Mathematical Folk Knowledge

The following are things that (almost) all mathematicians know, but few non-mathematicians know. They are rarely explicitly written down.

Bold (or sometimes italic) in a mathematics book or mathematics article means that the word or phrase is being defined. This is true regardless of whether the author has helpfully labeled it “Definition”.

The equals sign and the word “equals” mean that the two things are the same thing. This is all that they mean. For example, “x = y” means that “x” and “y” are two names for the same thing.

There are no such things as variables in mathematics in the sense that the word is used in high-school algebra. These “variables” are just names for numbers.

In advanced mathematics, numbers (natural numbers, integers, fractions, rational numbers, real numbers) are constructed from sets or constructed from the natural numbers or are not constructed and their properties (axioms) are simply given. But, for non-mathematicians, numbers should be points on the number line, and this is the way that mathematicians always think of them.

Mathematics is just “A implies B”. If you are reading a mathematics book or article and the sentence is not a definition (no bold) and is not a comment (i.e., not something like “Euler was the first to prove this”), then you (the reader) must verify that the sentence logically follows from what has gone before. This is true regardless of whether the author has helpfully put the sentence in a section labeled “Proof”. This makes reading a mathematics book a very slow endeavor.

Articles in mathematics journals have the authors listed alphabetically. (This is not true for mathematics books.)

Few mathematicians know this one: The natural numbers are an abstraction of counting. The real numbers are an abstraction of lines (and hence geometry). Sets are an abstraction of pointing.

Page published 2019-08-03. Revised 2023-12-10.