Hi Geremia. Welcome to the board. I guess your post is intentionally ambiguous perhaps because you’re looking for some focus. Your concern as I understand it is that there is a strong relationship between physics and mathematics and rather than suggest mathematics is used to describe physics, one might suggest that the reverse might somehow be more pertinent.
In his book “The Road to Reality”, Penrose writes,
It may be noted, with regard to the first of these mysteries – relating the Platonic mathematical world to the physical world – that I am allowing that only a small part of the world of mathematics need have relevance to the workings of the physical world. It is certainly the case that the vast preponderance of the activities of pure mathematicians today has no obvious connection with physics, nor with any other science, although we may frequently be surpised by unexpected important applications.
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In other words, physics can be described with mathematics. Anything objectively physical (ie: anything objectively measurable) follows some kind of mathematical rule. Penrose also points out that humans should, in principal, be able to access the entirety of mathematics, but not all mathematics is applicable to physics.
Regarding your point:
For example, many equations in physics take equivalent forms, with different symbols representing their variables and having different interpretations based upon context. … For example, that the wave equation for light has the same form as the wave equation for sound waves should tell us more than "These are just waves." It should tell us that light and matter are related at an even deeper level than currently proposed light-matter interaction theories like QED would suggest.
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Certainly there is a fundamental similarity between things that oscilate. Regardless of whether that thing that is oscilating is an electron, or a mass on a spring, the causal influences at work (in this case, forces acting on a mass) will produce analogous phenomena and therefore produce similar mathematical equations. Perhaps that tells us something about the intrinsic nature of forces, such as the linear nature of force (ie: doubling the force doubles acceleration?). I don't know if there's anything worthwhile there or not.
Does that help at all?
