This paper argues against a strong philosophical interpretation of the leading role of mathematics in all of physics. To do so the paper focuses on a specific case study, that of the truly astonishing success of symmetry groups in modern particle physics. Specifically, I analyze the case of one local gauge symmetry, that of the strong nuclear interaction. I would say this is an especially pertinent case study, as gauge symmetry applies throughout most of our current best fundamental physics and the intimate relation with the physics it describes is particularly astonishing. The paper advocates for an understanding of mathematics only as an (especially appropriate) language which does nothing but describe patterns, a subset of which are instantiated in Nature. With such an understanding I argue that the effectiveness of mathematics is not unreasonable; on the contrary, it is to be expected. Such an explanation undermines the viewpoint that takes gauge symmetry principles as a priori reasonable or as some sort of necessary meta-laws. Likewise, such an explanation weakens the reasons to endorse a strong ontological commitment to the mathematical entities (as the diverse variants that suggest that the universe is fundamentally mathematical, like [Tegmark, 2014] or [French, 2014]).
Aldo Filomeno