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Surfaces and Essences by Emmanuel Sander, Douglas Hofstadter

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CHAPTER 8

Analogies that Shook the World

The Royal Role of Analogies in the Realm of Rigor

In Chapter 7, we saw that students who encounter new mathematical ideas for the first time lean heavily on analogies at every step, doing their best to keep from taking tumbles in the abstract world into which they are tremulously treading. We also saw that naïve analogies have a way of sticking in people’s minds for a long time. Indeed, naïve mathematical analogies often last a lifetime in the minds of non-mathematicians, tending to lead their makers into dead ends, confusions, and errors. To avoid this fate, one has to gradually refine one’s category system ...

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