I was thinking about Plato's cave analogy, because I had forgotten what it was an analogy for, when I remembered that it was Plato saying, "If you have knowledge of the forms of the world, you know more than if you just knew about the things of the world." My gut reaction was, "Man, that is dumb. Sounds profound for about five seconds, and then you realize it's dumb." Trying to clarify exactly why I would say such a thing... my answer became a little more nuanced.
One interpretation is that if you understand what chairs are for, how they are used, and how they are built, you understand chairs much better, potentially in a way that you could understand how to design and engineer a new chair. But the question is whether this means knowing the form of the chair.
But this is not how Plato thinks of Forms. Instead, they are abstract things that exist apart from the material world. Sadly for Plato, much of the academic theorizing of the past century has been about exploring the social construction of our world, and how it has such little attachment to anything that is objective and universal. The study of history, for example, is about constructing a narrative to explain the causality of past events. Historical myths abound because of how much more we like an interesting story.
How deep does it go? Well, I heard a rumor at my college that one of the math teachers stopped teaching the 100-level Intro to Analysis because he had heard enough objections to set theory that he could no longer believe in it. This was a class in which the students went through all the proofs that numbers exist, and that you can do arithmetic with them, and it is all founded on set theory.
And really, at the end of the day, the universe is just a bunch of matter doing stuff. We happen to have good models and theories for how that matter behaves, but the universe happens to be completely indifferent to whatever we think about it. Even if there is objectivity in math and the natural sciences, though, any abstract knowledge of chairs is going to be knowledge about them as a social construction, because there just isn't any other form of it.