Question: If a spirit is non-physical, but the human body is physical, how does a spirit stay in our bodies? Also, experience seems to tell us that our memories are stored in our physical brains. How could a spirit store physical memories? This distinction would seem to disprove the whole idea of anything non-physical existing at all.
Good question, but the question does not disprove the existence of spirits or souls. First, this is merely a question posed about something we do not fully understand. When Christians claim God as a cause of things we do not understand, Atheists and physical naturalists are quick to point out that this is a logical fallacy, a type of God-of-the-gaps (deus ex machina): when we do not have sufficient explanation, claiming ‘God did it’ is a fallacy. Well, the sword cuts both ways. Merely because we do not have a full understanding of how a spiritual soul interacts with a physical body does not prove that it does not. Claiming so is the atheist equivalent, a type of atheist of the gaps theory.
Second, if nothing spiritual exists, we are faced with physical naturalism, which presents a host of other problems that are greater than any misunderstanding about spirits (see here, and here, and here).
Third, denial of the human soul would make the operation of understanding purely mechanistic, a product of nothing more than a complex machine. Such a view destroys the whole concept of understanding in the first place. For example, I could program a computer that, upon my arrival home after work, would say to me, “Good evening dear, I’m glad you’re home. I’ve missed you and I love you.” But the computer does not really have longing for anyone, nor is it glad about anything, nor does it love. The machine is merely doing what it is programmed to do, and it might as well be saying “Gob nortesk bire,” for the computer does not truly understand in the same way that humans do. For more on this, see the problem of the Chinese Room.)
Thomas Aquinas had a similar objection 750 years ago. He claimed that the understanding must be inherently different than that of any bodily organ, for the organs of the body are insufficient to develop abstraction, which is necessary for understanding. The problem can be expressed in several facets that have long been discussed in philosophical circles, such as the problem of the one and the many. Why, when we hear a tune, do we perceive a tune, and not merely a series of tones? We have the ability to recognize not just individual tones, but music, and even assign emotions to the music, recognizing music that is happy, sad, bold, or timid.
If no spirit or soul existed, we would be left with a purely mechanistic world. We would have much difficulty explaining abstract objects, our ability to perceive universals, or assign meaning whatsoever, including love.