Elaborating on Davis's remarks: what you have to consider is that mathematics is something of a game, something of an art, and something of a science - to become a mathematician is not to be declared such, but to develop the skill of mathematics.

On the point of zero: just to begin with, consider subtraction. Subtraction is a simple operation, the inversion of addition - because one plus one is two, two minus one is one. When we consider this operation, we discover that there is nothing to prevent us from subtracting a number from itself unless we make a rule such - and why should we? The only reason to make such a rule is to appease an idea of numbers which does not include those less than zero, and that is no reason at all.

So, we call this number - one less than one, two less than two, etc. - zero. We find it behaves as the other numbers do - it may be added, subtracted, multiplied just as other numbers, and that it cannot be divided by suggests a host of new ideas.

Calling zero a number pays dividends. Therefore zero is a number. Quod erat demonstrandum.