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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. Alice Ambrose (1982). Wittgenstein on Mathematical Proof. Mind 91 (362):264-272.
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  2. Scientific American, Randomness and Mathematical Proof.
    Almost everyone has an intuitive notion of what a random number is. For example, consider these two series of binary digits: 01010101010101010101 01101100110111100010 The first is obviously constructed according to a simple rule; it consists of the number 01 repeated ten times. If one were asked to speculate on how the series might continue, one could predict with considerable confidence that the next two digits would be 0 and 1. Inspection of the second series of digits yields no such comprehensive (...)
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  3. Sr Arthur H. Copeland (1966). Mathematical Proof and Experimental Proof. Philosophy of Science 33 (4):303-316.
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  4. Ferdinando Arzarello, Valeria Andriano, Federica Olivero & Ornella Robutti (1998). Abduction and Conjecturing in Mathematics. Philosophica 61 (1):77-94.
    The logic of discovering and that of justifying have been a permanent source of debate in mathematics, because of their different and apparently contradictory features within the processes of production of mathematical sentences. In fact, a fundamental unity appears as soon as one investigates deeply the phenomenology of conjecturing and proving using concrete examples. In this paper it is shown that abduction, in the sense of Peirce, is an essential unifying activity, ruling such phenomena. Abduction is the major ingredient in (...)
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  5. Jeremy Avigad, Computers in Mathematical Inquiry.
    In Section 2, I survey some of the ways that computers are used in mathematics. These raise questions that seem to have a generally epistemological character, although they do not fall squarely under a traditional philosophical purview. The goal of this article is to try to articulate some of these questions more clearly, and assess the philosophical methods that may be brought to bear. In Section 3, I note that most of the issues can be classified under two headings: some (...)
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  6. Jeremy Avigad (2010). Proof Theory. Gödel and the Metamathematical Tradition. In Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Kurt Gödel: Essays for His Centennial. Association for Symbolic Logic.
  7. Jeremy Avigad (2010). Understanding, Formal Verification, and the Philosophy of Mathematics. Journal of the Indian Council of Philosophical Research 27:161-197.
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  8. Jeremy Avigad (2009). Marcus Giaquinto. Visual Thinking in Mathematics: An Epistemological Study. Philosophia Mathematica 17 (1):95-108.
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  9. J. Azzouni (2013). The Relationship of Derivations in Artificial Languages to Ordinary Rigorous Mathematical Proof. Philosophia Mathematica 21 (2):247-254.
    The relationship is explored between formal derivations, which occur in artificial languages, and mathematical proof, which occurs in natural languages. The suggestion that ordinary mathematical proofs are abbreviations or sketches of formal derivations is presumed false. The alternative suggestion that the existence of appropriate derivations in formal logical languages is a norm for ordinary rigorous mathematical proof is explored and rejected.
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  10. J. Azzouni (2013). That We See That Some Diagrammatic Proofs Are Perfectly Rigorous. Philosophia Mathematica 21 (3):323-338.
    Mistaken reasons for thinking diagrammatic proofs aren't rigorous are explored. The main result is that a confusion between the contents of a proof procedure (what's expressed by the referential elements in a proof procedure) and the unarticulated mathematical aspects of a proof procedure (how that proof procedure is enabled) gives the impression that diagrammatic proofs are less rigorous than language proofs. An additional (and independent) factor is treating the impossibility of naturally generalizing a diagrammatic proof procedure as an indication of (...)
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  11. Jody Azzouni & Otavio Bueno, Critical Studies/Book Reviews 319.
    Ask a philosopher what a proof is, and you’re likely to get an answer hii empaszng one or another regimentationl of that notion in terms of a finite sequence of formalized statements, each of which is either an axiom or is derived from an axiom by certain inference rules. (Wecan call this the formal conception of proof) Ask a mathematician what a proof is, and you will rbbl poay get a different-looking answer. Instead of stressing a partic- l uar regimented (...)
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  12. O. Bradley Bassler (2006). The Surveyability of Mathematical Proof: A Historical Perspective. Synthese 148 (1):99 - 133.
    This paper rejoins the debate surrounding Thomas Tymockzko’s paper on the surveyability of proof, first published in the Journal of Philosophy, and makes the claim that by attending to certain broad features of modern conceptions of proof we may understand ways in which the debate surrounding the surveyability of proof has heretofore remained unduly circumscribed. Motivated by these historical reflections, I suggest a distinction between local and global surveyability which I believe has the promise to open up significant new advances (...)
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  13. P. R. Baxandall (ed.) (1978). Proof in Mathematics ("If", "Then" and "Perhaps"): A Collection of Material Illustrating the Nature and Variety of the Idea of Proof in Mathematics. University of Keele, Institute of Education.
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  14. Lev D. Beklemishev (2003). On the Induction Schema for Decidable Predicates. Journal of Symbolic Logic 68 (1):17-34.
    We study the fragment of Peano arithmetic formalizing the induction principle for the class of decidable predicates, $I\Delta_1$ . We show that $I\Delta_1$ is independent from the set of all true arithmetical $\Pi_2-sentences$ . Moreover, we establish the connections between this theory and some classes of oracle computable functions with restrictions on the allowed number of queries. We also obtain some conservation and independence results for parameter free and inference rule forms of $\Delta_1-induction$ . An open problem formulated by J. (...)
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  15. Donald C. Benson (1999). The Moment of Proof: Mathematical Epiphanies. Oxford University Press.
    When Archimedes, while bathing, suddenly hit upon the principle of buoyancy, he ran wildly through the streets of Syracuse, stark naked, crying "eureka!" In The Moment of Proof, Donald Benson attempts to convey to general readers the feeling of eureka--the joy of discovery--that mathematicians feel when they first encounter an elegant proof. This is not an introduction to mathematics so much as an introduction to the pleasures of mathematical thinking. And indeed the delights of this book are many and varied. (...)
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  16. James Robert Brown (2008). Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures. Routledge.
    1. Introduction : the mathematical image -- 2. Platonism -- 3. Picture-proofs and Platonism -- 4. What is applied mathematics? -- 5. Hilbert and Gödel -- 6. Knots and notation -- 7. What is a definition? -- 8. Constructive approaches -- 9. Proofs, pictures and procedures in Wittgenstein -- 10. Computation, proof and conjecture -- 11. How to refute the continuum hypothesis -- 12. Calling the bluff.
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  17. James Robert Brown (1999). Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures. Routledge.
    Philosophy of Mathematics is clear and engaging, and student friendly The book discusses the great philosophers and the importance of mathematics to their thought. Among topics discussed in the book are the mathematical image, platonism, picture-proofs, applied mathematics, Hilbert and Godel, knots and notation definitions, picture-proofs and Wittgenstein, computation, proof and conjecture.
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  18. Otávio Bueno & Jody Azzouni (2005). Review of D. Mac Kenzie, Mechanizing Proof: Computing, Risk, and Trust. Philosophia Mathematica 13 (3):319-325.
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  19. Gregory Chaitin, Less Proof, More Truth.
    MATHEMATICS is a wonderful, mad subject, full of imagination, fantasy and creativity that is not limited by the petty details of the physical world, but only by the strength of our inner light. Does this sound familiar? Probably not from the mathematics classes you may have attended. But consider the work of three famous earlier mathematicians: Leonhard Euler (18th century), Georg Cantor (19th century) and Srinivasa Ramanujan (20th century).
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  20. Justin Clarke-Doane (2013). What is Absolute Undecidability?†. Noûs 47 (3):467-481.
    It is often alleged that, unlike typical axioms of mathematics, the Continuum Hypothesis (CH) is indeterminate. This position is normally defended on the ground that the CH is undecidable in a way that typical axioms are not. Call this kind of undecidability “absolute undecidability”. In this paper, I seek to understand what absolute undecidability could be such that one might hope to establish that (a) CH is absolutely undecidable, (b) typical axioms are not absolutely undecidable, and (c) if a mathematical (...)
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  21. Edwin Coleman (2009). The Surveyability of Long Proofs. Foundations of Science 14 (1-2):27-43.
    The specific characteristics of mathematical argumentation all depend on the centrality that writing has in the practice of mathematics, but blindness to this fact is near universal. What follows concerns just one of those characteristics, justification by proof. There is a prevalent view that long proofs pose a problem for the thesis that mathematical knowledge is justified by proof. I argue that there is no such problem: in fact, virtually all the justifications of mathematical knowledge are ‘long proofs’, but because (...)
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  22. Sr Copeland (1966). Mathematical Proof and Experimental Proof. Philosophy of Science 33 (4):303-.
    In studies of scientific methodology, surprisingly little attention has been given to tests of hypotheses. Such testing constitutes a methodology common to various scientific disciplines and is an essential factor in the development of science since it determines which theories are retained. The classical theory of tests is a major accomplishment but requires modification in order to produce a theory that accounts for the success of science. The revised theory is an analysis of the nondeductive aspect of scientific reasoning. It (...)
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  23. John W. Dawson Jr (2006). Why Do Mathematicians Re-Prove Theorems? Philosophia Mathematica 14 (3):269-286.
    From ancient times to the present, the discovery and presentation of new proofs of previously established theorems has been a salient feature of mathematical practice. Why? What purposes are served by such endeavors? And how do mathematicians judge whether two proofs of the same theorem are essentially different? Consideration of such questions illuminates the roles that proofs play in the validation and communication of mathematical knowledge and raises issues that have yet to be resolved by mathematical logicians. The Appendix, in (...)
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  24. J. W. Dawson (2006). Why Do Mathematicians Re-Prove Theorems? Philosophia Mathematica 14 (3):269-286.
    From ancient times to the present, the discovery and presentation of new proofs of previously established theorems has been a salient feature of mathematical practice. Why? What purposes are served by such endeavors? And how do mathematicians judge whether two proofs of the same theorem are essentially different? Consideration of such questions illuminates the roles that proofs play in the validation and communication of mathematical knowledge and raises issues that have yet to be resolved by mathematical logicians. The Appendix, in (...)
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  25. Michael Detlefsen (1995). The Mechanization of Reason. Philosophia Mathematica 3 (1).
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  26. Michael Detlefsen (ed.) (1992). Proof and Knowledge in Mathematics. Routledge.
    Proof and Knowledge in Mathematics tackles the main problem that arises when considering an epistemology for mathematics, the nature and sources of mathematical justification. Focusing both on particular and general issues, these essays from leading philosophers of mathematics raise important issues for our current understanding of mathematics. Is mathematical justification a priori or a posteriori? What role, if any, does logic play in mathematical reasoning or inference? And how epistemologically important is the formalizability of proof? Michael Detlefsen has brought together (...)
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  27. Michael Detlefsen & Andrew Arana (2011). Purity of Methods. Philosophers' Imprint 11 (2).
    Throughout history, mathematicians have expressed preference for solutions to problems that avoid introducing concepts that are in one sense or another “foreign” or “alien” to the problem under investigation. This preference for “purity” (which German writers commonly referred to as “methoden Reinheit”) has taken various forms. It has also been persistent. This notwithstanding, it has not been analyzed at even a basic philosophical level. In this paper we give a basic analysis of one conception of purity—what we call topical purity—and (...)
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  28. Michael Detlefsen & Mark Luker (1980). The Four-Color Theorem and Mathematical Proof. Journal of Philosophy 77 (12):803-820.
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  29. Edward T. Dixon (1929). Mathematical Proof. Mind 38 (151):343-351.
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  30. Don Fallis, What Do Mathematicians Want? Probabilistic Proofs and the Epistemic Goals of Mathematicians.
    Several philosophers have used the framework of means/ends reasoning to explain the methodological choices made by scientists and mathematicians (see, e.g., Goldman 1999, Levi 1962, Maddy 1997). In particular, they have tried to identify the epistemic objectives of scientists and mathematicians that will explain these choices. In this paper, the framework of means/ends reasoning is used to study an important methodological choice made by mathematicians. Namely, mathematicians will only use deductive proofs to establish the truth of mathematical claims. In this (...)
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  31. J. Fang (1970). The Axiomatic Method in Exposition and Exploration. Philosophia Mathematica (1-2):13-24.
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  32. S. Feferman (2006). Are There Absolutely Unsolvable Problems? Godel's Dichotomy. Philosophia Mathematica 14 (2):134-152.
    This is a critical analysis of the first part of Go¨del’s 1951 Gibbs lecture on certain philosophical consequences of the incompleteness theorems. Go¨del’s discussion is framed in terms of a distinction between objective mathematics and subjective mathematics, according to which the former consists of the truths of mathematics in an absolute sense, and the latter consists of all humanly demonstrable truths. The question is whether these coincide; if they do, no formal axiomatic system (or Turing machine) can comprehend the mathematizing (...)
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  33. Solomon Feferman, Presentation to the Panel, “Does Mathematics Need New Axioms?” Asl 2000 Meeting, Urbana Il, June 5, 2000.
    The point of departure for this panel is a somewhat controversial paper that I published in the American Mathematical Monthly under the title “Does mathematics need new axioms?” [4]. The paper itself was based on a lecture that I gave in 1997 to a joint session of the American Mathematical Society and the Mathematical Association of America, and it was thus written for a general mathematical audience. Basically, it was intended as an assessment of Gödel’s program for new axioms that (...)
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  34. Solomon Feferman, The Impact of the Incompleteness Theorems on Mathematics.
    In addition to this being the centenary of Kurt Gödel’s birth, January marked 75 years since the publication (1931) of his stunning incompleteness theorems. Though widely known in one form or another by practicing mathematicians, and generally thought to say something fundamental about the limits and potentialities of mathematical knowledge, the actual importance of these results for mathematics is little understood. Nor is this an isolated example among famous results. For example, not long ago, Philip Davis wrote me about what (...)
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  35. J. Ferreiros (2009). 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. [REVIEW] Philosophia Mathematica 17 (3):378-381.
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  36. José Ferreirós (2009). 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. History of Science, Philosophy and Culture in Indian Civilization. [REVIEW] Philosophia Mathematica 17 (3):nkn028.
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  37. Janet Folina (1998). Church's Thesis: Prelude to a Proof. Philosophia Mathematica 6 (3):302-323.
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  38. Harvey Friedman, Adventures in the Verification of Mathematics.
    Mathematical statements arising from program verification are believed to be much easier to deal with than statements coming from serious mathematics. At least this is true for “normal programming”.
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  39. Harvey Friedman, Computer Assisted Certainty.
    Certainty (and the lack thereof) is a major issue in mathematics and computer science. Mathematicians strongly believe in a special kind of certainty for their theorems.
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  40. Harvey Friedman, Can Mathematics Be Formalized?
    It has been accepted since the early part of the Century that there is no problem formalizing mathematics in standard formal systems of axiomatic set theory. Most people feel that they know as much as they ever want to know about how one can reduce natural numbers, integers, rationals, reals, and complex numbers to sets, and prove all of their basic properties. Furthermore, that this can continue through more and more complicated material, and that there is never a real problem.
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  41. Harvey Friedman, Godel's Legacy in Mathematical Philosophy.
    Gödel's definitive results and his essays leave us with a rich legacy of philosophical programs that promise to be subject to mathematical treatment. After surveying some of these, we focus attention on the program of circumventing his demonstrated impossibility of a consistency proof for mathematics by means of extramathematical concepts.
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  42. Harvey Friedman, 1 the Formalization of Mathematics.
    It has been accepted since the early part of the Century that there is no problem formalizing mathematics in standard formal systems of axiomatic set theory. Most people feel that they know as much as they ever want to know about how one can reduce natural numbers, integers, rationals, reals, and complex numbers to sets, and prove all of their basic properties. Furthermore, that this can continue through more and more complicated material, and that there is never a real problem.
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  43. Harvey Friedman (2000). Does Mathematics Need New Axioms? Bulletin of Symbolic Logic 6 (4):401 - 446.
    Since about 1925, the standard formalization of mathematics has been the ZFC axiom system (Zermelo Frankel set theory with the axiom of choice), about which the audience needs to know nothing. The axiom of choice was controversial for a while, but the controversy subsided decades ago.
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  44. Mihai Ganea (2008). Epistemic Optimism. Philosophia Mathematica 16 (3):333-353.
    Michael Dummett's argument for intuitionism can be criticized for the implicit reliance on the existence of what might be called absolutely undecidable statements. Neil Tennant attacks epistemic optimism, the view that there are no such statements. I expose what seem serious flaws in his attack, and I suggest a way of defending the use of classical logic in arithmetic that circumvents the issue of optimism. I would like to thank an anonymous referee for helpful comments. CiteULike Connotea Del.icio.us What's this?
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  45. M. Giaquinto (2007). Visual Thinking in Mathematics: An Epistemological Study. Oxford University Press.
    Visual thinking -- visual imagination or perception of diagrams and symbol arrays, and mental operations on them -- is omnipresent in mathematics. Is this visual thinking merely a psychological aid, facilitating grasp of what is gathered by other means? Or does it also have epistemological functions, as a means of discovery, understanding, and even proof? By examining the many kinds of visual representation in mathematics and the diverse ways in which they are used, Marcus Giaquinto argues that visual thinking in (...)
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  46. Joanna Golińska-Pilarek & Ewa Orłowska (2007). Tableaux and Dual Tableaux: Transformation of Proofs. 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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  47. Rubin Gotesky (1965). Stray Thoughts on Formalization. Philosophia Mathematica (1):33-37.
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  48. G. H. Hardy (1929). Mathematical Proof. Mind 38 (149):1-25.
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  49. Reuben Hersh (1997). Prove—Once More and Again. Philosophia Mathematica 5 (2):153-165.
    There are two distinct meanings to ‘mathematical proof’. The connection between them is an unsolved problem. The first step in attacking it is noticing that it is an unsolved problem.
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  50. David Hilbert (1970). Axiomatic Thinking. Philosophia Mathematica (1-2):1-12.
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