SummaryWe present a systematic discussion of the structural and conceptual simplifications of proofs of standard theorems afforded by nonstandard methods and examine to what extent the resulting nonstandard proofs satisfy the informal criterion of “plausibility”. We introduce the concept of a “standard detour” and show that all nonstandard proofs considered avoid such detours. Among the proofs examined are proofs of the Intermediate Value Theorem, the Riemann Integration Theorem, the Spectral Theorem for compact Hermitian operators, and the Arzela‐Ascoli Theorem.RésuméNous discutons systématiquement (...) les simplifications structurelles et conceptuelles apportées par des méthodes non standard aux démonstrations de théorèmes standard et examinons jusQuà quel point de telles démonstrations non standard satisfont à un critère informel de plausibilityé. Nous introduisons la notion de détour standard et montrons que toutes les démonstrations non standard considérées évitent de tels détours. Parmi les démonstrations examinées: celles du théoréme de la valeur intermédiate, du théorème ?on;intégration de Riemann, du théorème spectral pour les opérateurs hermitiques compacts et du théorème ?on;Arzelà‐Arcoli.ZusammenfassungWir stellen eine systematische Betrachtung der durch die Nichtstandard‐Methoden gegebe‐nen strukturellen und begrifflichen Vereinfachungen von Beweisen von Standardsätzen an, und untersuchen, wie weit die entstandenen Nichtstandard‐Beweise das informelle Kriterium der Plausibilität erfüllen. Wir führen den Begriff eines Standard‐Umweges ein, und zeigen, dass alle betrachteten Nichtstandard‐Beweise solche Umwege vermeiden. Unter den betrachteten Beweisen befinden sich Beweise des Zwischenwertsatzes, des Riemannschen lntegralsatzes, des Spektralsatzes für kompakte, hermitische Operatoren, und des Theorems von Arzelà‐Ascoli. (shrink)

Provability, Computability and Reflection.

We introduce the notion of an alphabetic trace of a cut-free intuitionistic prepositional proof and show that it serves to characterize the equality of arrows in cartesian closed categories. We also show that alphabetic traces improve on the notion of the generality of proofs proposed in the literature. The main theorem of the paper yields a new and considerably simpler solution of the coherence problem for cartesian closed categories than those in [11, 14].

This is the Stanford Encyclopedia of Philosophy comprehensive article on J.M.E. MacTaggart, with special focus on his methodology for philosophy, his metaphysical system, and his ethics.

A recent analysis by de Barros and Suppes of experimentally realizable GHZ correlations supports the conclusion that these correlations cannot be explained by introducing local hidden variables. We show, nevertheless, that their analysis does not exclude local hidden variable models in which the inefficiency in the experiment is an effect not only of random errors in the detector equipment, but is also the manifestation of a pre-set, hidden property of the particles ("prism models"). Indeed, we present an explicit prism model (...) for the GHZ scenario; that is, a local hidden variable model entirely compatible with recent GHZ experiments. (shrink)

On the basis of what I call physico-formalist philosophy of mathematics, I will develop an amended account of the Kantian–Reichenbachian conception of constitutive a priori. It will be shown that the features attributed to a real object are not possessed by the object as a “thing-in-itself”; they require a physical theory by means of which these features are constituted. It will be seen that the existence of such a physical theory implies that a physical object can possess a property only (...) if other contingently existing physical objects exist; therefore, the intrinsic–extrinsic distinction is flawed. (shrink)

The Birkhoff-Maltsev problem asks for a characterization of those lattices each of which is isomorphic to the lattice L(K) of all subquasivarieties for some quasivariety K of algebraic systems. The current status of this problem, which is still open, is discussed. Various unsolved questions that are related to the Birkhoff-Maltsev problem are also considered, including ones that stem from the theory of propositional logics.

According to the standard realistic interpretation of classical electrodynamics, the electromagnetic field is conceived as a real physical entity existing in space and time. The problem we address in this paper is how to understand this spatiotemporal existence, that is, how to describe the persistence of a field-like physical entity like electromagnetic field. First, we provide a formal description of the notion of persistence: we derive an “equation of persistence” constituting a necessary condition that the spatiotemporal distributions of the fundamental (...) attributes of a persisting physical entity must satisfy. We then prove a theorem according to which the vast majority of the solutions of Maxwell’s equations, describing possible spatiotemporal distributions of the fundamental attributes of the electromagnetic field, violate the equation of persistence. Finally, we discuss the consequences of this result for the ontology of the electromagnetic field. (shrink)

This book seeks to critically expound and appraise the thoughts of the foremost British philosopher, J.M.E. McTaggart, with respect to three principal themes of his philosophy: substance, self, and immortality. Sharma draws on all of McTaggart’s major writings to provide a comprehensive exposition of his overall theory of reality.

J. M. E. McTaggart: Substance, Self and Immortality. By Sharma Ramesh K.