V. I. Arnold argues (http://pauli.uni-muenster.de/~munsteg/arnold.html) that math is most effectively learned in conjunction with its physical motivation. Are there good examples of when physics provides insight into math, and are there recommended books that approach mathematics in such a fashion?
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I think one of the most intriguing examples is mirror symmetry: physical intuition has suggested formulae for the number of curves of a fixed genus ona Calabi-Yau 3-fold that mathematicians did not expect