"Neuroscience reveals that the mind cannot be reduced to the brain, and that reductive materialism is a dead end. The whole realm of subjective experience lies outside its domain, and outside the domain of objective science altogether. Two features of the mind--specifically consciousness and free will--define the human soul. These features seem to operate outside the laws of nature and therefore are not subject to the laws governing the mortality of the body. The body dies, but the soul lives on." (Life After Death, 220)There are three arguments here, which D'Souza elaborates in his book. The first is that while neuroscience can explain the objective physical reality of the brain (the structure and functioning of the brain's neural circuitry), it cannot explain the subjective mental reality of the mind (thoughts, emotions, decisions, and so on). My brain states can be objectively observed by others. But my mental states cannot, because they are private experiences that only I have. Other people cannot have the direct access to my mental states that I have. My brain states are located in space. But my mental states are not. My mental states can intentionally refer to things external to them--they are "about" something. But my brain states don't refer to anything outside themselves. I can be infallibly sure about my mental states: my thoughts might be mistaken, but I know they are my thoughts. But I cannot be infallible about my brain states. In all of these ways, brain states and mental states differ. Thus they cannot be identical, and mind cannot be reduced to brain. Any attempt to reduce the mind to the brain is implausible, because it denies the self-evident subjective reality of mental experience.
It is true that neuroscience can show that brain states and mental states are correlated, but that does not show that brain states cause mental states. It could be that the brain receives or channels the mind, analogous to a radio receiving signals that are translated into sounds.
If this is so, then it's at least possible that mind could live on after the death of the brain.
The second argument is that consciousness has no physical or scientific explanation, because the subjective experience of self-awareness cannot be objectively observed. Consciousness is an irreducible element of reality like matter and energy. Even a scientific materialist like Steven Pinker must admit that "there is no scientific explanation of consciousness."
The third argument is that free will is another mysterious feature of mental experience that cannot be explained through natural laws of physical causality. We all know that we have the power to make mental decisions and then execute those decisions through our brains and bodies. Neuroscientists recognize this as neuroplasticity--that is, the mind can change the brain. For example, people suffering from obsessive compulsive disorder (OCD) can be taught through cognitive therapy how to refocus their minds away from the compulsion and redirect their thoughts towards more desirable behavior. When they succeed in doing this, neuroscientists can see that these people are changing the neurocircuitry of their brains. This free will seems to be free from the causal determinism of the physical brain and body, and therefore it seems reasonable that the mental capacity for free will could live on after the death of the brain and body.
D'Souza rightly notes that some of the founders of modern neuroscience--such as Charles Sherrington, Wilder Penfield, and John Eccles--were dualists who were open to the possibility of the mind existing as a spiritual reality separated from the body.
I agree that mental experience, self-consciousness, and free will are all mysterious in their correlation with the brain and body, because it's mysterious as to exactly how the brain acts on the mind or the mind acts on the brain.
We can explain the emergent evolution of the mind or soul by saying that with the increasing size and complexity of the primate brain--and particularly the higher parts of the brain in the prefrontal cortex that support conscious deliberation and choice--the human mind emerged once the human brain passed over a critical threshold of size and complexity. This still leaves us with a mystery because we do not now know exactly how the brain creates a mind that can then act on the brain itself.
The radical dualism of Kant and D'Souza--the claim that mind belongs to a transcendent world of spirit beyond the natural world of bodies--does not resolve this mystery. Rather, it tries to overcome one mystery with an even deeper mystery--the mystery of how a transcendent world interacts with our natural world. D'Souza even admits that if we adopt his dualism, "we still won't be able to fully understand how minds interact with bodies" (127).
And even if we recognize the mystery in this interaction of minds and bodies, it does not follow logically from this mystery that we must believe that minds without any interaction with living bodies can live forever in Heaven or Hell.