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.
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13bJsMyjiNF21qYTLK5tZizwEovpT6DNeu VerifiedIn 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.
Few mathematicians would have ever thought that until the 20th century. "Axioms" used to mean a self-evident or easily demonstrate truth. Now, because of the blunders modern math, "axioms" mean "any set of assumptions we take as a given and cannot prove."
There's no such thing as "self-evidence"
There are definitely self-evident truths.
"Every thing is exactly the way that it is, and it isn't however it isn't."
Self-evidently true.
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.
Is the distance along the diagonal of a square a quantity?
If we're talking about actually-existing geometric objects - not Euclidean nothings - then sure. A "square" is a composite object, made up of a finite number of points. The "diagonal" is another composite object whose units can be counted.
Ok then if one side of the square has distance 1, then the diagonal has distance sqrt(2). What else would you call the distance?