Paul M. Thompson
University College, Oxford University OX1 4BH, England
Now at the: Laboratory of Neuro Imaging, Department of Neurology, Division of Brain Mapping, UCLA School of Medicine, Los Angeles, California 90095
Note: This mathematical paper was written many years ago in 1991.
The manuscript was officially accepted for publication in 1993,
and it was finally published in 1998!
[Full paper, 333KB, .html]
Principia Mathematica (below), a milestone in the history of mathematics
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 Godel, 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.
|RESUME| E-MAIL ME| PERSONAL HOMEPAGE| PROJECTS|