The Matrix presents a version of an old philosophical fable: the brain in a vat. A disembodied brain is floating in a vat, inside a scientist's laboratory. The scientist has arranged that the brain will be stimulated with the same sort of inputs that a normal embodied brain receives. To do this, the brain is connected to a giant computer simulation of a world. The simulation determines which inputs the brain receives. When the brain produces outputs, these are fed back into the simulation. The internal state of the brain is just like that of a normal brain, despite the fact that it lacks a body. From the brain's point of view, things seem very much as they seem to you and me.
The brain is massively deluded, it seems. It has all sorts of false beliefs about the world. It believes that it has a body, but it has no body. It believes that it is walking outside in the sunlight, but in fact it is inside a dark lab. It believes it is one place, when in fact it may be somewhere quite different. Perhaps it thinks it is in Tucson, when it is actually in Australia, or even in outer space.
Neo's situation at the beginning of The Matrix is something like this. He thinks that he lives in a city, he thinks that he has hair, he thinks it is 1999, and he thinks that it is sunny outside. In reality, he is floating in space, he has no hair, the year is around 2199, and the world has been darkened by war. There are a few small differences from the vat scenario above: Neo's brain is located in a body, and the computer simulation is controlled by machines rather than by a scientist. But the essential details are much the same. In effect, Neo is a brain in a vat.
Let's say that a matrix (lower-case "m") is an artificially-designed computer simulation of a world. So the Matrix in the movie is one example of a matrix. And let's say that someone is envatted, or that they are in a matrix, if they have a cognitive system which receives its inputs from and sends its outputs to a matrix. Then the brain at the beginning is envatted, and so is Neo.
We can imagine that a matrix simulates the entire physics of a world, keeping track of every last particle throughout space and time. (Later, we will look at ways in which this set-up might be varied.) An envatted being will be associated with a particular simulated body. A connection is arranged so that whenever this body receives sensory inputs inside the simulation, the envatted cognitive system will receive sensory inputs of the same sort. When the envatted cognitive system produces motor outputs, corresponding outputs will be fed to the motor organs of the simulated body.
When the possibility of a matrix is raised, a question immediately follows. How do I know that I am not in a matrix? After all, there could be a brain in a vat structured exactly like my brain, hooked up to a matrix, with experiences indistinguishable from those I am having now. From the inside, there is no way to tell for sure that I am not in the situation of the brain in a vat. So it seems that there is no way to know for sure that I am not in a matrix.
Let us call the hypothesis that I am in a matrix and have always been in a matrix the Matrix Hypothesis. Equivalently, the Matrix Hypothesis says that I am envatted and have always been envatted. This is not quite equivalent to the hypothesis that I am in the Matrix, as the Matrix is just one specific version of a matrix. For now, I will ignore the some complications that are specific to the Matrix in the movie, such as the fact that people sometimes travel back and forth between the Matrix and the external world. These issues aside, we can think of the Matrix Hypothesis informally as saying that I am in the same sort of situation as people who have always been in the Matrix.
The Matrix Hypothesis is one that we should take seriously. As Nick Bostrom has suggested, it is not out of the question that in the history of the universe, technology will evolve that will allow beings to create computer simulations of entire worlds. There may well be vast numbers of such computer simulations, compared to just one real world. If so, there may well be many more beings who are in a matrix than beings who are not. Given all this, one might even infer that it is more likely that we are in a matrix than that we are not. Whether this is right or not, it certainly seems that we cannot be certain that we are not in a matrix.
Serious consequences seem to follow. My envatted counterpart seems to be massively deluded. It thinks it is in Tucson; it thinks it is sitting at a desk writing an article; it thinks it has a body. But on the face of it, all of these beliefs are false. Likewise, it seems that if I am envatted, my own corresponding beliefs are false. If I am envatted, I am not really in Tucson, I am not really sitting at a desk, and I may not even have a body. So if I don't know that I am not envatted, then I don't know that I am in Tucson, I don't know that I am sitting at a desk, and I don't know that I have a body.
The Matrix Hypothesis threatens to undercut almost everything I know. It seems to be a skeptical hypothesis: a hypothesis that I cannot rule out, and one that would falsify most of my beliefs if it were true. Where there is a skeptical hypothesis, it looks like none of these beliefs count as genuine knowledge. Of course the beliefs might be true — I might be lucky, and not be envatted — but I can't rule out the possibility that they are false. So a skeptical hypothesis leads to skepticism about these beliefs: I believe these things, but I do not know them.
To sum up the reasoning: I don't know that I'm not in a matrix. If I'm in a matrix, I'm probably not in Tucson. So if I don't know that I'm not in a matrix, then I don't know that I'm in Tucson. The same goes for almost everything else I think I know about the external world.