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.