What is the formal way to express the meaning of a variable?

Question

I would like to know what is the formal way (if any) of defining the meaning of variables. When I start writing a proof, or if I simply want to establish a formal definition, I usually follow the notation below, but I recently understood that this is most likely not correct:

$F_g \equiv\text{''Magnitude of the gravitational force applied to a body, measured in Newtons"}$

$g \equiv \text{''Average acceleration at Earth's surface, in meters per squared seconds, caused by gravity''}$

$m \equiv\text{''Mass of the body, measured in kilograms''}$

$F_g = g \times m\space,\space\space g=9.8$

How should I express this information in a formal way?

Answer 1

Open any book in the notation section:

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Answer 2

I live by the mantra that math should be written as though it is natural language, punctuation included. So, in your shoes, I would write:

Let $F_g$ be the magnitude of the gravitational force applied to a body, measured in Newtons; let $g$ be the average acceleration at Earth's surface, caused by gravity, measured in meters per squared seconds; and let $m$ be the mass of this body, measured in kilograms. Then $F_g=gm$, where $g\approx9.8$.

Answer 3

One can always nit pick about "formality", or indeed "verifiability", if you want to revive the failed philosophical project, called logical positivism, from the first half of the twentieth century. For example:

"average acceleration at [the] Earth's surface, caused by gravity"

This statement, in mathematical, physical and engineering terms, is quite a claim if you really think about it.

The acceleration measured at the earths surface varies with height above sea level, potentially has other measurable components other than the main (unspecified) vertical one, especially so if you live next to a mountain, and has a component due to the rotation of the earth which varies with latitude.

The question is:

What set of measurements is the acceleration you refer to the (mathematical) average of?