David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Philosophia 26 (3-4):279-319 (1998)
Great intuitions are fundamental to conjecture and discovery in mathematics. In this paper, we investigate the role that intuition plays in mathematical thinking. We review key events in the history of mathematics where paradoxes have emerged from mathematicians' most intuitive concepts and convictions, and where the resulting difficulties led to heated controversies and debates. Examples are drawn from Riemannian geometry, set theory and the analytic theory of the continuum, and include the Continuum Hypothesis, the Tarski-Banach Paradox, and several works by GÃ¶del, Cantor, Wittgenstein and Weierstrass. We examine several fallacies of intuition and determine how far our intuitive conjectures are limited by the nature of our sense-experience, and by our capacities for conceptualization. Finally, I suggest how we can use visual and formal heuristics to cultivate our mathematical intuitions and how the breadth of this new epistemic perspective can be useful in cases where intuition has traditionally been regarded as out of its depth
|Keywords||Philosophy Philosophy Epistemology Ethics Philosophy of Language Philosophy of Mind Philosophy of Science|
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References found in this work BETA
Kurt Gödel (1947). What is Cantor's Continuum Problem? In Solomon Feferman, John Dawson & Stephen Kleene (eds.), Journal of Symbolic Logic. Oxford University Press 176--187.
Philip Kitcher (1983). The Nature of Mathematical Knowledge. Oxford University Press.
Morris Kline (1980). Mathematics the Loss of Certainty. Monograph Collection (Matt - Pseudo).
Penelope Maddy (1988). Review: Saunders Mac Lane, Mathematics: Form and Function. [REVIEW] Journal of Symbolic Logic 53 (2):643-645.
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