Is "thousands of thousands" grammatically correct? Why does it seem that "thousands upon thousands" sounds better, even though the former is closer to the logical truth? Is there any difference at all?
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I realize this is an English language question site, but to answer this question in terms of math:
So in simple terms, "thousands of thousands" is a lot more than "thousands upon thousands", although I do agree with you the latter does sound better.
To my ear, thousands upon thousands could just mean "several thousands" or "many thousands".
I've never understood it to mean thousands of thousands, which would amount to millions — which the speaker would probably use, if that was meant.
To my ears, thousands of thousands is a really large number (possibly a million) at a specific moment (as already indicated in other answers)
However, thousands upon thousands might imply a progress in time: a thousand followed by another thousand later followed by yet another thousand even later and so on.
Thousands upon thousands is an undetermined number. Rather than multiplying which would be the case with the word "of" as in thousands of thousands we instead have thousands upon thousands which would indicate adding thousands upon thousands.
With thousands upon thousands we only know that the thousands are plural so the lowest number would be two thousand plus two thousand. From that we know that four thousand is the lowest number we can come up with for thousands upon thousands.
Unless a specific number of thousands is given we have NO idea how many thousands upon how many thousands are being added together. It could mean as little as four thousand to an infinitely higher number of thousands. We only know it is the plural usage of thousands upon thousands.
(thousands upon thousands is equal to thousands added to thousands)
Thousands of thousands is also an indeterminate number because we do not know how many thousands times how many thousands unless a specific number of thousands is given. The lowest number thousands of thousands could represent is two thousand times two thousands which equals four million. I say that is the lowest because of the plurality of "thousands."
Here again the number could be infinitely higher because there is no limitation on how many unlimited thousands of unlimited thousands. The only limitation is each number has to be at least two thousand because of plural usage.
Grammatically speaking, either one is fine. It depends upon what you are trying to say and how you want to say it or how you want it to sound. If the exact number is critical to your message then I suggest using exact numbers. i.e. ten thousands of ten thousands which is one hundred million or ten thousands upon ten thousands which is twenty thousand.
If you want to leave it open to the imagination of the reader either thousands of thousands or thousands upon thousands could be used because either "could be" an infinite number.