When studying electrodynamics do we assume Maxwell's Equations or derive them?

This question is because something made me confused. I always thought that the idea behind electrodynamics was to postulate some things, like Coulomb's law in electrostatics and so on, and then manipulate those things to derive Maxwell's Equations. After that we could then use Maxwell's Equations at will.

Now, I'm confused because I've seen people assuming the equations. Namely saying that the electromagnetic field is a pair of vector fields $(E,B)$ such that $E$ and $B$ satisfies Maxwell's equations. Then with this the consequences are derived. But this seems strange to me: if you start by Maxwell's Equations, is because you already know them, but to know them, you needed to start by assuming other things, instead of the equations themselves!

My thought was: historically we start really with Coulomb's law and so on, and then we find out that there's a system of $4$ differential equations that completely specify the magnetic and electric field. Then since we found this out, practically speaking, we simply forget about the rest and restart from these equations. Now, why do we do this? Why do we start from the equations rather than deriving them?

• Essentially a duplicate of physics.stackexchange.com/q/3618/2451 and links therein. – Qmechanic Sep 15 '13 at 14:36
• Because history is a winding path filled with accidents and coincidences. Teaching things in exactly the same order as history is simply misguided. – genneth Sep 15 '13 at 16:18
• @Qmechanic it's not a duplicate, the OP is asking why Maxwell's equations have replaced Coulombs law, even though Coulombs law was "postulated" to start with. – Larry Harson Sep 15 '13 at 17:38
• Axiomatizations are not unique. For example, you can take Euclid's postulates and prove the Pythagorean theorem. But you can also take Euclid's first four postulates, get rid of the parallel postulate, and add in the Pythagorean "theorem" as a postulate. Then the parallel postulate becomes a theorem. – Ben Crowell Sep 15 '13 at 20:00

So no, Maxwell's equations haven't replaced Coulomb's law, they've included it in Gauss's law which is equivalent to Coulomb's law: $$\nabla\cdot\vec E= \rho/\epsilon_0$$