Results for 'mathematical proof'

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  1.  6
    100% Mathematical Proof.Rowan Garnier & John Taylor - 1996 - John Wiley & Son.
    "Proof" has been and remains one of the concepts which characterises mathematics. Covering basic propositional and predicate logic as well as discussing axiom systems and formal proofs, the book seeks to explain what mathematicians understand by proofs and how they are communicated. The authors explore the principle techniques of direct and indirect proof including induction, existence and uniqueness proofs, proof by contradiction, constructive and non-constructive proofs, etc. Many examples from analysis and modern algebra are included. The exceptionally (...)
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  2. "A mathematical proof must be surveyable" what Wittgenstein meant by this and what it implies.Felix Mühlhölzer - 2006 - Grazer Philosophische Studien 71 (1):57-86.
    In Part III of his Remarks on the Foundations of Mathematics Wittgenstein deals with what he calls the surveyability of proofs. By this he means that mathematical proofs can be reproduced with certainty and in the manner in which we reproduce pictures. There are remarkable similarities between Wittgenstein's view of proofs and Hilbert's, but Wittgenstein, unlike Hilbert, uses his view mainly in critical intent. He tries to undermine foundational systems in mathematics, like logicist or set theoretic ones, by stressing (...)
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  3. Mathematical proof.G. H. Hardy - 1929 - Mind 38 (149):1-25.
  4.  58
    Mathematical Proofs: The Beautiful and The Explanatory.Marcus Giaquinto - unknown
    Mathematicians sometimes judge a mathematical proof to be beautiful and in doing so seem to be making a judgement of the same kind as aesthetic judgements of works of visual art, music or literature. Mathematical proofs are also appraised for explanatoriness: some proofs merely establish their conclusions as true, while others also show why their conclusions are true. This paper will focus on the prima facie plausible assumption that, for mathematical proofs, beauty and explanatoriness tend to (...)
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  5.  19
    Mathematical Proofs in Practice: Revisiting the reliability of published mathematical proofs.Joachim Frans & Laszlo Kosolosky - 2014 - Theoria: Revista de Teoría, Historia y Fundamentos de la Ciencia 29 (3):345-360.
    Mathematics seems to have a special status when compared to other areas of human knowledge. This special status is linked with the role of proof. Mathematicians often believe that this type of argumentation leaves no room for errors and unclarity. Philosophers of mathematics have differentiated between absolutist and fallibilist views on mathematical knowledge, and argued that these views are related to whether one looks at mathematics-in-the-making or finished mathematics. In this paper we take a closer look at (...) practice, more precisely at the publication process in mathematics. We argue that the apparent view that mathematical literature, given the special status of mathematics, is highly reliable is too naive. We will discuss several problems in the publication process that threaten this view, and give several suggestions on how this could be countered.Las matemáticas parecen tener un estatuto especial cuando se las compara con otras áreas del conocimiento humano. Este estatuto especial está conectado con el papel de la demostración. Los matemáticos creen con frecuencia que este tipo de argumentos no deja margen para el error o la falta de claridad. Los filósofos de la matemática han distinguido entre una concepción absolutista y una falibilista del conocimiento matemático, argumentando que estas concepciones están relacionadas con una consideración de las matemáticas-en-proceso o en tanto que matemáticas ya hechas. En este artículo examinamos más de cerca la práctica matemática, más en concreto el proceso de publicación en matemáticas. Argumentaremos que la idea preconcebida de que la literatura matemática, dado el estatuto especial de las matemáticas, es altamente fiable, es demasiado ingenua. Discutiremos algunos problemas del proceso de edición que amenazan esta visión y haremos algunas sugerencias sobre cómo enfrentarlos. (shrink)
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  6.  75
    On mathematical proofs of the vacuity of compositionality.dag westerståhl - 1998 - Linguistics and Philosophy 21 (6):635-643.
  7.  3
    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 (...)
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  8. 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 (...) objects. If we understand them as external entities whose 'existence' is independent of us and if we think that their enjoying certain properties is a fact, then we should argue that a theorem is a statement that claims that this fact occurs. If we also maintain that a mathematical proof is internal to a mathematical theory, then it becomes very difficult indeed to explain how a proof can be a warrant for such a statement. This is the essential content of a dilemma set forth by P. Benacerraf (cf. Benacerraf 1973). Such a dilemma, however, is dissolved if we understand mathematical objects as internal constructions of mathematical theories and think that they enjoy certain properties just because a mathematical theorem claims that they enjoy them. (shrink)
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  9.  6
    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 (...)
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  10.  34
    Mathematical Proof and Discovery Reductio ad Absurdum.Dale Jacquette - 2008 - Informal Logic 28 (3):242-261.
    The uses and interpretation of reductio ad absurdum argumentation in mathematical proof and discovery are examined, illustrated with elementary and progressively sophisticated examples, and explained. Against Arthur Schopenhauer’s objections, reductio reasoning is defended as a method of uncovering new mathematical truths, and not merely of confirming independently grasped mathematical intuitions. The application of reductio argument is contrasted with purely mechanical brute algorithmic inferences as an art requiring skill and intelligent intervention in the choice of hypotheses and (...)
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  11.  5
    Reverse mathematics: proofs from the inside out.John Stillwell - 2018 - Princeton: Princeton University Press.
    This book presents reverse mathematics to a general mathematical audience for the first time. Reverse mathematics is a new field that answers some old questions. In the two thousand years that mathematicians have been deriving theorems from axioms, it has often been asked: which axioms are needed to prove a given theorem? Only in the last two hundred years have some of these questions been answered, and only in the last forty years has a systematic approach been developed. In (...)
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  12.  49
    Ancient Greek Mathematical Proofs and Metareasoning.Mario Bacelar Valente - 2024 - In Maria Zack & David Waszek (eds.), Research in History and Philosophy of Mathematics. Annals of the Canadian Society for History and Philosophy of Mathematics. pp. 15-33.
    We present an approach in which ancient Greek mathematical proofs by Hippocrates of Chios and Euclid are addressed as a form of (guided) intentional reasoning. Schematically, in a proof, we start with a sentence that works as a premise; this sentence is followed by another, the conclusion of what we might take to be an inferential step. That goes on until the last conclusion is reached. Guided by the text, we go through small inferential steps; in each one, (...)
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  13.  66
    Mathematical proof theory in the light of ordinal analysis.Reinhard Kahle - 2002 - Synthese 133 (1/2):237 - 255.
    We give an overview of recent results in ordinal analysis. Therefore, we discuss the different frameworks used in mathematical proof-theory, namely "subsystem of analysis" including "reverse mathematics", "Kripke-Platek set theory", "explicit mathematics", "theories of inductive definitions", "constructive set theory", and "Martin-Löf's type theory".
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  14. Explanation in mathematics: Proofs and practice.William D'Alessandro - 2019 - Philosophy Compass 14 (11):e12629.
    Mathematicians distinguish between proofs that explain their results and those that merely prove. This paper explores the nature of explanatory proofs, their role in mathematical practice, and some of the reasons why philosophers should care about them. Among the questions addressed are the following: what kinds of proofs are generally explanatory (or not)? What makes a proof explanatory? Do all mathematical explanations involve proof in an essential way? Are there really such things as explanatory proofs, and (...)
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  15.  76
    Plans and planning in mathematical proofs.Yacin Hamami & Rebecca Lea Morris - 2020 - Review of Symbolic Logic 14 (4):1030-1065.
    In practice, mathematical proofs are most often the result of careful planning by the agents who produced them. As a consequence, each mathematical proof inherits a plan in virtue of the way it is produced, a plan which underlies its “architecture” or “unity”. This paper provides an account of plans and planning in the context of mathematical proofs. The approach adopted here consists in looking for these notions not in mathematical proofs themselves, but in the (...)
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  16.  44
    Mathematical Proofs, Gaps and Postulationism.Hugh Lehman - 1984 - The Monist 67 (1):108-114.
    In a recent paper, the mathematician Harold Edwards claimed that Euler’s alleged proof, that Fermat’s last theorem is true for the case n = 3, is flawed. Fermat’s last theorem is the conjecture that there are no positive integers x, y, z, or n, such that n is greater than two and such that xn + yn = zn. In this paper we shall first briefly explain the specific flaw to which Edwards called attention. After that we briefly explain (...)
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  17.  31
    Mathematical Proof as a Form of Appeal to a Scientific Community.Valentin A. Bazhanov - 2012 - Russian Studies in Philosophy 50 (4):56-72.
    The author analyzes proof and argumentation as a form of appeal to a scientific community with deep ethical meaning. He presents proof primarily as an effort to persuade a scientific community rather than a search for true knowledge, as an instrument by which responsibility is taken for the correctness of the thesis being proved, which usually originates in a sudden flash of insight.
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  18.  77
    The informal logic of mathematical proof.Andrew Aberdein - 2006 - In Reuben Hersh (ed.), 18 Unconventional Essays About the Nature of Mathematics. Springer Verlag. pp. 56-70.
    Informal logic is a method of argument analysis which is complementary to that of formal logic, providing for the pragmatic treatment of features of argumentation which cannot be reduced to logical form. The central claim of this paper is that a more nuanced understanding of mathematical proof and discovery may be achieved by paying attention to the aspects of mathematical argumentation which can be captured by informal, rather than formal, logic. Two accounts of argumentation are considered: the (...)
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  19. On the concept of proof in elementary geometry Pirmin stekeler-weithofer.Proof In Elementary - 1992 - In Michael Detlefsen (ed.), Proof and Knowledge in Mathematics. Routledge.
     
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  20.  16
    Rationality in Mathematical Proofs.Yacin Hamami & Rebecca Lea Morris - 2023 - Australasian Journal of Philosophy 101 (4):793-808.
    Mathematical proofs are not sequences of arbitrary deductive steps—each deductive step is, to some extent, rational. This paper aims to identify and characterize the particular form of rationality at play in mathematical proofs. The approach adopted consists in viewing mathematical proofs as reports of proof activities—that is, sequences of deductive inferences—and in characterizing the rationality of the former in terms of that of the latter. It is argued that proof activities are governed by specific norms (...)
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  21. The History of Mathematical Proof in Ancient Traditions.Karine Chemla (ed.) - 2012 - Cambridge University Press.
    This radical, profoundly scholarly book explores the purposes and nature of proof in a range of historical settings. It overturns the view that the first mathematical proofs were in Greek geometry and rested on the logical insights of Aristotle by showing how much of that view is an artefact of nineteenth-century historical scholarship. It documents the existence of proofs in ancient mathematical writings about numbers and shows that practitioners of mathematics in Mesopotamian, Chinese and Indian cultures knew (...)
     
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  22. Mathematical Proof and the Reliability of DNA Evidence.Don Fallis - 1996 - The American Mathematical Monthly 103 (6):491-497.
  23.  24
    Mathematical Proof.John Pollock - 1967 - American Philosophical Quarterly 4 (3):238 - 244.
  24.  33
    The dialectical tier of mathematical proof.Andrew Aberdein - 2011 - In Frank Zenker (ed.), Argumentation: Cognition & Community. Proceedings of the 9th International Conference of the Ontario Society for the Study of Argumentation (OSSA), May 18--21, 2011. OSSA.
    Ralph Johnson argues that mathematical proofs lack a dialectical tier, and thereby do not qualify as arguments. This paper argues that, despite this disavowal, Johnson’s account provides a compelling model of mathematical proof. The illative core of mathematical arguments is held to strict standards of rigour. However, compliance with these standards is itself a matter of argument, and susceptible to challenge. Hence much actual mathematical practice takes place in the dialectical tier.
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  25. Mathematical proof: Dedicated to the memory of A. Thomas Tymoczko (1943 9 1-1996 8 9).R. S. D. Thomas - 1999 - Philosophia Mathematica 7 (1):3-4.
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  26. On mathematical proofs and meaning.H. N. Castaneda - 1961 - Mind 70 (279):385-390.
  27.  50
    Mathematical proof.Edward T. Dixon - 1929 - Mind 38 (151):343-351.
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  28.  18
    Mathematical Proof and Experimental Proof.Arthur H. Copeland - 1966 - 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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  29.  13
    Mathematical Proof and Experimental Proof.Arthur H. Copeland - 1966 - Philosophy of Science 33 (4):303 - 316.
    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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  30.  11
    Arguing Around Mathematical Proofs.Michel Dufour - 2013 - In Andrew Aberdein & Ian J. Dove (eds.), The Argument of Mathematics. Dordrecht: Springer. pp. 61-76.
    More or less explicitly inspired by the Aristotelian classification of arguments, a wide tradition makes a sharp distinction between argument and proof. Ch. Perelman and R. Johnson, among others, share this view based on the principle that the conclusion of an argument is uncertain while the conclusion of a proof is certain. Producing proof is certainly a major part of mathematical activity. Yet, in practice, mathematicians, expert or beginner, argue about mathematical proofs. This happens during (...)
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  31.  42
    Audience role in mathematical proof development.Zoe Ashton - 2020 - Synthese 198 (Suppl 26):6251-6275.
    The role of audiences in mathematical proof has largely been neglected, in part due to misconceptions like those in Perelman and Olbrechts-Tyteca which bar mathematical proofs from bearing reflections of audience consideration. In this paper, I argue that mathematical proof is typically argumentation and that a mathematician develops a proof with his universal audience in mind. In so doing, he creates a proof which reflects the standards of reasonableness embodied in his universal audience. (...)
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  32.  77
    Acceptable gaps in mathematical proofs.Line Edslev Andersen - 2020 - Synthese 197 (1):233-247.
    Mathematicians often intentionally leave gaps in their proofs. Based on interviews with mathematicians about their refereeing practices, this paper examines the character of intentional gaps in published proofs. We observe that mathematicians’ refereeing practices limit the number of certain intentional gaps in published proofs. The results provide some new perspectives on the traditional philosophical questions of the nature of proof and of what grounds mathematical knowledge.
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  33.  43
    Strategic Maneuvering in Mathematical Proofs.Erik C. W. Krabbe - 2008 - Argumentation 22 (3):453-468.
    This paper explores applications of concepts from argumentation theory to mathematical proofs. Note is taken of the various contexts in which proofs occur and of the various objectives they may serve. Examples of strategic maneuvering are discussed when surveying, in proofs, the four stages of argumentation distinguished by pragma-dialectics. Derailments of strategies (fallacies) are seen to encompass more than logical fallacies and to occur both in alleged proofs that are completely out of bounds and in alleged proofs that are (...)
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  34.  77
    Mathematical proof and experimental proof.Sr Arthur H. Copeland - 1966 - Philosophy of Science 33 (4):303-316.
    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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  35.  76
    Granularity Analysis for Mathematical Proofs.Marvin R. G. Schiller - 2013 - Topics in Cognitive Science 5 (2):251-269.
    Mathematical proofs generally allow for various levels of detail and conciseness, such that they can be adapted for a particular audience or purpose. Using automated reasoning approaches for teaching proof construction in mathematics presupposes that the step size of proofs in such a system is appropriate within the teaching context. This work proposes a framework that supports the granularity analysis of mathematical proofs, to be used in the automated assessment of students' proof attempts and for the (...)
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  36.  13
    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 (...)
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  37.  23
    The Phenomenology of Mathematical Proof.Gian_carlo Rota - 1997 - Synthese 111 (2):183-196.
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  38.  65
    Wittgenstein on Mathematical Proof.Crispin Wright - 1990 - Royal Institute of Philosophy Supplement 28:79-99.
    To be asked to provide a short paper on Wittgenstein's views on mathematical proof is to be given a tall order . Close to one half of Wittgenstein's writings after 1929 concerned mathematics, and the roots of his discussions, which contain a bewildering variety of underdeveloped and sometimes conflicting suggestions, go deep to some of the most basic and difficult ideas in his later philosophy. So my aims in what follows are forced to be modest. I shall sketch (...)
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  39. The Beauty (?) of Mathematical Proofs.Catarina Dutilh Novaes - 2019 - In Andrew Aberdein & Matthew Inglis (eds.), Advances in Experimental Philosophy of Logic and Mathematics. London: Bloomsbury Academic. pp. 63-93.
  40. The Surveyability of Mathematical Proof: A Historical Perspective.O. Bradley Bassler - 2006 - 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 (...)
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  41.  3
    An introduction to mathematical proofs.Nicholas A. Loehr - 2020 - 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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  42.  2
    Fundamentals of mathematical proof.Charles A. 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 (...)
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  43.  90
    Intentional gaps in mathematical proofs.Don Fallis - 2003 - Synthese 134 (1-2):45 - 69.
  44.  6
    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. (...)
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  45.  68
    Evolution of mathematical proof.Marian Mrozek & Jacek Urbaniec - 1997 - Foundations of Science 2 (1):77-85.
    The authors present the main ideas of the computer-assisted proof of Mischaikow and Mrozek that chaos is really present in the Lorenz equations. Methodological consequences of this proof are examined. It is shown that numerical calculations can constitute an essential part of mathematical proof not only in the discrete mathematics but also in the mathematics of continua.
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  46.  13
    The Instructive Function of Mathematical Proof: A Case Study of the Analysis cum Synthesis method in Apollonius of Perga’s Conics.Linden Anne Duffee - 2021 - Axiomathes 31 (5):601-617.
    This essay discusses the instructional value of mathematical proofs using different interpretations of the analysis cum synthesis method in Apollonius’ Conics as a case study. My argument is informed by Descartes’ complaint about ancient geometers and William Thurston’s discussion on how mathematical understanding is communicated. Three historical frameworks of the analysis/synthesis distinction are used to understand the instructive function of the analysis cum synthesis method: the directional interpretation, the structuralist interpretation, and the phenomenological interpretation. I apply these interpretations (...)
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  47.  65
    Introduction to mathematical proofs: a transition.Charles E. Roberts - 2009 - Boca Raton: CRC Press.
    The book includes more than 75 examples and more than 600 problems. A solutions manual is available upon qualifying course adoptions.
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  48. Introduction to mathematical proof: a transition to advanced mathematics.Charles E. Roberts - 2015 - Boca Raton: CRC Press, Taylor & Francis Group.
  49.  4
    Granularity analysis for tutoring mathematical proofs.Marvin Schiller - 2011 - [Heidelberg]: AKA Verlag.
    Rigorous formal proof is one of the key techniques in the natural sciences, engineering, and of course also in the formal sciences. Progress in automated reasoning increasingly enables computer systems to support, and even teach, users to conduct formal a.
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  50.  5
    The science of learning mathematical proofs: an introductory course.Elana Reiser - 2021 - 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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