In "I walked past", "past" is an adverb. Correct me if I'm wrong.
Wouldn't it still be a preposition, just one with an implied object?
I'm at a loss for an explanation, Jane. But for what I can tell:
He allowed him through. Here, "though" is an adverb.
A fox slipped through a hole. Here it is a preposition.
Maybe it is in fact modifying the verb. We'll need an authority on the matter.
You summoned the grammar imp?
You're both right, but you're using different theories.
Eryl is using traditional grammar, where parts of speech are defined in an odd mixture of form and function. A preposition is defined by preceding a nounphrase and giving information as to the wehere, where to, when etc. (It's not an adjective, because an adjective precedes a noun and is "part" of the noun phrase: "the blue cat", not "blue the cat" vs. "for the cat" not "the for cat".)
If the word doesn't precede a noun-phrase, it's viewed an adverb. Adverbs - in traditional grammar - are a very large class. (This has implications for the semantics of "to be", too; for example, in "I am here," "here" is an adverb; this often leads traditional grammarians to interpret the sentence as "I exist here".)
Jane's theory is newer (but more convincing to me). In this theory, the wordclass "preposition" functions similarly to verbs - i.e. there are transitive prepositions and intransitive prepositions. So "up" in "Tim climbed up," and "Tim climbed up the tree," differ in that the former is intransitive, so that the direction is all you get, while the latter is transitive so that the direction is viewed relative to the prepositional object "tree". In the "up" might be an implied object, same as there could be one in "I am eating." This is not a syntactic question so much, as it is a semantic one. There is no implied object in "I am here," or "Come here," but "here" is still an intransitive preposition, in this theory.
Note that both approaches have advantages and disadvantages. The second approach is better at dealing with the "piling on" of prepositions:
"He went
out through the door." "He looked
back down into the void."
They generally frame it as a preposition phrase functioning as the object of a preposition:
He went [out [through the door]].
He looked [back[down[into the void]]].
The adverb theory has problems with this, as all the adverbs would have to tie back to the verb, but the problem is that in "through the door" "through" is a preposition, so you'd have a prepositional phrase modifying either an adverb (subordination with "out") or the verb (coordination with "out"). This complicates the theory, as now you make a difference between "adverbial" (as sentence function) and adverb (as word type) that you failed to make when the phrase (the "preposition/adverb") was just a word long. There are lots of papers about this.
The other theory, however, has troubles dealing with "prepositions of time":
For example, "now" looks like the temporal equivalent to "here", but in certain respects it behaves like an adjective, not like a preposition. Take word-order:
Adverb:
He quickly comes. -- Quickly he comes. -- He comes quickly.
Temporal adverb/preposition:
He now comes. -- Now he comes. -- He comes now.
Spatial adverb/preposition:
*He here comes. -- Here he comes. -- He comes here.
No theory pans out completely if you scrutinise them, but I do prefer the second theory that prepositions have transitivity, like verbs.
Grammar imp sings out.