grammar - Is "to" really part of the infinitive?

Consider this:

I like to eat here.

vs

I would eat here.

It appears to me that "to" has nothing to do with the infinitive form of the verb that follows. It is, in this example, an integral part of like to, not of to eat.

Is my thinking flawed? If you think it is, could you please explain which form of the verb "eat" is used in the second example, if not the infinitive?

Answer

It is probably a matter of definition, not of true inherent "belonging to". I believe the etymology of this kind of "to" is the usage of a preposition before the infinitive, in order to indicate a relation of direction or purpose between a finite verb and the infinitive: I went to school; I went to pick him up. This probably originated in predicates with verbs that have a direction. But that was long ago, when the infinitive still had a distinctive form in English or Proto-Germanic; I think it was something like *eatan (don't pin me down on this, I haven't looked it up). Later, when the infinitive became indistinguishable from other forms of the verb, this to evolved into a more general marker of the infinitive. So the most precise definition would probably be to say that to belongs neither to the finite verb ("like") nor to the infinitive.

That said, you could say "like often goes with to, so we say to belongs to like", or "to often introduces or links to infinitives, so we say it belongs to the infinitive". In lists of phrases, educational books often tend to emphasise the connection with like; in passages that describe the infinitive, they tend to treat it as part of the "full" infinitive, as opposed to the "bare" infinitive in "I will go". I believe this is the most traditional way to describe it.