2.6. Some Notes on Proofs and Abstraction 37 limx→a c = c for all constants c, and your recursive definition from Exercise 2.5.13, prove that every rational function is continuous on its entire domain. Exercise 2.5.19. Using the recursive definition of addition from the previous section (a(0, 0) = 0 a(m+1,n) = a(m, n+1) = a(m, n)+1), prove that addition is commutative (i.e., for all m and n, a(m, n) = a(n, m)). 2.6. Some Notes on Proofs and Abstraction 2.6.1. Definitions. Definitions in mathematics are somewhat dif- ferent from definitions in English. In natural language, the definition of a word is determined by the usage and may evolve. For example, “broadcasting” was originally just a way of sowing seed. Someone used it by analogy to mean spreading messages widely, and then it was adopted for radio and TV. For present-day speakers of English I doubt the original meaning is ever the first to come to mind. In contrast, in mathematics we begin with the definition and assign a term to it as a shorthand. That term then denotes exactly the objects that fulfill the terms of the definition. To say something is “by definition impossible” has a rigorous meaning in mathematics: if it contradicts any of the properties of the definition, it cannot hold of an object to which we apply the term. Mathematical definitions do not have the fluidity of natural lan- guage definitions. Sometimes mathematical terms are used to mean more than one thing, but that is a re-use of the term and not an evolution of the definition. Furthermore, mathematicians dislike that because it leads to ambiguity (exactly what is meant by this term in this context?), which defeats the purpose of mathematical terms in the first place: to serve as shorthand for specific lists of properties. 2.6.2. Proofs. There is no way to learn how to write proofs without actually writing them, but I hope you will refer back to this section from time to time. There are also a number of books available about learning to write proofs and solve mathematical problems that you can turn to for more thorough advice.

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