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  1. Don't Throw the Baby Out with the Math Water: Why Discounting the Developmental Foundations of Early Numeracy is Premature and Unnecessary.Kevin Muldoon, Charlie Lewis & Norman Freeman - 2008 - Behavioral and Brain Sciences 31 (6):663-664.
    We see no grounds for insisting that, because the concept natural number is abstract, its foundations must be innate. It is possible to specify domain general learning processes that feed into more abstract concepts of numerical infinity. By neglecting the messiness of children's slow acquisition of arithmetical concepts, Rips et al. present an idealized, unnecessarily insular, view of number development.
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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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  • 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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  • Computers as a Source of A Posteriori Knowledge in Mathematics.Mikkel Willum Johansen & Morten Misfeldt - 2016 - International Studies in the Philosophy of Science 30 (2):111-127.
    Electronic computers form an integral part of modern mathematical practice. Several high-profile results have been proven with techniques where computer calculations form an essential part of the proof. In the traditional philosophical literature, such proofs have been taken to constitute a posteriori knowledge. However, this traditional stance has recently been challenged by Mark McEvoy, who claims that computer calculations can constitute a priori mathematical proofs, even in cases where the calculations made by the computer are too numerous to be surveyed (...)
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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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  • 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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  • Philosophy of Mathematics.Jeremy Avigad - manuscript
    The philosophy of mathematics plays an important role in analytic philosophy, both as a subject of inquiry in its own right, and as an important landmark in the broader philosophical landscape. Mathematical knowledge has long been regarded as a paradigm of human knowledge with truths that are both necessary and certain, so giving an account of mathematical knowledge is an important part of epistemology. Mathematical objects like numbers and sets are archetypical examples of abstracta, since we treat such objects in (...)
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  • Numbers as Quantitative Relations and the Traditional Theory of Measurement.Joel Michell - 1994 - British Journal for the Philosophy of Science 45 (2):389-406.
    The thesis that numbers are ratios of quantities has recently been advanced by a number of philosophers. While adequate as a definition of the natural numbers, it is not clear that this view suffices for our understanding of the reals. These require continuous quantity and relative to any such quantity an infinite number of additive relations exist. Hence, for any two magnitudes of a continuous quantity there exists no unique ratio. This problem is overcome by defining ratios, and hence real (...)
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  • Modality and Anti-Metaphysics.Stephen K. McLeod - 2001 - Ashgate.
    Modality and Anti-Metaphysics critically examines the most prominent approaches to modality among analytic philosophers in the twentieth century, including essentialism. Defending both the project of metaphysics and the essentialist position that metaphysical modality is conceptually and ontologically primitive, Stephen McLeod argues that the logical positivists did not succeed in banishing metaphysical modality from their own theoretical apparatus and he offers an original defence of metaphysics against their advocacy of its elimination. -/- Seeking to assuage the sceptical worries which underlie modal (...)
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  • Experimental Philosophy and Apriority.Jonathan Jenkins Ichikawa - 2014 - In Al Casullo & Josh Thurow (eds.), The A Priori in Philosophy. Oxford: Oxford University Press. pp. 45-66.
    One of the more visible recent developments in philosophical methodology is the experimental philosophy movement. On its surface, the experimentalist challenge looks like a dramatic threat to the apriority of philosophy; ‘experimentalist’ is nearly antonymic with ‘aprioristic’. This appearance, I suggest, is misleading; the experimentalist critique is entirely unrelated to questions about the apriority of philosophical investigation. There are many reasons to resist the skeptical conclusions of negative experimental philosophers; but even if they are granted—even if the experimentalists are right (...)
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  • Scientific Pluralism and the Chemical Revolution.Martin Kusch - 2015 - Studies in History and Philosophy of Science Part A 49:69-79.
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  • The Modal Status of Contextually A Priori Arithmetical Truths.Markus Pantsar - 2016 - In Andrea Sereni & Francesca Boccuni (eds.), Objectivity, Realism, and Proof. Springer International Publishing. pp. 67-79.
    In Pantsar (2014), an outline for an empirically feasible epistemological theory of arithmetic is presented. According to that theory, arithmetical knowledge is based on biological primitives but in the resulting empirical context develops an essentially a priori character. Such contextual a priori theory of arithmetical knowledge can explain two of the three characteristics that are usually associated with mathematical knowledge: that it appears to be a priori and objective. In this paper it is argued that it can also explain the (...)
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  • Envisioning Transformations – The Practice of Topology.Silvia De Toffoli & Valeria Giardino - 2016 - In Brendan Larvor (ed.), Mathematical Cultures: The London Meetings 2012--2014. Zurich, Switzerland: Birkhäuser. pp. 25-50.
    The objective of this article is twofold. First, a methodological issue is addressed. It is pointed out that even if philosophers of mathematics have been recently more and more concerned with the practice of mathematics, there is still a need for a sharp definition of what the targets of a philosophy of mathematical practice should be. Three possible objects of inquiry are put forward: (1) the collective dimension of the practice of mathematics; (2) the cognitives capacities requested to the practitioners; (...)
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  • The Eightfold Way: Why Analyticity, Apriority and Necessity Are Independent.Douglas Ian Campbell - 2017 - Philosophers' Imprint 17:1-17.
    This paper concerns the three great modal dichotomies: (i) the necessary/contingent dichotomy; (ii) the a priori/empirical dichotomy; and (iii) the analytic/synthetic dichotomy. These can be combined to produce a tri-dichotomy of eight modal categories. The question as to which of the eight categories house statements and which do not is a pivotal battleground in the history of analytic philosophy, with key protagonists including Descartes, Hume, Kant, Kripke, Putnam and Kaplan. All parties to the debate have accepted that some categories are (...)
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  • Mathematical Alchemy.P. Maddy - 1986 - British Journal for the Philosophy of Science 37 (3):279-314.
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  • Review. [REVIEW]Donald A.: Gillies - 1992 - British Journal for the Philosophy of Science 43 (2):263-278.
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  • Mathematical Explanation and the Theory of Why-Questions.David Sandborg - 1998 - British Journal for the Philosophy of Science 49 (4):603-624.
    Van Fraassen and others have urged that judgements of explanations are relative to why-questions; explanations should be considered good in so far as they effectively answer why-questions. In this paper, I evaluate van Fraassen's theory with respect to mathematical explanation. I show that his theory cannot recognize any proofs as explanatory. I also present an example that contradicts the main thesis of the why-question approach—an explanation that appears explanatory despite its inability to answer the why-question that motivated it. This example (...)
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  • Maddy and Mathematics: Naturalism or Not.Jeffrey W. Roland - 2007 - British Journal for the Philosophy of Science 58 (3):423-450.
    Penelope Maddy advances a purportedly naturalistic account of mathematical methodology which might be taken to answer the question 'What justifies axioms of set theory?' I argue that her account fails both to adequately answer this question and to be naturalistic. Further, the way in which it fails to answer the question deprives it of an analog to one of the chief attractions of naturalism. Naturalism is attractive to naturalists and nonnaturalists alike because it explains the reliability of scientific practice. Maddy's (...)
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  • Literalism and the Applicability of Arithmetic.L. Luce - 1991 - British Journal for the Philosophy of Science 42 (4):469-489.
    Philosophers have recently expressed interest in accounting for the usefulness of mathematics to science. However, it is certainly not a new concern. Putnam and Quine have each worked out an argument for the existence of mathematical objects from the indispensability of mathematics to science. Were Quine or Putnam to disregard the applicability of mathematics to science, he would not have had as strong a case for platonism. But I think there must be ways of parsing mathematical sentences which account for (...)
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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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  • Mathematics and Fiction II: Analogy.Robert Thomas - 2002 - Logique Et Analyse 45:185-228.
    The object of this paper is to study the analogy, drawn both positively and negatively, between mathematics and fiction. The analogy is more subtle and interesting than fictionalism, which was discussed in part I. Because analogy is not common coin among philosophers, this particular analogy has been discussed or mentioned for the most part just in terms of specific similarities that writers have noticed and thought worth mentioning without much attention's being paid to the larger picture. I intend with this (...)
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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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  • 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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  • 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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  • Introduction: Scientific Explanation Beyond Causation.Alexander Reutlinger & Juha Saatsi - 2017 - In Alexander Reutlinger & Juha Saatsi (eds.), Explanation Beyond Causation: Philosophical Perspectives on Non-Causal Explanations. Oxford: Oxford University Press.
    This is an introduction to the volume "Explanation Beyond Causation: Philosophical Perspectives on Non-Causal Explanations", edited by A. Reutlinger and J. Saatsi (OUP, forthcoming in 2017). -/- Explanations are very important to us in many contexts: in science, mathematics, philosophy, and also in everyday and juridical contexts. But what is an explanation? In the philosophical study of explanation, there is long-standing, influential tradition that links explanation intimately to causation: we often explain by providing accurate information about the causes of the (...)
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  • The Creative Growth of Mathematics.Jean Paul Van Bendegem - 1999 - Philosophica 63.
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  • Aspects of Mathematical Explanation: Symmetry, Unity, and Salience.Marc Lange - 2014 - Philosophical Review 123 (4):485-531.
    Unlike explanation in science, explanation in mathematics has received relatively scant attention from philosophers. Whereas there are canonical examples of scientific explanations, there are few examples that have become widely accepted as exhibiting the distinction between mathematical proofs that explain why some mathematical theorem holds and proofs that merely prove that the theorem holds without revealing the reason why it holds. This essay offers some examples of proofs that mathematicians have considered explanatory, and it argues that these examples suggest a (...)
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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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  • How Not to Reject the a Priori.Célia Teixeira - 2018 - Kriterion: Journal of Philosophy 59 (140):365-384.
    According to one influential argument against the existence of a priori knowledge, there is no a priori knowledge because (i) no belief is immune to revision, and (ii) if there were a priori knowledge, at least some beliefs would be unrevisable. The aim of this paper is to examine and reject this argument against the a priori. It is given special attention to premise (ii) of the argument. Philip Kitcher has famously defended a version of this premise. His arguments are (...)
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  • Disregarding the 'Hole Argument'.Bryan W. Roberts - unknown
    Jim Weatherall has suggested that Einstein's hole argument, as presented by Earman and Norton, is based on a misleading use of mathematics. I argue on the contrary that Weatherall demands an implausible restriction on how mathematics is used. The hole argument, on the other hand, is in no new danger at all.
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  • O nadużywaniu twierdzenia Gödla w sporach filozoficznych.Krzysztof Wójtowicz - 1996 - Zagadnienia Filozoficzne W Nauce 19.
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