Platonic theory of the Soul
Plato thinks you are your soul, it’s where your thinking happens. (Plato’s theory of the soul is presented in the Phaedo and the Republic.)
Question: What is the soul?
Natural answer: the brain. (I think in ancient Greece people thought the heart does what we think the brain does. So they might have answered with “the heart.”) Isn’t this where your thoughts and memories and desires, etc. are?
Plato doesn’t think this is right.
Why not? One thing to notice, is that the nerves that make up your brain are constantly dying and being replaced by new nerves. So your brain today is not the same brain you were born with. But, arguably, you’re still the same person you were when you were born. So you can’t just be your brain.
This same sort of argument will work for any physical object: so you, your soul, is not a physical object.
Souls then aren’t the sort of thing that we can see. They’re “imperceptible and immaterial.”
Souls are immortal.
Difference between Phaedo and Republic
In the Phaedo, Plato argues that the soul is simple, that is, that it has no parts.
In the Republic, however, Plato seems to have changed his mind: he argues that the soul is complex: it has parts. Specifically, Plato argues that the soul has three parts: reason, spirit, and desire.
Nevertheless, in both dialogues, Plato holds that the soul is immaterial and imperceptible.