The four main branches of philosophy are logic, epistemology, metaphysics, and axiology. Logic, the study of the structure of arguments, examines the connection between evidence and conclusions which one wishes to draw from the evidence. Epistemology studies the nature of knowledge itself. Metaphysics studies the nature and existence of things. Axiology studies value, both ethical and non-ethical. Which of these branches is most important is a matter of perspective.
Logic is important because no matter what they are working on, philosophers construct and analyze arguments, and it is logic that tells them whether or not a given argument’s conclusion follows from the argument’s premises. Although it is possible to do philosophy without knowing any logic, it is a very bad idea: doing philosophy without logic is an invitation to muddled thinking.
Epistemology is important because philosophers pursue knowledge, and therefore presumably need a good idea of what knowledge is and what it takes to get it. It is possible to do philosophy without first definitively answering even the core epistemological questions, but one should at least have working answers.
Metaphysics is important in that it is the most expansive branch of philosophy, and covers most of the non-ethical philosophical questions people are interested in. It is nearly impossible to not take a stance on many metaphysical questions through the course of one’s life, and this stance cannot help but to have a great impact on how one lives. Few people would see much point to philosophy if it never asked metaphysical questions.
Axiology is important in particular because of its ethical component. Questions about what one ought or ought not do are just as pervasive as metaphysical questions in an individual’s everyday life. Ethical questions are even more pressing from a social perspective because although people with very different metaphysical beliefs can coexist as long as their most fundamental ethical beliefs align, the reverse is not the case. This makes axiology the branch that is the most pertinent to “real world” concerns.
So, which branch of philosophy is most important? Clearly, it is the foundations of quantum mechanics, since that is what I specialize in.