Neighbor says: >Therefore, a line is that which has only length, and each line can be considered made up of smaller lines and these of smaller and so on, and we imagine the extremities of a line as points,
Pick a number x as small as you please, a line segment of length x is a line segment, not a point, no matter how small. On the other hand, we can get a single point out of the intersection (as in set theory - it keeps coming up here) of a family of shrinking segments. But that's only because we consider a line segment to be a set of points in the first place.
>but cuts the line into two lines which can be taken separately without any trouble.
"taken separately" has no meaning mathematically, unless you can explain or define it in mathematical terms.
>So, the point is only something that we imagine on a line
Sure, but the line is imaginary too...
>I'm impressed, I'm going to contemplate the meaning of what has been said, but I think this to be satisfactory for out purpose unless other doubts will arise
You two can create your own branch of math... a perfect match.