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  1. Between Probability and Certainty: What Justifies Belief.Martin Smith - 2016 - Oxford, GB: Oxford University Press UK.
    This book explores a question central to philosophy--namely, what does it take for a belief to be justified or rational? According to a widespread view, whether one has justification for believing a proposition is determined by how probable that proposition is, given one's evidence. In this book this view is rejected and replaced with another: in order for one to have justification for believing a proposition, one's evidence must normically support it--roughly, one's evidence must make the falsity of that proposition (...)
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  • Groundwork for a Fallibilist Account of Mathematics.Silvia De Toffoli - 2021 - Philosophical Quarterly 7 (4):823-844.
    According to the received view, genuine mathematical justification derives from proofs. In this article, I challenge this view. First, I sketch a notion of proof that cannot be reduced to deduction from the axioms but rather is tailored to human agents. Secondly, I identify a tension between the received view and mathematical practice. In some cases, cognitively diligent, well-functioning mathematicians go wrong. In these cases, it is plausible to think that proof sets the bar for justification too high. I then (...)
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  • Group Knowledge and Mathematical Collaboration: A Philosophical Examination of the Classification of Finite Simple Groups.Joshua Habgood-Coote & Fenner Stanley Tanswell - 2023 - Episteme 20 (2):281-307.
    In this paper we apply social epistemology to mathematical proofs and their role in mathematical knowledge. The most famous modern collaborative mathematical proof effort is the Classification of Finite Simple Groups. The history and sociology of this proof have been well-documented by Alma Steingart (2012), who highlights a number of surprising and unusual features of this collaborative endeavour that set it apart from smaller-scale pieces of mathematics. These features raise a number of interesting philosophical issues, but have received very little (...)
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  • Epistemic injustice in mathematics.Colin Jakob Rittberg, Fenner Stanley Tanswell & Jean Paul Van Bendegem - 2020 - Synthese 197 (9):3875-3904.
    We investigate how epistemic injustice can manifest itself in mathematical practices. We do this as both a social epistemological and virtue-theoretic investigation of mathematical practices. We delineate the concept both positively—we show that a certain type of folk theorem can be a source of epistemic injustice in mathematics—and negatively by exploring cases where the obstacles to participation in a mathematical practice do not amount to epistemic injustice. Having explored what epistemic injustice in mathematics can amount to, we use the concept (...)
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  • The Role of Trust in Argumentation.Catarina Dutilh Novaes - 2020 - Informal Logic 40 (2):205-236.
    Argumentation is important for sharing knowledge and information. Given that the receiver of an argument purportedly engages first and foremost with its content, one might expect trust to play a negligible epistemic role, as opposed to its crucial role in testimony. I argue on the contrary that trust plays a fundamental role in argumentative engagement. I present a realistic social epistemological account of argumentation inspired by social exchange theory. Here, argumentation is a form of epistemic exchange. I illustrate my argument (...)
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  • 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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  • Open texture, rigor, and proof.Benjamin Zayton - 2022 - Synthese 200 (4):1-20.
    Open texture is a kind of semantic indeterminacy first systematically studied by Waismann. In this paper, extant definitions of open texture will be compared and contrasted, with a view towards the consequences of open-textured concepts in mathematics. It has been suggested that these would threaten the traditional virtues of proof, primarily the certainty bestowed by proof-possession, and this suggestion will be critically investigated using recent work on informal proof. It will be argued that informal proofs have virtues that mitigate the (...)
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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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  • Non-deductive methods in mathematics.Alan Baker - 2010 - Stanford Encyclopedia of Philosophy.
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  • When journal editors play favorites.Remco Heesen - 2018 - Philosophical Studies 175 (4):831-858.
    Should editors of scientific journals practice triple-anonymous reviewing? I consider two arguments in favor. The first says that insofar as editors’ decisions are affected by information they would not have had under triple-anonymous review, an injustice is committed against certain authors. I show that even well-meaning editors would commit this wrong and I endorse this argument. The second argument says that insofar as editors’ decisions are affected by information they would not have had under triple-anonymous review, it will negatively affect (...)
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  • Bullshit activities.Kenny Easwaran - forthcoming - Analytic Philosophy.
    Frankfurt gave an account of “bullshit” as a statement made without regard to truth or falsity. Austin argued that a large amount of language consists of speech acts aimed at goals other than truth or falsity. We don't want our account of bullshit to include all performatives. I develop a modification of Frankfurt's account that makes interesting and useful categorizations of various speech acts as bullshit or not and show that this account generalizes to many other kinds of act as (...)
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  • Computational Complexity Theory and the Philosophy of Mathematics†.Walter Dean - 2019 - Philosophia Mathematica 27 (3):381-439.
    Computational complexity theory is a subfield of computer science originating in computability theory and the study of algorithms for solving practical mathematical problems. Amongst its aims is classifying problems by their degree of difficulty — i.e., how hard they are to solve computationally. This paper highlights the significance of complexity theory relative to questions traditionally asked by philosophers of mathematics while also attempting to isolate some new ones — e.g., about the notion of feasibility in mathematics, the $\mathbf{P} \neq \mathbf{NP}$ (...)
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  • Rebutting and undercutting in mathematics.Kenny Easwaran - 2015 - Philosophical Perspectives 29 (1):146-162.
    In my () I argued that a central component of mathematical practice is that published proofs must be “transferable” — that is, they must be such that the author's reasons for believing the conclusion are shared directly with the reader, rather than requiring the reader to essentially rely on testimony. The goal of this paper is to explain this requirement of transferability in terms of a more general norm on defeat in mathematical reasoning that I will call “convertibility”. I begin (...)
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  • The Epistemological Subject(s) of Mathematics.Silvia De Toffoli - 2024 - In Bharath Sriraman (ed.), Handbook of the History and Philosophy of Mathematical Practice. Cham: Springer. pp. 2880-2904.
    Paying attention to the inner workings of mathematicians has led to a proliferation of new themes in the philosophy of mathematics. Several of these have to do with epistemology. Philosophers of mathematical practice, however, have not (yet) systematically engaged with general (analytic) epistemology. To be sure, there are some exceptions, but they are few and far between. In this chapter, I offer an explanation of why this might be the case and show how the situation could be remedied. I contend (...)
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  • Reasoning by Analogy in Mathematical Practice.Francesco Nappo & Nicolò Cangiotti - 2023 - Philosophia Mathematica 31 (2):176-215.
    In this paper, we offer a descriptive theory of analogical reasoning in mathematics, stating general conditions under which an analogy may provide genuine inductive support to a mathematical conjecture (over and above fulfilling the merely heuristic role of ‘suggesting’ a conjecture in the psychological sense). The proposed conditions generalize the criteria of Hesse in her influential work on analogical reasoning in the empirical sciences. By reference to several case studies, we argue that the account proposed in this paper does a (...)
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  • Proofs, Reliable Processes, and Justification in Mathematics.Yacin Hamami - 2021 - British Journal for the Philosophy of Science 74 (4):1027-1045.
    Although there exist today a variety of non-deductive reliable processes able to determine the truth of certain mathematical propositions, proof remains the only form of justification accepted in mathematical practice. Some philosophers and mathematicians have contested this commonly accepted epistemic superiority of proof on the ground that mathematicians are fallible: when the deductive method is carried out by a fallible agent, then it comes with its own level of reliability, and so might happen to be equally or even less reliable (...)
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  • Probabilistic Proofs, Lottery Propositions, and Mathematical Knowledge.Yacin Hamami - 2021 - Philosophical Quarterly 72 (1):77-89.
    In mathematics, any form of probabilistic proof obtained through the application of a probabilistic method is not considered as a legitimate way of gaining mathematical knowledge. In a series of papers, Don Fallis has defended the thesis that there are no epistemic reasons justifying mathematicians’ rejection of probabilistic proofs. This paper identifies such an epistemic reason. More specifically, it is argued here that if one adopts a conception of mathematical knowledge in which an epistemic subject can know a mathematical proposition (...)
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  • Teaching and Learning Guide for: Explanation in Mathematics: Proofs and Practice.William D'Alessandro - 2019 - Philosophy Compass 14 (11):e12629.
    This is a teaching and learning guide to accompany "Explanation in Mathematics: Proofs and Practice".
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  • Randomized arguments are transferable.Jeffrey C. Jackson - 2009 - Philosophia Mathematica 17 (3):363-368.
    Easwaran has given a definition of transferability and argued that, under this definition, randomized arguments are not transferable. I show that certain aspects of his definition are not suitable for addressing the underlying question of whether or not there is an epistemic distinction between randomized and deductive arguments. Furthermore, I demonstrate that for any suitable definition, randomized arguments are in fact transferable.
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  • Transferable and Fixable Proofs.William D'Alessandro - forthcoming - Episteme:1-12.
    A proof P of a theorem T is transferable when a typical expert can become convinced of T solely on the basis of their prior knowledge and the information contained in P. Easwaran has argued that transferability is a constraint on acceptable proof. Meanwhile, a proof P is fixable when it’s possible for other experts to correct any mistakes P contains without having to develop significant new mathematics. Habgood-Coote and Tanswell have observed that some acceptable proofs are both fixable and (...)
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  • Confirming Mathematical Conjectures by Analogy.Francesco Nappo, Nicolò Cangiotti & Caterina Sisti - 2024 - Erkenntnis 89 (6):2493-2519.
    Analogy has received attention as a form of inductive reasoning in the empirical sciences. Its role in mathematics has, instead, received less consideration. This paper provides a novel account of how an analogy with a more familiar mathematical domain can contribute to the confirmation of a mathematical conjecture. By reference to case-studies, we propose a distinction between an _incremental_ and a _non-incremental_ form of confirmation by mathematical analogy. We offer an account of the former within the popular framework of Bayesian (...)
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  • Empiricism, Probability, and Knowledge of Arithmetic.Sean Walsh - 2014 - Journal of Applied Logic 12 (3):319–348.
    The topic of this paper is our knowledge of the natural numbers, and in particular, our knowledge of the basic axioms for the natural numbers, namely the Peano axioms. The thesis defended in this paper is that knowledge of these axioms may be gained by recourse to judgements of probability. While considerations of probability have come to the forefront in recent epistemology, it seems safe to say that the thesis defended here is heterodox from the vantage point of traditional philosophy (...)
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  • Objective Probabilities in Number Theory.J. Ellenberg & E. Sober - 2011 - Philosophia Mathematica 19 (3):308-322.
    Philosophers have explored objective interpretations of probability mainly by considering empirical probability statements. Because of this focus, it is widely believed that the logical interpretation and the actual-frequency interpretation are unsatisfactory and the hypothetical-frequency interpretation is not much better. Probabilistic assertions in pure mathematics present a new challenge. Mathematicians prove theorems in number theory that assign probabilities. The most natural interpretation of these probabilities is that they describe actual frequencies in finite sets and limits of actual frequencies in infinite sets. (...)
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  • Statistical Data and Mathematical Propositions.Cory Juhl - 2015 - Pacific Philosophical Quarterly 96 (1):100-115.
    Statistical tests of the primality of some numbers look similar to statistical tests of many nonmathematical, clearly empirical propositions. Yet interpretations of probability prima facie appear to preclude the possibility of statistical tests of mathematical propositions. For example, it is hard to understand how the statement that n is prime could have a frequentist probability other than 0 or 1. On the other hand, subjectivist approaches appear to be saddled with ‘coherence’ constraints on rational probabilities that require rational agents to (...)
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  • John P. Burgess. Rigor and Structure. Oxford: Oxford University Press, 2015. ISBN: 978-0-19-872222-9 ; 978-0-19-103360-5 . Pp. xii + 215. [REVIEW]Richard Pettigrew - 2016 - Philosophia Mathematica 24 (1):129-136.
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  • Bernulf Kanitscheider. Natur und Zahl: Die Mathematisierbarkeit der Welt [Nature and Number: The Mathematizability of the World]. Berlin: Springer Verlag, 2013. ISBN: 978-3-642-37707-5 ; 978-3-642-37708-2 . Pp. vii + 385. [REVIEW]William Lane Craig - 2016 - Philosophia Mathematica 24 (1):136-141.
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