Good sources on the construction of mathematical definitions?

Question

Please forgive sloppiness/ignorance, I have no formal background and am basically a fan. I've recently run into some problems by carelessly including statements of uniqueness and existence in definitions for mathematical objects. I'm now wondering what kinds of sources are out there with good, detailed info about the construction of mathematical definitions generally.

Answer 1

In principle, you can define anything you want, but in real life we want to define only useful things. So rather than constructing it, a definition should be motivated.

One widespread pattern is that one discovers a theorem that shows that some a priori somewhat unrelated properties something can have are equivalent. This then motivates coining a new name for this property.

A quick abridged example:

Theorem 1. Let $V$ be a vector space and $S\subseteq V$ a subset. Then the following are equivalent

1. $S$ is linearly independent and no proper superset is linearly independent
2. $S$ generates $V$ and no proper subset generates $V$
3. $S$ is both linearly independent and generates $V$

Proof. We have $1\implies 2$ because yada yada yada. We have $2\implies 3$ becasue yada yada yada. We have $3\implies 1$ because yada yada yada. $\square$

Definition. A subset $S$ of a vector space $V$ that has either (and hence all) of the properties in theorem 1 is called a basis of $V$.

Another motivation is existence and uniqueness, which you mention. Again, we start with a theorem that states that for a given object some unique other object with a specific property exists. Essentially, this allows us to define a map that associates this other object with the former.

Example:

Theorem 2. Let $A$ be an $n\times n$ matrix with $\det A\ne 0$. Then there exists a unique matrix $B$ with $BA=AB=I$.

Proof. a) Uniqueness is clear because yada yada yada. b) Existence follows from yada yada yada. $\square$

Definition. Given an $n\times n$ matrix $A$ with $\det A\ne 0$, we call the unique matrix $B$ that exists according to theorem 2 the inverse matrix of $A$, written as $A^{-1}$.