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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?

cinico
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    "this is most likely not correct": can you elaborate ? By the way, there is nothing formal here, just natural language. –  Apr 04 '19 at 08:03
  • @YvesDaoust I assumed until recently that the symbol $\equiv$ meant "is defined as", when actually the notation ":=" is more correct. Furthermore, as you said, the fact that I'm mixing mathematical notation with natural language doesn't seem very consistent to me. I think I accept the answer that says that, for this type of intent, the best way is to use exclusivelly natural language, not mathematical notation. – cinico Apr 04 '19 at 09:23
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    Whenever you do write chunks of text in a LaTeX formular (which you shouldn't do in this case, but sometimes it can be appropriate), make sure you surround it with \text{ }. See $m \equiv \text{Force}$ vs $m \equiv Force$ (yuk). Also, quoting in LaTeX should be written ``Force'' (two backticks on the left, two single apostrophes on the right) to properly appear as “Force” in the rendered document. – leftaroundabout Apr 04 '19 at 12:00
  • @leftaroundabout Thanks for teaching me! :) – cinico Apr 04 '19 at 12:57
  • Thanks @ToddWilcox. It was a copy/paste thing. Fixed. – cinico Apr 04 '19 at 17:33

3 Answers3

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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 $g\approx9.8$.

Community
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wjmolina
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  • Can confirm. Scientific articles always (well, should always) explain the meanings of their variables in plain language. – 5xum Apr 04 '19 at 08:01
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    Agree with the natural-language mantra. However, I'd prefer leaving out the units in the description, after all the equation holds independent of the units: "Left $F$ be the [,..] force, let $g$ be the acceleration [...] then $F=gm$ where $g\approx 9.8 \mathrm{m}/\text{s}^2$..." – Toffomat Apr 04 '19 at 08:43
  • Thanks. While I am aware of this way of expressing the meaning, somehow I miss a more clean (almost bullet type) way of stating the definitions. I accept that it's best to use exclusivelly natural language for the definition, and nothing forbids me to format the text to more clean way :) – cinico Apr 04 '19 at 09:26
  • @cinico: There's nothing wrong with putting Cleric's answer into point form, one point for each definition. I don't think it's fair to claim that mathematics can ever be written as though it is natural language. It isn't natural and never will be. However, it is best expressed in semi-natural language, mixing natural language and mathematical symbols in a way that is most suitable for reader consumption. – user21820 Apr 04 '19 at 12:07
  • In the quoted section, did you mean to write "and let $m$ be the mass of the body in question, in kilograms"? Not only are kilograms units of mass, not force, but if $m$ represents mass and not force, then the formula stated makes more sense. Oh I just noticed you were quoting the question. Might make sense to fix it anyway to prevent confusion. – Todd Wilcox Apr 04 '19 at 15:15
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Open any book in the notation section:

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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?

James Arathoon
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