# Thread: Just for fun: Babylon Tower and Infinity

1. ## Just for fun: Babylon Tower and Infinity

Babylon Tower and Infinity (Linguomathematic Essay)

Many people in different countries learn foreign languages. However, very few of them can proudly say they are able to speak and understand a lot of languages indeed. And there is a question that often appears: What is the total number of the languages existing in our world?
No precise answer can be given. The matter is that it is not always possible to discriminate between languages and dialects. However, according to different scientific sources, the total number of languages varies from 3,000 to 6,000, and that is if not taking into account the so-called “dead” languages. So, is it too many?
Let’s see. The earth’s population is already more than 6,000,000,000 people. Even if we accept the number of languages to be about 6,000 (i.e. at its maximum limit), we’ll get the average quantity of speakers as 1,000,000 per each language. In fact, the quantity of language speakers varies from almost 1,000,000,000 people (Chinese) to less than a hundred and even less than a dozen people (hundreds of nearly extinct languages of Asia, Africa, indigenous peoples in Americas, Australia and so on).

But this article is dedicated to another interesting question which can appear in someone’s sore mind. What is the total number of all the theoretically possible human’s languages? Including all the real languages which do exist now, existed formerly and will exist some day in distant future, including all the artificial languages already created by somebody, or those which will be ever created in perspective, and even the ones which will be never created by anyone but however are possible! That means, what is the maximum mathematical number of human’s languages which could exist anytime in the Universe?
Someone may answer straight off: “Are you crazy? It’s infinity!” But stop! Wait for a while! Certainly, this is an idea that could come into someone’s sore mind only, but I’d like to prove this is not infinity. This is a really huge, super-astronomic number, but it is finite, however. Let’s see…

First, we are talking about human languages, i.e. the languages which can be (even theoretically) used by real human persons for real communication purposes. We are not talking about the languages of any hypothetic creatures and machines.
A human language is a language of sounds. So, let’s first ask ourselves, how many different sounds can be produced by a human throat and perceived by a human ear? It is the fact that the sphere of possible sounds produced by our articulation organs has its definite limits. Of course, each separate sound of speech is just a point in the acoustic space. Therefore, you can say the number of sound points is infinite in the limited sphere of the acoustic space. That is true, but the ability of our ear to distinguish sounds is limited as well. We should only take into consideration the sounds which are possible to be produced and merely distinguishable by human ear. And their number is undoubtedly finite. Moreover, it is not too large. The number of sounds functionally distinguishable in a separate language (i.e. phonemes) varies from 15 to 80 in different languages. The number of sounds in all the Earth’s languages is certainly greater, but I would not think it is more than, let’s say, one thousand. The International Phonetic Transcription chart http://www2.arts.gla.ac.uk/IPA/fullchart.html specifies all the types of phonemes existing in the world’s languages. However, there are more phonemes indeed, since the chart does not specify many possible additional features for each separate type of phoneme (as palatalization, velarization, glottalization, aspiration of consonants, length, nasalization, tension of vowels and many other features). All those features can also make phonemes distinguishable in some of the languages; therefore I suggest that we accept 1,000 possible sounds as a guide.
Well, let’s go on. Our speech consists of words which are composed of sounds. What is the possible length of a word? Let’s say, it is not usually greater than 15 sounds for a single word. We will not consider compound words here, which can be certainly longer, however they are nothing but a mechanical combination of two or more single words.

Thus, we have the ability to construct words of up to 15 sounds each from the total phoneme inventory of approximately 1,000. How many words can be constructed this way? The mathematics say, it will be 1,000^15, which equals to 10^45 (1000^15=(10^3)^15=10^(3*15)=10^45). Of course, not all the theoretic sound combinations are possible to pronounce. For example, we cannot imagine a word consisting of 15 consonants only. Therefore, the real number of possible words will be much fewer, let’s say, 1% of all the theoretical combination. Thus, we get 10^43 phonetically different words in all the possible languages can exist. This number is approximately equal to the number of atoms in the planet of Earth!

But we are talking about possible languages, not separate words. Well, let’s go on. The words are used to signify concepts of human life. How many concepts need to be signified? That is the most difficult question. In everyday’s conversation 5,000-10,000 concepts are quite enough to be expressed by words of a language. Another matter is scientific sphere, publicism, poetry and other special spheres of language application. Some linguists say the vocabulary of a developed literature language may include up to 1,000,000 words. But that amount involves such special things as proper names, professional terms, archaic words and so on, the most of which is completely unknown for an average language speaker. Therefore I suggest that we take 100,000 concepts to be expressed for a well educated speaker is enough.

Thus, we need express 100,000 human life’s concepts using an appropriate set of the total inventory of all the phonetically possible words, which is 10^43. How many ways are possible to do that? The combinatory theory gives the answer. It is Acw, where Acw is the number of allocations of c elements from the set of w elements, c is the number of concepts to be expressed using different words, and w is the number of possible words; c=100,000; w=10^43. Acw=Pc Ccw, where Pc is the number of permutations of c elements, Ccw is the number of combinations of c elements from the set of w elements.
Pc=c! , Ccw=w! / (c! (w-c)!),
so, Acw= c! w! / (c! (w-c)!) = w! /(w-c)! = 10^43! / (10^43-100,000)! ≈ (10^43)^100,000 = 10^4,300,000 ways to assign all the phonetically possible words to 100,000 necessary meanings.

Of course, we cannot consider two languages to be different, if only one thing is named differently in them. And even if there are a few things named differently, we still believe it is exactly the same language, not different ones. There are hundred of things called differently in the United Kingdom and in the United States, but we do call it the same language, English, however in its slightly different versions. Therefore, the number of combinations should be much less than 10^4,300,000 to consider the languages as different ones.

But we did not pay any attention to the grammar differences yet. It is much harder to calculate the number of all the possible grammatical systems. Let’s say there are G possible grammatical systems in total. Let’s accept the number of different vocabularies is 10^4,300,000/X, where X determines the “area of freedom” where we can vary a few words and still believe the languages are the same (see above). Thus, the total number of theoretically possible languages L = G*10^4,300,000/X.

We have no idea what do G and X equal to. Both of them are very large numbers, of course. Therefore, let’s believe G≈X. Thus, we finally get L≈10^4,300,000. We cannot even write this number in the ordinary way (as we write, for example, 10^6 like 1,000,000) since we will need 4,300,000 zeroes in it. It is a super-astronomic quantity which has no physical interpretation. But it is not infinity!

Imagine, there are infinity of people, and every one of them was given the task to create his own language. Everybody is working independently from others. When the task is finished, we will find that there will be people who worked independently, but they constructed exactly the same language, with the same phonetics, grammar and vocabulary! And there will be the infinite number of people constructed the same language. That’s because an infinite value divided by any finite quantity always gives infinity!

We all must be very lucky that the number of all the possible languages is finite. It means it’s possible to learn all of them! And I advise everybody to do it as soon as possible. Imagine you accidentally find yourself inside a blackhole. According to Mr. Einstein’s theory, after falling into the blackhole, you may get out in a completely different place of the Universe, and in an unknown time, somewhere in the past or in the future. Let’s hope some people will save your life there. But what will you do if you don’t understand their language? You will not be even able to explain what happened to you. Do you know what language those people speak? Of course, no, you have no way to foresee it. There is the only one way: learn all of the 10^4,300,000 possible languages beforehand. Do it right now! That’s my useful advice, friends.

2. Imagine, there are infinity of people, and every one of them was given the task to create his own language. Everybody is working independently from others. When the task is finished, we will find that there will be people who worked independently, but they constructed exactly the same language, with the same phonetics, grammar and vocabulary! And there will be the infinite number of people constructed the same language. That’s because an infinite value divided by any finite quantity always gives infinity!
Wha-a-a-at? Don't see a logical connection between the 1st and the 2nd parts of the paragraph. Has anyone comprehended this?

3. Did you write that Bob?

4. Originally Posted by detail
Imagine, there are infinity of people, and every one of them was given the task to create his own language. Everybody is working independently from others. When the task is finished, we will find that there will be people who worked independently, but they constructed exactly the same language, with the same phonetics, grammar and vocabulary! And there will be the infinite number of people constructed the same language. That’s because an infinite value divided by any finite quantity always gives infinity!
Wha-a-a-at? Don't see a logical connection between the 1st and the 2nd parts of the paragraph. Has anyone comprehended this?
The infinite number of people (∞) divided by a finite number of languages they can construct (10^4,300,000) equals the number of people who will construct exactly the same languge. And that equals infinity (∞), of course. That's the logical connection.

5. Originally Posted by Leof
Did you write that Bob?
Sure, just now.

The second part is coming soon: How many ways to say "I love you".

6. Originally Posted by Боб Уайтман
The infinite number of people (∞) divided by a finite number of languages they can construct (10^4,300,000) equals the number of people who will construct exactly the same languge. And that equals infinity (∞), of course. That's the logical connection.
That's a very disputable point of view, aren't you afraid of being blamed for numbers/words/judicions jugglery? Maybe the meanings are figurative? What about the probability of selecting the same basic elements out of such big amount of choices? It is neglible, close to 0. In other words I doubt very much that isolated people can invent the absolutely the same language. Any example? Analysis of constructed languages?

7. Originally Posted by detail
Originally Posted by Боб Уайтман
The infinite number of people (∞) divided by a finite number of languages they can construct (10^4,300,000) equals the number of people who will construct exactly the same languge. And that equals infinity (∞), of course. That's the logical connection.
That's a very disputable point of view, aren't you afraid of being blamed for numbers/words/judicions jugglery? Maybe the meanings are figurative? What about the probability of selecting the same basic elements out of such big amount of choices? It is neglible, close to 0. In other words I doubt very much that isolated people can invent the absolutely the same language. Any example? Analysis of constructed languages?
Have you read the entire article, Detail?
It looks like you have not got its sense. First of all, I prove that the total amount of ALL the possible human languages (including ANY languages that could be EVER constructed) is mathematically finite. It's a very large, super-astronomic number, but it's finite! Do you understand why?
Second, are you acquainted with the mathematical Set Theory? Do you know what is the difference between a finite and an infinite set? If we accept the number of people who construct languages to be infinite in our case, is it possible that everyone's constructed language will be unique? What do you think?

The amount of choices, as you noticed, is really very, very big. But it's finite! And I calculated it here. But the amount of hypothetical people making a choice is infinite. How, in your opinion, is it possible that everyone from the infinite set of people make a unique choice from the finite set of choices?

Well, here's a puzzle for you.
Someone, let's say Mr. X, has very many friends.
Once all of his friends came to see him, and everyone was wearing a hat. When they left him, one hat was left at Mr. X's home, but no one went without a hat.
Another day, they came wearing hats again. And when they left him, one man went without his hat, but no hat was left at Mr. X's home.
One more time, they came again, everyone wearing a hat. When leaving his home, they swapped their hats such way that every odd numbered person was wearing a hat, and every even numbered person was not. But no hat was left at Mr. X's home again.
The last time they came, only some of them wearing hats. But when leaving his home, they swapped their hats such way that everybody had his hat on.
Question: how many friends does Mr. X have?

8. Bob, I did read the article. I agree that the number of choices is finite.
The amount of choices, as you noticed, is really very, very big. But it's finite! And I calculated it here. But the amount of hypothetical people making a choice is infinite. How, in your opinion, is it possible that everyone from the infinite set of people make a unique choice from the finite set of choices?
But if you assume that this would be done isolately (independency of probabilities) and randomly, the probability of the latter event is

(1/<number of elements in the set>)^∞ = 0

So this is impossible.

9. Bob, now I stop disturbing you, and just want to post a link to an article that gives a good model of a similar issue:

Katz, Michael L
Shapiro, Carl
Network Externalities, Competition, and Compatibility
http://ideas.repec.org/a/aea/aecrev/v75 ... 24-40.html

10. Originally Posted by detail
Bob, I did read the article. I agree that the number of choices is finite.

But if you assume that this would be done isolately (independency of probabilities) and randomly, the probability of the latter event is

(1/<number of elements in the set>)^∞ = 0

So this is impossible.
Unfortunately, you still do not understand what I mean. You keep considering only two persons constructing two languages independently. The probability that they get exactly the same result is
0 < (1/<number of elements in the set>) << 1.
In our case, we have an infinite number of persons working simultaneously. That's a very different thing!

Let's take a simpler example. Suppose, every employee of a company was requested to write a number of, let's say, 7 digits. What is the probability that any two of them will write exactly the same number? It approaches zero, of course.
Now, suppose that every person of the world was requested to do the same. There are 6,000,000,000 people in total. There are 10,000,000 choices. Do you still think it is impossible that some people will write the same number?
I guess every number of 7 digits will be written by 600 different people in average! Do you get what I mean?
The matter is in the latter case the number of attempts is many times bigger than the number of all the possible choices!
Hope you should agree with this fact.

11. Okay. I don't want to pester you, but I understood the following piece as independent and isolated work. All right, then I agree.
Everybody is working independently from others. When the task is finished, we will find that there will be people who worked independently, but they constructed exactly the same language,

12. It's interesting, Боб Уайтман, thanks for your thoughts. However you take into account only INVARIANTS of all possible languages. (I temporarily agree that the number is finite). And one cannot just learn an invariant, it's way too impossible. It's great you didn't forget about phonemes. But what about different types of phonetic processes they take part in? What about prosody? Non-verbal systems? Such things can move mountains, and when you comes out a blackhole, I don't think it's possible to understand much of the spoken to you.

13. ok guys, you got me a headache but i have to admit, very interesting theory...

14. Haha, great, linguistic mathematics!

I love "G≈X".

15. At any rate, it's just theory. More than fifteen sounds could be used for any given word in any given made up language, which once again makes the number of possible permutations an infinite number (seeing as how you could use the same sounds and infinite number of times within individual words). Although, I guess actually if we're going with the capabilities of the human mind, that might be incorrect (maybe in the future we're all linguistic superhumans?).

It was a good read! You ARE going to write another one?

16. I agree. Decidedly... entertaining.

17. Originally Posted by джон
At any rate, it's just theory. More than fifteen sounds could be used for any given word in any given made up language, which once again makes the number of possible permutations an infinite number (seeing as how you could use the same sounds and infinite number of times within individual words). Although, I guess actually if we're going with the capabilities of the human mind, that might be incorrect (maybe in the future we're all linguistic superhumans?).

It was a good read! You ARE going to write another one?
I'm thinking about that. It's supposed to be about the collapsing Universe theory and possible ways of saying "I love you" by its inhabitants.
Would anybody like to read it?

18. Post it.

19. Боб Уайтман, спасибо за интересное чтение!

20. Yes, who knew so much hot air could be described in words

#### Posting Permissions

• You may not post new threads
• You may not post replies
• You may not post attachments
• You may not edit your posts
•