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This question's pretty simple really. A concept of the number 0 is a major landmark that's used when discussing advanced civilizations in pre-modern history. It was something that civilizations came to use after thousands of years and was a "major advancement".

But why? 3 - 1 = 2, 2 - 1 = 1, and 1 - 1 = ... well, 0. To be clear, I understand that before this, there still would've been concepts such as null or, probably much more universally, there isn't/aren't any. Furthermore a concept of negative numbers seems less natural and intuitive, so I don't put that in the same category. However, 0 is something that is quite visible to the naked eye (as when you run completely out of food, which is something Ancients were in routine fear of). Really, not having 0 seems like quite the elephant in the room, especially among civilizations with full-blown writing systems and reasonably sophisticated architecture, and the reason that this was so difficult to attain to scientifically seems like an elephant the room as well.

Why was the concept of zero so difficult to come by?

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  • $\begingroup$ @SteveBird Thanks! I didn't realize that site existed. I flagged this for migration. $\endgroup$
    – Panzercrisis
    May 1, 2023 at 22:30
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    $\begingroup$ Hindsight is 20/20. What is visible to the naked eye is not something but nothing, the absence of anything (relevant). Conceptualizing an absence as a presence is quite a conceptual leap requiring a high degree of abstraction, and specifically symbolic abstraction. Ancient Greeks, who used geometry as the foundation of mathematical reasoning, not symbolism, did not make it. If subtractive iteration worked so smoothly we should expect negative numbers spring up right after 0, after all, there are debts and shortfalls too. Yet they were still resisted in 19th century, by de Morgan no less. $\endgroup$
    – Conifold
    May 2, 2023 at 2:58

2 Answers 2

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Zero is not a number as long as a number is understood as a count of things. You can have one thing or two things or ten things, but if you have one and take it away, there are no things there to count.

As long as numbers were being used to record tallies of various things -- so many cattle, so much gold, so many enemies slaughtered -- only the positive integers were really needed.

Even today, if I show people my open hand holding two marbles and ask them "How many marbles am I holding?" they'll look at you oddly and say "two". If you hold up your empty hand and ask the same question, they'll look at you even more oddly and say something like "You don't have any marbles in your hand." (It's true that some people will say "Zero marbles" but we all tend to look at them a bit oddly...)

Modern mathematics has at its base the "Natural Numbers" a.k.a. the "counting numbers" which are the integers starting at one and going up. The non-negative integers are a superset of the natural numbers with zero added, and the integers are a superset of those with the negative numbers added. And more complicated numbers — rational, irrational, transcendental, complex, etc. — are built further on that base.

The idea that there should be a number for no objects turns out to be a fairly sophisticated concept.

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    $\begingroup$ Note that the ISO 80000-2 norm deifines the natural numbers as starting at zero : en.wikipedia.org/wiki/ISO/IEC_80000 . This is also how the set IN is defined in mathematics and the habit of not including zero in "positive integers" is probably specific to the English language: in French or German zero is positive (and a natural number)... $\endgroup$
    – Evargalo
    May 2, 2023 at 7:50
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    $\begingroup$ This is the answer I came here to give. The idea of an iterable subtraction operation as mentioned in the question is a long way ahead of the simple purpose of numbers: to count things. Zero raises all kinds of questions. If you're holding zero marbles then you're holding an infinity of zeros of every other conceivable thing! We actually haven't really got an answer for that one yet. So we just say dividing by zero isn't allowed. :) $\endgroup$ May 2, 2023 at 10:54
  • $\begingroup$ @Luke Sawczak Excellent point! As to the other comment, whether or not zeo is a member of the natural numbers is not well defined. You'll find both definitions used in mathematics, and most of the cases I've seen equate the NN to the counting numbers and use an explicit definition of them including zero if that is intended. But if you like you can replace "Natural number" with "counting number" in my answer $\endgroup$
    – Mark Olson
    May 2, 2023 at 12:44
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    $\begingroup$ @MarkOlson: In particular, North American math pedagogy has for at least the past 60 years explicitly defined the Natural Numbers N as just the positive integers, with the set of non-negative integers being defined as the Whole Numbers represented by the letter W. $\endgroup$ May 3, 2023 at 16:16
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The Zero was used as a pure addendum/non-stand alone symbol most of the centuries. F.e. the old Sumerer used a special sign which indicated a multiple of the base number 5000 years ago.

But it was not used as a stand-alone symbol indicating the abstract entity "nothing".

Just like a comma or a space is not used without other symbols - you need to sandwich it. The space is not even visible.

But the Indians used it already as a stand alone symbol for "nothing".

And in modern times some centuries ago, who would fight against the use of the Number Zero in Europe?

Of course the churches in Europe were a reliable censor for many scientific discoveries and progress, as usual.

The church was against the modern concept of the Zero and declared it's use to be blasphemy.

It stands for nothing, but if you write it at the end of a number, it means a multiplication by 10?

And how will you devide through zero?

And where do you find it in the Bible?

That Zero must be pure devil's work. Let's call the Spanish Inquisition.

Luckily the powerful Italian trading families used already modern bookkeeping/accounting in the 17th century, with Debit and Credit sheets.

They would control their financial assets by substracting each 2 corresponding entries - it must deliver zero. And they used a symbol for that non-existing difference.

And it was also in Italy where Fibonacci introduced the arabian/indian number system in the 13th century, replacing the old unusable Roman numeral system, including a first introduction of zero in Europe.

Furthermore, mathematicians developed the system of differentiation and integration in the 17th century, too, where zero can be interpreted as a number getting smaller and smaller in infinite number of steps.

In that aspect, the zero is a kind of counterpart of infinity.

Source: "The Nothing That Is: A Natural History of Zero" by Robert Kaplan.

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    $\begingroup$ What is the source of the Church's statements about zero quoted above? $\endgroup$ May 2, 2023 at 10:32
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    $\begingroup$ @MauroALLEGRANZA I hope it's The Monty Python ! $\endgroup$
    – Evargalo
    May 2, 2023 at 14:09
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    $\begingroup$ @MauroALLEGRANZA I'm pretty confident that's not actually a quote. It's imagined thinking of the church set in a humorous tone. $\endgroup$ May 2, 2023 at 23:53
  • $\begingroup$ @CareyGregory Absolutely correct. It's a kind of fictional quotes reflecting the opposition of the church. And always remember: Nobody expects the Spanish Inquisition! In the beginning, the church/clerical leaders did actually refuse to accept this symbol for zero. Source is mentioned. $\endgroup$
    – xeeka
    May 3, 2023 at 7:02

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