About this topic
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 (...)
  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 (...)
  3. Institutionism, Pluralism, and Cognitive Command.Stewart Shapiro & William W. Taschek - 1996 - Journal of Philosophy 93 (2):74.
  4. Looking for Busy Beavers. A Socio-Philosophical Study of a Computer-Assisted Proof.Liesbeth De Mol - unknown
  5. A Note on Wittgenstein’s “Notorious Paragraph” About the Gödel Theorem.Juliet Floyd & Hilary Putnam - 2000 - Journal of Philosophy 97 (11):624-632.
  6. 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”.
  7. 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 (...)
  8. Gödel’s Incompleteness Theorems and Artificial Life.John P. Sullins Iii - 1997 - Techné: Research in Philosophy and Technology 2 (3):185-195.
  9. Gödel’s Incompleteness Theorems and Physics.Newton C. A. Da Costa - 2012 - Principia: An International Journal of Epistemology 15 (3).
  10. 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 (...)
  11. A Note on Incompleteness and Heterologicality.P. M. Sullivan - 2003 - Analysis 63 (1):32-38.
  12. 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.
  13. Remarks on the Incompleteness Proof.Gerold Stahl - 1961 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 7 (11-14):164-170.
  14. 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 (...)
  15. Gödel’s Incompleteness Phenomenon—Computationally.Saeed Salehi - 2014 - Philosophia Scientae 18:23-37.
  16. 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 (...)
  17. 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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  18. 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 (...)
  19. 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.
  20. 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 (...)
  21. Godel's Theorem in Focus.S. G. Shanker (ed.) - 2012 - 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.
  22. Week-Long Tutorial: Gödel's Incompleteness Theorems.Bernd Buldt - unknown
  23. Gödel on Truth and Proof.Dan Nesher - unknown
  24. 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 (...)
  25. 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.
  26. 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.
  27. 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.
  28. On the Depth of Szemerédi'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 (...)
  29. 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 (...)
  30. 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 (...)
  31. Deduction Automated Logic.W. Bibel & Steffen Hölldobler - 1993
  32. Automated Deduction--Cade-17 17th International Conference on Automated Deduction, Pittsburgh, Pa, Usa, June 2000 : Proceedings. [REVIEW]David A. Mcallester - 2000
  33. Automated Deduction--Cade 16 16th International Conference on Automated Deduction, Trento, Italy, July 7-10, 1999 : Proceedings. [REVIEW]H. Ganzinger - 1999
  34. Automated Deduction, Cade-15 15th International Conference on Automated Deduction, Lindau, Germany, July 5-10, 1998 : Proceedings. [REVIEW]Claude Kirchner - 1998
  35. Automated Deduction, Cade-14 14th International Conference on Automated Deduction, Townsville, North Queensland, Australia, July 13-17, 1997 : Proceedings. [REVIEW]W. Mccune - 1997
  36. 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
  37. Isabelle a Generic Theorem Prover.Lawrence C. Paulson & Tobias Nipkow - 1994
  38. Automated Deduction Cade-12 : 12th International Conference on Automated Deduction : Nancy, France, June 26-July 1, 1994 : Proceedings. [REVIEW]Alan Bundy - 1994
  39. Automated Deduction-Cade-18 18th International Conference on Automated Deduction, Copenhagen, Denmark, July 27-30, 2002 ; Proceedings. [REVIEW]A. Voronkov - 2002
  40. Automated Deduction--Cade-19 19th International Conference on Automated Deduction, Miami Beach, Fl, Usa, July 28-August 2, 2003 : Proceedings. [REVIEW]Franz Baader - 2003
  41. 9th International Conference on Automated Deduction, Argonne, Illinois, Usa, May 23-26, 1988 Proceedings.Ewing Lusk & Ross A. Overbeek - 1988
  42. Automated Deduction, Cade-11 11th International Conference on Automated Deduction, Saratoga Springs, Ny, Usa, June 15-18, 1992, Proceedings. [REVIEW]Deepak Kapur - 1992
  43. 6th Conference on Automated Deduction, New York, Usa, June 7-9, 1982.Donald W. Loveland - 1982
  44. 5th Conference on Automated Deduction, les Arcs, France, July 8-11, 1980.W. Bibel & Robert Kowalski - 1980
  45. Proofs and Mathematical Experience.G. Lolli - 1982 - Scientia 76:501.
  46. From Consistency to Incompleteness: A Philosophical Study of Hilbert's Program and Goedel's Incompleteness Theorem.Byoung-il Choi - 1997 - Dissertation, University of California, Berkeley
    The main objective of this thesis is a philosophical study of Hilbert's Program and Godel's Incompleteness Theorem. For this purpose we pursue historical, metamathematical, and conceptual investigations of them. ;By tracing the historical origins and conceptual developments of Hilbert's Program and Godel's Incompleteness Theorem, we will argue that both have inherently philosophical motivations. Also, by considering the relevant metamathematical developments such as Reverse Mathematics, the Paris-Harrington Incompleteness Theorem and related materials, we will argue that Hilbert's Program and Godel's Incompleteness Theorem (...)
  47. The Web as A Tool For Proving.Ioannis M. Vandoulakis Petros Stefaneas - 2012 - Metaphilosophy 43 (4):480-498.
    The Web may critically transform the way we understand the activity of proving. The Web as a collaborative medium allows the active participation of people with different backgrounds, interests, viewpoints, and styles. Mathematical formal proofs are inadequate for capturing Web‐based proofs. This article claims that Web provings can be studied as a particular type of Goguen's proof‐events. Web‐based proof‐events have a social component, communication medium, prover‐interpreter interaction, interpretation process, understanding and validation, historical component, and styles. To demonstrate its claim, the (...)
  48. Automated Analysis of Scientific Reasoning.H. Koppelaar - 1985 - International Logic Review 32:112.
  49. Empirical Elements in Mathematicians' Proofs.L. O. Kattsoff - 1971 - International Logic Review 4:191.
  50. Lakatos on proof and on mathematics.J. Agassi - 1981 - Logique Et Analyse 24 (95):437.
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