A philosophical position is, broadly speaking, a belief that a particular statement is either true or false.
In the context of mathematics the term is usually reserved for particular fundamental questions whose truth values have not been explicitly determined in any chosen frame of reference or axiom schema. Hence, in order to progress down a particular logical path, one needs to make a decision as to which it is.
Usually for such questions, its answer is of far-reaching importance.
It having been necessary to take such a position, one then finds that diverging schools of thought arise: one which accepts and one which rejects the truth of the question. The answer to the question itself is added to the particular axiom schema for that school to form an expanded schema.
- Results about philosophical positions can be found here.