I don't dispute what happens now. I am, as you are, considered a biological organism. Biology is the study of biological organisms, and is partly a study in the area of chemistry.

But how does one define a biological organism? How is it different than studying non-biological, inanimate, matter?

: The only thing I can offer is we don't know exactly how the first
: self-replicating molecules occurred, but since that time, it's been
: chemistry.

No only the "first", but any. No has been able, as of today, to create a self-replicating molecule (that is, biological life, since there is plenty of work being put into creating artificial, non-biological life) from non-biological matter. It's not about when it "first" happened, it's that it has not happened since and hasn't been observed or repeated.

: We call what we refer to as life as biology as a separate
: division, but that's just a handy distinction, not something completely
: separate.

But what's the distinction?

: It's the same difference as algebra to math. One's a small
: subset of the other but it's all math.

Uh... not the same thing. Biology isn't just chemistry, for one, and vice versa, so neither is a subset of the other. They may intersect at the study of biology on a molecular level. You know, molecular biology.

Again, what's the difference between a biological organism and a non-biological organism? Is there a difference? Or is there no difference? (I sincerely hope you don't agree with the latter; but if you agree with the former, then I rest my case and we both hold valid opinions about the nature of life, and we can be free to move along from this unending thread)