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  1. Andrew Aberdein (2011). The Dialectical Tier of Mathematical Proof. 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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  2. Andrew Aberdein (2006). The Informal Logic of Mathematical Proof. In Reuben Hersh (ed.), 18 Unconventional Essays About the Nature of Mathematics. Springer-Verlag. 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 pioneering work of (...)
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  3. Andrew Arana (2009). On Formally Measuring and Eliminating Extraneous Notions in Proofs. Philosophia Mathematica 17 (2):208–219.
    Many mathematicians and philosophers of mathematics believe some proofs contain elements extraneous to what is being proved. In this paper I discuss extraneousness generally, and then consider a specific proposal for measuring extraneousness syntactically. This specific proposal uses Gentzen’s cut-elimination theorem. I argue that the proposal fails, and that we should be skeptical about the usefulness of syntactic extraneousness measures.
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  4. Andrew Arana (2008). Logical and Semantic Purity. Protosociology 25:36-48.
    Many mathematicians have sought ‘pure’ proofs of theorems. There are different takes on what a ‘pure’ proof is, though, and it’s important to be clear on their differences, because they can easily be conflated. In this paper I want to distinguish between two of them.
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  5. A. Baker (2010). Mathematical Induction and Explanation. Analysis 70 (4):681-689.
  6. John Corcoran (1973). A Mathematical Model of Aristotle's Syllogistic. Archiv für Geschichte der Philosophie 55 (2):191-219.
    In the present article we attempt to show that Aristotle's syllogistic is an underlying logiC which includes a natural deductive system and that it isn't an axiomatic theory as had previously been thought. We construct a mathematical model which reflects certain structural aspects of Aristotle's logic. We examine the relation of the model to the system of logic envisaged in scattered parts of Prior and Posterior Analytics. Our interpretation restores Aristotle's reputation as a logician of consummate imagination and skill. Several (...)
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  7. Cesare Cozzo (2011). Matematica e retorica. Paradigmi (3):59-72.
    The traditional opposition between mathematical proof and rhetorical argument is based on a non-contextual picture of proof, against which historical and theoretical objections have been raised. The author advocates a different opposition, between epistemic rhetoric and instrumental rhetoric. Instrumental rhetoric aims at persuasion without caring for truth. Epistemic rhetoric is a practice aimed at both persuasion and truth. Aiming at truth is a way of acting, which can be characterized in terms of epistemically virtuous behavioural traits. In this sense epistemic (...)
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  8. Cesare Cozzo (2005). Can a Proof Compel Us? In C. Cellucci D. Gillies (ed.), Mathematical Reasoning and Heuristics. King's College Publications. 191-212.
    The compulsion of proofs is an ancient idea, which plays an important role in Plato’s dialogues. The reader perhaps recalls Socrates’ question to the slave boy in the Meno: “If the side of a square A is 2 feet, and the corresponding area is 4, how long is the side of a square whose area is double, i.e. 8?”. The slave answers: “Obviously, Socrates, it will be twice the length” (cf. Me 82-85). A straightforward analogy: if the area is double, (...)
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  9. Harry Deutsch (2010). Diagonalization and Truth Functional Operators. Analysis 70 (2):215-217.
    (No abstract is available for this citation).
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  10. James Franklin (1996). Proof in Mathematics: An Introduction. Quakers Hill Press.
    Why do students take the instruction "prove" in examinations to mean "go to the next question"? Because they have not been shown the simple techniques of how to do it. Mathematicians meanwhile generate a mystique of proof, as if it requires an inborn and unteachable genius. True, creating research-level proofs does require talent; but reading and understanding the proof that the square of an even number is even is within the capacity of most mortals.
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  11. Miguel Hoeltje, Benjamin Schnieder & Alex Steinberg (2013). Explanation by Induction? Synthese 190 (3):509-524.
    Philosophers of mathematics commonly distinguish between explanatory and non-explanatory proofs. An important subclass of mathematical proofs are proofs by induction. Are they explanatory? This paper addresses the question, based on general principles about explanation. First, a recent argument for a negative answer is discussed and rebutted. Second, a case is made for a qualified positive take on the issue.
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  12. Kenneth Manders (2008). The Euclidean Diagram. In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. Oxford University Press. 80--133.
    This chapter gives a detailed study of diagram-based reasoning in Euclidean plane geometry (Books I, III), as well as an exploration how to characterise a geometric practice. First, an account is given of diagram attribution: basic geometrical claims are classified as exact (equalities, proportionalities) or co-exact (containments, contiguities); exact claims may only be inferred from prior entries in the demonstration text, but co-exact claims may be asserted based on what is seen in the diagram. Diagram control by constructions is necessary (...)
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  13. Jaroslav Peregrin, Diagonalization.
    It is a trivial fact that if we have a square table filled with numbers, we can always form a column which is not yet contained in the table. Despite its apparent triviality, this fact underlies the foundations of most of the path-breaking results of logic in the second half of the nineteenth and the first half of the twentieth century. We explain how this fact can be used to show that there are more sequences of natural numbers than there (...)
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  14. Marvin R. G. Schiller (2013). Granularity Analysis for Mathematical Proofs. 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 presentation of hints and (...)
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  15. David Sherry (2009). The Role of Diagrams in Mathematical Arguments. Foundations of Science 14 (1-2):59-74.
    Recent accounts of the role of diagrams in mathematical reasoning take a Platonic line, according to which the proof depends on the similarity between the perceived shape of the diagram and the shape of the abstract object. This approach is unable to explain proofs which share the same diagram in spite of drawing conclusions about different figures. Saccheri’s use of the bi-rectangular isosceles quadrilateral in Euclides Vindicatus provides three such proofs. By forsaking abstract objects it is possible to give a (...)
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  16. Petros Stefaneas & Ioannis M. Vandoulakis (2012). The Web as A Tool For Proving. 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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  17. Iulian D. Toader (2011). Objectivity Sans Intelligibility. Hermann Weyl's Symbolic Constructivism. Dissertation, University of Notre Dame