Angels and pinheads...

... and pre-sales marketing? In February 2014, someone asked at ResearchGate

Are there infinitely many possible sentences in a natural language? 

Most authors seem to think so. Frege, one of the few mathematicians who worked on that problem spoke of “unpredictably many” (or uncalculably many; “unabsehbar viele”) - not an infinite number of them. (Frege, Logische Untersuchungen, 3. Teil Gedankengefüge, 1993, p.72 ff (German) /see also Fodor/Lepore „Holism“ 1992,p 242).

Further discussion (much much further) here

 The answer, if you value your sanity, is 'Who cares?' Frege's  “unabsehbar viele” is good enough for me.

But not, it seems,  for Max Louwerse Ph.D., whose article in Psychology Today I just came across.  But, not without disingenuousness, he reframed the question to make it less intelligible:

How Many Words and Sentences Do We Know?

Honestly? Did he really ask that? What a stultifying, demeaning, dehumanizing, meaningless question 

<incoherent-rant>
Splutter, splutter. Wha...? No wonder academia gets a bad press sometimes. These people are caricatures of ...[I won't pursue that thought. Don't wrestle a pig. The pig enjoys it and you both get dirty. I shall just count to ten, take a deep breath and... See? Sweetness and light.]

</incoherent-rant>

And, in a way beloved of sub-editors everywhere, the article had a frankly risible corollary:

The number of words and sentences you know
is more impressive than you guessed.

Umm.... no, it's not, as a matter of fact. But, moving on. Dr Louwerse adduces quite a thought-provoking thought experiment

Let’s try to compute the numbers (bear with me). As a thought experiment, let’s first assume that we only have six words we know, rather than those 60,000 words. Let’s assume these six words consist of three nouns  (John, Mary, and Jane) and three verbs (hits, beats, and hugs). From this rather limited vocabulary of six words and a sentence structure English has (noun verb noun), we can generate 27 different sentences (Mary hugs Jane, Mary hits John, Jane beats Mary being three of them). Because the sentence structure could also consist of a noun-verb combination (John hugs), the number increases to 36.

But GIGO: garbage in: garbage out. Or, rather, "homely over-simplification in-[to an infinitely creative system (what Pinker called The Language Instinct): hopelessly incalculable number out":

This [I've ellipted loads of ifs, ans and buts here, but we started out with a thought experiment, so we're talking about an infinite number of back-of-an-envelope approximations: sue me]  means that the possible permutations of a 10-word sentence are over 4,741,000,000,000 sentences. If we now add the number of permutations from a three-word sentence to a 20-word sentence, we end up interpreting over 5,000,000,000,000,000,000,000 sentences. But so far we assume that the order of the word classes in a sentence remains constant. This is obviously not the case. Let’s simplify the situation again and assume that there are only two variations of word order in a sentence. If we only take two-word order variations into account, we can safely assume that we know at least 10,000,000,000,000,000,000,000 sentences or 10 sextillion sentences.

Gosh!  That many! Little me! Squillions and squillions! What's that in Terabytes? What a piece of work is man. And, speaking of pieces of work (in the non-Shakespearean 21st-century sense of "He's a piece of work") there's a moment of truth at the end of the article:

Before you object that it is unfair to count the number of words this way, let alone the number of sentences because once you know the rule system behind words and sentences one should not count words and sentences individually, you are right. But that’s not the point. The point here is to marvel at the impressive talent that we have that often goes unnoticed because it comes so naturally to us. 

Louwerse says that the marvellousness (I've done him the courtesy of repairing his syntax: a point has to be a noun...

<inline_PS>

(or a subordinate clause doing the work of a noun, or "NP" as linguists like to say) 

<inline_PS>

...of the system of natural language is 'the point', throwing dust in the eyes of his pursuers by claiming not to have wasted his time on pointless number-crunching, and finally admitting (cherchez le bouquin?...

Getting a sense of how many words and sentences we know, brings us to the question: How do we keep those words in mind. That’s a question I’ll save for a later post (and for a popular science book [HD: My emphasis] that will come out soon).

...) You'll have to forgive me if I don't join the queue at the bookseller's.

b

Update:2021.04.28.15.05 – Added <inline_PS />