We found that phase cone clustering patterns in EEG ripple bands demonstrate an increased turnover rate in epileptogenic zones compared to adjacent regions. We employed 256 channel EEG data collected in four adult subjects with refractory epilepsy. The analysis was performed in the 80–150 and 150–250 Hz ranges. Ictal onsets were documented with intracranial EEG recordings. Interictal scalp recordings, free of epileptiform patterns, of 240-s duration, were selected for analysis for each subject. The data was filtered, and the instantaneous phase (...) was extracted after the Hilbert transformation. Spatiotemporal contour plots of the unwrapped instantaneous phase with 1.0 ms intervals were constructed using a montage layout of the 256 electrode positions. Stable phase cone patterns were selected based on criteria that the sign of spatial gradient did not change for a minimum of three consecutive time samples and the frame velocity was consistent with known propagation velocities of cortical axons. These plots exhibited increased dynamical formation and dissolution of phase cones in the ictal onset zones, compared to surrounding cortical regions, in all four patients. We believe that these findings represent markers of abnormally increased cortical excitability. They are potential tools that may assist in localizing the epileptogenic zone. (shrink)

Feynman's sum-over-histories formulation of quantum mechanics has been considered a useful calculational tool in which virtual Feynman histories entering into a coherent quantum superposition cannot be individually measured. Here we show that sequential weak values, inferred by consecutive weak measurements of projectors, allow direct experimental probing of individual virtual Feynman histories, thereby revealing the exact nature of quantum interference of coherently superposed histories. Because the total sum of sequential weak values of multitime projection operators for a complete set of orthogonal (...) quantum histories is unity, complete sets of weak values could be interpreted in agreement with the standard quantum mechanical picture. We also elucidate the relationship between sequential weak values of quantum histories with different coarse graining in time and establish the incompatibility of weak values for nonorthogonal quantum histories in history Hilbert space. Bridging theory and experiment, the presented results may enhance our understanding of both weak values and quantum histories. (shrink)

This paper argues that Weyl's view that scientific objectivity requires that concepts be freely created, i.e., introduced via Hilbert-style axiomatizations, led him to abandon the phenomenological view of objectivity.

The language of manuals may be used to discuss inference in measurement in a general experimental context. Specializing to the context of the frame manual for Hilbert space, this inference leads to state dominance of the inferred state from partial measurements; this in turn, by Sakai's theorem, determines observables which are described by positive operator-valued measures. Symmetries are then introduced, showing that systems of covariance, rather than systems of imprimitivity, are natural objects to study in quantum mechanics. Experiments measuring different (...) polarization components simultaneously are reexamined in this language. Finally, implications of the Naimark extension theorem for the manual approach are investigated. (shrink)

The distinction between analytic and synthetic propositions, and with that the distinction between a priori and a posteriori truth, is being abandoned in much of analytic philosophy and the philosophy of most of the sciences. These distinctions should also be abandoned in the philosophy of mathematics. In particular, we must recognize the strong empirical component in our mathematical knowledge. The traditional distinction between logic and mathematics, on the one hand, and the natural sciences, on the other, should be dropped. Abstract (...) mathematical objects, like transcendental numbers or Hilbert spaces, are theoretical entities on a par with electromagnetic fields or quarks. Mathematical theories are not primarily logical deductions from axioms obtained by reflection on concepts but, rather, are constructions chosen to solve some collection of problems while fitting smoothly into the other theoretical commitments of the mathematician who formulates them. In other words, a mathematical theory is a scientific theory like any other, no more certain but also no more devoid of content. (shrink)

Here is a deft and new introduction to Gentzen proof techniques in axiom systems and to the analysis of formal axiom systems; in short, axiomatics inside and out. Treating of deduction in propositional and predicate logic, metatheoretical problems about both set theory and its paradoxes, the book is flexibly structured for selective use as a text. Yet the discussion is unified and motivated by the concept of the axiomatic system--the history of its use and analysis, and its present practical and (...) theoretical status. Well-chosen illustrations convey an historical overview of Euclid, non-Euclidean geometers, Peano, Boole, Hilbert and Gödel.--P. D. J. (shrink)

Starting with a Leninist thesis on the identity of dialectic and logic, the author concludes that the only authentic logic is dialectico-materialistic. Lenin predominates; though such logicians as Hilbert, Russell, and Carnap are mentioned, they are not seriously considered.--R. D. G.

We show that in quantum logic of closed subspaces of Hilbert space one cannot substitute quantum operations for classical (standard Hilbert space) ones and treat them as primitive operations. We consider two possible ways of such a substitution and arrive at operation algebras that are not lattices what proves the claim. We devise algorithms and programs which write down any two-variable expression in an orthomodular lattice by means of classical and quantum operations in an identical form. Our results show that (...) lattice structure and classical operations uniquely determine quantum logic underlying Hilbert space. As a consequence of our result, recent proposals for a deduction theorem with quantum operations in an orthomodular lattice as well as a, substitution of quantum operations for the usual standard Hilbert space ones in quantum logic prove to be misleading. Quantum computer quantum logic is also discussed. (shrink)

The trace formulation of quantum mechanical expectations is derived in a classical deterministic setting by averaging over an assembly of states. Interference of probabilities is discussed and its usual Hilbert space formulation is questioned. Nevertheless, it is shown that the observable predictions of quantum statics remain unchanged in the framework developed here.

Quantum entanglement is a key resource, which grants quantum systems the ability to accomplish tasks that are classically impossible. Here, we apply Feynman's sum-over-histories formalism to interacting bipartite quantum systems and introduce entanglement measures for bipartite quantum histories. Based on the Schmidt decomposition of the matrix comprised of the Feynman propagator complex coefficients, we prove that bipartite quantum histories are entangled if and only if the Schmidt rank of this matrix is larger than 1. The proposed approach highlights the utility (...) of using a separable basis for constructing the bipartite quantum histories and allows for quantification of their entanglement from the complete set of experimentally measured sequential weak values. We then illustrate the nonclassical nature of entangled histories with the use of Hardy's overlapping interferometers and explain why local hidden variable theories are unable to correctly reproduce all observable quantum outcomes. Our theoretical results elucidate how the composite tensor product structure of multipartite quantum systems is naturally extended across time and clarify the difference between quantum histories viewed as projection operators in the history Hilbert space and those viewed as chain operators and propagators in the standard Hilbert space. (shrink)

We present a solution to the ghost problem in fourth order derivative theories. In particular we study the Pais–Uhlenbeck fourth order oscillator model, a model which serves as a prototype for theories which are based on second plus fourth order derivative actions. Via a Dirac constraint method quantization we construct the appropriate quantum-mechanical Hamiltonian and Hilbert space for the system. We find that while the second-quantized Fock space of the general Pais–Uhlenbeck model does indeed contain the negative norm energy eigenstates (...) which are characteristic of higher derivative theories, in the limit in which we switch off the second order action, such ghost states are found to move off shell, with the spectrum of asymptotic in and out S-matrix states of the pure fourth order theory which results being found to be completely devoid of states with either negative energy or negative norm. We confirm these results by quantizing the Pais–Uhlenbeck theory via path integration and by constructing the associated first-quantized wave mechanics, and show that the disappearance of the would-be ghosts from the energy eigenspectrum in the pure fourth order limit is required by a hidden symmetry that the pure fourth order theory is unexpectedly found to possess. The occurrence of on-shell ghosts is thus seen not to be a shortcoming of pure fourth order theories per se, but rather to be one which only arises when fourth and second order theories are coupled to each other. (shrink)

The objective of this paper is to examine the thought of John Philoponus contra Aristotle, as it pertains to consciousness and its emergence, in light of both contemporary cosmology and philosophy. It will be argued that in an eternal universe the emergence of consciousness is an impossibility. The inspiration for this line of reasoning is found in Philoponus’ sixth century arguments against Aristotle on the eternity of the world. It will be shown that much of Philoponus’ argumentation is corroborated by (...) contemporary cosmology and philosophy. (shrink)

We introduce a model with a set of experiments of which the probabilities of the outcomes coincide with the quantum probabilities for the spin measurements of a quantum spin- $ \frac{1}{2} $ particle. Product tests are defined which allow simultaneous measurements of incompatible observables, which leads to a discussion of the validity of the meet of two propositions as the algebraic model for conjunction in quantum logic. Although the entity possesses the same structure for the logic of its experimental propositions (...) as a genuine spin- $ \frac{1}{2} $ quantum entity, the probability measure corresponding with the meet of propositions using the Hilbert space representation and quantum rules does not render the probability of the conjunction of the two propositions. Accordingly, some fundamental concepts of quantum logic, Piron-products, “classical” systems and the general problem of hidden variable theories for quantum theory are discussed. (shrink)

The formalism of abstracted quantum mechanics is applied in a model of the generalized Liar Paradox. Here, the Liar Paradox, a consistently testable configuration of logical truth properties, is considered a dynamic conceptual entity in the cognitive sphere (Aerts, Broekaert, & Smets, [Foundations of Science 1999, 4, 115–132; International Journal of Theoretical Physics, 2000, 38, 3231–3239]; Aerts and colleagues[Dialogue in Psychology, 1999, 10; Proceedings of Fundamental Approachs to Consciousness, Tokyo ’99; Mind in Interaction]. Basically, the intrinsic contextuality of the truth-value (...) of the Liar Paradox is appropriately covered by the abstracted quantum mechanical approach. The formal details of the model are explicited here for the generalized case. We prove the possibility of constructing a quantum model of the m-sentence generalizations of the Liar Paradox. This includes (i) the truth–falsehood state of the m-Liar Paradox can be represented by an embedded 2m-dimensional quantum vector in a (2m) m -dimensional complex Hilbert space, with cognitive interactions corresponding to projections, (ii) the construction of a continuous ‘time’ dynamics is possible: typical truth and falsehood value oscillations are described by Schrödinger evolution, (iii) Kirchoff and von Neumann axioms are satisfied by introduction of ‘truth-value by inference’ projectors, (iv) time invariance of unmeasured state. (shrink)

The traditional “realist” conception of physics, according to which human concepts, laws and theories can grasp the essence of a reality in our absence , seems incompatible with quantum formalism and it most fruitful interpretation. The proof rests on the violation by quantum mechanical formalism of some fundamental principles of the classical ontology. We discuss if the conception behind Einstein’s idea of a reality in our absence, could be still maintained and at which price. We conclude that quantum mechanical formalism (...) is not formulated on those terms, leaving for a separated paper the discussion about the terms in which it could be formulated and the onto-epistemological implications it might have. (shrink)

We carry out a systematic study of decidability for theories of real vector spaces, inner product spaces, and Hilbert spaces and of normed spaces, Banach spaces and metric spaces, all formalized using a 2-sorted first-order language. The theories for list turn out to be decidable while the theories for list are not even arithmetical: the theory of 2-dimensional Banach spaces, for example, has the same many-one degree as the set of truths of second-order arithmetic.We find that the purely universal and (...) purely existential fragments of the theory of normed spaces are decidable, as is the ∀∃ fragment of the theory of metric spaces. These results are sharp of their type: reductions of Hilbertʼs 10th problem show that the ∃∀ fragments for metric and normed spaces and the ∀∃ fragment for normed spaces are all undecidable. (shrink)

We provide a Hilbert-style axiomatization of the logic of , as well as a two-dimensional semantics with respect to which our logics are sound and complete. Our completeness results are quite general, pertaining to all such actuality logics that extend a normal and canonical modal basis. We also show that our logics have the strong finite model property and permit straightforward first-order extensions.