Demystifier said:
Einstein, Schrodinger, Bell, ... For example, Bell discovered his Bell inequalities by starting from quantum philosophy.
In mathematics, I would mention Frege, Russell, Godel, Quine, ...
For me Einstein is rather an example for the ineffectiveness (even danger) of philosophy in the natural sciences. There's no doubt that in his younger years Einstein was one of the most creative physicist with an amazing imagination about how nature works, and he was in this time always "close to experiment", i.e., he had the observed phenomena in mind when developing theories, which is an creative act rather than some machinery of rational derivation. In his later years, he fell however in the trap of a philosophical prejudice against the implications of quantum theory, particularly the "inseparability" which was his real trouble wrt. the infamous EPR paper, as he clarified some years later in 1948. That's why he was looking for almost 30 years for a unified classical field theory of gravitation and electromagnetism, ignoring the newer experimental facts, according to which there must be more "forces" (or rather "fundamental interactions") than just electromagnetic and gravitational interactions as well as the fact that the quantum-theoretical predictions all were confirmed.
Further for me Bell's is to the contrary an example for the successful exorcism of philosophical demons by finding a clearly defined scientific approach to the philosophical quibbles of EPR, i.e., he made the philosophical unclearly defined "problem" a scientifically decidable question, i.e., to a quantitative prediction for the outcome of experiments assuming "local realistic hidden-variable theories" (thereby clarifying EPR's vague philosophical formulations) contradicting the predictions of QT, and the result is well known in favor for QT and not EPR's philosophical prejudice of how a physical theory must look like. Though, of course, the motivation for Bell was some philosophical question, thus he ingeniously resolved it by bringing it to the realm of scientifically well-defined, quantitative and thus empirically testable/decidable questions.
Mathematics for me is neither a natural science nor a humanity. Nowadays it's put into the third category of the "structural sciences". The quoted mathematicians were of course also philosophers to some extend, but also mathematicians, and I'd put "mathematical logics" clearly in the realm of the structural sciences and not so much of philosophy.
For me the "effectiveness of mathematics" in the natural sciences is simply explained by the fact that math developed from applications to real-world problems by abstraction, and not the other way around. That's why math started with natural numbers, then inventing the 0 and negative numbers, the rational numbers, and finally the real numbers in some centuries, until it was formalized in the 19th-20th century with the demand for more rigorous formulations after some "foundational crisis of analysis". The same holds for geometry: Euclidean geometry was in a sense discovered from real-world practice. E.g., in Egypt it was important to get the areas of the land precisely measured after each flooding by the Nile every year, making use of Pythagoras's theorem.
