Is there any difference in definition of Area in maths and physics?

Question

I am a little bit confused after reading the definition of area :

The area of a shape can be measured by comparing the shape to squares of a fixed size.[2] In the International System of Units (SI), the standard unit of area is the square metre (written as m2), which is the area of a square whose sides are one metre long.[3] A shape with an area of three square metres would have the same area as three such squares. In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.

Is there any difference in definition of Area in maths and physics ?

Area is dimensionless quantity in mathematics, but as per SI unit, it appears it has dimension L^2 as it is metere sq. I am confused :(

Answer 1

This is a rare case where the usual practice in physics is more rigorous than the usual practice in mathematics. In mathematics, we allow both the standard metric and the standard measure in $\mathbb R^2$ to have the same value group, $\mathbb R$. In physics, we ensure that the metric and the measure take values in two different copies of the real numbers, $\mathbb R_{\mathrm m}$ and $\mathbb R_{\mathrm m^2}$, and if you want to follow an isomorphism from one to the other, you have to specify which one you want.

Answer 2

On a fixed Cartesian coordinate space, there is a unit distance specified. This is why units of distance and area are dimensionless in mathematics.

But dimensions must be specified in physics, because the physical universe does not have any particular natural unit distance, so we mere humans have to decide how to specify that unit.

Perhaps we have already specified our units of time, and then we decide to define a unit of distance as the length of the path travelled by light in a vacuum in $1/ 299792458$ seconds, affectionately known as a meter.

Or, someone else might decide that the unit is the distance between the tip of the thumb and the tip of the little finger of Noah's right hand when fully extended, otherwise known as a span.

There are many other possiblities too, hence the many different units of distance (and area, and volume ...).