> That makes mathematics just a subdivision in philosophy, in which statements about statements must be (axiomatically) derived exclusively from a completely explicit starting point.
The main problem with philosophy is exactly the reason which makes above wrong. A global definition of 'contradiction' is not possible, Nor is even something that resembles a consensus to a sufficiently strong degree to make it meaningful.
This is precisely the reason why you can have radically opposing schools of thought that are simultaneously existing. The trivial example being mathematical empiricism or coherentism, both of which directly oppose the axiomatic approach you've provided, from different angles.
A more absurd example is to say it's never the case that all philosophers would agree "P and not P" is not true (including the sentence itself), let alone it being false (i.e. negation as failure), because the value of "P and not P" is dependent upon your theory of truth and no global theory of truth exist without G.E. Moorean appeals to common-sense.