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Well, Newton's three laws talks about forces, but no definition is given. In truth, Newton's second law gives an idea of what total force is: the time change rate of momentum. But, if we have a force amongst others, this force alone isn't the time rate change of momentum, so that we cannot use Newton's second law to define force, but just to try defining total force.

Knowing all of this, is there a definition of force or we take force as a primitive concept like that of time?

Thanks very much in advance!

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It's a matter of taste. You can take $F=ma$ to be merely a definition of force, in which case what's testable in Newton's laws are the ancillary statements such as the claim that forces add like vectors. Or you can take force to be defined operationally, e.g., in terms of scales and balances, and then $F=ma$ is a law that can be tested empirically.

Knowing all of this, is there a definition of force or we take force as a primitive concept like that of time?

Time might or might not be a primitive notion. Again, it's a matter of taste. For example, one could define it operationally as what a clock measures (but if that is going to work, then clock comparison experiments had better give null results).

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    F = ma isn't a definition for force, but the expression of a relationship between three quantities. Force can exist without acceleration. stretch a rubber band to place it around something; there is tension (force) in the rubber band, but nothing is accelerating. Perhaps a better definition might be something along the lines of "capacity to accelerate a mass" or "capacity to perform mechanical work". – Anthony X Sep 16 '13 at 00:46
  • @AnthonyX You raise a good point. But there is much to be said for stepping back to the modern view of trading the idea of "force" for one of "interaction" between systems. Swapping of virtual particles if you like. Then Newton's second law simply becomes a definition at a macroscopic level for describing what happens when systems are interacting and the impulse exchange is unbalanced. Curiously, the modern QF idea is much nearer than the classical one to something I would think of as good for small children to grasp: ... – Selene Routley Sep 16 '13 at 00:58
  • @AnthonyX ... see answers to "Explaining Newton's Laws of motion to a 6 year old" – Selene Routley Sep 16 '13 at 00:58
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    @AnthonyX: Force can exist without acceleration The F in F=ma is the total force. –  Sep 16 '13 at 01:00
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Many physical science and introductory physics texts define a force as a push or pull on an object. While I have often thought this possibly too simple, it actually works pretty well. I have personally pondered the same question myself quite a bit, and I think the difficulty lies in the fact that we have an intuitive idea of what a force is in common language. It may have been the same for Newton, and remember that before Principia Mathematica there weren't a lot of resources or well-defined terms. As of Newton's day, the only recognized forces were contact forces, which is why the story of Newton's apple is so profound. That the earth could pull the apple down with touching it meant that the sun could pull the planets, and the earth could pull on the moon.

Presently, with centuries of continued study and new fields of research, depending on what you decide is the fundamental area of physics, one can find several "definitions" of force. It can be the rate of change of momentum over time ($F = \Delta p /\Delta t$) or the change of potential energy over space ($F = -\Delta U /\Delta x$ or $F = -\nabla U$).

I wonder what Newton himself would say. I imagine that if he were pressed for a definition, he could at least say "A force is something that causes acceleration." That definition would be well supported by his 1st and 2nd laws; nothing accelerates without a force, and when there is a force, the acceleration is proportional to it (and inversely proportional to mass).

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