136 research outputs found

    (b2023 to 2014) The UNBELIEVABLE similarities between the ideas of some people (2006-2016) and my ideas (2002-2008) in physics (quantum mechanics, cosmology), cognitive neuroscience, philosophy of mind, and philosophy (this manuscript would require a REVOLUTION in international academy environment!)

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    (b2023 to 2014) The UNBELIEVABLE similarities between the ideas of some people (2006-2016) and my ideas (2002-2008) in physics (quantum mechanics, cosmology), cognitive neuroscience, philosophy of mind, and philosophy (this manuscript would require a REVOLUTION in international academy environment!

    Kolmogorov's Calculus of Problems and Its Legacy

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    Kolmogorov's Calculus of Problems is an interpretation of Heyting's intuitionistic propositional calculus published by A.N. Kolmogorov in 1932. Unlike Heyting's intended interpretation of this calculus, Kolmogorov's interpretation does not comply with the philosophical principles of Mathematical Intuitionism. This philosophical difference between Kolmogorov and Heyting implies different treatments of problems and propositions: while in Heyting's view the difference between problems and propositions is merely linguistic, Kolmogorov keeps the two concepts apart and does not apply his calculus to propositions. I stress differences between Kolmogorov's and Heyting's interpretations and show how the two interpretations diverged during their development. In this context I reconstruct Kolmogorov's philosophical views on mathematics and analyse his original take on the Hilbert-Brouwer controversy. Finally, I overview some later works motivated by Kolmogorov's Calculus of Problems and propose a justification of Kolmogorov's distinction between problems and propositions in terms of Univalent Mathematics.Comment: 66 pages including Appendi

    Kolmogorov's Calculus of Problems and Its Legacy

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    Kolmogorov's Calculus of Problems is an interpretation of Heyting's intuitionistic propositional calculus published by A.N. Kolmogorov in 1932. Unlike Heyting's intended interpretation of this calculus, Kolmogorov's interpretation does not comply with the philosophical principles of Mathematical Intuitionism. This philosophical difference between Kolmogorov and Heyting implies different treatments of problems and propositions: while in Heyting's view the difference between problems and propositions is merely linguistic, Kolmogorov keeps the two concepts apart and does not apply his calculus to propositions. I stress differences between Kolmogorov's and Heyting's interpretations and show how the two interpretations diverged during their development. In this context I reconstruct Kolmogorov's philosophical views on mathematics and analyse his original take on the Hilbert-Brouwer controversy. Finally, I overview some later works motivated by Kolmogorov's Calculus of Problems and propose a justification of Kolmogorov's distinction between problems and propositions in terms of Univalent Mathematics

    Computability in constructive type theory

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    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

    Computing NP-complete problems in polynomial time by means of Physics

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    Can NP-complete problems be solved efficiently in the physical universe? Some researchers have claimed to be able to solve NP-complete problems in polynomial time by encoding the problem in the state of a physical system and letting it evolve naturally, according to the laws of physics. However, their proposals have not proven to be very effective in practice. Additionally, there are several reasons to believe that those methods would not work if P 6= NP. We present some physical assumptions (both from classical physics and quantum mechanics) that would allow us to provably solve NP-complete problems in polynomial time by means of Physics, even if P 6= NP and NP 6⊂ BQP. We also study if our proposals are consistent with currently known laws of Physics

    Sets, Logic, Computation: An Open Introduction to Metalogic

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    An introductory textbook on metalogic. It covers naive set theory, first-order logic, sequent calculus and natural deduction, the completeness, compactness, and Löwenheim-Skolem theorems, Turing machines, and the undecidability of the halting problem and of first-order logic. The audience is undergraduate students with some background in formal logic

    General Course Catalog [July-December 2020]

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    Undergraduate Course Catalog, July-December 2020https://repository.stcloudstate.edu/undergencat/1132/thumbnail.jp

    General Course Catalog [January-June 2020]

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    Undergraduate Course Catalog, January-June 2020https://repository.stcloudstate.edu/undergencat/1131/thumbnail.jp

    Wittgenstein’s Remarks on Mathematics, Turing and Computability

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    Typically, Wittgenstein is assumed to have been apathetic to the developments in computability theory through the 1930s. Wittgenstein’s disparaging remarks about Gödel’s incompleteness theorems, and mathematical logic in general, are well documented. It seems safe to assume the same would apply for Turing’s work. The chief aim of this thesis is to debunk this picture. I will show that: a) Wittgenstein read, understood and engaged with Turing’s proofs regarding the Entscheidungsproblem. b) Wittgenstein’s remarks on this topic are highly perceptive and have pedagogical value, shedding light on Turing’s work. c) Wittgenstein was highly supportive of Turing’s work as it manifested Wittgenstein’s prevailing approach to mathematics. d) Adopting a Wittgensteinian approach to Turing’s proofs enables us to answer several live problems in the modern literature on computability. Wittgenstein was notably resistant to Cantor’s diagonal proof regarding uncountability, being a finitist and extreme anti-platonist. He was interested, however, in the diagonal method. He made several remarks attempting to adapt the method to work in purely intensional, rule-governed terms. These are unclear and unsuccessful. Turing’s famous diagonal application realised this pursuit. Turing’s application draws conclusions from the diagonal procedure without having to posit infinite extensions. Wittgenstein saw this, and made a series of interesting remarks to that effect. He subsequently gave his own (successful) intensional diagonal proof, abstracting from Turing’s. He endorsed Turing’s proof and reframed it in terms of games to highlight certain features of rules and rule-following. I then turn to the Church-Turing thesis (CTT). I show how Wittgenstein endorsed the CTT, particularly Turing’s rendition of it. Finally, I show how adopting a family-resemblance approach to computability can answer several questions regarding the epistemological status of the CTT today
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