Your question is more intriguing than most people realize. Because these equations have been altered a lot from the way they were originally conceived by Maxwell - due to the efforts of Heaviside, Fitzgerald, Lodge and Hertz. In brief:

- There were initially
*twenty* equations, later reduced by Heaviside to the four nowadays known as the Maxwell equations. - Faraday's
**electrotonic state**, also called **A field** and **vector potential**, was originally understood to be a physical entity but was later regarded as a mere mathematical convenience by Heaviside. - However, a variety of phenomena known in modern physics, such as the Aharonov-Bohm and Altshuler-Aronov-Spivac effects, the Berry-Aharonov-Anandan phase, the Josephson effect, the quantum Hall effect, the De Haas-Van Alphen effect and the Sagnac effect show the vector potential to indeed be real. - The equations were originally written in
**quaternionic algebra**, not vector calculus. This was seen as an unnecessary complexity by the Maxwell's successors mostly because quaternions are tricky to calculate with (especially without having computers). However, covering aforesaid effects in fact calls for usage of such a system of higher order symmetry forms.

You can read the whole story in much more detail in

**Terrence W. Barrett: Topological Foundations of Electromagnetism**.

Hope this helps.

-Michael