About this topic
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. The Epistemology of Abstract Objects: Access and Inference.David Bell & W. D. Hart - 1979 - Proceedings of the Aristotelian Society 53:153-165.
  2. Malament–Hogarth Machines and Tait's Axiomatic Conception of Mathematics.Sharon Berry - 2014 - 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.
  3. Justified Concepts and the Limits of the Conceptual Approach to the A Priori.Darren Bradley - 2011 - 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 (...)
  4. C. S. Jenkins, Grounding Concepts: An Empirical Basis for Arithmetical Knowledge Reviewed By.Manuel Bremer - 2010 - Philosophy in Review 30 (3):205-207.
  5. Grounding Concepts: An Empirical Basis for Arithmetical Knowledge – C.S. Jenkins.James Robert Brown & James Davies - 2011 - Philosophical Quarterly 61 (242):208-211.
  6. Computer Proof, Apriori Knowledge, and Other Minds.Tyler Burge - 1998 - Noûs 32 (S12):1-37.
  7. Bolzano Versus Kant: Mathematics as a Scientia Universalis.Paola Cantù - 2011 - 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 (...)
  8. Grassmann’s Epistemology: Multiplication and Constructivism.Paola Cantù - 2010 - 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 (...)
  9. "7 + 5 = 12" as a Synthetic Proposition.Hector Neri Castaneda - 1960 - Philosophy and Phenomenological Research 21 (2):141-158.
  10. Grounding Concepts, by C. S. Jenkins.A. Casullo - 2010 - Mind 119 (475):805-810.
    (No abstract is available for this citation).
  11. What is the Benacerraf Problem?Justin Clarke-Doane - 2017 - 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 (...)
  12. Kant's Views on Non-Euclidean Geometry.Michael E. Cuffaro - 2012 - 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 (...)
  13. Carrie Jenkins, Grounding Concepts. An Empirical Basis for Arithmetical Knowledge. Oxford, Oxford University Press, 2008.Lorenz Demey - 2009 - Tijdschrift Voor Filosofie 71 (4):796-798.
  14. Proof and Knowledge in Mathematics.Michael Detlefsen (ed.) - 1992 - Routledge.
    These questions arise from any attempt to discover an epistemology for mathematics. This collection of essays considers various questions concerning the nature of justification in mathematics and possible sources of that justification. Among these are the question of whether mathematical justification is _a priori_ or _a posteriori_ in character, whether logical and mathematical differ, and if formalization plays a significant role in mathematical justification.
  15. The Four-Color Theorem and Mathematical Proof.Michael Detlefsen & Mark Luker - 1980 - 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 (...)
  16. A Framework for Implicit Definitions and the A Priori.Philip A. Ebert - 2016 - In Philip A. Ebert & Marcus Rossberg (eds.), Abstractionism. Oxford: Oxford University Press. pp. 133--160.
  17. A Puzzle About Ontological Commitments.Philip A. Ebert - 2008 - 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 (...)
  18. Introduction to Abstractionism.Philip A. Ebert & Marcus Rossberg - 2016 - In Philip A. Ebert & Marcus Rossberg (eds.), Abstractionism. Oxford: Oxford University Press. pp. 3-33.
  19. Recent Debates About the A Priori.Hartry Field - 2005 - In Tamar Szabo Gendler & John Hawthorne (eds.), Oxford Studies in Epistemology Volume 1. Oxford University Press.
  20. Grounding Concepts: The Problem of Composition.Gábor Forrai - 2011 - 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 (...)
  21. Modern Geometries and the “Transcendental Aesthetic”.Lawrence Foss - 1967 - Philosophia Mathematica (1-2):35-45.
  22. Synthetic a priori judgments.Alessandro Giordani - 2009 - Rivista di Filosofia Neo-Scolastica 17:297 - 313.
  23. Closure of A Priori Knowability Under A Priori Knowable Material Implication.Jan Heylen - 2015 - 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 (...)
  24. Descriptions and Unknowability.Jan Heylen - 2010 - 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 (...)
  25. Mathematics and Truth.R. J. Hirst - 1953 - Philosophical Quarterly 3 (12):211-224.
  26. Apriority and Applied Mathematics.Robert A. Holland - 1992 - 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 (...)
  27. Grounding Concepts: An Empirical Basis for Arithmetical Knowledge.C. S. Jenkins - 2008 - Oxford University Press.
    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.
  28. The Role of Perception in a Priori Knowledge: Some Remarks. [REVIEW]Jaegwon Kim - 1981 - Philosophical Studies 40 (3):339 - 354.
  29. Scientific Heritage.Zbigniew Król - 2012 - 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.
  30. Benacerraf's Dilemma and Informal Mathematics.Gregory Lavers - 2009 - 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 (...)
  31. Commentary: The Timing of Brain Events.Benjamin Libet - 2006 - Consciusness and Cognition 15:540--547.
  32. A Priori Knowledge Contextualised and Benacerraf's Dilemma.Maja Malec - 2004 - 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 (...)
  33. The Eleatic and the Indispensabilist.Russell Marcus - 2015 - Theoria : An International Journal for Theory, History and Fundations of Science 30 (3):415-429.
    The debate over whether we should believe that mathematical objects exist quickly leads to the question of how to determine what we should believe. Indispensabilists claim that we should believe in the existence of mathematical objects because of their ineliminable roles in scientific theory. Eleatics argue that only objects with causal properties exist. Mark Colyvan’s recent defenses of Quine’s indispensability argument against some contemporary eleatics attempt to provide reasons to favor the indispensabilist’s criterion. I show that Colyvan’s argument is not (...)
  34. A Priori.Edwin Mares - 2011 - Routledge.
    In recent years many influential philosophers have advocated that philosophy is an a priori science. Yet very few epistemology textbooks discuss a priori knowledge at any length, focusing instead on empirical knowledge and empirical justification. As a priori knowledge has moved centre stage, the literature remains either too technical or too out of date to make up a reasonable component of an undergraduate course. Edwin Mares book aims to rectify this. This book seeks to make accessible to students the standard (...)
  35. Russell on Logicism and Coherence.Conor Mayo-Wilson - 2011 - Russell: The Journal of Bertrand Russell Studies 31 (1):89-106.
    According to Quine, Charles Parsons, Mark Steiner, and others, Russell’s logicist project is important because, if successful, it would show that mathematical theorems possess desirable epistemic properties often attributed to logical theorems, such as aprioricity, necessity, and certainty. Unfortunately, Russell never attributed such importance to logicism, and such a thesis contradicts Russell’s explicitly stated views on the relationship between logic and mathematics. This raises the question: what did Russell understand to be the philosophical importance of logicism? Building on recent work (...)
  36. Does The Necessity of Mathematical Truths Imply Their Apriority?Mark McEvoy - 2013 - 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 (...)
  37. CS Jenkins, Grounding Concepts: An Empirical Basis for Arithmetical Knowledge.Nenad Miščević - 2009 - Croatian Journal of Philosophy 9 (2):252-257.
  38. Review of Space, Time, and Number in the Brain. [REVIEW]Carlos Montemayor & Rasmus Grønfeldt Winther - 2015 - 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). (...)
  39. 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 (...)
  40. Against Mathematical Convenientism.Seungbae Park - 2016 - 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 (...)
  41. Friedman on Implicit Definition: In Search of the Hilbertian Heritage in Philosophy of Science.Woosuk Park - 2012 - 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.
  42. The Paradox of Infinite Given Magnitude: Why Kantian Epistemology Needs Metaphysical Space.Lydia Patton - 2011 - 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 (...)
  43. Review of Discourse on a New Method: Reinvigorating the Marriage of History and Philosophy of Science. [REVIEW]Lydia Patton - 2011 - 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.
  44. Proof, Reliability, and Mathematical Knowledge.Anthony F. Peressini - 2003 - Theoria 69 (3):211-232.
    With respect to the confirmation of mathematical propositions, proof possesses an epistemological authority unmatched by other means of confirmation. This paper is an investigation into why this is the case. I make use of an analysis drawn from an early reliability perspective on knowledge to help make sense of mathematical proofs singular epistemological status.
  45. Kitcher, Mathematics, and Apriority.Jeffrey W. Roland - forthcoming - Erkenntnis:1-16.
    Philip Kitcher has argued against the apriority of mathematical knowledge in a number of places. His arguments rely on a conception of mathematical knowledge as embedded in a historical tradition and the claim that this sort of embedding compromises apriority. In this paper, I argue that tradition dependence of mathematical knowledge does not compromise its apriority. I further identify the factors which appear to lead Kitcher to argue as he does.
  46. The Four-Color Theorem Solved, Again: Extending the Extended Mind to Philosophy of Mathematics.Casey Rufener - 2011 - 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 (...)
  47. Review of Grounding Concepts by C. S. Jenkins. [REVIEW]Joshua Schechter - 2010 - Notre Dame Philosophical Reviews 2010 (5).
  48. Knowledge of Abstract Objects in Physics and Mathematics.Michael J. Shaffer - 2017 - Acta Analytica 32 (4):397-409.
    In this paper a parallel is drawn between the problem of epistemic access to abstract objects in mathematics and the problem of epistemic access to idealized systems in the physical sciences. On this basis it is argued that some recent and more traditional approaches to solving these problems are problematic.
  49. Lakatos’ Quasi-Empiricism in the Philosophy of Mathematics.Michael J. Shaffer - 2015 - Polish Journal of Philosophy 9 (2):71-80.
    Imre Lakatos' views on the philosophy of mathematics are important and they have often been underappreciated. The most obvious lacuna in this respect is the lack of detailed discussion and analysis of his 1976a paper and its implications for the methodology of mathematics, particularly its implications with respect to argumentation and the matter of how truths are established in mathematics. The most important themes that run through his work on the philosophy of mathematics and which culminate in the 1976a paper (...)
  50. Ontology, Understanding, and the a Priori.Ernest Sosa - 2003 - 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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