The Larson Research Center

In the previous post below, we discussed the accepted concept of the natural numbers (which now days includes zero), as arising from Cantor’s abstraction of sets. This entails conceiving of something (the symbol Φ), which represents nothing, and counting it as something.

However, there is no need to resort to these shananigins, as can be seen from Hamilton’s early essay, known as the Algebra of Pure Time. Instead of considering numbers stemming from countable objects, Hamilton begins with order in progression, where a number is represented by moments in time, which can be either coincident, earlier, or later, when one is compared to the other. As we’ve noted before, this is the very ancient idea of fundamental order: Given that two numbers (or magnitudes) exist, one greater than the other, there will always be a third number (or magnitude) greater than them both.

Unfortunately, Hamilton got sidetracked by his pursuit of algebraic geometry, but a century later, Larson discovered the efficacy of thinking in terms of nothing but motion, when motion is defined as two, reciprocal, progressions, one a progression of time, and the other a progression of space. Order in progression then becomes the basis for all physical magnitudes, in terms of space and time speed-displacements, and, coincidently, all numerical values, in terms of operationally interpreted, reciprocal numbers.

In the case of numbers, we begin by noting that n/n = 1/1 = 1, and n|n = 0. When we consider that the eternal progression of the reciprocal quantities has no beginning and no end, then we can pick a starting point for a reference in which all past progression is not counted, and all future progression has not yet begun; that is, we abstract out a starting point in the progression, which we can designate P₀, from which point, everything is past, or future, in terms of both space and time progression.

However, in terms of a 3D space progression, “past” necessarily means smaller (less space) than P₀, while “future” necessarily means larger (more space) than P₀, but this requires that P₀ be picked out of an infinite set of possibilities, relative to the previous point, P_-1, since any point on the expanded sphere, relative to the origin, at P_-1, may be selected for P₀ (which is the foundation for the weirdness in quantum mechanics). Nevertheless, when considering numbers only, this is the difference between the scalar and the pseudoscalar, in the tetraktys, which difference is not apparent at the beginning of the tetraktys, where the dimension of the number is zero, because any dimensionless number (a number raised to the power of 0) is equal to 1.

Thus, (n/n)⁰ = 1⁰/1⁰, and, in a progression ratio, this is how we represent the unit ratio, a ratio of unit scalars. However, when we choose a reference point in the progression, P₀, then, at this point, before the next step in the progression occurs, the progression count is zero. After one unit of progression occurs, the progression count is one, and so on giving us a series, P₀, P₁, P₂, …P_n, where P₀ = (0/0)⁰, P₁ = (1/1)⁰, P₂ = (2/2)⁰, …P_n = (n/n)⁰.

If this seems a little like spinning wheels, it is. There is no numerical difference between these steps, since any number, raised to the power of zero, has the same value as any other number raised to the power of zero. Yet, obviously, as the progression continues, the point, P_n, is much greater than the point, P₀, in terms of the number of steps of progression. 

The minimum number of steps of progression that we can perceive, from P₀, is one, but there is no maximum, because, in an eternal progression, there is always a next step, ad infinitum. But just as Hamilton based his numbers on “steps,” in the relative values of temporal progression, we can comprehend steps of reciprocal progression in terms of relative values. In one step, 1/1, there is no variability, but in two, or more, steps, there certainly is. That is, in one step of the progression, (1/1)⁰, there is no freedom, but in two steps, (2/2)⁰, there is an increased degree of freedom.  The question is, therefore, what are the consequences of this increase in freedom?

The answer is obvious, but profound, as Hamilton discovered, when he developed his time steps into numeric couples. Interestingly enough, though, as Hamilton developed the ideas of steps and couples, in the order of temporal progression, with its concept of change, as a central notion, Grassmann was discovering the meaning of dimension in numbers, which corresponds to dimension in space, with no concept of change involved. Neither one of these two geniuses understood their work in terms of the tetraktys, as we now do, but Hamilton loved the mystery of the tetraktys, even writing a poem to it, alluding to his quaternions.

THE TETRACTYS

Or high Mathesis, with her charm severe,
   Of line and number, was our theme; and we
   Sought to behold her unborn progeny,
And thrones reserved in Truth’s celestial sphere:
While views, before attained, became more clear;
   And how the One of Time, of Space the Three,
   Might, in the Chain of Symbol, girdled be:
And when my eager and reverted ear
Caught some faint echoes of an ancient strain,
   Some shadowy outlines of old thoughts sublime,
Gently he smiled to see, revived again,
   In later age, and occidental clime,
     A dimly traced Pythagorean lore,
     A westward floating, mystic dream of FOUR.

However, in the work of Clifford, the power of the tetraktys would begin to emerge, as he brought the work of Hamilton and Grassmann together, combining the two in a fashion that would eventually lead to a new way of viewing numbers, the power of which was not fully appreciated, until Hestenes empahsized it.

Yet all this seemed so irrelevant, as Dedekind looked to understand real numbers, and he and Cantor eventually took the path of formulating axioms, leading to set theory and modern mathematics. Even today, though geometric algebra (GA) seems to live in the fullness of the tetraktys, thereby subsuming the algebras of the reals, the complexes, and Hamilton’s quaternions, the preoccupation of mathematicians, with set theory and group theory and the associated notions of topology and category theory, appears to have blinded their minds, in part at least, to the power of the tetraktys, in spite of the tremendous clues, given by GA and Raul Bott.

To appreciate this power takes a willingness to understand the advantages of the intuitive notion of order in progression, as Hamilton (encouraged by the writings of Kant) first dared to entertain them.  Later, Larson (encouraged by the writings of Samuel Alexander), enlarged upon the ideas of Hamilton (though there’s no indication that he knew of them), by assuming that space, like time, is a natural progression, identical in every respect to the time progression, but the reciprocal of it.

When we take these ideas and apply them to the tetraktys, we can develop a set of natural numbers, where the first set is

{P₁ = (1/1)⁰}

and the next set is

{P₂ = (2/2)⁰}

which consists of two steps, identical to P₁. Hence,

P₂ = P₁+ P₁

P₁ being the progression step from P₀ to P₁ and P₂ being the progression step from P₁ to P₂.  

Applying this to the tetraktys, we get

{P₁}
{P₂ P₂}

or,

{1/1}
{{2/2} {2/2}}

the geometric interpretation being a point in the first set and two double points in the second set, or

{(P₁)⁰}
{(P₂)⁰ {(P₂)¹}}

since two, zero-dimensional, points are free to form a one-dimensional line.

However, here we are deviating from the usual interpretation, with regards to the scalar (on the left) and the pseudoscalar (on the right). In this interpretation, the difference in their dimensions indicates that one is the inverse of the other; that is, in the first line of the tetraktys the element of the set is its own inverse (1/1 = 1/1), but in the second line of the tetraktys, the inverse is one dimensional (2/2 = (1+1)/(1+1), as in reflection symmetry and the commutative law.

On the same basis, the third set is then

{P₃ = (3/3)⁰}

where

P₃ = P₂ + P₁

which, in one-dimension, is a combination of a point on a line, recognizing that two, independent, lines are required for two dimensions, which independence we don’t attain until P₄, forming the fourth set: 

P₄ = P₂ + P₂

or

{(P₁)⁰}
{(P₂)⁰} {(P₂)¹}
{(P₄)⁰ {(P₄)¹} (P₄)²}

Of course, all P_n, greater than P₁, are combinations of P₁, or multiples of P₁, but again, the independence, this time of both points and lines, enables the formulation of higher dimensions, in a progression of degrees of freedom, characteristic of the tetraktys. The freedom is the freedom in the order of progression, where one step is either coincident, earlier, or later than another. In other words, it increases as the power of 2, or 2ⁿ, beginning with 0. That this is nothing more than the inherent property of direction is easy enough to recognize, but what must be recognized now is that it is also the inherent property of “direction” (see previous post on this).

With this much understood, we can substitute the idea of “direction” for the idea of direction in the tetraktys. On this basis, P₂ represents two “directions” along a line, rather than two points, forming a line; that is, with two units of progression, two polarities of a single line can be represented, or the “distance” between two polarities is analogous of the distance between two points. Hence, 1/2, or 2/1, are two values, or “directions” that are inherent in P₂ = (2/2)¹, just as two directions (left or right, up or down, or back and forth) are inherent in a line between two points.

At the same time, these two, inherent, “directions” are not realized in P₂ = (2/2)⁰, and not realizable in P₁ = (1/1)⁰. Therefore, in P₃, the only “directional” possibility is 1/2 and 1/1, or 2/1 and 1/1, analogous to the “distance” from a point to the 1/2 value, or from a point to the 2/1 value, in the other “direction,” with the point located between the two values. This is reminiscent of the ancient Greek idea that three was the first number, it being the perfect number, representing the reunion, or perfection, of duality, which must split unity.

However, as Hamilton discovered, three does not have sufficient power to reach the next higher dimension.  Only at P₄, is there sufficient freedom to reach the next higher level, the third dimension, of the tetraktys (counting 0). We can see this as extending the freedom of P₃, where the “direction” from point to 1/2, or 2/1, goes from a point to 1/2 AND 2/1, because the total number of units now makes this possible, whereas before it didn’t:

P₄ : 4/4 - 1/2 = 3/2, and 3/2 - 2/1 = 1/1

Now, there are sufficient units to combine both the two separate “directions” and the point into one dual “direction,” relative to the point, in terms of reciprocal numbers, but at P₃ there is insufficient magnitude to join them together through a point in the middle, so-to-speak:

P₃ : 3/3 - 1/2 = 2/1 

Likewise, at P₅, P₆, and P₇, there are an insufficient number of P₁s to form the next level of the tetraktys, which requires two P₄s, which is predictable enough, since the second dimension requires two P₁s, and the third requires two P₂s. This is easier to track if we use actual numbers.  The full tetraktys looks like this:

(1/1)⁰

(2/2)⁰ (2/2)¹  

(4/4)⁰ {(2/2)¹ (2/2)¹} (4/4)²

(8/8)⁰ {(2/2)¹ (2/2)¹ (2/2)¹} {(4/4)² (4/4)² (4/4)²} (8/8)³

There is much more to be explained, but this will do for now.