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  1. 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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  2. Michael Detlefsen (2011). Discovery, Invention and Realism: Gödel and Others on the Reality of Concepts. In John Polkinghorne (ed.), Meaning in Mathematics. OUP Oxford
    The general question considered is whether and to what extent there are features of our mathematical knowledge that support a realist attitude towards mathematics. I consider, in particular, reasoning from claims such as that mathematicians believe their reasoning to be part of a process of discovery (and not of mere invention), to the view that mathematical entities exist in some mind-independent way although our minds have epistemic access to them.
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  3. Kenny Easwaran (2009). Probabilistic Proofs and Transferability. 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 have. I also consider what this (...)
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  4. Don Fallis (2002). What Do Mathematicians Want? Probabilistic Proofs and the Epistemic Goals of Mathematicians. Logique Et Analyse 45.
    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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  5. Don Fallis (1997). The Epistemic Status of Probabilistic Proof. Journal of Philosophy 94 (4):165-186.
  6. James Franklin (2014). Aristotelian Realist Philosophy of Mathematics. Palgrave MacMillan.
    An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, are parts of the physical world and (...)
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  7. James Franklin (2013). Non-Deductive Logic in Mathematics: The Probability of Conjectures. In Andrew Aberdein & Ian J. Dove (eds.), The Argument of Mathematics. Springer 11--29.
    Mathematicians often speak of conjectures, yet unproved, as probable or well-confirmed by evidence. The Riemann Hypothesis, for example, is widely believed to be almost certainly true. There seems no initial reason to distinguish such probability from the same notion in empirical science. Yet it is hard to see how there could be probabilistic relations between the necessary truths of pure mathematics. The existence of such logical relations, short of certainty, is defended using the theory of logical probability (or objective Bayesianism (...)
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  8. James Franklin (1987). Non-Deductive Logic in Mathematics. 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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  9. Catherine Legg & James Franklin (forthcoming). Perceiving Necessity. Pacific Philosophical Quarterly.
    In many diagrams one seems to perceive necessity – one sees not only that something is so, but that it must be so. That conflicts with a certain empiricism largely taken for granted in contemporary philosophy, which believes perception is not capable of such feats. The reason for this belief is often thought well-summarized in Hume’s maxim: ‘there are no necessary connections between distinct existences’. It is also thought that even if there were such necessities, perception is too passive or (...)
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  10. Mark McEvoy (2008). The Epistemological Status of Computer-Assisted Proofs. Philosophia Mathematica 16 (3):374-387.
    Several high-profile mathematical problems have been solved in recent decades by computer-assisted proofs. Some philosophers have argued that such proofs are a posteriori on the grounds that some such proofs are unsurveyable; that our warrant for accepting these proofs involves empirical claims about the reliability of computers; that there might be errors in the computer or program executing the proof; and that appeal to computer introduces into a proof an experimental element. I argue that none of these arguments withstands scrutiny, (...)
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  11. Mark McEvoy (2013). Experimental Mathematics, Computers and the a Priori. Synthese 190 (3):397-412.
    In recent decades, experimental mathematics has emerged as a new branch of mathematics. This new branch is defined less by its subject matter, and more by its use of computer assisted reasoning. Experimental mathematics uses a variety of computer assisted approaches to verify or prove mathematical hypotheses. For example, there is “number crunching” such as searching for very large Mersenne primes, and showing that the Goldbach conjecture holds for all even numbers less than 2 × 1018. There are “verifications” of (...)
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  12. E. Mendelson (2005). Selmer Bringsjord and Michael Zenzen. Superminds: People Harness Hypercomputation, and More. Studies in Cognitive Systems, Volume 29. Dordrecht: Kluwer Academic Publishers, 2003. Pp. Xxx + 339. ISBN 1-4020-1094-X. [REVIEW] Philosophia Mathematica 13 (2):228-230.
  13. Ryszard Stanisław Michalski (1977). Toward Computer-Aided Induction: A Brief Review of Currently Implemented Aqval Programs. Dept. Of Computer Science, University of Illinois at Urbana-Champaign.
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  14. Philip L. Peterson (1991). What is Empirical in Mathematics? Philosophia Mathematica (1):91-110.
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  15. George Polya (1990). Mathematics and Plausible Reasoning. Princeton University Press.
    Here the author of How to Solve It explains how to become a "good guesser." Marked by G. Polya's simple, energetic prose and use of clever examples from a wide range of human activities, this two-volume work explores techniques of guessing, inductive reasoning, and reasoning by analogy, and the role they play in the most rigorous of deductive disciplines.
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  16. George Pólya (1968). Mathematics and Plausible Reasoning. Princeton, N.J.,Princeton University Press.
  17. David Sherry (2006). Mathematical Reasoning: Induction, Deduction and Beyond. Studies in History and Philosophy of Science Part A 37 (3):489-504.
    Mathematics used to be portrayed as a deductive science. Stemming from Polya , however, is a philosophical movement which broadens the concept of mathematical reasoning to include inductive or quasi-empirical methods. Interest in inductive methods is a welcome turn from foundationalism toward a philosophy grounded in mathematical practice. Regrettably, though, the conception of mathematical reasoning embraced by quasi-empiricists is still too narrow to include the sort of thought-experiment which Mueller describes as traditional mathematical proof and which Lakatos examines in Proofs (...)
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  18. Petros Stefaneas & Ioannis M. Vandoulakis (2014). The Web as a Tool for Proving. In Harry Halpin & Alexandre Monnin (eds.), Philosophical Engineering: Toward a Philosophy of the Web. Wiley-Blackwell 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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  19. 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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  20. Ioannis M. Vandoulakis (2010). A Genetic Interpretation of Neo-Pythagorean Arithmetic. Oriens - Occidens 7:113-154.