During my studies I observed that while most lecturers try to explain theorems and their proofs, only very few of them try to explain definitions. However, in my opinion, definitions are the base of maths education. In principle, it is possible to take them as given and just work with them, but establishing a link between the content of a definition and the name of the object or property we define seems very important to me. However, while some people asked questions concerning theorems and proofs, I never saw anybody asking about the plausibilty of definitions.
Let me give you an example: The definition of linear (in)dependency. Quoting Wikipedia:
The vectors in a subset $S=\{\vec v_1,\vec v_2,\dots,\vec v_k\}$ of a vector space $V$ are said to be linearly dependent, if there exist scalars $a_1,a_2,\dots,a_k$, not all zero, such that: $$a_1\vec v_1+a_2\vec v_2+\cdots+a_k\vec v_k= \vec 0,$$ where $\vec 0$ denotes the zero vector.
Now, imagine a student asking you: Why do we actually call this property "linear dependency"? Please take a minute or two if you can't come up with an answer immediately before looking into the spoiler below. Hint: The answer "It's a definition and there is no point in asking why a definition is the way it is!" is not the one I had in mind.
A variable $y$ is called dependent on another variable $x$ if there is a function $f$ such that $y=f(x)$. This also works for several dependencies: A variable $y$ is called dependent on other variables $x_1, ..., x_n$, if there is a function $f$ such that $y=f(x_1, ..., x_n)$. This also works for a vector valued function which takes vectors as arguments: A vector variable $\vec y$ is dependent on other vector variables $\vec x_1, ..., \vec x_n$ if there is a function $f$ such that $\vec y=f(\vec x_1, ..., \vec x_n)$. If the function $f$ is linear, it can be written as $f(\vec x_1, ..., \vec x_n)=c_1\vec x_1 + ... + c_n \vec x_n$ for some coefficients $c_1, ..., c_n$. Then it is natural to say that $\vec y$ is linearly dependent on the vector variables $\vec x_1, ..., \vec x_n$, if there are coefficients $c_1, ..., c_n$ such that $\vec y=c_1\vec x_1 + ... + c_n \vec x_n$. Now, how does this relate to the definition quoted above? If there is a non-trivial solution of $a_1\vec v_1+a_2\vec v_2+\cdots+a_k\vec v_k= \vec 0$, there is an $a_i\neq 0$, so we can solve for the corresponding $\vec v_i$ and express it through the other $\vec v_j$, $j\neq i$ and get that $\vec v_i$ is linearly dependent on the other $\vec v_j$ in the sense as described above.
Now that I have written it down, the explanation looks trivial and lengthy to me, but I recall the time when I was wondering what the relation between linear dependency of vectors and dependency in the sense of a functional dependency of two variables is and had a tough time to really pinpoint it. Also, there are definitions (for example from algebra, say, the different kinds of field extensions) where I am still missing the link between the name and the content (hints to good literature for this topic are highly appreciated).
So I am wondering:
- Is it a common phenomenon that giving intuition for definitions, in particular, relating name and content is treated as an orphan, or is it just my subjective perception?
- Would it make sense to spend more time on giving intuition for definitions?
- What are possible reasons for the phenomenon?