What is a number? Seemingly popular approach I am using here does not reduce the strength and significant clarity of the definition. Bear with me, and listen carefully :-) You may find out many interesting things!
Here is the clearest approach to defining number.
Now, back to sculpture again. The sculpture can represent anything. The similar thing is with numbers. In mathematics we are dealing with numbers only, the same way sculptor deals only with clay! But, if we want to interpret math results and use math in some other fields then we will have to keep track what we have counted, measured, keep track which objects are numbers, counts associated with! That would be called applied math. And, again, the numbers can represent count of many, many different things.
In developing and understanding a subject, axioms come late. Then in the formal presentations, they come early. - Rueben Hersh.
The view that mathematics is in essence derivations from axioms is backward. In fact, it's wrong. - Rueben Hersh http://books.google.ca/books?id=R-qgdx2A5b0C&printsec=frontcover&dq=what+is+mathematics+really&hl=en&sa=X&ei=6O9BT5RIqdfRAcyqteMH&ved=0CDsQ6AEwAA#v=onepage&q=what%20is%20mathematics%20really&f=false