Zero can be conceptualized at several levels, including as an “absence,” a special category of emptiness, a quantity or as a number used in calculations. Although many animals have a number sense, Nieder, who has studied crows and monkeys, suspects that only humans use zero mathematically.Age 6? In my experience, most adults have trouble grasping the zero. Also negative numbers.Furthermore, numeric zero, as used in mathematics, is something that humans need to learn about from others—it’s not an innate concept. Children generally cannot understand it until about age six. That’s roughly two years later than other numbers.
The notion that zero is somehow distinct comes from studies of brain injury as well. About 14 percent of people who have had a stroke may be unable to read or process numbers that include a zero digit, points out Barnett.
The trouble with tracing the origin of zero is that not everyone agrees about what qualifies as understanding zero. There is zero in the sense of lacking whatever is being counted. There is zero as a placeholder in a positional notation, such as the number 105. And there is zero as the number between the positives and negatives.
I think credit for zero should require having a symbol for it, and having text that describes zero just like any other number.
I believe Ptolemy had tables of angles, where a zero angle was in the table just like positive angles. I don't think he had any negative angles, though.
Ancient people did measurements in directions of North, East, South, West. It seems as if they must have understood that if they go 10 miles east and then 20 miles west, they will be 10 miles west, and that is the same as -10 miles east. And if they go back to where they started, then the net displacement is zero. Maps could thus describe locations with positive and negative coordinates. As far as I know, there is no ancient text or map that does this.
All this must have been obvious to Kepler and Newton. But did they really treat zero just like other numbers? I don't know.
Here is a blog post from the world's smartest mathematician, defining the natural numbers as { 1, 2, 3, ... }. Why no zero? Including zero makes much more logical sense. Obviously Terry Tao understand zero as well as anyone, and yet he has a reluctance to call it a natural number.
One reason for there to be no zero is that zero is often defined as an equivalence class in the set of natural number pairs, (m,n), debit and credit, so to speak, for which m=n. But mathematical taste differs.
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