Mathematics vs. Physics

While my presentation of the AQM class, an interesting question came up. My professor claimed that since Mathematics is based on logic, the sole fact that a theory is mathematicaly correct does not imply that it is physicaly correct. Nature is not logic. For example, one can work on Klein-Gordon theory and get all the mathematics correct, but the numbers are wrong.

This made me think about who would be right, the theorist that belive that nature and mathematics have differences because of their incapacity of accurate measurements; or the experimentalist that claim that nature is not logic and some of it aspect cannot be explained by mathematics.

This, for me, is indeed a deep question