I have never seen an adequate definition of philosophy. It seems like a "family-resemblance" concept to me, to borrow Wittgenstein's famous phrase. It is easy to give definitions of, say, physics, or chemistry, or biology, or even mathematics. But philosophy seems like a more vague, imprecise concept. Why is that? And is there even a good definition of philosophy?
Simply put, there is no good definition of philosophy because in this sense you mean stipulative definition; for definition by stipulation you need agreement, and there is a considerable disagreement about what philosophy is and what it should be used for. Since philosophy began under the Pre-Socratics, philosophers have been commenting on this topic, however, recently the sub-field of metaphilosophy has begun to catalogue and marshal the debate. From the IEP article on metaphilosophy:
What is philosophy? What is philosophy for? How should philosophy be done? These are metaphilosophical questions, metaphilosophy being the study of the nature of philosophy. Contemporary metaphilosophies within the Western philosophical tradition can be divided, rather roughly, according to whether they are associated with (1) Analytic philosophy, (2) Pragmatist philosophy, or (3) Continental philosophy.
Thus, you will see that philosophers who make arguments about what philosophy is and what its methods should be fall into a number of schools, of which one can supplement the list above:
- Ancient philosophy
- Analytic philosophy
- Continential philosophy
- Process philosophy
- Pragmatic philosophy
The one introductory text I own on the topic is:
Note that the lack of consensus is not a bad thing necessarily; in fact, one can make the argument that the diversity inherent in metaphilosophical pluralism adds some value above and beyond the individual opinions by showing that philosophy itself seems subject to diversity as a function of the psychology of the philosophers themselves.
Demarcation, investigating definitions and boundaries between disciplines, are core concerns of philosophy. So when you turn it on itself in this way, of course you are going to have to do philosophy, which is hard to do well.
You assume those other subjects are easy to define. If you approach them not with their definitions from inside their practice, but philosophically, then I don't think they are. Scientific Method cannot be defined exhaustively and rigorously, it varies profoundly between disciplines in ways that don't always overlap, and so I'd defend that science fundamentally is a culture, it is what scientists do: Can one speak unambiguously of "the" scientific method? Similar to doing philosophy, in deciding what it is. And the status of mathematics is I'd say very high in the pecking order of most contentious subjects in philosophy, as discussed in this answer: The Unreasonable Ineffectiveness of Mathematics in most sciences
I would argue that in defining philisophy, we need to look at how it came to distinguish itself from previous similar practices, which we now group under 'wisdom traditions'. The word philosophy was used before Socrates (it's disputed but some argue it was coined by Pythagoras), but I'd argue that Socrates with the 'mutual enquiry into truth' of Socratic Dialogue, really codified the distinguishing quality of philosophy, and was himself paradigmatic in living the role by his example abd especially acceptance of 'martyrdom' for his views. Discussed here: Weren't there any philosophers from Africa, America or the Middle East before Socrates? Plato fused Socratic methods with the math-mysticism of Pythagoras to create the Academy and academia of professional scholars rather than sophists and rhetoriticians for politics and law that had been the previous educational elite. Aristotle developed the aspiration of universal enquiry and education, that became the ambition of universities. And the success and wealth of his most famous student, Alexander the Great, made sure the reputations were amplified and records survived (though it's thought none of Aristotle's own writing survives, only lecture notes of his students).
To understand Socrates, who the Oracle at Delphi called the wisest man in Athens, we need to enquire into wisdom, and what it does for people. My picture is here: Wisdom and John Vervaeke's awakening from the meaning crises? The TLDR is: It's the skill of dilemma solving, cultivated by finding the integrated centre of our concerns ('Know Thyself', the highest admonition of the Oracle). And what it does is maximise our relative freedom, from coercion, impulses, and self-deception, when they don't serve the self that the wise person has come to know. Crucially though, Socrates chose the mutual pursuit of truth over the wisdom of not dying (he could have gone into exile, as Aristotle was forced to), he talked about always listening to his daimon, his conscience, wjen it forbade him to do something. And this is why his philosophia distinguished itself from love-of-wisdom (an equally valid translation), for love-of-truth. His Euthyphro Dilemma had far reaching consequences, in placing even the gods under constraints of reason, a still controversial and uncompromising stance.
If you look at Kahneman's Thinking Fast and Slow, I'd call philosophy the slowest carefullest thinking. It is meta-thinking, which you might even go as far as to call 'system 3' thinking. We develop a personal practice, a toolbox, for stepping back from or outside of the rails our thinking has run on, and consider the general grounds of our thought, typically when progress has got stuck in some way, or we seems to face contradictions unsolvable in the current framing or paradigm. Discussed here: (Why) is this negative outlook on the concept of philosophy misguided?
What I take from this is, trying to fully define philosophy, is misguided. Define it by doing it, by taking up the endeavour, the culture, and practicing it ccording to your own understanding, by trying to do it well according to that. That will be the wise way, to pursue the truth of the matter.
Any student of philosophy should develop their own working picture of the subject. And it will develop, as their practice does. I would advocate to any aspiring student, enquire for yourself into what this subject is, and make sure to go about doing it in the terms you understand it should be done. A great philosopher doesn't do someone else's philosophy, they do their own. Let us all aspire to that.