Without working in a rigorous axiomatic system, how can you intuitively convince me that $a\cdot(b\cdot c) = (a\cdot b)\cdot c$ for all real numbers $a, b, c\in \mathbb R$? (Or, more generally, how can you convince me that the value of a product of real numbers $a\cdot b\cdot c\cdot\dots$ does not depend on how one puts the brackets?)
Remark on the use of axiomatic systems: Axiomatic systems of numbers are a very recent discovery, the first of them were discovered a little more than hundred years ago. At that time, the people already manipulated numbers with confident. So, how did they ensure that multiplication is associative and that the value of a product doesn't depend on how one puts the brackets? For instance, what geometric intuitions did they use?