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I'm interested in whether the common view of numbers as 'quantities' is mathematically/philosophically incorrect. If you search the definition of number you get 'quantity'.

Bertrand russell's definition of a 'number', based on the ideas of Frege is that a number is a 'property that must be shared by the same quantities'. This suggests to me that there is an bijection between real quantities under certain contexts and the numbers themselves, but that the numbers aren't themselves 'quantities'.

However as pointed out to me, Bertrand Russel states that:

"It is used to be said that mathematics is the science of 'quantity'. 'Quantity' is a vague word, but for the sake of argument we may replace it by the word 'number'."

In our language we will say thing like 'seven is one more than six', if our numbers are not quantities, how can such a statement make sense? I understand it is simply suggesting that 7=6+1 but the language itself suggests we view the numbers as quantities. If we view 'numbers' as a sort of 'amount' then the statement makes more sense, is this something fundamental about numbers as 'quantities' or simply language from a time when numbers really just represented how many things there are?

Is the statement that 'Numbers are quantities' correct, or is it that there is a 'correspondence' or bijection between them.

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    Does this answer your question? Relationship between real quantities and numbers
    – user14511
    Commented Nov 22, 2022 at 11:27
  • "Bertrand Russell's definition of a 'number' is that a number is a 'property that must be shared by the same quantities" ? See B.Russell, Introduction to Mathematical Philosophy, page 12: "A number is something that characterises certain collections, namely. those that have that number." Commented Nov 22, 2022 at 12:27
  • And see page 195: "It is used to be said that mathematics is the science of 'quantity'. 'Quantity' is a vague word, but for the sake of argument we may replace it by the word 'number'." Commented Nov 22, 2022 at 12:33
  • @MauroALLEGRANZA so russell says that they characterise quantity, which I would agreee with, I haven't seen that part on page 195, would you intepret this as number being quantity?
    – Confused
    Commented Nov 22, 2022 at 17:35

2 Answers 2

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‘Number’ is the quality of having a quantity. Saying I have 2 things and 3 doohickeys is no different in principle than saying I have red things and blue doohickeys. Both statements apply a quality to things and doohickeys. The quality of number has rules that apply to it (addition, subtraction, etc); the quality of color has rules that apply to it (blending, aesthetics, psychological impact, etc).

The confusion here is linguistic. Most of us are quite clear on the distinction between ‘color’ (as the abstract quality of having a color) and ‘color’ (as a specific hue like red or blue). But most of us tend to use the term ‘number’ in the second form only, as a reference to a concrete quantity. The abstract quality of having a number isn’t something we think about much, though we do use it implicitly: e.g., we realize it’s an error to talk about (say) 3 water or 2 grass without a specifier like ‘cup’, ‘serving’, ‘blade’, or ‘field’.

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  • So to say 'seven is one more than six' is another example of this? That kind of language only really makes sense with concrete quantities, '5 apples is one apple more than four apples', we mix the concrete with the abstract?
    – Confused
    Commented Nov 23, 2022 at 7:58
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    @Confused, one “quasi-realist” approach is to suggest that it’s perfectly fine to talk this way about abstract quantities as long as we’re conscious of what it means to consistently talk about the concrete instances. There is something objective about this kind of talk, even if we have reason to be skeptics about the independent reality of “The” number 4.
    – Paul Ross
    Commented Nov 23, 2022 at 8:35
  • @PaulRoss what are the 'concrete instances', every collection of '5 things'? Or just the concrete quantities themselves like '5 apples'? I guess either is fine, but I was just slightly confused by the meaning.
    – Confused
    Commented Nov 23, 2022 at 13:39
  • @Confused: Philosophically speaking, we'd want to say something like this: "For any object-type that has the quality of 'number' (let's call that numerosity), the rules of numerosity say that the quality of 'having 7' has the quality of 'having 1 more' than the quality of 'having 6'." It's atrocious English, which is why no one talks like that, but it's the strictly correct way of putting it, I think. Commented Nov 23, 2022 at 18:11
  • @Confused: It's not fundamentally different from saying: "A specific shade of red mixed with a specific shade of blue in specific proportions creates a specific shade of purple.' We can state that in the abstract as part of the rules of the quality 'color', but it's purely formal manipulation of symbols until we determine the objects and specifics involved. Commented Nov 23, 2022 at 18:17
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Numbers were invented to descibe and communicate the abstract concept of quantity. This invention has been so successful that number and quantity have become synonomous. But its better to say that numbers describe quantities rather than numbers are quantites.

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  • How would this work with fractions? It would seem to me that a fraction only makes sense when represented as in relation to something else (a whole). Conceiving of a fraction as a stand-alone quantity would be difficult, I think.
    – Yorick
    Commented Nov 23, 2022 at 11:45
  • @Yorick if I were to guess, I;d say it 'describes' 'incomplete' amounts, when the unit is known but it is difficult.
    – Confused
    Commented Nov 23, 2022 at 13:40
  • @Yorick Fractions are shorthand for a ratio of two numbers. This ratio allows for the comparison of two quantities or the expression of one quantity relative to another.
    – user59124
    Commented Nov 26, 2022 at 17:51

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