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Typically in natural language we will say something like 'I have two' to denote a group of two things, obviously using the 'mathematical' view of numbers as an abstract object with well defined 'names' that denote them. Being abstract in nature it is impossible for any person to 'have it'.

Do we draw any meaningful ontological distinction between number as used in mathematics and as used casually in natural language settings? For example 'one' can be used as an indefinite description like 'one thing', or simply imprecise language, talking this way is so widespread in everyday situations.

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  • Nominalists would take an issue with natural language "obviously" making numbers into abstract objects. And even many mathematicians favor formalism, where they are just manipulatives in a "game" that do not denote anything. The ontological distinction does not run across natural language vs mathematics, one can give both realist and nominalist interpretations in both. Moreover, casual uses do not implicate any interpretation whatsoever, one merely follows linguistic conventions for stringing words into sentences without a thought of anything ontological.
    – Conifold
    Feb 1, 2023 at 1:06

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Typically in natural language we will say something like 'I have two' to denote a group of two things

No, we don't.

We never say "I have two" as if we had the number 2.

We say "I have two apples" or "I have two oranges", with a noun after the number.

The only situation where we say "I have two" is when the noun is omitted but implicit, in which case "I have two" should really be understood as "I have two objects of the type that was just mentioned", not "I have the number two".

For instance, in a conversation: "A: How many apples do you have? B: I have two."

Here "apple" is omitted, but the sentence only makes sense because it's implicit, and B's response should really be understood as "I have two [apples]."

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  • You make a good point. Your answer would be much better if you explained how it relates to the question. It would be better still if you answered the actual question, at the beginning of the second paragraph.
    – Ludwig V
    Feb 2, 2023 at 12:38
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The implicit claim is that the things we have multiples of are equivalent, swappable. It declares a category. See Natural Kinds.

I would relate real objects to the domain of physics. And abstract objects to the domain of mathematics, with related tools that apply including numberline concepts, and transformation operations like operators. See The Unreasonable Ineffectiveness of Mathematics in most sciences

As per How The Laws of Physics Lie, we can make provable ststements about given abstractions. But, abstractions are not objects, and we must check carefully which abstractions have what validity, or be led into error.

Consider apples. How moldy can they be, before they don't fit the category? A pool of liquid? Digested by worms and mycelium? In which contexts is a peeled apple still 'one', or if it's cored? How thick a layer can you remove before it becomes less than one apple, half the weight? Does a crab apple count? How far can you hybridise the apple, or mutate it, before it stops being in the category 'apple'? Is a plastic apple in the category? This is all in the domain of checking abstractions.

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