TITLE: Applications of nonconstructive mathematics
QUESTION [17 upvotes]: What are good/interesting examples of theorems than can be proven classically, but not constructively, and have applications in e.g. physics?

REPLY [2 votes]: A famous Martin Gardner mathematical games column had a number of April Fool items, one of which was about a guy getting rich by duplicating gold spheres using the Banach-Tarski theorem.  I guess that doesn't count as a real application, but it's surely nonconstructive and I can't resist mentioning it.