As far as you and I are concerned, there is only 1 dimension and that is the third dimension, which isn't a bad dimension to reside in by the way.
3D.. three axis.. three dimensions to everything, 2D does not exist it is purely hypothetical and as for the fourth dimension, how can we define something that by definition is undefinable, if you subscribe to the popular science fiction version that is, they rarely go into detail but as an example, just imagine if 2D did exist, and our reality resided in it, then the fourth dimension would be like the third in that case we wouldn't know anything about the third let alone if it existed or not.
If we lived in a 2D world, we wouldn't be able to percieve the third and even trying to dream up what it would be like would be like staring into the sun.
I think I have a pretty good idea though, after interaction with a G FORCE 2 GTS with 3D (stoner) shutter glasses. We have recreated the third dimension, we understand our perception of it, we understand that our two eyes give use two angles which our minds blend together to make up a 3D world, we can even trick our own minds with technology that we have designed therefore, through technology comes new perception, and what more is perception than A DIMENSION, how we perceive the world defines our dimension THUS!!!
The fourth dimension is still out of our grasp because we dont know what it FEELS like to exist in the fourth dimension (heck we dont even know what the second dimension feels like) however we can say, quite easily, that all it is nothing more than, a lack of or a gain in, perception.
So the third dimension cant exist without at least the theory of the third (and stupid humans that reside in the third who dream up absurd theorys about perception) but we know for a fact that the third exists and we feel evidence of this everyday with our awareness of being, so or reality is the basis for the fourth!! hehe so now comes the cool bitty... we can recreate the third with our technology, and there is nothing stopping us from telling the difference from something generated by a computer system or something that actually exists, it is an addition to our reality, so the existance of a computer generated reality is now an addition to your current reality, you are aware that it exists.
Now we take a look at technology and the way it progresses, less than a hundred years ago the human race where making powered flight a reality and now we travel into space on a regular basis, so you KNOW technology will continue to speed up and confirm our reality, and even our theorys. THAT SAID...
The fourth dimension is a computer generated dimension...
It would be as valid as the third because the third would only confirm it's existance, we would take the perception of existence within the fourth back to the third, we know that it exists and confirm it with our own minds, you cant get more real than that now can you.
The fourth dimension is ANY ADDITIONAL PERCEPTION GENERATED BY A COMPUTER (or otherwise man made) what I have in mind is pure knoledge, you dont just see an object, feel an object, smell, taste or hear and object, you KNOW it.. making this easier to understand, it would be like knowing every angle and surface of the object, the weight size texture of every part, the smell taste and sound of every part.. in addition to that you would know what it's like to BE that object, this would be a residual effect from knowing every possible detail about it, you would not project your emotions into it, you would not worship it, you would not want to possess it or consume it.. BECAUSE YOU KNOW IT!!!

Postlude (thanks sarah):
so now you are aware of the fouth dimension, does it exist? I think so because technology has already given us another perception.. what do you think? not yet? maybe in the future? or does anything in the future exist already because without the present it couldn't exist, I think this is how we end up justifying our accumulation and, inevitable, loss of information about our environment, you may not even be aware though and if you dont understand any of this then you probably never will.