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Summary One does not have to look anywhere in order to come to know that 2+3=5. One merely has to think.  Considerations like this underlie the strong intuition that mathematical truths are apriori. That is, very roughly, that our canonical justification or knowledge of them does not essentially rest on experience. Although this claim may seem intuitive, empiricists such as Quine deny it, holding that there is no fundamental epistemological difference between mathematical and non-mathematical knowledge. The question also arises how apriori knowledge and justification in mathematics is possible at all. What exactly do the belief-forming and warrant-generating processes look like? Finally, the rise of computer proofs generates new interesting questions as to the epistemological status of relevant mathematical propositions.
Key works A classical defense of the thesis that arithmetic is apriori is Frege 1953. A sustained defense of a modern philosophy of mathematics combining Platonism with the claim that mathematics is apriori can be found in Hale 2001. As to the mentioned empiricist picture, a classical paper is Quine 1951. Jenkins 2008 defends the thesis that a version of empiricism can be combined with both mathematical realism and the claim that arithmetic is apriori.
Introductions For more information on the notion of apriority, consult the relevant category on philpapers. For a discussion of different notions of apriority in the philosophy of mathematics, consult e.g. Field 2005. A sustained discussion of the orthodoxy that mathematics is apriori, and the problems it raises in the context of other orthodoxies can be found in Jenkins 2008.
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  1. David Bell & W. D. Hart (1979). The Epistemology of Abstract Objects: Access and Inference. Proceedings of the Aristotelian Society 53:153-165.
  2. Sharon Berry (2014). Malament–Hogarth Machines and Tait's Axiomatic Conception of Mathematics. Erkenntnis 79 (4):893-907.
    In this paper I will argue that Tait’s axiomatic conception of mathematics implies that it is in principle impossible to be justified in believing a mathematical statement without being justified in believing that statement to be provable. I will then show that there are possible courses of experience which would justify acceptance of a mathematical statement without justifying belief that this statement is provable.
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  3. Darren Bradley (2011). Justified Concepts and the Limits of the Conceptual Approach to the A Priori. Croatian Journal of Philosophy 11 (3):267-274.
    Carrie Jenkins (2005, 2008) has developed a theory of the a priori that she claims solves the problem of how justification regarding our concepts can give us justification regarding the world. She claims that concepts themselves can be justified, and that beliefs formed by examining such concepts can be justified a priori. I object that we can have a priori justified beliefs with unjustified concepts if those beliefs have no existential import. I then argue that only beliefs without existential import (...)
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  4. Tyler Burge (1998). Computer Proof, Apriori Knowledge, and Other Minds. Noûs 32 (S12):1-37.
  5. Paola Cantù, Bolzano Versus Kant: Mathematics as a Scientia Universalis. Philosophical Papers Dedicated to Kevin Mulligan.
    The paper discusses some changes in Bolzano's definition of mathematics attested in several quotations from the Beyträge, Wissenschaftslehre and Grössenlehre: is mathematics a theory of forms or a theory of quantities? Several issues that are maintained throughout Bolzano's works are distinguished from others that were accepted in the Beyträge and abandoned in the Grössenlehre. Changes are interpreted as a consequence of the new logical theory of truth introduced in the Wissenschaftslehre, but also as a consequence of the overcome of Kant's (...)
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  6. Paola Cantù (2010). Grassmann’s Epistemology: Multiplication and Constructivism. In Hans-Joachim Petsche (ed.), From Past to Future: Graßmann's Work in Context.
    The paper aims to establish if Grassmann’s notion of an extensive form involved an epistemological change in the understanding of geometry and of mathematical knowledge. Firstly, it will examine if an ontological shift in geometry is determined by the vectorial representation of extended magnitudes. Giving up homogeneity, and considering geometry as an application of extension theory, Grassmann developed a different notion of a geometrical object, based on abstract constraints concerning the construction of forms rather than on the homogeneity conditions required (...)
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  7. Hector Neri Castañeda (1960). "7 + 5 = 12" as a Synthetic Proposition. Philosophy and Phenomenological Research 21 (2):141-158.
  8. Justin Clarke-Doane (forthcoming). What is the Benacerraf Problem? In Fabrice Pataut (ed.), New Perspectives on the Philosophy of Paul Benacerraf: Truth, Objects, Infinity.
    In "Mathematical Truth", Paul Benacerraf articulated an epistemological problem for mathematical realism. His formulation of the problem relied on a causal theory of knowledge which is now widely rejected. But it is generally agreed that Benacerraf was onto a genuine problem for mathematical realism nevertheless. Hartry Field describes it as the problem of explaining the reliability of our mathematical beliefs, realistically construed. In this paper, I argue that the Benacerraf Problem cannot be made out. There simply is no intelligible problem (...)
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  9. Michael E. Cuffaro (2012). Kant's Views on Non-Euclidean Geometry. Proceedings of the Canadian Society for History and Philosophy of Mathematics 25:42-54.
    Kant's arguments for the synthetic a priori status of geometry are generally taken to have been refuted by the development of non-Euclidean geometries. Recently, however, some philosophers have argued that, on the contrary, the development of non-Euclidean geometry has confirmed Kant's views, for since a demonstration of the consistency of non-Euclidean geometry depends on a demonstration of its equi-consistency with Euclidean geometry, one need only show that the axioms of Euclidean geometry have 'intuitive content' in order to show that both (...)
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  10. Michael Detlefsen (ed.) (1992). Proof and Knowledge in Mathematics. Routledge.
    Proof and Knowledge in Mathematics tackles the main problem that arises when considering an epistemology for mathematics, the nature and sources of mathematical justification. Focusing both on particular and general issues, these essays from leading philosophers of mathematics raise important issues for our current understanding of mathematics. Is mathematical justification a priori or a posteriori? What role, if any, does logic play in mathematical reasoning or inference? And how epistemologically important is the formalizability of proof? Michael Detlefsen has brought together (...)
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  11. Michael Detlefsen & Mark Luker (1980). The Four-Color Theorem and Mathematical Proof. Journal of Philosophy 77 (12):803-820.
    I criticize a recent paper by Thomas Tymoczko in which he attributes fundamental philosophical significance and novelty to the lately-published computer-assisted proof of the four color theorem (4CT). Using reasoning precisely analogous to that employed by Tymoczko, I argue that much of traditional mathematical proof must be seen as resting on what Tymoczko must take as being "empirical" evidence. The new proof of the 4CT, with its use of what Tymoczko calls "empirical" evidence is therefore not so novel as he (...)
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  12. Philip A. Ebert (2008). A Puzzle About Ontological Commitments. Philosophia Mathematica 16 (2):209-226.
    This paper raises and then discusses a puzzle concerning the ontological commitments of mathematical principles. The main focus here is Hume's Principle—a statement that, embedded in second-order logic, allows for a deduction of the second-order Peano axioms. The puzzle aims to put pressure on so-called epistemic rejectionism, a position that rejects the analytic status of Hume's Principle. The upshot will be to elicit a new and very basic disagreement between epistemic rejectionism and the neo-Fregeans, defenders of the analytic status of (...)
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  13. Hartry Field (2005). Recent Debates About the A Priori. In Tamar Szabo Gendler & John Hawthorne (eds.), Oxford Studies in Epistemology. OUP Oxford
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  14. Gábor Forrai (2011). Grounding Concepts: The Problem of Composition. Philosophia 39 (4):721-731.
    In a recent book C.S. Jenkins proposes a theory of arithmetical knowledge which reconciles realism about arithmetic with the a priori character of our knowledge of it. Her basic idea is that arithmetical concepts are grounded in experience and it is through experience that they are connected to reality. I argue that the account fails because Jenkins’s central concept, the concept for grounding, is inadequate. Grounding as she defines it does not suffice for realism, and by revising the definition we (...)
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  15. Lawrence Foss (1967). Modern Geometries and the “Transcendental Aesthetic”. Philosophia Mathematica (1-2):35-45.
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  16. Alessandro Giordani (2009). Synthetic a priori judgments. Rivista di Filosofia Neo-Scolastica 17:297 - 313.
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  17. Jan Heylen (2015). Closure of A Priori Knowability Under A Priori Knowable Material Implication. Erkenntnis 80 (2):359-380.
    The topic of this article is the closure of a priori knowability under a priori knowable material implication: if a material conditional is a priori knowable and if the antecedent is a priori knowable, then the consequent is a priori knowable as well. This principle is arguably correct under certain conditions, but there is at least one counterexample when completely unrestricted. To deal with this, Anderson proposes to restrict the closure principle to necessary truths and Horsten suggests to restrict it (...)
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  18. Jan Heylen (2010). Descriptions and Unknowability. Analysis 70 (1):50-52.
    In a recent paper Horsten embarked on a journey along the limits of the domain of the unknowable. Rather than knowability simpliciter, he considered a priori knowability, and by the latter he meant absolute provability, i.e. provability that is not relativized to a formal system. He presented an argument for the conclusion that it is not absolutely provable that there is a natural number of which it is true but absolutely unprovable that it has a certain property. The argument depends (...)
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  19. R. J. Hirst (1953). Mathematics and Truth. Philosophical Quarterly 3 (12):211-224.
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  20. Robert A. Holland (1992). Apriority and Applied Mathematics. Synthese 92 (3):349 - 370.
    I argue that we need not accept Quine's holistic conception of mathematics and empirical science. Specifically, I argue that we should reject Quine's holism for two reasons. One, his argument for this position fails to appreciate that the revision of the mathematics employed in scientific theories is often related to an expansion of the possibilities of describing the empirical world, and that this reveals that mathematics serves as a kind of rational framework for empirical theorizing. Two, this holistic conception does (...)
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  21. C. S. Jenkins (2008). Grounding Concepts: An Empirical Basis for Arithmetic Knowledge. Oxford University Press Uk.
    Grounding Concepts is a book about arithmetical knowledge, which puts forward a new and inventive theory. The book aims to do justice to three intuitions which have hitherto been considered impossible to accommodate simultaneously. The first is that arithmetic is knowable without the need to conduct tests or otherwise gather empirical evidence for it. The second is that arithmetic is a body of mind-independent fact; it is not created by us. The third is that our senses supply the basis for (...)
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  22. C. S. Jenkins (2008). Grounding Concepts: An Empirical Basis for Arithmetical Knowledge. OUP Oxford.
    Carrie Jenkins presents a new account of arithmetical knowledge, which manages to respect three key intuitions: a priorism, mind-independence realism, and empiricism. Jenkins argues that arithmetic can be known through the examination of empirically grounded concepts, non-accidentally accurate representations of the mind-independent world.
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  23. Jaegwon Kim (1981). The Role of Perception in a Priori Knowledge: Some Remarks. [REVIEW] Philosophical Studies 40 (3):339 - 354.
  24. Zbigniew Król (2012). Scientific Heritage. Dialogue and Universalism 22 (4):41-65.
    This paper presents sources pertinent to the transmission of Euclid’s Elements in Western medieval civilization. Some important observations follow from the pure description of the sources concerning the development of mathematics, e.g., the text of the Elements was supplemented with new axioms, proofs and theorems as if an “a priori skeleton” lost in Dark Ages was reconstructed and rediscovered during the late Middle Ages. Such historical facts indicate the aprioricity of mathematics.
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  25. Gregory Lavers (2009). Benacerraf's Dilemma and Informal Mathematics. Review of Symbolic Logic 2 (4):769-785.
    This paper puts forward and defends an account of mathematical truth, and in particular an account of the truth of mathematical axioms. The proposal attempts to be completely nonrevisionist. In this connection, it seeks to satisfy simultaneously both horns of Benacerrafs work on informal rigour. Kreisel defends the view that axioms are arrived at by a rigorous examination of our informal notions, as opposed to being stipulated or arrived at by trial and error. This view is then supplemented by a (...)
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  26. Benjamin Libet (2006). Commentary: The Timing of Brain Events. Consciusness and Cognition 15:540--547.
  27. Maja Malec (2004). A Priori Knowledge Contextualised and Benacerraf's Dilemma. Acta Analytica 19 (33):31-44.
    In this article, I discuss Hawthorne'€™s contextualist solution to Benacerraf'€™s dilemma. He wants to find a satisfactory epistemology to go with realist ontology, namely with causally inaccessible mathematical and modal entities. I claim that he is unsuccessful. The contextualist theories of knowledge attributions were primarily developed as a response to the skeptical argument based on the deductive closure principle. Hawthorne uses the same strategy in his attempt to solve the epistemologist puzzle facing the proponents of mathematical and modal realism, but (...)
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  28. Edwin David Mares (2011). A Priori. Acumen.
    Edwin Mares seeks to make the standard topics and current debates within a priori knowledge, including necessity and certainty, rationalism, empiricism and analyticity, Quine's attack on the a priori, Kantianism, Aristotelianism, mathematical knowledge, moral knowledge, logical knowledge, and philosophical knowledge, accessible to students.
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  29. Mark McEvoy (2013). Does The Necessity of Mathematical Truths Imply Their Apriority? Pacific Philosophical Quarterly 94 (4):431-445.
    It is sometimes argued that mathematical knowledge must be a priori, since mathematical truths are necessary, and experience tells us only what is true, not what must be true. This argument can be undermined either by showing that experience can yield knowledge of the necessity of some truths, or by arguing that mathematical theorems are contingent. Recent work by Albert Casullo and Timothy Williamson argues (or can be used to argue) the first of these lines; W. V. Quine and Hartry (...)
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  30. Carlos Montemayor & Rasmus Grønfeldt Winther (2015). Review of Space, Time, and Number in the Brain. [REVIEW] Mathematical Intelligencer 37 (2):93-98.
    Albert Einstein once made the following remark about "the world of our sense experiences": "the fact that it is comprehensible is a miracle." (1936, p. 351) A few decades later, another physicist, Eugene Wigner, wondered about the unreasonable effectiveness of mathematics in the natural sciences, concluding his classic article thus: "the miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve" (1960, p. 14). (...)
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  31. Seungbae Park (2016). Against Mathematical Convenientism. Axiomathes 26 (2):115-122.
    Indispensablists argue that when our belief system conflicts with our experiences, we can negate a mathematical belief but we do not because if we do, we would have to make an excessive revision of our belief system. Thus, we retain a mathematical belief not because we have good evidence for it but because it is convenient to do so. I call this view ‘ mathematical convenientism.’ I argue that mathematical convenientism commits the consequential fallacy and that it demolishes the Quine-Putnam (...)
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  32. Woosuk Park (2012). Friedman on Implicit Definition: In Search of the Hilbertian Heritage in Philosophy of Science. Erkenntnis 76 (3):427-442.
    Michael Friedman’s project both historically and systematically testifies to the importance of the relativized a priori. The importance of implicit definitions clearly emerges from Schlick’s General Theory of Knowledge . The main aim of this paper is to show the relationship between both and the relativized a priori through a detailed discussion of Friedman’s work. Succeeding with this will amount to a contribution to recent scholarship showing the importance of Hilbert for Logical Empiricism.
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  33. Lydia Patton (2011). The Paradox of Infinite Given Magnitude: Why Kantian Epistemology Needs Metaphysical Space. Kant-Studien 102 (3):273-289.
    Kant's account of space as an infinite given magnitude in the Critique of Pure Reason is paradoxical, since infinite magnitudes go beyond the limits of possible experience. Michael Friedman's and Charles Parsons's accounts make sense of geometrical construction, but I argue that they do not resolve the paradox. I argue that metaphysical space is based on the ability of the subject to generate distinctly oriented spatial magnitudes of invariant scalar quantity through translation or rotation. The set of determinately oriented, constructed (...)
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  34. Lydia Patton (2011). Review of Discourse on a New Method: Reinvigorating the Marriage of History and Philosophy of Science. [REVIEW] Notre Dame Philosophical Reviews.
    That the history and the philosophy of science have been united in a form of disciplinary marriage is a fact. There are pressing questions about the state of this union. Discourse on a New Method: Reinvigorating the Marriage of History and Philosophy of Science is a state of the union address, but also an articulation of compelling and well-defended positions on strategies for making progress in the history and philosophy of science.
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  35. Casey Rufener (2011). The Four-Color Theorem Solved, Again: Extending the Extended Mind to Philosophy of Mathematics. Res Cogitans 2 (1):215-228.
    In 1977 when Appel, Haken and Koch used a computer to mathematically solve the century old four-color-problem philosopher Thomas Tymoczko thought that the epistemic justification in mathematics had been changed. Essentially, Tymoczko, and others, argue we can now have mathematical epistemic justification through a posteriori means. This has obvious implication in philosophy of mathematics and epistemology because this would be the first case where mathematics isn’t justified through a priori means of investigation. However, I ultimately disagree with Tymoczko. I argue (...)
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  36. Joshua Schechter (2010). Review of Grounding Concepts by C. S. Jenkins. [REVIEW] Notre Dame Philosophical Reviews 2010 (5).
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  37. Michael J. Shaffer (forthcoming). Lakatos’ Quasi-Empiricism in the Philosophy of Mathematics. Polish Journal of Philosophy.
    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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  38. Ernest Sosa (2003). Ontology, Understanding, and the a Priori. Ratio 16 (2):178–188.
    How might one explain the reliability of one's a priori beliefs? What if anything is implied about the ontology of a certain realm of knowledge by the possibility of explaining one's reliability about that realm? Very little, or so it is argued here.
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  39. Ernest Sosa (2002). Reliability and the a Priori. In John Hawthorne & Tamar Gendler (eds.), Conceivability and Possibility. Oxford University Press 369--384.
  40. Ludovic Soutif (2010). Wittgenstein e as teorias semânticas do a priori visual: Observações Filosóficas , cap. XVI & XX. Doispontos 6 (1):13-34.
    Neste artigo, tentamos mostrar que um problema comum aos capítulos XVI e XX das Observações filosóficas é o da aplicabilidade dos conceitos e «proposições» da geometria à realidade física e perceptiva (visual, em particular), e que o modo pelo qual Wittgenstein aborda esse problema nessa obra difere radicalmente, a despeito de aparentes similitudes, daquele que caracteriza as teorias semânticas do a priori visual em termos de estipulações (notadamente, o de Carnap em 1922). O esclarecimento do estatuto dos enunciados sobre os (...)
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  41. Michael Starks, What Do Paraconsistent, Undecidable, Random, Computable and Incomplete Mean? A Review of Godel's Way: Exploits Into an Undecidable World by Gregory Chaitin, Francisco A Doria , Newton C.A. Da Costa 160p (2012).
    In ‘Godel’s Way’ three eminent scientists discuss issues such as undecidability, incompleteness, randomness, computability and paraconsistency. I approach these issues from the Wittgensteinian viewpoint that there are two basic issues which have completely different solutions. There are the scientific or empirical issues, which are facts about the world that need to be investigated observationally and philosophical issues as to how language can be used intelligibly (which include certain questions in mathematics and logic), which need to be decided by looking at (...)
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  42. Pirmin Stekeler-Weithofer (1987). Sind Die Urteile der Arithmetik Synthetisch a Priori? Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 18 (1-2):215-238.
    According to Kant, arithmetic judgements are not analytic since they are about our practice of operating with figures and things in a certain way. Hence the empiricist thesis that any meaningful assertion is either analytic or synthetic a posteriori seems to be refuted (§§ 1, 2). Using syntax and semantics of truth-conditional logic Frege nevertheless shows that arithmetic can be understood as a system of quasi-analytic sentences speaking about numbers as abstract entities (§§ 3, 4). Axiomatic set theory, however, conceals (...)
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  43. N. Tennant (2010). Review of C.S. Jenkins, Grounding Concepts: An Empirical Basis for Arithmetical Knowledge. [REVIEW] Philosophia Mathematica 18 (3):360-367.
    (No abstract is available for this citation).
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  44. Neil Tennant (2010). Review of C. S. Jenkins, Grounding Concepts: An Empirical Basis for Arithmetical Knowledge. [REVIEW] Philosophia Mathematica 18 (3):360-367.
    This book is written so as to be ‘accessible to philosophers without a mathematical background’. The reviewer can assure the reader that this aim is achieved, even if only by focusing throughout on just one example of an arithmetical truth, namely ‘7+5=12’. This example’s familiarity will be reassuring; but its loneliness in this regard will not. Quantified propositions — even propositions of Goldbach type — are below the author’s radar.The author offers ‘a new kind of arithmetical epistemology’, one which ‘respects (...)
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  45. Joshua C. Thurow (2013). The Defeater Version of Benacerraf's Problem for a Priori Knowledge. Synthese 190 (9):1587-1603.
    Paul Benacerraf’s argument that mathematical realism is apparently incompatible with mathematical knowledge has been widely thought to also show that a priori knowledge in general is problematic. Although many philosophers have rejected Benacerraf’s argument because it assumes a causal theory of knowledge, some maintain that Benacerraf nevertheless put his finger on a genuine problem, even though he didn’t state the problem in its most challenging form. After diagnosing what went wrong with Benacerraf’s argument, I argue that a new, more challenging, (...)
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  46. Jonathan Y. Tsou (2003). A Role for Reason in Science. Dialogue 42 (3):573-598.
    In "Dynamics of Reason" (2001), Michael Friedman advocates a neo-Kantian perspective for philosophy of science that addresses the problem of scientific change and opposes both Quine's naturalism and Kuhn's relativism. This critical notice of Friedman's book focuses on the "relativized a priori" principles articulated by Friedman. Friedman's arguments against Quine and Kuhn are subsequently evaluated. It is concluded that Friedman succeeds in illustrating deficiencies of Quine's naturalism, however, he fails to sufficiently establish a "rational" basis for theory-choice and, hence, his (...)
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  47. Jessica M. Wilson (2000). Could Experience Disconfirm the Propositions of Arithmetic? Canadian Journal of Philosophy 30 (1):55--84.
    Alberto Casullo ("Necessity, Certainty, and the A Priori", Canadian Journal of Philosophy 18, 1988) argues that arithmetical propositions could be disconfirmed by appeal to an invented scenario, wherein our standard counting procedures indicate that 2 + 2 != 4. Our best response to such a scenario would be, Casullo suggests, to accept the results of the counting procedures, and give up standard arithmetic. While Casullo's scenario avoids arguments against previous "disconfirming" scenarios, it founders on the assumption, common to scenario and (...)
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  48. Kai-Yee Wong, Computers, Mathematical Proof, and a Priori Knowledge.
    The computer played an essential role in the proof given by Kenneth Appel and Kenneth Henken of the Four-Color Theorem (4CT).1 First proposed in 1852 by Francis Guthrie, the four color problem is to determine whether four colors are sufficient to color any map on a plane so that no adjacent regions have the same color. Appel and Heken’s proof involves a lemma that a certain ‘avoidable’ set U of configurations is reducible. The proof of this critical lemma requires certain (...)
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