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Of course, "particles" are not some mathematical abstractions but real things that can be observed. E.g., take electrons. They were discovered at the end of the 19th century by J. J. Thomson in cathode ray tubes by studying their reflection in electric and magnetic fields. They were "visible" for him due to the excitation of the rest gas in the tube.
Of course, this and other discoveries at the time (like radioactivity) started a whole new world for the physicists, namely the world of atomism in a broad sense, i.e., the fact that matter is composed of particles. As it turned out, however, the meaning of the concept of "particle" had to be drastically revised, which lead to the discovery of quantum mechanics in the 1925 and very quickly thereafter also of quantum field theory. These theories are pretty abstract but necessary to fully describe the behavior of "particles" and to make sense out of this picture of matter at all. It is thus not so easy to answer the question, what a particle is. Strictly speaking, physics doesn't give an answer at all, and that's also not the purpose of physics. The natural sciences try to describe as good as one can and as accurately as possible all quantitative observations in Nature, but it doesn't tell you what it is what we observe.
From a theoretical point of view, you thus get sometimes answers like "an elementary particle is what can be described as the asymtotically free Fock states of elementary quantum fields". This is the most accurate answer you can get from the point of view of a theoretician using the most recent scientific knowledge about how to get an accurate description of what an elementary particle is.
This is, however a pretty incomplete picture, because physics is after all an empirical science about observable facts, and thus this very abstract "definition" of the theorist must be always seen as complemented by how experimentalists observe the "particles" and how they make sure that the theoretical description as an "elementary particle" is really right. So the theorists make a model, based on knowledge about observations concerning "particles" and provide the experimentalists with all kinds of observable properties of what they call "elementary particle". Then the experimentalist can build devices to test these hypotheses.
This endeavor went on from the early days of Thomson, the Curies, Rutherford et al until today, and with more and more technological progress (among them the ability to build accelerators that brought the "particles" to higher and higher energies, and very fancy and accurate dection methods) and more and more progress in describing the particles. From this endeavor the "Standard Model of Elementary Particle Physics" emerged as the so far most comprehensive model. It was finished in the early 1970ies with the discovery of asymptotic freedom of QCD, I'd say. Since then it was more and more confirmed, and this success is on the other hand also a kind of problem, because there are some reasons to hope for it to fail, i.e., to make some observation that contradicts the predictions made by it. That's why at the LHC, after having its discovery of the last building block of the Standard Model, the Higgs boson, one looks for "physics beyond the standard model" with the hope to find a hint, how to get to an even better description, hopefully also closer to an understanding what the famous "dark matter" or even "dark energy" might be. From the point of view of cosmology we know only about 5% of the energy content of the universe, which is the amount made up by the known "elementary particles" of the Standard Model. About 20% or so is "dark matter", which should exist because of the discrepancy between the motion of stars in our galaxy and the expectation of this motion given the amount of visible "Standard Model matter" and the theory of gravitation (general relativity). The rest of about 75% is "dark energy", which is the most mysterious piece of contemporary physics. It's described by the cosmological constant, but it's totally unknown, why it has the value observed by accurate measurements of the fluctuations of the cosmic microwave radiation and the redshift-distance relationship (Hubble law) measuring supernovae, now used as "standard candles" to the largest distances possible.
Of course, this and other discoveries at the time (like radioactivity) started a whole new world for the physicists, namely the world of atomism in a broad sense, i.e., the fact that matter is composed of particles. As it turned out, however, the meaning of the concept of "particle" had to be drastically revised, which lead to the discovery of quantum mechanics in the 1925 and very quickly thereafter also of quantum field theory. These theories are pretty abstract but necessary to fully describe the behavior of "particles" and to make sense out of this picture of matter at all. It is thus not so easy to answer the question, what a particle is. Strictly speaking, physics doesn't give an answer at all, and that's also not the purpose of physics. The natural sciences try to describe as good as one can and as accurately as possible all quantitative observations in Nature, but it doesn't tell you what it is what we observe.
From a theoretical point of view, you thus get sometimes answers like "an elementary particle is what can be described as the asymtotically free Fock states of elementary quantum fields". This is the most accurate answer you can get from the point of view of a theoretician using the most recent scientific knowledge about how to get an accurate description of what an elementary particle is.
This is, however a pretty incomplete picture, because physics is after all an empirical science about observable facts, and thus this very abstract "definition" of the theorist must be always seen as complemented by how experimentalists observe the "particles" and how they make sure that the theoretical description as an "elementary particle" is really right. So the theorists make a model, based on knowledge about observations concerning "particles" and provide the experimentalists with all kinds of observable properties of what they call "elementary particle". Then the experimentalist can build devices to test these hypotheses.
This endeavor went on from the early days of Thomson, the Curies, Rutherford et al until today, and with more and more technological progress (among them the ability to build accelerators that brought the "particles" to higher and higher energies, and very fancy and accurate dection methods) and more and more progress in describing the particles. From this endeavor the "Standard Model of Elementary Particle Physics" emerged as the so far most comprehensive model. It was finished in the early 1970ies with the discovery of asymptotic freedom of QCD, I'd say. Since then it was more and more confirmed, and this success is on the other hand also a kind of problem, because there are some reasons to hope for it to fail, i.e., to make some observation that contradicts the predictions made by it. That's why at the LHC, after having its discovery of the last building block of the Standard Model, the Higgs boson, one looks for "physics beyond the standard model" with the hope to find a hint, how to get to an even better description, hopefully also closer to an understanding what the famous "dark matter" or even "dark energy" might be. From the point of view of cosmology we know only about 5% of the energy content of the universe, which is the amount made up by the known "elementary particles" of the Standard Model. About 20% or so is "dark matter", which should exist because of the discrepancy between the motion of stars in our galaxy and the expectation of this motion given the amount of visible "Standard Model matter" and the theory of gravitation (general relativity). The rest of about 75% is "dark energy", which is the most mysterious piece of contemporary physics. It's described by the cosmological constant, but it's totally unknown, why it has the value observed by accurate measurements of the fluctuations of the cosmic microwave radiation and the redshift-distance relationship (Hubble law) measuring supernovae, now used as "standard candles" to the largest distances possible.