I posted below on the limits of math, and now I want to give a physical example.
Consider a simple photon. It would seem to be completely describable by its direction (velocity), frequency (energy, wavelength, color), and spin (polarization). The photon can then be considered synonymous with a few numbers. This is not really possible because of the uncertainty principle, but physicists like to pretend that there is an underlying mathematical reality.
Now suppose you have two entangled photons that are emitted in some process that assures that they are equal and opposite. So their.y frequencies are the same and their velocities and spins are opposite. Measuring one tells you that you will get the same outcome if you later do the opposite measurement on the other.
The trouble is that even if the photons are miles apart, there is no way to give separate mathematical descriptions of them.
These entangled photon experiments are sometimes cited as proof of nonlocality. But there is no physical nonlocality. There is nothing that you can do to one photon that will have any physical effect on the other photon. The apparent nonlocality is just an artifact of the mathematical models.
The problem is that when a quantum of energy splits into two opposing photons, the mathematical description of that quantum does not split into separate mathematical descriptions of those photons. The photons are physically separate, but not mathematically separate.
My conclusion from this is that the mathematical description of the photon is inadequate. Others conclude that there no such thing as causality, or that there is no reality, or that physics is spooky. These latter conclusions seem crazy to me.
The current RadioLab podcast explains how locality is essential to our everyday experiences. Life is like being at the end of a slinky, in the words of the show. Any locality violation would be huge news.
Usually when mathematical models fail, the solution is to look for more accurate models. But the failure to describe a photon is not an accuracy problem.
The uncertainty principle says that you cannot measure the position and momentum of a particle at the same time. A particle cannot be defined by its position and momentum. Your first reaction to this principle was probably that it was just a statement of our lack of cleverness in making measurements. But it is not. The principle is a statement that physical reality does not match our mathematical descriptions.
Likewise, entanglement shows that the physics does not match the mathematics.
There are physicists who argue that the whole universe has a completely mathematical description. In fact we have no such description for even a single photon. Or a single electron. And much of 20th century physics says that there can be no such description.