There are many procedural difficulties in removing the notion of mass, not the least being its intrinsic assumption in all experimental data which leads to our current understanding of the structure and behaviour of a related concept :matter. Because of the long use of the notion of matter, it is also a difficult concept to untangle.
The corpuscular theory of matter is really our main theory of matter from which our current quantum theory by degrees and adjustments has descended. The corpuscular theory has its roots in the atomic philosophy of Democritus and Leucippus, but really much empirical observational data had gone into its refinement. However, it was still a rational description of a non spiritual reality, that is it derives it's raisin d'être as a mechanism for explaining the divide between immaterial and material realities. Thus it brings embedded within it several per Socratic presuppositions regarding fundamental units and ultimate unity. The fundamental unit was thought to be a Pythagorean scholarly conception, enabling all forms and magnitudes in reality to be accounted for. As I have discussed this was a model of all realities based on the notions of the mosaic floor patterns in the shrines to the muses.
There was a reality beyond the unit, but it was unknowable by humans, and inexpressible by them, but devotion could bring a mystical experience of it, or so Parmenides revealed was his experience of The Monad. Thus the unit was of fundamental psychological as well as logical and philosophical importance. The work of Eudoxus, in tandem with Plato ensured that it's position would not be seriously challenged again, as reputedly had happened in the Pythagorean school.
The theory of corpuscles therefore was a scholarly attempt to satisfy a long history of debate about the nature and structure of reality as humans experienced it, and it avoided the clear fractal foundations of the immaterial reality with its various levels of experience. It really grounded human reality on a fundamental unit particle, to which all attributes of human experience could in principle be attached, and from combinations of which all human experience of human reality could be synthesised. It took a long time for this theoretical and philosophical model to find supporting empirical data, but when it did it typically lead to revision of the theory toward that new " standard" of describing it.
Along the way the influence of Newton cannot be discounted. His formulation of the mathematical principles of quantities, a philosophy of Quantity was so profound that Kant declared a truce between rationalism and those philosophies that were faith based could now stop the controversy and focus on the evidence that Newton demonstrated. God was a Divine Mechanic, and all could now acknowledge that with due awe.
Of corse that was not the end of the matter, but Newton provided empirical demonstrations of the Laws he proposed God had set in place, and amongst those principles was the quantity of matter.
Newton included magnetism among his principles, in acknowledgement of Gilbert's work, but the principle of occult fluids like electrostatic fluids he assiduously and definitely avoids from the outset. Because of this, the effect on the growing interest in electric phenomenon, defined in passing by Gilbert, on his way to a grand theory of magnetism, was profound. Electric fluids could not be admitted into the constitution of particulate matter, although of course magnetism could be. Boyle takes great pains to touch upon these occult forces with care so as to preserve his social reputation!
Matter or mass then enjoyed a privileged position across Europe and into the Americas. However, in the east matter and mass were generally considered as illusory conceptions, concealing of the fundamental reality which is spiritual, and related to Play!. The word matter itself in the indo European root is related to illusion, playful experiences, thought not object.
So gradually Newton's gravitational theory and his principles became the gold standard, because now a concept called mass could now be measured and understood, and a standard mass on earth could be proposed, on the presumption that gravity, whatever it was, was as constant as Newton's theory posited it to be. It also engendered, and was used to foster the notion of mass attraction.
Mass attraction is in fact a non Newtonian idea. Newton famously said he did not have a hypothesis for the nature of gravity. Even his book the Systems of the Worlds, which greatly influenced Cotes, did not purport to say what the nature of Gravity was. All Newton himself ever discusses is the application of certain quantitative measures he has defined in analysing the behaviours of planets. What Newton could not account for was the stickiness of matter, and the absurd action at a distance! Le Sage spent much of his life attempting to explain these things in terms of the collision models of the corpuscular theory, but again he relied on mass attraction because Newton had apparently demonstrated that! Newton, of course begged to differ, being very cautious about such claims even though closely following the argument of Le Sage and the various demonstrations of the inverse square law result for a large number of particles themselves obeying the inverse square law. The argument is clearly tautological, it assumes an inverse square law to demonstrate an inverse square law. While it is not obvious that a large number of particles should compound to an inverse square law, the mathematical demonstration thst it does, does not demonstrate any physical law that it should, nor does it preclude other possible explanations. All in all Newton remained to be convinced of mass attraction. Others were not so scrupulous!
Action at a distance is by every means of description, the most occult phenomenon. Cotes defends Newton against the charge of occult practices by smoke screening the argument, pointing to others with way more occult theories, and then focussing on the simplicity, clarity, scrupulousness of Newton philosophical method, and of course it's transparency and humility. By this means, he hoped to persuade the reader and the audience at large that Newton's method had produced the most astonishingly accurate and understandable explanation of yhe data, but furthermore would be the platform to reach forward to yet greater profundities of God's creation. The znewtonian philosophical method was thus recommended, and indeed percolates physics and general science to this day, but is his motion of matter that was the most ignored concept as it went through this process.
Dalton's atomic theory of matter, a reworking of the atomic theory of Democritus, and the corpuscular theory, was based on the "Newtonian" that is gravitational conception of mass. The British empire placed many British scientists in preeminent positions in the development of the chemical theory of matter, but it was an international collaboration on he basis of the gravitational concept of mass.
This rapid churning over of theories of matter left few people time to think clearly. Even when electric phenomenon became increasingly more popular to study, the structure of matter was beinfpg formulated on the mass, and atomic particle principles derived from modified corpuscular theories.. Thus the chemical success of the British Alchemists, along with their European counterparts lead to a periodic table of elements based on atomic weights. Medeleyevs curious periodicity had no explanation.
In the meantime Franklin, Priestly and Coulomb were making some experimental headway with electric phenomenon. But all accepted mass as the fundamental material basis, to which electric fluids would be added. It was proposed 2 fluids be added to explain the 2 effects. Franklin proposed one , called it charge, thought it was positive, and explained the two states by the absence or presence of charge. He kind of had positive charge pushing apart gold leaves, negative charge pulling apart gold leaves and the positive charge restoring equilibrium by going back to where it came from by being rubbed off!
Priestleys report on one of Franklin's experimental findings, triggered Coulomb to propose a inverse square charge law like Newtons gravitational one. At this stage and to this day positive and negative charge are always exposited as point charges, just as Newton used point masses. The reality could not be further from "the truth".
No one at this stage, not even Faraday thought of charge as anything other than an adjunct to mass. Franklin never proposed the proton as a substrate for his charge, neither did coulomb, it was down to Weber to propose an Ionic theory for electrolysis. This was the first inling that matter was charged, the difference is clear: mass does not carry charge, mass is charged.
Of course that was not acceptable because you then have 2 types of mass, and what does that mean? So mass was left to carry point charge until experimental data in the midv1800's made that untenable. Matter had to be differentiated into different types. An Irish scientist seized on the opportunity to reassertion a fundamental unit particle called the electron. This was the first charged particle to be named, but it had yet to be found. Actually some other charged particles were observed first and called canal or anode rays, but they were too difficult to isolate a pattern from.
This fracturing of matter meant a fracturing of mass. No need to rethink mass, just change the size of the units, because it was scientifically "obvious" that everything had mass. However mass had changed, it was now charged, not just carrying charge, even though it is still spoken of in the same way.. Charged matter had immediate implications for chemistry in terms of what held compounds together. Suddenly Dalton's atomic theory needed to be revised. At the same time as the experimental data fractured mass into different charged matter, the proposed electron excited international interest the leading European theoretician became Lorentz. Who proposed with his colleagues a detailed theory of the electron. His calculations based on faraday and Maxwells theoretical descriptions were detailed, but shown to be flawed, in correcting them he came up with a theoretical prediction which astonishes even today.
Meanwhile British theoreticians were also working on the problem of electrical partition of matter. It was on the back of this British theorising and experimental data that J J Thompson was able to set out a series of experiments that defined the physical smallest particle of charge. How did he know it was the smallest? By using the notion of mass again. The gravitational mass determined the size or rather the relative size/density of the particles.
In fact his was not the only calculation to establish the relative masses, besides Lorentz there were other independent confirmations.
Sounds convincing until you realise that this all depends on gravity, an unexplained force at a distance. Thus to obtain a concept of matter from these figures of relative mass/density one has to accept mass attraction. Thus by mass attraction the negative particle of charge , again proposed by an Irish scientist to be called the electron, was less massive than the material thst "carried" it, later called the proton, but equal in charge. This was just on the turn of the century into the 1900's and just at the time to pique Einsteins interest in the considerable theoretical problems this posed for our conception of matter and mass.
Rutherford and Bohr, rushed headlong into this virgin territory, redefining the concept of the atom, founding atomic physics and causing a total rewrite of Dalton's atomic theory explanation of chemical interaction. The periodic table could now be explained in terms of the electron proton interaction, and bond strengths could be proposed and measured. Spatial atomic arrangements could be proposed in crystal formations etc etc., but the problem of mass became apparent.
Electromagnetism had a consistent theoretical structure that gravity did not. Charge and charge fields and magnetic flux fields could be empirically demonstrated, gravity could not. Where was the gravitational field?
Gravity would have been quietly dropped were it not for one thing, the notion of mass depended on it. Get rid of gravity and you remove the foundation to empirical experimental data. Surprisingly no one wanted to tackle the problems with gravity until Einstein. He had earlier redefined a relationship between mass and energy, and in special relativity introduced the concept of space time to physics, with new rules to the game one constraint in the Lagrangian was the speed of light was the maximum allowed scalar speed.
Einstein then went on to tackle gravity in general relativity. Using the Lagrangian he made mass obsolete, that is interchangeable with energy. What he was doing was what Coulomb and others failed to do, redefining matter and mass in the light of a fundamental change in our model of space. In this conception the quantity of matter is allowed to be dependent on volume and particle density, but the force on that quantity of matter arises from the correct constraints on the behaviour of matter. By making that constraint a complex combination of space and time rather than some assumed force he was able to remove the concept of mass, and indeed charge. Now the behaviour was explained in terms of how spacetime was warped. Doing it this way was extremely hard work, but it is perhaps the only way ro get rid of the incipient presuppositions of long ago. However the results more than justify the assertion that replacing mass by charge, that is completely eliminating mass, provides the best explanation of gravitational attraction we can come up with. When you do that you pave the way for the realisation that we have a unified theory based on the two types of forces repulsion and attraction at all scales, and that is the signature of a charge field theory consisting in 2 types of charge field that interact dynamically.
What I propose to replace mass with is a vector quantity called Quantity of Charge Space, which is the conjunction of volume and the Charge Density vector. Elsewhere I have called this the Fractal Region Density.
This vector is complex including Twistors and field vectors, thus we have a lineal combinatorial description of this Quantity of Charge Space. The important thing is that the vector quantity has a magnitude, and it is this magnitude which sums to give the charge space count value. Thus the vector sum and products may be zero, but the charge space count may sum to a count of the regions. Thus providing a particle sum, like mass, while being dealt with as a vector whose sum may vary directionally and probabilistically depending on the interaction status.
Thus the net space charge quantity may be zero, but the count magnitude may be non zero, and a large quantity like that may be free to move in a field, but experience large forces in the very short range. It may exhibit dipole behaviours in an external field, even with an apparently zero charge density vector, and thus the twistorque force may also effect it's rotational motion in a field.
There will be many different types of space charge quantities as regions of space, and their interaction is what would be studied to explore the behaviour as a model of reality.
Finally the existence of a unit space charge could be used to develop other models of the atomic structure, like the proposed bag of unit charges model, in which the proton is a positive bag containing just enough electrons to only need one more! The dipole field for such a model should correct the current point charge model!