This paper is the second in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics (...) so as to shed new light on the relevant strengths and limits of higher-order logic. (shrink)

Steve Awodey and Erich H. Reck. Completeness and Categoricity: 19th Century Axiomatics to 21st Century Senatics.

Abstract Not much scholarly work is needed in order to stumble across many passages where Edmund Husserl seems to advocate an anti-realist attitude towards the natural sciences. This tendency, however, is not well-received within the secondary literature. While some commentators criticize Husserl for his alleged scientific anti-realism, others argue that Husserl's position is much more realist than the first impression indicates. It is against this background that I want to argue for the following theses: a) The basic outlook of Husserl's (...) epistemology as well as his more substantial comments regarding the natural sciences indeed result in a (sophisticated version of) scientific anti-realism which bears certain resemblances to Bas van Fraassen's constructive empiricism; b) This scientific anti-realism can be defended against the two most common objections raised in the secondary literature; c) It is only by means of this sophisticated version of scientific anti-realism that phenomenology can circumvent the problem of ?scientific objectivism? (shrink)

This paper is the first in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics (...) so as to shed new light on the relevant strengths and limits of higher-order logic. (shrink)

In this paper I discuss the version of predicative analysis put forward by Hermann Weyl in "Das Kontinuum". I try to establish how much of the underlying motivation for Weyl's position may be due to his acceptance of a phenomenological philosophical perspective. More specifically, I analyze Weyl's philosophical ideas in connexion with the work of Husserl, in particular "Logische Untersuchungen" and "Ideen I". I believe that this interpretation of Weyl can clarify the views on mathematical existence and mathematical intuition which (...) are implicit in "Das Kontinuum". (shrink)

Hermann Weyl was one of the early contributors to the mathematics of general relativity. This article argues that in 1929, for the formulation of a general relativistic framework of the Dirac equation, he both abolished and preserved in modified form the conceptual perspective that he had developed earlier in his “analysis of the problem of space.” The ideas of infinitesimal congruence from the early 1920s were aufgehoben in the general relativistic framework for the Dirac equation. He preserved the central idea (...) of gauge as a “purely infinitesimal” aspect of symmetries in a group extension schema. With respect to methodology, however, Weyl gave up his earlier preferences for relatively a-priori arguments and tried to incorporate as much empiricism as he could. This signified a clearly expressed empirical turn for him. Moreover, in this step he emphasized that the mathematical objects used for the representation of matter structures stood at the center of the construction, rather than interaction fields which, in the early 1920s, he had considered as more or less derivable from geometrico-philosophical considerations. (shrink)

The following definition of objective is proposed: A statement S is objective if and only if in S all parameters that are relevant to its truth value are made explicit. The objectivity of predicates and relations can be defined in a similar manner. This simple conception of objectivity-which could be called explicitness conception of objectivity-can be found in Hermann Weyl and plays a central part in the natural sciences. There are grades of objectivity depending on the quality and the number (...) of parameters our predicates are relativized to A relativistic Ockham principle has to be recognized: Relativization parameters are not to be multiplied beyond necessity. The explicitness conception of objectivity is accessible to mathematical specifications, is the core of the idea of invariance, has a lot of philosophical applications and leads to precise notions of subjectivity and a precise formulation of the problem of the limits of objectivity. (shrink)

The following definition of “objective” is proposed: A statement S is objective if and only if in S all parameters that are relevant to its truth value are made explicit. The objectivity of predicates and relations can be defined in a similar manner. This simple conception of objectivity-which could be called “explicitness conception of objectivity”-can be found in Hermann Weyl and plays a central part in the natural sciences. There are grades of objectivity depending on the ‘quality’ and the number (...) of parameters our predicates are relativized to A relativistic Ockham principle has to be recognized: Relativization parameters are not to be multiplied beyond necessity. The explicitness conception of objectivity is accessible to mathematical specifications, is the core of the idea of invariance, has a lot of philosophical applications and leads to precise notions of ‘subjectivity’ and a precise formulation of the problem of the limits of objectivity. (shrink)

The London and Bauer monograph occupies a central place in the debate concerning the quantum measurement problem. Gavroglu has previously noted the influence of Husserlian phenomenology on London's scientific work. However, he has not explored the full extent of this influence in the monograph itself. I begin this paper by outlining the important role played by the monograph in the debate. In effect, it acted as a kind of 'lens' through which the standard, or Copenhagen, 'solution' to the measurement problem (...) came to be perceived and, as such, it was robustly criticized, most notably by Putnam and Shimony. I then spell out the Husserlian understanding of consciousness in order to illuminate the traces of this understanding within the London and Bauer text. This, in turn, yields a new perspective on this 'solution' to the measurement problem, one that I believe has not been articulated before and, furthermore, which is immune to the criticisms of Putnam and Shimony. (shrink)

In der neuen Literatur tiber Leibniz' Spatphilosophie findet man zwei deutlich einander entgegengesetzte Theorien Uber die Realitat des Körpers. Auf der einen Seite gibt es Gesichtspunkte, die ihn mit einer Phänomenalismuslehre verbinden, nach welcher die Körper nichts anderes als koordinierte Perzeptionen unausgedehnter Monaden sind. Auf der anderen Seite gibt es Griinde, die dafur sprechen, daß Leibniz die Auffassung vertreten muß, daß Körper Aggregate von Monaden sind. In diesem Aufsatz suche ich zu zeigen, daß die phanomenalistische Interpretation aufgrund der starken Textzeugnisse, (...) die Leibniz beständig zur Aggregatthese hindrängen, zuriickzuweisen ist. Daruber hinaus weise ich jedoch die verbreitete Auffassung eben der Aggregatthese zurück, d. h. die Meinung, gewisse Vielheiten von Monaden riefen bloß die Illusion, Körper zu sein, hervor, wenn sie durch andere Monaden M falsch perzipiert" werden. Gegen diese Auffassung des falschen Perzipierens ftihre ich ins Feld, daß die Aggregatthese als ein Deutungsversuch der Natur oder des Wesens des Körpers verstanden werden mufi. Wenn Leibniz sagt, Körper seien Aggregate von Monaden, so ist das nicht als blofie Behauptung, gewisse Monaden erschienen anderen Monaden als Körper, zu verstehen, sondern als die These, daß jeder Körper seinem Wesen nach eine Vielheit von Monadenist. (shrink)