Why do positive and negative charges attract?

[f.wright]

If they didn't, there would be no existence. Atoms could't form. There would be no light. Subhan Allah wa bihamdihi

 

[Khashishi]

While it's meaningless to ask "why" on a fundamental concept, the truth is that we don't know what concepts are fundamental, and what we call fundamental physics gets more esoteric as we learn more. In the case of charges attracting and repelling; I think it's possible to answer that in terms of the stationary action principle. And the stationary action principle can be explained in terms of a Feynman path integral formulation. Beyond that, I don't think there are any good answers.

The problem that Feynman was talking about in the explanation is that the stationary action principle formulation isn't any more intuitive than just stating that charges repel. So you are answering one mystery with another. The more fundamental explanation is better because it matches experiment more precisely, but that doesn't mean it is any more understandable or satisfying to a questioning mind. When you ask why, you probably are searching for something more understandable, not something that gives more precise experimental results. But nothing is understandable below a certain level.

 

[USeptim]

For the moment, there is not a clear why but we can consider this behavior of charges as a "principle" of physics, and it will remain so until a better, unifying principle integrates many of the present "principles".

 

[jarekd]

Great arguments: we can't understand it just because we cannot ... the same as for quantum mechanics ...
Maybe just try changing attitude for a moment and really search for an answer ...

Do we have in mathematics something allowing only constructions of integer "number"?
Yes we have - so called topological singularities and we call this number as winding number or Conley index or ... topological charge.
So what would be dynamics of such topological solitons? Let us look at simple 2D field configuration for "-" and "+" singularities in different distances:
https://dl.dropboxusercontent.com/u/12405967/fig1_cr.jpg
the first observation is that the closer they are, the weaker stress of the field - down to zero when they finally annihilate.
Practically any field theory we would define here, its spatial derivatives correspond to stress of the field - fields have tendency to minimize energy, what means here that opposite charges attract.

And we are very close to Faber's model of electron, where due to using natural Lagrangian, such 3D analogues get dynamics described by Maxwell's equations ...