How to make a good definition

Question

The reason why I come up this idea may due to Banach–Tarski paradox. The process we make a definition may consist of several steps. First step is that we observe a phenomenon. Second is to make a definiton that can be included it in a rigorous mathematics system. For example, the axiomatic of geometry by D.Hilbert: The point, line, plane are not rigorous mathematical objects (something we see but can't say), so we use numbers to axiomatise them. However, as mentioned previously, we may come across something we don't expect it. In other words, when we establish a theory or make a definition it is rigorous or self-contained in mathematics, but may disobey our common sense in some particular example.

Question. How can we deal with a situation where our definitions and assumptions lead us to conclusions that contradict common sense?

Answer 1

This may be an instance of what Imre Lakatos called a "monster-barring" technique. In the case of the Banach-Tarski paradox, this consists in introducing a distinction between measurable sets and nonmeasurable sets, and declaring that only the former obey our usual intuitions of "conservation of mass", etc. But you should keep clearly in mind that the theory thus developed is not known to contain logical inconsistencies.

Answer 2

"A line is made from infinite points"

This is the intuition failing. The $\mathbb{R}^n$ space is perfect, in this sense. A line indeed is a set of infinite points. Hence you can introduce stretches from an object to another of bigger length, area or volume, under the proper assumptions.

But the physical space is limited. You cannot stretch, and when you try, no, the atomic forces do not allow you. Also, in the physical space there is no infiniteness. An atom, is indivisible once you reach it.