Can we take Mathematics too seriously?

Eugene Wigner famously commented on the "unreasonable effectiveness of mathematics" to paraphrase. Why he said this is pretty obvious. Looking at how precisely you can calculate things, like the energy levels of a hydrogen atom or using pure mathematics to come up with something like an atomic bomb, the deep connection between mathematics and the physical universe becomes all too apparent.

But sometimes I read articles like Big Brain Theory and think some physicists have gone off the deep end. Its all too tempting to observe that mathematics works, and works very well in explaining the physical world and in making predictions. Then jump to the conclusion that mathematics is reality. Many physicists (theoretical ones anyway) almost have a platonists view of the world.

This reminds me of the parable of the shadows in the cave. Plato came up with this idea to illustrate what we think of as reality is really just a shadow. I think this is how many theoretical physicists view the universe. The icky, wet, solid universe of things we experience is the shadow, and the precise and tidy world of mathematics is the real universe.

But maybe its the other way around. Maybe math, beautiful as it is, is nothing more than a tool. The math and theories of physics is the shadow. To make this idea more concrete, Einstein's theory of general relativity isn't reality, its just a tool that makes predictions about observations. This would be more of a positivist point of view. I think if you hold this point of view then the solid every day universe we experience has more reality than some mathematical theory. After all, death or injury is all too real, isn't it?

Honestly I am not sure what side of the argument I fall on. My point of view on this changes from time to time. This is a very deep question about the nature of physical science and there aren't any easy answers. Maybe this reflects the Yin-Yang nature of the universe, and both points of view are true.