The nature and role of intuition in mathematical epistemology

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 Gödel, 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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DOI 10.1007/BF02381494
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References found in this work BETA
Morris Kline (1980). Mathematics the Loss of Certainty. Monograph Collection (Matt - Pseudo).
Penelope Maddy (1988). Believing the Axioms. I. Journal of Symbolic Logic 53 (2):481-511.
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.

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