In the physical sciences, a particle is a small localized object to which can be ascribed several physical properties such as volume or mass. The word is rather general in meaning, and is refined as needed by various scientific fields.
Whether objects can be considered particles depends on the scale of the context; if an object's own size is small or negligible, or if geometrical properties and structure are irrelevant, then it can be considered a particle. For example, grains of sand on a beach can be considered particles because the size of one grain of sand (~1 mm) is negligible compared to the beach, and the features of individual grains of sand are usually irrelevant to the problem at hand. However, grains of sand would not be considered particles if compared to buckyballs (~1 nm).
The concept of particles is particularly useful when modelling nature, as the full treatment of many phenomena is complex. It can be used to make simplifying assumptions concerning the processes involved.
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I've quoted this same Wikipedia definition before in a previous attempt to discuss the way that this concept, and other models, are used in the physical sciences. It seems to me a good start. So let's try again! The particular physical science about which I am most comfortable speaking is physics, so I'll confine myself to that subject for now.
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The physical world is hugely complex and interactive. Everything has some kind of effect on everything else. And there are countless billions of "things". If I wanted to attempt to calculate the exact motion of any object then I would, in principle, have to account for the gravitational, electromagnetic and other influences on that object of every other piece of matter in the universe. Clearly this is impossible.
But, rather than give up and say that nothing is predictable at all, we use a method called "reductionism" to try to at least partially understand and predict the behaviour of nature. Reductionism means, essentially, ignoring some things. Actually, it means ignoring most things. Actually it means ignoring almost everything!.
Take, for example, the calculation of the motion of an object falling freely towards the surface of the Earth. In this calculation we typically ignore:
1. The gravitational, electromagnetic and every other force of every object in the entire Universe except the object being considered and the Earth.
2. All forces, except gravity, acting between the object and the Earth.
3. The shape and mass distribution of the Earth.
4. The shape and mass distribution of the object.
In other words, we treat the problem as one of two infinitesimally small points (particles) with, therefore, zero volume, infinite density and finite mass moving freely in a vacuum. When expressed like this, it may seem crazy. But it works! It gives an accurate enough result for many purposes. So reductionism is very useful. We literally could not live without it.
An essential feature of reductionism is the concept of "models". A model is an imaginary piece of the world which possesses only a small subset of the properties of the real thing. One such model is "particle". The most obvious piece of reality missing from the concept of a particle is physical extension. A key simplification in the particle concept is that it is used whenever we think that physical size is a small enough consideration to be ignored.
So, to answer the question. A particle is a model of some aspects of the natural world. It is very useful.
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I presume that any discussion which follows will go into the ways in which the particle concept has been used (and, some will no doubt claim, abused) in real life physics. But, in any such discussion, I think it's important to remember that this idea of a particle being a model is crucial. And it's important to remember that all physicists implicitly understand this, even if it is not reflected in their day-to-day language about particles and other models.