When professional philosophers investigate philosophical questions concerning mathematics, they are said to contribute to the philosophy of mathematics

It is broadly conceived as comprising proof theory, model theory, set theory, and computability theory as subfields

In the first decades of the twentieth century, three non-Platonistic accounts of mathematics were developed except one.

Frege's Basic Law V

Russell’s paradox

Hume’s Principle

Peano Arithmetic

mathematics is essentially an activity of construction

Resolutely rejects the existence of the actual (or completed) infinite; only potentially infinite collections are given in the activity of construction.

principle of excluded third

Heyting Arithmetic

natural numbers can be taken to be symbols

Primitive Recursive Arithmetic

Gödel’s first incompleteness theorem