Roger Schlafly over at Dark Buzz (and author of How Einstein Ruined Physics) has written an essay called Nature has no faithful mathematical representation for the FQXi Essay Contest 2012.
It is in my view a fascinating, must-read essay where Roger is saying that Nature cannot be represented faithfully by mathematics. It surely doesn't take much thought to see how important this issue is to modern physics and our understanding of the universe.
Mathematics dominates theoretical physics, and underlies the deepest realities of nature. It is “unreasonably effective” in the words of E. Wigner. It is widely believed that the most fundamental objects of physics will be perfectly describable by mathematical structures. The structures might be variants of quantum field theory, or string theory, or supergravity, or some other unified field theory, but they will be given precisely by mathematical constants, formulas, equations, and other structures. I believe that this is a profound mistake.
Concepts of physics like mass, electricity, gravity, and electrons can be represented by mathematical structures. That is what the formulas in physics books are all about. A representation is faithful if it perfectly characterizes the physics. In particular, it must allow calculations that predict physical outcomes to as many decimal places as desired.A faithful representation of the elementary particles (quarks, leptons, and bosons) is the holy grail of theoretical physics. I believe that there is no such thing, and that it is foolish to look for one.