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Science Without Numbers: A Defence of Nominalism

Princeton, NJ, USA: Princeton University Press (1980)

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  1. Three conceptions of explaining how possibly—and one reductive account.Johannes Persson - 2009 - In Henk W. de Regt (ed.), Epsa Philosophy of Science: Amsterdam 2009. Springer. pp. 275--286.
    Philosophers of science have often favoured reductive approaches to how-possibly explanation. This article identifies three alternative conceptions making how-possibly explanation an interesting phenomenon in its own right. The first variety approaches “how possibly X?” by showing that X is not epistemically impossible. This can sometimes be achieved by removing misunderstandings concerning the implications of one’s current belief system but involves characteristically a modification of this belief system so that acceptance of X does not result in contradiction. The second variety offers (...)
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  • Properly Extensive Quantities.Zee R. Perry - 2015 - Philosophy of Science 82 (5):833-844.
    This article introduces and motivates the notion of a “properly extensive” quantity by means of a puzzle about the reliability of certain canonical length measurements. An account of these measurements’ success, I argue, requires a modally robust connection between quantitative structure and mereology that is not mediated by the dynamics and is stronger than the constraints imposed by “mere additivity.” I outline what it means to say that length is not just extensive but properly so and then briefly sketch an (...)
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  • Know-How and Gradability.Carlotta Pavese - 2017 - Philosophical Review 126 (3):345-383.
    Orthodoxy has it that knowledge is absolute—that is, it cannot come in degrees. On the other hand, there seems to be strong evidence for the gradability of know-how. Ascriptions of know-how are gradable, as when we say that one knows in part how to do something, or that one knows how to do something better than somebody else. When coupled with absolutism, the gradability of ascriptions of know-how can be used to mount a powerful argument against intellectualism about know-how—the view (...)
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  • Two Criticisms against Mathematical Realism.Seungbae Park - 2017 - Diametros 52:96-106.
    Mathematical realism asserts that mathematical objects exist in the abstract world, and that a mathematical sentence is true or false, depending on whether the abstract world is as the mathematical sentence says it is. I raise two objections against mathematical realism. First, the abstract world is queer in that it allows for contradictory states of affairs. Second, mathematical realism does not have a theoretical resource to explain why a sentence about a tricle is true or false. A tricle is an (...)
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  • Ontic terms and metaontology, or: on what there actually is.T. Parent - 2014 - Philosophical Studies 170 (2):199-214.
    Terms such as ‘exist’, ‘actual’, etc., (hereafter, “ontic terms”) are recognized as having uses that are not ontologically committing, in addition to the usual commissive uses. (Consider, e.g., the Platonic and the neutral readings of ‘There is an even prime’.) In this paper, I identify five different noncommissive uses for ontic terms, and (by a kind of via negativa) attempt to define the commissive use, focusing on ‘actual’ as my example. The problem, however, is that the resulting definiens for the (...)
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  • Mathematical fictionalism.David Papineau - 1988 - International Studies in the Philosophy of Science 2 (2):151 – 174.
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  • The Enculturated Move From Proto-Arithmetic to Arithmetic.Markus Pantsar - 2019 - Frontiers in Psychology 10.
    The basic human ability to treat quantitative information can be divided into two parts. With proto-arithmetical ability, based on the core cognitive abilities for subitizing and estimation, numerosities can be treated in a limited and/or approximate manner. With arithmetical ability, numerosities are processed (counted, operated on) systematically in a discrete, linear, and unbounded manner. In this paper, I study the theory of enculturation as presented by Menary (2015) as a possible explanation of how we make the move from the proto-arithmetical (...)
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  • Objectivity in Mathematics, Without Mathematical Objects†.Markus Pantsar - 2021 - Philosophia Mathematica 29 (3):318-352.
    I identify two reasons for believing in the objectivity of mathematical knowledge: apparent objectivity and applications in science. Focusing on arithmetic, I analyze platonism and cognitive nativism in terms of explaining these two reasons. After establishing that both theories run into difficulties, I present an alternative epistemological account that combines the theoretical frameworks of enculturation and cumulative cultural evolution. I show that this account can explain why arithmetical knowledge appears to be objective and has scientific applications. Finally, I will argue (...)
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  • From Maximal Intersubjectivity to Objectivity: An Argument from the Development of Arithmetical Cognition.Markus Pantsar - 2022 - Topoi 42 (1):271-281.
    One main challenge of non-platonist philosophy of mathematics is to account for the apparent objectivity of mathematical knowledge. Cole and Feferman have proposed accounts that aim to explain objectivity through the intersubjectivity of mathematical knowledge. In this paper, focusing on arithmetic, I will argue that these accounts as such cannot explain the apparent objectivity of mathematical knowledge. However, with support from recent progress in the empirical study of the development of arithmetical cognition, a stronger argument can be provided. I will (...)
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  • An empirically feasible approach to the epistemology of arithmetic.Markus Pantsar - 2014 - Synthese 191 (17):4201-4229.
    Recent years have seen an explosion of empirical data concerning arithmetical cognition. In this paper that data is taken to be philosophically important and an outline for an empirically feasible epistemological theory of arithmetic is presented. The epistemological theory is based on the empirically well-supported hypothesis that our arithmetical ability is built on a protoarithmetical ability to categorize observations in terms of quantities that we have already as infants and share with many nonhuman animals. It is argued here that arithmetical (...)
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  • Fictionalism and the Problem of Universals in the Philosophy of Mathematics.Strahinja Đorđević - 2018 - Filozofija I Društvo 29 (3):415-428.
    Many long-standing problems pertaining to contemporary philosophy of mathematics can be traced back to different approaches in determining the nature of mathematical entities which have been dominated by the debate between realists and nominalists. Through this discussion conceptualism is represented as a middle solution. However, it seems that until the 20th century there was no third position that would not necessitate any reliance on one of the two points of view. Fictionalism, on the other hand, observes mathematical entities in a (...)
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  • Dummett and Frege on the philosophy of mathematics.Alex Oliver - 1994 - Inquiry: An Interdisciplinary Journal of Philosophy 37 (3):349 – 392.
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  • Semblance or similarity? Reflections on Simulation and Similarity: Michael Weisberg: Simulation and similarity: using models to understand the world. Oxford University Press, 2013. 224pp. ISBN 9780199933662, $65.00.Jay Odenbaugh - 2015 - Biology and Philosophy 30 (2):277-291.
    In this essay, I critically evaluate components of Michael Weisberg’s approach to models and modeling in his book Simulation and Similarity. First, I criticize his account of the ontology of models and mathematics. Second, I respond to his objections to fictionalism regarding models arguing that they fail. Third, I sketch a deflationary approach to models that retains many elements of his account but avoids the inflationary commitments.
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  • A Critique of Resnik’s Mathematical Realism.Timothy John Nulty - 2005 - Erkenntnis 62 (3):379 - 393.
    This paper attempts to motivate skepticism about the reality of mathematical objects. The aim of the paper is not to provide a general critique of mathematical realism, but to demonstrate the insufficiency of the arguments advanced by Michael Resnik. I argue that Resnik’s use of the concept of immanent truth is inconsistent with the treatment of mathematical objects as ontologically and epistemically continuous with the objects posited by the natural sciences. In addition, Resnik’s structuralist program, and his denial of relational (...)
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  • A Critique of Resnik’s Mathematical Realism.Timothy John Nulty - 2005 - Erkenntnis 62 (3):379-393.
    This paper attempts to motivate skepticism about the reality of mathematical objects. The aim of the paper is not to provide a general critique of mathematical realism, but to demonstrate the insufficiency of the arguments advanced by Michael Resnik. I argue that Resnik's use of the concept of immanent truth is inconsistent with the treatment of mathematical objects as ontologically and epistemically continuous with the objects posited by the natural sciences. In addition, Resnik's structuralist program, and his denial of relational (...)
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  • Reflexive fictionalisms.Daniel Nolan & J. O'Leary-Hawthorne - 1996 - Analysis 56 (1):23-32.
    There is a class of fictionalist strategies (the reflexive fictionalisms) which appear to suffer from a common problem: the problem that the entities which are supposedly fictional turn out, by the lights of the fictionalist theory itself, to exist. The appropriate solution is to reject so-called strong fictionalism in each case: that is, to reject the variety of fictionalism which takes appeal to the domain of fictional entities to provide an explanation or analysis of the operators or predicates with which (...)
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  • Moral fictionalism versus the rest.Daniel Nolan, Greg Restall & Caroline West - 2005 - Australasian Journal of Philosophy 83 (3):307 – 330.
    In this paper we introduce a distinct metaethical position, fictionalism about morality. We clarify and defend the position, showing that it is a way to save the 'moral phenomena' while agreeing that there is no genuine objective prescriptivity to be described by moral terms. In particular, we distinguish moral fictionalism from moral quasi-realism, and we show that fictionalism possesses the virtues of quasi-realism about morality, but avoids its vices.
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  • Mathematics is not the only language in the book of nature.James Nguyen & Roman Frigg - 2017 - Synthese 198 (Suppl 24):1-22.
    How does mathematics apply to something non-mathematical? We distinguish between a general application problem and a special application problem. A critical examination of the answer that structural mapping accounts offer to the former problem leads us to identify a lacuna in these accounts: they have to presuppose that target systems are structured and yet leave this presupposition unexplained. We propose to fill this gap with an account that attributes structures to targets through structure generating descriptions. These descriptions are physical descriptions (...)
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  • Neo-positivist metaphysics.Alyssa Ney - 2012 - Philosophical Studies 160 (1):53-78.
    Some philosophers argue that many contemporary debates in metaphysics are “illegitimate,” “shallow,” or “trivial,” and that “contemporary analytic metaphysics, a professional activity engaged in by some extremely intelligent and morally serious people, fails to qualify as part of the enlightened pursuit of objective truth, and should be discontinued” (Ladyman and Ross, Every thing must go: Metaphysics naturalized , 2007 ). Many of these critics are explicit about their sympathies with Rudolf Carnap and his circle, calling themselves ‘neo-positivists’ or ‘neo-Carnapians.’ Yet (...)
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  • Reality, Systems and Impure Systems.J. Nescolarde-Selva & J. L. Usó-Doménech - 2014 - Foundations of Science 19 (3):289-306.
    Impure systems contain Objects and Subjects: Subjects are human beings. We can distinguish a person as an observer (subjectively outside the system) and that by definition is the Subject himself, and part of the system. In this case he acquires the category of object. Objects (relative beings) are significances, which are the consequence of perceptual beliefs on the part of the Subject about material or energetic objects (absolute beings) with certain characteristics.The IS (Impure System) approach is as follows: Objects are (...)
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  • On the one hand: Reflections on enantiomorphy.Graham Nerlich - 1995 - Australasian Journal of Philosophy 73 (3):432 – 443.
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  • Incongruent counterparts and the reality of space.Graham Nerlich - 2009 - Philosophy Compass 4 (3):598-613.
    Left and right hands are incongruent counterparts. Yet each replicates the intrinsic properties of the other. This suggests that differing relations to space make the difference. Kant's and Weyl's discussions of the problem are critically discussed. It emerges that spatial relationism fails to explain how its relations may be interpreted. An excursion into visual geometry explains the basis of handedness in the orientable structure of space.
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  • Names of Attitudes and Norms for Attitudes.Inga Nayding - 2015 - Disputatio 7 (40):1-24.
    Fictionalists claim that instead of believing certain controversial propositions they accept them nonseriously, as useful make-believe. In this way they present themselves as having an austere ontology despite the apparent ontological commitments of their discourse. Some philosophers object that this plays on a distinction without a difference: the fictionalist’s would-be nonserious acceptance is the most we can do for the relevant content acceptance-wise, hence such acceptance is no different from what we ordinarily call ‘belief’ and should be so called. They (...)
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  • Folk intuitions, slippery slopes, and necessary fictions : an essay on Saul Smilansky's free will illusionism.Thomas Nadelhoffer - 2007 - In Peter A. French & Howard K. Wettstein (eds.), Philosophy and the Empirical. Blackwell. pp. 202–213.
    During the past two decades, an interest among philosophers in fictitious and illusory beliefs has sprung up in fields ranging anywhere from mathematics and modality to morality.1 In this paper, we focus primarily on the view that Saul Smilansky has dubbed “free will illusionism”—i.e., the purportedly descriptive claim that most people have illusory beliefs concerning the existence of libertarian free will, coupled with the normative claim that because dispelling these illusory beliefs would produce negative personal and societal consequences, those of (...)
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  • The metaphysics of quantity.Brent Mundy - 1987 - Philosophical Studies 51 (1):29 - 54.
    A formal theory of quantity T Q is presented which is realist, Platonist, and syntactically second-order (while logically elementary), in contrast with the existing formal theories of quantity developed within the theory of measurement, which are empiricist, nominalist, and syntactically first-order (while logically non-elementary). T Q is shown to be formally and empirically adequate as a theory of quantity, and is argued to be scientifically superior to the existing first-order theories of quantity in that it does not depend upon empirically (...)
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  • Scientific theory as partially interpreted calculus II.Brent Mundy - 1988 - Erkenntnis 28 (2):165 - 183.
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  • On the general theory of meaningful representation.Brent Mundy - 1986 - Synthese 67 (3):391 - 437.
    The numerical representations of measurement, geometry and kinematics are here subsumed under a general theory of representation. The standard theories of meaningfulness of representational propositions in these three areas are shown to be special cases of two theories of meaningfulness for arbitrary representational propositions: the theories based on unstructured and on structured representation respectively. The foundations of the standard theories of meaningfulness are critically analyzed and two basic assumptions are isolated which do not seem to have received adequate justification: the (...)
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  • Faithful representation, physical extensive measurement theory and archimedean axioms.Brent Mundy - 1987 - Synthese 70 (3):373 - 400.
    The formal methods of the representational theory of measurement (RTM) are applied to the extensive scales of physical science, with some modifications of interpretation and of formalism. The interpretative modification is in the direction of theoretical realism rather than the narrow empiricism which is characteristic of RTM. The formal issues concern the formal representational conditions which extensive scales should be assumed to satisfy; I argue in the physical case for conditions related to weak rather than strong extensive measurement, in the (...)
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  • Distant action in classical electromagnetic theory.Brent Mundy - 1989 - British Journal for the Philosophy of Science 40 (1):39-68.
    The standard mathematical apparatus of classical electromagnetic theory in Minkowski space-time allows an interpretation in terms of retarded distant action, as well as the standard field interpretation. This interpretation is here presented and defended as a scientifically significant alternative to the field theory, casting doubt upon the common view that classical electromagnetic theory provides scientific support for the physical existence of fields as fundamental entities. The various types of consideration normally thought to provide evidence for the existence of the electromagnetic (...)
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  • 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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  • What Can Our Best Scientific Theories Tell Us About The Modal Status of Mathematical Objects?Joe Morrison - 2023 - Erkenntnis 88 (4):1391-1408.
    Indispensability arguments are used as a way of working out what there is: our best science tells us what things there are. Some philosophers think that indispensability arguments can be used to show that we should be committed to the existence of mathematical objects (numbers, functions, sets). Do indispensability arguments also deliver conclusions about the modal properties of these mathematical entities? Colyvan (in Leng, Paseau, Potter (eds) Mathematical knowledge, OUP, Oxford, 109-122, 2007) and Hartry Field (Realism, mathematics and modality, Blackwell, (...)
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  • On the possibility of science without numbers.Chris Mortensen - 1998 - Australasian Journal of Philosophy 76 (2):182 – 197.
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  • Evidential holism.Joe Morrison - 2017 - Philosophy Compass 12 (6):e12417.
    Evidential holism begins with something like the claim that “it is only jointly as a theory that scientific statements imply their observable consequences.” This is the holistic claim that Elliott Sober tells us is an “unexceptional observation”. But variations on this “unexceptional” claim feature as a premise in a series of controversial arguments for radical conclusions, such as that there is no analytic or synthetic distinction that the meaning of a sentence cannot be understood without understanding the whole language of (...)
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  • Indispensability and explanation: an overview and introduction.Daniele Molinini, Fabrice Pataut & Andrea Sereni - 2016 - Synthese 193 (2):317-332.
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  • The Conditions of Realism.Christian Miller - 2007 - Journal of Philosophical Research 32:95-132.
    The concern of this paper is not with the truth of any particular realist or anti-realist view, but rather with determining what it is to be a realist or anti-realist in the first place. While much skepticism has been voiced in recent years about the viability of such a project, my goal is to articulate interesting and informative conditions whereby any view in any domain of experience can count as either a realist or an anti-realist position.
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  • Rule Following, Error Theory and Eliminativism.Alexander Miller - 2015 - International Journal of Philosophical Studies 23 (3):323-336.
    In this paper, I argue for three main claims. First, that there are two broad sorts of error theory about a particular region of thought and talk, eliminativist error theories and non-eliminativist error theories. Second, that an error theory about rule following can only be an eliminativist view of rule following, and therefore an eliminativist view of meaning and content on a par with Paul Churchland’s prima facie implausible eliminativism about the propositional attitudes. Third, that despite some superficial appearances to (...)
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  • On the individuation of words.J. T. M. Miller - 2020 - Inquiry: An Interdisciplinary Journal of Philosophy 63 (8):875-884.
    ABSTRACT The idea that two words can be instances of the same word is a central intuition in our conception of language. This fact underlies many of the claims that we make about how we communicate, and how we understand each other. Given this, irrespective of what we think words are, it is common to think that any putative ontology of words, must be able to explain this feature of language. That is, we need to provide criteria of identity for (...)
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  • Introduction.Eugene Mills - 2003 - Topoi 22 (2):105-109.
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  • The explainability of intuitions.Nenad Miščević - 2004 - Dialectica 58 (1):43–70.
    Explaining intuitions in terms of "facts of our natural history" is compatible with rationally trusting them. This compatibilist view is defended in the present paper, focusing upon nomic and essentialist modal intuitions. The opposite, incompatibilist view alleges the following: If basic modal intuitions are due to our cognitive make-up or "imaginative habits" then the epistemologists are left with a mere non-rational feeling of compulsion on the side of the thinker. Intuitions then cannot inform us about modal reality. In contrast, the (...)
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  • Ontology and logic: remarks on hartry field's anti-platonist philosophy of mathematics.Michael D. Resnik - 1985 - History and Philosophy of Logic 6 (1):191-209.
    In Science without numbers Hartry Field attempted to formulate a nominalist version of Newtonian physics?one free of ontic commitment to numbers, functions or sets?sufficiently strong to have the standard platonist version as a conservative extension. However, when uses for abstract entities kept popping up like hydra heads, Field enriched his logic to avoid them. This paper reviews some of Field's attempts to deflate his ontology by inflating his logic.
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  • How to apply mathematics.Ulrich Meyer - 2004 - Erkenntnis 61 (1):17-28.
    This paper presents a novel account of applied mathematics. It shows how we can distinguish the physical content from the mathematical form of a scientific theory even in cases where the mathematics applied is indispensable and cannot be eliminated by paraphrase.
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  • Extending Hartry field's instrumental account of applied mathematics to statistical mechanics.Glen Meyer - 2009 - Philosophia Mathematica 17 (3):273-312.
    A serious flaw in Hartry Field’s instrumental account of applied mathematics, namely that Field must overestimate the extent to which many of the structures of our mathematical theories are reflected in the physical world, underlies much of the criticism of this account. After reviewing some of this criticism, I illustrate through an examination of the prospects for extending Field’s account to classical equilibrium statistical mechanics how this flaw will prevent any significant extension of this account beyond field theories. I note (...)
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  • Ficcionalização da moral:um roteiro.Wilson Mendonça & Idia Laura Ferreira - 2010 - Ethic@ - An International Journal for Moral Philosophy 9 (2):175-198.
    O trabalho identifica algumas questões críticas em tentativas recentes de tratar o objeto da moralidade como uma ficção. Sobre o pano de fundo de uma distinção entre cognitivismo e não-cognitivismo moral exclusivamente em termos da natureza do estado mental expresso pelo enunciado moral, independentemente do conteúdo semântico do enunciado , o trabalho mostra um deficit estrutural no argumento da intransigência proposto por Kalderon a favor do ficcionalismo moral hermenêutico . Mediante a reconstrução dos passos que levam do cognitivismo irrealista ao (...)
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  • Weaseling away the indispensability argument.Joseph Melia - 2000 - Mind 109 (435):455-480.
    According to the indispensability argument, the fact that we quantify over numbers, sets and functions in our best scientific theories gives us reason for believing that such objects exist. I examine a strategy to dispense with such quantification by simply replacing any given platonistic theory by the set of sentences in the nominalist vocabulary it logically entails. I argue that, as a strategy, this response fails: for there is no guarantee that the nominalist world that go beyond the set of (...)
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  • The conservativeness of mathematics.J. Melia - 2006 - Analysis 66 (3):202-208.
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  • Field's programme: some interference.J. Melia - 1998 - Analysis 58 (2):63-71.
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  • Penelope Maddy. Defending the Axioms: On the Philosophical Foundations of Set Theory. Oxford: Oxford University Press, 2011. ISBN 978-0-19-959618-8 (hbk); 978-0-19-967148-9 (pbk). Pp. x + 150. [REVIEW]C. McLarty - 2013 - Philosophia Mathematica 21 (3):385-392.
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  • The holistic presumptions of the indispensability argument.Russell Marcus - 2014 - Synthese 191 (15):3575-3594.
    The indispensability argument is sometimes seen as weakened by its reliance on a controversial premise of confirmation holism. Recently, some philosophers working on the indispensability argument have developed versions of the argument which, they claim, do not rely on holism. Some of these writers even claim to have strengthened the argument by eliminating the controversial premise. I argue that the apparent removal of holism leaves a lacuna in the argument. Without the holistic premise, or some other premise which facilitates the (...)
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  • The (un)detectability of absolute Newtonian masses.Niels C. M. Martens - 2019 - Synthese 198 (3):2511-2550.
    Absolutism about mass claims that mass ratios obtain in virtue of absolute masses. Comparativism denies this. Dasgupta, Oxford studies in metaphysics, Oxford University Press, Oxford, 2013) argues for comparativism about mass, in the context of Newtonian Gravity. Such an argument requires proving that comparativism is empirically adequate. Dasgupta equates this to showing that absolute masses are undetectable, and attempts to do so. This paper develops an argument by Baker to the contrary: absolute masses are in fact empirically meaningful, that is (...)
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  • Does Semantic Deflationism Entail Meta-Ontological Deflationism?Benjamin Marschall & Thomas Schindler - 2021 - Philosophical Quarterly 71 (1):99-119.
    Deflationary positions have been defended in many areas of philosophy. Most prominent are semantic deflationism about truth and reference, and meta-ontological deflationism, according to which existence has no deep nature and the standard neo-Quinean approach to ontology is misguided. Although both kinds of views have generated much discussion, surprisingly little attention has been paid to the question of how they relate to each other. Are they independent, is it advisable to hold them all at once, or do they even entail (...)
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