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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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II.7 The Crisis in the Foundations of Mathematics

José Ferreirós

The foundational crisis is a celebrated affair among mathematicians and it has also reached a large nonmathematical audience. A well-trained mathematician is supposed to know something about the three viewpoints called “logicism,” “formalism,” and “intuitionism” (to be explained below), and about what GÖDEL’S INCOMPLETENESS RESULTS [V.15] tell us about the status of mathematical knowledge. Professional mathematicians tend to be rather opinionated about such topics, either dismissing the foundational discussion as irrelevant—and thus siding with the winning party—or defending, either as a matter of principle or as an intriguing option, some form of revisionist approach to mathematics. ...

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