As 3-dimensional beings, do we really have 3-dimensional vision? I was watching this video on YouTube of a high school student explaining perception in different dimensions, basically stuff he learned from reading the book Flatand.
At one point in the video, he says that as 3-dimensional beings, we can't really see our 3-dimensional world in the 3-dimensional way it actually is, because we have 2-dimensional sight. To illustrate this, he used a sphere, and explained how we can only know certain physical things about it by the way light reflects off of it. It seemed convincing, but thinking about it again right now I'm not sure that I'm really convinced.
He said in order to perceive our 3-dimensional world in the 3-dimensional way it actually is, we would have to be 4-dimensional beings, because then we would have 3-dimensional sight. But I always thought that as 3-dimensional beings, we have 3-dimensional sight and it's the 4th dimension that we can't see and have difficulty with mentally picturing.
But according to him, in order to see an $n$-dimensional world with $n$-dimensional vision, you'd have to be looking at it as an $(n+1)$ dimensional being in the $(n+1)$ dimensional world that encapsulates it.
Is this true? I was hoping on getting some clarification on this because it's kind of confusing.
 A: I'm not sure about four dimensional beings, but as for us it seems plausible. To parametrize image as we perceive it with one eye or picture taken with camera it's enough to use just two parameters. For eye it would be natural to use angles and for picture from camera distance from the border of image. We have two eyes, so we get slightly different images and our brain reconstructs pseudo-3D. But we can be easily fooled - there is quite a lot of illusions. 
    Now I have problem to grasp 4D but it seems we would need three parameters to describe image perceived by four dimensional beings, so probably they would see in full 3D, and their theoretical brains would fool them into thinking they see 4D.
A: It depends on what you mean by 3-D vision.
Assuming you have both eyes working then your vision is capable of establishing the distance of an object as well as it's horizontal and vertical position, so in this sense our vision is 3-D because using it we can establish all three position co-ordinates.
However we cannot see all of any object for the obvious reason that if an object is opaque we cannot see its far side. To take a trivial example we cannot see the Moon's far side. In order to simultaneously see all parts of the Moon's surface we would have to view it from a fourth dimension. The Flatland analogy is that to a Flatlander a polygon looks like a line because the Flatlander can only see the side nearest them. It's only when you move out of the plane in the third dimension that you simultaneously see the whole polygon.
The question of whether our vision can be called "3-D" or seems to me to be a matter of semantics and not terribly informative.
A: We live in a 4-dimensional universe with 3 spatial and 1 temporal dimension.
Our vision really isn't really 3-dimensional in the usual sense. If you fix your vision to a single point at something which is not moving, there's no way for you to be sure you're looking at something 3-dimensional. Of course, we learn to recognize shadows and we know that far away objects seem smaller, but that's not the point, you can do the same with a printed picture, but you know that it isn't 3-dimensional.
What allows us to infer that our world really does have 3-dimensions is movement, we can move around and we see things moving in a way that clearly must be representative of a 3D world.
Now think about this: What is it that allows movement to be perceived in the first place?
Of course, it is time, because movement doesn't make sense to us without time.
So, in a sense, we do experience a 3D world using our vision, but not directly a world with 3 spatial dimensions, instead we perceive directly a world with 2 spatial and 1 temporal dimension which allows us to infer the existence of the third spatial dimension.
P.S.
Consider a tesseract (4-cube). It's usually represented by an object changing shape in time and rotating around, because that's the only way you can represent it on a 2D screen. Why? It's completely analogous to what I said earlier, think about how time fits into it.
A: A surface is 2D because you can describe every location on it with just two numbers - even when the serface is not flat. Think longitude and latitude on earth. So the question for sight is, how many independetly derived "measurements" about a point we get.
I think you have to understand that "Something has X-Dimensions" first and foremost means that you can describe every point in Something using X different coordinates, and adding another one would not give you more useful information.
Our vision is, seen this way, 3D: when looking at a point on the monitor, each of my eyes sees it at a certain angle to the left or right of an arbitrarily chosen "straight ahead. Each eye also sees it somewhat downward or upward, but this angle is the same for both eyes, or at least not independent. So I have 3 indipendent measurements about this point I'm looking at - prinicipally enough to know where it is in 3D-Space.
A: My opinion is that the third dimension can only be made possible by hallucinatory perceptions of the mind. We can think about a third dimensional world, but not experience it. The senses are flawed because they are first filtered through the mind. This is why we have ambiguous questions, such as this one, where the only answers can be opinions that are based on passive observations. So I'm sorry, but I highly doubt you'll be able to get an objective answer. The only way to come to a fact scientifically is to be able to perceive everything about every particle in existence. This is because even a grain of sand existing light years away has a subtle impact on anything and everything we can experience on earth. So if you want a "real" answer, ask your Self.
"I" see a "basketball." Let's break this down.
My right eye sees an orange circle. My left eye sees a slightly different orange circle. These two barely varied 2-D images of orange circles are combined into one "3D" image by the mind which remembers that the context of this orange circle means it's a basketball. You now "see" the 3D image with a mental caption that says basketball.
A: This is not entirely a physics question.
We need to define what we mean by the term see.
Images forming on the retina might be 2D, but the brain then creates a sensation based on not just the stereoscopic image pair, but all sorts of other contextual information.
Ask yourself if there is any difference in your experience in viewing genuine 2D images and 3D scenes?
So for my money, yes we do see in 3D. 
However, the issue with the retinal image being a lower dimension that the space it receives light from is valid.
A: We actually have 2D vision, but we learn how to place things in 3D, either by context or by our rather limited binocular vision.  
The fact that we can be foiled by things like the wonky room illusion, and that when one has a new set of glasses, things are at a different distance to what one thinks they are, is that we are seeing 2D and reading the clues (which are usually right), to make depth.
Apparently, reading pictures on the page is a learnt art.  There is a story about Lawrence of Arabia giving some Arabs a painting of them, but they were unable to read the image because that art is foreign to them.
