The Larson Research Center

It’s been a long time since my last entry on this blog. Mostly that’s due to time constraints, but also because I’ve written about things on the new physics blog that probably should have gone here. Sometimes, though, it’s hard to separate the math from the physics topics.

However, there’s no doubt where this topic goes. I want to take the new math from the top, and lay out the new concepts from the beginning. I will be referring to them as I develop the physics theory on the other blog.

The first concept that must be clearly understood from the start is that the reason for calling it the new math is that there are two interpretations of number. the first interpretation of number is the usual quantitative one that is a measure of how much or how many of something there is. In the second, the operational number represents a relation between two quantities. 

We begin by viewing the familiar quantitative number line below in light of these two interpretations of number.

Figure 1. The Quantitative Interpretation of Number Line

In the quantitative interpretation of number, the whole numbers and proper fraction, rational, numbers lie to the right of 0 on the number line, in all cases. For instance, the number 1 occupies the first place to the right from 0, and 1/2 lies half way between 0 and 1 on the quantitative number line. The negative numbers and negative proper fractions to the left of 0 are somewhat problematic and were only accepted by mathematicians gradually and grudgingly. Wikipedia defines them as follows:

Negative integers can be regarded as an extension of the natural numbers, such that the expression x – y has a well-defined value for all values of x and y. Other number systems, such as the rational numbers, are then derived as progressively more elaborate extensions and generalizations from the integers.

On the other hand, in an operational interpretation of a rational number, we can take the relation of the numerator and denominator, say the difference between them, instead of the quotient, and it permits us to replace all the positive numbers on the number line with the reciprocal of proper fractions that replace all the negative numbers on the line, none of which are less than 1.

This way, we get a new number line,

1/n, …1/3, 1/2, 1/1, 2/1, 3/1, …n/1,

which is an operational equivalent of the quantitative number line in figure 1, above, but which is not based on integers, but constitutes a new generalization from which integers themselves are derived. In this case, however, instead of positive and negative numbers, we have a rational number and its inverse. To be sure, while the rational numbers are not the same as the quantitative numbers on the quantitative number line, their operational interpretation is; That is, 

1/n = 1-n; …1/3 = -2; 1/2 = -1; 1/1 = 0; 2/1 = 1, 3/1 = 2, …n/1 = n-1;

In Larson’s new system of physical theory (RST), as opposed to the legacy system of physical theory (LST), there are two, reciprocal, sectors of the physical universe, the sector where motion is above unity (the cosmic sector), and the sector where motion is below unity (the material sector.) Within each of these two sectors, there is an important sub-sector, the interior of unit distance, which Larson refers to as the time region (inside unit space) and the space region (inside unit time).

A complete mathematical analogy of this space-time structure can be reproduced by considering the quantitative and operational interpretations of number together. The operational interpretation extends outward from 0 (1/1) to infinity, in both “directions,” while the quantitative interpretation extends inward from 1 and -1 (i.e. 2/1 and 1/2 respectively) to 0 (i.e. 1/1), in both “directions.”

However, there is another important distinction between these two interpretations of number, besides their respective data of 0 and 1, and it must be understood as well. In the operational interpretation, we must pick a perspective; that is, we must view the reciprocal side of the datum from its inverse perspective, just as we must view a see-saw from one side or the other. We cannot view the operational interpretated number line from both sides at the same time, any more than we can view the see-saw profile from both sides of the fulcrum at the same time. In terms of motion, this means we must choose to interpret both views as above and below unit motion, or above and below unit inverse-motion (s/t or t/s, but not both together.)

On the other hand, in the quantitative interpretation of number, we must view the reciprocal side of the datum from its own perspective, where one side is motion, while the other side is inverse-motion (s/t and t/s, at the same time.) This difference is illustrated in figure 2 below.

Figure 2. Operationally and Quantitatively Interpreted Number Lines

The division operation of the quantitative (how much or how many) interpretation of number requires us to differentiate the positive and negative quantities, as if they were real, even though there is no such thing as a negative quantity. As Sir Rowland Hamilton observed:

it requires no peculiar scepticism to doubt, or even to disbelieve, the doctrine of Negatives
and Imaginaries, when set forth (as it has commonly been) with principles like these: that a
greater magnitude may be subtracted from a less, and that the remainder is less than nothing; that two negative numbers, or numbers denoting magnitudes each less than nothing, may be multiplied the one by the other, and that the product will be a positive number, or a number denoting a magnitude greater than nothing; and that although the square of a number, or the product obtained by multiplying that number by itself, is therefore always positive, whether the number be positive or negative, yet that numbers, called imaginary, can be found or conceived or determined, and operated on by all the rules of positive and negative numbers, as if they were subject to those rules, although they have negative squares, and must therefore be supposed to be themselves neither positive nor negative, nor yet null numbers, so that the magnitudes which they are supposed to denote can neither be greater than nothing, nor less than nothing, nor even equal to nothing.

Contemplating the arbitrary nature of the quantitative number line, Hamilton sought a better approach using the dynamic concept of order in progression, rather than the static concept of bounded magnitude. This was a good idea, as far as it went, but it requires two orders of progression to make it work, not just one. Hamilton’s idea was to use the flow of time to give algebra an intuitional foundation, but Larson’s idea was to use the flow of time, together with the flow of space, to put physics on an intuitional foundation.

At first Larson’s idea seems absurd, and it would never had ocurred to Hamilton, but today the flow of space has actually been observed. The logical conclusion is that the two should be considered together. The difficulty is recognizing that they are not separate quantities, but actually two aspects of the same quantity, motion. We start with unit motion and go in both “directions,” toward greater or less than unit motion, when the flow of one aspect is less than the flow of the other.

Larson’s conclusion was that the only possiblility of introducing a difference between the two flows, is to assume that one or the other of them periodically reverses its “direction.” He called this simple harmonic motion, and he pointed out that it was just as reasonable to believe that the flow of space, or of time, could oscillate as not, and that this is the basis of all physics.

Were this the summum and bonum of the subject, we would be home free, but it is not complete at this point, because the two inverse aspects of the universal motion, the two flowing quantities, if you will, do not have the same dimensions. The flow of space exists in three dimensions, while the flow of time has no dimensions. Mathematically, then, the natural progression is not linear. If:

s/t = 23/20

then it does not give us a natural progression of 0, 1, 2, 3, … but rather it gives us a progresssion of 0, 8, 216, 512, …, and, at first glance, it’s totally impractical to construct a number line from such a non-linear progression.

However, it turns out that, within this 3D progression, there is an associated 0D, 1D and 2D progression as well, and by recognizing that the natural progression contains all four numerical progressions, we can construct a new, composite, number line and with it a new number system to use in our investigations of the RST. We will take a look at it next time.