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Evidence, Proofs, and Derivations

ZDM 51 (5):825-834 (2019)

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  1. The Uses of Argument.Stephen E. Toulmin - 1958 - Philosophy 34 (130):244-245.
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  • Why Do We Prove Theorems?Yehuda Rav - 1998 - Philosophia Mathematica 6 (3):5-41.
    Ordinary mathematical proofs—to be distinguished from formal derivations—are the locus of mathematical knowledge. Their epistemic content goes way beyond what is summarised in the form of theorems. Objections are raised against the formalist thesis that every mainstream informal proof can be formalised in some first-order formal system. Foundationalism is at the heart of Hilbert's program and calls for methods of formal logic to prove consistency. On the other hand, ‘systemic cohesiveness’, as proposed here, seeks to explicate why mathematical knowledge is (...)
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  • Formalization, primitive concepts, and purity: Formalization, primitive concepts, and purity.John T. Baldwin - 2013 - Review of Symbolic Logic 6 (1):87-128.
    We emphasize the role of the choice of vocabulary in formalization of a mathematical area and remark that this is a particular preoccupation of logicians. We use this framework to discuss Kennedy’s notion of ‘formalism freeness’ in the context of various schools in model theory. Then we clarify some of the mathematical issues in recent discussions of purity in the proof of the Desargues proposition. We note that the conclusion of ‘spatial content’ from the Desargues proposition involves arguments which are (...)
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  • Knowledge of Mathematics without Proof.Alexander Paseau - 2015 - British Journal for the Philosophy of Science 66 (4):775-799.
    Mathematicians do not claim to know a proposition unless they think they possess a proof of it. For all their confidence in the truth of a proposition with weighty non-deductive support, they maintain that, strictly speaking, the proposition remains unknown until such time as someone has proved it. This article challenges this conception of knowledge, which is quasi-universal within mathematics. We present four arguments to the effect that non-deductive evidence can yield knowledge of a mathematical proposition. We also show that (...)
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  • Intuition.Elijah Chudnoff - 2013 - New York, NY: Oxford University Press.
    Elijah Chudnoff elaborates and defends a view of intuition according to which intuition purports to, and reveals, how matters stand in abstract reality by making us aware of that reality through the intellect. He explores the experience of having an intuition; justification for beliefs that derives from intuition; and contact with abstract reality.
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  • The parallel structure of mathematical reasoning.Andrew Aberdein - 2012 - In Alison Pease & Brendan Larvor (eds.), Proceedings of the Symposium on Mathematical Practice and Cognition Ii: A Symposium at the Aisb/Iacap World Congress 2012. Society for the Study of Artificial Intelligence and the Simulation of Behaviour. pp. 7--14.
    This paper proposes an account of mathematical reasoning as parallel in structure: the arguments which mathematicians use to persuade each other of their results comprise the argumentational structure; the inferential structure is composed of derivations which offer a formal counterpart to these arguments. Some conflicts about the foundations of mathematics correspond to disagreements over which steps should be admissible in the inferential structure. Similarly, disagreements over the admissibility of steps in the argumentational structure correspond to different views about mathematical practice. (...)
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  • Mathematical Wit and Mathematical Cognition.Andrew Aberdein - 2013 - Topics in Cognitive Science 5 (2):231-250.
    The published works of scientists often conceal the cognitive processes that led to their results. Scholars of mathematical practice must therefore seek out less obvious sources. This article analyzes a widely circulated mathematical joke, comprising a list of spurious proof types. An account is proposed in terms of argumentation schemes: stereotypical patterns of reasoning, which may be accompanied by critical questions itemizing possible lines of defeat. It is argued that humor is associated with risky forms of inference, which are essential (...)
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  • Five theories of reasoning: Interconnections and applications to mathematics.Alison Pease & Andrew Aberdein - 2011 - Logic and Logical Philosophy 20 (1-2):7-57.
    The last century has seen many disciplines place a greater priority on understanding how people reason in a particular domain, and several illuminating theories of informal logic and argumentation have been developed. Perhaps owing to their diverse backgrounds, there are several connections and overlapping ideas between the theories, which appear to have been overlooked. We focus on Peirce’s development of abductive reasoning [39], Toulmin’s argumentation layout [52], Lakatos’s theory of reasoning in mathematics [23], Pollock’s notions of counterexample [44], and argumentation (...)
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  • Mathematics and argumentation.Andrew Aberdein - 2009 - Foundations of Science 14 (1-2):1-8.
    Some authors have begun to appeal directly to studies of argumentation in their analyses of mathematical practice. These include researchers from an impressively diverse range of disciplines: not only philosophy of mathematics and argumentation theory, but also psychology, education, and computer science. This introduction provides some background to their work.
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  • Non-deductive logic in mathematics.James Franklin - 1987 - British Journal for the Philosophy of Science 38 (1):1-18.
    Mathematicians often speak of conjectures as being confirmed by evidence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long resisted proof, such as Fermat's Last Theorem and the Riemann Hypothesis, have had to be considered in terms of the evidence for and against them. It is argued here that it is not adequate to describe the relation of evidence to hypothesis as `subjective', `heuristic' or (...)
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  • The Epistemic Status of Probabilistic Proof.Don Fallis - 1997 - Journal of Philosophy 94 (4):165-186.
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  • Impact of Informatics on Mathematics and Its Teaching.Simon Modeste - 2016 - In F. Gadducci & M. Tavosanis (eds.), History and Philosophy of Computing. HaPoC 2015. IFIP Advances in Information and Communication Technology, vol 487. Springer. pp. 243-255.
    In this article, we come back to the seminal role of epistemology in didactics of sciences and particularly in mathematics. We defend that the epistemological research on the interactions between mathematics and informatics is necessary to feed didactical research on today’s mathematics learning and teaching situations, impacted by the development of informatics. We develop some examples to support this idea and propose some perspectives to attack this issue.
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  • Argumentation Schemes.Douglas Walton, Christopher Reed & Fabrizio Macagno - 2008 - Cambridge and New York: Cambridge University Press. Edited by Chris Reed & Fabrizio Macagno.
    This book provides a systematic analysis of many common argumentation schemes and a compendium of 96 schemes. The study of these schemes, or forms of argument that capture stereotypical patterns of human reasoning, is at the core of argumentation research. Surveying all aspects of argumentation schemes from the ground up, the book takes the reader from the elementary exposition in the first chapter to the latest state of the art in the research efforts to formalize and classify the schemes, outlined (...)
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  • The Argument Web: an Online Ecosystem of Tools, Systems and Services for Argumentation.Mark Snaith, Alison Pease, John Lawrence, Barbara Konat, Mathilde Janier, Rory Duthie, Katarzyna Budzynska & Chris Reed - 2017 - Philosophy and Technology 30 (2):137-160.
    The Argument Web is maturing as both a platform built upon a synthesis of many contemporary theories of argumentation in philosophy and also as an ecosystem in which various applications and application components are contributed by different research groups around the world. It already hosts the largest publicly accessible corpora of argumentation and has the largest number of interoperable and cross compatible tools for the analysis, navigation and evaluation of arguments across a broad range of domains, languages and activity types. (...)
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  • Why the Naïve Derivation Recipe Model Cannot Explain How Mathematicians’ Proofs Secure Mathematical Knowledge.Brendan Larvor - 2016 - Philosophia Mathematica 24 (3):401-404.
    The view that a mathematical proof is a sketch of or recipe for a formal derivation requires the proof to function as an argument that there is a suitable derivation. This is a mathematical conclusion, and to avoid a regress we require some other account of how the proof can establish it.
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  • What Students' Arguments Can Tell Us: Using Argumentation Schemes in Science Education.Fabrizio Macagno & Aikaterini Konstantinidou - 2013 - Argumentation 27 (3):225-243.
    The relationship between teaching and argumentation is becoming a crucial issue in the field of education and, in particular, science education. Teaching has been analyzed as a dialogue aimed at persuading the interlocutors, introducing a conceptual change that needs to be grounded on the audience’s background knowledge. This paper addresses this issue from a perspective of argumentation studies. Our claim is that argumentation schemes, namely abstract patterns of argument, can be an instrument for reconstructing the tacit premises in students’ argumentative (...)
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  • The Web as A Tool For Proving.Petros Stefaneas & Ioannis M. Vandoulakis - 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 (...)
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  • Is there a problem of induction for mathematics?Alan Baker - 2007 - In Mary Leng, Alexander Paseau & Michael D. Potter (eds.), Mathematical Knowledge. Oxford, England: Oxford University Press. pp. 57-71.
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  • Understanding Students’ Reasoning: Argumentation Schemes as an Interpretation Method in Science Education.Aikaterini Konstantinidou & Fabrizio Macagno - 2013 - Science & Education 22 (5):1069-1087.
    The relationship between teaching and argumentation is becoming a crucial issue in the field of education and, in particular, science education. Teaching has been analyzed as a dialogue aimed at persuading the interlocutors, introducing a conceptual change that needs to be grounded on the audience’s background knowledge. This paper addresses this issue from a perspective of argumentation studies. Our claim is that argumentation schemes, namely abstract patterns of argument, can be an instrument for reconstructing the tacit premises in students’ argumentative (...)
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  • Why Do We Prove Theorems?Yehuda Rav - 1999 - Philosophia Mathematica 7 (1):5-41.
    Ordinary mathematical proofs—to be distinguished from formal derivations—are the locus of mathematical knowledge. Their epistemic content goes way beyond what is summarised in the form of theorems. Objections are raised against the formalist thesis that every mainstream informal proof can be formalised in some first-order formal system. Foundationalism is at the heart of Hilbert's program and calls for methods of formal logic to prove consistency. On the other hand, ‘systemic cohesiveness’, as proposed here, seeks to explicate why mathematical knowledge is (...)
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  • Proofs and Arguments: The Special Case of Mathematics.Jean van BendegemPaul - 2005 - Poznan Studies in the Philosophy of the Sciences and the Humanities 84 (1):157-169.
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  • The Art of Logic: How to Make Sense in a World That Doesn't.Eugenia Cheng - 2018 - London, England: Profile Books.
    Emotions are powerful. In newspaper headlines and on social media, they have become the primary way of understanding the world. But strong feelings make it more difficult to see the reality behind the rhetoric. In The Art of Logic, Eugenia Cheng shows how mathematical logic can help us see things more clearly - and know when politicians and companies are trying to mislead us.First Cheng explains how to use black-and-white logic to illuminate the world around us, giving us new insight (...)
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  • Many-valued logic of informal provability: A non-deterministic strategy.Pawel Pawlowski & Rafal Urbaniak - 2018 - Review of Symbolic Logic 11 (2):207-223.
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  • 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 (...)
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  • 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 what (...)
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  • Mathematics, Form and Function.Saunders MacLane - 1986 - Journal of Philosophy 84 (1):33-37.
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  • Mathematics: Form and Function.Saunders Mac Lane - 1990 - Studia Logica 49 (3):424-426.
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  • Proof and proving.Oswaldo Chateaubriand - 2003 - O Que Nos Faz Pensar:41-56.
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  • The Web as a Tool for Proving.Petros Stefaneas & Ioannis M. Vandoulakis - 2014 - In Harry Halpin & Alexandre Monnin (eds.), Philosophical Engineering: Toward a Philosophy of the Web. Wiley-Blackwell. pp. 149-167.
    This is the first interdisciplinary exploration of the philosophical foundations of the Web, a new area of inquiry that has important implications across a range of domains. - Contains twelve essays that bridge the fields of philosophy, cognitive science, and phenomenology. - Tackles questions such as the impact of Google on intelligence and epistemology, the philosophical status of digital objects, ethics on the Web, semantic and ontological changes caused by the Web, and the potential of the Web to serve as (...)
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  • Is There a “Hilbert Thesis”?Reinhard Kahle - 2019 - Studia Logica 107 (1):145-165.
    In his introductory paper to first-order logic, Jon Barwise writes in the Handbook of Mathematical Logic :[T]he informal notion of provable used in mathematics is made precise by the formal notion provable in first-order logic. Following a sug[g]estion of Martin Davis, we refer to this view as Hilbert’s Thesis.This paper reviews the discussion of Hilbert’s Thesis in the literature. In addition to the question whether it is justifiable to use Hilbert’s name here, the arguments for this thesis are compared with (...)
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  • The Relationship of Derivations in Artificial Languages to Ordinary Rigorous Mathematical Proof.J. Azzouni - 2013 - 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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  • Lakatos-style collaborative mathematics through dialectical, structured and abstract argumentation.Alison Pease, John Lawrence, Katarzyna Budzynska, Joseph Corneli & Chris Reed - 2017 - Artificial Intelligence 246 (C):181-219.
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  • Argumentation Theory for Mathematical Argument.Joseph Corneli, Ursula Martin, Dave Murray-Rust, Gabriela Rino Nesin & Alison Pease - 2019 - Argumentation 33 (2):173-214.
    To adequately model mathematical arguments the analyst must be able to represent the mathematical objects under discussion and the relationships between them, as well as inferences drawn about these objects and relationships as the discourse unfolds. We introduce a framework with these properties, which has been used to analyse mathematical dialogues and expository texts. The framework can recover salient elements of discourse at, and within, the sentence level, as well as the way mathematical content connects to form larger argumentative structures. (...)
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  • Revealing Structures of Argumentations in Classroom Proving Processes.Christine Knipping & David Reid - 2013 - In Andrew Aberdein & Ian J. Dove (eds.), The Argument of Mathematics. Dordrecht, Netherland: Springer. pp. 119--146.
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  • “Inference versus consequence” revisited: inference, consequence, conditional, implication.Göran Sundholm - 2012 - Synthese 187 (3):943-956.
    Inference versus consequence , an invited lecture at the LOGICA 1997 conference at Castle Liblice, was part of a series of articles for which I did research during a Stockholm sabbatical in the autumn of 1995. The article seems to have been fairly effective in getting its point across and addresses a topic highly germane to the Uppsala workshop. Owing to its appearance in the LOGICA Yearbook 1997 , Filosofia Publishers, Prague, 1998, it has been rather inaccessible. Accordingly it is (...)
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  • Mathematics as the art of abstraction.Richard L. Epstein - 2013 - In Andrew Aberdein & Ian J. Dove (eds.), The Argument of Mathematics. Dordrecht, Netherland: Springer. pp. 257--289.
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