Physics is not philosophy. Physics is a branch of science.
Mathematics is a language.
The SYMBOLS in mathematics are a language. Mathematics itself is a closed functional system.
There are no 'mathematical laws'.
There certainly is.
Mathematics is about objective values.
and therefore limited by its definitions to circular reasoning.
Mathematics is not reasoning. It is a closed functional system...no more.
Physics is a branch of science, an open functional system.
Don't confuse the two. Physics isn't required to conform to mathematical definitions.
A theory of science MUST be transcribed to a closed functional system (such as mathematics) to gain the power of prediction (which comes with the power of proof).
Math can only be used to find patterns.
Mathematics is not a pattern matching device. It is a closed functional system.
And, it can't find all of them without determining some new repeatable definitions, which will be defined by the physics, not mathematics itself.
Physics theories are typically expressed in mathematical form. A theory of science that is transcribed into mathematical form becomes a 'law'.
It is still a theory. If it is falsified, the 'law' goes with it.
A closed functional system is simply one that is defined by a set of rules (or axioms). It has nothing outside those set of rules. Mathematics is one such system. Logic is another. Change a rule and you are playing a different game. Minor changes in the axioms for mathematics create the different Domains of mathematics (each with a completely different behavior). Theories of science in physics are typically translated into the Real Math Domain, the Domain taught to you in grade school.
An open functional system is one that does not define any boundary. Science is such an system. A theory of science may be literally about anything. It's only requirement is that the theory must be falsifiable, or testable to see if the theory itself is false. That test must be specific, available, and be objective (it must produce a specific result), and it must test the theory itself. As long as a theory of science can withstand such tests, it is automatically part of the body of science. It will be until the theory, if and when, is falsified.