On the RS2 site, here, Bruce Peret discusses the history of Larson's development of the fundamental postulates from which he deduced his universe of motion. Bruce makes some interesting observations on the evolution of the postulates, which started me thinking about some fundamental issues in geometry and mathematics.

Interestingly enough, Bruce points out that Larson did not use the word motion in the beginning, but the compound word space-time, indicating a ratio of space and time, without employing the mathematical symbol usually used to show an inverse relationship between two scalars quantites. Later, he decided to use the word motion instead, because, even though it is fraught with unwanted connotations, it captures the inverse relationship of the two changing scalar quantities involved, showing that it is the reciprocal relation of changing space and time that is central to the postulate, and that the two quantities are simply two aspects of one reciprocal change.

This is a crucial point to make, because it reveals the difficulty of conveying in symbols what the words of the postulate are meant to convey: While the assumption of the reciprocal relationship of space and time is crucial, the bounds of that relationship are not confined to one of reciprocal change conveyed by the word motion alone, but extend to an inverse change as well, an inverse motion if you will, which, mathematically, is not motion but energy. Yet, in legacy physics, energy is not strictly a mathematically scalar quantity, but a scalar quantity defined physically, through the use of a vector quantity (work).

Reconciling the symbols of mathematics with the new physical concepts was not the only challenge facing Larson in his attempt to fashion the fundamental postulates of the universe of motion. As Bruce points out, the challenge of defining what is meant by the word scalar was also part of the daunting task. How do you capture the idea that the fundamental change upon which the universe of change is founded is not vectorial change, but scalar change, without explicitly including words to that affect in the postulate?

Larson's solution was to add two more postulates to the system, one of which he later dropped, stating that, if need be, it could be restored, but that it was unnecessary, because "The scalar nature of space-time is a part of the system; the only question at issue is whether or not it needs to be expressed as an additional postulate." Ok, but then what is it that makes the scalar nature of the system's motion part of the system? Larson explains:

The space-time displacements which are necessary for the existence of physical phenomena originate because the reciprocal postulate involves something more than the equivalence of the individual units. If this were the extent of the relationship we would postulate that space and time are equivalent, not that they are reciprocal. The reciprocal postulate includes the further requirement that under certain conditions associations of n units of one component must exist and that under those conditions the n units of this kind are equivalent to 1/n units of the other component.

We are then led to inquire how it can be possible for n units of space or time to act as an association when each of the individual units in this association is required to Progress uniformly with a unit of the opposite kind as an integral part of the general space-time progression. A detailed consideration of this point discloses that it requires the existence of a difference between space (or time) as a constituent of space-time and space (or time) as a separate entity. The only such difference permitted by the Fundamental Postulates is a difference in direction; hence we arrive at the conclusion that space-time is scalar and that direction is a property of space and time individually.

It is this directional difference, the difference between the changing scalar quantities constituting scalar motion and their existence as separate quantities, which makes the explicit assumption of scalar motion unnecessary; that is, the only way to obtain a non-unit ratio of space and time (different speeds, if you will) is if there exists a directional difference in the changing entites (one increasing, the other decreasing in value, for instance.) Larson explains it this way:

From the foregoing it is apparent that where n units of one component replace a single unit in association with one unit of the other kind in a linear progression, the direction of the multiple component must reverse at each end of the single unit of the opposite variety. Since space-time is scalar the reversal of direction is meaningless from the space-time standpoint and the uniform progression, one unit of space per unit of time, continues just as if there were no reversals. From the standpoint of space and time individually the progression has involved n units of one kind but only one of the other, the latter being traversed repeatedly in opposite directions. It is not necessary to assume any special mechanism for the reversal of direction. In order to meet the requirements of the First Postulate the multiple units must exist, and they can only exist by means of the directional reversals. It follows that these reversals are required by the Postulate itself.

That this idea is a deep and fundamental revision of our concept of space and time is an understatement of colossal proportions. It not only has profound implications for physics, but mathematics as well, and the long sought unification of geometry and algebra, way beyond that which Descartes is often attributed with having achieved.

The idea is so simple, but like the idea of eternity, it teases us out of thought, as it were. If the progression continues forever, as a three dimensional expansion, yet each accumulating aspect, at any moment in time, or at any location in space, has the potential to reverse the "direction" of its increasing numbers, and begin to oscillate, so that in one sense it continues to increase, but in another it doesn't, because, like a soldier marching in place, the increase is not an effective increase, a whole new universe of thought is suddenly opened up to our view.

With this simple observation, that each physical dimension has two "directions" (I use quotes to distinguish direction as Larson uses it in the above quotes and direction as it's normally used in a coordinate system,) Larson cuts the gordian knot that has forever perplexed mankind's exertions to understand how nature integrates the discrete quantities of numbers with the continuous magnitudes of geometry.

Of course, volumes and volumes will be written on this subject for centuries to come, but for our present purposes, I want to focus on just the implications for mathematics and geometry. As Bruce points out, the motion consisting of these changing, reciprocal, units defined by Larson in the postulates, can not include concepts of Euclidean geometry ("such as a point, line, circle or plane") in their definition; that is, scalar quantities have no direction ("scalar motion has no geometry!") Yet, the consequences of postulating this "magnitude only" scalar motion, a priori, can be shown to lead to an Euclidean universe, consisting of "right lines and circles," as Newton described geometry.

And just as Larson concluded that the first postulate requires scalar motion, and so a third postulate stipulating it is not really necessary, as long as students of the system understand the underlying logic of it, it is clear that the second postulate is also not necessary, on the same conditions. However, in both cases, it requires some sophistication in mathematics and geometry to follow the logic, and avoid straying off the path.

In the case of the second postulate, the idea that the universe of motion "conforms" to "ordinary, commutative, mathematics" does not mean that non-commutative mathematics is not useful, or that because its magnitudes are absolute that relative magnitudes are not useful, or that because its geometry is Euclidean that non-Euclidean geometry is not useful.

On the contrary, these concepts have been shown to be very useful in legacy science. However, what the second postulate assumes is that the application of the first postulate produces a physical universe that can be described in terms of scalar units, the magnitudes of which are necessarily absolute (i.e. its algebra is ordered), commutative (i.e. its algebra is commutative), and Euclidean (i.e. its algebra is associative.)

In legacy science, this is not possible. Scientists in the legacy community must abandon ordered quantities (i.e use non-ordered, complex, numbers), forsake commutativity (i.e. employ non-commutative concepts of quantum spin), and replace Euclidean geometry (i.e. seek a counterpart to the observed geometry of the universe in a non-associative algebra that corresponds to hyperbolic geometry), in order to describe a universe consisting of elementary, interacting, particles.

In this way, the RST distinguishes itself dramatically from the LST. For this reason, I think it's a good idea to keep the second postulate and continue to make the implications of the postulated scalar motion explicit. In the next post, I will make some specific observations regarding the Euclidean geometry of the universe and how it relates to scalar mathematics.