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Here is my question from the title:

Given a (natural) language with its writing system based on an alphabet, are there any theories giving (quantitative) estimates on the number of words the language can (and should) have before the invention of its alphabet?

For example, it seems intuitive that a spoken language cannot hold too many words without having a way to write them down (imagine having to memorize 100000 words without the possibility of saving them for later reference). On the other extreme, if the number of words is too small (say a few hundred), i.e. the language is not rich enough, then it would make less sense to invent an alphabet.

So given the above, there should be some threshold (depending on the language) on the number of words which would necessitate an invention of an alphabet.

I assume my question above should be something well-known, but being an outsider to the field it is a bit hard to locate the correct keywords, so any reference, clarification, and correct keywords will be much appreciated.

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    I don't think your intuitions are likely to be correct. Consider that early civilisations preserved long texts through oral transmission. The Iliad and the Odyssey are almost 28 thousand lines, with about nine thousand unique words. The human memory is good when your train it. And in an oral society you'll never be in a situation when you hear a word without being able to ask someone what it means, because they said it to you! It's not like written language where we can read words we don't know and have to consult a different text to find out the meaning. – curiousdannii Oct 24 '19 at 12:36
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    And if your question is strictly about alphabets and not any writing system, then Chinese and Japanese adults will typically know at least 3-4000 characters, while some Chinese dictionaries have over 100,000 characters. It seems there is no limit that forces the adoption of an alphabetic script over a logographic one. – curiousdannii Oct 24 '19 at 12:47
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    According to this site even five year olds have a typical vocabulary of 10 thousand words. Most 20 year old Americans have a vocab of 42,000 words, but they would be literate. So I have no doubt there would be many non literate languages with vocabs of more than 10 thousand words. I don't know if anyone has studied the exact question you're asking, but through looking at dictionaries for undocumented languages we could get an idea of typical vocab sizes. – curiousdannii Oct 24 '19 at 13:03
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    The alphabet was invented only once in the history of humanity, and users of different languages just borrowed the idea. There are several languages with millions of speakers and centuries of literature which do not use an alphabet as the primary writing system. And it is generally found that languages do not differ in the number of things they are capable to express (if something new become known in the speech community of a language, they just borrow or make up a word for it). So I don't think one can come up with such a threshold. – Nardog Oct 24 '19 at 14:42
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    In the past few centuries, a number of alphabetic writing systems were devised for languages, but that's because missionaries and anthropologists came in contact with them, not because they saw a sudden increase in the number of words. – Nardog Oct 24 '19 at 14:47
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I'm afraid I'm going to have to frame-challenge this one.

For example, it seems intuitive that a spoken language cannot hold too many words without having a way to write them down (imagine having to memorize 100000 words without the possibility of saving them for later reference).

Perhaps surprisingly, this does not seem to be the case! Writing systems don't really affect spoken language as much as one might expect, which is why some linguists consider them separate from the languages themselves. The Homeric epics, the Vedas, the Avestas, the Oral Torah, and many other enormous works were composed centuries before they were written down; in the intervening time, they were passed down through oral tradition. Human memory is capable of amazing things!

On the other extreme, if the number of words is too small (say a few hundred), i.e. the language is not rich enough, then it would make less sense to invent an alphabet.

As far as we know, a true alphabet was invented only once (by the Greeks, modifying the Phoenician abjad). Writing in general has been invented a decent handful of times, but it's usually associated with a need for record-keeping as opposed to any change in the language itself. Most of the time, when a language gains a writing system, it's because missionaries, traders, or other emissaries from literate cultures came in and introduced the idea—again, not usually associated with any change in the language itself.

So given the above, there should be some threshold (depending on the language) on the number of words which would necessitate an invention of an alphabet.

Perhaps most surprisingly of all, there's an axiom in linguistics (as in, it's almost universally assumed to be true, and nobody's ever found a counterexample to it (*)) that no languages are more complex or more complete than others. Anything that can be expressed in English can be expressed in French, Japanese, Swahili, Sumerian, Nahuatl, or any other. Where language came from in the first place is an open question, but nobody's ever found a "primitive" language in the wild.

How many words a given language has is a complicated question—in part because it's really hard to determine what a "word" actually is! But if there's a measurable level of "how many concepts can this language express?", it seems to be a universal constant.


(*) There's one extremely controversial "exception", the Pirahã language spoken in South America, which supposedly violates everything everyone has ever thought was true about linguistics. But only one linguist has made these claims, he hasn't provided much data or evidence for them, and nobody else has been able to replicate his results. So many linguists just ignore that.

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    I like your new (to me) verb 'frame-challenge' and hope you don't mind if I borrow it! – Gaston Ümlaut Oct 24 '19 at 20:50
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    @GastonÜmlaut Go for it! I picked it up on some SE or another, I don't remember which. – Draconis Oct 24 '19 at 20:54
  • @Draconis, thanks for your very informative comments (+1). In the light of the discussion on this thread, I agree that it seems more likely that different cultures became aware of the Greek alphabet (derived from Phoenician abjad and that derived from Egyptian hieroglyphs) and implemented the idea for their own languages. – Hayk Oct 25 '19 at 6:08
  • According to what Dan Everett says, Pirahã doesn't "violate everything everyone has ever thought was true about linguistics". That's just wingnut criticism, and very few take it seriously. It's surprising, but he's demonstrated that there's no reason to assume recursive structures in the language. They also don't have numbers or counting, but that's cultural more than linguistic. Other than that, it's a regular language with all the bells and tones and whistles. – jlawler Oct 25 '19 at 23:30
  • @jlawler He's also claimed that, not only do they not have numbers, they're unable to comprehend numbers, and same with writing. To me, those extraordinary claims require extraordinary evidence, and he hasn't really given extraordinary evidence (plus the fact that nobody else can verify his claims). – Draconis Oct 26 '19 at 17:15

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