Why I can’t walk through walls, d’oh
In the 18th and 19th centuries, Newton’s mathematical description of motion using calculus and his model for the gravitational force were extended very successfully to the emerging science and technology of electromagnetism. Calculus evolved into classical field theory.
Once electromagnetic fields were thoroughly described using mathematics, many physicists felt that the field was finished, that there was nothing left to describe or explain.
Then the electron was discovered, and particle physics was born. Through the mathematics of quantum mechanics and experimental observation, it was deduced that all known particles fell into one of two classes: bosons or fermions. Bosons are particles that transmit forces. Many bosons can occupy the same state at the same time. This is not true for fermions, only one fermion can occupy a given state at a given time, and this is why fermions are the particles that make up matter. This is why solids can’t pass through one another, why we can’t walk through walls — because of Pauli repulsion — the inability of fermions (matter) to share the same space the way bosons (forces) can.
While particle physics was developing with quantum mechanics, increasing observational evidence indicated that light, as electromagnetic radiation, travelled at one fixed speed (in a vacuum) in every direction, according to every observer. This discovery and the mathematics that Einstein developed to describe it and model it in his Special Theory of Relativity, when combined with the later development of quantum mechanics, gave birth to the rich subject of relativistic quantum field theory. Relativistic quantum field theory is the foundation of our present theoretical ability to describe the behavior of the subatomic particles physicists have been observing and studying in the latter half of the 20th century.
But Einstein then extended his Special Theory of Relativity to encompass Newton’s theory of gravitation, and the result, Einstein’s General Theory of Relativity, brought the mathematics called differential geometry into physics.
General relativity has had many observational successes that proved its worth as a description of Nature, but two of the predictions of this theory have staggered the public and scientific imaginations: the expanding Universe, and black holes. Both have been observed, and both encapsulate issues that, at least in the mathematics, brush up against the very nature of reality and existence.
To understand why the repulsion arises, consider two helium ions, and assume that you put them right on top of each other. Of course, with the nuclei right on top of each other, the nuclear repulsion will be infinite, but ignore that for now. There is another effect, and that is the interesting one here. There are now 4 electrons in the 1s shell.
Without the Pauli exclusion principle, that would not be a big deal. The repulsion between the electrons would go up, but so would the combined nuclear strength double. However, Pauli says that only two electrons may go into the 1s shell. The other two 1s electrons will have to divert to the 2s shell, and that requires a lot of energy.