F=ma is not a definition of force, because force has much more to it than just a number. What you can do is say,
"Let me define a variable x, which I will call force as mass*acceleration."
Now you come in the real world, see a body accelerating, you will say I can calculate the variable x for this body. Moreover if you give me x for this body, I can calculate its acceleration or mass. This is what mathematicians do.
Now the physicist comes and says let's observe a rope pulling a block !
The mathematician quickly calculates the value of variable x , observing the acceleration of the block, and is satisfied.
But, the physicist is interested in the "cause" of the motion of block. He asks what is similar in this block moving, and any other mass's motion?
He thinks and observes a lot of blocks moving due to different reasons and comes up with the following :-
I have seen a lot of blocks moving. Although all had different sources cause their motion, I could generalize something out of these cases. It seemed that the motion did not depend on the source , but something else that the source produced at the location of the block. Let's call this thing a dorce.
And just for the sake of calculating acceleration, let me assign a number to this thing that I have perceived as existing alongside all moving blocks that I observed.
One that produces an acceleration of $1 m/s^2$ in 1kg block gets the number 1. one that produces same acceleration in opposite direction get the number -1. One that produces an acceleration of $2 m/s^2$ is assigned the number 2 and so on.. (Note how this numbering would not exist , if he had not observed that there is a thing called dorce..in fact there would be nothing to number! )
Now the patterns he observes in the world and his numbering :-
- When 2 dorces which are assigned a number x and -x are made to act on a particle at the same time, it does not move! No matter what x was.
- When the force numbered x is made to act on 2 $1 kg$ masses tied together , half the acceleration is observed; when the force numbered x is applied to 3 $1 kg$ masses tied together , acceleration produced was $1/3$ original one, and so on. No matter what x was.
- When 2 dorces numbered x and y are made to act together on a particle, the acceleration produced was sum of that produced in case when each one acted alone.
Until and unless, you don't see what the physicist has perceived, you don't know what a dorce is. Nor will you understand the fact that what he has done is classified all the dorces of the universe and numbered each group.
Now seeing the properties 1,2,3 and his numbering he announces, To any dorce, is assigned the number d=ma where m is mass of the particle and a is the acceleration produced in the particle when the force to be numbered is applied on it and by my experiments, I propose that properties 1,2,3 hold in all cases of any dorce being applied on any body.
It is the result of the way he chose to number as well as the inherent properties of the dorce that $d=ma$ holds$*$(and is not defined) and that dorces can be summed to get net dorce.
$*$in all the cases we have observed.
Now as soon as we observe $d!=ma$ ,i.e. that the number we assigned to the dorce doesnot equal the product of mass and acceleration produced by that dorce, we can change the number assigned to that dorce !
NOTE :- The physicist could have said, let's just define d=ma and say that this is the dorce we have perceived and let's use properties 1,2,3. Now he'd have to prove that this definition of dorce which assigns the number ma to the every dorce
that produces a acceleration in body of mass m , is consistent with rules 1,2,3.(unlike earlier case where it was proposed that all 3 properties hold) THIS CANNOT BE PROVED. (Think why?)
Hence , the physicist will never define dorce as ma.
There have been 2 kinds of approximations in doing as in 4th blockquote :-
$1.)$The physicist doesnot measure all possible cases.(He certainly will never be able to).
$2.)$Each time he measured acceleration and mass he measured only upto certain decimal places.(He never will be able to infinite decimal places)
So all physical laws he will propose , will be some kind of approximations.