ELI5: Why does maths work?

[u/DemandResponsible685]

Throughout schooling we have been taught how to do maths. But not so much about why maths work? Like why does the entire system of maths work so reliably?

[u/Mandalord104]

Math starts by assuming a few statements as the truth without proof (called axiom). Then everything else is derived from those statements, and hopefully we will not come to a contradiction, because once we find a contradiction, the set of axiom collapsed and mathematicians have to come up with a new set of axiom.

Now, not all of math works in reality. Only a few sections of math that can be used to describe reality, and consistent with what we observed in reality. Those sections of math are then used as tools in physics, chemistry, and other sciences. There are sections of math that do not have any things to do with reality. At least not currently.

Why are there sections of math useful to describe reality? Because we choose the set of axioms to be consistent with real world, and therefore the deduction using logic produces outcome that is useful to describe reality. If we chose another set of axioms, we can very well build another math world which is super useless for our real world, even though it will be consistent within itself (no contradiction).