As far as I know, mathematics consists not of science but of a logic. A logic meaning a grammar of decidability. And in the case of mathematics, the grammar of decidability consists of reduction of all references to positional names, and therefore all relations to positional relations. And we can do so with an unlimited number of dimensions,
A science is necessary when we do not know the first principles (causal relations) of phenomenon and seek to identify them. Science therefore consists of theories and laws.
A logic is necessary when we do know the first principles (causal relations). Ergo, logics consist of axioms.
You can declare an axiom, but only identify a law.
Once a law is known you may model it with axioms.
That I know of there are only two assumptions in mathematics, and both are necessary for the simple reason that independent of context (applied mathematics) we have no means of decidability in matters of scale independence.
The law of the excluded middle.
The need for choice.
Mathematics is actually quite simple. Its that because it is so simple, consisting only of positional relations, we can describe any set of constant relations with it.