If experimentation is not your thing, the presence of magnetism can be deduced from the concepts of relativity.
Magnetism and Relativity
The existence of magnetism allowed for the derivation of special relativity. In fact, Einstein's famous 1905 paper was entitled "On the Electrodynamics of Moving Bodies". The strange phenomena that only appeared with moving charges allowed Einstein to show that magnetism was simply electric interactions in different reference frames, and to use this obsevation to prove the existence of time dilation, length contraction, and all the other mind-boggling concepts that accompany relativity.
Since empirical derivation often fails to bring full understanding of a topic, a popular way of teaching magnetism is to derive it from relativity, reversing Einstein's logic. Starting with relativity and electrostatic theory, one may derive quite precisely the experimentally calculated properties of magnetism. The mathematical derivation is quite complicated, but we can give a brief qualitative description.
Imagine that you are a negatively charged particle moving as part of a current in a wire. You rush past a bunch of positively charged particles, which seem to move rapidly behind you. You look over and observe another wire, and wave to another negative charge moving at the same velocity as you. Using Einstein's principle of the equivalence of reference frames, you assume that the frame you are in is stationary. Looking over at the other wire, you notice that in that wire there are a bunch of positive charges, moving behind you in a manner similar to those in your wire. According to Einstein's length contraction, these positive particles seem closer together than the negative charges, which are stationary in your reference frame. Thus, according to your frame, there are more positive charges per unit length of the other wire than negative charges. Since you are a negative charge, you are naturally attracted to these positive charges, and that attraction of yours overcomes the repulsion you feel toward the less tightly packed negative charges. Now consider the reverse: you are a positive charge in the other wire, with negative charges rushing past you. You see the first wire, with what seems like a higher concentration of negative charges, and are attracted to it. In this way both wires are attracted to each other, accounting for the magnetic phenomenon we encountered empirically earlier.
Don't worry if this derivation is a little confusing. This section is not essential to the study of magnetism. We have merely provided an informal treatment of the subject before diving into the mathematics because many times students simply plug values into equations when studying this complicated topic, and have no idea of the concepts behind it. Equipped with a practical understanding of what magnetism is, the sections to come will become much easier.