Mathematics

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

Comments

Yes, in my view. Suppose there were no difference between the number 3 as an abstract object and the number 3 as used to express a certain length or volume. This would mean that there is no difference between 3 meters and 3, and no difference between 3 and 3 liters. Would it then not follow (by transitivity of no difference) that there is no difference between 3 meters and 3 liters?

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. Where cardinals are used to say how many, ordinals are used to pick out an object by its position in a sequence: first, second, third, and so forth. So here's a somewhat more contentious question: Is the cardinal number one the same as or different from the ordinal number first? In the usual set theoretic constructions, these would be the same; but in some other ways of doing the construction, they would come out different, and for good reason. And in so-called neo-logicist developments of mathematics, they again come out different, though neo-logicist treatments of the ordinals are a bit lacking, so it's hard to say exactly how it is all supposed to go.

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