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I think that for the majority of people throughout history, numbers were inseparable from numerals, i.e. notation for numbers. This would explain why people are

I think that for the majority of people throughout history, numbers were inseparable from numerals, i.e. notation for numbers. This would explain why people are far more comfortable with the notion of real numbers (which despite their name are very very strange in a lot of ways) than imaginary and complex numbers. Even their names betray the difference. However, one has a common notation that everyone has learned whereas the other has a more confusing and less well-known notation. Therefore I think ancient arguments over whether 0 is a number (and acceptance thereof) are representative of a greater paradigmatic shift, similar in essence to the arguments over whether the square root of -1 is truly a "number." Viewed that way 0 is the first step in a journey of an understanding of numbers from purely counting discrete entities, to abstract parts of computation. So basically I would posit that it is in fact a great conceptual leap (just as the negative numbers are) that only seems like an obvious fluke of notation when every schoolchild has learned it.

crazygringo 6y ago
I would clarify that to say numbers were inseparable from words for them. Writing has only existed a short time of our history, so numerals are pretty recent. And the "meaning" of zero as a number like others along a number line, rather than as mere notation for "nothing", I assume only ever became necessary with the invention of negative numbers. With addition, multiplication and division, zero simply does nothing or annihilates a number, and so doesn't need to be treated like other numbers. AFAIK, zero as a number arises first in figuring out how to "get to" negative numbers, e.g. what is two minus four (one, zero, negative one, negative two), where zero is required as a numeric concept. Negative numbers were a big step forwards. Zero, I still don't see it -- either it was just convenient notation for "nothing", or part and parcel of the shift to negative numbers. Unless I'm missing something in the historical record?
- IIAOPSW 6y ago
  You need zero for place based notation. Eg 210 is "two in the hundreds place, one in the tens place, none in the ones place." Place based notation gets you representation of multiples all the way to infinity. Otherwise you need to represent multiples with unique symbols like in Roman Numerals. Eg X means 10 no matter how many symbols are to the right of it.
- jcranmer 6y ago
  From my understanding (and I well could be wrong here), Europeans inherited from Greek a mathematics system that favored geometry and tended to abhor algebra. Concepts like integers, rationals, and irrational numbers are all pretty easy to explore and explain with geometry. By contrast, zero, negative numbers, and imaginary numbers create absurdities in geometry (how can a line have length 0? -2? 3 + 4i?). Moreover, even as algebra is introduced to Europeans via the Arabs, I can see people resisting algebra in part because it introduces these absurdities and paradoxes that need explanation. As far as I can tell from the historical record (and it doesn't help that modern histories tends to describe historical mathematical discoveries in modern terms, meaning it's difficult to work out as a lay person in what terms the historical discoverer understood their own work), it looks like the acceptance of zero, negative numbers, and complex numbers are more or less concurrent, and this also seems to coincide with the shift in mathematics from being predominantly geometric to algebraic.

crazygringo 6y ago

I would clarify that to say numbers were inseparable from words for them. Writing has only existed a short time of our history, so numerals are pretty recent. And the "meaning" of zero as a number like others along a number line, rather than as mere notation for "nothing", I assume only ever became necessary with the invention of negative numbers. With addition, multiplication and division, zero simply does nothing or annihilates a number, and so doesn't need to be treated like other numbers. AFAIK, zero as a number arises first in figuring out how to "get to" negative numbers, e.g. what is two minus four (one, zero, negative one, negative two), where zero is required as a numeric concept. Negative numbers were a big step forwards. Zero, I still don't see it -- either it was just convenient notation for "nothing", or part and parcel of the shift to negative numbers. Unless I'm missing something in the historical record?

IIAOPSW 6y ago
You need zero for place based notation. Eg 210 is "two in the hundreds place, one in the tens place, none in the ones place." Place based notation gets you representation of multiples all the way to infinity. Otherwise you need to represent multiples with unique symbols like in Roman Numerals. Eg X means 10 no matter how many symbols are to the right of it.
jcranmer 6y ago
From my understanding (and I well could be wrong here), Europeans inherited from Greek a mathematics system that favored geometry and tended to abhor algebra. Concepts like integers, rationals, and irrational numbers are all pretty easy to explore and explain with geometry. By contrast, zero, negative numbers, and imaginary numbers create absurdities in geometry (how can a line have length 0? -2? 3 + 4i?). Moreover, even as algebra is introduced to Europeans via the Arabs, I can see people resisting algebra in part because it introduces these absurdities and paradoxes that need explanation. As far as I can tell from the historical record (and it doesn't help that modern histories tends to describe historical mathematical discoveries in modern terms, meaning it's difficult to work out as a lay person in what terms the historical discoverer understood their own work), it looks like the acceptance of zero, negative numbers, and complex numbers are more or less concurrent, and this also seems to coincide with the shift in mathematics from being predominantly geometric to algebraic.