Are 'units' mathematical objects?

Question

I have been studying dimensional analysis and my first question is whether the 'units' i.e the symbols we use are proper mathematical objects, such that the concept of equality makes sense mathematically.

Secondly if we have a unit of metres, $m$, and $2m+2m=4m$ do we see it as a mathematical statement of objects, or as a sort of loose statement like what you would learn at school.

$1apple+1 apple=2 apples$

In this case 'one apple' is a sort of description more than a mathematical object, and the idea that it holds in general is something we learn but theres no specific 'logical rules' here.

When we see something like $1L=1000ml$ is this a statement about mathematical objects as all mathematical formulas are or some kind of loose statement about what happens when you mix certain quantities of things?

You can certainly make units of measure mathematical objects, formalized like any other math. But a lot of math ignores units. So, for example, mathematicians will treat areas and lengths as just numbers. — Thomas Andrews, Nov 18 '22 at 12:59
An amusing way to think of the formulas for the area and circumference of a circle, $A=\pi r^2, C=2\pi r,$ is to think of the radius $r$ as a unit of length, and thus $r^2$ is a unit of area. — Thomas Andrews, Nov 18 '22 at 13:03
@JeanMarie: The OP started out by saying he had been studying that. — Henrik supports the community, Nov 18 '22 at 13:37
Unit is a mathematical term with many different meanings. The units of the ring $\Bbb Z$ are the invertible elements, i.e., ${\pm 1}$. So concerning your title question, definitiely "yes". — Dietrich Burde, Nov 18 '22 at 14:28
@Henriksupportsthecommunity does $2\times apple$ slightly different than the description 'two apples'? In that $2\times apple$ is a mathematical object that we use to represent the idea of having 'two apples'? — , Nov 18 '22 at 22:01
Somehow related to https://math.stackexchange.com/q/3974144. — Paul Frost, Nov 24 '22 at 14:01
Of course they are precise and meaningful mathematical objects: WP has it all. Is there lack of clarity anywhere there? — Cosmas Zachos, Nov 30 '22 at 21:56

Are 'units' mathematical objects?

0 Answers0