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Babylon Tower and Infinity (Linguomathematic Essay)
Weak-nerved people, do not read, please.
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.
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Originally Posted by detail 
It approaches zero, of course.
