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.
  Show all references
Related categories
Siblings:
36 found
Search inside:
(import / add options)   Sort by:
  1. David Bell & W. D. Hart (1979). The Epistemology of Abstract Objects: Access and Inference. Proceedings of the Aristotelian Society 53:153-165.
  2. Tyler Burge (1998). Computer Proof, A Priori Knowledge, and Other Minds. Philosophical Perspectives 12 (S12):1-37.
  3. 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 (...)
    Remove from this list | Direct download (6 more)  
     
    My bibliography  
     
    Export citation  
  4. 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 (...)
    Remove from this list |
    Translate to English
    | Direct download (2 more)  
     
    My bibliography  
     
    Export citation  
  5. Hector Neri Castañeda (1960). "7 + 5 = 12" as a Synthetic Proposition. Philosophy and Phenomenological Research 21 (2):141-158.
  6. 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 (...)
    Remove from this list |
    Translate to English
    | Direct download (2 more)  
     
    My bibliography  
     
    Export citation  
  7. 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 (...)
    Remove from this list | Direct download (4 more)  
     
    My bibliography  
     
    Export citation  
  8. 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 (...)
    Remove from this list | Direct download (7 more)  
     
    My bibliography  
     
    Export citation  
  9. Hartry Field (2005). Recent Debates About the A Priori. In Tamar Szabo Gendler & John Hawthorne (eds.), Oxford Studies in Epistemology Volume 1. Oup Oxford.
    Remove from this list | Direct download (2 more)  
     
    My bibliography  
     
    Export citation  
  10. 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 (...)
    Remove from this list | Direct download (7 more)  
     
    My bibliography  
     
    Export citation  
  11. Lawrence Foss (1967). Modern Geometries and the “Transcendental Aesthetic”. Philosophia Mathematica (1-2):35-45.
    Remove from this list | Direct download (4 more)  
     
    My bibliography  
     
    Export citation  
  12. Alessandro Giordani (2009). Synthetic a priori judgments. Rivista di Filosofia Neo-Scolastica 17:297 - 313.
    Remove from this list |
    Translate to English
    |
     
    My bibliography  
     
    Export citation  
  13. Jan Heylen (2010). Descriptions and Unknowability. Analysis 70 (1):50-52.
    (No abstract is available for this citation).
    Remove from this list | Direct download (10 more)  
     
    My bibliography  
     
    Export citation  
  14. R. J. Hirst (1953). Mathematics and Truth. Philosophical Quarterly 3 (12):211-224.
    Remove from this list | Direct download (5 more)  
     
    My bibliography  
     
    Export citation  
  15. 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 (...)
    Remove from this list | Direct download (6 more)  
     
    My bibliography  
     
    Export citation  
  16. C. S. Jenkins (2008). Grounding Concepts: An Empirical Basis for Arithmetical Knowledge. OUP Oxford.
    Grounding Concepts tackles the issue of arithmetical knowledge, developing a new position which respects three intuitions which have appeared impossible to satisfy simultaneously: a priorism, mind-independence realism, and empiricism. -/- Drawing on a wide range of philosophical influences, but avoiding unnecessary technicality, a view is developed whereby arithmetic can be known through the examination of empirically grounded concepts. These are concepts which, owing to their relationship to sensory input, are non-accidentally accurate representations of the mind-independent world. Examination of such concepts (...)
    Remove from this list | Direct download  
     
    My bibliography  
     
    Export citation  
  17. Jaegwon Kim (1981). The Role of Perception in a Priori Knowledge: Some Remarks. [REVIEW] Philosophical Studies 40 (3):339 - 354.
  18. 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.
    Remove from this list | Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
  19. 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 (...)
    Remove from this list | Direct download (8 more)  
     
    My bibliography  
     
    Export citation  
  20. 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 (...)
    Remove from this list | Direct download (7 more)  
     
    My bibliography  
     
    Export citation  
  21. 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.
    Remove from this list |
     
    My bibliography  
     
    Export citation  
  22. 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 (...)
    Remove from this list | Direct download (6 more)  
     
    My bibliography  
     
    Export citation  
  23. Carlos Montemayor & Rasmus Grønfeldt Winther (forthcoming). Review of Space, Time, and Number in the Brain. [REVIEW] Mathematical Intelligencer.
    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). (...)
    Remove from this list | Direct download  
     
    My bibliography  
     
    Export citation  
  24. 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.
    Remove from this list | Direct download (8 more)  
     
    My bibliography  
     
    Export citation  
  25. 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.
    Remove from this list |
    Translate to English
    | Direct download  
     
    My bibliography  
     
    Export citation  
  26. 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 (...)
    Remove from this list | Direct download (4 more)  
     
    My bibliography  
     
    Export citation  
  27. Joshua Schechter (2010). Review of Grounding Concepts by C. S. Jenkins. [REVIEW] Notre Dame Philosophical Reviews 2010 (5).
    Remove from this list |
    Translate to English
    | Direct download  
     
    My bibliography  
     
    Export citation  
  28. 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.
    Remove from this list | Direct download (7 more)  
     
    My bibliography  
     
    Export citation  
  29. Ernest Sosa (2002). Reliability and the a Priori. In John Hawthorne & Tamar Gendler (eds.), Conceivability and Possibility. Oxford University Press. 369--384.
  30. Pirmin Stekeler-Weithofer (1987). Sind Die Urteile der Arithmetik Synthetisch a Priori? Journal for General Philosophy of Science 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 (...)
    Remove from this list | Direct download (5 more)  
     
    My bibliography  
     
    Export citation  
  31. 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).
    Remove from this list | Direct download (11 more)  
     
    My bibliography  
     
    Export citation  
  32. Neil Tennant (2010). Review of C. S. Jenkins, Grounding Concepts: An Empirical Basis for Arithmetical Knowledge. [REVIEW] Philosophia Mathematica 18 (3):360-367.
    Remove from this list | Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
  33. 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, (...)
    Remove from this list | Direct download (5 more)  
     
    My bibliography  
     
    Export citation  
  34. 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 (...)
    Remove from this list | Direct download (4 more)  
     
    My bibliography  
     
    Export citation  
  35. 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 (...)
    Remove from this list | Direct download (6 more)  
     
    My bibliography  
     
    Export citation  
  36. 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 (...)
    Remove from this list | Direct download  
     
    My bibliography  
     
    Export citation