Hi Merle,
Thanks for the good question.
While philosophy underlies all sciences I wouldnt really confine myself to it, I would say that inquiry adapts itself to the question at hand.
I know some might recoil at the thought that philosophy underlies science, so let me remind them that science is permeated with unprovable assumptions:
Except in that these assumptions are in turn validated by their success. The results of these validations are tentative and subject to revision or are outright discarded if these assumptions wind up not working.
An example would be logical & mathematical truths cannot be proven by science.
Science doesn't "prove" anything. Proof is the domain of Mathematics and Alcohol. Logical and mathematical axioms (or assumptions if you insist) can be validated by their use, in the same way that a pen you've never seen before can be assumed to work, and then validated by its use.
Science presupposes logic & math so that to try and prove them by science would be arguing in a circle.
No, to "presuppose" something as absolute and incontrovertible would be to err in your ways. we can adopt axioms for those things we can't "prove", but only as far as needed and the results of any such axioms are still tentative and not in anyway incontrovertible.
Another example: in the special theory of relativity—the whole theory hinges on the assumption that the speed of light is constant in a one way direction between any two points A & B—but that strictly cannot be proven, we simply have to assume that in order to hold to the theory.
But as discussed, the results of these axioms adopted beforehand are validated by their use - that is, we have GPS that relies on bi-directional speed of light in order to interoperate to give us the accuracy we enjoy - this is one such successful validation example, and there are many more like it, such as communicating with probes across the solar system, LIGO Gravitational wave detectors across continents and space telescopes in orbit, etc.
None of these beliefs can be scientifically proven, and yet they are accepted by all of us as rational deductions of the world in which we live.
well, yes, we can accept them as rational deductions because they work and continue to work, giving us meaningful and useful results the more we test it.
The law of uniformity and law of causality and the speed of light being constant are also unprovable assumptions of science.
well, they've been demonstrated to work very well thanks all the same. We have a global communications network that demonstrates the uniformity of bi-directional speed of light - we're communicating over it now. You choosing not to accept the facts, doesn't make the facts go away.
So really, philosophy gave you and I our scientific method.
In fact the scientific method is a combination of philosophy (inductive reasoning for hypothesis formulation) and methodological naturalism(deductive reasoning).
Well, probably true - Science and the scientific method have really come unto its own these days and the scientific method is the single most reliable method to come about facts of reality. You're right in that it has its roots firmly planted in philosophy - It isn't perfect to be sure, but no other method has been demonstrated as being any more effective. Feel free to speak up if you think otherwise.
However when doing forensic science for example, you can’t use the scientific method, yet they are still doing science.
This applies to all historical sciences including evolutionary biology.
What?? How are they not doing Science? It's exactly the same scientific method used anywhere else in science.
At the same time when doing those sciences you can appeal to all sorts of results that come from the scientific method.
...well, yes, because it's science.
Trying to confine yourself to a sub domain of inquiry is limiting.
What other domain of inquiry should be included in, or used instead of science, and perhaps we can have a chat about it?
I hope this helps, I’m happy to give some examples of what I mean if you want me to.
Will they be any different to the examples you gave above that were mistaken? I'm interested to hear more examples...