Do We Need Axioms?

The title above is quite clear enough to understand. For many student on economics, especially who are taking advance micro economics shall frustating why they need so many axiom just to explain a demand (or supply) curve? It is true, necessary?

I got a good explaination from Prof. Iwan Jaya Azis, Cornell Unversity, when he though about General Equilibrium, as written on his article 'Antara Teori dan Intuisi' (between theory and intuition). One day a student of his class questioning why he must learn Debreu axioms. Let me sum up their conversation:

Student (S) : I think a simple economic model is quiet enough to explain our daily economic phenomena.
Iwan J. Azis (A): What do you mean about "daily economic phenomena"?
(S) : Suppose that US government canceled such a tariff import policy on clothing from Asia. It easily predicted that cloth price will fall, since increasing on supply (caused by import flow). I just use demand-supply model without confusing on axioms, right?
(A) : How?
(S) : It simply because demand curve has negative slope.
(A) : Who said that? Since you accept a postulate that demand curve has negative slope and supply has positive slope, you are using some assumption. For example, cloth is a commodity that its substitution effect is less than income effect, so consumen behaviour is normal. It implies you assumed that cloth definetly not a giffen goods?
(S) : Yes, I am
(A) :So, what kind of assumption we need to draw an indifference curve for this case?
(S) : Curve must be convex, continous, and has monotonicity
(A) : You just already said Debreu axioms.