Topics Mathematics

Question Are 3 and √9 the same mathematical object (in light of the fact that they have the same numerical value), or are they distinct mathematical objects? In other words, are the expressions '3' and '√9' co-referential names (both referring to the number 3), or do they refer to different objects?

Accepted:
March 21, 2013

Comments

If "√9" refers to the positive square root of 9 (I'm not sure what the convention is concerning the square-root symbol), then I'd say that 3 and √9 are the same object, just as Mark Twain and Samuel Clemens are the same object. (Indeed, the plural verb "are" in each case is a bit of loose talk.) Leibniz's Law (the Indiscernibility of Identicals) therefore implies that everything true of 3 is true of √9, and everything true of Twain is true of Clemens, which seems right to me.

Using Gottlob Frege's theory of sense and reference, you might say that '3' is the name of the natural number that is the third successor of 0, and that 'the (positive) square root of 9' is a (definite) description of that very same number. The name and the description have different senses, but the same referent; the senses "get at" the same mathematical object in two different ways.