If you were in a cave you could likely, with enough time, through looking at the shadows, learn that they are shadows. Secondly, any person would feel chains, they would be aware of senses beyond that of the cave. 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. Asides, you can make assumptions all day "oh, this could exist!", and you can make them as well put as you want, but until you have reason to suspect that the things are true, all you are is someone making up stories.
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.
Geez. If it were up to you we'd be stuck in that cave forever. Making up stories is what makes us human. Wondering is what keeps us evolving.
If it were up to me we would leave the cave the moment you see the physical knife. Making up stories is awesome, but it's when you decide they have any effect on reality or you think that you are describing reality that they become nothing of worth. If the analogy is "we sense a reality that exists" than, yeah, I would have figured that is obvious. All we see or do is just sensing a reality. The whole description of things changing, though, is where that all falls apart. Things change in the "higher reality" in this case, and the things we see are directly connected to reality.
Your splitting hairs on the analogy. Plato's argument isn't "there could be a reality beyond our own", his argument is that there is an ultimate reality that is constant and directly influences our own. We know this because the world we experience changes. So like how the fire and sword effect what the shadow looks like, a higher reality is effecting everything we can experience. This takes the term "reality" pretty lightly, plato believed that the highest reality were mathematical proofs and laws.