After several schools of the philosophy of mathematics were abandoned one after the other in the 20th century, the assumption that any foundation that could be stated within mathematics itself began to be heavily challenged.
Today one refers more ambiguously to the foundations of mathematics to avoid giving the impression that there is a 'problem' that can be solved in the sense of a science or mathematics problem, with a single right answer that is checked by means describable in proof theory.
The term 'foundations problem' only occurs in literature that makes the assumption that there is such a provable and single foundation. In the 1970s, many arguing the prevalence of quasiempiricism in mathematics denied that assumption  and sought to focus on mathematical practice and later a cognitive science of mathematics  an outright empirical basis for why mathematics works, and "Where Mathematics Comes From".
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