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Blast into Math! Pure mathematics: the proof of the pudding is in the eating

Please don’t worry if you don’t yet know the meaning of the upside-down A, “ ,” or the rounded-out
∀
E, “ ∈ ” or the funny looking N “N.” These logical symbols will soon be part of your mathematical
vocabulary. Mathematical sentences may be formed using mathematical symbols like + and =, logical
symbols like ∀ and ∈, numbers, and ordinary words. For example, the following is a mathematical
sentence known as the Prime Number Theorem.

The number of primes less than or equal to a positive number is asymptotically equal to that
number divided by its natural logarithm.

Please don’t be intimidated by the above sentence if you do not yet understand it; you will be able to
understand the meaning of the Prime Number Theorem by the end of this book.

2.2 Theorems, propositions, and lemmas

This chapter is called “pure mathematics.” What exactly makes mathematics pure?

A theorem consists of one or more true mathematical sentences which have been proven.

A theorem is infallible: it is pure, and in pure mathematics we prove theorems. The reason a theorem
is true is called a proof.

Definition 2.2.1. A proof is an unbreakable logical argument in which every statement follows immediately

from the preceding statements, definitions, previously proven statements, and hypotheses.

In pure mathematics, we prove theorems rigorously. What makes a proof “rigorous? ” When we prove
a theorem rigorously, there are no logical gaps, no missing steps and no cases left out. In addition to
proving theorems, we also prove propositions and lemmas. A proposition, like a theorem, is a true
mathematical statement, but it is usually easier to prove than a theorem, so we do not endow it with
the hefty and distinguished title “theorem.” A lemma is a true mathematical statement which helps to
prove a big hefty theorem; the German word for lemma is Hilfsatz which literally means helper-theorem.
Lemmas are not usually very interesting by themselves; their purpose is to accomplish a certain step in
the proof of a theorem. On the other hand, a proposition is a true mathematical statement which could

be used for a variety of purposes and is interesting all by itself.

2.3 Logic

To prove theorems we often use logical statements of the form, “if this is true, then that is true,” where
this and that are two statements. We could also say, “this implies that,” which means exactly the same

thing. Just like in algebra, it is convenient to use letters rather than this and that. For example, we could
say, “if A is true, then B is true,” where A and B are used to represent the two statements “this” and
“that.” We can shorten this further to, “if A then B,” or equivalently, A implies B.

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