How advanced can a civilization get without zero?

Question

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.

Answer 1

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.

Answer 2

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

Answer 3

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.

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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.