Bertrand Russell and Alfred North Whitehead took the position known as “logicism,” endeavoring to show in their massive Principia Mathematica that mathematics was really logic in disguise. And “platonism”—the idea that mathematics describes a perfect and eternal realm of mind-independent objects, like Plato’s world of Forms—was championed by Kurt Gödel. All of these mathematical figures were passionately engaged in what Harris slights as philosophy of Mathematics-with-a-capital-M. The debate among them and their partisans was fierce in the 1920s, often spilling over into personal animus. And no wonder: mathematics at the time was undergoing a “crisis” that had resulted from a series of confidence-shaking developments, like the emergence of non-Euclidean geometries and the discovery of paradoxes in set theory. If the old ideal of certainty was to be salvaged, it was felt, mathematics had to be put on a new and secure foundation. At issue was the very way mathematics would be practiced: what types of proof would be accepted as valid, what uses of infinity would be permitted. For reasons both technical and philosophical, none of the competing foundational programs of the early twentieth century proved satisfactory. (Gödel’s “incompleteness theorems,” in particular, created insuperable problems both for Hilbert’s formalism and for Russell and Whitehead’s logicism: they showed—roughly speaking—that the rules of Hilbert’s mathematical “game” could never be proved consistent, and that a logical system like that of Russell and Whitehead could never capture all mathematical truths.) The issues of mathematical existence and truth remain unresolved, and philosophers have continued to grapple with them, if inconclusively—as witness the frank title that Hilary Putnam gave to a 1979 paper: “Philosophy of Mathematics: Why Nothing Works.” - http://www.nybooks.com/articles/2015/12/03/mountains-mathematics/