Philosophical ideas have inspired some ideas in science and mathematics indeed. New solutions to problems in science and mathematics, such as the incompleteness theorems, complexity theory, etc, have shown ways to begin answering age-old philosophical questions.

But can you give an example where a new philosophical solution, has driven advances in existing mathematical or scientific problems? My point is that this happens the other way around, much more often.

Philosophy is useful, but useless by itself. It can help you towards learning certain skills such as logical reasoning, or methods such as how to begin to formulate a problem - but if we want to correctly assign truth-values to propositions, the only tool that will allow us to do that is science and mathematics - including being able to precisely define "truth".