You’re heading deeper into the philosophy/study of knowledge which is called Epistemology.

What I think you’re trying to say is you think there’s a divide between sciences which deal with making models about “reality” and testing them versus the sciences like logic and abstract mathematics which constructively discover truth by deriving it from previous truths (axioms/theorems).

However, in a very general sense, any logic whatsoever—written symbolically or in natural language—is mathematics. Making the assumption “matter/energy is not created nor destroyed” and then using that assumption (axiom) to derive “momentum must be conserved in a collision” is mathematics just as much as it is physics. Any time you build ideas on/using a “scientific model” you are using logic/mathematics.

The symbols/notations were invented and axioms are arguably arbitrarily chosen, but the consequences of those axioms are undeniably discovered not invented

Oh yeah I meant to say that some sciences have origins that predate empiricism itself. Astronomy for example. Also, some may argue that not all sciences are rooted on evidence in the same way (e.g. positivism would call psychoanalysis a pseudoscience even though there're powerful conclusions that were drown without evidence and have helped people).

Actually, all hypothesis start with a certain lack of evidence. Some hold for their own value, as potential explanations... Yes, O am not talking about maths anymore but about all sciences. For example many subatomic particles that are investigated by CERN. That's why I believe your approach was shortsighted.