verbs - If Vて+いる isn't a gerund, then what is it?

I always thought that a verb ending in the て form along with the いる suffix was the English equivelent of the "ing" form of a verb.

Thus:

see = 見{み}る, seeing = 見{み}ている

do = する, doing = している

However, according to the Wikipedia entry on gerunds, a gerund in Japanese is when you add the particle の.

Say what?

To be honest, I'm not actually that interested in grammatical technicalities, such as what label is applied to what verb form.

What I am very interested in, though, is usage.

So what, precisely, is the difference in meaning and implication between:

見{み}ている

見{み}ているの

見{み}るの

(This question is a spin off from "What is the difference between は and のは?")

Answer

I think the confusion here arises from the fact that English can use the "-ing" form of a verb in two different ways: using a verb as a noun (gerund), or expressing a continuous action (progressive tense).

In plain language, adding の to a verb in Japanese transforms it into a noun and makes it suitable to be followed by は, が, or various other particles that need to have a noun preceding them:

映画【えいが】を見【み】る。 I will watch a movie.

映画【えいが】を見【み】るの watching a movie (or, "the act of watching a movie")

映画【えいが】を見【み】るのが好【す】きだ。 I like watching movies. (Slightly different translation since the plural is more natural in English.)

映画【えいが】を見【み】るが好【す】きだ。 (incorrect, since が can't follow a verb; it needs a noun.)

So let's throw ～ている into the mix. ～ている shows the continuation of an action, for which English uses the "-ing" form plus a helper verb:

映画【えいが】を見【み】ている。 I am watching a movie.

Translating 見【み】ているの gets a little weird in English, but it can be used like this:

見【み】ているの [the act of] being in the middle of watching

彼【かれ】が映画【えいが】を見【み】ているのを見【み】た。 I saw him [when he was in the middle of] watching a movie.

And of course, remember that ～ている does not always show the continuation of an action in Japanese. It can also show the continuation of a state:

このいすは壊【こわ】れている。 This chair is broken. (not "is breaking")

かばんの中【なか】に携帯【けいたい】が入【はい】っている。 A cell phone is in the bag. (not "is entering")

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Postscript:

こと is the other "nominalizer" (read: noun-making machine) that can be used like の to turn a verb into a noun. Sometimes (but not always!) you can interchange the two:

ピアノを弾【ひ】くのが好【す】きだ。 I like playing the piano.

ピアノを弾【ひ】くことが好【す】きだ。 I like playing the piano.

The full differentiation of の and こと as nominalizers is beyond the scope of this question, but as a quick rule, の is generally used when the outer action happens at the same place or time as the inner action, while こと is generally used when the two can be considered from a removed standpoint lacking immediacy. (For further explanation, please see What is the difference between the nominalizers こと and の?.)