Mathematicians worry about justifying such approximations and spend a lot of effort coping with paranoid delusions, e.g., in section 4.2 that a sequence of numbers all of which lie between 1 and 2 might not converge.
Mathematicians cherish the rare moments where physicists’ leaps of faith get them into trouble.... While it is fun to point out physicists’ errors, it is much more satisfying when we discover something that they don’t know.... Despite remarks in the last few paragraph, our goal is not to lift ourselves up by putting other people down. As Mark Newman said in an email to me “I think there’s room in the world for people who have good ideas but don’t have the rigor to pursue them properly – makes more for mathematicians to do.
The first chapter is a nice overview of the theory of random graphs. The rest of the book has been rather interesting to flip through as well. Durrett doesn't shy away from using the sort of heuristics that physicists use in order to help give intuition, which I think is tremendously useful when thinking about probability. When one is doing "rigorous" probability it's too easy to get lost in a maze of calculations. (This is true for any sort of rigorous mathematics, but it's especially a shame in probability precisely because the correct intuitions are pretty close to the surface.)