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I know the second law of Newton says $F=ma$ which seems to me the most fundamental expression of the force. In other words, force cannot exist without a "forced" one. It is somehow a "secondary" being, a relation maybe. In addition, the rate of change of velocity of this forced object and its mass determine the quantity of force being exerted.
If we can define mass with some other terms(if the second law is not the definition of mass), then this will seem to be a good definition of force.
However, why the second derivative of position, why not third or something else? Is this just a case of naming? I mean can we say, "oh yes we have a useful quantity for calculations that is $ma$ and it really resembles intuitively the force concept so let's name it as force and denote it by $F$."