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A cornerstone of modern physics is mathematics. Like it or not, without the tools provided to physicists by mathematics, physics would be dead in the water. But (and this is something that all of us forget on occasion) solving equations is not the same as understanding the physics. A critical step in the development of physical insight is to recognize which solutions to an equation might correspond to reality, and which do not.
To give a concrete example, the equations of physics are blind to the direction of time, yet we know that solutions that involve time going backward are usually (but not always) invalid.
Unfortunately, as a recent publication in Physical Review Letters shows, even the brightest and best can get this wrong, and do so repeatedly over the course of years. A team of physicists has shown that light with a negative frequency (thought to be a quirk of the equations) actually, in some sense, exists.
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