Demystifier said:
To use such a method, one doesn't need philosophy. But to discover such a method, philosophy may be of a great help.
Exactly, this cannot be stressed enough! If one starts with mathematical methods alone then there are simply too many possible roads to choose from and progress halts, while if one starts with a purely intuitive conceptual description constrained by logical truths then there are far fewer roads to choose from; choosing the right road quickly leads to progress. All of them have to be taken individually and at the beginning of each road is where the mathematical method begins anew.
Moreover, the correct route to choose prior to using any mathematical methods can actually be known in advance, purely as a matter of having a highly developed intuition based on the experience of having done this many times before and then recognizing how to proceed through the correct usage of analogy in this purely rational endeavor of logical exploration.
Poincaré describes this method of reasoning of the scientist masterfully in
Science and Method, specifically in Book 1 The Scientist and Science, Chapter 1 The Selection of Facts. The book is
available here by courtesy of Google Books.
vanhees71 said:
Theoretical physics is a mixture of mathematics and knowledge about the empirical facts and a lot of intuition about how to use the former to describe the latter.
The bolded part represents your misunderstanding: physics is not knowledge of empirical facts, physics is instead a form of reasoning using principles about empirical facts. Knowledge and reasoning while related are not the same thing, since the former is the thought content while the latter is the thought process.
In fact, physics is the only science which can not primarily be characterized by being merely a collection of facts but instead by being a collection of methods, i.e. not the content of knowledge but the method does physics make. No other science has this property (biology, chemistry, psychology, economics, etc) and when they do attain it they are usually reduced so completely to physics that they are called physics (cf. biophysics, econophysics, sociophysics, etc).
This is what Rutherford meant when he said
"All science is either physics or stamp collecting.", with stamp collecting referring to the mere collecting of facts, i.e. the collection of knowledge. This is also what Einstein meant when he said:
"[I do not] carry such information in my mind since it is readily available in books. ... The value of a college education is not the learning of many facts but the training of the mind to think."
Concrete example to drive home the point: merely knowing what the speed of sound is does not make one a physicist; instead knowing how to determine the speed of sound - especially if done purely by reasoning using intuition alone without having any prior knowledge of this topic - is what makes one a physicist.