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.
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.