Is there a definition of the relationship between real quantities and the numbers we relate to them, generally we use 'numbers' as mathematical objects with a 'proper' nouns, but we associate them with a unit to give a quantity like '5 apples' or '5 metres'. We use numbers like adjectives, which suggests we could define the numbers as a 'state' of quantity.

The issue with this becomes that the 'object' nature of numbers means that we can differentiate between a number as a real or natural. The number '2' that is an element of the real numbers is technically a different object to the number '2' as a natural number. The question arises, when we use a statement such as 'there are two apples', which '2' are we using?

If there are different object we can call '2', we cannot have them as states of quantity, as the same 'state of quantity' can be represented by two different numbers.

In particular if we consider numbers as a level of quantity or state then each quantity can 'instantiate' the number, however if the numbers are distinct from the state of quantity then we struggle with whether a quantity can be a physical token of it.

How do we relate numbers and quantity, especially when giving them in natural language?

  • 1
    Is there any practical difference between treating 2 in 'there are 2 apples' as a natural or a real number? If not, then no problem arises with its use in natural language. In fact, the formal relationship between the natural and the real numbers is constructed in such a way that there be no such problem. Real numbers are a conservative extension of natural ones, whatever is true about 2 as a natural number is retained when it is treated as a real one, we just gain more uses we can make of it. We are free to use whichever is more convenient in a context, there is no need to spell it out.
    – Conifold
    Oct 25, 2022 at 23:28
  • @Conifold perhaps the natural number '2' is used for discrete. and the real number for continuous?
    – Confused
    Oct 30, 2022 at 17:07
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    Does this answer your question? What do quantities describe? Nov 17, 2022 at 11:13

1 Answer 1


I'd relate numberlines to an abstraction of continuous symmetries under transformation, as discussed here: The Unreasonable Ineffectiveness of Mathematics in most sciences

"In this sort of predicament, always ask yourself: How did we learn the meaning of this word ("good", for instance)? From what sort of examples? In what language-games? Then it will be easier for you to see that the word must have a family of meanings." -Wittgenstein, in Philosophical Investigations

We can look through the history of maths, the emergence of zero in Egypt as a baseline that construction could happen above or below, and in India as a placeholder in a decimal system, and then algebra as a further systemising of abstractions, and imaginary numbers as an extension of real numberlines to include orientation. We begin as children with blocks and dice, we teach a series of abstractions, and we have language games around particular mathematical purposes.

What is quantity? It's a way of usefully systemising and abstracting comparisons. We relate it to numberlines, and geometry, and tensors and so on, by agreements and definitions, based on what those allow us to achieve as mathematical purposes. We seek a coherent meaning-cosmology, justified by the uses we put it to. It's like the 'journey of unification' in physics: Is the idea that "Everything is energy" even coherent?

A language is made more powerful by addressing additional topics, and finding interfaces in the modules for different topics that shows consistency between them. We expect math to be unified, like we expect physics to be unified, because our subjectivity is unified. When we find we are unable to speak dolphin, it is because we don't have a subjectively coherent experience of one subjectivity encompassing the dolphin and human modes of life. I see it as a huge source of error to think mathematics is 'just out there'. It crucially depends on intersubjective symmetry, on similar modes of life, in similar kinds of bodies, using similar language games. That is all the 'objectivity' they can have - the reification of intersubjective commonalities.

Tools like rings in algebraic number theory, allow a step back in a similar way that formal logic does for the inferences we draw from the structures of statements, encompassing many different modes of expression and identifying what they share and how they differ, and so what is possible.

  • To describe numbers as a 'level of quantity' is incorrect as abstractly they can exist without reference to quantity? Also as said before they are among other objects (numbers in other number systems) that can define the quantities?
    – Confused
    Oct 26, 2022 at 18:52
  • @user1007028: I said quantity is about systemising and abstracting comparisons, & number is about abstracting continuous symmetries. Imaginary numbers are an extension of real numbers to include rotations, that can also conveniently represent phase. Abstract algebra like rings helps systemise & compare properties of different types of numberlines: they are sets, with certain properties analogous to or including addition & multiplication.
    – CriglCragl
    Oct 26, 2022 at 20:10

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