This cuts it rather close to a religious debate, don't you think? So perhaps we should just stay on philosophical ground by debating if a human has an immortal soul or not (but not from a religious viewpoint or with religious arguments).
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As for the question, of course we're not immortal. You live for a few years, and then you die and become plant food. If you doubt this theory, simply shoot yourself in the head and discover that you are not immortal.
If you refer to your body, then yes, our bodies are not immortal or indestructible. But then the question arrises, if we are only bodies or if human beings have something like a spirit or soul, which could be immortal.
Now, I am aware that I do not have any proof that we do have a soul. But then again, there is nothing to prove that we haven't, so it is still possible. There are certain things which could be better explained if humans had a spirit/soul, like what our psyche really is, how the process of thinking really works etc. The symptoms of feelings are not feelings themselves, they are only reactions of the bodies caused by the actual feelings. So feelings in themselves cannot be thoroughly explained chemically. When you feel for example "anger" then what does happen exactly? Your body certainly reacts, but where does the actual "feeling" occur? This feeling is not defined by its symptoms. You feel angry, but you do not define this feeling by the symptomes seen in your body. The feeling itself is something deeper, more in the inside. But what is it and what causes it?
Questions, questions... Take the DNA. DNA is information. Information, on which your body is built on. But how is this information used? How does it come that out of a pure string of information, a living being comes into existence? What causes life to exist? Information cannnot do anything by itself. Information must be used. But by what? Take a computer. On your harddisk, the information is stored. So you could compare your harddisk to the DNA. But the harddisk alone cannot produce anything at all, it only provides information.
Do we have answer? I don't believe so. And perhaps it is good like it is. If everything could be explained, what mysteries would life hold for us? What could we wonder about? What could we admire with puzzlement?