I was thinking about how we define numbers with respect to their uses, and came up with the definition of 'a level of quantity' which can have a different physical consequence for each quantity measured, for example for counting objects '0' has a certain quantity, as does '1' etc, if we are measuring guage pressure '0pa' has a certain property, but so does -10,000pa. How is this as a defintion of 'numbers' or are they simply abstracted to a sense they can only be used to 'define' a level of quantity by association? Does their various uses and definition as 'sets' give them a more diverse existence?

Numbers can be used as ordinals, or to specify rotations or as themselves an algebraic structure that forms a vector space or ring. We can however, consider the operations acting on my 'levels'.

Is this definition entirely wide of the mark, for example if we take the view that the object named '2' is a different object in the set of reals vs the set of naturals this gives the issue that the two distinict objects yield the same 'quantity'.

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    Does this answer your question? Are numbers real?
    – user14511
    Oct 30, 2022 at 14:06
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    I'm not an expert, but 'level of quantity' seems tautological. It seems reducible to simply 'quantity', which is what numbers are designed to calculate. Oct 30, 2022 at 14:51
  • What about complex numbers? Quaternions? They don't represent quantities but they're numbers, or at least they're considered numbers in contemporary math.
    – user4894
    Oct 30, 2022 at 18:33
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    I don't think anyone says that the definitions of numbers as sets in set theory are metaphysically what numbers actually are. There are other metaphysical reductions such as Frege's reduction of a number to an equivalence class of equinumerous sets, but that's a bit different. No one thinks that 0 really is the empty set, 1 really is the set that contains nothing but the empty set, etc. Oct 30, 2022 at 18:51


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