The Larson Research Center

One of the things that the FQXI contest highlights is just how much mathematics, geometry and physics enter into philosophical discussions! There is no way to get a handle on anything other than a small fraction of the discussions the contest generates, and the mix of professionals, semi-professionals and amateurs makes for a unique and stimulating experience. I encourage all ISUS members and interested non-members to participate.

I mentioned one of the effects the contest has had on me in the New Physics blog: It forced me to recognize that the number line is sensitive to perspective. With respect to the unit progression, or the physical datum of the physical system, the RST, there is only a difference in “direction” between less than unity and greater than unity speeds, while from the perspective of one or the other, the inverse is alway greater. 

In other words, from the perspective of 0 (i.e. 0 displacement from unity), a unit space/time displacement of 1/2 is no different than a unit time/space displacement of 2/1, except in “direction.” They are separated by two units, one in one “direction” and the other in the opposite “direction.” However, from the perspective of 1/2, 2/1 is four times as great, or it is one-fourth as big. On the other hand, from the perspective of 2/1, the same perception holds. An observer in the t/s sector of the universe would regard his time (our space) and his space (our time) exactly the same way we do.

But, from the perspective of a unit speed, a slower speed than unit speed is not the same as a higher speed than unit speed, just as .5 is not the same as 2, though they both are one unit of displacement removed from unity, in opposite “directions.” There is a quantitative difference as well as a qualitative difference, in the latter case.

Hence, in considering the mathematics of the new number line, there are these two aspects of the same relationship to wrestle with. How do we add, subtract, multiply and divide with these 3D numbers? If we add two s/t units, is the sum greater or less than one t/s unit? If less, then four s/t units are equivalent to one t/s unit. If greater, then one s/t unit is equivalent to one t/s unit. Since the universe of motion deals with speeds, I have always thought that the unequal relation held, but when I realized that the 3D inverse of space is required for 3D oscillation, then the equal relation is required.

This leads me to think harder about rational numbers. When a rational number is equated with the infinite parts of a whole, a fraction of the whole, then these fractions and multiples of the whole reside entirely within the realm of positive real numbers: 0 —> infinity. But when a rational number is equated with two, reciprocal, aspects of one component, such as two orthogonal dimensions of space, then both magnitudes are multiples of the whole, residing entirely within the realm of 0 —> infinity, because they are completely independent variables.

Of course, we can add fractions of the whole to the accumulated total of units, in each orthogonal dimension, in order to obtain greater precision in specifying these positive magnitudes of space, but we can clearly see that the meaning of the rational number, as a fraction of a positive magnitude, and its meaning as the ratio of the magnitudes of the two orthogonal dimensions, are quite distinct.

In the context of the space/time ratios, where space is taken to be the inverse of time, we need to make the same type of distinction between the two meanings of rational number. Larson’s conclusion was that the discrete unit postulate prevents fractions of units, in all but the effective sense. In other words, when the limit of a discrete unit is reached in the relations between motions, then motion, s/t, limited by the discrete unit of space, can revert to motion, t/s, which is to say, motion in time, something Larson called “equivalent space.” He writes in “New Light on Space and Time”:

Let us consider an atom A in motion toward another atom B through free space…. According to accepted ideas, atom A will continue to move in the direction AB until the atoms, or the force fields surrounding them, if such fields exist, are in contact. The postulates of the Reciprocal System specify, however, that space exists only in units, hence when atom A reaches point x, one unit of space distant from B. it cannot move any closer to B in space. It is, however, free to change its position in time relative to the time location occupied by atom B. The reciprocal relation between space and time makes an increase in time separation equivalent to a decrease in space separation, and while atom A cannot move any closer to atom B in space, it can move to the equivalent of a spatial position that is closer to B by moving outward in coordinate time. When the time separation between the two atoms has increased to n units, space remaining unchanged, the equivalent space separation, the quantity that will be determined by the usual methods of measurement, is then 1/n units. In this way the measured distance, area, or volume may be a fraction of a natural unit, even though the actual one, two, or three-dimensional space cannot be less than one unit in any case.

This is an astounding, but perfectly consistent concept. It means that the only way a unit radius ball of space can contract to zero is for an inverse ball of time to increase to unit radius and vice-versa, but Larson never envisions this idea of equivalent space (time) in any other sense than that of relative positions, the non-progressing locations of space and time occupied by atoms. Clearly, however, the consequences of this concept ought to manifest themselves much earlier in the development of his RSt. The reason they don’t, I suspect, is that Larson’s initial progression reversals are 1D not 3D, as are ours, and the requirement for the contraction of 1D units to zero, needing to be accompanied by the expansion of 1D units of the reciprocal aspect, is not as apparent in the 1D case as it is in the 3D case.

Regardless, the idea that 3D time, or 1/s3, must increase from 0 to 1, if 3D space, or s3/1, is to decrease from 1 to 0, is a fundamental consequence of the RST postulates. The fact that it is mathematically consistent is shown by the 2D analogy of rotation, when we describe rotation by the changing angle of the radius, together with the changing angle of its inverse, or the two changing angles of the rotating diameter of the unit circle. As one end of the diameter rotates the last degree, say inward from 179 degrees toward 180 degrees (or 1), the inverse end MUST rotate inward from 359 toward 360 degrees (or 0), and as the rotation of the diameter reverses “direction” at 180 degrees, heading away from 180 degrees outward toward 181 degrees, the inverse end must also reverse “direction” heading outward from 360 degrees (0) toward 1 degree. There is no other way.

So this is a major distinction between the rational numbers of true inverses, and the rational numbers of orthogonal variables. In the latter case, we can change the magnitude of one, without affecting the other, but not so in the case of the former. At least in the case of the space/time progression, where it serves as the datum of the physical universe, an increase in space has the same affect on the magnitude of the motion, as an increase in time, just as the magnitude of an area is affected equally, regardless of which of its two, orthogonal, dimensions is increased or decreased.

The difference is that the magnitude of an area is not normally required to be held constant, while the magnitude of the natural progression of the RST is. Therefore, we cannot always treat the numbers in the space/time ratios that pertain to the order of progression, in the same manner that we treat the numbers of the x/y dimensions that pertain to bounded magnitudes.

For instance, we cannot just add (subtract) quantities of space (s/1), or quantities of time (t/1), to/from existing units of motion, changing their magnitudes. In order to change the magnitude of motion (s/t or t/s), we have to add (subtract) units of motion to/from units of motion.

So this difference requires a different algebra than the one we use with the notion of bounded magnitudes. It is an algebra restricted to rational numbers, where the two units that form the numbers of the number system that constitute the unit ratio cannot be sub-divided, as with a knife. The range of sub-divisions of the bounded magnitudes of traditional algebra is unlimited, but no such concept is possible in the new algebra.

This has many consequences, some of which we will try to explore here soon.