As the title says, can anyone help me to find any book that shows how physical reasoning using concepts from classical/quantum mechanics and physics in general can enlighten us about mathematical problems/theorems? Second question,can you list some contemporary mathematician who subscribe to this view ,I know many russian mathematicians do like Vladimir Arnold, any others?
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The book you want is Mark Levi, The Mathematical Mechanic: Using Physical Reasoning to Solve Problems.
I recommend "The Geometrical Language of Continuum Mechanics" by Marcelo Epstein. I still have some way to go to understand everything this book says, but he makes a strong case that differential geometry and the mechanics of a continuum dovetail nearly perfectly with each other.
Epstein states (I quote) "...the presence of continuum mechanics as a materialization of many of the important geometric notions is of invaluable help."