I've been seeing the phrase "for definiteness, ..." used in a few proofs recently. It almost seems like a filler phrase. Does this have a general meaning? I had never seen it before and am not sure.
Here is a statement from a proof in which I saw it used:
In that case, either $f(x_3)$ lies between $f(x_1)$ and $f(x_2)$ or $f(x_1)$ lies between $f(x_2)$ and $f(x_3)$. For definiteness assume that the latter is the case.
If more context than this is needed, let me know.
"For definiteness," generally means that a choice is made to further an argument that has branching possibilities.
Notice the subtle difference with WLOG. One makes a choice WLOG to simplify an argument. For instance, one has reduced his proof and he can claim that can make a choice WLOG. On the other hand, a choice for definiteness is necessary, when following an argument presents multiple possibilities. Then one has to consider each possibility so he chooses one possibility "for definiteness," and continues the argument. In my experience, the argument for the remaining possibilities is analogous.
Maybe a choice WLOG is equivalent to a choice for definiteness under some circumstances but I am not aware of this.
e.g. Hooke's Law, $F = kx$ describes a the extension $x$ of a simple helical spring with one end attached, while a force $F$ is "pulling" (for definiteness) on the other end. But Hooke's law for a spring is conventionally (or for definiteness) expressed from the point of view of the restorative force $F^\prime=-F$ the spring exerts on whatever is "pulling" it. In that case, the law becomes $F^\prime = -kx$. Hooke's Law is true up to a sign depending on the point of view and we choose the sign for definiteness.
The term also has technical use as (positive or negative) definite quadratic form, the sign definiteness lemma and possibly others. The technical use should not be confused with the colloquial use.