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How multiplication is defined in Peano arithmetic

A familiar joke in mathematical logic (the field I worked in for the first twenty years of my career) is the following hypothetical entry in...

Having taught set theory and the foundations of mathematics for many years at various universities, I long ago learned that many students never do fully grasp it. In this column I want to explain the recursion principle, its importance, its power, its need in mathematics, and how it is used, so you too will know why it is flat wrong to tell kids that multiplication is repeated addition. Remember, even if you don’t see the purpose of all the mathematical navel-inspection that led to the formulation of the principle, some of the students in your class may well find themselves wanting or needing to understand it in a few years time.

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