The Larson Research Center

In discussing the concept of scalar mathematics, we’ve discussed its aspects of geometry and algebra, which is to say the relationship of numbers to geometric magnitudes in the tetraktys and the Clifford algebras, and how the scalar concepts of n-dimensional OI numbers relate to the vector concepts of complex and hypercomplex numbers (the complexes, the quaternions, and the octonions.)

In the previous post, I introduced the four numbers of the tetraktys, as four bases of motion, with four magnitudes of motion (I really messed it up too, sorry to say, but I think it makes sense now that I’ve corrected it). The motion chart, as I will call it, is extremely important in understanding the new mathematics of OI number, even though I didn’t explicitly use OI numbers to designate the bases of motion. If I had, the chart would have looked like this:

1. (1/1)⁰, (2/2)⁰, (3/3)⁰, (4/4)⁰
2. (1/1)¹, (2/2)¹, (3/3)¹, (4/4)¹
3. (1/1)², (2/2)², (3/3)², (4/4)²
4. (1/1)³, (2/2)³, (3/3)³, (4/4)³

But then the exponents of the bases aren’t OI numbers, so to be consistent, we would have to write these as OI numbers as well, like this:

1. (1/1)^1/1, (2/2)^1/1, (3/3)^1/1, (4/4)^1/1
2. (1/1)^2/1, (2/2)^2/1, (3/3)^2/1, (4/4)^2/1
3. (1/1)^3/1, (2/2)^3/1, (3/3)^3/1, (4/4)^3/1
4. (1/1)^4/1, (2/2)^4/1, (3/3)^4/1, (4/4)^4/1

Which means that we can then have another table of inverse bases with inverse dimensions, like this:

1. (1/1)^1/1, (2/2)^1/1, (3/3)^1/1, (4/4)^1/1
2. (1/1)^1/2, (2/2)^1/2, (3/3)^1/2, (4/4)^1/2
3. (1/1)^1/3, (2/2)^1/3, (3/3)^1/3, (4/4)^1/3
4. (1/1)^1/4, (2/2)^1/4, (3/3)^1/4, (4/4)^1/4

The space/time dimensions of the units of motion in the third table are t/s, not s/t, and the units are negative magnitudes, which will come in handy at some future point down the road, but would only serve to confuse things at this point, even though it shows the wonderful symmetry and completeness of the tetraktys. Nevertheless, it brings up another, very important, point that we have not directly addressed to this point: that bugaboo of ancient and modern mathematics and physics, the harmonizing of descrete versus continuous magnitudes.

In base 2 motion (vector motion), the change of position between two points means that ds/dt has a different meaning than it does in base 4 motion. This is because there is a continuum of values that ds and dt can take in vector motion, whereas in base 4 motion, there are only certain, fixed, values that ds and dt can take, because they are set by the clocks of the universe of motion, in which we assume that the magnitudes of the universe are absolute, not relative.

This seems to fly in the face of special relativity theory that maintains that time and space are not fixed, which experience has borne out, over a century of empirical tests. However, all these tests are conducted within the domain of base 2 motion, which is necessary, if we are to conduct experiments exploiting the laws of conservation and symmetry, since base 2 motion is the domain of observation of the motion of mass, or objects, where the conservation laws, based on mass, appear. Yet, clearly, the n-dimensional magnitudes of base 3 and base 4 motion are distinct in certain, important, respects, since changes in the position of massive objects are not relevant in these domains.

The ancient Greeks were never able to describe problems involving motion as speed, or velocity, because, while their mathematical language worked for problems in geometry and algebra, it was not adequate for problems of velocity. Again, the major difficulty is dealing with the notions of discrete and continuous. Base 2 speed, as a change of distance over time, depends on what interval of distance and time is referred to. If you ignore this, then Zeno’s paradox can confuse the definition of speed.

In the race of Achilles and the tortoise, if the tortoise has a head start, then Achilles can never catch it, because, in the time it takes him to close the distance, the corresponding time aspect of the tortoise’s motion is associated with a quantity of space that represents a new position, separated by a certain distance from its initial position, and, since it will always take some, finite, amount of time for Achilles to move the distance that separates them, no matter how small the distance, the tortise will always have that much time to move a little further away.

This is just another way of saying that space does not exist independently of its association with time in motion. We cannot measure space without time, or time without space; regardless of the magnitude of the ratio of space/time, it always has two, reciprocal, aspects. The only reason that Achilles, can in fact, catch up with the tortoise, and surpass it, is found in the speed of the tortoise, which in real life, is fixed. The speed of the tortoise in Zeno’s paradox is actually changing, because as the interval of time gets smaller and smaller, there’s always an interval of space associated with it by definiton, regardless of how small it is. Thus, the speed of the tortoise, in the paradox, reaches the speed of light, when the interval of time, and the interval of space reach the natural units of space and time.

At that point, neither Achilles nor the tortoise can go any faster, and the distance between their locations becomes absolute. So, we conclude that, while the time and distance separating two locations is infinitely divisable, the ratio of time and distance, as the reciprocal unit change of space per unit change of time, is limited to a natural unit value, one natural unit of space, per natural unit of time, the speed of light.

However, in LST physics, the fact that both distance and time, as quantities between two locations, are infinitely divisable, as Zeno’s paradox seems to show,  this idea is used to define velocity, not in the reciprocal limit of unity, but in the quantitative limit of zero; that is, velocity is defined “in the limit” of delta s over delta t, when delta t gets “vanishingly small.”  It is written as

v = lim (Δt -> 0) Δs/Δt,

where the symbols in the parentheses are usually written under the letters “lim,” which I can’t do in html.  This means that space, for purposes of base 2 motion calculations, is defined as

Δs = v Δt,

provided that the velocity does not change during the quantity Δt, which means Δt has to be very close to zero, or “vanishingly small.”  This way of thinking, derives from the fact that base 2 motion can be any value at any time, so, in order to define it precisely enough, it is necessary to “pin it down” like this, to the smallest possible interval of time.  Thus, the mathematics of motion, must become the mathematics of change first and foremost, and the ancient Greeks just didn’t have the required mathematical concepts. 

The mathematics of change, called the differential calculus, came from Newton and Leibniz, and forms the basis of LST physics. The concept of quantity, in the definition of motion, Δs/Δt, where Δt is zero, but not really zero, that is, it is infintesimally small, was the beginning of the use of a principle in LST physics called the “to a good approximation” principle.  It is something that the ancient Greeks would have found abhorent, and it has come back to haunt modern physics too, in the end.

However, it works, and to the degree that it works, it can support technological applications that change society radically. So, even though the philosophers object, the practicioners are delighted.  Yet, when the practicioners want more from our understanding of nature, the practicality of “to a good approximation,” becomes a barrier to further progress, and it’s at that point that we must turn to the philosophers for help.  “What’s wrong with our understanding?” we want to know, “Why can’t we understand what mass is and where it comes from?  Why don’t we have a better theory of charge? Why can’t we understand the common origins of electrical, magnetic, and gravitational forces?”

Well, at the LRC, we believe that we can answer these kinds of questions, if we redefine motion, which means we need a better understanding of the nature of space and time. To define space in terms of the product of velocity and an infinitesimal interval of time is not good enough, even though it works “to a good approximation.”   Now, with the knowledge of base 3 and base 4 motion of the tetraktys, as seen in the motion chart, we can begin to see that we can’t use the concepts of differential calculus, as it was developed for applications of base 2 motion, which depends on an object’s change of position.  We need a different calculus in dealing with base 3 and base 4 motion, because these motions are not based on arbitrary changes of position over time, but on fixed changes of interval and size over time.