Subject: What is a number?
Date: Mon, 01 Mar 2000 17:00:45 -0800
From: Michael A. Smith

I have a question which seems fairly simple at first but turns out to be really nasty philosophically: What is the definition of a "number"? I have a Dictionary of Mathematics Terms, but its "definition" starts with the claim that "everybody know about the natural, or counting, numbers." This "everybody knows" method of ascerting truth is a big no-no in math, so the very basis of the book's "definition" is screwy.

I've heard that a number can be defined as an idea which represents a quantity. However, to me this just seems like begging the question: What is a quantity? "Quantity" is an aspect of reality describing how much, yet here it seems the borders between number and non-number get awfully fuzzy (e.g., you have an idea of a chicken, and your idea has a certain amount of chicken-ness; with a little philosophical twisting, it seems that "chicken" is a quantity.)

I have two reasons for asking this insane question. The first is just curiosity. The second has to do with an explanation of division by zero: Because zero isn't really a number, division by zero has as much meaning as division by chicken-ness. This makes some sort of twisted sense to me, but it also grinds against my intuition.