From F=ma to ma=mg
The Newtonian formulas for gravity are sometimes used to justify the fact that gravity works the same for all objects, regardless of their mass. 
Here is the mathematical reasoning:

1) F    = ma
2) ma = mg 
3) F    = GMm/R²
4) ma = GMm/R²
5) a    = GM/R² = g

This way the mass of the objects being pulled by the earth becomes irrelevant.
Is that a valid reasoning?
In (1), F is a force while ma is the quantity attributed to that force. But neither m, mass, nor a, acceleration, can be considered as a force. So what does (2) mean? Both the quantities are equal. Here is the big question: is g a force? That would be quite strange, because that would mean that we have two forces of gravitation, g, and G. Or maybe even three, if we count F as we should.
In fact, only F can be considered as a force, G and g being the values F can have depending on our perspective.
(2) would therefore be the same as

20) 4 times 9 = 4 times 3². 

Which would be of course right. We would be then justified in positing

21) 9 = 3²

We have gotten rid of the value of the mass. But did we get rid of the mass itself? Did we explain why gravity works the same for all masses? Or did we just express mathematically what we had found out empirically? The values of all factors concerned, m, a, g, G and R, are all empirical. It is therefore not surprising that our calculations give the same result we had already established empirically. Otherwise we would have had serious reasons to doubt the validity of these calculations.
But now, this mathematical reasoning is getting a life of its own. It seems to sanctify our empirical experience with the blessing of mathematical logic. More importantly, it reinforces us in our false conviction that we have found the ultimate reason behind the workings of gravity. In fact, all we have done is express an empirical fact mathematically, without explaining it any further. Newtonians still owe us an explanation as to why gravity does not care about the mass of an object. Empiricists might not be interested in this why question, but it is certainly a fundamental one for cosmologists.