Inverse units, inverse algebra

The use of space (stance) as an independent variable and time as a dependent variable leads to inverse ratios. There is pace instead of speed, that is, change in time per unit of length instead of change in length per unit of time. But a faster pace is a smaller number, which is counterintuitive and contrary to speed, for which faster speeds are larger numbers. There are two approaches to dealing with this:

  1. Use two kinds of algebra: the usual one, in which zero signifies the smallest quantity, and an algebra in which zero signifies the largest quantity (or infinity). Then they are isomorphic, with their extremes corresponding inversely. For example, speed and pace both measure how fast a body is going, but they use different algebras. A large number for speed corresponds to a small number for pace, and vice versa.

  2. Use the same algebra for both but invert one when making a comparison. For example, speed and pace are effectively inverses (apart from which is the independent and which the dependent variable). Given a speed and a pace for some body, to compare them requires inverting either the speed or the pace. An algebraic means of one corresponds to the harmonic mean of the other.

The first approach inverts the algebra, whereas the second approach inverts the units. The second approach is preferable because we are so accustomed to ordinary algebra that introducing an alternative would be unnecessarily difficult. It’s much easier to change units than to change algebra.

The second approach is found in the comparison of time mean speed and space mean speed: the space mean speed is the algebraic mean of speeds with a common time unit, and the time mean speed is the harmonic mean of speeds with a common length unit. The time mean speed is the mean pace inverted, which is a reciprocal speed, which uses reciprocal arithmetic.

Relativity uses a quotient, β, which is the speed of a body divided by the speed of light. Since the speed of light is the highest speed, β is always between zero and one, inclusive, in which zero signifies rest and one signifies the speed of light. What corresponds to β for pace? At first it seems to be the pace of a body divided by the pace of light. But such a quotient is the inverse of β, which would require an inverse algebra. The second approach is to take the inverse of this, which equals β. This is consistent with the inverse correspondence between space and time.