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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