I am placing this question on philosophy stack exchange because a mathematician wouldn't care, and a physicist would be extremely insulted.

Consider Newton's Law F=ma. First, I am observing this as a definition of force. from a philosophical standpoint (assuming that we have developed a presumably accurate understanding of a coordinate system - and thus acceleration) this begs the question, what is mass? Moreover, it is objectively impossible to verify; it is defined through the invention of undefined quantities. Therefore, any result produced will be true (assuming we use sound logic, of course) because our method of verification will be circular. Say we simply assume it is a characteristic of a body. What, then, is the objective value of this?

We often hail Newtonian mechanics as turning point in physics; but why is it any less brilliant or valid to simply "invent" a new relationship and call it a "law." I could say, for instance, that every body has a quantity called "blablabla" and that from now on, force is the quantity: F=blablabla^(a). Obviously, this is a bit of a silly example... just an illustration.

  • First, physical theories are not just mathematics, quantities in them have operational definitions external to the theory. So F=ma is not a "definition" of force, although it may look that way when the physical part is detached. All three can be independently measured, and the relation verified. Second, Newtonian mechanics does not equal F=ma, or even all three laws put together (Newton did not even claim credit for them). It is a framework that connected measurements to a mathematical apparatus jointly able to make predictions of a previously unparalleled range and precision. – Conifold Apr 26 '20 at 2:45
  • how would one measure force independently from mass. Could you please outline a way of "verifying" the relation? – user46399 Apr 26 '20 at 2:54
  • Measure forces using springs, measure masses using balances, measure accelerations using clocks and rulers. You can even be bolder and eliminate forces from equations you test, as when colliding pendulums, or even from the theory altogether, as in Mach's later reformulations. It is theory as a whole that makes predictions, not individual equations in it, and it is allowed to have quantities with no operational definitions at all, whose role is to properly string together equations that do. – Conifold Apr 26 '20 at 3:10
  • If you try to measure force with a spring, hooks law will simply lead you to newton... We use a balance from newtons law of gravity... – user46399 Apr 26 '20 at 3:24
  • You do not need Hooke's law if your operational definition of force uses springs, it is baked in. The role of force in Newton's mechanics is exactly to string together equations related to different types of measurements, and the second law is supplemented by separate expressions for Hooke's, friction, gravity and other forces to gain substance. One can pick one of them to "define" force, or none, it makes little difference to the theory as a predictive apparatus. – Conifold Apr 26 '20 at 3:32

Although mass measuring devices do make use of the F=ma relationship, that doesn't make the relationship tautological. Once you have the masses of the objects, you can make predictions for an infinite number of novel situations using F=ma again. So F=ma is a predictive tool.

So for example... say we measure to mass m1 and m2 using a spring balance... making use of Hooke's law, Newton's law of gravity and F=ma.

So we have two known masses m1 and m2. So a couple of simple novel situations:

  1. Say you hang the two masses on either side of a pulley. What is the acceleration of the system? We can fully predict this using Newton's laws. Note this is information about the kinematics of the system that was fully measurable prior to our introduction to Newton's laws, but impossible to predict without Newton's laws.

  2. Say attach the two masses by a thread. Place one on a table and let the other hang over the edge. What's the acceleration of the system? Again this is fully predictable using Newton's laws.

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