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We are discussing the sentence "We can obtain the result by analytic inversion." It's from a math context.

  1. Is this an OK sentence, or does it have to be "an inversion"? If so, why? If not, why? Is this sentence particularly problematic because "inversion" can have both a non-countable (the process of inverting) and a countable (if you have different ways of inverting) interpretation?

  2. If this is an OK sentence, why is the sentence so much different if you replace "inversion" by "algorithm"? Clearly, in this case, you do need "an".

  • If you put "analytic" before "inversion", you should not focus on "inversion" itself. There are many types of inversions, even a "grammatical inversion". That's why you can use it in a plural form. If "analytic inversion" is used as an abstract concept, you should not use the article. – user140086 Nov 19 '15 at 15:38
  • Forget the beginners' rule about countable and uncountable nouns. You can drop the article whenever the noun group without article is as understandable as with the article. – rogermue Dec 17 '15 at 18:12
  • Wow, what does 'understandable' mean? Like in this example: "I would like to rent a vehicle." - "What kind, a car or a bike?" - "I would prefer car." The last sentence is certainly as understandable without an article as with an article, but I would not say that you can drop the article here. – bers Dec 17 '15 at 20:39
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You've more or less answered your own question:

"inversion" can have both a non-countable (the process of inverting) and a countable (if you have different ways of inverting) interpretation

With the first interpretation, "We can obtain the result by analytic inversion" is correct. With the second, "We can obtain the result by an analytic inversion" is correct.
The latter would only be used if you're particularly interested in which analytic inversion is being used. If the point is that analytic inversion is being performed, and you don't care whether it's Fourier inversion, Laplace inversion, etc., then the first version is normally used.

"Algorithm", on the other hand, does not have a non-countable interpretation, so "We can obtain the result by an analytic algorithm" is correct while "We can obtain the result by analytic algorithm" is not.

Source: I am a mathematician, specialising in analysis, and my research involves analytic inversion!

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You can obtain the result by an analytic algorithm for the same reason you can obtain it by an analytic axe.

Likewise, you can obtain the result by analytic inversion for the same reason you can obtain it by analytic cheese.

The former are countable, the latter are not.

Should the latter indeed be countable in your particular context, then you do use an article. If you have many cheeses, then one of them is a cheese. If you have many inversions, then one of them is an inversion.

So, how many cheeses do you have? How many inversions? Are you talking about the uncountable concept or its countable implementations?

  • Ninja'd! But my answer addresses the question in a way more specific to the OP's sentence - I have the advantage of being very familiar with the topic of the sentence :-) – Rand al'Thor Nov 19 '15 at 15:51
  • @randal'thor: what do you mean, "more specific"? Your answer says the exact same thing as mine. It is in no way, shape, or form more specific. And what do you mean, "advantage"? This is a question purely about language, not maths. Lastly, why do you assume you are the only one here who is a mathematician specializing in analysis? Can you tell me where you got the idea from that I am not? – RegDwigнt Nov 19 '15 at 16:46
  • Sorry if I came across as rude - I didn't mean to be! (Re "advantage" - yes, it's about language, but familiarity with the OP's sentence in context is surely helpful. And it'd be a massive coincidence if you're also a mathematician specialising in analysis and inversion ... but if you are, good on you! :-) ) – Rand al'Thor Nov 19 '15 at 16:50
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The sentence is fine as written. Analytic inversion describes a process or procedure, which frequently are not preceded by an article. An algorithm, however is a distinct, countable thing, not a procedure, so it should include an article.

As an example, you would say

He's going in for brain surgery.

rather than a brain surgery.

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