As well as mathematics I teach IB Theory of Knowledge, which includes mathematics as an area of knowledge. The class are mainly not students with a maths focus, although they all study at least some for the IB, and all studied mathematics at GCSE level.
One of the key concepts that I have struggled to get across to them is mathematics as a process of making logical deductions from a set of axioms, which has implications for the relationship between mathematics and reality. For example, I have talked to them about Euclidean versus non-Euclidean geometry, and how for a long time non-Euclidean geometry was considered an abstract curiosity but eventually turned out to apply to reality as a consequence of the non-Euclidean axioms applying to reality, but the example is fairly far from their frame of reference.
Does anyone have any suggestions for a simple illustration, which ideally the students could try for themselves, of how starting from different axiom sets and following the same deductive rules leads to different consequences, that lends itself to a discussion of how reality could turn out to make one or the other set of axioms true?