We state that electrons are subatomic particles with no known subcomponents.
We discover that these electrons behave as waves.
We also discover that sometimes, these electrons behave as point-particles.
We conclude that they're the both at the same time.
This, to me, sounds nonsensical. If this happened in math, and one discovered that 1 = 2, one would not go around saying that we have one-two duality, would we? No, we would question the axioms that we used to perform our calculations. If one-two duality is an outcome of those axioms, we should not accept one-two duality, rather we should reject the axioms!
Yet, in physics, this does not seem to be the case. Why? Why is wave-particle duality accepted, rather than the axioms rejected?
And what are the axioms in this case? Well, it's the first line in this question. We assumed electrons have no known subcomponents. Well, clearly our experimens show we're wrong. They do have subcomponents, and when the wave-like behavior of electrons occurs, it occurs precisely because of interactions between those subcomponents that we are not familiar with and therefore cannot understand.
So what am I missing? It seems like physicists have accepted this bizarre notion of wave-particle duality that even Einstein thought was an embarrassment of physics, rather than consider that some premises may be wrong and it is those wrong premises that lead to the wave-particle duality.