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>given what we know about neuroscience and cognitive science, how can I believe there is a soul?
I am not intimately familiar with neuroscience and cognitive science, but I can't imagine how they would cause trouble for believing in the soul. The soul of a thing is what makes that thing that thing. You are a person because you have a human soul. The soul also contains various functions that do not exist in any physical organ. People often argue that all thought occurs in the brain, but this is no more true than saying all sight occurs in the eyes. The brain, like the eyes, are a certain physical necessity. You cannot think without a brain, but the brain does not, and in fact cannot, account for all thought. There is a mindset that says "Science is always getting better. Maybe we can't explain it using the brain today, but maybe tomorrow science will know more". This idea is optimistic but misguided. At each point in history the "science of today" has been proven wrong, often in large ways, by the "science of tomorrow". The scientist then has no reason to believe his claims that the brain can account for all thought will be proven or supported or believed tomorrow, even though they are today. For an easy example, compare Newton's gravity with modern gravity. Isn't it clear that they are two entirely different beasts? And in fact, Newton's law of gravitation doesn't hold everywhere at all levels, so such a simple and powerful law that explains so much, almost all, of gravitational relationships, was still insufficient to explain all of them. It is an act of faith, not reason, to say the same will not happen to neuroscience and its claims.
>given the article “Physics and the Immortality of the Soul” by Sean Carroll, how can I believe there is an afterlife?
I have not read this, so I cannot respond at all.
>given that almost all philosophers of note are materialists, and that Swinburne -assumes- dualism in his proof of god, how can I believe that there is anything about man that isn’t physical?
Materialism is certainly a more popular position in philosophy today. This is because materialism is easy, straightforward, and has a few practical benefits. Materialism in modern philosophy however is not based on the philosophical reasons to support it. They are still debating nominalism and realism after all. Materialism in modern philosophy stems, I think, from a sacrifice philosophers made to enable science to quantify the world in a simple and practical manner. This starts with Galileo and his peers and continues onwards. As Francis Bacon says, science should concern itself only with the material and efficient causes of things, not formal and final causes which belong to philosophy. However, seeing the "great strides" of science, many philosophers were more interested in defending sciences limited view of the world as the view of the world, rather than pointing out the obvious: No amount of success with a strategy proves it is the correct strategy. Hume correctly pointed out with his analysis of causation and induction that the world view science has taken up is a house of cards ready to fall over.
To be clear: Science does make great strides, I think that fact is clear. What I am saying is science rests its work on the sand, it is made by conveniently ignoring, and then forgetting, the existence of things it cannot measure. Science can never explain the whole of reality because science was made so productive by ignoring large parts of reality.