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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.

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  • 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.
    – eigengrau
    Apr 26 at 10:50
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    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.
    – J Kusin
    Apr 26 at 15:45
  • A number is a specific instance of value. A quantity is any value.
    – RodolfoAP
    Apr 26 at 17:50

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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.

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".
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  • So number is an instance of value because every number is a value? Or because every number has the property of value? A number can be associated to a quantity? So we can say for any quantity at a particular time there's an associated value but for a particular value, there does not exist necessarily a quantity for that value.
    – user58502
    Apr 27 at 8:45
  • Number: written symbol or figure of arithmetic value (most common use). So, yes. Number can be "value". "An instance of value" is to remark the feature in the ontological sense. Your last phrase: correct, because, for example, 100 concepts (a qty.) is not the same as 100 (a number, which is a single concept, that can be part of multiple quantities to provide different senses). Units matter.
    – RodolfoAP
    Apr 27 at 9:10
  • so in essence a quantity is a property to be associated with a number (we may even add a unit as part of this 'association') to say cardinal number = 5 is just associating '5' with the number of elements in the set.
    – user58502
    Apr 27 at 9:11

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