Is there a technical term for "meaningfulness" of mathematical operations? Is there a technical term for "meaningfulness" of mathematical operations? 
For example, adding vectors that represent forces has a meaning regardless of the coordinate frame, but an elementwise product does not. 
Multiplying quaternions that represent rotations has a meaning, while adding quaternions does not.
 A: Actually I think "Meaningful" or "Physically Meaningful" in many cases is as good as you're going to get, although the word can split up into finer meanings. If we think of mathematics as a language, then think of words that describe how well the description meets its intended purpose. Does the mathematical description evoke the "right" ideas? So words you might like to think about are:


*

*"Well-formed" or "syntatically correct", i.e. is the expression even "legal" as defined by the relevant axiom system: non-well-formed expressions might be $x\,-$ or $y\,\times$;

*"tautological" (in the logical sense) i.e. is the expression true in every possible interpretation: it may achieve tautology by being a theorem in a consistent axiom system for example. Often the expressions won't be fully tautological, but at least it can be construed as such in the very restricted context of the problem or physical situation at hand.

*"well motivated" or "well supported": there may be a physical argument as to why the proposed expression is meaningful in a given context. There may also be experimental motivation: if you used quaternion addition restricted to the $<i,\,j,\,k>$ subspace to do statics calculations with, experiment would back you up.

*"sound": very often mathematicians make explicit definitions of objects and how they are to be manipulated, so the only meaningful expressions are those that flow from the particular definition / axiom system: this is very like "well formed". As an example, a Hilbert space is a complete inner product space / vector space equal to its topological dual (two equivalent definitions), so arguments flowing from Hilbert spaces behest forming inner products, formation of Cauchy sequences, construction of linear functionals and testing them for continuity and so forth: my point is that the "meaningful" expressions are very tightly, explicitly and obviously defined at the outset. Physicists often do similar things: define an "axiom system" to represent some physical phenomenon and then ferret out "theorems" in the system: in physics there is the further required step that we must test that our inferred theorems are in keeping with experimental observation, but in principle it is very like, at least in principle, the "axiom, definition, lemma, proposition...." flow of a mathematics argument: the "meaningful" ways of putting statements together is decided upon at the beginning. The process that Feynman described as (1) Guess a theory (i.e. lay down your axioms and work out all their implications) and (2) test whether it models reality and (3) if you're wrong, as you almost always are in physics, go back to step (1) and iterate again! 
Keep in mind that mathematics is very much a language, and the kind if ideas you're groping for are not unlike the kinds of words a teacher, child developmental clinician or behaviouralist might use to describe the language they witness as the baby grows, acquires his or her mother tongue, firstly as "non-legal" (at least in our restricted logical way of thinking) sequences and babbles, then as simple sentences, then as accurate descriptions of the immediate world around him or her leading onto complicated stories and allegories wherein many concepts are grasped and wielded at once and accurately woven together to evoke precise ideas in the minds of the other social animals that make up the child's social world. There are many different shades of meaning for "meaningful".
