force

Force is a human-devised concept. It encapsulates how the universe's interaction (be it Coulomb's law, Lorentz force, etc.) alter an object's motion.

Source (Feynman lectures).

The equation \[F=ma\] is a very fundamental relationship in physics. However, using this equation as the sole definition of force can be very misleading, then stating that "if an object accelerates, there is a force on it, and that force is precisely what causes the acceleration" becomes a circular argument.

Instead, the true significance of Newton's laws of motion lies not in providing a rigid definition of force, but in implying that force possesses some independent properties that can be discovered through empirical observation and experimentation. That is, if we only define force by \[F=ma\] and nothing else, we learn nothing new, as the power of the concept lies in linking a measurable acceleration to other laws (gravity, electromagnetism, etc.) that predict the magnitude and direction of \[F\].

Now, the first example of such forces was the law of gravitation given by Newton himself. If there were nothing but gravitation force, then the gravitation and second law of motion would be a complete theory, but we all know that isn't the case, as force is used in virtually every field of physics. Thus, to extend this concept to many other situations, we must understand a very crucial characteristic of force itself. That is, if we find a force that is not equal to zero, we will find something in the neighbourhood that acts as the source of said force. This tells us that force has a material origin.

Newton also gave one more rule about force, i.e. the forces between interacting bodies are equal and opposite, or more commonly known as action equals reaction. It turns out that this isn't exactly true, but a very outstanding approximation (with spectacular accuracy) for all that matters. Same goes with \[F=ma\], it isn't exactly true either (therefore we regard this as a relationship rather than a definition as a definition must always be true).

So, how is it not true? Consider an object, but wait, what even constitutes an object? Someone might say, well, a chair is an object. However, physics also tells us that the chair has atoms evaporating or added to it from time to time, for instance a speck of dirt falls and a minuscule amount of it gets absorbed into the paint. Then, to define a chair precisely, or to say exactly which of the atoms belong to the chair, paint, dirt, or air is very much impossible. Consequently, for all practical purposes, the mass of said chair can only be defined approximately. In the same way, it is impossible to define the mass of any single object because there is not a single, left-alone object in this world, as every object is a mixture of many things. We can only deal with these objects with a series of approximations and idealizations.

In fact, all of physics rely on a whole lot of idealizations and approximations, where its accuracy is sufficient for all we are concerned about. One may be dissatisfied with the approximate view of nature that physics tries to obtain, and may prefer a mathematical definition, but mathematical definitions can never work in the real world. In the same way, we cannot just call \[F=ma\] a definition, deduce everything purely mathematically and make mechanics a mathematical theory, when mechanics is in fact a description of nature.