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

2.1 A universal language

Where do you live? Where were you born? What is the first language that you learned to speak? Do
you speak other languages? Now, you’re probably thinking, “Why does it matter where I was born or
what languages I speak. This is a math book!” Well, you’re right: it doesn’t matter where you live or what
language you speak, because this is a math book, and mathematics is a universal language.

Mathematics is a language based on universal concepts. Like any language it has words.

Definition 2.1.1. A mathematical word is a mathematical concept with a precise technical definition.

Everything we discuss in mathematics has a definition. The first step in learning mathematics is to

learn definitions. Just like any language, you must first memorize its vocabulary. It is not sufficient in
mathematics to memorize a definition vaguely. You must memorize its exact meaning. There are however
many correct ways to state a definition, and it is helpful to think about the definition in your own words
in your native language. The goal in memorizing a mathematical definition should not be to memorize
all the words, but rather to memorize the concise meaning. The next step in learning mathematics is to
practice using the definitions to make mathematical sentences.

Definition 2.1.2. A sentence in the language of mathematics is a sentence in the usual grammatical sense
which is about mathematical concepts and is definitely either true or false.

For example,

1 + 1 = 2,

is a true mathematical sentence, whereas

5 + 6 = 8

is a false mathematical sentence. In both of these sentences, the mathematical verb is equals. However,
1 + 2 + 3 is not a mathematical sentence, because there is no verb. The sentence:

Mathematicsiscool!

is not a mathematical sentence because, although it is about mathematics, it is not definitely true or
false; it is a matter of opinion. By the end of Chapter Three, you will be able to understand the following
true mathematical sentence:

0+ x = x, ∀x ∈ N.

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