Results for 'proofs'

996 found
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  1. Proofs and Refutations: The Logic of Mathematical Discovery.Imre Lakatos (ed.) - 1976 - Cambridge University Press.
    Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. (...)
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  2. Philosophical Proofs Against Common Sense.Bryan Frances - forthcoming - Analysis.
    Many philosophers are sceptical about the power of philosophy to refute commonsensical claims. They look at the famous attempts and judge them inconclusive. I prove that even if those famous attempts are failures, there are alternative successful philosophical proofs against commonsensical claims. After presenting the proofs I briefly comment on their significance.
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  3. Proofs, Pictures, and Euclid.John Mumma - 2010 - Synthese 175 (2):255 - 287.
    Though pictures are often used to present mathematical arguments, they are not typically thought to be an acceptable means for presenting mathematical arguments rigorously. With respect to the proofs in the Elements in particular, the received view is that Euclid's reliance on geometric diagrams undermines his efforts to develop a gap-free deductive theory. The central difficulty concerns the generality of the theory. How can inferences made from a particular diagrams license general mathematical results? After surveying the history behind the (...)
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  4.  9
    Short Proofs for Slow Consistency.Anton Freund & Fedor Pakhomov - 2020 - Notre Dame Journal of Formal Logic 61 (1):31-49.
    Let Con↾x denote the finite consistency statement “there are no proofs of contradiction in T with ≤x symbols.” For a large class of natural theories T, Pudlák has shown that the lengths of the shortest proofs of Con↾n in the theory T itself are bounded by a polynomial in n. At the same time he conjectures that T does not have polynomial proofs of the finite consistency statements Con)↾n. In contrast, we show that Peano arithmetic has polynomial (...)
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  5.  79
    New Proofs for the Existence of God: Contributions of Contemporary Physics and Philosophy.Robert J. Spitzer - 2010 - William B. Eerdmans.
    New Proofs for the Existence of God responds to these glaring omissions. / From universal space-time asymmetry to cosmic coincidences to the intelligibility of ...
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  6.  51
    Proofs and Countermodels in Non-Classical Logics.Sara Negri - 2014 - Logica Universalis 8 (1):25-60.
    Proofs and countermodels are the two sides of completeness proofs, but, in general, failure to find one does not automatically give the other. The limitation is encountered also for decidable non-classical logics in traditional completeness proofs based on Henkin’s method of maximal consistent sets of formulas. A method is presented that makes it possible to establish completeness in a direct way: For any given sequent either a proof in the given logical system or a countermodel in the (...)
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  7.  13
    Proofs and Refutations.Imre Lakatos - 1980 - Noûs 14 (3):474-478.
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  8.  75
    Socratic Proofs.Andrzej Wiśniewski - 2004 - Journal of Philosophical Logic 33 (3):299-326.
    Our aim is to express in exact terms the old idea of solving problems by pure questioning. We consider the problem of derivability: "Is A derivable from Δ by classical propositional logic?". We develop a calculus of questions E*; a proof (called a Socratic proof) is a sequence of questions ending with a question whose affirmative answer is, in a sense, evident. The calculus is sound and complete with respect to classical propositional logic. A Socratic proof in E* can be (...)
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  9.  30
    Motivated Proofs: What They Are, Why They Matter and How to Write Them.Rebecca Lea Morris - 2020 - Review of Symbolic Logic 13 (1):23-46.
    Mathematicians judge proofs to possess, or lack, a variety of different qualities, including, for example, explanatory power, depth, purity, beauty and fit. Philosophers of mathematical practice have begun to investigate the nature of such qualities. However, mathematicians frequently draw attention to another desirable proof quality: being motivated. Intuitively, motivated proofs contain no "puzzling" steps, but they have received little further analysis. In this paper, I begin a philosophical investigation into motivated proofs. I suggest that a proof is (...)
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  10. Probabilistic Proofs and Transferability.Kenny Easwaran - 2009 - Philosophia Mathematica 17 (3):341-362.
    In a series of papers, Don Fallis points out that although mathematicians are generally unwilling to accept merely probabilistic proofs, they do accept proofs that are incomplete, long and complicated, or partly carried out by computers. He argues that there are no epistemic grounds on which probabilistic proofs can be rejected while these other proofs are accepted. I defend the practice by presenting a property I call ‘transferability’, which probabilistic proofs lack and acceptable proofs (...)
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  11. Why Proofs by Mathematical Induction Are Generally Not Explanatory.Marc Lange - 2009 - Analysis 69 (2):203-211.
    Philosophers who regard some mathematical proofs as explaining why theorems hold, and others as merely proving that they do hold, disagree sharply about the explanatory value of proofs by mathematical induction. I offer an argument that aims to resolve this conflict of intuitions without making any controversial presuppositions about what mathematical explanations would be.
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  12.  97
    Constructions, Proofs and the Meaning of Logical Constants.Göran Sundholm - 1983 - Journal of Philosophical Logic 12 (2):151 - 172.
  13.  94
    Proofs and Refutations (I).Imre Lakatos - 1963 - British Journal for the Philosophy of Science 14 (53):1-25.
  14.  80
    Proofs and Refutations (III).Imre Lakatos - 1963 - British Journal for the Philosophy of Science 14 (55):221-245.
  15. Proofs for Eternity, Creation, and the Existence of God in Medieval Islamic and Jewish Philosophy.Herbert A. Davidson - 1987 - Oxford University Press.
    The central debate of natural theology among medieval Muslims and Jews concerned whether or not the world was eternal. Opinions divided sharply on this issue because the outcome bore directly on God's relationship with the world: eternity implies a deity bereft of will, while a world with a beginning leads to the contrasting picture of a deity possessed of will. In this exhaustive study of medieval Islamic and Jewish arguments for eternity, creation, and the existence of God, Herbert Davidson provides (...)
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  16.  65
    Proofs and Refutations (IV).I. Lakatos - 1963 - British Journal for the Philosophy of Science 14 (56):296-342.
  17. Proofs and Refutations (II).Imre Lakatos - 1963 - British Journal for the Philosophy of Science 14 (54):120-139.
  18.  40
    Normal Proofs, Cut Free Derivations and Structural Rules.Greg Restall - 2014 - Studia Logica 102 (6):1143-1166.
    Different natural deduction proof systems for intuitionistic and classical logic —and related logical systems—differ in fundamental properties while sharing significant family resemblances. These differences become quite stark when it comes to the structural rules of contraction and weakening. In this paper, I show how Gentzen and Jaśkowski’s natural deduction systems differ in fine structure. I also motivate directed proof nets as another natural deduction system which shares some of the design features of Genzen and Jaśkowski’s systems, but which differs again (...)
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  19.  25
    Uniform Proofs as a Foundation for Logic Programming.Dale Miller, Gopalan Nadathur, Frank Pfenning & Andre Scedrov - 1991 - Annals of Pure and Applied Logic 51 (1-2):125-157.
    Miller, D., G. Nadathur, F. Pfenning and A. Scedrov, Uniform proofs as a foundation for logic programming, Annals of Pure and Applied Logic 51 125–157. A proof-theoretic characterization of logical languages that form suitable bases for Prolog-like programming languages is provided. This characterization is based on the principle that the declarative meaning of a logic program, provided by provability in a logical system, should coincide with its operational meaning, provided by interpreting logical connectives as simple and fixed search instructions. (...)
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  20.  44
    Explanatory Proofs and Beautiful Proofs.Marc Lange - unknown
    This paper concerns the relation between a proof’s beauty and its explanatory power – that is, its capacity to go beyond proving a given theorem to explaining why that theorem holds. Explanatory power and beauty are among the many virtues that mathematicians value and seek in various proofs, and it is important to come to a better understanding of the relations among these virtues. Mathematical practice has long recognized that certain proofs but not others have explanatory power, and (...)
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  21.  48
    Wellordering Proofs for Metapredicative Mahlo.Thomas Strahm - 2002 - Journal of Symbolic Logic 67 (1):260-278.
    In this article we provide wellordering proofs for metapredicative systems of explicit mathematics and admissible set theory featuring suitable axioms about the Mahloness of the underlying universe of discourse. In particular, it is shown that in the corresponding theories EMA of explicit mathematics and KPm 0 of admissible set theory, transfinite induction along initial segments of the ordinal φω00, for φ being a ternary Veblen function, is derivable. This reveals that the upper bounds given for these two systems in (...)
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  22.  51
    Proofs of Strong Normalisation for Second Order Classical Natural Deduction.Michel Parigot - 1997 - Journal of Symbolic Logic 62 (4):1461-1479.
    We give two proofs of strong normalisation for second order classical natural deduction. The first one is an adaptation of the method of reducibility candidates introduced in [9] for second order intuitionistic natural deduction; the extension to the classical case requires in particular a simplification of the notion of reducibility candidate. The second one is a reduction to the intuitionistic case, using a Kolmogorov translation.
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  23.  38
    Socratic Proofs and Paraconsistency: A Case Study.Andrzej Wiśniewski, Guido Vanackere & Dorota Leszczyńska - 2005 - Studia Logica 80 (2-3):431-466.
    This paper develops a new proof method for two propositional paraconsistent logics: the propositional part of Batens' weak paraconsistent logic CLuN and Schütte's maximally paraconsistent logic Φv. Proofs are de.ned as certain sequences of questions. The method is grounded in Inferential Erotetic Logic.
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  24. Proofs for the Existence of God.Lawrence Nolan & Alan Nelson - 2006 - In Stephen Gaukroger (ed.), The Blackwell to Descartes’ Meditations. Blackwell. pp. 104--121.
    We argue that Descartes’s theistic proofs in the ’Meditations’ are much simpler and straightforward than they are traditionally taken to be. In particular, we show how the causal argument of the "Third Meditation" depends on the intuitively innocent principle that nothing comes from nothing, and not on the more controversial principle that the objective reality of an idea must have a cause with at least as much formal reality. We also demonstrate that the so-called ontological "argument" of the "Fifth (...)
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  25.  49
    Algebraic Proofs of Cut Elimination.Jeremy Avigad - manuscript
    Algebraic proofs of the cut-elimination theorems for classical and intuitionistic logic are presented, and are used to show how one can sometimes extract a constructive proof and an algorithm from a proof that is nonconstructive. A variation of the double-negation translation is also discussed: if ϕ is provable classically, then ¬(¬ϕ)nf is provable in minimal logic, where θnf denotes the negation-normal form of θ. The translation is used to show that cut-elimination theorems for classical logic can be viewed as (...)
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  26.  35
    Socratic Proofs for Quantifiers★.Andrzej Wiśniewski & Vasilyi Shangin - 2006 - Journal of Philosophical Logic 35 (2):147-178.
    First-order logic is formalized by means of tools taken from the logic of questions. A calculus of questions which is a counterpart of the Pure Calculus of Quantifiers is presented. A direct proof of completeness of the calculus is given.
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  27.  49
    On Proofs of the Incompleteness Theorems Based on Berry's Paradox by Vopěnka, Chaitin, and Boolos.Makoto Kikuchi, Taishi Kurahashi & Hiroshi Sakai - 2012 - Mathematical Logic Quarterly 58 (4-5):307-316.
    By formalizing Berry's paradox, Vopěnka, Chaitin, Boolos and others proved the incompleteness theorems without using the diagonal argument. In this paper, we shall examine these proofs closely and show their relationships. Firstly, we shall show that we can use the diagonal argument for proofs of the incompleteness theorems based on Berry's paradox. Then, we shall show that an extension of Boolos' proof can be considered as a special case of Chaitin's proof by defining a suitable Kolmogorov complexity. We (...)
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  28.  37
    Proofs and Refutations: The Logic of Mathematical Discovery.Imre Lakatos, John Worrall & Elie Zahar (eds.) - 1976 - Cambridge University Press.
    Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre Lakatos is concerned throughout to combat the classical picture (...)
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  29.  16
    Models & Proofs: LFIs Without a Canonical Interpretations.Eduardo Alejandro Barrio - 2018 - Principia: An International Journal of Epistemology 22 (1):87-112.
    In different papers, Carnielli, W. & Rodrigues, A., Carnielli, W. Coniglio, M. & Rodrigues, A. and Rodrigues & Carnielli, present two logics motivated by the idea of capturing contradictions as conflicting evidence. The first logic is called BLE and the second—that is a conservative extension of BLE—is named LETJ. Roughly, BLE and LETJ are two non-classical logics in which the Laws of Explosion and Excluded Middle are not admissible. LETJ is built on top of BLE. Moreover, LETJ is a Logic (...)
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  30. Proofs and Pictures.James Robert Brown - 1997 - British Journal for the Philosophy of Science 48 (2):161-180.
    Everyone appreciates a clever mathematical picture, but the prevailing attitude is one of scepticism: diagrams, illustrations, and pictures prove nothing; they are psychologically important and heuristically useful, but only a traditional verbal/symbolic proof provides genuine evidence for a purported theorem. Like some other recent writers (Barwise and Etchemendy [1991]; Shin [1994]; and Giaquinto [1994]) I take a different view and argue, from historical considerations and some striking examples, for a positive evidential role for pictures in mathematics.
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  31.  7
    Proofs and Refutations: The Logic of Mathematical Discovery.Daniel Isaacson - 1978 - Philosophical Quarterly 28 (111):169-171.
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  32. Informal Proofs and Mathematical Rigour.Marianna Antonutti Marfori - 2010 - Studia Logica 96 (2):261-272.
    The aim of this paper is to provide epistemic reasons for investigating the notions of informal rigour and informal provability. I argue that the standard view of mathematical proof and rigour yields an implausible account of mathematical knowledge, and falls short of explaining the success of mathematical practice. I conclude that careful consideration of mathematical practice urges us to pursue a theory of informal provability.
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  33.  46
    Parity Proofs of the Bell-Kochen-Specker Theorem Based on the 600-Cell.Mordecai Waegell, P. K. Aravind, Norman D. Megill & Mladen Pavičić - 2011 - Foundations of Physics 41 (5):883-904.
    The set of 60 real rays in four dimensions derived from the vertices of a 600-cell is shown to possess numerous subsets of rays and bases that provide basis-critical parity proofs of the Bell-Kochen-Specker (BKS) theorem (a basis-critical proof is one that fails if even a single basis is deleted from it). The proofs vary considerably in size, with the smallest having 26 rays and 13 bases and the largest 60 rays and 41 bases. There are at least (...)
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  34.  59
    Proofs as Acts and Proofs as Objects: Some Questions for Dag Prawitz.Goran Sundholm - 1998 - Theoria 64 (2-3):187-216.
  35. Proofs and Arguments: The Special Case of Mathematics.Jean Paul Van Bendegem - 2005 - Poznan Studies in the Philosophy of the Sciences and the Humanities 84 (1):157-169.
    Most philosophers still tend to believe that mathematics is basically about producing formal proofs. A consequence of this view is that some aspects of mathematical practice are entirely lost from view. My contention is that it is precisely in those aspects that similarities can be found between practices in the exact sciences and in mathematics. Hence, if we are looking for a (more) unified treatment of science and mathematics it is necessary to incorporate these elements into our view of (...)
     
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  36.  10
    Two Proofs of the Algebraic Completeness Theorem for Multilattice Logic.Oleg Grigoriev & Yaroslav Petrukhin - 2019 - Journal of Applied Non-Classical Logics 29 (4):358-381.
    Shramko [. Truth, falsehood, information and beyond: The American plan generalized. In K. Bimbo, J. Michael Dunn on information based logics, outstanding contributions to logic...
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  37. Principles and Proofs: Aristotle's Theory of Demonstrative Science.Richard D. McKirahan (ed.) - 1992 - Princeton University Press.
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  38. Completeness proofs for the systems RM3 and BN4.Ross T. Brady - 1982 - Logique Et Analyse 25 (97):9.
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  39.  47
    Optimal Proofs of Determinacy.Itay Neeman - 1995 - Bulletin of Symbolic Logic 1 (3):327-339.
  40.  73
    Understanding Proofs.Jeremy Avigad - manuscript
    “Now, in calm weather, to swim in the open ocean is as easy to the practised swimmer as to ride in a spring-carriage ashore. But the awful lonesomeness is intolerable. The intense concentration of self in the middle of such a heartless immensity, my God! who can tell it? Mark, how when sailors in a dead calm bathe in the open sea—mark how closely they hug their ship and only coast along her sides.” (Herman Melville, Moby Dick, Chapter 94).
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  41.  75
    The Proofs of the Grundgedanke in Wittgenstein's Tractatus.Leo K. C. Cheung - 1999 - Synthese 120 (3):395-410.
    The Tractatus contains twodifferent proofs of the Grundgedanke, or thenonreferentiality of logical constants. In thispaper, I explicate the first proof in TLP 5.4s andreconstruct the less explicitly stated second proof. My explication of the first proof shows it to beelegant but based on an invalid inference. In myreconstruction of the second proof, the main argumentis that the sign of a logical constant does not denotebecause it possesses the punctuation-mark-nature. Andit possesses the punctuation-mark-nature because,given the analyticity thesis in TLP 5, (...)
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  42.  38
    Explanatory Proofs in Mathematics.Erik Weber & Liza Verhoeven - 2002 - Logique Et Analyse 179:299-307.
  43.  8
    Simple Proofs of $${\Mathsf{SCH}}$$ SCH From Reflection Principles Without Using Better Scales.Hiroshi Sakai - 2015 - Archive for Mathematical Logic 54 (5-6):639-647.
    We give simple proofs of the Singular Cardinal Hypothesis from the Weak Reflection Principle and the Fodor-type Reflection Principle which do not use better scales.
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  44.  28
    Proofs and Retributions, Or: Why Sarah Can’T Take Limits.Vladimir Kanovei, Karin U. Katz, Mikhail G. Katz & Mary Schaps - 2015 - Foundations of Science 20 (1):1-25.
    The small, the tiny, and the infinitesimal have been the object of both fascination and vilification for millenia. One of the most vitriolic reviews in mathematics was that written by Errett Bishop about Keisler’s book Elementary Calculus: an Infinitesimal Approach. In this skit we investigate both the argument itself, and some of its roots in Bishop George Berkeley’s criticism of Leibnizian and Newtonian Calculus. We also explore some of the consequences to students for whom the infinitesimal approach is congenial. The (...)
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  45.  64
    Short Proofs of Normalization for the Simply- Typed Λ-Calculus, Permutative Conversions and Gödel's T.Felix Joachimski & Ralph Matthes - 2003 - Archive for Mathematical Logic 42 (1):59-87.
    Inductive characterizations of the sets of terms, the subset of strongly normalizing terms and normal forms are studied in order to reprove weak and strong normalization for the simply-typed λ-calculus and for an extension by sum types with permutative conversions. The analogous treatment of a new system with generalized applications inspired by generalized elimination rules in natural deduction, advocated by von Plato, shows the flexibility of the approach which does not use the strong computability/candidate style à la Tait and Girard. (...)
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  46. Meaning Approached Via Proofs.Dag Prawitz - 2006 - Synthese 148 (3):507-524.
    According to a main idea of Gentzen the meanings of the logical constants are reflected by the introduction rules in his system of natural deduction. This idea is here understood as saying roughly that a closed argument ending with an introduction is valid provided that its immediate subarguments are valid and that other closed arguments are justified to the extent that they can be brought to introduction form. One main part of the paper is devoted to the exact development of (...)
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  47.  72
    The Logic of Proofs, Semantically.Melvin Fitting - 2005 - Annals of Pure and Applied Logic 132 (1):1-25.
    A new semantics is presented for the logic of proofs (LP), [1, 2], based on the intuition that it is a logic of explicit knowledge. This semantics is used to give new proofs of several basic results concerning LP. In particular, the realization of S4 into LP is established in a way that carefully examines and explicates the role of the + operator. Finally connections are made with the conventional approach, via soundness and completeness results.
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  48.  16
    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 (...)
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  49.  18
    On Proofs of Rejection.Walenty Staszek - 1971 - Studia Logica 29 (1):17 - 25.
  50. Mathematical Proofs.Marco Panza - 2003 - Synthese 134 (1-2):119 - 158.
    The aim I am pursuing here is to describe some general aspects of mathematical proofs. In my view, a mathematical proof is a warrant to assert a non-tautological statement which claims that certain objects (possibly a certain object) enjoy a certain property. Because it is proved, such a statement is a mathematical theorem. In my view, in order to understand the nature of a mathematical proof it is necessary to understand the nature of mathematical objects. If we understand them (...)
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