Full Idea
Shapiro's structuralism champions model theory as the branch of mathematics that best describes mathematics. The essence of mathematical activity is seen as an exercise in comparing mathematical structures to each other.
Gist of Idea
Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics
Source
report of Stewart Shapiro (Philosophy of Mathematics [1997], 4.4) by Michèle Friend - Introducing the Philosophy of Mathematics
Book Reference
Friend,Michèle: 'Introducing the Philosophy of Mathematics' [Acumen 2007], p.96
A Reaction
Note it 'best describes' it, rather than being foundational. Assessing whether propositional logic is complete is given as an example of model theory. That makes model theory a very high-level activity. Does it capture simple arithmetic?