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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. Andrew Arana (2009). Visual Thinking in Mathematics • by Marcus Giaquinto. Analysis 69 (2):401-403.
    Our visual experience seems to suggest that no continuous curve can cover every point of the unit square, yet in the late 19th century Giuseppe Peano proved that such a curve exists. Examples like this, particularly in analysis received much attention in the 19th century. They helped to instigate what Hans Hahn called a ‘crisis of intuition’, wherein visual reasoning in mathematics came to be thought to be epistemically problematic. Hahn described this ‘crisis’ as follows : " Mathematicians had for (...)
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  2. Laurence BonJour (1998). In Defense of Pure Reason. Cambridge University Press.
    A comprehensive defence of the rationalist view that insight independent of experience is a genuine basis for knowledge.
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  3. John Burgess, Review of Charles Parsons: Mathematical Thought and its Objects. [REVIEW]
    This long-awaited volume is a must-read for anyone with a serious interest in\nphilosophy of mathematics. The book falls into two parts, with the primary focus of\nthe first on ontology and structuralism, and the second on intuition and\nepistemology, though with many links between them. The style throughout involves\nunhurried examination from several points of view of each issue addressed, before\nreaching a guarded conclusion. A wealth of material is set before the reader along\nthe way, but a reviewer wishing to summarize the author’s views (...)
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  4. John P. Burgess (2008). Charles Parsons. Mathematical Thought and its Objects. Philosophia Mathematica 16 (3):402-409.
    This long-awaited volume is a must-read for anyone with a serious interest in philosophy of mathematics. The book falls into two parts, with the primary focus of the first on ontology and structuralism, and the second on intuition and epistemology, though with many links between them. The style throughout involves unhurried examination from several points of view of each issue addressed, before reaching a guarded conclusion. A wealth of material is set before the reader along the way, but a reviewer (...)
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  5. Emily Carson (1988). The Role of Intuition in Mathematics. Dissertation, McGill University
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  6. Carlo Cellucci & Donald Gillies (eds.) (2005). Mathematical Reasoning and Heuristics. College Publications.
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  7. Stefania Centrone (2014). Richard Tieszen, After Gödel. Platonism and Rationalism in Mathematics and Logic. [REVIEW] 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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  8. Colin Cheyne (1997). Getting in Touch with Numbers: Intuition and Mathematical Platonism. Philosophy and Phenomenological Research 57 (1):111-125.
    Mathematics is about numbers, sets, functions, etc. and, according to one prominent view, these are abstract entities lacking causal powers and spatio-temporal location. If this is so, then it is a puzzle how we come to have knowledge of such remote entities. One suggestion is intuition. But `intuition' covers a range of notions. This paper identifies and examines those varieties of intuition which are most likely to play a role in the acquisition of our mathematical knowledge, and argues that none (...)
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  9. Elijah Chudnoff (2014). Intuition in Mathematics. 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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  10. Elijah Chudnoff (2013). Awareness of Abstract Objects. Noûs 47 (4):706-726.
    Awareness is a two-place determinable relation some determinates of which are seeing, hearing, etc. Abstract objects are items such as universals and functions, which contrast with concrete objects such as solids and liquids. It is uncontroversial that we are sometimes aware of concrete objects. In this paper I explore the more controversial topic of awareness of abstract objects. I distinguish two questions. First, the Existence Question: are there any experiences that make their subjects aware of abstract objects? Second, the Grounding (...)
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  11. Elijah Chudnoff (2013). Intuitive Knowledge. Philosophical Studies 162 (2):359-378.
    In this paper I assume that we have some intuitive knowledge—i.e. beliefs that amount to knowledge because they are based on intuitions. The question I take up is this: given that some intuition makes a belief based on it amount to knowledge, in virtue of what does it do so? We can ask a similar question about perception. That is: given that some perception makes a belief based on it amount to knowledge, in virtue of what does it do so? (...)
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  12. Richard Cobb-Stevens (1992). Husserl on Eidetic Intuition and Historical Interpretation. American Catholic Philosophical Quarterly 66 (2):261-275.
  13. Antoine Danchin (1983). Permanence and Change. Substance 12 (40: Determinism):61-71.
    Determinism/indeterminism, permanence/change, global/local — these have been the occasion for disputes that have persisted for ages. Combined in every conceivable fashion, these three pairs have given rise to theories of reality which, though incompatible, nevertheless possess some degree of adequacy. Accounting for the properties of the inorganic world, on invariably confronts several opposing attitudes, each of which questions the pertinence of the continuous/discontinuous pair, which underlies any discussion of the three pairs noted above. Thinkers of Antiquity sought to deal with (...)
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  14. Helen De Cruz (2007). An Enhanced Argument for Innate Elementary Geometric Knowledge and its Philosophical Implications. In Bart Van Kerkhove (ed.), New perspectives on mathematical practices. Essays in philosophy and history of mathematics. World Scientific.
    The idea that formal geometry derives from intuitive notions of space has appeared in many guises, most notably in Kant’s argument from geometry. Kant claimed that an a priori knowledge of spatial relationships both allows and constrains formal geometry: it serves as the actual source of our cognition of principles of geometry and as a basis for its further cultural development. The development of non-Euclidean geometries, however, seemed to definitely undermine the idea that there is some privileged relationship between our (...)
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  15. Helen De Cruz & Johan De Smedt (2010). The Innateness Hypothesis and Mathematical Concepts. Topoi 29 (1):3-13.
    In historical claims for nativism, mathematics is a paradigmatic example of innate knowledge. Claims by contemporary developmental psychologists of elementary mathematical skills in human infants are a legacy of this. However, the connection between these skills and more formal mathematical concepts and methods remains unclear. This paper assesses the current debates surrounding nativism and mathematical knowledge by teasing them apart into two distinct claims. First, in what way does the experimental evidence from infants, nonhuman animals and neuropsychology support the nativist (...)
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  16. Antony Eagle (2008). Mathematics and Conceptual Analysis. Synthese 161 (1):67–88.
    Gödel argued that intuition has an important role to play in mathematical epistemology, and despite the infamy of his own position, this opinion still has much to recommend it. Intuitions and folk platitudes play a central role in philosophical enquiry too, and have recently been elevated to a central position in one project for understanding philosophical methodology: the so-called ‘Canberra Plan’. This philosophical role for intuitions suggests an analogous epistemology for some fundamental parts of mathematics, which casts a number of (...)
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  17. Eva-Maria Engelen (forthcoming). 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. 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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  18. Eva-Maria Engelen, Kurt Gödels mathematische Anschauung und John P. Burgess’ mathematische Intuition. XXIII Deutscher Kongress für Philosophie Münster 2014, Konferenzveröffentlichung.
  19. Solomon Feferman (2000). Mathematical Intuition Vs. Mathematical Monsters. Synthese 125 (3):317-332.
    Geometrical and physical intuition, both untutored andcultivated, is ubiquitous in the research, teaching,and development of mathematics. A number ofmathematical ``monsters'', or pathological objects, havebeen produced which – according to somemathematicians – seriously challenge the reliability ofintuition. We examine several famous geometrical,topological and set-theoretical examples of suchmonsters in order to see to what extent, if at all,intuition is undermined in its everyday roles.
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  20. Janet Folina (2008). Intuition Between the Analytic-Continental Divide: Hermann Weyl's Philosophy of the Continuum. Philosophia Mathematica 16 (1):25-55.
    Though logical positivism is part of Kant's complex legacy, positivists rejected both Kant's theory of intuition and his classification of mathematical knowledge as synthetic a priori. This paper considers some lingering defenses of intuition in mathematics during the early part of the twentieth century, as logical positivism was born. In particular, it focuses on the difficult and changing views of Hermann Weyl about the proper role of intuition in mathematics. I argue that it was not intuition in general, but his (...)
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  21. David Galloway (1999). Seeing Sequences. Philosophy and Phenomenological Research 59 (1):93-112.
    This article discusses Charles Parsons’ conception of mathematical intuition. Intuition, for Parsons, involves seeing-as: in seeing the sequences I I I and I I I as the same type, one intuits the type. The type is abstract, but intuiting the type is supposed to be epistemically analogous to ordinary perception of physical objects. And some non-trivial mathematical knowledge is supposed to be intuitable in this way, again in a way analogous to ordinary perceptual knowledge. In particular, the successor axioms are (...)
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  22. Sergio Galvan (2010). 12 The Emergence of the Intuition of Truth in Mathematical Thought. In Antonella Corradini & Timothy O'Connor (eds.), Emergence in Science and Philosophy. Routledge. 6--233.
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  23. Valeria Giardino (2010). Intuition and Visualization in Mathematical Problem Solving. Topoi 29 (1):29-39.
    In this article, I will discuss the relationship between mathematical intuition and mathematical visualization. I will argue that in order to investigate this relationship, it is necessary to consider mathematical activity as a complex phenomenon, which involves many different cognitive resources. I will focus on two kinds of danger in recurring to visualization and I will show that they are not a good reason to conclude that visualization is not reliable, if we consider its use in mathematical practice. Then, I (...)
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  24. Terry F. Godlove Jr (2009). Poincaré, Kant, and the Scope of Mathematical Intuition. Review of Metaphysics 62 (4):779-801.
    Today it is no news to point out that Kant’s doctrine of space as a form of intuition is motivated by epistemological considerations independent of his commitment to Euclidean geometry. These considerations surface—apparently without his own recognition—in Poincaré’s, Science and Hypothesis, the very work that helped turn analytically-minded philosophers away from the Critique. I argue that we should view Poincaré as refining Kant’s doctrine of space as the form of intuition, even as we see both views as arbitrarily limited—in Kant’s (...)
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  25. Michael Hallett (2006). Gödel, Realism and Mathematical 'Intuition'. In Emily Carson & Renate Huber (eds.), Intuition and the Axiomatic Method. Springer. 113--131.
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  26. Richard Heck (2000). Cardinality, Counting, and Equinumerosity. Notre Dame Journal of Formal Logic 41 (3):187-209.
    Frege, famously, held that there is a close connection between our concept of cardinal number and the notion of one-one correspondence, a connection enshrined in Hume's Principle. Husserl, and later Parsons, objected that there is no such close connection, that our most primitive conception of cardinality arises from our grasp of the practice of counting. Some empirical work on children's development of a concept of number has sometimes been thought to point in the same direction. I argue, however, that Frege (...)
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  27. Jeremy Heis (2011). Ernst Cassirer's Neo-Kantian Philosophy of Geometry. British Journal for the History of Philosophy 19 (4):759 - 794.
    One of the most important philosophical topics in the early twentieth century and a topic that was seminal in the emergence of analytic philosophy was the relationship between Kantian philosophy and modern geometry. This paper discusses how this question was tackled by the Neo-Kantian trained philosopher Ernst Cassirer. Surprisingly, Cassirer does not affirm the theses that contemporary philosophers often associate with Kantian philosophy of mathematics. He does not defend the necessary truth of Euclidean geometry but instead develops a kind of (...)
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  28. Luca Incurvati (forthcoming). On the Concept of Finitism. Synthese:1-24.
    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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  29. Daniel Isaacson (1994). Mathematical Intuition and Objectivity. In Alexander George (ed.), Mathematics and Mind. Oxford University Press. 118--140.
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  30. René Jagnow (2007). Lisa A. Shabel. Mathematics in Kant's Critical Philosophy: Reflections on Mathematical Practice. Studies in Philosophy Outstanding Dissertations, Robert Nozick, Ed. New York & London: Routledge, 2003. ISBN 0-415-93955-0. Pp. 178 (Cloth). [REVIEW] Philosophia Mathematica 15 (3):366-386.
    In this interesting and engaging book, Shabel offers an interpretation of Kant's philosophy of mathematics as expressed in his critical writings. Shabel's analysis is based on the insight that Kant's philosophical standpoint on mathematics cannot be understood without an investigation into his perception of mathematical practice in the seventeenth and eighteenth centuries. She aims to illuminate Kant's theory of the construction of concepts in pure intuition—the basis for his conclusion that mathematical knowledge is synthetic a priori. She does this through (...)
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  31. Rene Jagnow (2003). Geometry and Spatial Intuition: A Genetic Approach. Dissertation, Mcgill University (Canada)
    In this thesis, I investigate the nature of geometric knowledge and its relationship to spatial intuition. My goal is to rehabilitate the Kantian view that Euclid's geometry is a mathematical practice, which is grounded in spatial intuition, yet, nevertheless, yields a type of a priori knowledge about the structure of visual space. I argue for this by showing that Euclid's geometry allows us to derive knowledge from idealized visual objects, i.e., idealized diagrams by means of non-formal logical inferences. By developing (...)
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  32. F. Janet (2007). Review of J. Norman, After Euclid: Visual Reasoning and the Epistemology of Diagrams. [REVIEW] Philosophia Mathematica 15 (1):116-121.
    This monograph treats the important topic of the epistemology of diagrams in Euclidean geometry. Norman argues that diagrams play a genuine justificatory role in traditional Euclidean arguments, and he aims to account for these roles from a modified Kantian perspective. Norman considers himself a semi-Kantian in the following broad sense: he believes that Kant was right that ostensive constructions are necessary in order to follow traditional Euclidean proofs, but he wants to avoid appealing to Kantian a priori intuition as the (...)
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  33. Robin Jeshion (2014). Intuiting the Infinite. 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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  34. Joongol Kim (2006). Concepts and Intuitions in Kant's Philosophy of Geometry. Kant-Studien 97 (2):138-162.
    This paper is an exposition and defense of Kant’s philosophy of geometry. The main thesis is that Euclidean geometry investigates the properties of geometrical objects in an inner space that is given to us a priori (pure space) and hence is a priori and synthetic. This thesis is supported by arguing that Euclidean geometry requires certain intuitive objects (Sect. 1), that these objects are a priori constructions in pure space (Sect. 2), and finally that the role of geometrical construction is (...)
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  35. L. Kvasz (2011). Kant's Philosophy of Geometry--On the Road to a Final Assessment. Philosophia Mathematica 19 (2):139-166.
    The paper attempts to summarize the debate on Kant’s philosophy of geometry and to offer a restricted area of mathematical practice for which Kant’s philosophy would be a reasonable account. Geometrical theories can be characterized using Wittgenstein’s notion of pictorial form . Kant’s philosophy of geometry can be interpreted as a reconstruction of geometry based on one of these forms — the projective form . If this is correct, Kant’s philosophy is a reasonable reconstruction of such theories as projective geometry; (...)
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  36. Catherine Legg & James Franklin (forthcoming). Perceiving Necessity. Pacific Philosophical Quarterly.
    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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  37. Dieter Lohmar (2010). Intuition in Mathematics : On the Function of Eidetic Variation in Mathematical Proofs. In Mirja Hartimo (ed.), Phenomenology and Mathematics. Springer. 73--90.
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  38. Giuseppe Longo (1999). The Mathematical Continuum, From Intuition to Logic. In Jean Petitot, Franscisco J. Varela, Barnard Pacoud & Jean-Michel Roy (eds.), Naturalizing Phenomenology. Stanford University Press.
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  39. Giuseppe Longo & Arnaud Viarouge (2010). Mathematical Intuition and the Cognitive Roots of Mathematical Concepts. Topoi 29 (1):15-27.
    The foundation of Mathematics is both a logico-formal issue and an epistemological one. By the first, we mean the explicitation and analysis of formal proof principles, which, largely a posteriori, ground proof on general deduction rules and schemata. By the second, we mean the investigation of the constitutive genesis of concepts and structures, the aim of this paper. This “genealogy of concepts”, so dear to Riemann, Poincaré and Enriques among others, is necessary both in order to enrich the foundational analysis (...)
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  40. Penelope Maddy (1985). Charles Parsons, Mathematics in Philosophy Reviewed By. Philosophy in Review 5 (3):125-126.
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  41. Penelope Maddy (1985). Charles Parsons, Mathematics in Philosophy. [REVIEW] Philosophy in Review 5:125-126.
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  42. Penelope Maddy (1980). Perception and Mathematical Intuition. Philosophical Review 89 (2):163-196.
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  43. Mark McEvoy (2007). Kitcher, Mathematical Intuition, and Experience. Philosophia Mathematica 15 (2):227-237.
    Mathematical apriorists sometimes hold that our non-derived mathematical beliefs are warranted by mathematical intuition. Against this, Philip Kitcher has argued that if we had the experience of encountering mathematical experts who insisted that an intuition-produced belief was mistaken, this would undermine that belief. Since this would be a case of experience undermining the warrant provided by intuition, such warrant cannot be a priori.I argue that this leaves untouched a conception of intuition as merely an aspect of our ordinary ability to (...)
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  44. Iris Merkač (2013). Parsons' Mathematical Intuition. Croatian Journal of Philosophy 13 (1):99-107.
    The paper offers one of Parsons’ main themes in his book Mathematical Thought and Its Objects of 2008 : the role of intuition in our understanding of arithmetic. Our discussion does not cover all of the issues that have relevance for Parsons’ account of mathematical intuition, but we focus on the question: whether our knowledge that there is a model for arithmetic can reasonably be called intuitive. We focus on this question because we have some concerns about that.
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  45. J. M. B. Moss (1985). The Mathematical Philosophy of Charles Parsons. [REVIEW] British Journal for the Philosophy of Science 36 (4):437-457.
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  46. Felix Mühlhölzer (2010). Mathematical Intuition and Natural Numbers: A Critical Discussion. [REVIEW] Erkenntnis 73 (2):265–292.
    Charles Parsons’ book “Mathematical Thought and Its Objects” of 2008 (Cambridge University Press, New York) is critically discussed by concentrating on one of Parsons’ main themes: the role of intuition in our understanding of arithmetic (“intuition” in the specific sense of Kant and Hilbert). Parsons argues for a version of structuralism which is restricted by the condition that some paradigmatic structure should be presented that makes clear the actual existence of structures of the necessary sort. Parsons’ paradigmatic structure is the (...)
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  47. Felix Muhlholzer (2010). Mathematical Intuition and Natural Numbers: A Critical Discussion: Charles Parsons, Mathematical Thought and Its Objects, Cambridge University Press, New York, 2008, Xx+ 378 Pp. [REVIEW] Erkenntnis 73 (2):265-292.
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  48. Jennifer Wilson Mulnix (2008). Reliabilism, Intuition, and Mathematical Knowledge. Filozofia 62 (8):715-723.
    It is alleged that the causal inertness of abstract objects and the causal conditions of certain naturalized epistemologies precludes the possibility of mathematical know- ledge. This paper rejects this alleged incompatibility, while also maintaining that the objects of mathematical beliefs are abstract objects, by incorporating a naturalistically acceptable account of ‘rational intuition.’ On this view, rational intuition consists in a non-inferential belief-forming process where the entertaining of propositions or certain contemplations results in true beliefs. This view is free of any (...)
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  49. Anne Newstead & Franklin James, The Epistemology of Geometry I: The Problem of Exactness. ASCS09: Proceedings of the 9th Conference of the Australasian Society for Cognitive Science (pp. 254-260). Sydney: Macquarie Centre for Cognitive Science.
    We show how an epistemology informed by cognitive science promises to shed light on an ancient problem in the philosophy of mathematics: the problem of exactness. The problem of exactness arises because geometrical knowledge is thought to concern perfect geometrical forms, whereas the embodiment of such forms in the natural world may be imperfect. There thus arises an apparent mismatch between mathematical concepts and physical reality. We propose that the problem can be solved by emphasizing the ways in which the (...)
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  50. Keith K. Niall (2002). Visual Imagery and Geometric Enthymeme: The Example of Euclid I. Behavioral and Brain Sciences 25 (2):202-203.
    Students of geometry do not prove Euclid's first theorem by examining an accompanying diagram, or by visualizing the construction of a figure. The original proof of Euclid's first theorem is incomplete, and this gap in logic is undetected by visual imagination. While cognition involves truth values, vision does not: the notions of inference and proof are foreign to vision.
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