Neighbor says: >If a beginner starts to learn math, the first concepts he comes upon are very likely points and lines. How should we explain to him these? (of course, not only giving the definition, but also explaining to him the why of this by means of rational arguments) That is to say, telling him In what way he should approach these sorts of problems.
Now *that* is a very good question. (I think children learn counting, and other concepts such as fluid quantities, long before they contemplate zero dimensional points and 1 dimensional lines and such.)
The "number line" comes to mind, as a possible introduction -- something you more or less explicitly rejected in this thread. At first, concepts like "infinitely thin" don't need to be mentioned at all. The first step is to illustrate numbers as lying on the line, and numbers also get associated with lengths that way. As the grades progress through negative numbers and fractions, the number line is a steady referent. By the time real numbers are explored, the notion of "completion" (as in closed under the limiting operation) could be broached in a very naive and informal way, along with many other Alice in Wonderland notions such as decimal expansions "that go on forever without repeating" etc. Some set theory along the way would be useful.
I think one thing lacking today is a bit of frank discussion about the big, big leap from say, rationals, to a proper recognition of what is required to understand what the real number system *is* (which is more or less a set of rules for playing with equations and abstract operations on completely abstracted quantities, because one can no longer perform the actual operations in reality.) Given two fractions, I can "really" multiply them (write down the fractions that represents their product.) Given two real numbers, in general, I cannot. (One might object that if the fractions are big enough I'm also stymied!)
Anyway, such a discussion has a lot to say about what "mathematical thinking" is, and has a lot to do with algebra and logic.
Without algebra and logic, thinking about infinitely thin lines and zero dimensional points really does lead into a labyrinth. Hell, it leads into a labyrinth even with those tools.