What is a number?

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What is a number?

In mathematics, there is no formal definition for "number".

Really?

We have definition for natural number, negative number, rational number, complex number, real number, p-adic number, hyperreal number, cardinal number, ordinal number, etc. We just don't have the definition for "number".

Isn't mathematics a subject of vigor? How come such an extensively used term have no formal definition?

Mathematics is a subject of vigor, but the use of language is not. Indeed, at the very beginning, "number" just refers to the counting number: 1,2,3,... Latter, the need to record different types of quantities force humans to extend the number system. Just as other branch of knowledge, we like to expand the meaning of the existing names, other than invent a new name-- we do this because we would have a ridiculous large dictionary otherwise.

Indeed, even if we don't mind to have an infinitely large dictionary, we still cannot afford to have one name for one mathematical object. There are much more real numbers than all possible finite combinations of alphabets!