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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.
  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. Emily Carson (1988). The Role of Intuition in Mathematics. Dissertation, McGill University
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  4. Carlo Cellucci (forthcoming). Explanatory and Non-Explanatory Demonstrations. In P.-E. Bour & P. Schroeder-Heister (eds.), Proceedings of the 14th Congress of Logic, Methodology and Philosophy of Science Nancy, July 19-26, 2011. College Publications.
    This paper concerns the question whether there exists an objective distinction between explanatory and non-explanatory demonstrations. It distinguishes between a static and a dynamic approach to explanatory demonstration, it discusses the relevance of this distinction to mathematical practice, and considers the relation of mathematical explanation to mathematical understanding.
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  5. Carlo Cellucci (2005). Mathematical Discourse Vs. Mathematical Intuition. In Carlo Cellucci & Donald Gillies (eds.), Mathematical Reasoning and Heuristics, 137-165. College Publications. 137--165.
    In this paper it is argued that the opposition between the two main methods of mathematics, the axiomatic and the analytic method, is first of all an opposition between intuition and <span class='Hi'>discourse</span>, and, in addition, an opposition between the socalled demonstrative and non-demonstrative reasoning. These two methods, however, are not on a par because the view that the method of mathematics is the axiomatic method is refuted by Goedel's incompleteness results, which on the contrary do not affect the view (...)
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  6. Carlo Cellucci & Donald Gillies (eds.) (2005). Mathematical Reasoning and Heuristics. College Publications.
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  7. Stefania Centrone (forthcoming). Richard Tieszen, After Gödel. Platonism and Rationalism in Mathematics and Logic. [REVIEW] Husserl Studies:1-10.
  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 (forthcoming). 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. 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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  14. 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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  15. 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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  16. 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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  17. 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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  18. 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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  19. 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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  20. 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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  21. 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 (...)
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  22. 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.
  23. René Jagnow, Geometry and Spatial Intuition : A Genetic Approach.
    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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  24. F. Janet (2007). Review of J. Norman, After Euclid: Visual Reasoning and the Epistemology of Diagrams. [REVIEW] Philosophia Mathematica 15 (1):116-121.
  25. 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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  26. 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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  27. 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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  28. 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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  29. Penelope Maddy (1980). Perception and Mathematical Intuition. Philosophical Review 89 (2):163-196.
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  30. 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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  31. 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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  32. 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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  33. 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.
  34. 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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  35. John E. Nolt (1983). Mathematical Intuition. Philosophy and Phenomenological Research 44 (2):189-211.
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  36. James Page (1993). Parsons on Mathematical Intuition. Mind 102 (406):223-232.
    Charles Parsons has argued that we have the ability to apprehend, or "intuit", certain kinds of abstract objects; that among the objects we can intuit are some which form a model for arithmetic; and that our knowledge that the axioms of arithmetic are true in this model involves our intuition of these objects. I find a problem with Parson's claim that we know this model is infinite through intuition. Unless this problem can be resolved. I question whether our knowledge that (...)
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  37. Matthew W. Parker (2013). Set Size and the Part-Whole Principle. Review of Symbolic Logic (4):1-24.
    Recent work has defended “Euclidean” theories of set size, in which Cantor’s Principle (two sets have equally many elements if and only if there is a one-to-one correspondence between them) is abandoned in favor of the Part-Whole Principle (if A is a proper subset of B then A is smaller than B). It has also been suggested that Gödel’s argument for the unique correctness of Cantor’s Principle is inadequate. Here we see from simple examples, not that Euclidean theories of set (...)
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  38. Charles Parsons (2008). Mathematical Thought and its Objects. Cambridge University Press.
    In Mathematical Thought and Its Objects, Charles Parsons examines the notion of object, with the aim to navigate between nominalism, denying that distinctively mathematical objects exist, and forms of Platonism that postulate a transcendent realm of such objects. He introduces the central mathematical notion of structure and defends a version of the structuralist view of mathematical objects, according to which their existence is relative to a structure and they have no more of a “nature” than that confers on them.
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  39. Charles Parsons (1995). Platonism and Mathematical Intuition in Kurt Gödel's Thought. Bulletin of Symbolic Logic 1 (1):44-74.
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  40. Lydia Patton (2011). The Paradox of Infinite Given Magnitude: Why Kantian Epistemology Needs Metaphysical Space. Kant-Studien 102 (3):273-289.
    Kant's account of space as an infinite given magnitude in the Critique of Pure Reason is paradoxical, since infinite magnitudes go beyond the limits of possible experience. Michael Friedman's and Charles Parsons's accounts make sense of geometrical construction, but I argue that they do not resolve the paradox. I argue that metaphysical space is based on the ability of the subject to generate distinctly oriented spatial magnitudes of invariant scalar quantity through translation or rotation. The set of determinately oriented, constructed (...)
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  41. Giuseppe Primiero (2010). Charles Parsons: Mathematical Thought and its Objects. [REVIEW] Minds and Machines 20 (2):311-315.
  42. Lyudmyla Pustelnyk (2009). Intuition. [REVIEW] Teaching Ethics 10 (1):119-122.
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  43. Andrei Rodin (2010). How Mathematical Concepts Get Their Bodies. Topoi 29 (1):53-60.
    When the traditional distinction between a mathematical concept and a mathematical intuition is tested against examples taken from the real history of mathematics one can observe the following interesting phenomena. First, there are multiple examples where concepts and intuitions do not well fit together; some of these examples can be described as “poorly conceptualised intuitions” while some others can be described as “poorly intuited concepts”. Second, the historical development of mathematics involves two kinds of corresponding processes: poorly conceptualised intuitions are (...)
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  44. Michael J. Shaffer (2006). Some Recent Existential Appeals to Mathematical Experience. Principia 10 (2):143-170.
    Some recent work by philosophers of mathematics has been aimed at showing that our knowledge of the existence of at least some mathematical objects and/or sets can be epistemically grounded by appealing to perceptual experience. The sensory capacity that they refer to in doing so is the ability to perceive numbers, mathematical properties and/or sets. The chief defense of this view as it applies to the perception of sets is found in Penelope Maddy’s Realism in Mathematics, but a number of (...)
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  45. Peter Smith (2009). Critical Notice of C. Parsons, Mathematical Thought and its Objects. [REVIEW] Analysis 69 (3):549-557.
    Needless to say, Charles Parsons’s long awaited book1 is a must-read for anyone with an interest in the philosophy of mathematics. But as Parsons himself says, this has been a very long time in the writing. Its chapters extensively “draw on”, “incorporate material from”, “overlap considerably with”, or “are expanded versions of” papers published over the last twenty-five or so years. What we are reading is thus a multi-layered text with different passages added at different times. And this makes for (...)
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  46. Mark Steiner (2000). Mathematical Intuition and Physical Intuition in Wittgenstein's Later Philosophy. Synthese 125 (3):333-340.
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  47. W. W. Tait (2010). Gödel on Intuition and on Hilbert's Finitism. In Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Kurt Gödel: Essays for His Centennial. Association for Symbolic Logic.
    There are some puzzles about G¨ odel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the contrary, G¨ odel’s writings represent a smooth evolution, with just one rather small double-reversal, of his view of finitism. He used the term “finit” (in German) or “finitary” or “finitistic” primarily to (...)
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  48. Richard Tieszen (1984). Mathematical Intuition and Husserl's Phenomenology. Noûs 18 (3):395-421.
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  49. Richard L. Tieszen (2011). After Gödel: Platonism and Rationalism in Mathematics and Logic. Oxford University Press.
    Gödel's relation to the work of Plato, Leibniz, Kant, and Husserl is examined, and a new type of platonic rationalism that requires rational intuition, called ...
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  50. D. van Dalen (1993). Richard L. Tieszen. 'Mathematical Intuition: Phenomenology and Mathematical Knowledge'. [REVIEW] Husserl Studies 10 (3):249-252.