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More or less exactly what it says on the tin. Start with a group of cavemen on prehistoric earth, discovering fire, language, wheels, etc, and walk them along the path to civilization, but with one major difference: at no point will anyone ever think of the concept of mathematical zero.

How big of an obstacle would this be? Exactly how many technologies could have been invented without zero-inclusive math? You could have things like pottery, smelting, and agriculture all just fine. You might even be able to stumble into some of the more basic technologies such as the printing press or the cotton gin without any awareness of zero. But things like physics or economics would be tricky, or maybe even impossible. Exactly how much of our modern society is reliant in some way upon zero, and how far could we have gotten without it?

Some details:

  • Base 10 doesn't exist, we're probably running off base 9
  • Anything that involves math more complicated than basic algebra is completely out of the question (orbital physics, advanced electronics, etc)
  • Anything that could feasibly be figured out without doing the math first is fair game, but not if it has a prerequisite discovery that would require math. So for example, you could probably invent a steam engine just by knowing enough about steam pressure and how it behaves in a confined space, but you probably couldn't invent a radio. Also, it might not be a very good steam engine, because you might not be easily able to calculate the forces acting upon the metal that makes the boiler, so the whole thing might explode.
  • A lot of this will probably have to do with larger organizational structures. In other words, it's not inventing the train, it's making sure the train runs on time.
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    $\begingroup$ Any standard place value system needs a zero, whether it's base 10 or base 9 or base 27. (In standard base 9, you use the symbols "10" to represent 8+1.) $\endgroup$ – Draconis Jul 6 '18 at 4:48
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    $\begingroup$ One thing is not having a symbol representing zero, and another thing is not having the abstract concept of "nothing". Even a toddler knows that if he has a toy, and you take that toy away from him, he's left with no toys. And he cries. All math starts with counting, and even the simplest animals know the difference between "something" and "nothing". $\endgroup$ – Rekesoft Jul 6 '18 at 7:38
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    $\begingroup$ Every base is base 10. $\endgroup$ – jkej Jul 6 '18 at 8:35
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    $\begingroup$ This is quite an interesting question, but several of your premises are incorrect. Base 10 was developed without a zero (Greeks, at least.) They also developed maths more complex than arithmetic: geometry, trigonometry, and "method of exhaustion" which is proto-calculus. The Greeks also invented a steam engine before they had a zero -- one of the big questions of history is why they never developed it further. Incidentally, as others have commented, most ancient civilizations did have a concept of nullity; the thing that had to be invented was positional zero. $\endgroup$ – Securiger Jul 6 '18 at 11:51
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    $\begingroup$ One of the less well-known causes of the fall of Roman Civilization was that, lacking zero, they had no way for their C programs to exit successfully (This joke is almost as old as C or the Romans; I can't claim credit for it) $\endgroup$ – MSalters Jul 6 '18 at 14:38
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I think the question is not so much what we can do without zero, but how zero could remain undiscovered when humans begin to advance.

One of my favourite quotes from one of my favourite contemporary mathematicians (Roger Penrose) is that it's always possible to create an equations from numbers of a given type whose answer falls beyond that type:

  • Positive Integers: $1-1=a$ or $1-6=a$
  • All Integers: $\frac12=a $
  • Rationals: $\sqrt2 =a$
  • Irrationals: $\sqrt{-1}=a$

Before science, the primary driver of mathematics was commerce. How can we possibly pay off an account if there is no concept of zero? How do we record balances as fully paid?

It's not so much that we couldn't advance without the concept of zero, it's more that discovering the concept of zero was always going to be a byproduct of advancement. Conceptually, it was always going to appear in mathematics because it's a necessary concept on which we build additional foundations.

I'd argue that the Greeks' initial struggle with zero as a concept only pushed back western civilisation several centuries, rather than truly impeding it. When it was (re)introduced to European culture during the middle ages by the Spanish Moors, it was embraced as a necessity. That Egypt (for example) had the concept of zero nearly 2 millenia before the time of Jesus should identify it as a concept that was always inevitable.

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    $\begingroup$ For discovering the irrationals, sqrt(2) is an easier one that stays within the bounds of algebra. $\endgroup$ – Draconis Jul 6 '18 at 4:41
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    $\begingroup$ About the last paragraph: it would be debatable that they had no concept of zero. They probably knew pretty well that there can exist zero amount of something. It's how they symbolize it within their writing system and their numeral representations which was the big problem, not that they had no idea about the concept that there can be zero amount of something. $\endgroup$ – vsz Jul 6 '18 at 7:05
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    $\begingroup$ @vsz No, they precisely had no idea of 'zero amount of something'. They had the idea that there was 'an amount of something' and 'nothing'. To them, it was a category error, not a point on a unified gradient. $\endgroup$ – lly Jul 6 '18 at 9:09
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    $\begingroup$ Similarly, 'how do we record balances as fully paid?' You erase, cross, stamp, etc. the ledger entry. There is no debt, not there is zero amount of some debt. $\endgroup$ – lly Jul 6 '18 at 9:13
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    $\begingroup$ Those equations should actually read: ● “Positive integers: 1 = 1 + a or 1 = 6 + a”, ● “All integers: 1 = 2 · a”, ● “Rationals: let a be the shortest length a closed path around some point p can have without ever approaching p to a distance less than ½”, ● “Irrationals: a ² = -1”. Because, concepts like the division operator or the square root on negatives are just not meaningful before you define what properties this is actually supposed to fulfill. $\endgroup$ – leftaroundabout Jul 6 '18 at 16:11
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The lack of a zero would not limit mathematics as much as you would think.

Contrary to popular belief, it is possible to have a place-sensitive number system without any digit to represent zero. It's slightly cumbersome, but can represent any rational number except zero itself.

How it works:

(Note: I'm going to first give examples in base ten, and then show how the concept works equally well in base nine. I think the examples would be too hard to read otherwise.)

If you eliminate zero, you just need to introduce an additional digit to represent the number base. For example, in base ten, the digit for ten could be "X".

So, to count from one to twenty-five (in base ten), you would have:

1, 2, 3, 4, 5, 6, 7, 8, 9, X, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1X, 21, 22, 23, 24, 25

Note that "1X" represents twenty, even though it starts with a 1. Specifically, "1X" means you have ten ones ("X" in the ones column), and one ten ("1" in the tens column). If we grew up with this system, we would probably call this number "ten-teen" or the like.

Counting from 95 to 115 would look like this:

95, 96, 97, 98, 99, 9X, X1, X2, X3, X4, X5, X6, X7, X8, X9, XX, 111, 112, 113, 114, 115

"9X" represents one hundred ("nine tens and ten ones"), and would probably be pronounced something like "ninety-ten".

You could represent non-integer values as simple fractions:

$\frac{195X1}{1385X}$

You could also use a decimal point with this system, but it's a little cumbersome, because you would have to use scientific notation for numbers less than one:

1.234567 = 1.234567
2.01 = 1.X1
0.00201 = 2.01×10-3 = 1.X1×X-3

All of the above examples are in base ten, but of course you can use any number base you wish. For example, in base nine, the first twenty-three positive integers would be written as:

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25

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    $\begingroup$ It's difficult to imagine scientific notation (which requires exponents) being developed without the concept of zero. $\endgroup$ – Draconis Jul 6 '18 at 4:47
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    $\begingroup$ @Draconis I don't find it too hard to imagine. Exponents are just an extension of the concept of multiplication, and multiplication works fine without zero. Plus you don't really need to understand exponents to use the idea behind scientific notation. All you really need to know is that this number needs to be shifted over three places when you add it to that number. $\endgroup$ – plasticinsect Jul 6 '18 at 5:09
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    $\begingroup$ I don't see a need to force a number system similar to our own, where a zero digit would more naturally fit... consider something based on a loose analogue of Roman numerals in base 9... I, II, III, IIII, IIIIN, IIIN, IIN, IN, N, NI, NII, NIII, NIIII, NIIIIN etc. "nothing" / "empty" is just a blank, and isn't a number. Powers of 9 (or maybe 3) get unique symbols. Simple fractions with powers of 9 as the denominator are equivalent to "scientific notation". As zero cannot appear as the demoninator of a simple fraction, this further confirms the concept does not correspond with a number. $\endgroup$ – Steve Jul 6 '18 at 8:01
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    $\begingroup$ @Steve The major disadvantage of roman-style numerals is that they are not place-sensitive. Because of that, it is very difficult to do arithmetic with them. Place sensitive numerals (including both the system we are familiar with, and the system described in my answer) make it possible to do things like long multiplication, division, addition, subtraction, etc. This is a huge advantage. $\endgroup$ – plasticinsect Jul 6 '18 at 8:37
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    $\begingroup$ @Steve: "Roman-style" numerals have never been used for mathematics. They were used for writing numerical values in text. For mathematics the Romans used the Greek system, which is very similar to what this answer describes. (And which included a symbol for "nothing", used in mathematical tables.) $\endgroup$ – AlexP Jul 6 '18 at 11:41
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One way in which the concept of zero could remain undiscovered: early availability of computers based on a floating point system. Easiest explanation would be for the computers to come from either an alien civilization or long lost, more advanced human population.

Our current standard for floating point, IEE 754, has a special case for zero values. The normal representation can handle numbers down to 10⁻³⁸ = 0.000 000 000 000 000 000 000 000 000 000 000 000 001. Numbers smaller than this are stored as denormal numbers, of which zero is a further special case.

The lack of zero as a concept is inherent in floating point representations that try to minimize the required storage space. Because all other numbers have at least one 1 bit, the first 1 bit is not stored in the format. Note however that zero as a digit still exists.

Now, if people didn't have pre-existing concept of zero, would they discover it from the computers? It seems likely that at least for a while, they'd just consider anything that small insignificant to discuss, and the difference between 10⁻³⁸ and 0 would be lost on them.


With this background, how far could they advance?

I'd say they could manage even complex calculations fine, by delegating them to the machine. There would be a lack of interest in developing manual computation techniques when computers do it so much faster.

However, deep research into mathematics would cause zero to be discovered eventually. Thus any technology that requires deep knowledge of mathematics or physics to construct would be ruled out, though like computers, they could operate it if they got it ready-made.

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    $\begingroup$ This becomes a chicken-egg dilemma. How would they get floating point computers without zero, is not disclosed. -1 $\endgroup$ – Mindwin Jul 6 '18 at 13:52
  • $\begingroup$ If we assume these computers were designed by some other civilization that did have a concept of zero (which is, I think, a necessary assumption for them to have invented computers) why wouldn't the computers be designed to represent and display zero? It seems like the young civilization that discovers/inherits these computers would be forced to learn about zero in order to use them at all. $\endgroup$ – plasticinsect Jul 6 '18 at 16:48
  • $\begingroup$ I think floating point can be easily represented as heads/tails, hands/feet, left/right. So the concept of zero would not need to be used as a concept for the mechanical manipulation of the system. The "Paddy-cake" game could be a calculation. $\endgroup$ – Yorik Jul 6 '18 at 17:49
  • $\begingroup$ @plasticinsect: The idea is that the computers would represent zero as denormal value / special case, but for a user that doesn't understand zero, any value so small would seem insignificant and they wouldn't bother to learn the difference between true zero and very small. $\endgroup$ – jpa Jul 6 '18 at 18:08
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    $\begingroup$ While floating point calculations can sidestep the issue of zero for some time, the concept of zero is absolutely there. "1.-1." should print "0.", and "1./(1.-1.)" should result in error or "infinity", if software is handling this condition. However, software can be deliberately designed to hide 0 from the user, but if requires someone who understands "zero" very well to do that. $\endgroup$ – Alexander Jul 6 '18 at 18:23

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