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Comment by hencq | Hacker News Reader

It's actually somewhat surprising to me that the Romans didn't invent the concept of zero, when they used the abacus for daily calculations. With the benefit of 20:20 hindsight you'd think that the concept of zero would follow quite naturally from the concept of an empty column on the abacus.

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But that's the thing about Roman numerals: you don't need a placeholder number to represent empty columns. And for "what is XVI subtracted from XVI" they could just use a word meaning "nothing", such as nihil or nihilum. The need for the concept of zero as we understand it really only arises together with a place-value system.

Thanks for explaining that clearly. I've always been so baffled by people who claim that some society didn't have a concept for zero, as if "inventing" zero marks some major advance in intelligence.

Every culture has a concept of "nothing" which works for zero. The ancient Greeks debated over whether nothing was a number or not, but that's just a semantic splitting of hairs.

At some point a symbol for nothing becomes useful so you invent a number-like notation for it. But that's just a matter of convenience. It's not some great conceptual leap.

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.

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?

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.

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.

This is interesting to me.

I suspect most HN readers probably live in a world of binary logic where things are either "true" or "false", but I come from data-land, where ternary logic is the norm, and where "zero" is a very different concept than "null".

"Zero" is something that can be compared to other numbers, mathematical operations can be performed on it, etc., but "nothing" is just that... nothing.

I'm sure the ancient Greeks used one term to refer to both concepts, and that's exactly what people are pointing out when they refer to other cultures "inventing" the concept of zero. These cultures correctly realized that these are two distinct concepts, and so they created a new, separate term for zero.

If you can debate whether or not zero is a number then you haven't invented it.

If zero might not be a number then you can't state that 0 * n = 0 and 0 + n = n and n - n = 0, etc

> Every culture has a concept of "nothing" which works for zero. The ancient Greeks debated over whether nothing was a number or not, but that's just a semantic splitting of hairs.

Nonsense. Zero being a number is the conceptual leap. It's important.

Is zero a number isn’t just a semantic splitting of hairs but has an important outcome. If zero is a number then you can do arithmetic to it. In particular having the concept of zero as a number that you can do maths with is a prerequisite for algebra.

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