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Is there a difference between a number as an abstract object and as a metric unit used to measure things?

I would put the question slightly differently, if I understand it right: The question is whether the cardinal number 3, used to say how many of something there are, is the same or different from the real number 3, which is used to report the results of measurement. There is of course a different between the cardinal number 3 and a length of three meters, but the question is whether, when one says, "There are three apples" and "This board is three meters long", we refer to the same number three both times. Mathematicians and people who work on foundations of mathematics tend to have different views about this, at least in practice. The way one defines the cardinal numbers in set theory, for example, is very different from how one defines the reals. But working mathematicians will often speak of "identifying" the cardinal with the real and often seem impatient with such niceties as whether they are really the same. A more difficult question, I think, concerns cardinals and ordinals....

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