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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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Introductions Horsten 2008
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  1. 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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  2. 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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  3. 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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  4. Institutionism, Pluralism, and Cognitive Command.Stewart Shapiro & William W. Taschek - 1996 - Journal of Philosophy 93 (2):74.
  5. 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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  6. 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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  7. 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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  8. 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 &Plus; 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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  9. Some Recent Developments in Complete Strategies for Theorem-Proving by Computer.Bernard Meltzer - 1968 - Mathematical Logic Quarterly 14 (25-29):377-382.
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  10. 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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  11. Michael Detlefsen , Proof, Logic and Formalization. Michael Detlefsen , Proof and Knowledge in Mathematics.Luiz Carlos Pereira - 1997 - Erkenntnis 47 (2):245-254.
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  12. Looking for Busy Beavers. A Socio-Philosophical Study of a Computer-Assisted Proof.Liesbeth De Mol - unknown
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  13. 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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  14. 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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  15. 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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  16. 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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  17. Gödel’s Incompleteness Theorems and Physics.Newton C. A. Da Costa - 2011 - Principia: An International Journal of Epistemology 15 (3).
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  18. 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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  19. A Note on Incompleteness and Heterologicality.P. M. Sullivan - 2003 - Analysis 63 (1):32-38.
  20. 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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  21. Remarks on the Incompleteness Proof.Gerold Stahl - 1961 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 7 (11-14):164-170.
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  22. Proofs as Spatio-Temporal Processes.Petros Stefaneas & Ioannis M. Vandoulakis - 2014 - Philosophia Scientae 18:111-125.
    The concept of proof can be studied from many different perspectives. Many types of proofs have been developed throughout history such as apodictic, dialectical, formal, constructive and non-constructive proofs, proofs by visualisation, assumption-based proofs, computer-generated proofs, etc. In this paper, we develop Goguen’s general concept of proof-events and the methodology of algebraic semiotics, in order to define the concept of mathematical style, which characterizes the proofs produced by different cultures, schools or scholars. In our view, style can be defined as (...)
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  23. Gödel’s Incompleteness Phenomenon—Computationally.Saeed Salehi - 2014 - Philosophia Scientae 18:23-37.
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  24. On Bolzano’s Alleged Explicativism.Jacques Dubucs & Sandra Lapointe - 2006 - Synthese 150 (2):229-246.
    Bolzano was the first to establish an explicit distinction between the deductive methods that allow us to recognise the certainty of a given truth and those that provide its objective ground. His conception of the relation between what we, in this paper, call "subjective consequence", i.e., the relation from epistemic reason to consequence and "objective consequence", i.e., grounding however allows for an interpretation according to which Bolzano advocates an "explicativist" conception of proof: proofs par excellence are those that reflect the (...)
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  25. Corcoran Reviews Boute’s 2013 Paper “How to Calculate Proofs”.John Corcoran - 2014 - MATHEMATICAL REVIEWS 14:444-555.
    Corcoran reviews Boute’s 2013 paper “How to calculate proofs”. -/- There are tricky aspects to classifying occurrences of variables: is an occurrence of ‘x’ free as in ‘x + 1’, is it bound as in ‘{x: x = 1}’, or is it orthographic as in ‘extra’? The trickiness is compounded failure to employ conventions to separate use of expressions from their mention. The variable occurrence is free in the term ‘x + 1’ but it is orthographic in that term’s quotes (...)
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  26. Mathematical Method and Proof.Jeremy Avigad - 2006 - Synthese 153 (1):105-159.
    On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that there are many criteria by which ordinary proofs are valued. I argue that at least some of these criteria depend on the methods of inference the proofs employ, and that (...)
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  27. Philosophy of Mathematics: An Introduction to a World of Proofs and Pictures.James Robert Brown - 1999 - Routledge.
    _Philosophy of Mathematics_ is an excellent introductory text. This student friendly book discusses the great philosophers and the importance of mathematics to their thought. It includes the following topics: * the mathematical image * platonism * picture-proofs * applied mathematics * Hilbert and Godel * knots and nations * definitions * picture-proofs and Wittgenstein * computation, proof and conjecture. The book is ideal for courses on philosophy of mathematics and logic.
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  28. Proof Analysis: A Contribution to Hilbert's Last Problem.Sara Negri & Jan von Plato - 2011 - Cambridge University Press.
    Machine generated contents note: Prologue: Hilbert's Last Problem; 1. Introduction; Part I. Proof Systems Based on Natural Deduction: 2. Rules of proof: natural deduction; 3. Axiomatic systems; 4. Order and lattice theory; 5. Theories with existence axioms; Part II. Proof Systems Based on Sequent Calculus: 6. Rules of proof: sequent calculus; 7. Linear order; Part III. Proof Systems for Geometric Theories: 8. Geometric theories; 9. Classical and intuitionistic axiomatics; 10. Proof analysis in elementary geometry; Part IV. Proof Systems for Nonclassical (...)
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  29. Godel's Theorem in Focus.S. G. Shanker (ed.) - 1990 - Routledge.
    A layman's guide to the mechanics of Gödel's proof together with a lucid discussion of the issues which it raises. Includes an essay discussing the significance of Gödel's work in the light of Wittgenstein's criticisms.
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  30. Week-Long Tutorial: Gödel's Incompleteness Theorems.Bernd Buldt - unknown
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  31. A Problem with the Dependence of Informal Proofs on Formal Proofs.Fenner Tanswell - 2015 - Philosophia Mathematica 23 (3):295-310.
    Derivationists, those wishing to explain the correctness and rigour of informal proofs in terms of associated formal proofs, are generally held to be supported by the success of the project of translating informal proofs into computer-checkable formal counterparts. I argue, however, that this project is a false friend for the derivationists because there are too many different associated formal proofs for each informal proof, leading to a serious worry of overgeneration. I press this worry primarily against Azzouni's derivation-indicator account, but (...)
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  32. Gödel on Truth and Proof.Dan Nesher - unknown
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  33. There's Something About Gdel: The Complete Guide to the Incompleteness Theorem.Francesco Berto - 2009 - Wiley-Blackwell.
    Berto’s highly readable and lucid guide introduces students and the interested reader to Gödel’s celebrated _Incompleteness Theorem_, and discusses some of the most famous - and infamous - claims arising from Gödel's arguments. Offers a clear understanding of this difficult subject by presenting each of the key steps of the _Theorem_ in separate chapters Discusses interpretations of the _Theorem_ made by celebrated contemporary thinkers Sheds light on the wider extra-mathematical and philosophical implications of Gödel’s theories Written in an accessible, non-technical (...)
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  34. Proof and Knowledge in Mathematics.Michael Detlefsen (ed.) - 1992 - Routledge.
    These questions arise from any attempt to discover an epistemology for mathematics. This collection of essays considers various questions concerning the nature of justification in mathematics and possible sources of that justification. Among these are the question of whether mathematical justification is _a priori_ or _a posteriori_ in character, whether logical and mathematical differ, and if formalization plays a significant role in mathematical justification.
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  35. Gödel’s First Incompleteness Theorem.Bernd Buldt - unknown
    Slides for the second tutorial on Gödel's incompleteness theorems, held at UniLog 5 Summer School, Istanbul, June 24, 2015.
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  36. Gödel’s Second Incompleteness Theorem.Bernd Buldt - unknown
    Slides for the third tutorial on Gödel's incompleteness theorems, held at UniLog 5 Summer School, Istanbul, June 24, 2015.
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  37. On the Depth of Szemeredi's Theorem.Andrew Arana - 2015 - Philosophia Mathematica 23 (2):163-176.
    Many mathematicians have cited depth as an important value in their research. However, there is no single widely accepted account of mathematical depth. This article is an attempt to bridge this gap. The strategy is to begin with a discussion of Szemerédi's theorem, which says that each subset of the natural numbers that is sufficiently dense contains an arithmetical progression of arbitrary length. This theorem has been judged deep by many mathematicians, and so makes for a good case on which (...)
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  38. Interpreting Gödel: Critical Essays.Juliette Kennedy (ed.) - 2014 - Cambridge: Cambridge University Press.
    The logician Kurt Gödel published a paper in 1931 formulating what have come to be known as his 'incompleteness theorems', which prove, among other things, that within any formal system with resources sufficient to code arithmetic, questions exist which are neither provable nor disprovable on the basis of the axioms which define the system. These are among the most celebrated results in logic today. In this volume, leading philosophers and mathematicians assess important aspects of Gödel's work on the foundations and (...)
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  39. Arithmetical Reflection and the Provability of Soundness.Walter Dean - 2015 - Philosophia Mathematica 23 (1):31-64.
    Proof-theoretic reflection principles are schemas which attempt to express the soundness of arithmetical theories within their own language, e.g., ${\mathtt{{Prov}_{\mathsf {PA}} \rightarrow \varphi }}$ can be understood to assert that any statement provable in Peano arithmetic is true. It has been repeatedly suggested that justification for such principles follows directly from acceptance of an arithmetical theory $\mathsf {T}$ or indirectly in virtue of their derivability in certain truth-theoretic extensions thereof. This paper challenges this consensus by exploring relationships between reflection principles (...)
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  40. Deduction Automated Logic.W. Bibel & Steffen Hölldobler - 1993
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  41. Automated Deduction--Cade-17 17th International Conference on Automated Deduction, Pittsburgh, Pa, Usa, June 2000 : Proceedings. [REVIEW]David A. Mcallester - 2000
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  42. Automated Deduction--Cade 16 16th International Conference on Automated Deduction, Trento, Italy, July 7-10, 1999 : Proceedings. [REVIEW]H. Ganzinger - 1999
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  43. Automated Deduction, Cade-15 15th International Conference on Automated Deduction, Lindau, Germany, July 5-10, 1998 : Proceedings. [REVIEW]Claude Kirchner - 1998
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  44. Automated Deduction, Cade-14 14th International Conference on Automated Deduction, Townsville, North Queensland, Australia, July 13-17, 1997 : Proceedings. [REVIEW]W. Mccune - 1997
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  45. Automated Deduction Cade-13 : 13th International Conference on Automated Deduction, New Brunswick, Nj, Usa, July 30-August 3, 1996 : Proceedings. [REVIEW]M. A. Mcrobbie & J. K. Slaney - 1996
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  46. Isabelle a Generic Theorem Prover.Lawrence C. Paulson & Tobias Nipkow - 1994
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  47. Automated Deduction Cade-12 : 12th International Conference on Automated Deduction : Nancy, France, June 26-July 1, 1994 : Proceedings. [REVIEW]Alan Bundy - 1994
  48. Automated Deduction-Cade-18 18th International Conference on Automated Deduction, Copenhagen, Denmark, July 27-30, 2002 ; Proceedings. [REVIEW]A. Voronkov - 2002
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  49. Automated Deduction--Cade-19 19th International Conference on Automated Deduction, Miami Beach, Fl, Usa, July 28-August 2, 2003 : Proceedings. [REVIEW]Franz Baader - 2003
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  50. 9th International Conference on Automated Deduction, Argonne, Illinois, Usa, May 23-26, 1988 Proceedings.Ewing Lusk & Ross A. Overbeek - 1988
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