The Larson Research Center

As mentioned before, one the most surprising (and gratifying!) discoveries made in connection with the new system of mathematics (the RSM), is the work of Sir William Rowan Hamilton, in regards to his early efforts to develop a science of progression, which he called a science of pure time. It turns out that, while this was very significant in the history of mathematics, because it lead to a new understanding of complex numbers, the concept of a vector (even the word), and the discovery of non-commutative quaternions, the novel ideas of Hamilton, regarding the advantage of the concept of order in temporal progression, rather than the concept of increasing and diminishing magnitude, as a basis for natural numbers, it was never accepted by the mathematical world, and remains an obscure footnote of history to this day.

In fact, Hamilton himself abandoned the conviction that this “strange theory” was a promising approach to an a priori intuition, revealing the natural fount of mathematical truth. In the end, he gave up the quest for an intuitive science of pure time, and turned to the more popular mathematical formalism, which found its ultimate expression in the work of the younger Dedekind and Cantor, who established the basis of modern set theory and set the course for the runaway formalism that is the bane of modern theoretical mathematics and physics, in my opinion.

Nevertheless, Hamilton’s use of “couples,” based on what he called “steps” in the order of progression, enabled him to validate the concept of negative numbers, algebraically, without recourse to geometry (which he regarded as the science of space) and the idea of rotation. In his words, his algebra of pure time “[removed] the difficulties of the usual theory of negative and imaginary quantities…”:

As already pointed out, though, Hamilton gave up the promise of this early intuition-based approach, and joined the ranks of formalists, like Peacock, after his discovery of quaternions. However, had he had the benefit of Larson’s insight, that his preferred basis of numbers, as intuitive-based values, emerging from the order in the temporal progression of nature, belongs not only to pure time, but to pure space as well, in a reciprocal relation, as pure motion, the subsequent history of mathematics and science would most assuredly be radically different.

We have seen that the simple operational interpretation (OI) of rational numbers, when combined with the idea of two, reciprocal, progressions, does indeed lead to an intuitive concept of positive and contrapositive numbers that are “free from those difficulties…derived from the bounded notion of magnitude.” We have discussed the two groups, one under addition, and one under multiplication, which these positives and contrapositives form, corresponding to magnitudes, which at once have both discrete and continuous properties.

We have identified the algebraic properties of these numbers and identified the correspondence with physical magnitudes of geometry, in the tetraktys, wherein the geometric points, lines, areas, and volumes correspond to numerical values, at least in terms of the scalar spaces that can be mapped to vector spaces.

We have also discussed Hestenes’ views regarding the Clifford algebras of these Grassmann spaces, especially with respect to his Geometric Algebra (GA). Through all this, it seemed clear that we were on the verge of uncovering a new, scalar, algebra that would be a complement to the vector algebra of GA, an exciting prospect indeed. However, trying to fit the three dimensions of OI numbers, into the tetraktys, while numerically revealing in many ways (see Chart of Motion), has proven difficult from a geometric standpoint, because while the red, green,and blue colors that we employed for this purpose, are good scalar analogs of discrete scalar motion magnitudes, they are not good vector analogs of points lines, areas, and volumes.

Since there was no immediate need to bridge this gap, it was more or less pushed into the background, but it comes to the foreground in trying to work out the mathematics of the standard model preons.  The question is, how do we map the three scalar magnitudes, of the scalar tetraktys, represented as red, blue and green “directions” of the scalar reference system, to the three vector magnitudes, represented as the x, y, and z directions of the spatial reference system?

It seems, intuitively, that this is a necessary task, but the advantages, algebraically, are marvelous, since such a correspondence would give us a map to a space|time algebra, based on known principles of vector algebra, but free from the pathologies in those algebras that stem from the loss of algebraic properties such as order, commutativity, and associativity, which make the complex numbers philosophically difficult, the quaternions neurotic, and the octonions, the “crazy uncles” of the mathematical family, so unfit for polite company that the mathematicians keep them locked up in the attic, as described in John Baez’s account of the four normed division algebras:

The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still respectable younger brother: not ordered, but algebraically complete. The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative.

The answer, seems to be in understanding the definition of independence, or orthogonality, both in terms of numbers and in terms of physical magnitudes. This is the meaning of dimension. Each independent dimension has two directions in the vector space, while each independent dimension has two “directions” in the scalar space, where the difference between direction and “direction” is the difference between direction of distance, ds/dt * t, and “direction” of polarity, ds/dt:dt/ds. However, direction has no meaning in the scalar space of positive and contra-positive magnitudes, and “direction” has no meaning in the vector space of distance between two points. Thus, directional orthogonality is not the same as “directional” orthogonality.

However, in vector space, there are three, and only three, orthogonal dimensions, each with its two directions, which can be interpreted in polar terms, but, in scalar space, there is one, and only one, dimension, with two polar “directions.” Nevertheless, there is an additional, orthogonal, dimension of sorts in scalar magnitudes, with one “direction.” It is the total number of positives and negatives. We have been considering these three scalar “directions,” as three scalar dimensions, and mapping them to the tetraktys, but this proves to be problematic, ultimately, when we consider the tetraktys in terms of the properties of physical magnitudes; that is, the binomial expansion of the tetraktys reflects the directional duality of the three dimensions of physical magnitudes, not the “directional” triality of two scalar dimensions! 

It’s true that we can assign three primary colors to each of these scalar “directions,” as if they were independent dimensions, and when we do the colors combine in an impressive way, as we’ve seen, but in the end, we sense that there is something wrong with identifying the third “dimension” with the total units of the scalar entity, even though it seemed the only reasonable alternative, until recently.

However, with the emergence of the S|T triplets, and the ideas of meshed gear ratios, which have become the center of so much attention here at the LRC, during the Summer and Fall months, it has become obvious that another identification of three scalar dimensions is possible, which has none of the former difficulties. These are the three nodes, the three “gear ratios,” of the S|T triplets, designated A, B, C (and D, the internal mode, which we haven’t discussed much yet). 

It’s somewhat ironic, I think, that the best way to understand the mathematical content of all this might be to consider it from what might be loosely called a set theoretical point of view; that is, if we start at the beginning, with the initial set of numbers, and work our way forward, to develop the idea of number, as a set, the power of the logic will be clearly evident. This is the way it has historically been approached, but from the standpoint of quantity and increasing/diminishing magnitudes, which Hamilton found so wanting.

For instance, in his book Road to Reality, Roger Penrose, explaining what we mean by natural numbers, integers, rationals, etc, asks the question, “Do natural numbers need the physical world?” His answer begins by noting that zero is included among the natural numbers today, but was not included by the ancient Greeks. He writes:

The natural numbers are the quantities that we now denote by 0, 1, 2, 3, 4, etc., i.e. they are the non-negative whole numbers….the role of the natural numbers is clear and unambiguous. They are indeed the most elementary ‘counting numbers,” which have a basic role whatever the laws of geometry or physics might be. Natural numbers are subject to certain familiar operations, most particularly the operation of addition and multiplication, which enable pairs of natural numbers to be combined together to produce new natural numbers. These operations are independent of the nature of the geometry of the world. 

But this definition would have irked Hamilton to no end. It was his conviction that, while the science of numbers is “independent of the nature of the geometry of the world,” all right, it still has to have a connection with the properties of the physical universe, which is just as essential to the science of numbers, as is the connection of physical space to the world, in the science of geometry. If numbers have a “basic role in whatever the laws of geometry or physics might be,” that role had to somehow be conceived, as rational and intuitively derived, starting from observation of the structure of the physical universe, as it is in geometry.  

But Penrose doesn’t think so: “There are various ways in which natural numbers can be introduced in pure mathematics and these do not seem to depend upon the actual nature of the physical universe at all,” he asserts. In other words, the formalist worldview, omnipresent in the minds of mathematicians today, doesn’t demand the intuitive basis that Hamilton sought. The only requirement is to think in terms of an abstract notion, Penrose explains. Although fraught with philosophical difficulties, “natural numbers can be introduced, merely using the abstract notion of set,” he writes.

Penrose goes on to show how Cantor, the father of modern set theory, developed the concept of natural numbers from the notion of set, which leads to definitions of set properties such as cardinality and ordinality. The simplest set of all, explains Penrose, is the empty set:

It is characterized by the fact that it contains no members whatever! The empty set is usually denoted by the symbol Φ, and we can write this definition

Φ = { }

where the curly brackets delineate a set, the specific set under consideration having, as its members, the quantities indicated within the brackets, so the set being described is indeed an empty set.

It’s interesting to note that, while physicists used to employ the notion of an empty vacuum, they now need to define a vacuum as teaming with virtual particles, but mathematicians, who used to begin with something (the Father of all was the number 1 for the ancient Greeks), they now need to define something as nothing. We can easily imagine the extent of consternation Hamilton would have no doubt felt over this modern perplexity of fundamental notions in physics and mathematics.

Penrose goes on to define natural numbers, beginning with 0, using the notion of set:

Let us associate Φ with the natural number 0. We can now proceed further and define the set whose only member is Φ; i.e. the set {Φ}. It is important to realize that {Φ} is not the same set as the empty set Φ. The set {Φ} has one member (namely Φ), whereas Φ itself has none at all.

Then, by combining these initial two sets of Φ and {Φ}, equivalent to 0 and 1, into a third set, the number 2 is defined as a set with two members. In the same way, by combining these three sets into a new set, a set of three is defined, and so on, ad infinitum. Penrose then observes:

This may not be how we usually think of natural numbers, as a matter of definition, but it is one of the ways that mathematicians can come to the concept….Moreover, it shows us at least, that things like the natural numbers can be conjured literally out of nothing, merely by employing the abstract notion of ‘set.’ We get an infinite sequence of abstract (Platonic) mathematical entities - sets containing, respectively, zero, one, two, three, etc., elements, one set for each of the natural numbers, quite independently of the actual physical nature of the universe.

To Penrose, the idea that an infinite sequence of numbers can be “conjured” up out of nothing like this, purely by means of the human mind, without recourse to the structure of the physical universe, is quite mysterious, because it is obvious that “‘real numbers’ indeed seem to have a direct revelance to the real structure of the world.” He notes in this context that in his preface to the book he discusses the subtle impact the mysterious nature of numbers may have on those who are convinced that they have no ability to comprehend math. That many times they believe that they cannot even fathom fractions, “with all that cancelling,” gives Penrose pause. He points out that there may be more to this difficulty than meets the eye, because, after all, there is something very mysterious about the fact that n/n = 1, although he is not able to go beyond merely noting it.

This actually paints a most dramatic scene before us, given our knowledge of the importance of n/n = 1 and how easily the number 1 can be developed into a complete system of reciprocal numbers, with both discrete and continuous properties, and given Hamilton’s work, where a world-class genius acknowledges the power of its intuitive connection to nature, through its central concept of order in temporal progression.

So with this scene as our prologue, we will proceed next to use this idea of sets in a new way, as directly connected to the structure of the physical world, instead of conjuring it up out of the imaginations of our minds, turning the science of mathematics from the inventive science of the Formalists, or, as Hamilton would have called them, the Philologists, to the inductive science of the Intuitionists, or, as Hamilton would have called them, the theoriticians.