.....

According to DBJG (pp. 316-317), there are 3 different uses of の.
1. Particle: 私はトムの（ペン）がほしい = I want Tom's (pen)  (ペン can be omitted if it is obvious from context)  
2. Indefinite pronoun: 私は黒いのがほしい = I want the black one (here, ペン cannot be inserted)
3. Nominalizer (like in your first example).

Interestingly, DBJG gives the following example:
高田さんが使っていたのを覚えていますか。
which can be interpreted as (2) or (3) with different meanings (do you remember that Mr. Takada was using [it]  / do you remember the one Mr. Takada was using)

AFAIK の can stand in for 事 or 物.

In the second example it is doing the latter.

Here is the entry for の in All About the Particles

Quote:

1. Used between two nouns, indicating that the first possession or is modifying the second; also used in place of が to indicate the subject in modifying clauses.
   
2. Used to nominalize verbs and adjectives
   
3. Used at the end of a sentence
   

In your case, as everyone agrees it is case 2: nominalize verbs. The book divides it to 2 types: a)Simple nominalizer "-ing, what", and b) Used as a nominalizer before verbs of perception (e.g. mieru or kikoeru)

example sentence for a)

Quote:外国語を学ぶのは、むずかしいですね。
Learning a foreign language is difficult, isn't it?

example sentence for b)

Quote:女の人が歌っているのが聞こえますね。
You can hear a woman singing, can't you. (Lit., A woman's singing is aubible...)

This info can be found in pg. 70 of the book

You'll also see this with のこと and のもの a lot (meaning that こと and もの get dropped). I think of it as an abbreviation where の is attributing a quality, phrase, or whatever to something that is understood through context.

In both of your example sentences, the の is nominalizing the the clause.
In both 読むのを見ました and 右へ行くのと左へ行くのと
the の is grammatically the same in both of those japanese clauses. You might choose to translate them differently, but how you choose to translate something has no relevance on the original grammar use.
A simple example 「本が好きだ」 and "I like books"; in the japanese 好き is an adjectival-noun, but it's translated into the english verb "to like." The fact that one might choose to translate 好き as a verb has no effect on the original language.

You could interpret the の as being the indefinite pronoun, which then would be modified by the clauses. That would be something like "the ones that go to the right" and "the ones that go to the left."

(2016-01-01, 11:16 pm)RandomQuotes Wrote: Whereas I agree with your general point, I would like to reiterate the clear distinction made in DBJG between "nominalizer" (of the preceding clause) and "indefinite pronoun". This is from a book written by Japanese authors (admittedly, for an English speaking readership), so it is likely not biased by the translation problem you are pointing out.
Could it just be that this is one of those language issues for which different views are possible, even among specialists?

Actually silly me--I forgot what I had previously studied in the book I first used to learn Japanese ("Teach Yourself Japanese").  I went and looked it up.  It says (Lesson 13, p. 64):

Instead of Watasi ga katta hon wa omosiroi desu-- "The book I bought is interesting", we may say Watashi ga katta no wa omosiroi desu--"The one I bought is interesting".  That is to say, if the topic of conversation, here "book", is already known, it may be replaced by the pronoun no, just as in English in these circumstances we can use "one".  This pronoun no is distinct from the particle no, and can refer to things abstract and concrete, time, place, persons.

［準体助］
１ （体言に付いて）下の名詞を表現せず、「のもの」「のこと」の意を表す。「この本、君―だろう」「自分―には記名しておく」

２ （活用語に付いて）その語を名詞と同じ資格にすることを表す。「読む―が速い」「彼を行かせる―はまずい」「こんな―が欲しい」
http://dictionary.goo.ne.jp/jn/171157/me...%E3%81%AE/

So this is の's use as 準体助詞, not 格助詞. But you can't say it's not 'particle' の since it's a type of 助詞 (= particle).

じゅんたい‐じょし【準体助詞】
助詞の種類の一。種々の語に付いてある意味を添え、それの付いた語句を全体として体言と同じ働きをもつものとする。ほとんどが格助詞からの転用。「私のがない」「きれいなのがほしい」「行くのをやめる」の「の」や、「30キロからの重さ」「こうなったからは一歩もひけない」「向こうに着いてからが心配だ」の「から」など
https://kotobank.jp/word/%E6%BA%96%E4%BD...%9E-530025