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

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V.20 The Insolubility of the Halting Problem

What does it mean to understand a certain area of mathematics completely? One possible answer is that you understand it when you can solve its problems mechanically. Consider, for instance, the following question. Jim is half the age of his mother, and in twelve years’ time he will be three-fifths of her age. How old is his mother now? For a child who is just old enough to understand the concept of “three-fifths,” this is likely to be an impossibly difficult problem. A bright and slightly older child may be able to solve it after some hard thought, which will probably include a certain amount of trial and error. But for anybody who has learned how to translate such problems into equations and who ...

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