Sure, assuming you somehow figured out projective geometry you could figure out what was casting the shadows. That doesn't contradict Plato, because that's what he thought philosophy was. Compare what the invention of the calculus did for physics. The real numbers can't be seen, observed, and do not affect the physical world in any shape or way. Real analysis is pretty import all the same. Objects not existing in that sense doesn't necessarily mean they're not worth studying. There are a lot of problems with Plato's metaphysics, and Plato himself pointed out the more glaring ones (see Parmenides ), but this is not one of them.If you were in a cave you could likely, with enough time, through looking at the shadows, learn that they are shadows.
The question, I guess, is "is there a reality beyond ours we don't know about". Honestly, it doesn't really matter. It apparently can't be seen, observed, and doesn't appear to effect our physical world in any shape or way, so it may as well not exist, even if it does.
"Real" numbers do not exist in any state or form aside to describe constructs or states of objects in reality. It's a symbol, a shorthand, for something that does exist. Yes the word "pipe" does not exist, but pipes do. The analysis of numbers analyzes them through applying rules to them that co-relate to those in real life.The real numbers can't be seen, observed, and do not affect the physical world in any shape or way.
More math vs. science. Mathematical objects don't exist in the way the physical objects exist, but you can study mathematical objects, and doing so turns out to help you understand the world. Think of Plato's forms as being like mathematical objects. Plato thought that spheres and rectangles were more real than the Earth and the page you're reading, which sounds a little weird with 2000 years of progress between us and him; we think of applied math as modelling real things, rather than real things being shadows cast by math. He had it backwards from our perspective, but we have the benefit of 2000 years of figuring out the relationship between relationships between maps and territories, and he was mostly treading new ground.
Most mathematicians will tell you math is just a game you play with symbols and that we don't really mean it when we say "these exists an x such that...", but there are mathematical platonists. Probably the most well-known modern mathematical platonists was Gödel. I don't know of any who claim the kind of relationship exists between mathematical objects and physical objects Plato claimed existed between the forms and physical objects. I'm hardly an expert though, they may very well be out there somewhere.