John Stillwell
University of San Francisco

"Yearning for the Impossible"

Abstract:
Many of the most important concepts in mathematics were once thought to be impossible; for example, irrational and imaginary numbers, infinitesimals, points at infinity, the fourth dimension, and curved space. Thus it seems that "yearning for the impossible" can be fruitful, but why? Kolmogorov once wrote (in his diary, 14 September, 1943): At a given moment there is only a fine layer between the "trivial" and the impossible. Mathematical discoveries are made in this layer. As a mathematician, I find this view congenial, but perhaps philosophers can make it clearer (or debunk it). To provide food for thought, I will present a survey of the "impossible" in mathematics, with illustrations from exponents of the "impossible" in art, such as Escher and Magritte.

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