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The Nature of Mathematical Knowledge

Oxford University Press (1983)

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  1. Great Philosophy: Discovery, Invention, and the Uses of Error.Christopher Norris - 2014 - International Journal of Philosophical Studies 22 (3):349-379.
    In this essay I consider what is meant by the description ‘great’ philosophy and then offer some broadly applicable criteria by which to assess candidate thinkers or works. On the one hand are philosophers in whose case the epithet, even if contested, is not grossly misconceived or merely the product of doctrinal adherence on the part of those who apply it. On the other are those – however gifted, acute, or technically adroit – to whom its application is inappropriate because (...)
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  • Mathematics as a Science of Patterns. [REVIEW]Mark Steiner - 2000 - Philosophical Review 109 (1):115-118.
    For the past hundred years, mathematics, for its own reasons, has been shifting away from the study of “mathematical objects” and towards the study of “structures”. One would have expected philosophers to jump onto the bandwagon, as in many other cases, to proclaim that this shift is no accident, since mathematics is “essentially” about structures, not objects. In fact, structuralism has not been a very popular philosophy of mathematics, probably because of the hostility of Frege and other influential logicists, and (...)
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  • The Story About Propositions.Bradley Armour-Garb & James A. Woodbridge - 2012 - Noûs 46 (4):635-674.
    It is our contention that an ontological commitment to propositions faces a number of problems; so many, in fact, that an attitude of realism towards propositions—understood the usual “platonistic” way, as a kind of mind- and language-independent abstract entity—is ultimately untenable. The particular worries about propositions that marshal parallel problems that Paul Benacerraf has raised for mathematical platonists. At the same time, the utility of “proposition-talk”—indeed, the apparent linguistic commitment evident in our use of 'that'-clauses (in offering explanations and making (...)
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  • The Current Epistemic Status of the Indispensability Arguments in the Philosophy of Science.Catalin Barboianu - 2016 - Analele Universitatii Din Craiova 36 (2):108-132.
    The predisposition of the Indispensability Argument to objections, rephrasing and versions associated with the various views in philosophy of mathematics grants it a special status of a “blueprint” type rather than a debatable theme in the philosophy of science. From this point of view, it follows that the Argument has more an epistemic character than ontological.
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  • Does the Counterfactual Theory of Explanation Apply to Non-Causal Explanations in Metaphysics?Alexander Reutlinger - 2016 - European Journal for Philosophy of Science:1-18.
    In the recent philosophy of explanation, a growing attention to and discussion of non-causal explanations has emerged, as there seem to be compelling examples of non-causal explanations in the sciences, in pure mathematics, and in metaphysics. I defend the claim that the counterfactual theory of explanation (CTE) captures the explanatory character of both non-causal scientific and metaphysical explanations. According to the CTE, scientific and metaphysical explanations are explanatory by virtue of revealing counterfactual dependencies between the explanandum and the explanans. I (...)
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  • Wie Individuell Sind Intentionale Einstellungen Wirklich?Ralf Stoecker - 2000 - Metaphysica 1:107-119.
    So selbstverständlich es klingt, vom Geist, der Psyche oder auch der Seele eines Menschen zu reden, und so vertraut uns wissenschaftliche Disziplinen sind, die sich philosophisch oder empirisch damit beschäftigen, so schwer fällt es, ein einheitliches Merkmale dafür anzugeben, wann etwas ein psychisches Phänomen ist. Viele der potentiellen Merkmale decken eben nur einen Teil des Spektrums dessen ab, was wir gewöhnlich als psychisch bezeichnen würden, und sind damit bestenfalls hinreichende, aber sicher keine notwendigen Bedingungen des Psychischen. Im Mittelpunkt des folgenden (...)
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  • Conceptual Metaphors and Mathematical Practice: On Cognitive Studies of Historical Developments in Mathematics.Dirk Schlimm - 2013 - Topics in Cognitive Science 5 (2):283-298.
    This article looks at recent work in cognitive science on mathematical cognition from the perspective of history and philosophy of mathematical practice. The discussion is focused on the work of Lakoff and Núñez, because this is the first comprehensive account of mathematical cognition that also addresses advanced mathematics and its history. Building on a distinction between mathematics as it is presented in textbooks and as it presents itself to the researcher, it is argued that the focus of cognitive analyses of (...)
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  • On Mathematicians' Different Standards When Evaluating Elementary Proofs.Matthew Inglis, Juan Pablo Mejia-Ramos, Keith Weber & Lara Alcock - 2013 - Topics in Cognitive Science 5 (2):270-282.
    In this article, we report a study in which 109 research-active mathematicians were asked to judge the validity of a purported proof in undergraduate calculus. Significant results from our study were as follows: (a) there was substantial disagreement among mathematicians regarding whether the argument was a valid proof, (b) applied mathematicians were more likely than pure mathematicians to judge the argument valid, (c) participants who judged the argument invalid were more confident in their judgments than those who judged it valid, (...)
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  • Coherence as Constraint Satisfaction.Paul Thagard & Karsten Verbeurgt - 1998 - Cognitive Science 22 (1):1-24.
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  • The Role of Intuition and Formal Thinking in Kant, Riemann, Husserl, Poincare, Weyl, and in Current Mathematics and Physics.Luciano Boi - 2019 - Kairos 22 (1):1-53.
    According to Kant, the axioms of intuition, i.e. space and time, must provide an organization of the sensory experience. However, this first orderliness of empirical sensations seems to depend on a kind of faculty pertaining to subjectivity, rather than to the encounter of these same intuitions with the real properties of phenomena. Starting from an analysis of some very significant developments in mathematical and theoretical physics in the last decades, in which intuition played an important role, we argue that nevertheless (...)
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  • Explication Et Pertinence : Du Sel Ensorcelé À la Loi des Aires.Cyrille Imbert - 2011 - Dialogue 50 (4):689-723.
    ABSTRACT: Whereas relevance in scientific explanations is usually discussed as if it was a single problem, several criteria of relevance will be distinguished in this paper. Emphasis is laid upon the notion of intra-scientific relevance, which is illustrated using explanation of the law of areas as an example. Traditional accounts of explanation, such as the causal and unificationist accounts, are analyzed against these criteria of relevance. Particularly, it will be shown that these accounts fail to indicate which explanations fulfill the (...)
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  • Frege on the Foundation of Geometry in Intuition.Jeremy Shipley - 2015 - Journal for the History of Analytical Philosophy 3 (6).
    I investigate the role of geometric intuition in Frege’s early mathematical works and the significance of his view of the role of intuition in geometry to properly understanding the aims of his logicist project. I critically evaluate the interpretations of Mark Wilson, Jamie Tappenden, and Michael Dummett. The final analysis that I provide clarifies the relationship of Frege’s restricted logicist project to dominant trends in German mathematical research, in particular to Weierstrassian arithmetization and to the Riemannian conceptual/geometrical tradition at Göttingen. (...)
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  • Numerals and Neural Reuse.Max Jones - 2020 - Synthese 197 (9):3657-3681.
    Menary OpenMIND, MIND Group, Frankfurt am Main, 2015) has argued that the development of our capacities for mathematical cognition can be explained in terms of enculturation. Our ancient systems for perceptually estimating numerical quantities are augmented and transformed by interacting with a culturally-enriched environment that provides scaffolds for the acquisition of cognitive practices, leading to the development of a discrete number system for representing number precisely. Numerals and the practices associated with numeral systems play a significant role in this process. (...)
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  • Top-Down and Bottom-Up Philosophy of Mathematics.Carlo Cellucci - 2013 - Foundations of Science 18 (1):93-106.
    The philosophy of mathematics of the last few decades is commonly distinguished into mainstream and maverick, to which a ‘third way’ has been recently added, the philosophy of mathematical practice. In this paper the limitations of these trends in the philosophy of mathematics are pointed out, and it is argued that they are due to the fact that all of them are based on a top-down approach, that is, an approach which explains the nature of mathematics in terms of some (...)
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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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  • Mathematical Diagrams in Practice: An Evolutionary Account.Iulian D. Toader - 2002 - Logique Et Analyse 179:341-355.
    This paper analyzes some examples of diagrammatic proofs in elementary mathematics. It suggests that the cognitive features that allow us to understand such proofs are extensions of those responsible for our navigating the physical world.
     
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  • Pasch's Empiricism as Methodological Structuralism.Dirk Schlimm - 2020 - In Erich Reck & Georg Schiemer (eds.), The Pre-History of Mathematical Structuralism. New York: Oxford University Press. pp. 80-105.
  • Recognizing Mathematics Students as Creative: Mathematical Creativity as Community-Based and Possibility-Expanding.Meghan Riling - 2020 - Journal of Humanistic Mathematics 10 (2).
    Although much creativity research has suggested that creativity is influenced by cultural and social factors, these have been minimally explored in the context of mathematics and mathematics learning. This problematically limits who is seen as mathematically creative and who can enter the discipline of mathematics. This paper proposes a framework of creativity that is based in what it means to know or do mathematics and accepts that creativity is something that can be nurtured in all students. Prominent mathematical epistemologies held (...)
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  • Pierre Duhem’s Good Sense as a Guide to Theory Choice.Milena Ivanova - 2010 - Studies in History and Philosophy of Science Part A 41 (1):58-64.
    This paper examines Duhem’s concept of good sense as an attempt to support a non rule-governed account of rationality in theory choice. Faced with the underdetermination of theory by evidence thesis and the continuity thesis, Duhem tried to account for the ability of scientists to choose theories that continuously grow to a natural classification. I will examine the concept of good sense and the problems that stem from it. I will also present a recent attempt by David Stump to link (...)
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  • Who's Afraid of Undermining?Peter B. M. Vranas - 2002 - Erkenntnis 57 (2):151-174.
    The Principal Principle (PP) says that, for any proposition A, given any admissible evidence and the proposition that the chance of A is x%, one's conditional credence in A should be x%. Humean Supervenience (HS) claims that, among possible worlds like ours, no two differ without differing in the spacetime-point-by-spacetime-point arrangement of local properties. David Lewis (1986b, 1994a) has argued that PP contradicts HS, and the validity of his argument has been endorsed by Bigelow et al. (1993), Thau (1994), Hall (...)
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  • Justification and the Growth of Error.Sherrilyn Roush - 2013 - Philosophical Studies 165 (2):527-551.
    It is widely accepted that in fallible reasoning potential error necessarily increases with every additional step, whether inferences or premises, because it grows in the same way that the probability of a lengthening conjunction shrinks. As it stands, this is disappointing but, I will argue, not out of keeping with our experience. However, consulting an expert, proof-checking, constructing gap-free proofs, and gathering more evidence for a given conclusion also add more steps, and we think these actions have the potential to (...)
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  • Consideraciones sobre la noción de intuición matemática.Lina María Peña-Páez - 2020 - Agora 39 (2):127-141.
    La historia de la matemática muestra como la intuición matemática ha estado presente en la invención y desarrollo de conceptos, teorías y procedimientos matemáticos. Así mismo, ha permeado el debate filosófico, los fundamentos de la matemática y los discursos educativos; otorgándole vigencia al estudio de este tema. En el presente artículo, se exponen los argumentos bajo los cuales es posible sustentar que la intuición es un proceso, que toma ideas que se presentan, inicialmente de manera “desordenada”, y que gracias al (...)
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  • Frege, Dedekind, and the Modern Epistemology of Arithmetic.Markus Pantsar - 2016 - Acta Analytica 31 (3):297-318.
    In early analytic philosophy, one of the most central questions concerned the status of arithmetical objects. Frege argued against the popular conception that we arrive at natural numbers with a psychological process of abstraction. Instead, he wanted to show that arithmetical truths can be derived from the truths of logic, thus eliminating all psychological components. Meanwhile, Dedekind and Peano developed axiomatic systems of arithmetic. The differences between the logicist and axiomatic approaches turned out to be philosophical as well as mathematical. (...)
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  • Kuhn, Lakatos, and the Image of Mathematics.Eduard Glas - 1995 - Philosophia Mathematica 3 (3):225-247.
    In this paper I explore possibilities of bringing post-positivist philosophies of empirical science to bear on the dynamics of mathematical development. This is done by way of a convergent accommodation of a mathematical version of Lakatos's methodology of research programmes, and a version of Kuhn's account of scientific change that is made applicable to mathematics by cleansing it of all references to the psychology of perception. The resulting view is argued in the light of two case histories of radical conceptual (...)
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  • Animal Cognition, Species Invariantism, and Mathematical Realism.Helen De Cruz - 2019 - In Andrew Aberdein & Matthew Inglis (eds.), Advances in Experimental Philosophy of Logic and Mathematics. London: Bloomsbury Academic. pp. 39-61.
    What can we infer from numerical cognition about mathematical realism? In this paper, I will consider one aspect of numerical cognition that has received little attention in the literature: the remarkable similarities of numerical cognitive capacities across many animal species. This Invariantism in Numerical Cognition (INC) indicates that mathematics and morality are disanalogous in an important respect: proto-moral beliefs differ substantially between animal species, whereas proto-mathematical beliefs (at least in the animals studied) seem to show more similarities. This makes moral (...)
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  • Lakatos’ Quasi-Empiricism in the Philosophy of Mathematics.Michael J. Shaffer - 2015 - Polish Journal of Philosophy 9 (2):71-80.
    Imre Lakatos' views on the philosophy of mathematics are important and they have often been underappreciated. The most obvious lacuna in this respect is the lack of detailed discussion and analysis of his 1976a paper and its implications for the methodology of mathematics, particularly its implications with respect to argumentation and the matter of how truths are established in mathematics. The most important themes that run through his work on the philosophy of mathematics and which culminate in the 1976a paper (...)
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  • Philosophy of Mathematical Practice: A Primer for Mathematics Educators.Yacin Hamami & Rebecca Morris - forthcoming - ZDM Mathematics Education.
    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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  • From Numerical Concepts to Concepts of Number.Lance J. Rips, Amber Bloomfield & Jennifer Asmuth - 2008 - Behavioral and Brain Sciences 31 (6):623-642.
    Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe that these abilities set the stage for later mathematics: natural numbers and arithmetic. However, the connection between these early and later skills is far from obvious. We evaluate two possible routes to mathematics and argue that neither is sufficient: (1) We first sketch what we think is the most likely model for infant abilities in this domain, and we examine proposals for extrapolating the natural number concept (...)
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  • Geometry and Generality in Frege's Philosophy of Arithmetic.Jamie Tappenden - 1995 - Synthese 102 (3):319 - 361.
    This paper develops some respects in which the philosophy of mathematics can fruitfully be informed by mathematical practice, through examining Frege's Grundlagen in its historical setting. The first sections of the paper are devoted to elaborating some aspects of nineteenth century mathematics which informed Frege's early work. (These events are of considerable philosophical significance even apart from the connection with Frege.) In the middle sections, some minor themes of Grundlagen are developed: the relationship Frege envisions between arithmetic and geometry and (...)
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  • Frege’s ‘On the Foundations of Geometry’ and Axiomatic Metatheory.Günther Eder - 2016 - Mind 125 (497):5-40.
    In a series of articles dating from 1903 to 1906, Frege criticizes Hilbert’s methodology of proving the independence and consistency of various fragments of Euclidean geometry in his Foundations of Geometry. In the final part of the last article, Frege makes his own proposal as to how the independence of genuine axioms should be proved. Frege contends that independence proofs require the development of a ‘new science’ with its own basic truths. This paper aims to provide a reconstruction of this (...)
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  • Non-Ontological Structuralism†.Michael Resnik - 2019 - Philosophia Mathematica 27 (3):303-315.
    ABSTRACT Historical structuralist views have been ontological. They either deny that there are any mathematical objects or they maintain that mathematical objects are structures or positions in them. Non-ontological structuralism offers no account of the nature of mathematical objects. My own structuralism has evolved from an early sui generis version to a non-ontological version that embraces Quine’s doctrine of ontological relativity. In this paper I further develop and explain this view.
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  • Labyrinth of Continua†.Patrick Reeder - 2018 - Philosophia Mathematica 26 (1):1-39.
    This is a survey of the concept of continuity. Efforts to explicate continuity have produced a plurality of philosophical conceptions of continuity that have provably distinct expressions within contemporary mathematics. I claim that there is a divide between the conceptions that treat the whole continuum as prior to its parts, and those conceptions that treat the parts of the continuum as prior to the whole. Along this divide, a tension emerges between those conceptions that favor philosophical idealizations of continuity and (...)
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  • XI- Naturalism and Placement, or, What Should a Good Quinean Say About Mathematical and Moral Truth?Mary Leng - 2016 - Proceedings of the Aristotelian Society 116 (3):237-260.
    What should a Quinean naturalist say about moral and mathematical truth? If Quine’s naturalism is understood as the view that we should look to natural science as the ultimate ‘arbiter of truth’, this leads rather quickly to what Huw Price has called ‘placement problems’ of placing moral and mathematical truth in an empirical scientific world-view. Against this understanding of the demands of naturalism, I argue that a proper understanding of the reasons Quine gives for privileging ‘natural science’ as authoritative when (...)
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  • A Cognitive Approach to Benacerraf's Dilemma.Luke Jerzykiewicz - 2009 - Dissertation, University of Western Ontario
    One of the important challenges in the philosophy of mathematics is to account for the semantics of sentences that express mathematical propositions while simultaneously explaining our access to their contents. This is Benacerraf’s Dilemma. In this dissertation, I argue that cognitive science furnishes new tools by means of which we can make progress on this problem. The foundation of the solution, I argue, must be an ontologically realist, albeit non-platonist, conception of mathematical reality. The semantic portion of the problem can (...)
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  • Benacerraf on Mathematical Knowledge.Vladimir Drekalović - 2010 - Prolegomena 9 (1):97-121.
    Causal theory of knowledge has been used by some theoreticians who, dealing with the philosophy of mathematics, touched the subject of mathematical knowledge. Some of them discuss the necessity of the causal condition for justification, which creates the grounds for renewing the old conflict between empiricists and rationalists. Emphasizing the condition of causality as necessary for justifiability, causal theory has provided stimulus for the contemporary empiricists to venture on the so far unquestioned cognitive foundations of mathematics. However, in what sense (...)
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  • Searching for Pragmatism in the Philosophy of Mathematics: Critical Studies / Book Reviews.Steven J. Wagner - 2001 - Philosophia Mathematica 9 (3):355-376.
  • Mathematics Dealing with 'Hypothetical States of Things'.Jessica Carter - 2014 - Philosophia Mathematica 22 (2):209-230.
    This paper takes as a starting point certain notions from Peirce's writings and uses them to propose a picture of the part of mathematical practice that consists of hypothesis formation. In particular, three processes of hypothesis formation are considered: abstraction, generalisation, and an abductive-like inference. In addition Peirce's pragmatic conception of truth and existence in terms of higher-order concepts are used in order to obtain a kind of pragmatic realist picture of mathematics.
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  • Stefano Donati. I Fondamenti Della Matematica Nel Logicismo di Bertrand Russell [the Foundations of Mathematics in the Logicism of Bertrand Russell].Gianluigi Oliveri - 2009 - Philosophia Mathematica 17 (1):109-113.
    Bertrand Russell's contributions to last century's philosophy and, in particular, to the philosophy of mathematics cannot be overestimated.Russell, besides being, with Frege and G.E. Moore, one of the founding fathers of analytical philosophy, played a major rôle in the development of logicism, one of the oldest and most resilient1 programmes in the foundations of mathematics.Among his many achievements, we need to mention the discovery of the paradox that bears his name and the identification of its logical nature; the generalization to (...)
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  • A New Perspective on the Problem of Applying Mathematics.Chistroper Pincock - 2004 - Philosophia Mathematica 12 (2):135-161.
    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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  • Naturalism, Notation, and the Metaphysics of Mathematics.Madeline M. Muntersbjorn - 1999 - Philosophia Mathematica 7 (2):178-199.
    The instability inherent in the historical inventory of mathematical objects challenges philosophers. Naturalism suggests we can construct enduring answers to ontological questions through an investigation of the processes whereby mathematical objects come into existence. Patterns of historical development suggest that mathematical objects undergo an intelligible process of reification in tandem with notational innovation. Investigating changes in mathematical languages is a necessary first step towards a viable ontology. For this reason, scholars should not modernize historical texts without caution, as the use (...)
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  • Lakatos as Historian of Mathematics.Brendan P. Larvor - 1997 - Philosophia Mathematica 5 (1):42-64.
    This paper discusses the connection between the actual history of mathematics and Lakatos's philosophy of mathematics, in three parts. The first points to studies by Lakatos and others which support his conception of mathematics and its history. In the second I suggest that the apparent poverty of Lakatosian examples may be due to the way in which the history of mathematics is usually written. The third part argues that Lakatos is right to hold philosophy accountable to history, even if Lakatos's (...)
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  • The Legacy of Lakatos: Reconceptualising the Philosophy of Mathematics.Paul Ernest - 1997 - Philosophia Mathematica 5 (2):116-134.
    Kitcher and Aspray distinguish a mainstream tradition in the philosophy of mathematics concerned with foundationalist epistemology, and a ‘maverick’ or naturalistic tradition, originating with Lakatos. My claim is that if the consequences of Lakatos's contribution are fully worked out, no less than a radical reconceptualization of the philosophy of mathematics is necessitated, including history, methodology and a fallibilist epistemology as central to the field. In the paper an interpretation of Lakatos's philosophy of mathematics is offered, followed by some critical discussion, (...)
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  • In Support of Significant Modernization of Original Mathematical Texts (in Defense of Presentism).A. G. Barabashev - 1997 - Philosophia Mathematica 5 (1):21-41.
    At their extremes, the modernization of ancient mathematical texts (absolute presentism) leaves nothing of the source and the refusal to modernize (absolute antiquarism) changes nothing. The extremes exist only as tendencies. This paper attempts to justify the admissibility of broad modernization of mathematical sources (presentism) in the context of a socio-cultural (non-fundamentalist) philosophy of mathematics.
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  • Argument and Explanation in Mathematics.Michel Dufour - 2013 - In Dima Mohammed and Marcin Lewiński (ed.), Virtues of Argumentation. Proceedings of the 10th International Conference of the Ontario Society for the Study of Argumentation (OSSA), 22-26 May 2013. pp. pp. 1-14..
    Are there arguments in mathematics? Are there explanations in mathematics? Are there any connections between argument, proof and explanation? Highly controversial answers and arguments are reviewed. The main point is that in the case of a mathematical proof, the pragmatic criterion used to make a distinction between argument and explanation is likely to be insufficient for you may grant the conclusion of a proof but keep on thinking that the proof is not explanatory.
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  • The Because of Because Without Cause†.Daniele Molinini - 2018 - Philosophia Mathematica 26 (2):275-286.
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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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  • Mathematical Knowledge is Context Dependent.Benedikt LÖWE & Thomas MÜLLER - 2008 - Grazer Philosophische Studien 76 (1):91-107.
    We argue that mathematical knowledge is context dependent. Our main argument is that on pain of distorting mathematical practice, one must analyse the notion of having available a proof, which supplies justification in mathematics, in a context dependent way.
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  • Understanding the Revisability Thesis.Célia Teixeira - 2018 - Grazer Philosophische Studien 95 (2):180-195.
    W. V. Quine famously claimed that no statement is immune to revision. This thesis has had a profound impact on twentieth century philosophy, and it still occupies centre stage in many contemporary debates. However, despite its importance it is not clear how it should be interpreted. I show that the thesis is in fact ambiguous between three substantially different theses. I illustrate the importance of clarifying it by assessing its use in the debate against the existence of a priori knowledge. (...)
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  • A Priori and A Posteriori: A Bootstrapping Relationship.Tuomas E. Tahko - 2011 - Metaphysica 12 (2):151-164.
    The distinction between a priori and a posteriori knowledge has been the subject of an enormous amount of discussion, but the literature is biased against recognizing the intimate relationship between these forms of knowledge. For instance, it seems to be almost impossible to find a sample of pure a priori or a posteriori knowledge. In this paper, it will be suggested that distinguishing between a priori and a posteriori is more problematic than is often suggested, and that a priori and (...)
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  • Lakatos and Hersh on Mathematical Proof.Hossein Bayat - 2015 - Journal of Philosophical Investigations at University of Tabriz 9 (17):75-93.
    The concept of Mathematical Proof has been controversial for the past few decades. Different philosophers have offered different theories about the nature of Mathematical Proof, among which theories presented by Lakatos and Hersh have had significant similarities and differences with each other. It seems that a comparison and critical review of these two theories will lead to a better understanding of the concept of mathematical proof and will be a big step towards solving many related problems. Lakatos and Hersh argue (...)
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