In addition to defining force, and showing its relationship with mass and acceleration, the formula is interesting because of few other aspects. It is a given way of thinking, a specific direction of thinking we can use in solving some important questions in real world. This value of directional thinking has its merit even if you can’t tell where the formula comes from. Given formula like this is comparable to an invention, it’s a postulate, an affirmation, a statement, a premise, given a priori, for your use.
So, let’s see what is the additional significance of this formula.
Since you can count almost anything, and conversely, can specify count of anything, finding relationships between counts can be very interesting. For instance you can initially specify that mass count will be 5 kg, i.e. m = 5kg. You specify that it is the mass we want to count or measure (measure is counting how many units of certain amount is in target quantity) and then you specify actual number, in this case 5. Note how you specify the starting number.
Then, acceleration. Pick any value, set any value, say 3 m/s^2.
Now it’s the time to find out how much of force there will be. But, look, the count of force, the amount of force is determined by the counts relationship between the counts of mass and counts of acceleration. Of course, if this relationship is not specified we could give force either arbitrary value or the value implied by specific requirement or application. In both cases we will give the starting number for force. However, since the quantity of force is bound by the relationship between mass and acceleration, we will use these two to obtain the amount of force. The magnitude of force is linked to the counts obtained from mass and acceleration. This is very elegant point! We are allowed to count objects. We are allowed to specify initial, starting quantitative value for objects. We manipulate counts obtained from two objects, and then use that count to obtain amount of the third object. We are allowed to specify a starting number for mass and acceleration apparently from thin air. But, also, if the physical law, which I would call law of quantitative relationships of quantifiable concepts, objects, or processes, is specified, then, instead of us to give the count for force, the multiplication of counts for mass and acceleration will dictate and give us that number! In this case it will be 15 N. Note how, before we specified mass and acceleration values, we did not know at all what the value for force will be. Also, note how counts of mass and acceleration dictate how much force we will count, then possibly apply. Of course, the genius of Newton is to know which quantities to measure and put in quantitative relationship. What to measure and what quantitative relationship to establish between physical objects is the major task of Physics.
At this point role in analysis of mathematics and physics ends. They fulfilled their roles. Even more precisely, mathematics role ends when the multiplication is done. Math did not care whether you multiply 5 apples in 3 baskets, or, as in this case, 5 kg with 3 m/s^2. Math did its job by multiplying the numbers 5 and 3 and giving back the result to you. It is you who kept track of what is multiplied and to what the multiplication result refers to. And then at that point the role of physics ends, in this application. How? Once you obtained a count, amount of a certain quantity, a number, in this case force, the role of physics ends. Physics of course had a part in deciding what to count. The very selection of mass and acceleration, and decision that you are going to quantify, i.e. count them, defines physics. So, physics can be defined in specifying what to count and trying to establish numerical inter relationships between such obtained counts. Logical methods are, of course, used throughout physics and math, but initial assumptions are left to geniuses to discover and postulate.
So, again, once you obtained the number, an amount, a count of certain property, in this case force, the role of Physics ends in this analysis. Everything else, what are you going to do with this force of 15 N, why, when, is matter of other sciences. Is it a force that has some value for human experience? Is it ethical to apply this force? How about moral value? Is it a force that will be apply to start a boat motor that can save lives later? Is it a tangential force on the bomber's engine pylon? All these questions are started after the physics did its job, in selecting force as a property of interest and specifying an amount for it. The interpretation of what that force is and can be is a matter of human experience, how our brain process and interpret the effect of the force applied, how biochemical triggers and neurological reactions process it, which neural path configurations in our brain, fuelled by oxygen and ATP are activated. What the force of 15N did can be subject of social analysis, cognition, psychology, maybe war tactics analysis, it may trigger our emotional, moral response, it can have legal consequences, economic consequences. And this is a boundary and a limitation of physics and mathematics. It will give a quantitative relationship inside certain system, in this case physical, but it stops there, because it can not define and measure our human experience, our human valuation of it. Is there a way to connect physics and our human experience? There is. The way is mapping, associating the physical actions with what do they mean to us from moral, ethical, emotional point of view (for instance, is physics used to develop nuclear and conventional weapons, are we going to build wind farms on the shore, are we going to construct a dam and possibly change ecosystems with the artificial lake). But, this map is not created by physics laws. It’s created by us.
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