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Important branches include proof theory, model theory, and mathematical logic. The original meaning of David Hilbert is closest to proof theory.
Many issues regarding the foundations of mathematics (there is no longer necessarily considered to be any one "problem") and the philosophy of mathematics touch on or use ideas from metamathematics.
On the other hand, quasiempiricism in mathematics, the cognitive science of mathematics, and ethnocultural studies of mathematics, which focus on scientific method, quasiempirical methods or other empirical methods used to study mathematics and mathematical practice by which such ideas become accepted, are nonmathematical ways to study mathematics.
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