Leon Horsten

Proof-theoretic reflection principles have been discussed in proof theory ever since Gödel’s discovery of the incompleteness theorems. But these reflection principles have not received much attention in the philosophical community. The present chapter aims to survey some of the principal meta-mathematical results on the iteration of proof-theoretic reflection principles and investigate these results from a logico-philosophical perspective; we will concentrate on the epistemological significance of these technical results and on the epistemic notions involved in the proofs. In particular, we will focus on the notions of commitment to and acceptance of a theory. Special attention is given to the connection between proof-theoretic reflection and axiomatic truth theories. After distinguishing between different types of proof-theoretic reflection principles, we review some proof-theoretic results concerning extensions of formal theories by (iterated) reflection principles. As basis theories, we concentrate on standard arithmetical and elementary axiomatic truth theories. We then go on to explore the epistemological significance of these results. In this investigation, we aim to show that the epistemic notion of acceptance of (or commitment to) a theory plays a crucial role in the philosophical argumentation for reflection principles and their iteration.

Building on the seminal work of Kit Fine in the 1980s, Leon Horsten here develops a new theory of arbitrary entities. He connects this theory to issues and debates in metaphysics, logic, and contemporary philosophy of mathematics, investigating the relation between specific and arbitrary objects and between specific and arbitrary systems of objects. His book shows how this innovative theory is highly applicable to problems in the philosophy of arithmetic, and explores in particular how arbitrary objects can engage with the nineteenth-century concept of variable mathematical quantities, how they are relevant for debates around mathematical structuralism, and how they can help our understanding of the concept of random variables in statistics. This fully worked through theory will open up new avenues within philosophy of mathematics, bringing in the work of other philosophers such as Saul Kripke, and providing new insights into the development of the foundations of mathematics from the eighteenth century to the present day.

This article discusses Peter Galison's views on the structure and evolution of experimental and instrumental cultures in 20th century particle physics, which are unfolded in his recent book Image and Logic. A Material Culture of Microphysics. First a description is given of the uncomfortable predicament in which the Kuhnian tradition finds itself in the past two decades. It is then explained how Galison distinguishes a layered structure in the practice of modern particle physics. Physics as a practice consists of three relatively autonomous levels: the experimental level, the instrumental level and the theoretical level. This paper focuses mainly (but not exclusively) on the instrumental level in particle physics. Subsequently an analysis is given of Galison's metaphor of the wall, which Galison offers as an alternative to the Kuhnian image of paradigms and scientific revolutions. Galison's model is also compared with Laudan's reticulational model of the evolution of scientific research traditions. To conclude, an analysis is made of Galison's attempt to solve the incommensurability problem

Contents: Introduction. NATURALIZED EPISTEMOLOGY. Ton DERKSEN: Naturalistic Epistemology, Murder and Suicide? But what about the Promises! Christopher HOOKWAY: Naturalism and Rationality. Mia GOSSELIN: Quine's Hypothetical Theory of Language Learning. A Comparison of Different Conceptual Schemes of Their Logic. THE NATURE OF PERCEPTUAL KNOWLEDGE. Jaap van BRAKEL: Quine and Innate Similarity Spaces. Dirk KOPPELBERG: Quine and Davidson on the Structure of Empirical Knowledge. Eva PICARDI: Empathy and Charity. ONTOLOGY. Sandra LAUGIER: Quine: Indeterminacy, ‘Robust Realism', and Truth. Roger VERGAUWEN: Quine and Putnam on Conceptual Relativity and Reference: Theft or Honest Toil? Igor DOUVEN: Empiricist Semantics and Indeterminism of Reference. Lieven DECOCK: Domestic Ontology and Ideology. Paul GOCHET: Canonical Notation, Predication and Ontology. About the Authors

In his Gibbs lecture, Gödel argued for the thesis that either the human mind is not a Turing machine, or there exist absolutely undecidable mathematical propositions. He believed that this disjunction can be deduced with mathematical certainty from certain results in mathematical logic. He thought that his disjunctive thesis is of great philosophical importance. First, Gödel's argument for his disjunctive thesis is discussed. It is argued that thisargument contains an ambiguity. But when it is made precise in a certain way, the argument is seen to be ultimately inescapable. Second, the disjuncts of the thesis are considered in their own right. It is argued that there are today no convincing arguments in favor of the first disjunct, nor are there at present convincing arguments against it. But there do exist today strong reasons to believe the second disjunct to be true, although these reasons do not establish the disjunct with anything like mathematical certainty

The concept of truth is now a major research subject in analytic philosophy. At the same time, working in different areas, mathematical logicians have developed sophisticated theories of truth and its formal paradoxes. Recent developments of semantical paradoxes in logical theories are highly relevant for philosophical research on the notion of truth. And conversely, philosophical guidance is necessary for the development of logical theories of truth and the paradoxes. From this perspective, this volume intends to reflect and promote deeper interaction and collaboration between philosophers and logicians investigating the concept of truth than has existed so far. In addition to an extended introductory overview of recent work in the theory of truth, the volume includes articles by leading philosophers and logicians on subjects and debates that interface logical and philosophical theories of truth. The volume is intended for graduate students in philosophy and in logic who want an introduction to contemporary research in this area, as well as for professional philosophers and logicians. Topics covered include a defense of minimalism, metaworlds, the role of correspondence theory, and truth, provability and its criteria. The volume includes contributions by John P. Burgess, Paul Horwich, and Michael Sheard. The contributors are united in their belief that at present the connections between truth and intentional notions are poorly understood, and the entire area is in an unsatisfactory state. Controversy abounds: Is truth a logical-mathematic concept or a properly philosophical one? Does truth pertain to temporal events or possible worlds? Axiomatic theories do not seem to possess the naturalness and elegance intentional notions of truth. The editors intend this volume to offer semantical guidance to those who would construct interesting axiomatic systems concerning intentional notions. Volker Halbach is on the faculty of philosophy of the Universitt Konstanz in Germany. He has contributed articles for Mind, Notre Dame Journal of Formal Logic, Journal of Symbolic Logic, Synthese, and other leading publications in philosophy. Leon Horsten is in the Department of Philosophy, University of Leuven. His work on the subject of this book has appeared in the Journal of Philosophical Logic.

This book contains ten papers that were presented at the symposium about the realism debate, held at the Center for Logic, Philosophy of Science and Philosophy of Language of the Institute of Philosophy at the Katholieke Universiteit Leuven on 10 and 11 March 1995. The first group of papers are directly concerned with the realism/anti-realism debate in the general philosophy of science. This group includes the articles by Ernan McMullin, Diderik Batens/Joke Meheus, Igor Douven and Herman de Regt. The papers of the second group concentrate on specific problems arising from the realism/anti-realism debate. Theo Kuipers' contribution discusses the problem of truth-approximation. Roger Vergauwen's article pertains to the issue of realism in the philosophy of mind and semantics. Jaap van Brakel's article focusses on the relation between everyday concepts and scientific concepts, and on the theory-dependence of observation. Paul Cortois investigates the relation between the question of realism and Kuhn's concept of incommensurability between scientific theories. The final group contains two papers on the realism/anti-realism debate in the special sciences. James Cushing discusses the problem of underdetermination in quantum mechanics and Jean Paul van bendegem addresses the question of the possiblity of an empiricist philosophy of mathematics.

Given any finite graph, which transitive graphs approximate it most closely and how fast can we find them? The answer to this question depends on the concept of “closest approximation” involved. In [8,9] a qualitative concept of best approximation is formulated. Roughly, a qualitatively best transitive approximation of a graph is a transitive graph which cannot be “improved” without also going against the original graph. A quantitative concept of best approximation goes back at least to [10]. A quantitatively best transitive approximation is a transitive graph that makes the minimal number of mistakes against the original graph. In other words, the sum of the edges that are removed from and are added to the original graph is minimal.