Books that integrate physical reasoning with mathematical reasoning? mathematicians? 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?
 A: The book you want is Mark Levi, The Mathematical Mechanic: Using Physical Reasoning to Solve Problems. 
A: A good one is Vladimir Arnold's Mathematical Understanding of Nature.
A: 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." 
A: Peter G. Doyle and J. Laurie Snell, Random Walks and Electric Networks. The blurb says, "Probability theory, like much of mathematics, is indebted to physics as a source of problems and intuition for solving these problems. Unfortunately, the level of abstraction of current mathematics often makes it difficult for anyone but an expert to appreciate this fact. Random Walks and Electric Networks looks at the interplay of physics and mathematics in terms of an example — the relation between elementary electric network theory and random walks — where the mathematics involved is at the college level." 
It appears that the book is freely available online. 
