That’s a very good question.
So, consciousness is obviously provided by the brain. So that’s a good first guess. But consciousness is assuredly not the brain. If you took a look at a running CPU processor, the actual instructions being executed are going to give you absolutely no guidance on what the computer is actually doing. It won’t tell you whether it’s drawing a pixel on the screen or calculating a game action.
Sure, you might be able to watch it long enough to where you can make some halfway decent guesses. But you’re missing a huge piece of the picture.
If you want to know what the computer is actually doing, you need to understand how computers work. Where is that knowledge located? In books. Or other forms of documentation.
Here’s where we can inject a bit of math. Math has the concept of a field. A field is simply an abstract space in which mathematics “takes place.” An example might be a cartesian number line.
When you’re doing math problems, where is the number line on which the problems are taking place? That’s essentially the question of where consciousness is. The line is nowhere, it’s abstract. It doesn’t have a physical referent. Maybe your brain, but that doesn’t really help. Just because your brain is processing information doesn’t mean that looking at the brain will help you understand what it’s doing.
So where is consciousness? It’s nowhere. It’s abstract.
That said, that doesn’t have to be the end of the question. Just because it’s abstract doesn’t mean you can’t study it. In the case of consciousness, you can draw out the levels of abstraction.
So you have the brain. Then you have the mind. The mind is software running on top of the brain. You can study how minds work scientifically by looking at patterns and genetics. The problem with this approach is that progress in the field is understandably slow. You can divide the effort into neurology, psychiatry and psychology. Neurology studies the way that the nervous system works. Psychiatry tries to understand the brain. Psychology tries to understand the mind.
So you have a neat division of abstraction. Neurology wants to know about the physical, psychiatry wants to abstract up a level to the mind, and psychology deals with what is most germane to us as individuals, our ongoing thought processes. If you want to study how consciousness works, it’s good to learn the fundamentals of all three.