Numbers themselves are simply conceptual objects, but when does number become a quantity? Is the 'cardinality' of a set a 'quantity'? it is a count but we represent it with just a number that we simply state, is the true 'quantity' actually '5 elements'? in the set, or is the cardinal number itself a quantity? We will say that the cardinality of the set [1,2,3,4,5] is 5 but is this a quantity itself or an associated mathematical object? As numbers can be continuous, ordinals, cardinals and have many uses.
The problem here is etymological, not metaphysical (logical/mathematical).
Quanti-(from quantus) -ty(suffix meaning state of) takes any value, the etymological meaning is "a state of accounting", that is, some state within a universe (e.g. a set of numbers). Number is a specific instance of value, e.g. 3.
So, it can be said that quantity means "number of" (there you have both terms). Then, mathematically, numbers are not associated with units, but quantities do. E.g. volume is a thermodynamic quantity, which corresponds to the number of....
Notice that a number can correspond to a quantity but a quantity can't correspond to a number.
∀q∃n(n∈R) but not
Also notice that the fact that for all quantity there is a number..., does not imply necessarily that the number is known: the quantity of stars in the universe. The reference is to the object, not to its knowledge.
UPDATES, to @user37577 additional questions, introducing "value":
- Values are subjective appreciations. That is, the appreciation every subject grants to the object which holds it.
- So, values are subjective. I will not pay 5 dollars for a Justin Bieber signed record, but many would do it.
- I've said "number can be value". A number is an arithmetic value, which is not the same as the quantitative value. Arithmetically, 100 is not the same as 1, but 100 cents have the same commercial value of 1 dollar.
- Although arithmetic values are also subjective, they can be considered objective, since we all agree on its arithmetic value. They don't exist outside of our heads: numbers would not exist if humans would have never appeared. So, they are subjective. What we call objective is usually "subjectively shared".