# When does 'number' become 'quantity'?

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.

• Not precisely sure what you're asking/suggesting, but my interpretation would be that a 'quantity' is a number plus 'physical units', e.g., 5 is a number whereas 5kilograms is a quantity. Of course, the units needn't be quite so physical as kilograms. 5 loaves-of-bread would also be a quantity. Apr 26 at 10:50
• In Eulid's Elements we have numbers (natural ones) and magnitudes: they are distinct. The first clear understanding that we can use numbers (and real ones) to measure every magnitude is due to Stevin. Apr 26 at 11:16
• @eigengrau essentially this, if we have just a number, it can't be a quantity, so associating the number '5' with a sets number of elements then the '5' itself cant be a quantity.
– user58502
Apr 26 at 12:43
• Is this a more general question about how can formal mathematical concepts are used outside their syntactical/purely formal nature? Using them for quantities being one case. Apr 26 at 15:45
• A number is a specific instance of value. A quantity is any value. Apr 26 at 17:50

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....

`q={n, u}`

Notice that a number can correspond to a quantity but a quantity can't correspond to a number.

`∀q∃n(n∈R)` but not `∀n∃q`.

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.