It is understood as an axiomatical deductive method, organized like Euclid's Elements , but independent from any intuitive base and sufficiently justified by meta-axiomatical investigations on completeness, independence and consistency. A closer look on Hilbert's own writings reveals the mythical character of this view. The axiomatic method can thus be identified with the analytical, regressive or critical method of traditional philosophy. This is a dynamical process in the sense that the foundations are deepened as far as necessary for providing the consistency of the respective branch of mathematics.
Philosophy of mathematics is reduced to a description of settled practice.
What seems to be open for Maddy is clear for Hilbert. He knew that his formalism was a mathematical program which had to be supplemented by a philosophical foundation. In his early writings he preferred a logicistic foundation based on a Kantian epistemology, later he founded his proof theory on strict finitism coming close to Brouwer's intuitionism.