Since ancient times, philosophers and later physicists debated whether matter is continuous and, therefore, infinitely divisible into smaller pieces, which are just smaller parts of the same substance. Or is matter discrete, composed of small building blocks (according to Democritus – atoms) that cannot be further divided? The philosophy of atomism was developed in Classical Greece by philosophers such as Leucippus, Democritus, and later in Rome (Lucretius), but during the Middle Ages, it was forgotten or rejected. Newton's model of the world did not raise questions about the divisibility of matter, so philosophers had no reason to doubt that matter is continuous and infinitely divisible. The ideas of atomism were revived in the 18th century, and by the end of the 19th century, atoms were already experimentally observed. However, the atom turned out to be not the smallest indivisible building block of nature. Scientists soon managed to split the atom into a nucleus and electrons. But this did not end the search for elementary particles (those that cannot be further divided).
The deeper we delve into matter, the more effort is required to open it up. From experience, we know that a lot of effort or, as physics says, a considerable amount of energy is needed to tear a polymer strip or break a steel strip because it requires breaking molecules or interatomic bonds. Breaking one atom requires 14 eV of energy. Considering the size of an atom, this is a significant amount. However, to break the nucleus of an atom into elementary particles requires several million eV, almost a million times more energy. But even "elementary" particles are not truly elementary. They can also be broken down into sub-particles called quarks, requiring even greater energies reached by particle accelerators. The world's largest accelerators can accelerate particles up to trillions of eV, a hundred thousand times more than needed for breaking the nucleus and a hundred billion times more than needed for breaking an atom. Therefore, we have not reached the end of matter's divisibility, and we have not found the indivisible building blocks of nature. So, it cannot be said that Atomism has triumphed. On the other hand, as we penetrate deeper into matter, the forces holding matter together become stronger, and the energy required to overcome them grows exponentially. Even if matter can be infinitely divided into ever smaller particles in principle, it would require energy that we cannot attain. Hence, even if there is no fundamental limit to the division of matter, the practical limit does exist.
When we break down matter into parts, the parts remember each other as long as their memory is not erased by external influences (so called de-coherence). Breaking down matter into composite parts, we find that some composite parts do not fit the definition of matter. Many elementary or quantum particles have no mass and no defined volume. Heisenberg's uncertainty prevents us from precisely determining either the particle's position or size. Volume becomes meaningless. In the so-called Standard Model, matter is composed of elementary particles obeying Fermi-Dirac statistics, called fermions. Fermions interact by exchanging particles subject to Bose-Einstein statistics, called bosons. Neither fermions nor bosons correspond to the classical definition of matter. In quantum field theory, all particles interact by exchanging virtual particles. However, a virtual particle does not differ much from a "real particle." If you asked a quantum field theorist whether the world is made of "real particles," you would get a negative answer. It is just a mathematical formalism describing our observations. Mathematical formalism does not reveal new truths. It is just another language. In quantum field theory, even the number of particles does not persist – particles are born, particles disappear, particles transform into each other. This contradicts the definition of matter and the principles of substance permanence. Should we expand the definition of matter to include fundamental force fields? But this would only further complicate the definition of matter and make it non-intuitive.
Quantum mechanics teaches us that these elementary particles are also waves. This explains why an atom which according to now discarded planetary model is 99.9999999999999 per cent empty is not empty at all. It is full of standing-waves/particles and force fields. Copenhagen interpretation of quantum mechanics goes even further – it states that we do not know reality beyond how it reacts to certain experimental measurements. In other words, the "deep" reality does not exist. Quantum particles (electrons, photons etc.) are just possibilities which are actualized (become real) by observation or measurement. Quantum particles are reality's response to scientists' experiments. It’s amazing that all solid matter which we can touch, feel, and weight (as well as light, energy, forces and fields, interactions, and communications) is made of these ephemeral particles.