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.
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.
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.
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.