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Summary Mathematical proof concerns itself with a demonstration that some theorem, lemma, corollary or claim is true. Proofs rely upon previously proven statements, logical inferences, and a specified syntax, which can usually trace back to underlying axioms and definitions. Many of the issues in this area concern the use of purely formal proof, informal proof, language, empirical methodologies, and everyday practice. 
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  1. The Epistemological Subject(s) of Mathematics.Silvia De Toffoli - 2023 - In B. Sriraman (ed.), Handbook of the History and Philosophy of Mathematical Practice. Switzerland: Springer Nature. pp. 1-27.
    Paying attention to the inner workings of mathematicians has led to a proliferation of new themes in the philosophy of mathematics. Several of these have to do with epistemology. Philosophers of mathematical practice, however, have not (yet) systematically engaged with general (analytic) epistemology. To be sure, there are some exceptions, but they are few and far between. In this chapter, I offer an explanation of why this might be the case and show how the situation could be remedied. I contend (...)
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  2. Understanding mathematical proof.John Taylor - 2014 - Boca Raton: Taylor & Francis. Edited by Rowan Garnier.
    The notion of proof is central to mathematics yet it is one of the most difficult aspects of the subject to teach and master. In particular, undergraduate mathematics students often experience difficulties in understanding and constructing proofs. Understanding Mathematical Proof describes the nature of mathematical proof, explores the various techniques that mathematicians adopt to prove their results, and offers advice and strategies for constructing proofs. It will improve students’ ability to understand proofs and construct correct proofs of their own. The (...)
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  3. The story of proof: logic and the history of mathematics.John Stillwell - 2022 - Princeton, New Jersey: Princeton University Press.
    How the concept of proof has enabled the creation of mathematical knowledge. The Story of Proof investigates the evolution of the concept of proof--one of the most significant and defining features of mathematical thought--through critical episodes in its history. From the Pythagorean theorem to modern times, and across all major mathematical disciplines, John Stillwell demonstrates that proof is a mathematically vital concept, inspiring innovation and playing a critical role in generating knowledge. Stillwell begins with Euclid and his influence on the (...)
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  4. The meaning of proofs: mathematics as storytelling.Gabriele Lolli - 2022 - Cambridge, Massachusetts: The MIT Press. Edited by Bonnie McClellan-Broussard & Matilde Marcolli.
    This book introduces readers to the narrative structure of mathematical proofs and why mathematicians communicate that way, drawing examples from classic literature and employing metaphors and imagery.
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  5. An introduction to proof via inquiry-based learning.Dana C. Ernst - 2022 - Providence, Rhode Island: MAA Press, an imprint of the American Mathematical Society.
    An Introduction to Proof via Inquiry-Based Learning is a textbook for the transition to proof course for mathematics majors. Designed to promote active learning through inquiry, the book features a highly structured set of leading questions and explorations. The reader is expected to construct their own understanding by engaging with the material. The content ranges over topics traditionally included in transitions courses: logic, set theory including cardinality, the topology of the real line, a bit of number theory, and more. The (...)
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  6. Who's Afraid of Mathematical Diagrams?Silvia De Toffoli - 2023 - Philosophers' Imprint 23 (1).
    Mathematical diagrams are frequently used in contemporary mathematics. They are, however, widely seen as not contributing to the justificatory force of proofs: they are considered to be either mere illustrations or shorthand for non-diagrammatic expressions. Moreover, when they are used inferentially, they are seen as threatening the reliability of proofs. In this paper, I examine certain examples of diagrams that resist this type of dismissive characterization. By presenting two diagrammatic proofs, one from topology and one from algebra, I show that (...)
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  7. Beweistheorie.K. Schütte - 1960 - Berlin,: Springer.
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  8. Formalization of Mathematical Proof Practice Through an Argumentation-Based Model.Sofia Almpani, Petros Stefaneas & Ioannis Vandoulakis - 2023 - Axiomathes 33 (3):1-28.
    Proof requires a dialogue between agents to clarify obscure inference steps, fill gaps, or reveal implicit assumptions in a purported proof. Hence, argumentation is an integral component of the discovery process for mathematical proofs. This work presents how argumentation theories can be applied to describe specific informal features in the development of proof-events. The concept of proof-event was coined by Goguen who described mathematical proof as a public social event that takes place in space and time. This new meta-methodological concept (...)
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  9. Wahrheit und Beweisbarkeit: e. Unters. über d. Verhältnis von Denken u. Anschauung in d. Mathematik.Johann Glöckl - 1976 - Bonn: Bouvier.
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  10. Syllogistic Logic and Mathematical Proof.Paolo Mancosu & Massimo Mugnai - 2023 - Oxford, GB: Oxford University Press.
    Does syllogistic logic have the resources to capture mathematical proof? This volume provides the first unified account of the history of attempts to answer this question, the reasoning behind the different positions taken, and their far-reaching implications. Aristotle had claimed that scientific knowledge, which includes mathematics, is provided by syllogisms of a special sort: 'scientific' ('demonstrative') syllogisms. In ancient Greece and in the Middle Ages, the claim that Euclid's theorems could be recast syllogistically was accepted without further scrutiny. Nevertheless, as (...)
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  11. Beweisen im Mathematik-Unterricht: didakt, Anwendungen d. Lehre vom log. Schliessen.Peter Zahn - 1979 - Darmstadt: Wissenschaftliche Buchgesellschaft, [Abt. Verl.].
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  12. Lehren des Beweisens im Mathematikunterricht.Walter Witzel - 1981 - Freiburg (Breisgau): Hochschulverlag.
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  13. Mathematical Proving as Multi-Agent Spatio-Temporal Activity.Ioannis M. Vandoulakis & Petros Stefaneas - 2016 - In Modelling, Logical and Philosophical Aspects of Foundations of Science. Lambert Academic Publishing. pp. 183-200.
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  14. Objectivity and Rigor in Classical Italian Algebraic Geometry.Silvia De Toffoli & Claudio Fontanari - 2022 - Noesis 38:195-212.
    The classification of algebraic surfaces by the Italian School of algebraic geometry is universally recognized as a breakthrough in 20th-century mathematics. The methods by which it was achieved do not, however, meet the modern standard of rigor and therefore appear dubious from a contemporary viewpoint. In this article, we offer a glimpse into the mathematical practice of the three leading exponents of the Italian School of algebraic geometry: Castelnuovo, Enriques, and Severi. We then bring into focus their distinctive conception of (...)
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  15. Rigor and the Context-Dependence of Diagrams: The Case of Euler Diagrams.David Waszek - 2018 - In Peter Chapman, Gem Stapleton, Amirouche Moktefi, Sarah Perez-Kriz & Francesco Bellucci (eds.), Diagrammatic Representation and Inference. Cham: Springer. pp. 382-389.
    Euler famously used diagrams to illustrate syllogisms in his Lettres à une princesse d’Allemagne [1]. His diagrams are usually seen as suffering from a fatal “ambiguity problem” [11]: as soon as they involve intersecting circles, which are required for the representation of existential statements, it becomes unclear what exactly may be read off from them, and as Hammer & Shin conclusively showed, any set of reading conventions can lead to erroneous conclusions. I claim that Euler diagrams can, however, be used (...)
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  16. Introduction to mathematical proof: a transition to advanced mathematics.Charles E. Roberts - 2015 - Boca Raton: CRC Press, Taylor & Francis Group.
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  17. Building proofs: a practical guide.Suely Oliveira - 2015 - New Jersey: World Scientific. Edited by David Stewart.
    This book introduces students to the art and craft of writing proofs, beginning with the basics of writing proofs and logic, and continuing on with more in-depth issues and examples of creating proofs in different parts of mathematics, as well as introducing proofs-of-correctness for algorithms. The creation of proofs is covered for theorems in both discrete and continuous mathematics, and in difficulty ranging from elementary to beginning graduate level. Just beyond the standard introductory courses on calculus, theorems and proofs become (...)
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  18. Proof theory: sequent calculi and related formalisms.Katalin Bimbo - 2014 - Boca Raton: CRC Press, Taylor & Francis Group.
    Sequent calculi constitute an interesting and important category of proof systems. They are much less known than axiomatic systems or natural deduction systems are, and they are much less known than they should be. Sequent calculi were designed as a theoretical framework for investigations of logical consequence, and they live up to the expectations completely as an abundant source of meta-logical results. The goal of this book is to provide a fairly comprehensive view of sequent calculi -- including a wide (...)
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  19. Fundamentals of mathematical proof.Charles Matthews - 2018 - [place of publication not identified]: [Publisher Not Identified].
    This mathematics textbook covers the fundamental ideas used in writing proofs. Proof techniques covered include direct proofs, proofs by contrapositive, proofs by contradiction, proofs in set theory, proofs of existentially or universally quantified predicates, proofs by cases, and mathematical induction. Inductive and deductive reasoning are explored. A straightforward approach is taken throughout. Plenty of examples are included and lots of exercises are provided after each brief exposition on the topics at hand. The text begins with a study of symbolic logic, (...)
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  20. Mathematical proofs: a transition to advanced mathematics.Gary Chartrand - 2018 - Boston: Pearson. Edited by Albert D. Polimeni & Ping Zhang.
    For courses in Transition to Advanced Mathematics or Introduction to Proof. Meticulously crafted, student-friendly text that helps build mathematical maturity Mathematical Proofs: A Transition to Advanced Mathematics, 4th Edition introduces students to proof techniques, analyzing proofs, and writing proofs of their own that are not only mathematically correct but clearly written. Written in a student-friendly manner, it provides a solid introduction to such topics as relations, functions, and cardinalities of sets, as well as optional excursions into fields such as number (...)
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  21. An introduction to mathematical proofs.Nicholas A. Loehr - 2019 - Boca Raton: CRC Press, Taylor & Francis Group.
    This book contains an introduction to mathematical proofs, including fundamental material on logic, proof methods, set theory, number theory, relations, functions, cardinality, and the real number system. The book is divided into approximately fifty brief lectures. Each lecture corresponds rather closely to a single class meeting.
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  22. The science of learning mathematical proofs: an introductory course.Elana Reiser - 2020 - New Jersey: World Scientific.
    College students struggle with the switch from thinking of mathematics as a calculation based subject to a problem solving based subject. This book describes how the introduction to proofs course can be taught in a way that gently introduces students to this new way of thinking. This introduction utilizes recent research in neuroscience regarding how the brain learns best. Rather than jumping right into proofs, students are first taught how to change their mindset about learning, how to persevere through difficult (...)
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  23. Idéaux de preuve : explication et pureté.Andrew Arana - 2022 - In Andrew Arana & Marco Panza (eds.), Précis de philosophie de la logique et des mathématiques. Volume 2, philosophie des mathématiques. Paris, France: pp. 387-425.
    Why do mathematics often give several proofs of the same theorem? This is the question raised in this article, introducing the notion of an epistemic ideal and discussing two such ideals, the explanatoriness and purity of proof.
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  24. Reconciling Rigor and Intuition.Silvia De Toffoli - 2020 - Erkenntnis 86 (6):1783-1802.
    Criteria of acceptability for mathematical proofs are field-dependent. In topology, though not in most other domains, it is sometimes acceptable to appeal to visual intuition to support inferential steps. In previous work :829–842, 2014; Lolli, Panza, Venturi From logic to practice, Springer, Berlin, 2015; Larvor Mathematical cultures, Springer, Berlin, 2016) my co-author and I aimed at spelling out how topological proofs work on their own terms, without appealing to formal proofs which might be associated with them. In this article, I (...)
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  25. Neues System der philosophischen Wissenschaften im Grundriss. Band II: Mathematik und Naturwissenschaft.Dirk Hartmann - 2021 - Paderborn: Mentis.
    Volume II deals with philosophy of mathematics and general philosophy of science. In discussing theoretical entities, the notion of antirealism formulated in Volume I is further elaborated: Contrary to what is usually attributed to antirealism or idealism, the author does not claim that theoretical entities do not really exist, but rather that their existence is not independent of the possibility to know about them.
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  26. Mathematical Inference and Logical Inference.Yacin Hamami - 2018 - Review of Symbolic Logic 11 (4):665-704.
    The deviation of mathematical proof—proof in mathematical practice—from the ideal of formal proof—proof in formal logic—has led many philosophers of mathematics to reconsider the commonly accepted view according to which the notion of formal proof provides an accurate descriptive account of mathematical proof. This, in turn, has motivated a search for alternative accounts of mathematical proof purporting to be more faithful to the reality of mathematical practice. Yet, in order to develop and evaluate such alternative accounts, it appears as a (...)
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  27. Poincaré and Prawitz on mathematical induction.Yacin Hamami - 2015 - In Pavel Arazim & Michal Dancak (eds.), Logica Yearbook 2014. London: College Publications. pp. 149-164.
    Poincaré and Prawitz have both developed an account of how one can acquire knowledge through reasoning by mathematical induction. Surprisingly, their two accounts are very close to each other: both consider that what underlies reasoning by mathematical induction is a certain chain of inferences by modus ponens ‘moving along’, so to speak, the well-ordered structure of the natural numbers. Yet, Poincaré’s central point is that such a chain of inferences is not sufficient to account for the knowledge acquisition of the (...)
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  28. A Note on Wittgenstein’s “Notorious Paragraph” About the Gödel Theorem.Juliet Floyd & Hilary Putnam - 2000 - Journal of Philosophy 97 (11):624-632.
    A look at Wittgenstein's comments on the incompleteness theorem with an inter-pretation that is consistent with what Gödel proved.
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  29. Institutionism, Pluralism, and Cognitive Command.Stewart Shapiro & William W. Taschek - 1996 - Journal of Philosophy 93 (2):74.
  30. Why Do We Prove Theorems?Yehuda Rav - 1999 - Philosophia Mathematica 7 (1):5-41.
    Ordinary mathematical proofs—to be distinguished from formal derivations—are the locus of mathematical knowledge. Their epistemic content goes way beyond what is summarised in the form of theorems. Objections are raised against the formalist thesis that every mainstream informal proof can be formalised in some first-order formal system. Foundationalism is at the heart of Hilbert's program and calls for methods of formal logic to prove consistency. On the other hand, ‘systemic cohesiveness’, as proposed here, seeks to explicate why mathematical knowledge is (...)
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  31. Explanatory Circles, Induction, and Recursive Structures.Tomasz Wysocki - 2016 - Thought: A Journal of Philosophy 6 (1):13-16.
    Lange offers an argument that, according to him, “does not show merely that some proofs by mathematical induction are not explanatory. It shows that none are […]”. The aim here is to present a counterexample to his argument.
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  32. Proof, rigour and informality : a virtue account of mathematical knowledge.Fenner Stanley Tanswell - 2016 - St Andrews Research Repository Philosophy Dissertations.
    This thesis is about the nature of proofs in mathematics as it is practiced, contrasting the informal proofs found in practice with formal proofs in formal systems. In the first chapter I present a new argument against the Formalist-Reductionist view that informal proofs are justified as rigorous and correct by corresponding to formal counterparts. The second chapter builds on this to reject arguments from Gödel's paradox and incompleteness theorems to the claim that mathematics is inherently inconsistent, basing my objections on (...)
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  33. Why Do We Prove Theorems?Yehuda Rav - 1998 - Philosophia Mathematica 6 (3):5-41.
    Ordinary mathematical proofs—to be distinguished from formal derivations—are the locus of mathematical knowledge. Their epistemic content goes way beyond what is summarised in the form of theorems. Objections are raised against the formalist thesis that every mainstream informal proof can be formalised in some first-order formal system. Foundationalism is at the heart of Hilbert's program and calls for methods of formal logic to prove consistency. On the other hand, ‘systemic cohesiveness’, as proposed here, seeks to explicate why mathematical knowledge is (...)
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  34. C.K. Raju. Cultural Foundations of Mathematics: The Nature of Mathematical Proof and the Transmission of the Calculus from India to Europe in the 16th c. CE.: Critical Studies/Book Reviews. [REVIEW]José FerreiróS. - 2009 - Philosophia Mathematica 17 (3):378-381.
    This book is part of a major project undertaken by the Centre for Studies in Civilizations , being one of a total of ninety-six planned volumes. The author is a statistician and computer scientist by training, who has concentrated on historical matters for the last ten years or so. The book has very ambitious aims, proposing an alternative philosophy of mathematics and a deviant history of the calculus. Throughout, there is an emphasis on the need to combine history and philosophy (...)
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  35. Francesco Berto. There's Something about Gödel. Malden, Mass., and Oxford: Wiley-Blackwell, 2009. ISBN 978-1-4051-9766-3 ; 978-1-4051-9767-0 . Pp. xx + 233. English translation of Tutti pazzi per Gödel! : Critical Studies/Book Reviews. [REVIEW]Vann Mcgee - 2011 - Philosophia Mathematica 19 (3):367-369.
    There's Something about Gödel is a bargain: two books in one. The first half is a gentle but rigorous introduction to the incompleteness theorems for the mathematically uninitiated. The second is a survey of the philosophical, psychological, and sociological consequences people have attempted to derive from the theorems, some of them quite fantastical.The first part, which stays close to Gödel's original proofs, strikes a nice balance, giving enough details that the reader understands what is going on in the proofs, without (...)
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  36. Some Recent Developments in Complete Strategies for Theorem‐Proving by Computer.Bernard Meltzer - 1968 - Mathematical Logic Quarterly 14 (25-29):377-382.
  37. Proof and Knowledge in Mathematics.Janet Folina - 1996 - Philosophical Quarterly 46 (182):125-127.
  38. Pi on Earth, or Mathematics in the Real World.Bart Van Kerkhove & Jean Paul Van Bendegem - 2008 - Erkenntnis 68 (3):421-435.
    We explore aspects of an experimental approach to mathematical proof, most notably number crunching, or the verification of subsequent particular cases of universal propositions. Since the rise of the computer age, this technique has indeed conquered practice, although it implies the abandonment of the ideal of absolute certainty. It seems that also in mathematical research, the qualitative criterion of effectiveness, i.e. to reach one’s goals, gets increasingly balanced against the quantitative one of efficiency, i.e. to minimize one’s means/ends ratio. Our (...)
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  39. Michael Detlefsen (ed.), Proof, Logic and Formalization. Michael Detlefsen (ed.), Proof and Knowledge in Mathematics. [REVIEW]Luiz Carlos Pereira - 1997 - Erkenntnis 47 (2):245-254.
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  40. Looking for Busy Beavers. A socio-philosophical study of a computer-assisted proof.Liesbeth de Mol - unknown
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  41. God, Human Memory, and the Certainty of Geometry: An Argument Against Descartes.Marc Champagne - 2016 - Philosophy and Theology 28 (2):299–310.
    Descartes holds that the tell-tale sign of a solid proof is that its entailments appear clearly and distinctly. Yet, since there is a limit to what a subject can consciously fathom at any given moment, a mnemonic shortcoming threatens to render complex geometrical reasoning impossible. Thus, what enables us to recall earlier proofs, according to Descartes, is God’s benevolence: He is too good to pull a deceptive switch on us. Accordingly, Descartes concludes that geometry and belief in God must go (...)
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  42. What Godel's Incompleteness Result Does and Does Not Show.Haim Gaifman - 2000 - Journal of Philosophy 97 (8):462.
    In a recent paper S. McCall adds another link to a chain of attempts to enlist Gödel’s incompleteness result as an argument for the thesis that human reasoning cannot be construed as being carried out by a computer.1 McCall’s paper is undermined by a technical oversight. My concern however is not with the technical point. The argument from Gödel’s result to the no-computer thesis can be made without following McCall’s route; it is then straighter and more forceful. Yet the argument (...)
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  43. What is a Proof?Reinhard Kahle - 2015 - Axiomathes 25 (1):79-91.
    In this programmatic paper we renew the well-known question “What is a proof?”. Starting from the challenge of the mathematical community by computer assisted theorem provers we discuss in the first part how the experiences from examinations of proofs can help to sharpen the question. In the second part we have a look to the new challenge given by “big proofs”.
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  44. Gödel's Third Incompleteness Theorem.Timothy McCarthy - 2016 - Dialectica 70 (1):87-112.
    In a note appended to the translation of “On consistency and completeness” (), Gödel reexamined the problem of the unprovability of consistency. Gödel here focuses on an alternative means of expressing the consistency of a formal system, in terms of what would now be called a ‘reflection principle’, roughly, the assertion that a formula of a certain class is provable in the system only if it is true. Gödel suggests that it is this alternative means of expressing consistency that we (...)
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  45. Gödel’s Incompleteness Theorems and Artificial Life.John P. Sullins Iii - 1997 - Techné: Research in Philosophy and Technology 2 (3):185-195.
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  46. Gödel’s Incompleteness Theorems and Physics.Newton C. A. Da Costa - 2011 - Principia: An International Journal of Epistemology 15 (3):453–459.
    This paper is a summary of a lecture in which I presented some remarks on Gödel’s incompleteness theorems and their meaning for the foundations of physics. The entire lecture will appear elsewhere.
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  47. Tableaux and Dual Tableaux: Transformation of Proofs.Joanna Golińska-Pilarek & Ewa Orłowska - 2007 - Studia Logica 85 (3):283-302.
    We present two proof systems for first-order logic with identity and without function symbols. The first one is an extension of the Rasiowa-Sikorski system with the rules for identity. This system is a validity checker. The rules of this system preserve and reflect validity of disjunctions of their premises and conclusions. The other is a Tableau system, which is an unsatisfiability checker. Its rules preserve and reflect unsatisfiability of conjunctions of their premises and conclusions. We show that the two systems (...)
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  48. A note on incompleteness and heterologicality.P. M. Sullivan - 2003 - Analysis 63 (1):32-38.
  49. Gödel's Second Incompleteness Theorem for General Recursive Arithmetic.William Ryan - 1978 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 24 (25-30):457-459.
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  50. Remarks on the Incompleteness Proof.Gerold Stahl - 1961 - Mathematical Logic Quarterly 7 (11-14):164-170.
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