What is required of a foundation for mathematics?

Philosophia Mathematica 2 (1):16-35 (1994)
The business of mathematics is definition and proof, and its foundations comprise the principles which govern them. Modern mathematics is founded upon set theory. In particular, both the axiomatic method and mathematical logic belong, by their very natures, to the theory of sets. Accordingly, foundational set theory is not, and cannot logically be, an axiomatic theory. Failure to grasp this point leads obly to confusion. The idea of a set is that of an extensional plurality, limited and definite in size, composed of well defined objects.It is the extension of Greek notion of 'number' (arithmos) into Cantor's 'transfinite'.
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DOI 10.1093/philmat/2.1.16
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Does Homotopy Type Theory Provide a Foundation for Mathematics?James Ladyman & Stuart Presnell - forthcoming - British Journal for the Philosophy of Science:axw006.

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