This "formal definition" of force is vague and unsatisfactory--really no better than the intuitive definition of "a push or a pull".
It is difficult to define the word "force" precisely. Some people think "F=ma" is a definition of force, but Feynman said that's not true and like all good scientists we accept what Feynman says without question. (I'm being sarcastic.) Most textbooks completely gloss over this important issue, and that is disappointing.
Here's an idea for how force may be defined. It needs some work.
Somehow, we need to be able to take something (like a spring) that pushes or pulls, and assign to that thing a vector which represents how hard and in what direction it pushes or pulls. Our definition must be specific enough that we could use it to actually measure the components of this vector numerically in a lab. Step 1: Declare that two things exert the same force when, if you make them push against each other, they balance each other exactly. Step 2: Pick some specific thing that pushes or pulls, like a particular spring in a particular lab, and declare that it exerts a force of 1. (In whatever direction it is pushing.) Step 3: If something exerts the same force as two unit force springs combined, then it exerts a force of 2. If something exerts the same force as three unit force springs combined, then it exerts a force of 3. And so on. If force A combined with a force equivalent to A exerts the same force as a unit spring, then force A is . If three things that exert force A combine to exert a force equivalent to that of a unit spring, then force A is . If three things that exert force A combine to exert a force equivalent to the combined force of two unit springs, then force A is . And so on. The idea is that any force may be defined in this way.
What I have said here gives an idea of how one may go about defining the word "force" precisely. I believe a major weakness of what I've said here is that it could not really be implemented in a lab for the various types of objects one would like to study.
A related issue is that it is tricky to define mass. I wish that textbooks handled these issues more carefully. Singularitarian 04:18, 25 February 2008 (EST)