The idea of the "immortal soul" is that the Mind can exist completely intact in the absence of a physical brain. Thinking, remembering, and perceiving will all be completely unaffected by the disintegration of this vital organ. The study of brain trauma however, reveals startling examples that this concept is false.
Prosopagnosia (the inability to recognize faces including your own) is one such example of a mind-brain dependence. Alzheimer's Diesease is another dramatic example of mind-brain dependence. Not only is the brain slowly destroyed neurologically - but the personality contained within is slowly destroyed in corresponding fashion. The creation of multiple personalities in split-brain patients whose brain is severed down the corpus callosum to reduce severe epileptic seizures, is yet another dramatic example. Left brain no longer communicates with right brain and causes the emergence of two independent persons operating in the same body - each unaware that the other exists.
Thus, your mind is not immune to damage and decay. When your physical brain begins to die off, that person known as *you* dies off as well. Your memories, your experiences, your emotions - are directly caused by the dendrites in your head. Destroy some of those dendrites and those memories will be lost forever - never to return. Destroy enough dendrites and you will become imbecillic - unable to recognize family, friends and favorite places. If all of us had "souls", then the only thing that brain damage would do is incapacitate our ability to move the body. Our memories, emotions and experiences should all remain indestructible and able to survive.
This is now known to be false.
Destruction to your brain parts results in the corresponding destruction to the memories and personality it contains. When your physical brain dies, your personhood and everything that makes you *you* - dies with it.