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I've been pondering the question of how we talk about variables '$x$ is a number' is a statement made about $x$ as a number, but we often call "the variable $x$'" as well, for example:

$'x'$ is of type real'

If we use a phrase with '$x$' in a sentence, it implies we are talking about it at a symbolic level, so "$x$' is of type real" means that, "the symbol $x$ is a variable in our language, varying over the reals"

This seems likely as a 'variable' assignment function seems to map from a 'set of symbols' is the 'variable' defined as a symbol that performs a certain role?

If we use '$x$' to refer to something on a higher level to the symbol, but not the value, seems to be confusing, as we can put $x$ inside quotation marks to specify the symbol and not to refer to where objects may be substituted.

Mauro ALLEGRANZA
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  • Relevant questions: MSE Q1703442, MSE Q133572, MSE 1474511, Phil SE Q90054, Phil SE Q36249. – Jam Oct 12 '22 at 19:49
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    The answers posted to the questions I've linked above should provide you with various points of view that answer your question. I think your characterization of a variable as a 'symbol' is a good one, and a view taken by some others on the linked pages. Others similarly liken variables to names. The variable is not an object itself, but it is a placeholder for one, and it is marked the fact that we can substitute an object in its place. – Jam Oct 12 '22 at 20:00
  • I'd also advise, whenever considering questions of "what really is an X?" or "why is X defined that way?", to consider the historical background and where X originally came from. Math is so often based on very palpable geometric and numerical facts despite how they can be obscured by abstractification and philosophizing. In the case of a variable, as user21820's answer tells us, it originated from the symbol for a geometric object (e.g. a line segment) that was dependent on something else. And in that case, I wouldn't see any difference between the name and the object itself. – Jam Oct 12 '22 at 20:08

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