There is no reason to posit "getting closer to a quantity."
Imagine we're talking about English sentences. I can give you a formula to construct a sentence. After each iteration, it contains more words. That doesn't imply it's "approaching" anything.
Sentences aren't associated with quantities, so that's an invalid analogy.
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13bJsMyjiNF21qYTLK5tZizwEovpT6DNeu VerifiedSentences can be associated with quantities, sure. Words are just symbols with a different form.
"If you take three apples and add them to three apples, you'll have a total of six apples." That concept can be expressed, "3a + 3a = 6a".
Mathematical language just uses a different set of symbols. Like sentences, some constructions are valid and some are invalid.
I can write down the symbols "15/0" and say the words "fifteen divided by zero." Doesn't mean that refers to any real quantity.
Ultimately people aren't going to agree on what is valid or not. We are talking about ideas, not about real things. I don't think there is such a thing as a "real" quantity.
In math we define things in terms of axioms. That defines what is valid or not. We can have opinions about whether we like a given set of axioms or not but I don't think we have some automatic shared idea on what is valid or not in mathematics.
A theory that doesn't talk about real things, or makes no attempt at connecting fictions to the world, is a bad theory.
Math can talk about real things, and it can say important things about the world.
"Quantity" is a real feature of the universe.
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