The scientific method explains and predicts physical phenomena. However, it is only a partner to, and not fundamental, with regards to areas of philosophy or humanistic (and theological) pursuits.
The problem is that there is no way to know in an
a priori fashion which entities are physical. A popular misconception is the belief that “If something is physical, it lends itself to scientific explanation”. But in practice, it’s really the converse that comes about: “If something can be explained scientifically, it is physical”. There is no method or algorithm we can use to determine whether something is physical before employing the scientific method, thus a Christian who says “This isn’t physical, so we can’t use science anyway” is jumping the gun.
Take energy for example. Energy used to be regarded as something nearly magical. But once we discovered that it could be used in naturalistic explanations in a consistent manner, it became the subject matter of physics. It wouldn’t have done any good for someone to insist that we shouldn’t have started investigating energy scientifically because it’s not physical. We classified it as physical after using science, not before.
There are a whole host of things the scientific method presupposes that themselves cannot be demonstrated through science, for example, the intelligibility of language, mathematics or human consciousness. Moreover, causality is itself unobservable.
Math, at least, also presupposes concepts it cannot demonstrate (such as the axioms of the deductive system you’re using). No one ever seems to make a fuss about this, so why be so critical of science?
Also, strictly speaking, science doesn’t assume what most people mean by “causality”. Quantum mechanics is an example of a discipline where what counts as a “cause” is questionable.
And while we may appeal to causation in our explanations, cause and effect are irrelevant to the math involved. Newtonian physics would work just as well from a mathematical point of view if we instead regarded motion as causing forces and not the other way around.