Adventures in Numbers, Measurement, and Math

Why TGM isn't Based on "Fundamental Constants"

In the dozenal system, there are perennially efforts to base measurement systems upon what are called the "fundamental constants." Typically these will include the Planck length, the gravitational constant, the fine structure constant, the charge of the electron, and similar constants.

TGM, of course, eschews this approach, instead basing its system upon the fundamental realities of the world around us. There are good reasons for this approach; a system based upon fundamental constants simply can't work. The reasons it can't work are simple and direct.

Accuracy of our Measures

There's no way to determine when the constants are accurately measured. We've had this problem with SI, as well; the meter, designed to be a very specific fraction of the earth's circumference, just isn't due to measurement error, instead being defined based on a platinum bar sitting in a vault in Paris. Systems based on the fundamental constants will inevitably wind up the same way.

Our best measurements of these fundamental constants are constantly improving; indeed, the Committee on Data for Science and Technology releases new, more accurate values for these constants every few years, most recently in 11E6. These constants are tiny, and getting them right is difficult; their accuracy improves constantly, and consequently the values we use for them changes constantly.

With a system based on these constants, every time that happens, the precise values of the units of measure must also be changed. Given how tiny these constants are, and that the values being used on a daily basis are typically very large multiples of those constants, the necessary changes can be significant. In a system based on the charge of an electron (4;16908EE233e-15 Quel), for example, narrowing that measurement to 4;1690900e-15 Quel, a very small increase, would result in a unit of 4;16909 instead of 4;16908, which is close enough for daily work but quite different in terms of scientific accuracy.

This is, of course, assuming that we make our human-sized units by multiplying by 1015, an assumption which may not be justified, and which brings us to our next point.

Scaling the Constants

These fundamental constants are obviously not suitable for direct human use in most situations. No one wants to measure out their room in square Planck lengths. Human-sized units must be produced in fundamental constant systems by multiplying those constants by some power of twelve.

But how do we select the power of twelve by which to multiply it? We may choose 1015 for electrical units, but we can't choose the same for distance units, because the Planck length is much smaller; we'd need something more like 1028 for that; or, to use the speed of light, we'd need something like 10-8 or 10-7!

We can do these things, of course; but they are fundamentally arbitrary; that is, they cannot be reduced to a system, but instead rely entirely on the judgment of the system's creator. But we want something systematic, not arbitrary; so this is a difficult problem for a metric system based on these constants. Which brings us further to the next problem.

Selecting Appropriate Constants

It's not at all clear which constants should be selected for such a system, and once selected, how to ensure that it fits in well with other quantities. To demonstrate, let's pick two of the most fundamental units in any metric system: length and speed.

To select a unit of length, at least two constants spring immediately to the fore: the Planck length and the speed of light. (One could also derive one from the gravitational constant, along with a unit of mass and time; but let's stick to these two for now, just for simplicity's sake.) Which do we select, and why? Ultimately, this choice is arbitrary, as well. But some amount of judgment is unavoidable in metrics, so let's skip ahead for a moment.

Presume we select the Planck length, and multiply it by 1028 to produce our human-sized length unit; we'll call that l. Also presume that we've already selected a time unit based on something else, which we'll call t. Our speed unit is therefore l/t. But this will not be equal to c, the speed of light, nor evenly divisible by it; yet the speed of light is unquestionably a fundamental constant.

Reverse the situation, if you like; choosing our units of length and time based on the speed of light will produce a unit of speed easily divisible by the speed of light, obviously; but our length unit so based will not be compatible with the Planck length, nor will our unit of time be compatible with whatever unit of time we've selected based on some other fundamental constant.

And here we've not even proceeded past units of length and time! We've not even concerning ourselves with relating these units to acceleration, force, electrical units, light units; and already we've arrived at a point where we're having to favor some fundamental constants over others, with no clear criteria for which should be favored over which. There is no way to base a system on fundamental constants without sacrificing easy use of some of those fundamental constants. In the end, one has gained nothing.

Much better to select some human-sized unit familiar to all, as TGM does, and allow the fundamental constants to be irregular units.