The proper explanation of intuitionistic logic: on Brouwer's demonstration of the Bar Theorem

Brouwer's demonstration of his Bar Theorem gives rise to provocative questions regarding the proper explanation of the logical connectives within intuitionistic and constructivist frameworks, respectively, and, more generally, regarding the role of logic within intuitionism. It is the purpose of the present note to discuss a number of these issues, both from an historical, as well as a systematic point of view

Computability and analysis: the legacy of Alan Turing

We discuss the legacy of Alan Turing and his impact on computability and analysis.Comment: 49 page

Computability in constructive type theory

We give a formalised and machine-checked account of computability theory in the Calculus of Inductive Constructions (CIC), the constructive type theory underlying the Coq proof assistant. We first develop synthetic computability theory, pioneered by Richman, Bridges, and Bauer, where one treats all functions as computable, eliminating the need for a model of computation. We assume a novel parametric axiom for synthetic computability and give proofs of results like Rice’s theorem, the Myhill isomorphism theorem, and the existence of Post’s simple and hypersimple predicates relying on no other axioms such as Markov’s principle or choice axioms. As a second step, we introduce models of computation. We give a concise overview of definitions of various standard models and contribute machine-checked simulation proofs, posing a non-trivial engineering effort. We identify a notion of synthetic undecidability relative to a fixed halting problem, allowing axiom-free machine-checked proofs of undecidability. We contribute such undecidability proofs for the historical foundational problems of computability theory which require the identification of invariants left out in the literature and now form the basis of the Coq Library of Undecidability Proofs. We then identify the weak call-by-value λ-calculus L as sweet spot for programming in a model of computation. We introduce a certifying extraction framework and analyse an axiom stating that every function of type ℕ → ℕ is L-computable.Wir behandeln eine formalisierte und maschinengeprüfte Betrachtung von Berechenbarkeitstheorie im Calculus of Inductive Constructions (CIC), der konstruktiven Typtheorie die dem Beweisassistenten Coq zugrunde liegt. Wir entwickeln erst synthetische Berechenbarkeitstheorie, vorbereitet durch die Arbeit von Richman, Bridges und Bauer, wobei alle Funktionen als berechenbar behandelt werden, ohne Notwendigkeit eines Berechnungsmodells. Wir nehmen ein neues, parametrisches Axiom für synthetische Berechenbarkeit an und beweisen Resultate wie das Theorem von Rice, das Isomorphismus Theorem von Myhill und die Existenz von Post’s simplen und hypersimplen Prädikaten ohne Annahme von anderen Axiomen wie Markov’s Prinzip oder Auswahlaxiomen. Als zweiten Schritt führen wir Berechnungsmodelle ein. Wir geben einen kompakten Überblick über die Definition von verschiedenen Berechnungsmodellen und erklären maschinengeprüfte Simulationsbeweise zwischen diesen Modellen, welche einen hohen Konstruktionsaufwand beinhalten. Wir identifizieren einen Begriff von synthetischer Unentscheidbarkeit relativ zu einem fixierten Halteproblem welcher axiomenfreie maschinengeprüfte Unentscheidbarkeitsbeweise erlaubt. Wir erklären solche Beweise für die historisch grundlegenden Probleme der Berechenbarkeitstheorie, die das Identifizieren von Invarianten die normalerweise in der Literatur ausgelassen werden benötigen und nun die Basis der Coq Library of Undecidability Proofs bilden. Wir identifizieren dann den call-by-value λ-Kalkül L als sweet spot für die Programmierung in einem Berechnungsmodell. Wir führen ein zertifizierendes Extraktionsframework ein und analysieren ein Axiom welches postuliert dass jede Funktion vom Typ N→N L-berechenbar ist

Sofia A. Yanovskaya: The Marxist Pioneer of Mathematical Logic in the Soviet Union

K. Marx’s 200th jubilee coincides with the celebration of the 85 years from the first publication of his “Mathematical Manuscripts” in 1933. Its editor, Sofia Alexandrovna Yanovskaya (1896–1966), was a renowned Soviet mathematician, whose significant studies on the foundations of mathematics and mathematical logic, as well as on the history and philosophy of mathematics are unduly neglected nowadays. Yanovskaya, as a militant Marxist, was actively engaged in the ideological confrontation with idealism and its influence on modern mathematics and their interpretation. Concomitantly, she was one of the pioneers of mathematical logic in the Soviet Union, in an era of fierce disputes on its compatibility with Marxist philosophy. Yanovskaya managed to embrace in an originally Marxist spirit the contemporary level of logico-philosophical research of her time. Due to her highly esteemed status within Soviet academia, she became one of the most significant pillars for the culmination of modern mathematics in the Soviet Union. In this paper, I attempt to trace the influence of the complex socio-cultural context of the first decades of the Soviet Union on Yanovskaya’s work. Among the several issues I discuss, her encounter with L. Wittgenstein is striking

Experiments in integrating constraints with logical reasoning for robotic planning within the twelf logical framework and the prolog language

Ankara : The Department of Computer Engineering and the Institute of Engineering and Sciences of Bilkent University, 2008.Thesis (Master's) -- Bilkent University, 2008.Includes bibliographical references leaves 92-96.The underlying domain of various application areas, especially real-time systems and robotic applications, generally includes a combination of both discrete and continuous properties. In robotic applications, a large amount of different approaches are introduced to solve either a discrete planning or control theoretic problem. Only a few methods exist to solve the combination of them. Moreover, these methods fail to ensure a uniform treatment of both aspects of the domain. Therefore, there is need for a uniform framework to represent and solve such problems. A new formalism, the Constrained Intuitionistic Linear Logic (CILL), combines continuous constraint solvers with linear logic. Linear logic has a great property to handle hypotheses as resources, easily solving state transition problems. On the other hand, constraint solvers deal well with continuous problems defined as constraints. Both properties of CILL gives us powerful ways to express and reason about the robotics domain. In this thesis, we focus on the implementation of CILL in both the Twelf Logical Framework and Prolog. The reader of this thesis can find answers of why classical aspects are not proper for the robotics domain, what advantages one can gain from intuitionism and linearity, how one can define a simple robotic domain in a logical formalism, how a proof in logical system corresponds to a plan in the robotic domain, what the advantages and disadvantages of logical frameworks and Prolog have and how the implementation of CILL can or cannot be done using both Twelf Logical Framework and Prolog.Duatepe, MertM.S

A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography

We analyze the developments in mathematical rigor from the viewpoint of a Burgessian critique of nominalistic reconstructions. We apply such a critique to the reconstruction of infinitesimal analysis accomplished through the efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy's foundational work associated with the work of Boyer and Grabiner; and to Bishop's constructivist reconstruction of classical analysis. We examine the effects of a nominalist disposition on historiography, teaching, and research.Comment: 57 pages; 3 figures. Corrected misprint