Clarification on meaning of scalar in math and scalar in physics When a mathematician says something is a scalar, say on the plane, they mean that it associates to points on the plane real numbers. When a physicist says something is a scalar, they mean that if we put a cartesian coordinate system on the plane and look at the value of this thing, and then rotate the coordinate system and look at the value again, the two values agree. But in the math definition there would be no reason to mention rotations, or the metric, or isometries of the metric, these are irrelevant to the definition of a scalar. 
Now I know there are some things that are scalars under rotations but not under rotations + reflections, which would seem to mean that such a thing couldn't be considered a scalar in the math sense, because scalars in math are manifestly coordinate independent. However if we were only interested in rotations of the coordinate system, then a physicist would call it a scalar.
So it seems to me that physicists only talk about scalars with respect to certain group actions, which don't enter the picture when a mathematician talks about scalars, and this leaves me a bit confused as to in what sense they are the same.
 A: When talking about scalars, mathematicians usually use your definition, that is, something which doesn't vary with coordinate changes. (Basically, that there's some mapping to the actual points in space, in which the scalar is well defined)
When physicists talk about scalars, we usually refer to Lorentz scalars, which requires two things:


*

*Invariance under Lorentz transformations (Rotations and boosts)

*Invariance under parity transformation


If an object satisfies requirement #1, but not #2, it's not a scalar, it's a pseudoscalar.
These definitions of scalars in mathematics and physics are the same, but I've never seen a mathematician talk about pseudoscalars.
For some info on this, see this and this and this 
A: Mathematicians don't talk about rotations or isometries or whatnot because they already know if they're talking about a scalar or something else; a physicist has to determine whether a physical quantity has the properties of a scalar or a vector or something else.
The easiest way to do that is to look at transformation laws--to devise some experiment (real or conceptual) that would distinguish between a scalar field and a vector field, or between a scalar field and a pseudoscalar field, and so on.  Rotations and reflections are such conceptual "experiments."
It's true that physicists sometimes conflate true scalars and pseudoscalars (or gloss over the differences), but the underlying pure mathematics definition still works fine here.  Physicists just tend to emphasize or focus on certain logical consequences of that concept.
