The Larson Research Center

In the previous post, I wrote about Sir William Rowan Hamilton’s lamentation of the fact that algebra lacks an inductive basis.  There can be no satisfactory science of algebra, in his view, if it is to be based on the magnitudes of substance.  We see how he felt that while geometry is an inductive science of space, based on Euclidean geometry and the postulate of parallel lines, algebra is not, because negative and imaginary quantities, so essential to algebra, simply make no sense. He wrote:

…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.
 

Today, hardly anyone shares these misgivings publicly. We teach negative numbers in grade school (about the fifth grade these days) without fussing over the philosophical torture to young minds that we are inflicting, but still, no doubt, causing a few “crying jags” along the way.  Then we go on to teach kids imaginary numbers in middle school, blissfully unaware that we are teaching them a mathematical conundrum, where numbers denote magnitudes that “can neither be greater than nothing, nor less than nothing, nor even equal to nothing.”

Yet, we never teach why negative magnitudes are useful, even though they don’t make sense intuitively, nor why negative squares are useful, even though there is no escape from the conundrum they represent. Students are simply expected to learn the rules that are known to work, in spite of these philosophical contradictions, and they do learn them, for the most part, but at what cost?

Actually, even if we could measure and appreciate the cost, there’s not much that can be done about it, because no one really knows why these concepts of numbers are so useful as an art, and so consistent as a language, in spite of the complete lack of intuition concerning them.  Clearly, the only thing we could do is admit this failure to the rising generations,  but instead of doing that, we just don’t think about it any more, letting it sink deeper and deeper into obscurity. 

Well, at least we don’t think about it as Hamilton thought about it. He thought about it in terms of advancing science.  He wrote:

The thing aimed at, is to improve the Science, not the Art nor the Language of Algebra. The imperfections sought to be removed, are confusions of thought, and obscurities or errors of reasoning; not difficulties of application of an instrument, nor failures of symmetry in expression. And that confusions of thought, and errors of reasoning, still darken the beginnings of Algebra, is the earnest and just complaint of sober and thoughtful men, who in a spirit of love and honour have studied Algebraic Science, admiring, extending, and applying what has been already brought to light, and feeling all the beauty and consistence of many a remote deduction, from principles which yet remain obscure, and doubtful.

Of course, he articulated these thoughts, because he believed that there was a glimmer of hope that an inductive science of numbers could be developed from intuition, even though he was somewhat apologetic about it.  He refers to this as a “rudiment” of hope and asked indulgence from his contemporaries to explore it:

Indulgence, therefore, may be hoped for, by any one who would inquire, whether existing Algebra, in the state to which it has been already unfolded by the masters of its rules and of its language, offers indeed no rudiment which may encourage a hope of developing a Science of Algebra: a Science properly so called: strict, pure and independent; deduced by valid reasonings from its own intuitive principles; and thus not less an object of priori contemplation than Geometry, nor less distinct, in its own essence, from the Rules which it may teach or use, and from the Signs by which it may express its meaning.

He called this “rudiment” of hope that he believed in the “Intuition of Time.”  He explains:

This belief involves the three following as components: First, that the notion of Time is connected with existing Algebra; Second, that this notion or intuition of Time may be unfolded into an independent Pure Science; and Third, that the Science of Pure Time, thus unfolded, is co-extensive and identical with Algebra, so far as Algebra itself is a Science. The first component judgment is the result of an induction; the second of a deduction; the third is the joint result of the deductive and inductive processes.

Really? The notion of time is connected with existing algebra? Just try to find that conclusion in a modern textbook of algebra!  Yet, Hamilton understood that this is the essential difference between algebra and geometry:

The argument for the conclusion that the notion of Time is connected with existing Algebra, is an induction of the following kind. The History of Algebraic Science shows that the most remarkable discoveries in it have been made, either expressly through the medium of that notion of Time, or through the closely connected (and in some sort coincident) notion of Continuous Progression. It is the genius of Algebra to consider what it reasons on as flowing, as it was the genius of Geometry to consider what it reasoned on as fixed.
…Lagrange considered Algebra to be the Science of Functions,* and…it is not easy to conceive a clearer or juster idea of a Function in this Science, than by regarding its essence as consisting in a Law connecting Change with Change. But where Change and Progression are, there is Time. The notion of Time is, therefore, inductively found to be connected with existing Algebra.

Thus, he begins to reason on the concept of a continuous progression in earnest, with the aim of developing an inductive science of algebra. He perceives that the continuous march of time, as counted by successive thoughts, if nothing else, means that there is past, present, and future: It means that counting moments of time implies that one moment is later, or earlier, than another, and that, taken together, this relationship is mathematical, because it leads to the following equations:

1) A = B,
2) B > A,
3) A < B,

where 1) can be compared to a point, and 2) and 3) to a line, or an interval, with “direction,” which he calls a step, in the order of a numeric progression.  The interval is B - A, but the “direction” is increasing from A to B. That this concept of number is distinct from the concept of fixed magnitudes of objects, where the number of fixed quantities in one collection may be equal to, greater than, or less than, another collection of objects, is of great significance to Hamilton.  He writes:

That a moment of time respecting which we inquire, as compared with a moment which we know, must either coincide with or precede or follow it, is an intuitive truth, as certain, as clear, and as unempirical as this, that no two straight lines can comprehend an area. The notion or intuition of Order in Time is not less but more deep-seated in the human mind, than the notion of intuition of Order in Space; and a mathematical Science may be founded on the former, as pure and as demonstrative as the science founded on the latter. There is something mysterious and transcendent involved in the idea of Time; but there is also something definite and clear: and while Metaphysicians meditate on the one, Mathematicians may reason from the other.

Consequently, he proceeds to reason in this essay, which he calls a “Preliminary and Elementary Essay on Algebra as the Science of Pure Time.” In it, he develops the mathematical entities of a group representation under addition, though, of course, he doesn’t have any idea of that concept as it later develops. Yet, he introduces a new symbol to denote the inverse elements of the group, and one for the reciprocal operation (an upside down R, for Reciprocatio, which defines the operation for transforming an element of the group into its inverse.)  He also identifies the identity element of the group and the property of closure and associativity.

Moreover, he goes on to develop ratios of these numbers, as multiples and submultiples, defining the operations in a way very similar to the group under multiplication, which we have developed, although at this point I can only follow his development enough to see the general outlines of this.  Significantly, however, he refers to these numbers as “reciprocal numbers,” something which really blew me away, as anyone who has been following this blog would readily understand.

To say that the discovery of Hamilton’s “essay” on the mathematics of progression, as opposed to the mathematics of magnitude, is an important milestone in the work of the LRC, is an understatement of gigantic proportions. It is so startling that I’ve had to shift my priorities in order to devote time to studying it in detail, which is why the blogging has slowed down so much recently.

However, I promise to share more as soon as I can manage it.

One of the most interesting of all biographies in math and science is that of Sir William Rowan Hamilton. David Hestenes relates how his life’s work in the 18th Century is misunderstood, because it is often painted in terms of his fascination with his quaternions, which were historically eclipsed by Heaviside’s and Gibb’s vector analysis. Indeed, until recently, quaternions had sunk into obscurity. Hestenes writes:

…because for the last twenty years of his life, Hamilton concentrated all his enormous mathematical powers on the study of quaternions in, as [E .T.] Bell would have it, the quixotic belief that quaternions would play a central role in the mathematics of the future. Hamilton’s Judgement was based on a new and profound insight into the relation between algebra and geometry. Bell’s evaluation [see his sketch entitled “An Irish Tragedy”] was made by surveying the mathematical literature nearly a century later. But union with Grassman’s algebra puts quaternions in a different perspective. It may yet prove true that Hamilton looking ahead saw further than Bell looking back.¹

However, the applications of quaternions, in 3D computer graphics and robotics, as an alternate, more efficient, method of image rotation and manipulation in the 21st Century, does not fulfill that destiny. The new perspective Hestenes has is that Hamilton found something fundamentally important, the “profound insight into the relation between algebra and geometry.” This new insight is that Hamilton found more than a use for four dimensional numbers, “he found a system of numbers to represent rotations in three dimensions. He was looking for a way to describe geometry by algebra, so he found a geometric algebra.”

Combining Hamilton’s operational interpretation of number, with Grassmann’s quantitative interpretation of number, led Clifford to the powerful geometric product, though his premature death almost took the secret with him. Hestenes observes:

Quaternions today [late 20th Century] reside in a kind of mathematical limbo, because their place in a more general geometric algebra is not recognized, …[but] Clifford observed that Grassmann developed the idea of a directed number from the quantitiative point of view, while Hamilton emphasized the operational interpretation. The two approaches are brought together by the geometric product. Either a quantitative or an operational interpretation can be given to any number, yet one or the other may be more important in most applications. Thus, vectors are usually interpreted quantitatively, while spinors are usually interpreted operationally.

Of course, it was Hestenes that brought Geometric Algebra, and the quaternions that are part of it, out of obscurity, during the last half of the 20th Century, but by then the institutionalization of academia and science, which might have benefited most from Hamilton’s “profound insight” into the fundamental relation between geometry and algebra, manifest in quaternions, had long since been set in the language of complex numbers and vector analysis.

Hestenes realized this, of course, and mostly turned his efforts to convincing the mathematical and academic communities of the pedagogical advantages of Geometric Algebra, due to its almost complete generalization of number with geometric magnitude. However, he also developed his 4D spacetime algebra (STA) and was able to show that it applies to a wide variety of problems in physics. Today, there are many approaches to physics based on Clifford algebras, but all are connected, more or less directly, to quaternions, emphasizing the geometric significance of vector products, while avoiding the much less intuitive concepts of matrices and tensors elements.

Nevertheless, it’s clear that Hamilton’s “belief that quaternions would play a central role in the mathematics of the future” was founded in a more fundamental vision than a basis for alternate formalisms, even if that includes providing a geometrical component to replace otherwise abstract concepts of multi-dimensional rotation. He really believed that since geometry is the science of fixed magnitudes, it follows that algebra should be the science of numbers, not just a tool, or a language, for science, but a science in and of itself.

In the preface to his “On Algebra as the Science of Pure Time,” he explains the difference between algebra as art, or language, and algebra as science, and then he explains why he feels algebra must be much more than a formulation of symbolic ideas:

Yet a natural regret might be felt, if such were the destiny of Algebra; if a study, which is continually engaging mathematicians more and more, and has almost superseded the Study of Geometrical Science, were found at last to be not, in any strict or proper sense, the Study of a Science at all: and if, in thus exchanging the ancient for the modern Mathesis, there were a gain only of Skill or Elegance, at the expense of Contemplation and Intuition.

If, in the end, Hamilton’s greatest achievement was simply the invention, or discovery, of quaternions, which form the basis for a more elegant formalism of quantum physics, or relativity theory, then, even with the new-found utility of quaternions that Hestenes has helped bring to light, the pathos of the “Irish Tragedy” is only blunted. In the final analysis, because it was not these more skillful and elegant applications Hamilton sought to gain, but the far more lofty objects of contemplation and intuition - algebraic truth - as it were, he would probably have agreed with Bell that his career was a failure. He lamented:

So useful are those rules, so symmetrical those expressions, and yet so unsatisfactory those principles from which they are supposed to be derived, that a growing tendency may be perceived to the rejection of that view which regarded Algebra as a Science, in some sense analogous to Geometry, and to the adoption of one or other of those two different views, which regard Algebra as an Art, or as a Language: as a System of Rules, or else as a System of Expressions, but not as a System of Truths, or Results having any other validity than what they may derive from their practical usefulness, or their logical or philological coherence. Opinions thus are tending to substitute for the Theoretical question,—“Is a Theorem of Algebra true?” the Practical question,—“Can it be applied as an Instrument, to do or to discover something else, in some research which is not Algebraical?” or else the Philological question,—“Does its expression harmonise, according to the Laws of Language, with other Algebraical expressions?”

Thus, he sets out on the quest to find aglebraic truth, and what we discover is that the famous moment, under the bridge, when he carves the quaternion equation in the stone, after years and years of searching, what he thought he found was not so much a geometric algebra, as Hestenes describes it, but a key to understanding the science of algebra that had started with a successful development of numeric couples, but mysteriously couldn’t be extended to numeric triplets, but only numeric quartos.

Given the cursory treatment of quaternions in academia, one is simply led to view them as natural extensions of complex numbers. If one imaginary number works such magic in the complex plane, then it’s only logical to think that adding more imaginary numbers would be even better.  Of course, this isn’t the case.  Adding one more imaginary number to a 2D complex number, to form a 3D number, doesn’t work.  You have to add two more imaginary numbers to the one of complex numbers to get the algebra to work, and even then it’s got a huge defect, in a sense, even though it’s perfectly understandable that a 4D system is non-commutative, given the directions of 3D space in terms of locations defined with four dimensions (3 space and 1 time dimension.)  Yet, this couldn’t have been pleasing to Hamilton.  Before the moment under the bridge, he wrote:

For it has not fared with the principles of Algebra as with the principles of Geometry. No candid and intelligent person can doubt the truth of the chief properties of Parallel Lines, as set forth by Euclid in his Elements, two thousand years ago; though he may well desire to see them treated in a clearer and better method. The doctrine involves no obscurity nor confusion of thought, and leaves in the mind no reasonable ground for doubt, although ingenuity may usefully be exercised in improving the plan of the argument.

Of course, this was before the days of non-Euclidean geometry (Hamilton probably turned over in his grave!), but his point is still valid: geometry is inductive. It’s an inductive science.  Algebra, on the other hand, is not nearly as satisfactory in this regard.  Hamilton goes on:

But 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. It must be hard to found a Science on such grounds as these, though the forms of logic may build up from them a symmetrical system of expressions, and a practical art may be learned of rightly applying useful rules which seem to depend upon them.

My guess is that the reason Hamilton spent the rest of his life, working with quaternions, is not because he sought to improve “the practical art” of vector analysis, but because he was convinced that there was a “science of algebra;” that is, he searched in vain for a way to make algebra an inductive science, as geometry is an inductive science, or was in his time.

As it turns out, he was really, really, close, but it wasn’t until Larson shed the light of reciprocity on mathematics and physics that we could really appreciate why.  We’re still trying to get our pea brains around it.

In The New Math Blog, we’ve been discussing the discovery of how the discrete and continuous faces of nature are two, reciprocal, aspects of the same reciprocal number (RN). For example, the 1|2 RN has a value of -1, but the 1/2 RN has a value of 0.5, and which numerator-denominator relation we select, inverse multiplication, represented by the slash symbol, “/”, or inverse addition, represented by the pipe symbol, “|”, depends on whether we are dealing with discrete numbers, or continuous numbers.

We discovered that the essence of nature’s concept of discrete and continuous values comes down to one of interpretation, like the changing image in that famous picture that is transformed in our perception, from a young girl, to an old woman, simply by how we associate the lines and colors of the same image in our minds. It’s not that one image is more valid than the other. It’s that they are two aspects of the same thing, just as space and time are two aspects of motion. You cannot have one without the other. They are the opposite sides of the same coin.

In applying this principle to our new physics, we run into exactly the same thing. As a function, the numerical representation of the reciprocal expansion-contraction of the SUDR and TUDR combo, can be interpreted in either of two ways. It can be viewed as a discrete number, where

S|T = 1|1 = 0,

or it can be interpreted as a continuous number, where

S/T = 1/1 = 1.

As a discrete number, we are using 0 as our datum in a discrete reference system, where the numerical set of RNs is interpreted as

-n, …-1, 0, 1, ,,,n

which set is the set of positive and negative integers that is a group under addition. As a continuous number, on the other hand, we are using 1 as our datum in a continuous reference system, where the numerical set of the same RNs is interpreted as

-.n, …-.5, 1, 2, …n

which set is the set of non-zero rationals that is conventionally regarded as a field under addition and multiplication.

When we first started using RNs, as numerical representations of discrete units of motion, it was clear that we could add and subtract them, but it wasn’t clear at all how we could multiply and divide them, or even when we would want to do so. It wasn’t until we had to describe the SUDR and TUDR connections, in the fermion triplets, that we found ourselves confronted with the need to multiply and divide, because the gear ratios in a gear train are multiplied together, not added together.

However, there is more to this situation than meets the eye, because it’s not clear why the discrete numbers, called integers, should be a group, under addition, while the continuous numbers, called non-zero rationals, are a field, under both addition and multiplication. If they are indeed two, reciprocal, aspects of the same thing, why aren’t their mathematical properties reciprocally related? In other words, we seem to have lost the symmetry of reciprocity, when we consider the conventional mathematical properties of the set of RNs, as two, reciprocal, aspects of the same number.

Happily, we seem to have found the answer, and perhaps discovered along the way that the reason why this has happened goes right to the heart of the difference between the RST, as a system of physical theory, based on scalars, and the LST, as system of physical theory, based on vectors. The symmetry in the mathematical properties of discrete and continuous numbers is preserved, when one realizes that the datum, and the associated reference system, of the continuous number, is the inverse of the datum, and the associated reference system, of the discrete number.

We can easily see this when we diagram the two, reciprocal, reference systems:

Reference system of discrete RNs:     -∞ <——————-> 0 <———————> +∞
Reference system of continuous RNs: -0 <——————-> 1 <———————> +0

Clearly, one is the reciprocal of the other. The problem is that, historically, the role of reciprocity and the dual nature of RNs, has not been appropriately understood. Rational numbers, as fractions of a whole, were not recognized as having inverses, because the inverses were not interpreted as inverse fractions, but as whole numbers, in the number line. However, when we recognize that 1/2 is the inverse of 2/1 and that they have exactly the same absolute value relative to the continuous reference system, then the broken symmetry that arises in the conventional mathematical properties, is restored.

Specifically, it turns out that the discrete numbers are a group under addition, while the continuous numbers are a group under multiplication, but the group properties of continuous numbers can only be clearly recognized when we employ the RNs. Moreover, only when we join the two groups together, do we get a field of numbers under addition and multiplication! That is to say, we can add and multiply fractions and integers like we normally do, precisely because, not having the insight of the duality of RNs, we use the two groups of numbers as a field, not realizing the underlying principles involved.

To get around this, mathematicians use the idea of a ring, which is a set of integers under addition and multiplication, which is not fully a field, but almost. Essentially, this is possible in the conventional understanding of integers because they are formed in a non-reciprocal manner, such that negative numbers are defined as the mirror reflection of positive numbers: -1 is -1/1, -2 is -2/1, and so on, ad infinitum.

Nevertheless, when we employ the RNs, we see that the set of integers constitutes a group under addition, and the set of rationals constitutes a group under multiplication. Then, combining the two sets, forms a field under addition and multiplication, without the need for an ad hoc invention, such as a ring.

For example, normally we multiply .5 times 2 to get 1, but we must interpret the 2 as a discrete number and the .5 as a portion of a discrete number, or a continuous number. If we don’t do this, and instead take .5 as -.5 and 2 as +.5, which are their values, in the continuous reference system, then multiplying them would result in -.25, and confusion would be the result. In other words, we have to interpret one multiplicand as the “old woman,” and the other as the reciprocal “young woman,” in order to get consistent results, even when we don’t recognize explicitly that we are using the two, reciprocal, aspects of number, in the multiplication operation!

The same implicit use of the two groups as a field happens under the sum operation. If we add .5 and 2, we normally get 2.5, but we don’t explicitly recognize that the sum operation of the field requires one summand to be a discrete number and the other to be a continuous number. As a result, we are adding a continuous number to a discrete number, without realizing that, in doing so, we are operating in the field of the two, reciprocal, groups. However, If we stay within the group, and we add two continuous numbers, we would be adding -.5 and +.5 and getting 0, instead of 2.5, for the sum of .5 and 2!

Also, you probably noticed that 0 is not the identity element of the continuous group . The datum of the continuous reference system is 1, not 0, so the sum of (-.n) + (.n) = 0 doesn’t yield the identity element of the continuous group, which is to be expected, because the continuous group is not a group under addition, but only under multiplication, when the numerical set is the set of RNs.

Another important point is that, unlike the unit of the elements in the discrete group, which are multiples of the fixed discrete unit, |1|, the unit of the elements of the continuous group is not fixed, but it is variable, |1/n|. Consequently, the continuous group is a finite group of order n, where n is the number of elements in the group; that is, there is an infinite number of orders of this group, but each order has a finite number of elements, meaning that we can divide up a whole into as many parts as we want, ad infinitum.

Of course, there is much more to say about all this, but to discover that the principle of reciprocity leads us to the unification of the discrete and continuous aspects of nature, unified in the concept of numerical magnitude that is so analogous to the concept of physical magnitude, leaves us breathless. 

The biggest mystery of nature, as far as physicists are concerned, is how she can be both discrete and non-discrete, or continuous, at the same time. In the development of the LRC’s RST-based physical theory, we are using the Reciprocal System of Mathematics (RSM) to develop the consequences of the fundamental postulates of the RST, which introduces the science of discrete units of scalar motion to the existing science of continuous, or vector, motion. The motivation for developing a new math here is to improve the language of the LRC’s new science, which now includes both discrete and continuous magnitudes of motion, so that it can be more effectively used in the LRC’s theoretical development efforts.

To undertake this task requires an intimate understanding of how the language is to be used and how it refers to the physical world, but now that our theoretical world consists of magnitudes of motion that are simultaneously discrete and continuous, it necessarily implies solving the biggest mystery of nature. “How can we be so pretentious?” one might ask. Well, for one thing, when we started, we didn’t understand that what we were doing could be characterized like this. It’s only in hindsight that we can articulate the effort in these terms. For most of the time, the left hand didn’t know what the right hand was doing.

The first big breakthrough was the development of the Progression Algorithm (PA), which was adopted from rule 254 in Stephen Wolfram’s list of cellular automata rules, the most uninteresting of all 256 rules, as far as Wolfram is concerned. However, by reinterpreting the progression of space and time in this rule, in order to incorporate them as two, reciprocal, aspects of motion, we were able, for the first time, to graph the scalar progression and the “direction” reversals, which enabled us to visualize the meaning of these concepts and think about their implications in terms of numbers.

From this, it soon became apparent that only two space|time ratios were possible, 1|2 and 2|1, and that these had to be combined to get anything interesting, but combining them only led us back to the unit progression, 1|1! However, we eventually recognized that the sum of the numbers 1|2 and 2|1 was actually the sum of 2|2 and 2|2 in terms of total units of progression; that is, there is one unit of inward motion, for every unit of outward motion, in a cycle of two “direction” reversals. Hence, the complete equation for summing two, reciprocal, units of scalar space|time oscillation had to be

ds|dt = 1|2 + 1|1 + 2|1 = 4|4 num,

where num is an acronym for “natural units of motion.” However, at that point in time, we were still using the division symbol “/” in the equation, so the normal mathematics didn’t work! Using normal rules for adding rationals, the sum equation would be

ds/dt = 1/2 + 1/1 + 2/1
= (1/2 + 1/1) + 2/1
= (1+2)/2) + 2/1
= 3/2 + 2/1
= (3+4)/2
= 7/2 = 3.5 num.

Nevertheless, it was apparent that the normal rules for adding rationals was incorrect, and that the new rules, adding numerators to numerators, and denominators to denominators, gave the correct answer in every case. We knew what the correct answer had to be, given the PA. Therefore, we proceeded with the development with the new rules for adding rational numbers, but having no idea why this strange way of adding fractions worked.

With this new math, we were able to add (and subtract) SUDRs and TUDRs in various numbers, which led to SUDR|TUDR combinations like

ds/dt = (1/2 + 1/1 + 2/1) + 1/2 = 5/6 num, and

ds/dt = (1/2 + 1/1 + 2/1) + 2/1 = 6/5 num,

where there is an unequal number of SUDRs and TUDRs. We immediately recognized that this is just like adding integers, except that 0 is replaced by 1, in the form of 1/1. However, in terms of Larson’s concept of speed-displacement, 1/1 is 0 speed-displacement! Duh, clearly we were working with a reciprocal, or rational, form of integers here! The conventional rational rules didn’t work, precisely because this is a rational form of positive and negative integers, not a rational form of a decimal number, or a fraction, of a whole.

Nevertheless, while this enabled us to explore sums of SUDRs and TUDRs, as discrete combinations of motion, there was no way we could find to multiply these reciprocal numbers (RNs), which was strange, because integers can certainly be multiplied. Well, we could multiply RNs in a sense, but it was equivalent to repeatedly adding a unit of 1/2 or 2/1 motion. For example,

3/1 * 1/2 = (1/2 + 1/2 + 1/2) = (-1 + -1 + -1) = -3,

but this required interpreting 3/1 as 3 not -2. In other words, we could conceptualize what it meant to multiply three units of -1, and that the result was equivalent to the sum of three negative units, but we couldn’t show how to consistently employ the RN form to do it. Then we realized that we could multiply RNs by positive integers, if we multiplied both the numerator and the denominator by the number. For example,

ds/dt = 3(1/2+1/1+2/1) = 3 * 4/4 = 12/12,

which is the same result as summing three S|T units: 4/4 + 4/4 + 4/4 = 12/12,

but this seems like cheating and imbues the new RN equations with an invalid connotation. Thus, we didn’t like it and seldom used it to express sums of RNs. Not only that, but we didn’t really need a multiplication operation, as long as we were satisfied that it was a shorthand method for repeated addition. Along the way, we had discovered that the RN form of integers were what David Hestenes would call an operational interpretation (OI) of number, as opposed to the quantitative interpretation (QI) of number, which gave us a legitimacy that we didn’t expect in the beginning, so we sort of hid the multiplication issue under the rug, so-to-speak, while we enjoyed our legitimate status as having found a new form of integers that can be summed.

It was not long after this phase that we began studying the properties of groups and this seemed to offer us insight into why we couldn’t multiply our OI RNs: integers form a group under addition; They don’t form a group under multiplication, and they don’t form a field under addition and multiplication. The importance of group properties is that they enable combinations. Combinations of a group elements are mathematically sound, in that the elements of the group can be combined into elements that can be combined into other elements of the group mathematically, in all the ways that combining is important. So, we were very happy campers to discover that our set of SUDRs and TUDRs were a group under addition.

Then we came across the preon model of Sundance Bilson-Thompson, wherein he employs the concept of twisted ribbons, as a topological approach to explaining the properties of the standard model entities. It was almost immediately clear that we could implement this same model in the form of S|T units, and we quickly did so, but, while this constituted a major breakthrough in identifying the triple combinations of S|T units that would serve the purpose, it also introduced a need for a multiplication operation to represent the relation between the constituent S|T units, and not only this, but the operation had to be in the other sense of multiplication, the one that raises and lowers (through its inverse) the degree of freedom, not the one that merely expresses repeated addition.

It’s not like we went out to discover such an operation directly, but we were studying how to form the triplet preons. We took our clue from Sundance that the bosons were simply stacks of S|T units, representing superimposed S|T units, but the fermions were quite different. In his twisted ribbon model, the two possible directions of the twists of the ribbons correspond to the two possible polarities of our RNs, positive and negative. An untwisted ribbon corresponds to our neutral RN, so this was really good news, but then Sundance represents the different configurations of these twisted ribbons, which correspond to the different entities of the standard model, by braiding three of them together in unique ways. Of course, we couldn’t braid three S|T units!

What we soon discovered, however, is that S|T units could combine in the form of triangles, which was doubly fortunate, because in this form, the associated space|time locations of the S|T units cannot be superimposed, but they must remain as partially merged, adjacent, locations! Wow. This means that the theoretical bosons and the theoretical fermions, constructed in this way, have the same properties that distinguish physical bosons from physical fermions! That was really the headline of the good news, but the caveat in the fine print was that coupling adjacent locations raises the dimensions, or the degree of freedom, of the triplet, from the one-dimensional set of a stack of S|T units (really a line of points), to the two-dimensional set of a triangle of S|T units (really an area of adjacent points).

This means that the binary relation between the constituent S|T units of the fermion triplet is fundamentally different than the binary relation of the constituent S|T units of the boson triplet, which is just the sum relation of the OI RNs that we already understand fairly well. In other words, we were looking for another group to use with the fermions, but this time it had to be a group under multiplication.

It seems remarkable in retrospect, but the conventional rational numbers didn’t come to mind immediately. I guess this is because we sort of felt that we had to have a set of OI RNs, because QI RNs, the set of rationals, weren’t discrete, and we more or less relegated them to the domain of the LST community and vector motion. They can take any value, and thus are non-discrete, and we were thinking in terms of combining the discrete units of motion, the SUDRs and TUDRs at the apexes, or nodes of the triplets, so non-discrete values didn’t’ seem appropriate.

However, no matter how hard we tried, we couldn’t see how to multiply, or divide, OI RNs in a way that raises or lowers a dimension of the number. Then, once again, something remarkable happened that opened the way. We came across a CAT scan of a photon! This is the work of Lvovsky et al, wherein they do something that is charmingly described by Rebecca Slayton in her article on it entitled “Golfing with a Single Photon.”

The connection here was not immediately apparent, but it involved a very important aspect of SUDRs, TUDRs, and SUDR|TUDR combos, which we had never really discussed in detail before, because there was never a good opportunity to do so. It has to do with the weirdness of quantum mechanics. Slayton writes:

Where quantum mysteries are concerned, Schrödinger’s cat has nothing on a single photon—at least you’d have some chance of finding the feline, whether dead or alive. In contrast, if you looked for a photon in a small space, within a limited range of momentum, you’d seem to have a negative chance of finding it.

Lvovsky shows that the photon’s wave property, expressed as a probability of locating its position, or determining its momentum, is actually a combination of positive and negative probability, but what is a negative probability? The only answer Lvovsky has for this question is that it is “a strongly non-classical character” of a photon.

However, the scalar motion of the S|T combo is just this sort of property; that is, it’s an oscillation of locality to non-locality, in that it’s a transformation of a point (localized point) into a sphere (non-localized point) and back again. There are also two, reciprocal, aspects of this oscillation in the S|T unit, the SUDR and the TUDR. In other words, the S|T unit is precisely a combination of a positive and negative oscillation of probability amplitudes, where one is the inverse of the other.

This means that these two oscillations are instances of two, reciprocal, harmonic motions, which can be represented by two, reciprocal, rotations, the same rotation that is present in two meshed gears. The bottom line of all this is that the ratios of gear rotations, in a set called a gear train, are multiplied straight across, from numerator to denominator, just like we add OI RNs! In fact, if we attribute the size of the gear to the number of SUDRs and TUDRs in a given S|T unit, coupled with another S|T unit, to the number of SUDRs or TUDRs contributed by the respective S|T units, we get a new RN, that is quantitatively interpreted; that is, we get a QI RN that we can multiply in the same way that we add OI RNs, straight across. Thus,

ds/dt = 2:1 * 1:2 = 2:2 = 1:1 = 1

is the coupling of two QI RNs, where each QI RN is composed of two OI RNs. In other words, a QI RN is a ratio of ratios, but since the OI RNs, in the numerator and denominator of the QI RN, are equivalent to positive and negative integers, the set of QI RNs is isomorphic to the conventional rationals, the fractions of ordinary mathematics!

This is a remarkable and welcome insight, because it not only will permit us to handle the mathematics of the fermion’s nodes, but it also sheds a great light on the fundamental issue of how nature can be both discrete and non-discrete at the same time. To understand this, we need only see that the set of OI RNs is the inverse of the set of the QI RNs. This may seem strange and impossible at first, because it’s tantamount to saying that the integers are the inverse of the fractions, but we can see that, as a set, or group, this is true. To show it, we simply compare the respective elements of the sets.

Clearly, the elements in the set of integers extend from 0 to infinity in the positive “direction,” and from 0 to infinity in the negative “direction.” At the same time, the non-zero rationals extend from 1 to 0 in the positive “direction,” and from 1 to 0 in the negative “direction.” Illustrating this graphically, we can see the inverse relationship of the two sets:

1. OI RNs (discrete): -infinity <——————-> 0 <——————> + infinity
2. QI RNs (non-discrete): -0 <————————-> 1 <————————> + 0

Since the only difference in the OI RNs and the QI RNs is the quantitative and the operational interpretations of the reciprocally related quantities, we are free to choose either interpretation, depending on the situation. In other words, the RNs are both discrete and continuous, depending on our perspective. Thus, it seems that the perfect symmetry of reciprocity explains the dual nature of reality through chirality.

Selah

Peter Woit gave a talk at the University of Central Florida (UCF) recently, entitled “The Challenge of Unifying Particle Physics.” He does a real good job of explaining what the Standard Model (SM) is, and then, in the last slide, he makes the point that is the most interesting of all: the challenge of unifying particle physics, he insists, is really the challenge of unifying the mathematics of the SM. Specifically, he makes three important observations:

1. The mathematics of the SM is poorly understood in many ways.
2. The representation theory of gauge groups is not understood.
3. The unification of physics may require the unification of mathematics.

But in the midst of these he puts another bullet that states:

• One indication of the problem with string theory: [It is] not formulated in terms of a fundamental symmetry principle. What is the group?

In other words, the trouble with the SM is that it works very well, but we don’t know why, and the trouble with string theory is that it doesn’t work very well, and this may be because it doesn’t work the way the SM works. The answer to the question, “What is the group?” is, of course, that there isn’t a group that corresponds to the vibrations of a string, even though there are two ends of the string, one positive and the other negative, which is the basic form of a group.

Thus, it seems that theoretical physics is caught on the horns of a mathematical dilemma. On reading this, it’s only natural to ask what the new mathematics, the reciprocal system of mathematics (RSM), has to say, if anything, about this predicament? Well, there are so many aspects to the problem that the most difficult challenge in answering it is the decision of where to begin. After contemplating it for some time, I think the best thing is to just start over, at the very beginning, with the so-called fixed coordinate system of three dimensions.

After all, the fixed, 3D, coordinate system is our basic framework of numbers and magnitudes. Indeed, it’s the framework upon which we have built our modern technology. Like women in a worldwide sewing circle, constructing various quilts by means of quilting frames, mathematicians and physicists have worked together to sew the intricate patterns of modern physical concepts into frames of the 3D fixed reference system.

Interestingly enough, is the fact that, for the scientists with their functions and variables, as well as for the ladies with their needles and threads, the most important and central principle of their work is the principle of symmetry. So, here is Peter Woit, a passionate member of the inner circle of theorists, standing up at the edge of the frame to complain about the lack of a clear symmetry in the latest work of the group. “I don’t get it,” we can almost hear him saying, “This new pattern sucks!” We can imagine him walking around the frame, pointing out the how the intricate needle work, gorgeous and elaborate as it is, in places, is just not symmetrical overall, as a whole!

“But,” we can imagine someone like David Gross protesting, “the overall symmetry doesn’t matter, because you can’t see the damn thing as a whole anyway. The only thing that really matters is the local symmetry, which everyone must admit is breathtaking.” Of course, Peter, and the many others who agree with him, are not convinced at all, by this argument. Yet, what can they do? To go back now and start over is unthinkable, but, at the same time, to find a principle which would serve to unify all the independent patterns into one seems impossible, at this advanced stage of the work.

However, this is not the case for amateur investigators like Larson. As he so eloquently pointed out, the amateur investigator has no vested interest in the current fabric of mainstream theory and is free to start from the beginning. Certainly, Larson is not the only one to recognize this. In fact, so many amateur have tried their hand at coming up with their own alternative that their ranks have become legion, but they are usually summarily and derisively dismissed by the professional craftsmen, as naive, with a hopeless lack of proficiency in the essential skills to work on the elaborate quilt, and Larson is no exception.

Nevertheless, and notwithstanding the lack of interest of the professionals in amateur suggestions, Larson’s approach should not be lightly dismissed, and for one reason: because his suggestion is completely and totally different than all others. In effect, he suggests that the sought for global symmetry, which can accommodate the desired local symmetry, can be found, if we modify the fixed, 3D, reference frame first. Since, historically, modifying this frame itself, in appropriate ways, has always led to the greatest advances in physical theory, this alone ought to get the attention of the pros.

Unfortunately, however, Larson never clearly understood how his modification of the frame worked mathematically. Indeed, he always insisted that his new frame was consonant with existing mathematics, but, as we have discovered, here at the LRC, this is far from the case. The truth is, his new, fixed, 3D, frame of reference constitutes a revolutionary new approach to the mathematics of the frame that completely transcends the transformations of the past.

The fundamental reason that this is so is that Larson’s recognition of the reciprocity of space and time, wherein he redefines space and time, as nothing more or less than the reciprocal aspects of changing magnitudes in the equation of motion, applies equally well to rational numbers in general: What we call scalar quantities, or zero-dimensional numbers, are best understood as reciprocal aspects of positive and negative magnitudes. In other words, the observed properties of physical magnitudes, quantity, duality, and dimension, are also the properties of abstract numbers, when they are understood, as the ancients understood them, as two, reciprocal, aspects of magnitude.

With this exceeding simple change in our most fundamental abstract concept, so obvious, but so revolutionary, we can construct a new, totally different, fixed, 3D, frame of reference upon which to build beautifully symmetric patterns that contain all the local variations that enchant us so, but we can do so without sacrificing the overall symmetry of the quilt.

The key to understanding how this works is found in understanding what we call the operational interpretation (OI) of the rational number, where the operation is the reciprocal relation of rational numbers just described. Given the OI number, we can form both negative and positive numbers, and zero, without the use of imaginary numbers. One of the most important things that this then does for us, is to give us a better handle on the abstraction of infinity and infinitude, which promises to help us to eventually extricate ourselves from the horns of the continuous-discrete dilemma, which has so exacerbated our difficulties in constructing consistent physical theory.

We start with the infinite OI number,

∞|∞ = 0,

where “|” is the operator symbol used to indicate the reciprocal relation of the OI rational number. Clearly,

∞|∞ = 0,

because of the perfect symmetry of the unit reciprocal - nothing is perfect. In other words, the value of the OI rational number is the relation between the reciprocal quantities, and the unit relation means that there is zero difference between the two, reciprocal, quantities. On the other hand, the only other possibility, the non-unit relation, is where

n|m < 0 and m|n > 0,

when m > n. This means that there is a non-zero difference between two, reciprocal, unequal quantities. It follows from this that if ∞|∞ is a group, then m|m, regardless of the value of m, is an identity element of the group. Thus, we conclude that both

∞|∞ = 1|1, and m|m = 1|1, a subset of ∞|∞,

are valid, but different, equations of OI numbers. The number of elements in the group, ∞|∞, is infinite, but a subset of these elements, s, is finite and determined by the selected identity element, m|m, where

s = 2(m-1).

Hence, in a 3D system, combining three, orthogonal, subsets of these OI rational numbers, x, y, and z, we have

1. s_x = 2(m-1)
2. s_y = 2(m’-1)
3. s_z = 2(m”-1)

and, when m = m’ = m”, we have a finite set of 3D OI numbers = (m|m)³, which is a subset of the infinite group of 3D OI numbers, (∞|∞)³, which is perfectly symmetrical.

For example, when m = m’ = m” = 1, there are 0 elements in the 3D set; When they are all equal to 2, there are 8 elements in the set, and so on:

1. (1|1)³ = (2(1-1))_x * (2(1-1))_y * (2(1-1))_z = 0³ = 0
2. (2|2)³ = (2(2-1))_x * (2(2-1))_y * (2(2-1))_z = 2³ = 8
3. (3|3)³ = (2(3-1))_x * (2(3-1))_y * (2(3-1))_z = 4³ = 64
4. (4|4)³ = (2(4-1))_x * (2(4-1))_y * (2(4-1))_z = 6³ = 216
5. (5|5)³ = (2(5-1))_x * (2(5-1))_y * (2(5-1))_z = 8³ = 512
6. …

Now, the question is, what can we do with these symmetrical sets of positive and negative numbers? The answer is, of course, we can make a 3D, fixed, “coordinate” system out of them, but where the “coordinates” represent potential discrete units of magnitude, rather than potential discrete points of space. For example, If we combine the eight elements of the (2|2)³ = 2x2x2 = 8 set, as follows:

1. Q1 = (2|1)_x + ( 2|1)_y + ( 1|2)_z = (5|4) (right, top, front )
2. Q2 = (1|2)_x + ( 2|1)_y + ( 1|2)_z = (4|5) (left, top, front)
3. Q3 = (1|2)_x + ( 1|2)_y + ( 1|2)_z = (3|6) (left, bottom, front)
4. Q4 = (2|1)_x + ( 1|2)_y + ( 1|2)_z = (4|5) (right, bottom, front)
5. Q5 = (2|1)_x + ( 2|1)_y + ( 2|1)_z = (6|3) (right, top, back)
6. Q6 = (1|2)_x + ( 2|1)_y + ( 2|1)_z = (5|4) (left, top, back)
7. Q7 = (1|2)_x + ( 1|2)_y + ( 2|1)_z = (4|5) (left, bottom, back)
8. Q8 = (2|1)_x + ( 1|2)_y +( 2|1)_z = (5|4) (right, bottom, back)

then we can form four OI sets along the diagonals, as follows:

1. Q5|Q3 (right, top, back) | (left, bottom, front)
2. Q6|Q4 (left, top, back) | (right. bottom, front)
3. Q2|Q8 (left, top, front) | (right, bottom, back)
4. Q1|Q7 (right, top, front) | (left, bottom, back)

Clearly, the three elements in the diagonally opposed sets are qualitative inverses of each other, as can be seen by comparing their textual descriptions; that is each element in the front, top, left and right is the inverse of an element in the back, bottom, right and left, but to be a mathematical group, the numbers have to conform to the rest of the rules of the group, which means that there also has to be an identity element in each diagonal set, and the binary operation (addition in this case) on the elements of the group has to be closed and associative.

Well, we know that there must be an identity element in each diagonal set, because the OI numbers from which they are constructed, being elements of the ∞|∞ group, each have an identity element, in this case the number is (2|2) = (1|1), at the intersection of x, y and z. Consequently, adding the identity element to any element of the set doesn’t change the value of the element. This condition of the mathematical group is obviously met in the set, since

(n|m) + (m|m) = (n+m)|(m+m)

doesn’t effect the relative value of the two, reciprocal, quantities, even though it changes the quantities themselves; that is, in the equation

4|5 + 1|1 = 5|6,

the difference between 4 and 5 is the same as the difference between 5 and 6.

To meet the closure condition, if group elements a and b are combined, then the combined element (a + b) must also be an element of the group. In this case, there are eight numbers in the set of (2|2)³, and combining any two of them clearly does not result in another member of the set. Nevertheless, the sum is an OI number in the infinite group, which seems to fulfill the requirement, even though it is not part of the finite set. If we combine the two inverses of the diagonals, however, the result is the identity element, the only other member of the set in this case, so, in that sense, the set is closed, I think. For example,

Q1 + Q2 = [(2|1)_x + ( 2|1)_y + ( 1|2)_z] + [(1|2)_x + ( 1|2)_y + ( 2|1)_z] = (3|3 + 3|3 + 3|3) = 9|9 = 1|1 = 0.

Consequently, we kill two birds with one stone, so-to-speak, showing that the inverse and the closure condition are both met in the diagonal sets; that is, an element plus its inverse equals the identity element (inverse condition), and the sum of two elements in the group is also in the group (closure), even though it’s, again, the identity element:

1. Q5|Q3 = 6|3 + 3|6 = 9|9 = 1|1
2. Q6|Q4 = 5|4 + 4|5 = 9|9 = 1|1
3. Q2|Q8 = 4|5 + 5|4 = 9|9 = 1|1
4. Q1|Q7 = 3|6 + 6|3 = 9|9 = 1|1

As m increases, however, the two tests can be performed separately, even though the result will always include one case where the sum is the identity element, regardless of the value of m. For example, in the case of (3|3)³, there are two non-identity numbers and their inverses, 2|3 and 1|3. The sum of these,

(1|3) + (2|3) = 3|6,

is the inverse of the sum on the other side,

(3|1) + (3|2) = 6|3,

which, combined together, is

(3|6) + (6|3) = 9|9 = 1|1.

However, remember these numbers are nothing but the potential magnitudes, or well-defined sets of static magnitudes, that we can form out of the number, our metric, we might say. It’s the same with the LST fixed coordinate system. The numbers of coordinate pairs, or triplets, and the patterns they form, are a framework to which the numbers of physical magnitude relate, not equate. Hence, there may not be a reason that they should constitute a group.

Fortunately, when we consider the units of this metric, or the interval between the numbers, the set of OI numbers does form a group. This is the set of numbers that conforms to actual physical magnitudes. To illustrate, if we assume only one and two are fundamental quantities, we can form all other magnitudes from these two quantities with OI numbers, just as all integers can be formed with the three fundamental QI numbers, the integers

-1, 0, 1.

The corresponding OI numbers,

1|2, 1|1, 2|1,

are the three fundamental OI numbers of the RSM; that is, all other OI numbers can be formed from them. The operation of the group that this set of numbers forms is addition, and any element of the group is a sum of multiple instances of either 1|2 or 2|1. For example,

(1|2) + (1|2) + (1|2) + (1|2) = 4|8,

which is equivalent to

(-1) + (-1) + (-1) + (-1) = -4.

The inverse of this,

(2|1) + (2|1) + (2|1) + (2|1) = 8|4,

is equivalent to

1 + 1 + 1 + 1 = 4.

Hence, while subtraction, multiplication and division are operations that are not used in the definition of the group, they are nevertheless valid operations that can be defined on elements of the group. For example,

5 x (4|8) = (4|8) + (4|8) + (4|8) + (4|8) + (4|8) = 20|40,

which is equivalent to

5 x (-4) = -20,

is really only another way of specifying the number of elements to be summed. Likewise, the other operations can be seen as an alternate form of specifying the elements to be summed. The important thing is that this set qualifies as a group. In other words, it’s a numerical expression of symmetry, a symmetrical system of numbers.

Now, combining the “coordinate” system of OI numbers, as magnitudes of “distance,” with OI numbers, as multiple units of unit magnitude, we can assign magnitudes to the coordinate “positions” of a given identity element, but since the available “coordinate positions” are limited, by the magnitude of the coordinate identity element, according to the identity element selected, the unit magnitudes, specifying those “positions,” are limited. For example, in a 2x2x2 framework, only one “position,” on either side of unity, in four diagonals, are available. Thus, if we order these, according to some arbitrary scheme, as shown above, then

Q1 = (2|1)_x + ( 2|1)_y + ( 1|2)_z = (5|4),

is the “coordinate position” of one unit of 3D magnitude, the absolute value of which is one unit, but the sign of which is determined by its Q index, indicating its “coordinate position.” In other words, the OI number, designating the magnitude (number of discrete units), at that position, is independent of the OI number designating the position of the magnitude itself. The magnitude has a coordinate position, or a numerical characteristic, we might say, which has both “direction” and limit.

Since these magnitudes form a group, they possess the local symmetry of the finite group that they are part of, but, at the same time, they are also elements of the infinite group and its global symmetry.  Isn’t this exactly what is needed?