About this topic
Summary What philosophers of mathematics usually have in mind when speaking of intuition in mathematics is the epistemological claim that there is a faculty of rational mathematical intuition providing us with (basic) belief-forming methods delivering knowledge of (basic) mathematical truths. Many philosophers of mathematics believe that no one has yet presented a defensible ground-level epistemology endorsing a faculty of rational intuition.
Key works The view that knowledge of basic mathematical truths can be obtained by some form of rational intuition is often ascribed to Kurt Gödel (see Gödel 1964). A sustained and modern defense of such a view can be found in BonJour 1998.
Introductions BonJour 1998 provides a good introduction. For an interpretation of Gödel’s claims, consult Parsons 1995.
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  1. On The Rabbinical Exegesis of an Enhanced Biblical Value of Pi.Edward G. Belaga - 1991 - In Hardi Grant, Israel Kleiner & Abe Shenitzer (eds.), Proc. of the 17th Congress of the Canadian Society of History and Philosophy of Mathematics. Kingston.
    We present here a biblical exegesis of the value of Pi, PI_{Hebrew} = 3.141509 ..., from the well known verse 1 Kings 7:23. This verse is then compared to 2 Chronicles 4:2; the comparison provides independent supporting evidence for the exegesis.
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  2. A Modal Logic and Hyperintensional Semantics for Gödelian Intuition.Timothy Bowen - manuscript
    This essay aims to provide a modal logic for rational intuition. Similarly to treatments of the property of knowledge in epistemic logic, I argue that rational intuition can be codified by a modal operator governed by the modal $\mu$-calculus. Via correspondence results between fixed point modal propositional logic and the bisimulation-invariant fragment of monadic second-order logic, a precise translation can then be provided between the notion of 'intuition-of', i.e., the cognitive phenomenal properties of thoughts, and the modal operators regimenting the (...)
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  3. On what Hilbert aimed at in the foundations.Besim Karakadılar - manuscript
    Hilbert's axiomatic approach was an optimistic take over on the side of the logical foundations. It was also a response to various restrictive views of mathematics supposedly bounded by the reaches of epistemic elements in mathematics. A complete axiomatization should be able to exclude epistemic or ontic elements from mathematical theorizing, according to Hilbert. This exclusion is not necessarily a logicism in similar form to Frege's or Dedekind's projects. That is, intuition can still have a role in mathematical reasoning. Nevertheless, (...)
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  4. The Eleatic and the Indispensabilist.Russell Marcus - manuscript
    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 (...)
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  5. Brouwer's Intuition of Twoity and Constructions in Separable Mathematics.Bruno Bentzen - forthcoming - History and Philosophy of Logic:1-21.
    My first aim in this paper is to use time diagrams in the style of Brentano to analyze constructions in Brouwer's separable mathematics more precisely. I argue that constructions must involve not only pairing and projecting as basic operations guaranteed by the intuition of twoity, as sometimes assumed in the literature, but also a recalling operation. My second aim is to argue that Brouwer's views on the intuition of twoity and arithmetic lead to an ontological explosion. Redeveloping the constructions of (...)
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  6. Observation and Intuition.Justin Clarke-Doane & Avner Ash - forthcoming - In Carolin Antos, Neil Barton & Venturi Giorgio (eds.), Palgrave Companion to the Philosophy of Set Theory.
    The motivating question of this paper is: ‘How are our beliefs in the theorems of mathematics justified?’ This is distinguished from the question ‘How are our mathematical beliefs reliably true?’ We examine an influential answer, outlined by Russell, championed by Gödel, and developed by those searching for new axioms to settle undecidables, that our mathematical beliefs are justified by ‘intuitions’, as our scientific beliefs are justified by observations. On this view, axioms are analogous to laws of nature. They are postulated (...)
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  7. What is the Link between Aristotle’s Philosophy of Mind, the Iterative Conception of Set, Gödel’s Incompleteness Theorems and God? About the Pleasure and the Difficulties of Interpreting Kurt Gödel’s Philosophical Remarks.Eva-Maria Engelen - forthcoming - In Gabriella Crocco & Eva-Maria Engelen (eds.), Kurt Gödel: Philosopher-Scientist. Presses Universitaires de Provence.
    It is shown in this article in how far one has to have a clear picture of Gödel’s philosophy and scientific thinking at hand (and also the philosophical positions of other philosophers in the history of Western Philosophy) in order to interpret one single Philosophical Remark by Gödel. As a single remark by Gödel (very often) mirrors his whole philosophical thinking, Gödel’s Philosophical Remarks can be seen as a philosophical monadology. This is so for two reasons mainly: Firstly, because it (...)
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  8. Intuition, Iteration, Induction.Mark van Atten - 2024 - Philosophia Mathematica 32 (1):34-81.
    Brouwer’s view on induction has relatively recently been characterised as one on which it is not only intuitive (as expected) but functional, by van Dalen. He claims that Brouwer’s ‘Ur-intuition’ also yields the recursor. Appealing to Husserl’s phenomenology, I offer an analysis of Brouwer’s view that supports this characterisation and claim, even if assigning the primary role to the iterator instead. Contrasts are drawn to accounts of induction by Poincaré, Heyting, and Kreisel. On the phenomenological side, the analysis provides an (...)
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  9. Recalcitrant Disagreement in Mathematics: An “Endless and Depressing Controversy” in the History of Italian Algebraic Geometry.Silvia De Toffoli & Claudio Fontanari - 2023 - Global Philosophy 33 (38):1-29.
    If there is an area of discourse in which disagreement is virtually absent, it is mathematics. After all, mathematicians justify their claims with deductive proofs: arguments that entail their conclusions. But is mathematics really exceptional in this respect? Looking at the history and practice of mathematics, we soon realize that it is not. First, deductive arguments must start somewhere. How should we choose the starting points (i.e., the axioms)? Second, mathematicians, like the rest of us, are fallible. Their ability to (...)
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  10. Diagrams, Visual Imagination, and Continuity in Peirce's Philosophy of Mathematics.Vitaly Kiryushchenko - 2023 - New York, NY, USA: Springer.
    This book is about the relationship between necessary reasoning and visual experience in Charles S. Peirce’s mathematical philosophy. It presents mathematics as a science that presupposes a special imaginative connection between our responsiveness to reasons and our most fundamental perceptual intuitions about space and time. Central to this view on the nature of mathematics is Peirce’s idea of diagrammatic reasoning. In practicing this kind of reasoning, one treats diagrams not simply as external auxiliary tools, but rather as immediate visualizations of (...)
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  11. Showing, Telling, Understanding.Emmanuel Ordóñez Angulo - 2022 - Mathematical Intelligencer 44 (2):123–129.
    I begin by proposing a notion of mathematical understanding; then I take a brief look at four approaches to the task of teaching infinity to mathematical novices—four approaches, that is, to “popular” philosophy and mathematics—and assess whether they provide such an understanding. But I don’t mean this to be a recommendation for or against any author. Rather, I want to submit the idea that mathematical understanding is an important dimension of popularizing efforts, all of which can be valuable, to be (...)
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  12. Mary Shepherd on the role of proofs in our knowledge of first principles.M. Folescu - 2022 - Noûs 56 (2):473-493.
    This paper examines the role of reason in Shepherd's account of acquiring knowledge of the external world via first principles. Reason is important, but does not have a foundational role. Certain principles enable us to draw the required inferences for acquiring knowledge of the external world. These principles are basic, foundational and, more importantly, self‐evident and thus justified in other ways than by demonstration. Justificatory demonstrations of these principles are neither required, nor possible. By drawing on textual and contextual evidence, (...)
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  13. The Unreasonable Effectiveness of Mathematics: From Hamming to Wigner and Back Again.Arezoo Islami - 2022 - Foundations of Physics 52 (4):1-18.
    In a paper titled, “The Unreasonable Effectiveness of Mathematics”, published 20 years after Wigner’s seminal paper, the mathematician Richard W. Hamming discussed what he took to be Wigner’s problem of Unreasonable Effectiveness and offered some partial explanations for this phenomenon. Whether Hamming succeeds in his explanations as answers to Wigner’s puzzle is addressed by other scholars in recent years I, on the other hand, raise a more fundamental question: does Hamming succeed in raising the same question as Wigner? The answer (...)
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  14. Five dogmas of logic diagrams and how to escape them.Jens Lemanski, Andrea Anna Reichenberger, Theodor Berwe, Alfred Olszok & Claudia Anger - 2022 - Language & Communication 87 (1):258-270.
    In the vein of a renewed interest in diagrammatic reasoning, this paper challenges an opposition between logic diagrams and formal languages that has traditionally been the common view in philosophy of logic and linguistics. We examine, from a philosophical point of view, what we call five dogmas of logic diagrams. These are as follows: (1) diagrams are non-linguistic; (2) diagrams are visual representations; (3) diagrams are iconic, and not symbolic; (4) diagrams are non-linear; (5) diagrams are heterogenous, and not homogenous. (...)
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  15. The Price of Mathematical Scepticism.Paul Blain Levy - 2022 - Philosophia Mathematica 30 (3):283-305.
    This paper argues that, insofar as we doubt the bivalence of the Continuum Hypothesis or the truth of the Axiom of Choice, we should also doubt the consistency of third-order arithmetic, both the classical and intuitionistic versions. -/- Underlying this argument is the following philosophical view. Mathematical belief springs from certain intuitions, each of which can be either accepted or doubted in its entirety, but not half-accepted. Therefore, our beliefs about reality, bivalence, choice and consistency should all be aligned.
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  16. Core knowledge of geometry can develop independently of visual experience.Benedetta Heimler, Tomer Behor, Stanislas Dehaene, Véronique Izard & Amir Amedi - 2021 - Cognition 212 (C):104716.
    Geometrical intuitions spontaneously drive visuo-spatial reasoning in human adults, children and animals. Is their emergence intrinsically linked to visual experience, or does it reflect a core property of cognition shared across sensory modalities? To address this question, we tested the sensitivity of blind-from-birth adults to geometrical-invariants using a haptic deviant-figure detection task. Blind participants spontaneously used many geometric concepts such as parallelism, right angles and geometrical shapes to detect intruders in haptic displays, but experienced difficulties with symmetry and complex spatial (...)
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  17. Gödelian platonism and mathematical intuition.Wesley Wrigley - 2021 - European Journal of Philosophy 30 (2):578-600.
    European Journal of Philosophy, Volume 30, Issue 2, Page 578-600, June 2022.
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  18. In Search of Intuition.Elijah Chudnoff - 2020 - Australasian Journal of Philosophy 98 (3):465-480.
    What are intuitions? Stereotypical examples may suggest that they are the results of common intellectual reflexes. But some intuitions defy the stereotype: there are hard-won intuitions that take d...
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  19. Ptolemy’s Philosophy: Mathematics as a Way of Life. By Jacqueline Feke. Princeton: Princeton University Press, 2018. Pp. xi + 234. [REVIEW]Nicholas Danne - 2020 - Metaphilosophy 51 (1):151-155.
  20. Reconciling Rigor and Intuition.Silvia De Toffoli - 2020 - Erkenntnis 86 (6):1783-1802.
    Criteria of acceptability for mathematical proofs are field-dependent. In topology, though not in most other domains, it is sometimes acceptable to appeal to visual intuition to support inferential steps. In previous work :829–842, 2014; Lolli, Panza, Venturi From logic to practice, Springer, Berlin, 2015; Larvor Mathematical cultures, Springer, Berlin, 2016) my co-author and I aimed at spelling out how topological proofs work on their own terms, without appealing to formal proofs which might be associated with them. In this article, I (...)
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  21. A Critique of Hintikka’s Reconstruction of Kantian Intuition In Logical and Mathematical Reasoning.Aran Arslan - 2019 - Dissertation, Bogazici University
    This thesis is a critique of Jaakko Hintikka’s reconstruction of Kantian intuition in logical and mathematical reasoning. I argue that Hintikka’s reconstruction of Kantian intuition in particular and his reconstruction of Kant's philosophy of mathematics in general fails to be successful in two ways: First, the logical formula which contains an instantiated term (henceforth, instantial term) that is introduced by the rule of existential instantiation in the ecthesis part of a proof of an argument is not even a proper singular (...)
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  22. The Three Formal Phenomenological Structures: A Means to Assess the Essence of Mathematical Intuition.A. Van-Quynh - 2019 - Journal of Consciousness Studies 26 (5-6):219-241.
    In a recent article I detailed at length the methodology employed to explore the reflective and pre-reflective contents of singular intuitive experiences in contemporary mathematics in order to propose an essential structure of intuition arousal in mathematics. In this paper I present the phenomenological assessment of the essential structure according to the three formal structures as proposed by Sokolowski's scheme and show their relevance in the description of the intuitive experience in mathematics. I also show that this essential structure acknowledges (...)
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  23. Kierkegaard on Variation and Thought Experiment.Eleanor Helms - 2018 - Kierkegaard Studies Yearbook 23 (1):33-54.
  24. The Epistemology of Mathematical Necessity.Catherine Legg - 2018 - In Peter Chapman, Gem Stapleton, Amirouche Moktefi, Sarah Perez-Kriz & Francesco Bellucci (eds.), Diagrammatic Representation and Inference10th International Conference, Diagrams 2018, Edinburgh, UK, June 18-22, 2018, Proceedings. Berlin: Springer-Verlag. pp. 810-813.
    It seems possible to know that a mathematical claim is necessarily true by inspecting a diagrammatic proof. Yet how does this work, given that human perception seems to just (as Hume assumed) ‘show us particular objects in front of us’? I draw on Peirce’s account of perception to answer this question. Peirce considered mathematics as experimental a science as physics. Drawing on an example, I highlight the existence of a primitive constraint or blocking function in our thinking which we might (...)
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  25. Perceiving Necessity.Catherine Legg & James Franklin - 2017 - Pacific Philosophical Quarterly 98 (3).
    In many diagrams one seems to perceive necessity – one sees not only that something is so, but that it must be so. That conflicts with a certain empiricism largely taken for granted in contemporary philosophy, which believes perception is not capable of such feats. The reason for this belief is often thought well-summarized in Hume's maxim: ‘there are no necessary connections between distinct existences’. It is also thought that even if there were such necessities, perception is too passive or (...)
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  26. Philosophy of Mathematics.Øystein Linnebo - 2017 - Princeton, NJ: Princeton University Press.
    Mathematics is one of the most successful human endeavors—a paradigm of precision and objectivity. It is also one of our most puzzling endeavors, as it seems to deliver non-experiential knowledge of a non-physical reality consisting of numbers, sets, and functions. How can the success and objectivity of mathematics be reconciled with its puzzling features, which seem to set it apart from all the usual empirical sciences? This book offers a short but systematic introduction to the philosophy of mathematics. Readers are (...)
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  27. The Psychology and Philosophy of Natural Numbers.Oliver R. Marshall - 2017 - Philosophia Mathematica (1):nkx002.
    ABSTRACT I argue against both neuropsychological and cognitive accounts of our grasp of numbers. I show that despite the points of divergence between these two accounts, they face analogous problems. Both presuppose too much about what they purport to explain to be informative, and also characterize our grasp of numbers in a way that is absurd in the light of what we already know from the point of view of mathematical practice. Then I offer a positive methodological proposal about the (...)
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  28. Universal intuitions of spatial relations in elementary geometry.Ineke J. M. Van der Ham, Yacin Hamami & John Mumma - 2017 - Journal of Cognitive Psychology 29 (3):269-278.
    Spatial relations are central to geometrical thinking. With respect to the classical elementary geometry of Euclid’s Elements, a distinction between co-exact, or qualitative, and exact, or metric, spatial relations has recently been advanced as fundamental. We tested the universality of intuitions of these relations in a group of Senegalese and Dutch participants. Participants performed an odd-one-out task with stimuli that in all but one case display a particular spatial relation between geometric objects. As the exact/co-exact distinction is closely related to (...)
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  29. Philosophy of Mathematics for the Masses : Extending the scope of the philosophy of mathematics.Stefan Buijsman - 2016 - Dissertation, Stockholm University
    One of the important discussions in the philosophy of mathematics, is that centered on Benacerraf’s Dilemma. Benacerraf’s dilemma challenges theorists to provide an epistemology and semantics for mathematics, based on their favourite ontology. This challenge is the point on which all philosophies of mathematics are judged, and clarifying how we might acquire mathematical knowledge is one of the main occupations of philosophers of mathematics. In this thesis I argue that this discussion has overlooked an important part of mathematics, namely mathematics (...)
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  30. A Virtue-Based Defense of Mathematical Apriorism.Noel L. Clemente - 2016 - Axiomathes 26 (1):71-87.
    Mathematical apriorists usually defend their view by contending that axioms are knowable a priori, and that the rules of inference in mathematics preserve this apriority for derived statements—so that by following the proof of a statement, we can trace the apriority being inherited. The empiricist Philip Kitcher attacked this claim by arguing there is no satisfactory theory that explains how mathematical axioms could be known a priori. I propose that in analyzing Ernest Sosa’s model of intuition as an intellectual virtue, (...)
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  31. Numerical cognition and mathematical realism.Helen De Cruz - 2016 - Philosophers' Imprint 16.
    Humans and other animals have an evolved ability to detect discrete magnitudes in their environment. Does this observation support evolutionary debunking arguments against mathematical realism, as has been recently argued by Clarke-Doane, or does it bolster mathematical realism, as authors such as Joyce and Sinnott-Armstrong have assumed? To find out, we need to pay closer attention to the features of evolved numerical cognition. I provide a detailed examination of the functional properties of evolved numerical cognition, and propose that they prima (...)
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  32. Two Weak Points of the Enhanced Indispensability Argument – Domain of the Argument and Definition of Indispensability.Vladimir Drekalović - 2016 - Organon F: Medzinárodný Časopis Pre Analytickú Filozofiu 23 (3):280-298.
    The contemporary Platonists in the philosophy of mathematics argue that mathematical objects exist. One of the arguments by which they support this standpoint is the so-called Enhanced Indispensability Argument (EIA). This paper aims at pointing out the difficulties inherent to the EIA. The first is contained in the vague formulation of the Argument, which is the reason why not even an approximate scope of the set objects whose existence is stated by the Argument can be established. The second problem is (...)
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  33. Parsons and I: Sympathies and Differences.Solomon Feferman - 2016 - Journal of Philosophy 113 (5/6):234-246.
    In the first part of this article, Feferman outlines his ‘conceptual structuralism’ and emphasizes broad similarities between Parsons’s and his own structuralist perspective on mathematics. However, Feferman also notices differences and makes two critical claims about any structuralism that focuses on the “ur-structures” of natural and real numbers: it does not account for the manifold use of other important structures in modern mathematics and, correspondingly, it does not explain the ubiquity of “individual [natural or real] numbers” in that use. In (...)
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  34. Ortega y Gasset on Georg Cantor’s Theory of Transfinite Numbers.Lior Rabi - 2016 - Kairos (15):46-70.
    Ortega y Gasset is known for his philosophy of life and his effort to propose an alternative to both realism and idealism. The goal of this article is to focus on an unfamiliar aspect of his thought. The focus will be given to Ortega’s interpretation of the advancements in modern mathematics in general and Cantor’s theory of transfinite numbers in particular. The main argument is that Ortega acknowledged the historical importance of the Cantor’s Set Theory, analyzed it and articulated a (...)
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  35. Review of The Art of the Infinite by R. Kaplan, E. Kaplan 324p(2003).Michael Starks - 2016 - In Suicidal Utopian Delusions in the 21st Century: Philosophy, Human Nature and the Collapse of Civilization-- Articles and Reviews 2006-2017 2nd Edition Feb 2018. Michael Starks. pp. 619.
    This book tries to present math to the millions and does a pretty good job. It is simple and sometimes witty but often the literary allusions intrude and the text bogs down in pages of relentless math--lovely if you like it and horrid if you don´t. If you already know alot of math you will still probably find the discussions of general math, geometry, projective geometry, and infinite series to be a nice refresher. If you don´t know any and don´t (...)
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  36. Why Did Weyl Think that Dedekind’s Norm of Belief in Mathematics is Perverse?Iulian D. Toader - 2016 - In William Demopoulos (ed.), The Western Ontario Series in Philosophy of Science. Springer. pp. 445-451.
    This paper argues that Weyl's criticism of Dedekind’s principle that "In science, what is provable ought not to be believed without proof." challenges not only a logicist norm of belief in mathematics, but also a realist view about whether there is a fact of the matter as to what norms of belief are correct.
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  37. Anthony Robert Booth and Darrell P. Rowbottom, eds., Intuitions. Reviewed by. [REVIEW]Eran Asoulin - 2015 - Philosophy in Review 35 (5):238-240.
  38. On the Concept of Finitism.Luca Incurvati - 2015 - Synthese 192 (8):2413-2436.
    At the most general level, the concept of finitism is typically characterized by saying that finitistic mathematics is that part of mathematics which does not appeal to completed infinite totalities and is endowed with some epistemological property that makes it secure or privileged. This paper argues that this characterization can in fact be sharpened in various ways, giving rise to different conceptions of finitism. The paper investigates these conceptions and shows that they sanction different portions of mathematics as finitistic.
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  39. The Eleatic and the Indispensabilist.Russell Marcus - 2015 - Theoria 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 (...)
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  40. Autonomy Platonism and the Indispensability Argument.Russell Marcus - 2015 - Lanham: Lexington Books.
    This book includes detailed critical analysis of a wide variety of versions of the indispensability argument, as well as a novel approach to traditional views about mathematics.
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  41. Arithmetic, Mathematical Intuition, and Evidence.Richard Tieszen - 2015 - Inquiry: An Interdisciplinary Journal of Philosophy 58 (1):28-56.
    This paper provides examples in arithmetic of the account of rational intuition and evidence developed in my book After Gödel: Platonism and Rationalism in Mathematics and Logic . The paper supplements the book but can be read independently of it. It starts with some simple examples of problem-solving in arithmetic practice and proceeds to general phenomenological conditions that make such problem-solving possible. In proceeding from elementary ‘authentic’ parts of arithmetic to axiomatic formal arithmetic, the paper exhibits some elements of the (...)
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  42. Necker’s smile: Immediate affective consequences of early perceptual processes.Sascha Topolinski, Thorsten M. Erle & Rolf Reber - 2015 - Cognition 140 (C):1-13.
    Current theories assume that perception and affect are separate realms of the mind. In contrast, we argue that affect is a genuine online-component of perception instantaneously mirroring the success of different perceptual stages. Consequently, we predicted that the success (failure) of even very early and cognitively encapsulated basic visual Processing steps would trigger immediate positive (negative) affective responses. To test this assumption, simple visual stimuli that either allowed or obstructed early visual processing stages without participants being aware of this were (...)
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  43. Richard Tieszen, After Gödel. Platonism and Rationalism in Mathematics and Logic.: Oxford University Press, Oxford, 2011. [REVIEW]Stefania Centrone - 2014 - Husserl Studies 30 (2):153-162.
    It is well known that Husserl, together with Plato and Leibniz, counted among Gödel’s favorite philosophers and was, in fact, an important source and reference point for the elaboration of Gödel’s own philosophical thought. Among the scholars who emphasized this connection we find, as Richard Tieszen reminds us, Gian-Carlo Rota, George Kreisel, Charles Parsons, Heinz Pagels and, especially, Hao Wang. Right at the beginning of After Gödel we read: “The logician who conducted and recorded the most extensive philosophical discussions with (...)
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  44. Intuition in Mathematics.Elijah Chudnoff - 2014 - In Barbara Held & Lisa Osbeck (eds.), Rational Intuition. Cambridge University Press.
    The literature on mathematics suggests that intuition plays a role in it as a ground of belief. This article explores the nature of intuition as it occurs in mathematical thinking. Section 1 suggests that intuitions should be understood by analogy with perceptions. Section 2 explains what fleshing out such an analogy requires. Section 3 discusses Kantian ways of fleshing it out. Section 4 discusses Platonist ways of fleshing it out. Section 5 sketches a proposal for resolving the main problem facing (...)
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  45. Intuição e Conceito: A Transformação do Pensamento Matemático de Kant a Bolzano.Humberto de Assis Clímaco - 2014 - Dissertation, Universidade Federal de Goiás, Brazil
  46. Forms and Roles of Diagrams in Knot Theory.Silvia De Toffoli & Valeria Giardino - 2014 - Erkenntnis 79 (4):829-842.
    The aim of this article is to explain why knot diagrams are an effective notation in topology. Their cognitive features and epistemic roles will be assessed. First, it will be argued that different interpretations of a figure give rise to different diagrams and as a consequence various levels of representation for knots will be identified. Second, it will be shown that knot diagrams are dynamic by pointing at the moves which are commonly applied to them. For this reason, experts must (...)
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  47. Kurt Gödels mathematische Anschauung und John P. Burgess’ mathematische Intuition.Eva-Maria Engelen - 2014 - XXIII Deutscher Kongress Für Philosophie Münster 2014, Konferenzveröffentlichung.
    John P. Burgess kritisiert Kurt Gödels Begriff der mathematischen oder rationalen Anschauung und erläutert, warum heuristische Intuition dasselbe leistet wie rationale Anschauung, aber ganz ohne ontologisch überflüssige Vorannahmen auskommt. Laut Burgess müsste Gödel einen Unterschied zwischen rationaler Anschauung und so etwas wie mathematischer Ahnung, aufzeigen können, die auf unbewusster Induktion oder Analogie beruht und eine heuristische Funktion bei der Rechtfertigung mathematischer Aussagen einnimmt. Nur, wozu benötigen wir eine solche Annahme? Reicht es nicht, wenn die mathematische Intuition als Heuristik funktioniert? Für (...)
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  48. 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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  49. Intuition et déduction en mathématiques: retour au débat sur la "crise des fondements".Bruno Leclercq - 2014 - Fernelmont: EME.
    A la fin du XVIIIe siècle, Emmanuel Kant pouvait encore voir dans les mathématiques le modèle même des jugements synthétiques a priori, c'est-à-dire dotés d'un contenu intuitif propre quoique non dérivé de l'expérience sensible. Des géométries non-euclidiennes à la théorie des transfinis de Cantor, les mathématiques du XIXe siècle vont cependant faire triompher des systèmes mathématiques résolument déductifs et non plus intuitifs. Sur fond d'interrogations quant à la légitimité de ces développements récents, interrogations renforcées par la découverte de paradoxes, d'âpres (...)
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  50. Hilbert’s Program.Richard Zach - 2014 - In Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy. Stanford, CA: The Metaphysics Research Lab.
    In the early 1920s, the German mathematician David Hilbert (1862–1943) put forward a new proposal for the foundation of classical mathematics which has come to be known as Hilbert's Program. It calls for a formalization of all of mathematics in axiomatic form, together with a proof that this axiomatization of mathematics is consistent. The consistency proof itself was to be carried out using only what Hilbert called “finitary” methods. The special epistemological character of finitary reasoning then yields the required justification (...)
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