The way I've always thought that philosophy worked is that philosophers have a certain set of tools (deduction, laws of thought, basic sources of knowledge) which they use to come to reasoned answers to questions. Most importantly, these tools are taken as axiomatic. That is, they are seen as starting points from which all reasoning must proceed. To question these axioms wouldn't be possible.
However, I've recently seen an attitude that has puzzled me. Many philosophers state that very rarely does reasoning in philosophy rely on axioms. Axioms are things to be avoided and go against the spirit of philosophy.
What am I misunderstanding here? If philosophers don't take their tools of reasoning as axiomatic, how do they go about doing philosophy? More importantly, if philosophical reasoning is so pervasive that it questions its own tools, from what framework does the questioning occur? What tools does the philosopher use to question their own tools?
When this question was posed to a philosophy student, they responded that:
Philosophers don't tend to think of human thought or reasoning in terms of strict "axioms". Axioms are part of a formal logical system and it's not clear that a lot of our reasoning is like that. We hold many beliefs that we might typically think of as taken for granted. Philosophy is really about trying to understand what those are, whether they really fit together properly, and what properties of those beliefs we might want to look at to determine whether we can trust them or we ought to abandon them . . . [philosphers] generally share the idea that we take seriously our basic intuitions about cases of reasoning and we determine general rules and principles from them.
Is this the case? Is this how professional philosophers typically go about doing philosophy I've simply held a naive view?