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Mathematical truth

Journal of Philosophy 70 (19):661-679 (1973)

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  1. Inference to the best explanation as supporting the expansion of mathematicians’ ontological commitments.Marc Lange - 2022 - Synthese 200 (2):1-26.
    This paper argues that in mathematical practice, conjectures are sometimes confirmed by “Inference to the Best Explanation” as applied to some mathematical evidence. IBE operates in mathematics in the same way as IBE in science. When applied to empirical evidence, IBE sometimes helps to justify the expansion of scientists’ ontological commitments. Analogously, when applied to mathematical evidence, IBE sometimes helps to justify mathematicians' in expanding the range of their ontological commitments. IBE supplements other forms of non-deductive reasoning in mathematics, avoiding (...)
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  • Excursus on Wittgenstein's Rule-Following Considerations.Elek Lane - 2017 - Nordic Wittgenstein Review 6 (1):53-83.
    In this essay, I seek to demonstrate the interplay of philosophical voices – particularly, that of a platonist voice and a community-agreement-view voice – that drives Wittgenstein’s rule-following dialectic forward; and I argue that each voice succumbs to a particular form of dialectical oscillation that renders its response to the problem of rule-following philosophically inadequate. Finally, I suggest that, by seeing and taking stock of the dilemma in which these responses to the skeptical problem are caught, we can come to (...)
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  • Fictionalism in Metaphysics.Frederick Kroon - 2011 - Philosophy Compass 6 (11):786-803.
    This is a survey of contemporary work on ‘fictionalism in metaphysics’, a term that is taken to signify both the place of fictionalism as a distinctive anti‐realist metaphysics in which usefulness rather than truth is the norm of acceptance, and the fact that philosophers have given fictionalist treatments of a range of specifically metaphysical notions.
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  • Set-theoretic realism and arithmetic.Michael Kremer - 1991 - Philosophical Studies 64 (3):253 - 271.
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  • Deflationary Nominalism and Puzzle Avoidance.David Mark Kovacs - 2019 - Philosophia Mathematica 27 (1):88-104.
    In a series of works, Jody Azzouni has defended deflationary nominalism, the view that certain sentences quantifying over mathematical objects are literally true, although such objects do not exist. One alleged attraction of this view is that it avoids various philosophical puzzles about mathematical objects. I argue that this thought is misguided. I first develop an ontologically neutral counterpart of Field’s reliability challenge and argue that deflationary nominalism offers no distinctive answer to it. I then show how this reasoning generalizes (...)
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  • Debunking arguments.Daniel Z. Korman - 2019 - Philosophy Compass 14 (12):e12638.
    Debunking arguments—also known as etiological arguments, genealogical arguments, access problems, isolation objec- tions, and reliability challenges—arise in philosophical debates about a diverse range of topics, including causation, chance, color, consciousness, epistemic reasons, free will, grounding, laws of nature, logic, mathematics, modality, morality, natural kinds, ordinary objects, religion, and time. What unifies the arguments is the transition from a premise about what does or doesn't explain why we have certain mental states to a negative assessment of their epistemic status. I examine (...)
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  • And they ain't outside the head either.John Koethe - 1992 - Synthese 90 (1):27-53.
    According to a classical view in the philosophy of language, the reference of a term is determined by a property of the term which supervenes on the history of its use. A contrasting view is that a term's reference is determined by how it is properly interpreted, in accordance with certain constraints or conditions of adequacy on interpretations. Causal theories of reference of the sort associated with Hilary Putnam, Saul Kripke and Michael Devitt are versions of the first view, while (...)
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  • The uncanny accuracy of God's mathematical beliefs.Robert Knowles - 2021 - Religious Studies 57 (2):333-352.
    I show how mathematical platonism combined with belief in the God of classical theism can respond to Field's epistemological objection. I defend an account of divine mathematical knowledge by showing that it falls out of an independently motivated general account of divine knowledge. I use this to explain the accuracy of God's mathematical beliefs, which in turn explains the accuracy of our own. My arguments provide good news for theistic platonists, while also shedding new light on Field's influential objection.
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  • Old wine in new bottles: Evolutionary debunking arguments and the Benacerraf–Field challenge.Michael Klenk - 2017 - Ethical Theory and Moral Practice 20 (4):781-795.
    Evolutionary debunking arguments purport to show that robust moral realism, the metaethical view that there are non-natural and mind-independent moral properties and facts that we can know about, is incompatible with evolutionary explanations of morality. One of the most prominent evolutionary debunking arguments is advanced by Sharon Street, who argues that if moral realism were true, then objective moral knowledge is unlikely because realist moral properties are evolutionary irrelevant and moral beliefs about those properties would not be selected for. However, (...)
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  • Pragmatism and Moral Objectivity.Michael Klenk - 2021 - European Journal of Pragmatism and American Philosophy 13 (2).
    Most non-robust-realist metaethical theories, such as expressivism, constructivism, and non-robust forms of realism, claim to retain a sense of objectivity in ethics. A persistent issue for these theories is to identify an objective criterion for moral truth that meets their objectivist aspiration. Objectivist aspirations are often probed by confronting non-realists with abject normative positions, such as those of rational racists, which are licensed by the framework of the respective non-realist theory but nevertheless strike us a wrong. In such cases, non-realist (...)
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  • Can Moral Realists Deflect Defeat Due to Evolutionary Explanations of Morality?Michael Klenk - 2017 - Pacific Philosophical Quarterly 98 (S1):227-248.
    I address Andrew Moon's recent discussion (2016, this journal) of the question whether third-factor accounts are valid responses to debunking arguments against moral realism. Moon argues that third-factor responses are valid under certain conditions but leaves open whether moral realists can use his interpretation of the third-factor response to defuse the evolutionary debunking challenge. I rebut Moon's claim and answer his question. Moon's third-factor reply is valid only if we accept externalism about epistemic defeaters. However, even if we do, I (...)
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  • Real realism: The galilean strategy.Philip Kitcher - 2001 - Philosophical Review 110 (2):151-197.
    This essay aims to disentangle various types of anti-realism, and to disarm the considerations that are deployed to support them. I distinguish empiricist versions of anti-realism from constructivist versions, and, within each of these, semantic arguments from epistemological arguments. The centerpiece of my defense of a modest version of realism - real realism - is the thought that there are resources within our ordinary ways of talking about and knowing about everyday objects that enable us to extend our claims to (...)
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  • Hilbert's epistemology.Philip Kitcher - 1976 - Philosophy of Science 43 (1):99-115.
    Hilbert's program attempts to show that our mathematical knowledge can be certain because we are able to know for certain the truths of elementary arithmetic. I argue that, in the absence of a theory of mathematical truth, Hilbert does not have a complete theory of our arithmetical knowledge. Further, while his deployment of a Kantian notion of intuition seems to promise an answer to scepticism, there is no way to complete Hilbert's epistemology which would answer to his avowed aims.
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  • Epistemology Without History is Blind.Philip Kitcher - 2011 - Erkenntnis 75 (3):505-524.
    In the spirit of James and Dewey, I ask what one might want from a theory of knowledge. Much Anglophone epistemology is centered on questions that were once highly pertinent, but are no longer central to broader human and scientific concerns. The first sense in which epistemology without history is blind lies in the tendency of philosophers to ignore the history of philosophical problems. A second sense consists in the perennial attraction of approaches to knowledge that divorce knowing subjects from (...)
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  • Bolzano's ideal of algebraic analysis.Philip Kitcher - 1975 - Studies in History and Philosophy of Science Part A 6 (3):229-269.
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  • What Would a Phenomenology of Logic Look Like?James Kinkaid - 2020 - Mind 129 (516):1009-1031.
    The phenomenological movement begins in the Prolegomena to Husserl’s Logical Investigations as a philosophy of logic. Despite this, remarkably little attention has been paid to Husserl’s arguments in the Prolegomena in the contemporary philosophy of logic. In particular, the literature spawned by Gilbert Harman’s work on the normative status of logic is almost silent on Husserl’s contribution to this topic. I begin by raising a worry for Husserl’s conception of ‘pure logic’ similar to Harman’s challenge to explain the connection between (...)
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  • 1995–1996 Annual Meeting of the Association for Symbolic Logic.H. Jerome Keisler - 1996 - Bulletin of Symbolic Logic 2 (4):448-472.
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  • On Field’s Epistemological Argument Against Platonism.Ivan Kasa - 2010 - Studia Logica 96 (2):141-147.
    Hartry Field's formulation of an epistemological argument against platonism is only valid if knowledge is constrained by a causal clause. Contrary to recent claims (e.g. in Liggins (2006), Liggins (2010)), Field's argument therefore fails the very same criterion usually taken to discredit Benacerraf's earlier version.
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  • Realistic rationalism. [REVIEW]Mark Eli Kalderon - 2000 - Philosophical Review 109 (3):456-459.
    Philosophy of mathematics is in an alienated state. While regarded by the profession as a serious and legitimate subdiscipline, a passing knowledge of its subject matter is considered something of a luxury—or at least not required of a conscientious philosopher the way a passing knowledge of logic is. Philosophy of mathematics is thus regarded with a benign neglect: best left to the experts, whose opinions should be deferred to, but mostly irrelevant to the central concerns of the working philosopher. Its (...)
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  • Abstract Objects, Causal Efficacy, and Causal Exclusion.Tim Juvshik - 2018 - Erkenntnis 83 (4):805-827.
    objects are standardly taken to be causally inert, but this claim is rarely explicitly argued for. In the context of his platonism about musical works, in order for musical works to be audible, Julian Dodd argues that abstracta are causally efficacious in virtue of their concrete tokens participating in events. I attempt to provide a principled argument for the causal inertness of abstracta by first rejecting Dodd’s arguments from events, and then extending and generalizing the causal exclusion argument to the (...)
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  • Relativity and the Causal Efficacy of Abstract Objects.Tim Juvshik - 2020 - American Philosophical Quarterly 57 (3):269-282.
    Abstract objects are standardly taken to be causally inert, however principled arguments for this claim are rarely given. As a result, a number of recent authors have claimed that abstract objects are causally efficacious. These authors take abstracta to be temporally located in order to enter into causal relations but lack a spatial location. In this paper, I argue that such a position is untenable by showing first that causation requires its relata to have a temporal location, but second, that (...)
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  • Objectivity and reliability.Justin Clarke-Doane - 2017 - Canadian Journal of Philosophy 47 (6):841-855.
    Scanlon’s Being Realistic about Reasons (BRR) is a beautiful book – sleek, sophisticated, and programmatic. One of its key aims is to demystify knowledge of normative and mathematical truths. In this article, I develop an epistemological problem that Scanlon fails to explicitly address. I argue that his “metaphysical pluralism” can be understood as a response to that problem. However, it resolves the problem only if it undercuts the objectivity of normative and mathematical inquiry.
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  • Possibility. [REVIEW]Jason Turner - 2010 - Analysis 70 (1):184-186.
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  • On Mathematical and Religious Belief, and on Epistemic Snobbery.Silvia Jonas - 2016 - Philosophy 91 (1):69-92.
    In this paper, I argue that religious belief is epistemically equivalent to mathematical belief. Abstract beliefs don't fall under ‘naive’, evidence-based analyses of rationality. Rather, their epistemic permissibility depends, I suggest, on four criteria: predictability, applicability, consistency, and immediate acceptability of the fundamental axioms. The paper examines to what extent mathematics meets these criteria, juxtaposing the results with the case of religion. My argument is directed against a widespread view according to which belief in mathematics is clearly rationally acceptable whereas (...)
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  • Mathematical Pluralism and Indispensability.Silvia Jonas - 2023 - Erkenntnis 1:1-25.
    Pluralist mathematical realism, the view that there exists more than one mathematical universe, has become an influential position in the philosophy of mathematics. I argue that, if mathematical pluralism is true (and we have good reason to believe that it is), then mathematical realism cannot (easily) be justified by arguments from the indispensability of mathematics to science. This is because any justificatory chain of inferences from mathematical applications in science to the total body of mathematical theorems can cover at most (...)
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  • Mathematical and Moral Disagreement.Silvia Jonas - 2020 - Philosophical Quarterly 70 (279):302-327.
    The existence of fundamental moral disagreements is a central problem for moral realism and has often been contrasted with an alleged absence of disagreement in mathematics. However, mathematicians do in fact disagree on fundamental questions, for example on which set-theoretic axioms are true, and some philosophers have argued that this increases the plausibility of moral vis-à-vis mathematical realism. I argue that the analogy between mathematical and moral disagreement is not as straightforward as those arguments present it. In particular, I argue (...)
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  • Access Problems and explanatory overkill.Silvia Jonas - 2017 - Philosophical Studies 174 (11):2731-2742.
    I argue that recent attempts to deflect Access Problems for realism about a priori domains such as mathematics, logic, morality, and modality using arguments from evolution result in two kinds of explanatory overkill: the Access Problem is eliminated for contentious domains, and realist belief becomes viciously immune to arguments from dispensability, and to non-rebutting counter-arguments more generally.
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  • Book Review: Kit Fine. The Limits of Abstraction. [REVIEW]John P. Burgess - 2003 - Notre Dame Journal of Formal Logic 44 (4):227-251.
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  • Intuiting the infinite.Robin Jeshion - 2014 - Philosophical Studies 171 (2):327-349.
    This paper offers a defense of Charles Parsons’ appeal to mathematical intuition as a fundamental factor in solving Benacerraf’s problem for a non-eliminative structuralist version of Platonism. The literature is replete with challenges to his well-known argument that mathematical intuition justifies our knowledge of the infinitude of the natural numbers, in particular his demonstration that any member of a Hilbertian stroke string ω-sequence has a successor. On Parsons’ Kantian approach, this amounts to demonstrating that for an “arbitrary” or “vaguely represented” (...)
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  • Book Review: Michael Resnik. Mathematics as a Science of Patterns. [REVIEW]Janet Folina - 1999 - Notre Dame Journal of Formal Logic 40 (3):455-472.
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  • Mathematical instrumentalism meets the conjunction objection.Hawthorne James - 1996 - Journal of Philosophical Logic 25 (4):363-397.
    Scientific realists often appeal to some version of the conjunction objection to argue that scientific instrumentalism fails to do justice to the full empirical import of scientific theories. Whereas the conjunction objection provides a powerful critique of scientific instrumentalism, I will show that mathematical instnrunentalism escapes the conjunction objection unscathed.
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  • Quantifying over the reals.Philip Hugly & Charles Sayward - 1994 - Synthese 101 (1):53 - 64.
    Peter Geach proposed a substitutional construal of quantification over thirty years ago. It is not standardly substitutional since it is not tied to those substitution instances currently available to us; rather, it is pegged to possible substitution instances. We argue that (i) quantification over the real numbers can be construed substitutionally following Geach's idea; (ii) a price to be paid, if it is that, is intuitionism; (iii) quantification, thus conceived, does not in itself relieve us of ontological commitment to real (...)
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  • Platonistic formalism.L. Horsten - 2001 - Erkenntnis 54 (2):173-194.
    The present paper discusses a proposal which says,roughly and with several qualifications, that thecollection of mathematical truths is identical withthe set of theorems of ZFC. It is argued that thisproposal is not as easily dismissed as outright falseor philosophically incoherent as one might think. Some morals of this are drawn for the concept ofmathematical knowledge.
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  • Lowe on Modal Knowledge.Joachim Horvath - 2014 - Thought: A Journal of Philosophy 3 (3):208-217.
    In recent work, E. J. Lowe presents an essence-based account of our knowledge of metaphysical modality that he claims to be superior to its main competitors. I argue that knowledge of essences alone, without knowledge of a suitable bridge principle, is insufficient for knowing that something is metaphysically necessary or metaphysically possible. Yet given Lowe's other theoretical commitments, he cannot account for our knowledge of the needed bridge principle, and so his essence-based modal epistemology remains incomplete. In addition to that, (...)
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  • Experiments in Thought.Walter Hopp - 2014 - Perspectives on Science 22 (2):242-263.
    . What are thought experiments, and how do they generate knowledge? More specifically, what sorts of intentional acts must one perform in order to carry out a thought experiment, what sorts of objects are such acts directed toward, and how are those objects made present in such acts? I argue on phenomenological grounds that the proper objects of thought experiments are, in certain cases, uninstantiated universals and relations among them. I will also argue that, in the best of cases, we (...)
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  • The Semantic Theory of Truth: Field’s Incompleteness Objection.Glen A. Hoffmann - 2007 - Philosophia 35 (2):161-170.
    According to Field’s influential incompleteness objection, Tarski’s semantic theory of truth is unsatisfactory since the definition that forms its basis is incomplete in two distinct senses: (1) it is physicalistically inadequate, and for this reason, (2) it is conceptually deficient. In this paper, I defend the semantic theory of truth against the incompleteness objection by conceding (1) but rejecting (2). After arguing that Davidson and McDowell’s reply to the incompleteness objection fails to pass muster, I argue that, within the constraints (...)
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  • Paraphrasing away properties with pluriverse counterfactuals.Jack Himelright - 2020 - Synthese 198 (11):10883-10902.
    In this paper, I argue that for the purposes of ordinary reasoning, sentences about properties of concrete objects can be replaced with sentences concerning how things in our universe would be related to inscriptions were there a pluriverse. Speaking loosely, pluriverses are composites of universes that collectively realize every way a universe could possibly be. As such, pluriverses exhaust all possible meanings that inscriptions could take. Moreover, because universes necessarily do not influence one another, our universe would not be any (...)
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  • A Lewisian Argument Against Platonism, or Why Theses About Abstract Objects Are Unintelligible.Jack Himelright - 2023 - Erkenntnis 88 (7):3037–3057.
    In this paper, I argue that all expressions for abstract objects are meaningless. My argument closely follows David Lewis’ argument against the intelligibility of certain theories of possible worlds, but modifies it in order to yield a general conclusion about language pertaining to abstract objects. If my Lewisian argument is sound, not only can we not know that abstract objects exist, we cannot even refer to or think about them. However, while the Lewisian argument strongly motivates nominalism, it also undermines (...)
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  • What Are the Odds that Everyone is Depraved?Scott Hill - 2020 - American Philosophical Quarterly 57 (3):299-308.
    Why does God allow evil? One hypothesis is that God desires the existence and activity of free creatures but He was unable to create a world with such creatures and such activity without also allowing evil. If Molinism is true, what probability should be assigned to this hypothesis? Some philosophers claim that a low probability should be assigned because there are an infinite number of possible people and because we have no reason to suppose that such creatures will choose one (...)
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  • Theism and Realism: A Match Made in Heaven?Simon Thomas Hewitt - 2018 - European Journal for Philosophy of Religion 10 (4):27-53.
    There is no interesting entailment either way between theism and various forms of realism. Taking its cue from Dummett’s characterisation of realism and his discussion of it with respect to theistic belief, this paper argues both that theism does not follow from realism, and that God cannot be appealed to in order to secure bivalence for an otherwise indeterminate subject matter. In both cases, significant appeal is made to the position that God is not a language user, which in turn (...)
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  • Tuples all the Way Down?Simon Thomas Hewitt - 2018 - Thought: A Journal of Philosophy 7 (3):161-169.
    We can introduce singular terms for ordered pairs by means of an abstraction principle. Doing so proves useful for a number of projects in the philosophy of mathematics. However there is a question whether we can appeal to the abstraction principle in good faith, since a version of the Caesar Problem can be generated, posing the worry that abstraction fails to introduce expressions which refer determinately to the requisite sort of object. In this note I will pose the difficulty, and (...)
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  • Platonism and metaphor in the texts of mathematics: Gödel and Frege on mathematical knowledge.Clevis Headley - 1997 - Man and World 30 (4):453-481.
    In this paper, I challenge those interpretations of Frege that reinforce the view that his talk of grasping thoughts about abstract objects is consistent with Russell's notion of acquaintance with universals and with Gödel's contention that we possess a faculty of mathematical perception capable of perceiving the objects of set theory. Here I argue the case that Frege is not an epistemological Platonist in the sense in which Gödel is one. The contention advanced is that Gödel bases his Platonism on (...)
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  • Immoral realism.Max Khan Hayward - 2019 - Philosophical Studies 176 (4):897-914.
    Non-naturalist realists are committed to the belief, famously voiced by Parfit, that if there are no non-natural facts then nothing matters. But it is morally objectionable to conditionalise all our moral commitments on the question of whether there are non-natural facts. Non-natural facts are causally inefficacious, and so make no difference to the world of our experience. And to be a realist about such facts is to hold that they are mind-independent. It is compatible with our experiences that there are (...)
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  • Perception, Intuition, and Reliability.Kai Hauser & Tahsİn Öner - 2018 - Theoria 84 (1):23-59.
    The question of how we can know anything about ideal entities to which we do not have access through our senses has been a major concern in the philosophical tradition since Plato's Phaedo. This article focuses on the paradigmatic case of mathematical knowledge. Following a suggestion by Gödel, we employ concepts and ideas from Husserlian phenomenology to argue that mathematical objects – and ideal entities in general – are recognized in a process very closely related to ordinary perception. Our analysis (...)
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  • Objectivity over objects: A case study in theory formation.Kai Hauser - 2001 - Synthese 128 (3):245 - 285.
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  • Is Cantor's continuum problem inherently vague?Kai Hauser - 2002 - Philosophia Mathematica 10 (3):257-285.
    I examine various claims to the effect that Cantor's Continuum Hypothesis and other problems of higher set theory are ill-posed questions. The analysis takes into account the viability of the underlying philosophical views and recent mathematical developments.
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  • Gödel's program revisited part I: The turn to phenomenology.Kai Hauser - 2006 - Bulletin of Symbolic Logic 12 (4):529-590.
    Convinced that the classically undecidable problems of mathematics possess determinate truth values, Gödel issued a programmatic call to search for new axioms for their solution. The platonism underlying his belief in the determinateness of those questions in combination with his conception of intuition as a kind of perception have struck many of his readers as highly problematic. Following Gödel's own suggestion, this article explores ideas from phenomenology to specify a meaning for his mathematical realism that allows for a defensible epistemology.
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  • On rules and practice.Jagdish Hattiangadi - 2002 - Social Epistemology 16 (4):311 – 347.
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  • The music of modality.W. D. Hart - 2003 - Topoi 22 (2):135-142.
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  • Modal scepticism, Yablo-style conceivability, and analogical reasoning.Peter Hartl - 2016 - Synthese 193 (1):269-291.
    This paper offers a detailed criticism of different versions of modal scepticism proposed by Van Inwagen and Hawke, and, against these views, attempts to vindicate our reliance on thought experiments in philosophy. More than one different meaning of “ modal scepticism” will be distinguished. Focusing mainly on Hawke’s more detailed view I argue that none of these versions of modal scepticism is compelling, since sceptical conclusions depend on an untenable and, perhaps, incoherent modal epistemology. With a detailed account of modal (...)
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