Heap's Law

[Anton]

On infinity, remember that proper names and numbers are also words which can appear in a text. Also, a language like Dutch theoretically allows for infinite compounding, though in practice this is just as impossible as an infinite text. Still, with these three tools, language can in fact generate an infinite amount of different words.

Well, it possibly can, but it does not. Of course there are neologisms appearing et cetera, but I understand that the spirit of the "Heaps law" is to consider a "snapshot" of a (vocabulary of a) language at a time when the text under test was created, is not it?

Good point about numbers, especially if we consider them words in themselves, but also without that, one can infinitely invent new terms to designate large numbers - like thousand, million, billion etc. Well, this is also valid for neologisms terming new objects. Philosophically speaking, the variety of material objects is infinite, and so could be infinite language constructs to designate them.

But alas the lifetime of a language is necessarily limited at least to that of mankind , so the vocabulary is fundamentally limited in terms of neologisms and internet usernames. While the size of a text is not fundamentally limited by anything.

But, apart from that philosophy, consider an arbitrary text (of a sufficient length) in a given language. Next, construct another text, say, twice as long as that, but using exactly the same vocabulary. The simplest way to do that would be just to copy and paste. And then, what would the Heaps law tell?  Nay, there's some fundamental flaw about it.

Anton,

there is a relationship between F and D as well.

Ah yes, surely a very good point. Must be where they manage to get their "asymptotiс equivalence" or whatever.

Heap's suggests that the dictionary size continues to increase as the collection adds documents, and that no maximum dictionary size is ever reached.

Qualitatively it's logical to a certain extent (see above), but the Heaps goes beyond and brings forward a qualitative dependance which IMHO is not very well defined to serve as a "law" quantitatively (twiddling K on the order of magnitude!) and is somewhat flawed from the fundamental side (as I suggested above).

So maybe one can make useful conclusions observing that a text does not fit the Heaps law, but maybe one cannot say anything definite if a text does fit it.