In the day-to-day practice of science i.e. when empirical inquiry 'works', there certainly are underpinning philosophical assumptions, whether or not they are reconsidered or appreciated with every experiment. The implicit in the act of hypothesis testing, some variant of which most scientists in day-to-day practice, are assumptions about the nature of probabilities, inference etc. The NHST framework that is typically used came about after extensive battles over philosophical considerations that apply to significance/hypothesis testing between Neyman and Pearson vs Fisher. The fact that I never write that a hypothesis is (as a research biologist) is 'proven', but some variant that of it having withstood an attempt at falsification, is loaded with Popperian critical rationalism.
> Math is the precise language needed to speak about these methods and the knowledge.
Except math can't map directly onto reality, or data generating processes that are studied, unless you are presupposing some kind of logical positivism (and I doubt you are). We need probabilities, statistics, and frameworks to map all of this uncertainty, and they must be underpinned by some sort of philosophical assumptions that can't be derived from science itself.
> Look at the achievements of science. That is how you get convinced it gives us a grip on reality.
But that in and of itself is a philosophy of science, one of instrumentalism. However, it only extends to whether science can be useful, but not whether it is accurately describing reality or is true.