The long career of the triad "logicism, formalism, and intuitionism" was launched by Felix Klein in 1893, actually a few years before "logic" gained an "ism". Klein meant none of these terms the way we understand them today. Brouwer refined and later reversed Klein's meaning of "intuitionism"--creating a tension in Brouwer's own thought and pulling a red herring across the path of later Poincari interpreters. Brouwer also shifted the meaning of "formalism". During walks along a broad North Sea beach he showed Hilbert how to combine this shift with the earlier meaning by passing to the meta-level, though he meant to show why not to do this. The history of this "ruling clichi in the philosophy of mathematics" (Howard Stein) clarifies the philosophy of early 20th century mathematicians and its relation to their practice.

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