It is certain that no other language ever used by man has the integrity, efficiency or sheer elegance of the language of mathematics. And from a perspective that values these attributes, likely no other language has ever had its beauty. (Footnote: One can hope that the fact that mathematics has the unpleasant reputation it has in much of our culture is due to our propensity to miscommunicate the point of what we are doing when we describe it to other people. The only compelling alternative would be to assume that we are afraid of mathematics precisely Because it gives us the ability to state things definitely, soundly and well, along with criteria to understand when we have done so. This carries with it the inescapable ability to tell when we are wrong or simply do not know enough to make statements of such quality. If we who use mathematics have not simply failed to demonstrate the way that it makes us capable of stating what we always wanted to but could not without it, it is hard to imagine another motivation than fear of our own ignorance that could cause us to dislike the most enabling and beautiful creation of language that the species has yet attained.) But that is a question for psychology. As pragmatists, we will introduce a few of the most basic notations from the mathematical languge simply because in what we have done so far we have already used the concepts, and it will be very nice to have a good way of talking about them.
We have already, through answering a need for it, developed a way to state physical laws, which is really a clumsily presented mathematical statement. Now we will proceed to show how the same thing can be said more elegantly.
Everything that we measure about a system is eventually represented by a number, which usually has units. Saying that a rock fell 3 meters is, in fact, tantamount to describing the whole process of marking the place it left and the place it arrived and counting the three metersticks that can be laid end to end to bridge that gap. After all, since the meter is itself an arbitrary unit of length, presumably chosen for its convenience, the only meaning of such a number is in the context of the prescriptions for action that give it meaning. So already, these numbers with units are compact descriptions of what is measured and what result is obtained.
We then combine those numbers into other expressions, like or , to give other numbers with units. While these constructed numbers and the units that accompany them may not have been actually measured, or correspond to particular standards that have special names, they have the same use in describing motions. (For example, while the ``meter'' is a special arbitrary name given to a special arbitrary unit of length, and similarly a ``second'' a name given to a time, there is no special name given to the unit for velocity (meters/seconds), except just ``meters per second''. But as a unit, it is in all respects on the same footing as meters or seconds in the way it encodes a prescription for measurement. And in the next chapter, we will find that there is a special standard with units of velocity that is in fact far more fundamental and far less arbitrary than the meter or the second.)
So, considering our combinations or as essentially statements of the results obtained from measuring certain characteristics of a system, our laws state that ``if you measure this characteristic, in any way, the result you get will be (some particular number with dimensions)''. Therefore it is natural to give such special numbers that encode patterns special names, and to write the previous statement in the form
If we denote the whole process of measurement of the acceleration, together with whatever number is obtained, by some letter (a), where the letter represents the units as well as the numerical value, and we simply name the corresponding expected result on the right side of the equal sign (g) to remind us that we are describing the pattern for gravity, our whole statement collapses to the notation:
A similar relation for springs would then be cast as
where is the particular constant, associated with a particular spring, which we have not bothered to name explicitly before now.
These are the kind of mathematical equations that physicists use to state laws. Note that they are not definitions of anything, but rather predictions that encode particular patterns, which give meaning to the ``='' sign. For instance, the first law states that when certain measurements are performed to find some number-with-dimensions (a), the answer found will always take a value that can be compared to a certain number (g) of ``meters per ''. Because meters and seconds are standard values that can be transferred simply by transporting metersticks and clocks, the content of our physical law can be transferred from one person to another even if the acceleration of falling objects cannot. Thus it is easy, if we find ourselves in circumstances where objects do not fall according to the expected pattern, such as on the moon, to simply transport metersticks and clocks there and measure the accelerations with which they do fall, compare to the number (g) which we have measured on the earth ahead of time, and see if they are the same, and if not, by how much they differ. Thus we have obtained the precision, non-ambiguity and communicability we desired from our description of the pattern. We have also discovered, proceeding from the assumption that patterns exists as repetitions in the events of nature, that whenever we can formulate a description of what happens in the form of a physical law, we have also made in the same statement a prediction of what will happen in future occurrances of the same kind.