Expressions Versus Sentences
People sometimes have trouble understanding mathematical ideas: not necessarily because the ideas are difficult, but because they are being presented in a foreign language—the language of mathematics.
The language of mathematics makes it easy to express the kinds of thoughts that mathematicians like to express. It is:
Every language has its vocabulary (the words) and its rules for combining these words into complete thoughts (the sentences). Mathematics is no exception.
As a first step in studying the mathematical language, we will make a very broad classification between the ‘nouns’ of mathematics (used to name mathematical objects of interest) and the ‘sentences’ of mathematics (which state complete mathematical thoughts).
An expression does not state a complete thought; it does not make sense to ask if an expression is true or false.
The most common expression types are numbers, sets, and functions.
Numbers have lots of different names: for example, the expressions
all look different, but are all just different names for the same number. This simple idea—that numbers have lots of different names—is extremely important in mathematics!
Sentences have verbs. In the mathematical sentence ‘$\,3 + 4 = 7\,$’ , the verb is ‘$\,=\,$’.
A sentence can be (always) true, (always) false, or sometimes true/sometimes false.
For example, the sentence ‘$1 + 2 = 3$’ is true.
The sentence ‘$1 + 2 = 4$’ is false.
The sentence ‘$x = 2$’ is sometimes true/sometimes false: it is true when $\,x\,$ is $\,2\,$, and false otherwise.
The sentence ‘$x + 3 = 3 + x$’ is (always) true, no matter what number is chosen for $\,x\,$.
So, $\,x\,$ is to mathematics as cat is to English: hence the title of the book,