The philosopher, Colin McGinn has listed five characteristics of mental states, which purportedly prevent them from being identified with brain states. Mental states are, he claims:
1. Unobservable — in the sense that they are not perceptible by means of the senses.
2. Asymmetrically accessible — in the sense that the owner of a mental state has a kind of immediate access to it that other people do not.
3. Subjective — in the sense that the nature of a mental state is knowable only from a single 'point of view'.
4. Non-spatial — in the sense that mental states do not take up a well-defined region of space.
5. Subject-dependent — in the sense that mental states only exist for a subject of awareness.
Now, as I argued a few weeks ago, non-commutative geometry is the best candidate for a mathematical structure which provides a common generalisation of the structures employed in quantum theory and relativity. And the crucial thing to note about non-commutative geometry is that it provides an algebraic representation of space, rather than a manifold of spatial points. Relationships between points in a manifold, such as spatial extension, are rendered obsolescent, and replaced with purely algebraic relationships. Hence, non-commutative geometry has the potential to dissolve one of the apparent barriers to resolving the mind-brain issue. McGinn's fourth point, that mental states are not spatially extended, is only a problem if brain states are held to be the states of spatially extended systems. Non-commutative geometry suggests that the spatial extension of any physical system, including the brain, is an illusion.