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The medieval Emperor is up to his tricks again. He has decided that the number seven is unlucky. Despite advice from his courtiers that it can't be done, he bans its use on penalty of death by any of his subjects.

This is a medieval society so they can get by without terribly complex mathematics. They mostly do add, subtract, multiply and divide. They don't have a concept of negative numbers. Apart from scribes and sages, most people do arithmetic by lining up groups of pebbles.

Here are some of the dilemmas the subjects face.

If you have eight sheep and one dies then you must immediately kill another one.

The local coinage is the Grundy. You must never be caught with exactly 7 Grundies on your person.

Counting and adding appear to be inconsistent: When children learn to count on their fingers, it goes as follows: 1, 2, 3, 4, 5, 6, 8, 9, 10, 11 ... Thus they all come to the conclusion that they have eleven digits. Except that when they count their hands separately and add, they get 5 + 5 = 10.

Similarly if you add 3 + 4 then the answer is not allowed to be 7, but how to get around this?

Question

Is there any consistent way that arithmetic can legally be done or will numbers just descend into chaos, thus disrupting trade and commerce?

Notes

If you think this kind of legislation is implausible then look at this (thanks to Alexis for drawing my attention): http://www.indianalegalarchive.com/journal/2015/3/14/legislating-pi

You cannot simply invent a new word or symbol for 7. That is just as bad. It is forbidden to have exactly seven of something regardless of what you call that number.

Most people do arithmetic by lining up groups of pebbles. Scribes and sages can use quill and paper for keeping accounts but no-one has yet mastered mental arithmetic.

The Emperor has decreed that from now on there are 6 days in a week.


EDIT

Just to illustrate that the Emperor isn't alone in this sort of madness, have a look at this article: 7 Modern Dictators Way Crazier Than You Thought Possible http://www.cracked.com/article_18850_7-modern-dictators-way-crazier-than-you-thought-possible.html

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closed as off-topic by user535733, Mołot, L.Dutch Dec 10 '18 at 13:59

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  • 1
    $\begingroup$ You might have already heard of this similar case: indianalegalarchive.com/journal/2015/3/14/legislating-pi $\endgroup$ – Alexis Dec 9 '18 at 14:24
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    $\begingroup$ I think there's a fault in the question, at least with your finger-counting example. If npne of your children knows what seven is, how do you enforce that they don't use it? $\endgroup$ – Jedediah Dec 9 '18 at 14:54
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    $\begingroup$ @Giter - Both. Any symbol or combination of symbols that signify 7 is banned and having seven of anything is also banned. $\endgroup$ – chasly from UK Dec 9 '18 at 14:59
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    $\begingroup$ Who is enforcing this? Are there sheepcounters travelling the empire? Realistically, people wouldn't do much different until someone gives them clear rules that are enforced and understandable + can be carried out by the average person, you mentioned a couple. It will take a lot of time to work out those rules. Soon the emperor will pass on and things will be back to normal - unless your people are irrational. I don't think the link adds any "this could happen"to the case. If your people are however irrational, you have a problem: We cannot tell you what will happen anymore $\endgroup$ – Raditz_35 Dec 9 '18 at 14:59
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    $\begingroup$ Just how is this going to be enforced? "You, varlet have seven sheep!" "Eh? Wot you talking about?" "You have seven sheep, in violation of the Emperor's edict! One has to go!" "Wot's 'seven', then?" $\endgroup$ – nzaman Dec 9 '18 at 15:06

11 Answers 11

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Yes - you use a Base 6 system which has been done before

Quite a few cultures used Base 6 counting systems in the past, and it is actually quite logical because it is the natural outcome of counting on one hand.

As well as being popular a long time ago, it is even in use today with some native cultures, such as in Papua New Guinea, Congo and Ural Mountains.

The basic premise is one of your hands has 6 positions for which to count, zero, and then five. Using this method, you never get to the number 7, but instead scroll over to the number 10.

As an example, a counting sequence is like this: 0, 1, 2, 3, 4, 5 .next group. 10, 11, 12, 13, 14, 15 .next group. 20, 21, 22, 23, 24, 25, and so on.

A monk in England called Saint Bede demonstrated the full range of this by counting to 10,000 using this technique. Because of it's ubiquity it is also common in Chinese number gestures.

So you never actually have 7 of anything, you have 10 instead. You don't even have two 7's (14), you have 21. Your emperor simply changes the counting system to suit Base 6, and as has been shown in the past, the system will work out fine (until you get to SI/metric units, hundreds of years later, where it gets very complicated).

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  • $\begingroup$ I thought of this but the question says you can't create another definition for '7' to avoid this. You would just have 12 in base 6 count as an illegal number now too. $\endgroup$ – Lio Elbammalf Dec 9 '18 at 15:24
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    $\begingroup$ @LioElbammalf But it isn't a 'number 7', as you never got to 7 in the first place. Just as we don't get to the 'number A' in a hexadecimal system because we use a decimal system. Just think of it as the number 7 just hasn't been invented yet. Let's say we alter the question to 'Emperor bans the number A in a hexadecimal system, can we still count to 10 perfectly well' then changing to a Decimal system will be the same as we do now, which works out fine, and we don't go around counting 'A'. $\endgroup$ – flox Dec 9 '18 at 15:28
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    $\begingroup$ ‘Natural consequence of counting on one hand’ - So why don’t we use base 11?? :-p $\endgroup$ – Joe Bloggs Dec 9 '18 at 19:01
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Nothing happens, because most folks are smarter than that.

What happens is that everybody at the court simply lies to the Mad Emperor about how great the new law is going over. They probably have long experience spinning such fantasies already, and lots of previous mad edicts probably never made it out the gate either.

Either the Emperor is sane enough to understand that the edict is mad...and so doesn't do it in the first place.

Or the Emperor is barking mad, in which case nobody pays him mind when out of his sight. Usually, Emperors like this get overthrown or usurped rather quickly since they cannot comprehend reality enough to defend themselves, so it might be a very short-term problem (like a spot of poison at tonight's dinner).

In some situations, a coalition at court wants to keep a mad monarch in power for their own purposes. However, since their own wealth and power rely upon accurate accounting, they will certainly not actually implement such foolishness. After all, they have underlings, too. And they eat dinner, too.

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I'm going to attempt an answer to my own question.

Ban all odd numbers

Thus you are allowed to have: 0, 2, 4, 6, 8, 10, etc. sheep.

Similarly count your fingers in pairs - then everything will add up correctly.

This will always work for addition, subtraction and multiplication. Unfortunately there is a problem with division. Let us just hope that the empire can get by without it.

Note - If you lose a finger you'll have to chop one off to match.

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Could they use something like "four-and-three" or "five-and-two," thus technically breaking it up into two different numbers? You haven't counted to seven fingers, you counted to four and then to three.

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  • $\begingroup$ I don't think the Emperor would take kindly to this. He would soon cotton on that everyone was saying "three'n'four" or whatever instead of "seven" and would count that as a new symbol. $\endgroup$ – chasly from UK Dec 9 '18 at 14:56
  • $\begingroup$ Welcome to Worldbuilding.SE. We usually recommend to take the tour. We might also interest you in the help center and Worldbuilding Chat if you have any question. $\endgroup$ – bilbo_pingouin Dec 9 '18 at 15:13
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Strictly speaking, no. There is no possible way of doing this the way you've specified without chaos.

Imagine you have fourteen sheep — officious local lord looks at them, counts them, and once they reach seven, declare one of that set has to die. They start again, get to seven again, and by the end of the process you've lost 8 sheep... except the slaughter house had to stop at six, so two sheep that can't be killed or not-killed are now wandering around like giant wooly Schrödinger's cats.

However, the lack of precise mathematical rigour in your society may save you: those within it wouldn't call it this, but it would effectively be switching to base-6 counting.

You can't have 7 Grundies, but if 6 Grundies makes a Höefer, then everyone who used to have 7 Grundies will now have 1 Höefer and 1 Grundy, which isn't much weirder than pounds/shillings/pence in the UK before 1971.

Likewise, 9 sheep would be {1 {group-of-6} and 3} sheep — this reminds me a bit of French names for numbers.

This doesn't solve the problem of there still being 7 fingers if you hold up 4 on one hand and 3 on the other, but nothing can deal with that.

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  • $\begingroup$ Yes but the lord isn't allowed to say "seven" so he will go from 6 to 8. Counting to base 6 is definitely a useful concept. I think that might be the answer. Would you care to elaborate? $\endgroup$ – chasly from UK Dec 9 '18 at 15:12
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    $\begingroup$ If he's not allowed to say "seven", but there are 7, then he's just renamed 7 as "eight", which you explicitly didn't want to permit. As I write this, I realise I should've called it base-7, for the same reason that 10 is not a digit in base-10. What about base-6 (/base-7) would you like me to elaborate about? $\endgroup$ – BenRW Dec 9 '18 at 19:20
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Let 6 people get caught

The emperor cannot kill the next person or, at some point, he will have created a law which has caught seven people.

The law cannot go on for more than six days

The law itself must be banned on the seventh.

For that matter it can't go on for more than seven seconds (or milliseconds)

What would happen:

Either everyone would follow it and 7 seconds in, one of his advisors would have him arrested for creating a law that had been around for seven seconds.

Or everyone would be aware that the law was terrible and pretend to the emperor that the rule is in place and everyone follows it when, in fact, they just avoid using the number 7 in his presence.

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  • $\begingroup$ Yes but when they are counted it it will go: 1,2,3,4,5,6,8. $\endgroup$ – chasly from UK Dec 9 '18 at 15:14
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    $\begingroup$ But then people are just replacing 7 with 8 and you've got the problem you mentioned earlier with 'four and three' as a replacement $\endgroup$ – Lio Elbammalf Dec 9 '18 at 15:16
  • $\begingroup$ @ Lio Elbammalf - I know. That's the problem! $\endgroup$ – chasly from UK Dec 9 '18 at 15:17
  • $\begingroup$ So all that will happen is he will say '8 actually means 7 so whether we're calling it 7 dead people or 8 dead people the law is in conflict with itself' so the law gets outlawed under its own rules. $\endgroup$ – Lio Elbammalf Dec 9 '18 at 15:20
  • $\begingroup$ The Emperor was advised not to do it and he is hopeless at maths. I'm trying to get my head around what will happen and hence my question. $\endgroup$ – chasly from UK Dec 9 '18 at 15:30
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In a historically accurate feudal society, this would not be the disruption that we think it is in modern day. There are many workarounds to this there were actually used at some point. Some still are in places today.

Bartering and Credit

Many feudal societies did not use cash for trade. They either traded goods directly or kept a ledger of what each customer owed. In a time where it was costly and rare for people to travel far, local small business owners established trust with customers and kept track of debts for long periods of time. When most people worked in agriculture, it was not unusual to wait until the annual harvest to pay off debts.

Bargaining and Haggling

Merchants were wealthy skilled salesmen who convinced customers to buy their goods at the highest price they were willing, they performed many of the roles a multimillion dollar advertising industry does today. In a society where haggling and bargaining is widespread, a ban on a number would be a trivial inconvenience.

“Those are $8 each”

“I’ll give you $14 for 2”

Split Bills or Tipping

It would also be fairly straight forward to split or combine bills to avoid explicitly charging 7 units of currency. Either pay for each item separately if a bill comes to a total with 7 or wait until bill doesn’t contain one to settle the tab. In a society with tipping, it would be fairly easy to round up the bill to avoid charging 7 units as well.

Using valuable goods as currency

It was also common for valuable items, rather than cash, to be used as currency. In the past, gold, silver, rice, and salt have all been used as currency. I’ll pay with this piece of gold that happens to weigh 7 grams but the shopkeeper will weight it discretely.

Using foreign currency

When a country becomes economically unstable, it is commonplace for foreign currencies to be used in place of or alongside the local currency. The US dollar, Euro, and Japanese Yen are often still used in Asian and African countries where hyperinflation is an issue. A dual currency system or black market for foreign currency would be simple to implement if it became impractical to use the local currency. The 2 currencies could also be used interchangeably to avoid using 7s.

“That’ll be ¥13,000”

pays ¥20,000

“Here’s $100 change” (not ¥7000)

Lack of enforcement

A King or Emperor does not have as much power as many believe. Often they were only a figurehead. Even those with real power, required a committee or cabinet of advisors and the loyalty of local Lords to effectively rule. Before modern transport and communications infrastructure and technology, it would be unthinkable for one person to run an entire country, especially one composed of many diverse states. Even an Emperor could convince their own government to enact such a whimsical law, it would be difficult to enforce.

Many states, provinces, and prefectures had far more autonomy than today. This was necessary to prevent revolts in a large disparate Empire. Many Emperors struggled to even collect taxes from the outer regions of their dominions, for example the Tokugawa Shogunate of Japan and the British Colonies in America. This was not a trivial issue to solve. Feudal states often had their own currencies, units of measurement, dialects, and militias. Feudal Lords were in charge of collecting taxes from their domains and passing on a share to the Emperor. Currency was either issued by local banks or measured in local units of weight. It was not uncommon for Lords to be dishonest about how much tax revenue they collected to avoid paying more to the Emperor. (This was one of the issues with Imperial units, they differed between regions). When each region has their own language, it would be nearly impossible to ban a word either. They’ll just use a different one that the imperial authorities don’t know. It would be far to difficult to enforce to be worth the effort for many regional authorities.

Predicted Outcome

If each state is following their own laws and the Emperor has little control over them, how would they even know if they regional Lords were complying with the law or enforcing it. What is likely to happen is that they Emperors decree would largely be ignored. Perhaps a few people will be made an example of but it won’t be widely enforced. For those in the provinces, it will be just another day of politics in the Capital: a moment of vanity of an Emperor with a short-lived reign to be quickly overturned by their successor or a flagrant excuse for the Emperor to purge any who oppose them.

It would be a few days of chaos in the capital before workarounds are in place with little to no impact of trade or the economy as a whole. Feudal societies were already a chaotic patchwork or regional systems and it was already difficult to trade between regions, even within a Empire (so exotic goods were very expensive). A minor inconvenience would not impact such an economic system that is already so complex and inefficient.

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Your question is: "Is there any consistent way that arithmetic can legally be done or will numbers just descend into chaos?"

There is a difference between having 7 stones, sheep or anything else and doing formal arithmetics, unless you use an abacus with 7 pearls on the string. Basically, you have already given the answer to your question yourself, when presenting the loop hole in the argumentation: children/people count "1, 2, 3, 4, 5, 6, 8, 9, 10, 11". This is just counting in an arithmetic system with 9 digits instead of 10, using 9 as its basis - a nonal system, if you want. Therefore, children have 11 (in the 9-system) = 1*9 + 1*1 fingers which is arithmetically perfectly consistent. Our most commonly used number systems are decimal, because humans naturally use 10 fingers for counting. If we had less fingers or like the mayas counted our fingers and toes and consistently used 20 as a basis for the number system, we would consider any other basis than 10 as the natural one. Therefore, there is no reason why an arithmetic number system with 0, 1, 2, 3, 4, 5, 6, 8, 9 as its digits and 10 meaning 1*9 + 0*1 = 9 (in our decimal number system) would not work. Most people can barely count or know any calculations and therefore will not notice the difference. Therefore, you would mainly have to train your money changers and people dealing with foreign currency or lists of goods to transform numbers given in the decimal system into your nonal system. From then on, everything handled within the realm of your kingdom can be handled consistently with your own arithmetical number system.

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Except that when they count their hands separately and add, they get 5 + 5 = 10.

I think you should reconsider that requirement. Because then the answer is just a plain no. The children of your example already found a contradiction, so the system can not possibly be made consistent. In what follows I work an answer that bends a bit the question.


It is just about labels

Let's call this system the hepto-phobic numeral system or HPNS for short. It is unwise for merchants to venture in your emperor's country if they are not aware of the conversions between HPNS and our more familiar system.

Luckily, the Encyclopedia Debilica provides the following guidelines


HPNS works similarly to the numeral system we are used to. Any statement made in one system can also be made in the other. The difference is purely a matter of vocabulary.

Converting numbers from one system to another

As far as naming goes integers from 0 to 6 are the same as usual. Integers from 7 onwards (7, 8, 9, 10, ...) are simply re-labelled 8, 9, 10, 11, etc.

This is why when a child from the Empire tells you that they have 11HPNS fingers, you should understand that they mean 10 fingers.

Below is a python implementation (Encyclopedia Debilica is targeted at educated readers learning about not-so-gifted people) to ease the conversion both ways

def regular_to_HPNS(regular_number):
    if regular_number < 7:
        return regular_number
    else:
        return regular_number + 1

def HPNS_to_regular(HPNS_number):
    if HPNS_number == 7:
       raise PermissionError("This number is not allowed in HPNS.")
    elif HPNS_number < 7:
        return HPNS_number
    else:
        return HPNS_number - 1 

Other operations

When a citizen of the empire ask you to perform an operation (say $3_{HPNS} \times 11_{HPNS} = ? $), the easiest way to go is to convert all the numbers involved to the system you are familiar to, get the result, and lastly convert it back to HPNS.

$$ 3_{HPNS} + 11_{HPNS} = 3 + 10 = 13 = 14_{HPNS}$$

$$ 8_{HPNS} - 1_{HPNS} = 7 - 1 = 6 = 6_{HPNS} $$

$$ 14_{HPNS} - 3_{HPNS} = 13 - 3 = 10 = 11_{HPNS} $$

$$ 3_{HPNS} \times 11_{HPNS} = 3 \times 10 = 30 = 31_{HPNS}$$

$$ 31_{HPNS} / 11_{HPNS} = 30 / 10 = 3 = 3_{HPNS}$$

Of course, citizens of the Empire do not resort to our numeral system. They just never talk about 7 and have learned their multiplication table that way. Notice how the common identities are preserved ($a + b - a = b$, etc.) so formal calculations are not a problem either.

Python implementation

def HPNS_function(regular_function):
    def wrapper(*HPNS_arguments):
        arguments = [HPNS_to_regular(hpns_arg) for hpns_arg in HPNS_arguments]
        result = regular_function(*arguments)
        return regular_to_HPNS(result)
    return wrapper

# Example usage:
def addition(a, b):
    return a + b

@HPNS_function
def HPNS_additon(a_HPNS, b_HPNS):
    return a_HPNS + b_HPNS
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  • $\begingroup$ It'll take me a while to get to grips with this! I'll have to come back to it. $\endgroup$ – chasly from UK Dec 9 '18 at 15:23
  • $\begingroup$ Reading tip: the python is mostly for showing off - though it demonstrate any operation can be made sensical. $\endgroup$ – Alexis Dec 9 '18 at 15:24
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So there's all sorts of fun solutions, like base 6, or mapping Z to Z/2 (that's the formal way of saying "ban all odd numbers" from chasly's self answer). We could also operate in environments which aren't fields.

But this question caught my eye, because I've already asked it. Well, I asked something similar. On Music Theory, I asked whether Chinese music avoids counting in 4's, given that 4 is bad luck (it is a homophone for the word for "death"). It seemed tricky to my Western mind, given that 4/4 music is so popular. The answer showed the kind of creativity that shows up.

One way of expressing meter in traditional Chinese music is in terms of ban and yan - 'beats' and 'eyes'. The 'ban' represents the main beat, or the pulse of the bar, while the 'yan' (eye) represents a weak beat. Some common meters were

One ban followed by three yans : ban - yan - yan - yan - ban - yan - yan - yan
Alternating : ban - yan - ban - yan
Constant strong beat : ban - ban - ban - ban

You can probably see the similarity of the first one to a time signature involving the dreaded unlucky number you mentioned - but expressing it in this way, we've been able to avoid mentioning it!

Now I find this interesting for a few reasons. First off, it's not just fantasy -- they actually count that way. The second is that it looks an awful lot like "counting in base 4," but it points out a subtle detail. They don't think of it that way. They think of it as counting something larger, rather than thinking of it as counting a fixed-size collection. It doesn't really matter how many eyes fit under a beat (as long as it's not four!)

This seems no more unreasonable than using the phrase "He Who Shall Not Be Named" to get around naming someone. The engineer in me thinks this is silly, because you just gave him a name that happens to have 6 words in it... but linguistically, we find people just don't work that way.

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If it's just about the word and symbol there's no real problem. If it's about the visual alignement of equal or equivalent objects then you can resort to gratuity (destruction of goods is not necessary). They can give things away and establish relationships based on mutual favors. But first you will try to storage things in another place. Very poor people will have a hard time having legitimate storage alternatives, but they can bury objects in the common lands. At the market place, traders will round up by pricing either 6 or 8 but better off will always offer to by two for the price of forteen. Of course if it's about the idea or concept of seven then you have to get rid of the decimal system and use hexadecimal described by flux.

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