It is a rather indisputable fact for physicists that mathematics really is the correct language of physics. Without mathematics one could not formulate physical theories and then make prediction to be tested against nature. Indeed, the formulation of physical theories has required the development of new mathematics. Theoretical physics is really the construction of mathematical models to describe nature.
Even the experimentalist cannot avoid mathematics. One has a lot of analysis of results and statistics to preform in order to make sense of the experiments.
It is rather clear then, that without mathematics one will not go very far in physics. Any understanding of nature is going to be rather superficial without some mathematics.
A little deeper than this I believe that mathematics is more than just a language for physics, or indeed all science. The structures, patterns and rules of mathematics can guide one in constructing/analysing theories. The notion of symmetry is so fundamental in modern theoretical physics and at its heart is group theory. Understanding physics can be driven my mathematical beauty. Given a new theory the first question to ask is what are the symmetries?
One has to ask why mathematics is the language of the physical sciences? Can we understand why mathematics has been just so useful and powerful in structuring our understanding of the Universe?
Eugene Wigner in 1960 wrote an article The Unreasonable Effectiveness of Mathematics in the Natural Sciences which was published in Communications on Pure and Applied Mathematics. Wigner argues that mathematics has guided many advances in the physical sciences and that this suggests some deep link between mathematics and physics far beyond mathematics simply being a language.
A very extreme version of this deep interconnection is Max Tegmark’s mathematical universe hypothesis, which basically states that all mathematics is realised in nature. What this hypothesise also suggests is that the Universe really is mathematical. We uncover this mathematical structure rather than impose it on nature. This would explain Wigner’s “unreasonable effectiveness”.
We are now close to having to think about the philosophy of mathematics and in particular Platonism. I am certainly no big thinker on philosophy and so will postpone discussion about the philosophy of mathematics.
I would not go as far as to say I believe in Tegmark’s hypothesis, but it is for sure an interesting and provocative idea. It certainly makes one think about the relation between mathematics, physics and the nature of our Universe.