We experiment with objects that are undergoing changes of velocity. We mathematically model the patterns found in the data with the equation f=ma. The data consists only of measurements of distance and time. We do not know what force is or what resistance to force is. We deduce from the patterns that both of these properties exist. However, we cannot learn their natures from the data. We have only distance and time to work with.
We experience distance and time directly. We move across distances during periods of time. We do not, at this early stage, see a connection between the two. So far as we can tell, they are unique, fundamental, properties of the universe. At this point, they are the properties of communication with us.
We give distance and time names and adopt unique units of measurement for each of them. The units of distance are called meters, and, those of time are seconds. Their units cannot be defined in terms of pre-existing units. There are no pre-existing units at this beginning point. Therefore, distance and time are accepted as fundamental, indefinable properties.
Even though we do not know their natures, it is from them that we learn all else. What we observe from the experiments is acceleration, and, acceleration consists of ratios of distance with respect to time squared. We deduce that the experiments indicate four properties. They are: distance, time, force, and resistance to force. We see variations in the motion of objects; but, we deduce just these four properties.
We cannot determine from the evidence that force and resistance to force are fundamentally unique. They may be related to each other. They may even be related to distance and/or time. In other words, unity may exist at this fundamental level. There may be a single cause that is manifesting itself in different ways. At this early stage of scientific learning, we cannot tell whether or not this is the case.
We decide to proceed into the unknown. We begin to theorize about the meaning of f=ma. There are different possibilities of choice. If we guess that the natures of force and resistance to force are fundamentally unique, then, in order to proceed with using the equation, we must, because of not knowing otherwise, theorize into existence a unique nature for one of them. We give them names. One is called force, and, the other is called mass. We choose mass to be assigned an indefinable nature. By this act, it joins distance and time as indefinable properties. Force is then definable in terms of mass, distance and time.
So far, our act has consisted of words and ideas; the equation is not directly affected. Now though there has been created a need to invent a unit of measurement for mass. Its units, to accurately represent the indefinable status assigned to mass, cannot be defined in terms of pre-existing units. Mass is assigned units of kilograms. There are now three indefinable units of measurement. By this act, the theoretically invented uniqueness of mass becomes solidified into the equation. The units of force are named Newtons and are defined in terms of kilograms, meters and seconds.
The equation that began as an empirical mathematical expression has been transformed into part of a theory. The theory is that mass is a fundamental indefinable property. In other words, it is set in stone in the equation that we do not know what mass is. From this point on, the equation becomes subservient to the ideas of the theory. That theory will infect itself into all higher level theory that makes use of this definition of mass.
What difference does this make? If unity exists at the fundamental level, we have not only missed the opportunity to discover it, but, we have introduced theoretical disunity into our analysis at its very beginning. If fundamental unity exists, we will not be able to discover that unless we determine how to define mass. No other correction will work.
It will be possible to theorize about the existence of unseen properties that when inserted into physics equations, will give the appearance of achieving unity. But, unity is not something that can be patched onto theory that has been developed on the premise that fundamental unity does not exist. That premise will remain solidified into physics equations until we define mass. When we learn what mass is, physics theory could change radically.
James Putnam
