Occasionally I have seen various papers, blog posts, and comments referencing the famous 44 year French civil servant who lives a normal life seemingly without a brain. Sometimes the writers demand an explanation from consciousness theories. Other times arguments are made that substantial parts of mental processing must actually be occurring outside the brain (in the ether or mind at large, I suppose). The case was reported in The Lancet originally and was accompanied by this image of a brain scan.
The patient suffers from non-communicating hydrocephalus that likely has developed over the course of many years. The man is not missing a cortex. Rather the cortex has been compressed into a thin sheet.
A paper in 2018 Revisiting Paradoxical Situations Associated with Hydrocephalus explains how people with hydrocephalus can seemingly have normal brain functions.
Presumably, functioning of the neural network of the brain does not depend on the volume of fluid that surrounds its structures provided that certain physical parameters are met. The main requirement is that the behavior of this fluid does not interfere with the function of the brain structures, including the cortical neural network.
If the behavior of cerebrospinal fluid in a hydrocephalic subject meets all three conditions, that is constancy of velocity, normal cerebrocranial pressure, and comparatively low density (in the range of 1.003–1.008 g/mL) of the fluid (or specific weight, which is the same), then the overfilling of the brain with CSF does not limit the efficient functioning of all parts of the brain, including the cortex. This explains the fact that the patients with hydrocephalus who meet these conditions can exist and develop normally.
What cases like this and others tell us is that the brain is very flexible. It can function in a variety of shapes and forms. It can compensate for damages and defects within a fairly wide range. That is amazing but by itself doesn’t present any special paradox to brain and mind theories.