Klaus Frovin Jørgensen

Visualization, Explanation and Reasoning Styles in Mathematics

This book contains groundbreaking contributions to the philosophical analysis of mathematical practice. Several philosophers of mathematics have recently called for an approach to philosophy of mathematics that pays more attention to mathematical practice. Questions concerning concept-formation, understanding, heuristics, changes in style of reasoning, the role of analogies and diagrams, etc. have become the subject of intense interest. The historians and philosophers in this book agree that there is more to understanding mathematics than a study of its logical structure. How are mathematical objects and concepts generated? How does the process tie up with justification? What role do visual images and diagrams play in mathematical activity? What are the different epistemic virtues (explanatoriness, understanding, visualizability, etc.) which are pursued and cherished by mathematicians in their work? The reader will find here systematic philosophical analyses as well as a wealth of philosophically informed case studies ranging from Babylonian, Greek, and Chinese mathematics to nineteenth century real and complex analysis.

Proof Theory, History and Philosophical Significance

Preface. Contributing Authors. Introduction. Part 1: Review of Proof Theory. Highlights in Proof Theory S. Feferman. Part 2: The Background of Hilberts Proof Theory. The Empiricist Roots of Hilberts Axiomatic Approach L. Corry. The Calm Before the Storm: Hilberts Early Views on Foundations D. Rowe. Toward Finitist Proof Theory W. Sieg. Part 3: Brouwer and Weyl on Proof Theory and Philosophy of Mathematics. The Development of Brouwers Intuitionism D. van Dalen. Did Brouwers Intuitionistic Analysis Satisfy its own Epistemological Standards? M. Epple. The Significance of Weyls Das Kontinuum S. Feferman. Herman Weyl on the Concept of Continuum E. Scholz. Part 4: Modern Views and Results from Proof Theory. Relationships between Constructive, Predicative and Classical Systems of Analysis S. Feferman. Index.

Contextual Validity in Hybrid Logic

Hybrid tense logic is an extension of Priorean tense logic in which it is possible to refer to times using special propositional symbols called nominals. Temporal indexicals are expressions such as now, yesterday, today, tomorrow and four days ago that have highly context-dependent interpretations. Moreover, such indexicals give rise to a special kind of validity--contextual validity--that interacts with ordinary logical validity in interesting and often unexpected ways. In this paper we model these interactions by combining standard techniques from hybrid logic with insights from the work of Hans Kamp and David Kaplan. We introduce a simple proof rule, which we call the Kamp Rule, and first we show that it is all we need to take us from logical validities involving now to contextual validities involving now too. We then go on to show that this deductive bridge is strong enough to carry us to contextual validities involving yesterday, today and tomorrow as well.

Functional interpretation and the existence property

It is shown that functional interpretation can be used to show the existence property of intuitionistic number theory. On the basis of truth variants a comparison is then made between realisability and functional interpretation showing a structural difference between the two. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Arthur Prior and ‘Now’

On the 4th of December 1967, Hans Kamp sent his UCLA seminar notes on the logic of ‘now’ to Arthur N. Prior. Kamp’s two-dimensional analysis stimulated Prior to an intense burst of creativity in which he sought to integrate Kamp’s work into tense logic using a one-dimensional approach. Prior’s search led him through the work of Castañeda, and back to his own work on hybrid logic: the first made temporal reference philosophically respectable, the second made it technically feasible in a modal framework. With the aid of hybrid logic, Prior built a bridge from a two-dimensional UT calculus to a one-dimensional tense logic containing the ‘now’ operator J. Drawing on material from the Prior archive, and the paper “‘Now”’ that detailed Prior’s findings, we retell this story. We focus on Prior’s completeness conjecture for the hybrid system and the role played by temporal reference.

A Seligman-Style Tableau System

Proof systems for hybrid logic typically use @-operators to access information hidden behind modalities; this labeling approach lies at the heart of most resolution, natural deduction, and tableau systems for hybrid logic. But there is another, less well-known approach, which we have come to believe is conceptually clearer. We call this Seligman-style inference, as it was first introduced and explored by Jerry Seligman in the setting of natural deduction and sequent calculus in the late 1990s. The purpose of this paper is to introduce a Seligman-style tableau system.

Reichenbach, Prior and hybrid tense logic

In this paper we argue that Prior and Reichenbach are best viewed as allies, not antagonists. We do so by combining the central insights of Prior and Reichenbach in the framework of hybrid tense logic. This overcomes a well-known defect of Reichenbach’s tense schema, namely that it gives multiple representations to sentences in the future perfect and the future-in-the-past. It also makes it easy to define an iterative schema for tense that allows for multiple points of reference, a possibility noted by Prior and demanded by Comrie, and we sketch how this schema can be generalized to a shift-and-restrict pattern in which special propositional symbols (for adverbials and indexicals) act as restrictors on the range of tense operators.

Henkin and Hybrid Logic

Leon Henkin was not a modal logician, but there is a branch of modal logic that has been deeply influenced by his work. That branch is hybrid logic, a family of logics that extend orthodox modal logic with special propositional symbols (called nominals) that name worlds. This paper explains why Henkin’s techniques are so important in hybrid logic. We do so by proving a completeness result for a hybrid type theory called HTT, probably the strongest hybrid logic that has yet been explored. Our completeness result builds on earlier work with a system called BHTT, or basic hybrid type theory, and draws heavily on Henkin’s work. We prove our Lindenbaum Lemma using a Henkin-inspired strategy, witnessing ◊-prefixed expressions with nominals. Our use of general interpretations and the construction of the type hierarchy is (almost) pure Henkin. Finally, the generality of our completeness result is due to the first-order perspective, which lies at the heart of both Henkin’s best known work and hybrid logic.