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  1. 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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  • Philosophy of mathematics.Leon Horsten - 2008 - Stanford Encyclopedia of Philosophy.
    If mathematics is regarded as a science, then the philosophy of mathematics can be regarded as a branch of the philosophy of science, next to disciplines such as the philosophy of physics and the philosophy of biology. However, because of its subject matter, the philosophy of mathematics occupies a special place in the philosophy of science. Whereas the natural sciences investigate entities that are located in space and time, it is not at all obvious that this is also the case (...)
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  • Mathematics, indispensability and scientific progress.Alan Baker - 2001 - Erkenntnis 55 (1):85-116.
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  • Phenomenology and mathematical practice.Mary Leng - 2002 - Philosophia Mathematica 10 (1):3-14.
    A phenomenological approach to mathematical practice is sketched out, and some problems with this sort of approach are considered. The approach outlined takes mathematical practices as its data, and seeks to provide an empirically adequate philosophy of mathematics based on observation of these practices. Some observations are presented, based on two case studies of some research into the classification of C*-algebras. It is suggested that an anti-realist account of mathematics could be developed on the basis of these and other studies, (...)
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  • Ontology and mathematical practice.Jessica Carter - 2004 - Philosophia Mathematica 12 (3):244-267.
    In this paper I propose a position in the ontology of mathematics which is inspired mainly by a case study in the mathematical discipline if-theory. The main theses of this position are that mathematical objects are introduced by mathematicians and that after mathematical objects have been introduced, they exist as objectively accessible abstract objects.
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  • Science, method and critical thinking.Antoine Danchin - 2023 - Microbial Biotechnology 16 (10):1888-1894.
    Science is founded on a method based on critical thinking. A prerequisite for this is not only a sufficient command of language but also the comprehension of the basic concepts underlying our understanding of reality. This constraint implies an awareness of the fact that the truth of the World is not directly accessible to us, but can only be glimpsed through the construction of mod- els designed to anticipate its behaviour. Because the relationship between models and reality rests on the (...)
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  • Frameworks, models, and case studies: a new methodology for studying conceptual change in science and philosophy.Matteo De Benedetto - 2022 - Dissertation, Ludwig Maximilians Universität, München
    This thesis focuses on models of conceptual change in science and philosophy. In particular, I developed a new bootstrapping methodology for studying conceptual change, centered around the formalization of several popular models of conceptual change and the collective assessment of their improved formal versions via nine evaluative dimensions. Among the models of conceptual change treated in the thesis are Carnap’s explication, Lakatos’ concept-stretching, Toulmin’s conceptual populations, Waismann’s open texture, Mark Wilson’s patches and facades, Sneed’s structuralism, and Paul Thagard’s conceptual revolutions. (...)
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  • Computing, Modelling, and Scientific Practice: Foundational Analyses and Limitations.Filippos A. Papagiannopoulos - 2018 - Dissertation, University of Western Ontario
    This dissertation examines aspects of the interplay between computing and scientific practice. The appropriate foundational framework for such an endeavour is rather real computability than the classical computability theory. This is so because physical sciences, engineering, and applied mathematics mostly employ functions defined in continuous domains. But, contrary to the case of computation over natural numbers, there is no universally accepted framework for real computation; rather, there are two incompatible approaches --computable analysis and BSS model--, both claiming to formalise algorithmic (...)
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  • Structural Relativity and Informal Rigour.Neil Barton - 2022 - In Gianluigi Oliveri, Claudio Ternullo & Stefano Boscolo (eds.), Objects, Structures, and Logics, FilMat Studies in the Philosophy of Mathematics. Springer. pp. 133-174.
    Informal rigour is the process by which we come to understand particular mathematical structures and then manifest this rigour through axiomatisations. Structural relativity is the idea that the kinds of structures we isolate are dependent upon the logic we employ. We bring together these ideas by considering the level of informal rigour exhibited by our set-theoretic discourse, and argue that different foundational programmes should countenance different underlying logics (intermediate between first- and second-order) for formulating set theory. By bringing considerations of (...)
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  • The Prospects for a Monist Theory of Non-causal Explanation in Science and Mathematics.Alexander Reutlinger, Mark Colyvan & Karolina Krzyżanowska - 2020 - Erkenntnis 87 (4):1773-1793.
    We explore the prospects of a monist account of explanation for both non-causal explanations in science and pure mathematics. Our starting point is the counterfactual theory of explanation for explanations in science, as advocated in the recent literature on explanation. We argue that, despite the obvious differences between mathematical and scientific explanation, the CTE can be extended to cover both non-causal explanations in science and mathematical explanations. In particular, a successful application of the CTE to mathematical explanations requires us to (...)
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  • (1 other version)Maddy On The Multiverse.Claudio Ternullo - 2019 - In Stefania Centrone, Deborah Kant & Deniz Sarikaya (eds.), Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts. Springer Verlag. pp. 43-78.
    Penelope Maddy has recently addressed the set-theoretic multiverse, and expressed reservations on its status and merits ([Maddy, 2017]). The purpose of the paper is to examine her concerns, by using the interpretative framework of set-theoretic naturalism. I first distinguish three main forms of 'multiversism', and then I proceed to analyse Maddy's concerns. Among other things, I take into account salient aspects of multiverse-related mathematics , in particular, research programmes in set theory for which the use of the multiverse seems to (...)
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  • Representational unification in cognitive science: Is embodied cognition a unifying perspective?Marcin Miłkowski & Przemysław Nowakowski - 2019 - Synthese 199 (Suppl 1):67-88.
    In this paper, we defend a novel, multidimensional account of representational unification, which we distinguish from integration. The dimensions of unity are simplicity, generality and scope, non-monstrosity, and systematization. In our account, unification is a graded property. The account is used to investigate the issue of how research traditions contribute to representational unification, focusing on embodied cognition in cognitive science. Embodied cognition contributes to unification even if it fails to offer a grand unification of cognitive science. The study of this (...)
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  • Reliability of mathematical inference.Jeremy Avigad - 2020 - Synthese 198 (8):7377-7399.
    Of all the demands that mathematics imposes on its practitioners, one of the most fundamental is that proofs ought to be correct. It has been common since the turn of the twentieth century to take correctness to be underwritten by the existence of formal derivations in a suitable axiomatic foundation, but then it is hard to see how this normative standard can be met, given the differences between informal proofs and formal derivations, and given the inherent fragility and complexity of (...)
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  • Philosophy of Mathematical Practice — Motivations, Themes and Prospects†.Jessica Carter - 2019 - Philosophia Mathematica 27 (1):1-32.
    A number of examples of studies from the field ‘The Philosophy of Mathematical Practice’ (PMP) are given. To characterise this new field, three different strands are identified: an agent-based, a historical, and an epistemological PMP. These differ in how they understand ‘practice’ and which assumptions lie at the core of their investigations. In the last part a general framework, capturing some overall structure of the field, is proposed.
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  • Against logical generalism.Nicole Wyatt & Gillman Payette - 2019 - Synthese 198 (Suppl 20):4813-4830.
    The orthodox view of logic takes for granted the central importance of logical principles. Logic, and thus logical reasoning, is to be understood as a system of rules or principles with universal application. Let us call this orthodox view logical generalism. In this paper we argue that logical generalism, whether monist or pluralist, is wrong. We then outline an account of logical consequence in the absence of general logical principles, which we call logical particularism.
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  • A Role for Representation Theorems†.Emiliano Ippoliti - 2018 - Philosophia Mathematica 26 (3):396-412.
    I argue that the construction of representation theorems is a powerful tool for creating novel objects and theories in mathematics, as the construction of a new representation introduces new pieces of information in a very specific way that enables a solution for a problem and a proof of a new theorem. In more detail I show how the work behind the proof of a representation theorem transforms a mathematical problem in a way that makes it tractable and introduces information into (...)
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  • Scientific Discovery Reloaded.Emiliano Ippoliti - 2020 - Topoi 39 (4):847-856.
    The way scientific discovery has been conceptualized has changed drastically in the last few decades: its relation to logic, inference, methods, and evolution has been deeply reloaded. The ‘philosophical matrix’ moulded by logical empiricism and analytical tradition has been challenged by the ‘friends of discovery’, who opened up the way to a rational investigation of discovery. This has produced not only new theories of discovery, but also new ways of practicing it in a rational and more systematic way. Ampliative rules, (...)
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  • Fitting Feelings and Elegant Proofs: On the Psychology of Aesthetic Evaluation in Mathematics.Cain Todd - 2017 - Philosophia Mathematica:nkx007.
    ABSTRACT This paper explores the role of aesthetic judgements in mathematics by focussing on the relationship between the epistemic and aesthetic criteria employed in such judgements, and on the nature of the psychological experiences underpinning them. I claim that aesthetic judgements in mathematics are plausibly understood as expressions of what I will call ‘aesthetic-epistemic feelings’ that serve a genuine cognitive and epistemic function. I will then propose a naturalistic account of these feelings in terms of sub-personal processes of representing and (...)
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  • Mohan Ganesalingam. The Language of Mathematics: A Linguistic and Philosophical Investigation. FoLLI Publications on Logic, Language and Information. [REVIEW]Andrew Aberdein - 2017 - Philosophia Mathematica 25 (1):143–147.
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  • Theological Underpinnings of the Modern Philosophy of Mathematics.Vladislav Shaposhnikov - 2016 - Studies in Logic, Grammar and Rhetoric 44 (1):147-168.
    The study is focused on the relation between theology and mathematics in the situation of increasing secularization. My main concern in the second part of this paper is the early-twentieth-century foundational crisis of mathematics. The hypothesis that pure mathematics partially fulfilled the functions of theology at that time is tested on the views of the leading figures of the three main foundationalist programs: Russell, Hilbert and Brouwer.
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  • What can the Philosophy of Mathematics Learn from the History of Mathematics?Brendan Larvor - 2008 - Erkenntnis 68 (3):393-407.
    This article canvasses five senses in which one might introduce an historical element into the philosophy of mathematics: 1. The temporal dimension of logic; 2. Explanatory Appeal to Context rather than to General Principles; 3. Heraclitean Flux; 4. All history is the History of Thought; and 5. History is Non-Judgmental. It concludes by adapting Bernard Williams’ distinction between ‘history of philosophy’ and ‘history of ideas’ to argue that the philosophy of mathematics is unavoidably historical, but need not and must not (...)
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  • International Handbook of Research in History, Philosophy and Science Teaching.Michael R. Matthews (ed.) - 2014 - Springer.
    This inaugural handbook documents the distinctive research field that utilizes history and philosophy in investigation of theoretical, curricular and pedagogical issues in the teaching of science and mathematics. It is contributed to by 130 researchers from 30 countries; it provides a logically structured, fully referenced guide to the ways in which science and mathematics education is, informed by the history and philosophy of these disciplines, as well as by the philosophy of education more generally. The first handbook to cover the (...)
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  • Non-deductive Logic in Mathematics: The Probability of Conjectures.James Franklin - 2013 - In Andrew Aberdein & Ian J. Dove (eds.), The Argument of Mathematics. Dordrecht, Netherland: Springer. pp. 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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  • Mathematical reasoning with higher-order anti-unifcation.Markus Guhe, Alison Pease, Alan Smaill, Martin Schmidt, Helmar Gust, Kai-Uwe Kühnberger & Ulf Krumnack - 2010 - In S. Ohlsson & R. Catrambone (eds.), Proceedings of the 32nd Annual Conference of the Cognitive Science Society. Cognitive Science Society.
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  • The Argument of Mathematics.Andrew Aberdein & Ian J. Dove (eds.) - 2013 - Dordrecht, Netherland: Springer.
    Written by experts in the field, this volume presents a comprehensive investigation into the relationship between argumentation theory and the philosophy of mathematical practice. Argumentation theory studies reasoning and argument, and especially those aspects not addressed, or not addressed well, by formal deduction. The philosophy of mathematical practice diverges from mainstream philosophy of mathematics in the emphasis it places on what the majority of working mathematicians actually do, rather than on mathematical foundations. -/- The book begins by first challenging the (...)
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  • Toward a constructivist epistemology of thought experiments in science.Kristian Camilleri - 2014 - Synthese 191 (8):1697-1716.
    This paper presents a critical analysis of Tamar Szabó Gendler’s view of thought experiments, with the aim of developing further a constructivist epistemology of thought experiments in science. While the execution of a thought experiment cannot be reduced to standard forms of inductive and deductive inference, in the process of working though a thought experiment, a logical argument does emerge and take shape. Taking Gendler’s work as a point of departure, I argue that performing a thought experiment involves a process (...)
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  • Hilary Putnam on Meaning and Necessity.Anders Öberg - 2011 - Dissertation, Uppsala University
    In this dissertation on Hilary Putnam's philosophy, I investigate his development regarding meaning and necessity, in particular mathematical necessity. Putnam has been a leading American philosopher since the end of the 1950s, becoming famous in the 1960s within the school of analytic philosophy, associated in particular with the philosophy of science and the philosophy of language. Under the influence of W.V. Quine, Putnam challenged the logical positivism/empiricism that had become strong in America after World War II, with influential exponents such (...)
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  • Managing Informal Mathematical Knowledge: Techniques from Informal Logic.Andrew Aberdein - 2006 - Lecture Notes in Artificial Intelligence 4108:208--221.
    Much work in MKM depends on the application of formal logic to mathematics. However, much mathematical knowledge is informal. Luckily, formal logic only represents one tradition in logic, specifically the modeling of inference in terms of logical form. Many inferences cannot be captured in this manner. The study of such inferences is still within the domain of logic, and is sometimes called informal logic. This paper explores some of the benefits informal logic may have for the management of informal mathematical (...)
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  • Argumentation in Mathematical Practice.Andrew Aberdein & Zoe Ashton - 2024 - In Bharath Sriraman (ed.), Handbook of the History and Philosophy of Mathematical Practice. Cham: Springer. pp. 2665-2687.
    Formal logic has often been seen as uniquely placed to analyze mathematical argumentation. While formal logic is certainly necessary for a complete understanding of mathematical practice, it is not sufficient. Important aspects of mathematical reasoning closely resemble patterns of reasoning in nonmathematical domains. Hence the tools developed to understand informal reasoning, collectively known as argumentation theory, are also applicable to much mathematical argumentation. This chapter investigates some of the details of that application. Consideration is given to the many contrasting meanings (...)
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  • Progressive and degenerative journals: on the growth and appraisal of knowledge in scholarly publishing.Daniel J. Dunleavy - 2022 - European Journal for Philosophy of Science 12 (4):1-27.
    Despite continued attention, finding adequate criteria for distinguishing “good” from “bad” scholarly journals remains an elusive goal. In this essay, I propose a solution informed by the work of Imre Lakatos and his methodology of scientific research programmes (MSRP). I begin by reviewing several notable attempts at appraising journal quality – focusing primarily on the impact factor and development of journal blacklists and whitelists. In doing so, I note their limitations and link their overarching goals to those found within the (...)
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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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  • Why Finance Needs Philosophy (and Vice Versa): Some Epistemic and Methodological Issues.Emiliano Ippoliti - 2021 - Foundations of Science 27 (3):957-974.
    As the world economy has for better or worse become more and more dependent on the financial markets, a rethinking of the role of finance in both theory and practice is necessary. I argue that such a rethinking requires a new look at the theories of finance that is philosophical in kind. In effect, as Martha Nussbaum claims, if the absence of philosophy in economics is arguably one of the main reasons for the flaws in certain economic theories, the absence (...)
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  • The "Artificial Mathematician" Objection: Exploring the (Im)possibility of Automating Mathematical Understanding.Sven Delarivière & Bart Van Kerkhove - 2017 - In B. Sriraman (ed.), Humanizing Mathematics and its Philosophy. Birkhäuser. pp. 173-198.
    Reuben Hersh confided to us that, about forty years ago, the late Paul Cohen predicted to him that at some unspecified point in the future, mathematicians would be replaced by computers. Rather than focus on computers replacing mathematicians, however, our aim is to consider the (im)possibility of human mathematicians being joined by “artificial mathematicians” in the proving practice—not just as a method of inquiry but as a fellow inquirer.
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  • Definition in mathematics.Carlo Cellucci - 2018 - European Journal for Philosophy of Science 8 (3):605-629.
    In the past century the received view of definition in mathematics has been the stipulative conception, according to which a definition merely stipulates the meaning of a term in other terms which are supposed to be already well known. The stipulative conception has been so absolutely dominant and accepted as unproblematic that the nature of definition has not been much discussed, yet it is inadequate. This paper examines its shortcomings and proposes an alternative, the heuristic conception.
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  • Validations of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem?Annie Selden - 2003 - Journal for Mathematics Education Research 34 (1):4-36.
    We report on an exploratory study of the way eight mid-level undergraduate mathematics majors read and reflected on four student-generated arguments purported to be proofs of a single theorem. The results suggest that mid-level undergraduates tend to focus on surface features of such arguments and that their ability to determine whether arguments are proofs is very limited -- perhaps more so than either they or their instructors recognize. We begin by discussing arguments (purported proofs) regarded as texts and validations of (...)
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  • Apriorist self-interest: How it embraces altruism and is not vacuous.J. C. Lester - 1997 - Journal of Social and Evolutionary Systems 20 (3):221-232.
    This essay is part of an attempt to reconcile two extreme views in economics: the (neglected) subjective, apriorist approach and the (standard) objective, scientific (i.e., falsifiable) approach. The Austrian subjective view of value, building on Carl Menger’s theory of value, was developed into a theory of economics as being entirely an a priori theory of action. This probably finds its most extreme statement in Ludwig von Mises’ Human Action (1949). In contrast, the standard economic view has developed into making falsifiable (...)
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  • Leonard Nelson: A Theory of Philosophical Fallacies: Translated by Fernando Leal and David Carus Springer, Cham, Switzerland, 2016, vi + 211 pp. [REVIEW]Andrew Aberdein - 2017 - Argumentation 31 (2):455-461.
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  • Commentary on Hitchcock, Mcburney & Parsons.Walid Saba - unknown
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  • Computing as a Science: A Survey of Competing Viewpoints. [REVIEW]Matti Tedre - 2011 - Minds and Machines 21 (3):361-387.
    Since the birth of computing as an academic discipline, the disciplinary identity of computing has been debated fiercely. The most heated question has concerned the scientific status of computing. Some consider computing to be a natural science and some consider it to be an experimental science. Others argue that computing is bad science, whereas some say that computing is not a science at all. This survey article presents viewpoints for and against computing as a science. Those viewpoints are analyzed against (...)
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  • Objects and Processes in Mathematical Practice.Uwe V. Riss - 2011 - Foundations of Science 16 (4):337-351.
    In this paper it is argued that the fundamental difference of the formal and the informal position in the philosophy of mathematics results from the collision of an object and a process centric perspective towards mathematics. This collision can be overcome by means of dialectical analysis, which shows that both perspectives essentially depend on each other. This is illustrated by the example of mathematical proof and its formal and informal nature. A short overview of the employed materialist dialectical approach is (...)
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  • Mathematical models and reality: A constructivist perspective. [REVIEW]Christian Hennig - 2010 - Foundations of Science 15 (1):29-48.
    To explore the relation between mathematical models and reality, four different domains of reality are distinguished: observer-independent reality, personal reality, social reality and mathematical/formal reality. The concepts of personal and social reality are strongly inspired by constructivist ideas. Mathematical reality is social as well, but constructed as an autonomous system in order to make absolute agreement possible. The essential problem of mathematical modelling is that within mathematics there is agreement about ‘truth’, but the assignment of mathematics to informal reality is (...)
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  • Intentional gaps in mathematical proofs.Don Fallis - 2003 - Synthese 134 (1-2):45 - 69.
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  • Matter, motion and irreversibility. [REVIEW]Peter Clark - 1982 - British Journal for the Philosophy of Science 33 (2):165-185.
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  • The derivation-indicator view of mathematical practice.Jody Azzouni - 2004 - Philosophia Mathematica 12 (2):81-106.
    The form of nominalism known as 'mathematical fictionalism' is examined and found wanting, mainly on grounds that go back to an early antinominalist work of Rudolf Carnap that has unfortunately not been paid sufficient attention by more recent writers.
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  • Definitions in practice: An interview study.V. J. W. Coumans & L. Consoli - 2023 - Synthese 202 (1):1-32.
    In the philosophy of mathematical practice, the aim is to understand the various aspects of this practice. Even though definitions are a central element of mathematical practice, the study of this aspect of mathematical practice is still in its infancy. In particular, there is little empirical evidence to substantiate claims about definitions in practice. In this article, we address this gap by reporting on an empirical investigation on how mathematicians create definitions and which roles and properties they attribute to them. (...)
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  • Logic, Reasoning, and Rationality.Erik Weber, Joke Meheus & Dietlinde Wouters (eds.) - 2014 - Dordrecht, Netherland: Springer.
    This book contains a selection of the papers presented at the Logic, Reasoning and Rationality 2010 conference in Ghent. The conference aimed at stimulating the use of formal frameworks to explicate concrete cases of human reasoning, and conversely, to challenge scholars in formal studies by presenting them with interesting new cases of actual reasoning. According to the members of the Wiener Kreis, there was a strong connection between logic, reasoning, and rationality and that human reasoning is rational in so far (...)
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  • Coordination in theory extension: How Reichenbach can help us understand endogenization in evolutionary biology.Michele Luchetti - 2021 - Synthese (3-4):1-26.
    Reichenbach’s early solution to the scientific problem of how abstract mathematical representations can successfully express real phenomena is rooted in his view of coordination. In this paper, I claim that a Reichenbach-inspired, ‘layered’ view of coordination provides us with an effective tool to systematically analyse some epistemic and conceptual intricacies resulting from a widespread theorising strategy in evolutionary biology, recently discussed by Okasha (2018) as ‘endogenization’. First, I argue that endogenization is a form of extension of natural selection theory that (...)
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  • Computing, Modelling, and Scientific Practice: Foundational Analyses and Limitations.Philippos Papayannopoulos - 2018 - Dissertation,
    This dissertation examines aspects of the interplay between computing and scientific practice. The appropriate foundational framework for such an endeavour is rather real computability than the classical computability theory. This is so because physical sciences, engineering, and applied mathematics mostly employ functions defined in continuous domains. But, contrary to the case of computation over natural numbers, there is no universally accepted framework for real computation; rather, there are two incompatible approaches --computable analysis and BSS model--, both claiming to formalise algorithmic (...)
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  • Philosophy of mathematical practice: A primer for mathematics educators.Yacin Hamami & Rebecca Morris - 2020 - ZDM Mathematics Education 52:1113–1126.
    In recent years, philosophical work directly concerned with the practice of mathematics has intensified, giving rise to a movement known as the philosophy of mathematical practice . In this paper we offer a survey of this movement aimed at mathematics educators. We first describe the core questions philosophers of mathematical practice investigate as well as the philosophical methods they use to tackle them. We then provide a selective overview of work in the philosophy of mathematical practice covering topics including the (...)
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  • Requiem for logical nihilism, or: Logical nihilism annihilated.Bogdan Dicher - 2020 - Synthese 198 (8):7073-7096.
    Logical nihilism is the view that the relation of logical consequence is empty: there are counterexamples to any putative logical law. In this paper, I argue that the nihilist threat is illusory. The nihilistic arguments do not work. Moreover, the entire project is based on a misguided interpretation of the generality of logic.
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