Is abstract knowledge possible without free will?

Do not confuse knowledge and certainty. Cats do not have a conceptual faculty, but they do have knowledge. They also don't care about certainty.

It is having a volitional faculty of knowing that gives rise to both the possibility of, and the need for, certainty. It is our ability of self-consciousness, and our awareness of our volition which makes certainty valuable and possible to us.

This is, of course, a counterfactual, but I think that if we could have abstract ideas without volition, then we would simply be like cats, but more capable. We'd be pretty sophisticated in our abilities, but we would have no cares about things like truth and certainty. We would have knowledge, but no desire to validate it.

I think, though, that a conceptual faculty as such is inherently volitional. For every concept, someone comes up with it first, while others have not. I cannot imagine forming concepts without choice or effort.

Without volition, we could still have knowledge, but no certainty -- but we wouldn't care.

For man, the answer is no, abstract knowledge would be impossible without free will, because man's conceptual faculty is fallible as well as volitional. Man can err in his conclusions. If he were predetermined to persist in his errors, with no means of correcting them, he could never be certain of whether his conclusions were correct or erroneous. That would be equivalent to not knowing.

If abstract knowledge could somehow be reached by some other type of consciousness or intelligence that is conceptual and infallible, the role of free will (if present) would be different. But man currently has no context other than the human conceptual faculty for understanding the meaning and requirements of abstract knowledge.

Additional insights on why belief in determinism (in man) leads to skepticism can be found in The Ayn Rand Lexicon under the topic of "Skepticism."