Fundamental Physics - The Search for New Answers to Foundational Questions

[dbundy]

Note: In 2016, four years before he passed away, Bruce Peret had intended to write a book on his contribution to the development of a Reciprocal System theory (RSt), he called the RS2. However, he was never able to accomplish it. I aspire to the same lofty goal, but I know how difficult it is to write a book, any book, let alone one dealing with as esoteric a subject as this one.

What follows is a draft of an introduction to what I have written to this point. Comments are welcome.

Introduction
The full title of Lee Smolin's book, The Trouble With Physics: The Rise of String Theory, The Fall of a Science, and What Comes Next, implies that the rise of string theory and the fall of "a science" go together; that is, Smolin asserts that string theory is responsible for the lack of real progress in fundamental physics for the past quarter century, and that the string theorists are willing to sacrifice the science of physics, rather than admit failure.

However, the most relevant message of the book is not the failure of string theory, but the fundamental crisis in theoretical physics that the pursuit of string theory has exacerbated. Smolin and others, like Peter Woit, author of the anti-string theory book, Not Even Wrong: The Failure of String Theory and the Search for Unity in Physical Law, are worried that, based on ideas coming from string theory, physicists are abandoning hope that a unique, unified, solution to five fundamental problems in theoretical physics actually exists, while they are disregarding, or denying, the severe impacts, which the implications of this thinking have on the science of physics itself.

Smolin discusses each of these five problems in the first part of his book, entitled the "Unfinished Revolution," referring to what is actually the second major revolution in the science of physics and mathematics. The first revolution, though well known, is so ancient that Smolin and other physicists rarely mention it. Modern physicists prefer to focus on the second scientific revolution, precipitated by Max Planck’s findings at the end of the 19th Century, which turned the attention of mathematicians and physicists from the mature physics of the continuum, based on the invention of real numbers, to the new physics of the quantum, based on the invention of complex numbers.

The crisis that precipitated the first revolution occurred around 330 BC, when the Pythagorean secret of the square root of 2 got out and turned the attention of mathematicians and geometers from the quantum of integers to the continuum of real numbers, which finally took off in a big way with the advent of the calculus, many centuries later, ushering in the modern marvels of continuous science and technology, based on the limits of differential equations. Yet, as magical as the invention of a solution to the problem of the square root of 2 proved to be, the invention of a solution to the problem of the square root of –1 was almost mystical, stealing the thunder away from Pythagoras’ right triangle, and endowing its grandchild, the unit circle, with almost a god-like status, ushering in the promise of quantum science and technology, based on the probability amplitudes of wave equations.

Today, these two inventions of the human mind, the real numbers of the continuum and the complex numbers of the quantum, are hailed as great milestones in the advancement of civilization. Yet, it is also clear that the key issue, at the heart of the modern trouble with physics, is not the development of string theory per se, but rather, the challenge of uniting these two pillars of modern science and technology, the quantum (or digital) concept, with the continuous (or analog) concept, the way nature unites them, as two aspects of the same thing - a duality.

The modern physics of the continuum concept is embodied in the principles of classical mechanics, via the mathematics of calculus, where the most erudite and exotic form is found in Einstein's theory of general relativity, while the modern physics of the quantum concept is now embodied in the theories of quantum mechanics, and the much heralded theories of quantum fields, and quantum colors, presented as the standard model (SM) of particle physics, which are all based on the invention of the imaginary square root of -1, the number ‘i,’ used to form complex numbers.

Unfortunately, however, while quantum theorists managed to successfully identify the infinite set of continuous magnitudes, with the infinite set of discrete complex numbers, through the magic of the unit circle, capturing the relation of the two through the rotation of the unit circle's radius, this approach has not lead to a complete understanding of the observed structure and behavior of nature.

Consequently, we are left with two theories, one continuous, the other discrete, and the philosophical trouble with these two physical theories, one based on the continuous magnitudes of a smooth manifold, and one based on the discrete magnitudes of particle fields, is that they are fundamentally incompatible with each other. Moreover, the practical trouble with these two, incompatible theories is that while the hope of reconciling them in string theory grows fainter in the minds of some, it’s more and more disconcerting that the seemingly futile pursuit of this approach continues to soak up the vast majority of the theoretical community’s manpower and resources.

Hence, in the view of those who have concluded that another approach is needed, the entrenched influence and the stubborn tenacity of those who remain optimistic that string theory can eventually reconcile the differences between the new physics of the quantum and the conventional physics of the continuous, and heal the schism of modern physics, are prejudicial to the scientific community. This has led to growing confrontation and polarization within their ranks over the lack of theoretical unity in fundamental physics.

A few years ago, Smolin and Brian Green, a popular string theorist, discussed the trouble with physics with Ira Flatow, the host of the National Public Radio series, Science Friday. After discussing Smolin et al's criticism of string theory's failure to predict new physical phenomena, and the effective suppression of new ideas due to its thorough domination of academia, they finally turned to discussing the physics crisis itself.

As Green explained how tests of the consistency of string theory calculations, and comparisons with the established concepts of physics, show that the "theory comes through with flying colors every step of the way and keeps us thinking that things are at least headed in the right direction," Flatow turned to Smolin and asked, "Well, Lee, what would be wrong with that, if things are working like that?"

Smolin's answer was very telling, and it's well known in the community: In spite of these favorable things that one can say about string theory, there are some very important things that "it doesn't come close to doing," asserted Smolin. Then he hit the nail squarely on the head:

If you really put quantum mechanics together with the description of space, then we know, from general considerations, that the notion of space should disappear. Just like the notion of the trajectory of a particle disappears in quantum mechanics, ...the same thing should happen to space and the geometry of space.

"So far, string theory doesn't address this very directly," Smolin said, "while other approaches do," referring, of course, to his own continuum-based theory of quantum gravity, called loop quantum gravity (LQG), but before he could explain this further, Flatow took a call from a listener who suggested that "thinking outside the box," is what's required, which momentarily distracted the conversation away from the idea that “the notion of space should disappear,” in the union of quantum mechanics and the description of space (the spacetime of relativity theory).

Smolin replied to the listener’s comment, stating that, while he agrees with her, he definitely feels that it has to be the trained minds of professional physicists that do the "out of the box" thinking, who "go back in the decision tree," looking for new answers to foundational questions, because only they are prepared to readily scale the true mountain of knowledge, once the location of the highest peak in the landscape is discovered. Whereupon Flatow interjected with the obvious conclusion, in the form of a penetrating question:

Are we at a point now, where you just have to sit and scratch your head and think, "We need some revolution, don't we?" I mean, we need a revolution in physics; maybe, we need a new physics!

However, Smolin's reply to this conclusion reveals just how difficult it is for the minds of professionals, trained from the start in what Thomas Kuhn termed “normal science,” to think "outside of the box." Instead of agreeing with Flatow’s conclusion that a “new physics” is required, he demurred. "Nothing can happen without experiments," he asserted laconically, deftly inferring a different meaning of the phrase “new physics,” which is a phrase that today is commonly used to refer to new experimental anomalies, not a new foundation for theoretical physics, something that is literally inconceivable to the professional physicist. Yet, the truth is, the trouble with physics is not due to a lack of available, inexplicable, empirical data, but to the fact that there is no satisfactory explanation of the existing data from many, many experiments, including the anomalous results behind the so-called dark energy and dark matter enigmas, and the famous Yang-Mills mass-gap problem, to name just a few.

Clearly, however, Smolin revealed his hand with his comment: While he’s certainly dismayed with the emphasis on string theory research, which seeks to unify the discrete with the continuous through modification of the current discrete paradigm, with which the string theorists are most familiar, Smolin and company prefer to approach the problem from within the context of the current continuous paradigm, with which they are most familiar. Smolin’s argument is not that a new foundation for theoretical physics is required, but that a shift in academic research emphasis is required, from the “let’s modify the existing discrete theory” to solve the problem (string theory), to the “let’s modify the existing continuous theory” to solve the problem (loop quantum gravity)! The pressing need, from Smolin’s point of view, is to complete the “unfinished revolution,” which Planck and Einstein started, by exploiting Einstein’s concept of the continuum, in order to unite the disparate theories, instead of modifying Planck's concept of the quantum, in order to unite them.

Yet, the most logical conclusion that naturally occurs to the non-professional, didn’t escape Flatow: “We need a revolution in physics; maybe, we need a new physics!” he interjected, implying the need for a completely new foundation for theoretical physics, which doesn’t require the reconciliation of two, incompatible, theoretical concepts of space and time, one static and fixed, the other dynamic and changing, but finds a new concept of space and time that works as nature works; that is, a new concept that works as the dual properties of one component, where the discrete and continuous realities are simply two aspects of the same thing.

Truly, as unpalatable, as unlikely, and as inconceivable, as the prospect is to the minds of today’s practitioners of “normal science,” the possibility that a totally new solution exists that would revolutionize existing discrete and continuous concepts, and that would explain the dual quantum and continuum nature of reality, as two aspects of the same entity, has to be regarded as a legitimate alternative that needs to be seriously considered, though it may seem iconoclastic to today’s scientists.

Evidently, as the NPR interview continued, since Smolin had dismissed the possibility of a "new [theoretical] physics," which he had suggested, Flatow turned to Green to get his comments, and Green seemed more willing to admit that something truly revolutionary in physical concepts is needed in our conception of space and time:

I full well believe that we will, when we do complete this revolution that Lee is referring to, have a completely different view of the universe. I totally agree with Lee, that everything we know points to space and time not even being fundamental entities...We think that space and time...rely upon more fundamental ideas...What those fundamental entities are...that make up space and time, we don't know yet, ...but, when we get there, I think that we will learn that space and time are not what we thought they are. They are going to morph into something completely unfamiliar, and we'll find that, in certain circumstances, space and time appear in the way we humans interpret those concepts, but fundamentally the universe is not built out of these familiar notions of space and time that we experience.

Flatow stumbled a little, trying to get his head around an idea of what this might mean in terms of changes to existing concepts, which prompted Green to add:

It would change the very notion of reality...We all think about reality existing in a region of space and taking place through some duration of time, but we've learned that those basic ideas of the arena of space and the duration of time are not concepts that even apply, in certain realms, and if the notions of space and time evaporate, then our whole conception of reality, the whole container of reality will have evaporated, and we'll have to learn to think about physics and the universe completely differently.

In contemplating the implications of this idea, one’s thoughts turn to the meaning of the mysterious relation of mathematics and physics, something still not understood after centuries of intellectual struggle. In fact, this seems to be mathematician Peter Woit’s perspective. Had he been in the Interview with Ira Flatow, Flatow’s conclusion might well have been, “We need a revolution in mathematics; maybe, we need a new mathematics!”

Woit’s view is that the formalism of quantum mechanics is not well understood. In a talk he gave at the University of Central Florida (UCF) recently, entitled “The Challenge of Unifying Particle Physics.” He does a very good job of explaining what the SM of particle physics is, and then, in the last slide, he makes the point that is the most relevant: 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 [first].

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 formalism of the SM (the mathematics of compact Lie groups) 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 formalism of the SM works. The answer to the string theory 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 an Abelian group.

Thus, it seems that, again, science is caught on the horns of a mathematical dilemma, but to understand the dilemma this time, it is necessary to understand the role that symmetry now plays in the drama. Woit explains this, by first delineating the difference between the concept of “observables” in continuous mechanics and “observables” in quantum mechanics. This difference casts the principle of symmetry into a more significant role in the discrete system than that which it enjoys in the continuous system. Woit writes:

The remarkable thing about observables is that they often correspond to
symmetries. This connection is much more direct in quantum mechanics
than in classical mechanics.

But the remarkable thing about symmetries is that they form groups, so Woit explains that groups of symmetries can be found in physical laws of conservation: conservation of momentum (space translation symmetry), conservation of energy (time translation symmetry), and conservation of angular momentum (rotational symmetry), and also in the complex numbers of the unit circle (conservation of unit length). This natural property of symmetry is important, explains Woit, because mathematicians were able to invent something called a group representation, in the 19th Century, that “turns out to be exactly what quantum mechanics is about.”

The quantum observables, which describe the states, or physical magnitudes, of the discrete system, much like the continuous observables, such as energy and momentum, which describe the states, or physical magnitudes, of the continuous system, form a group representation of the set of symmetries associated with them. Woit explains:

Whenever a physical system has a group of symmetries G, its quantum
mechanical state space H is a representation of G. These symmetries
leave invariant the length (t) of state vectors, so mathematicians call
this kind of representation “unitary”.

Which is just another way of saying that a one-to-one correspondence between the complex numbers of size one, the numbers that identify the continuum of points on the circumference of the unit circle, and the quantum observables that determine the physical magnitudes in the discrete system of modern physics, is accomplished by relating the common symmetry of the two, which is called a unitary representation.

In many ways, this 20th Century discrete system of physics is not all that different from the 19th Century continuous system of physics, except that by extending the invention of real numbers, which established the original one-to-one correspondence of rational numbers to the physical continuum, to the complex numbers of size one, an additional degree of freedom is attained, which enables two new, discrete, observables, the observables of electric charge and spin, to be added to the familiar observables of the continuous system. Woit explains:

1. The mathematical structure of quantum mechanics is closely connected to what mathematicians call representation theory. This is a central, unifying theme in mathematics.
2. The Hilbert space of quantum mechanical states H of a system is a representation of the groups of symmetries of the system.
3. Much of the physics of the system is determined by this, including the behavior of four of the most important observables (energy, momentum, spin and charge), which correspond to four different symmetries (time translation, space translation, spatial rotations, complex plane rotations).

In short, the new invention of complex numbers in the unit circle was initially very successful, and its central concept, based on the principle of groups of symmetries and their representations, led to the dramatic success of the discrete physics of the 20th Century that would have been impossible with only the continuum physics of the 19th Century.

However, as is common knowledge now, this remarkable invention, based on the earlier invention of the imaginary number, which the late Sir Michael Atiyah called “the biggest single invention of the human mind in history,” has become “a victim of its own success,” to use the words of Woit, and he describes quite succinctly the quagmire of theoretical confusion that it has led to. In the final analysis, Woit, reluctantly, draws the inevitable conclusion that most people likely would draw, once they have clearly understood the nature of the present perplexity: “Maybe future progress will require not just unification of physics, but unification [of physics] with mathematics...”

Indeed, but bringing the focus back to mathematics revives the issue of formalism versus intuitionism. Atiyah emphasized the role of simplicity and elegance in nature’s secrets. It’s crucial. From this perspective, he indicted string theory with one devastating observation:

If a final theory emerges soon from string theory, we will discover a universe built on fantastically intricate mathematics.

His point is made in the context of the “conundrum” of imaginary numbers that he describes. This conundrum is perplexing to him because, if, fundamentally, the origins of mathematics are found in nature, then the “fantastically intricate mathematics” of string theory reflects something ugly and unsatisfying in nature, which would be so surprising, given humanity’s historical experience with her.

On the other hand, if mathematics is just a mundane tool for studying the physical structure of the world, and is no more than an invention of the human mind, how is it that its “biggest, single, invention …in history,” the imaginary number, shows up in observed physical phenomena?

He said that while his position was more moderate than Kronecker’s, he still believed that the origins of math are to be found in nature’s fundamentals, which then man develops and elaborates upon. Clearly, he’s implying that we have strayed too far from the origins of mathematics; that the vastness of string theory’s mathematical complexity has now taken on a life of its own, which abandons the vital symbiotic relation of math and physics.

The thesis of this work is that the current crisis in fundamental physics is a direct result of the lack of this unity between our understanding of physics and our understanding of mathematics, and that Dewey B. Larson, the author of a new system of physical theory, called the Reciprocal System of Physical Theory, has discovered that the way forward, in clarifying the confusion that now exists, is to be found in the recognition of the role of reciprocity in nature, in particular the fundamental reciprocity of space and time.

However, even though Larson didn’t explicitly recognize that the fundamental role of reciprocity in the concepts of physics, which he discovered, applies with equal force to corresponding concepts of numbers, it is clear that he used these concepts in the development of his RST-based theory, i.e. the universe of motion, to the extent that it was possible to do so, without an explicit development of the numerical concepts involved.

What is now recognized is that these new numerical concepts of reciprocity give rise to the symmetry and chirality found in the logic, as well as in the structure, of the theoretical universe of motion, which logic and structure correspond to the observed laws of conservation and the observed structure of the SM of particle physics, found in natural phenomena, as far as has been determined.

Hence, a new system of physical theory emerges from these new insights. One that consists of a method for exploiting the principles of symmetry, in the development of physical theory, just as the 20th Century physicists have so brilliantly exploited it, but in the new system it is accomplished by means of the symmetry and chirality inherent in the character of the reciprocity of numbers, the foundation of nature’s duality, not by means of the ad hoc numerical inventions of the human mind, as ingenious as these are.

Unfortunately, the task of rebuilding the foundation of mathematics, upon intuitive principles of reciprocity, symmetry and chirality, in order to unify mathematics with physics, threatens the vast superstructure of elaborate and complex formalisms, currently hosting the community of professional mathematicians and physicists of academia. Consequently, it is a concept that they are reluctant to entertain.

For this reason, the task of identifying the highest mountain in the landscape of theoretical possibilities has been mostly the bailiwick of amateurs, up to this point, but, as Smolin pointed out to Flatow, eventually, the skills of the professionals will be required to scale the peak. The purpose of this work, then, is to try to entice them to do so.

Therefore, we begin at the beginning, at the foundations of modern science, where the seeds for today’s trouble with modern physical theory were sown. However, the compound lens through which these fresh perspectives will be focused is the relatively new light in the world of mathematics and physics, the principles of reciprocity, symmetry, and chirality (RSC), the inherent properties of the mathematical structure called a group, and its older brother the field.

The surprising, perhaps delightful, thing is that these groups will not be the esoteric type of group like a compact Lie group, or any group of continuous symmetry. They will not be the unitary or special unitary groups of the SM, but only the simple Abelian groups of integers and non-zero rationals. The magic is in the RSC properties of the elements of these simple and elementary groups.

Accordingly, Section I of this paper begins, not with the second revolution of the science of physics, but with the first revolution, when the ancient Greeks’ discovery of the geometric “incommensurable,” associated with the continuum of the length of the hypotenuse of a right triangle, precipitated the first theoretical crisis.

After analyzing the relevant developments that followed the first crisis, in Section I, the discussion in Section II turns to the second revolution, when the 19th Century scientists’ discovery of the recalcitrant incompatibility of the intellectual products of the first and second revolutions, associated with their respective mathematical approaches, incorporating the ad hoc inventions of real and complex numbers, precipitated the third, or present, theoretical crisis.

Sections II through IV divide the third revolution into three parts. Section II delineates and analyzes part one of the revolution, which constitutes the beginning of the new, intuitive, reconciliation of the incommensurables, based on the fundamental assumption of the RST that space is the reciprocal of time.

Section III explains part two of the revolution, which pertains to the recognition of the role of symmetry in the logical deduction of the properties of positive and negative units of reciprocal quantities.

Section IV deals with the final part of the third revolution, laying out the last phase in the developments that lead to the new, intuitive, reconciliation of the incommensurables, where it is shown how the principle of reciprocity, with its properties of symmetry and chirality, underlies the encoding of the inevitable logic of a reciprocal system of numbers, and a corresponding reciprocal system of physical magnitudes, where the properties of their respective elements are: quantity, dimension, and polarity, enabling the intuitive construction of a general physical theory of the universe, consisting of nothing but motion, existing in three dimensions, and in discrete units.

Finally, in Section V, the findings of the book are summarized and their implications discussed.

Note: I already have some ideas for revisions, but for a good orientation in the general approach I want to take, see this excellent video by Neil Turek: "The Astonishing Simplicity of Everything" It's gratifying to note that his presentation follows the historical path of the LST development that I think we need to understand. Also, I have not included the many footnotes in this online preview that will eventually be provided, if and when the book is published.

[Djchrismac]

Note: In 2016, four years before he passed away, Bruce Peret had intended to write a book on his contribution to the development of a Reciprocal System theory (RSt), he called the RS2. However, he was never able to accomplish it.

Hi Doug, I need to correct you there, Bruce, Gopi and K.V.K. Nehru were working on an RS2 book for some time and I have a copy, i'm not sure if it is the final version or not, i'll contact Gopi to check with him as if it is finished then it would be a good time to get it posted.

However, a lot of recent developments would be worthwhile including. This was one of the stumbling blocks with the book, the more they learned and wrote down, the more the theory evolved and content had to be re-written as it was refined. What I have goes up to chapter 3 and looks like it still needs work done, i'll get back to you once Gopi confirms.

[dbundy]

Ok, sounds good. I wasn't aware that it had progressed that far. Thanks for the info. Looking forward to learning more.

[dbundy]

Note: This is the second post of the draft:

The Long and Vain Struggle to Unify Numbers and Magnitudes

Mathematics and Physics

In his book, New Foundations for Classical Mechanics, David Hestenes observes:

There is a tendency among physicists to take mathematics for granted, to regard the development of mathematics as the business of mathematicians. However, history shows that most mathematics of use in physics has origins in successful attacks on physical problems. The advance of physics has gone hand in hand with the development of a mathematical language to express and exploit the theory…The task of improving the language of physics…is one of the fundamental tasks of theoretical physics.

Yet, the most common criticism of Larson's works is that they lack a "mathematical language to express and exploit the theory." As he reports in his book Beyond Newton:

One of the most frequent comments offered by those who have become acquainted with the gravitational theory of this work through previous publications concerns the relatively minor use of mathematics in the development. “I am particularly puzzled about the lack of mathematics associated with your methods,” writes a British correspondent, “surely in order to show the superiority of your theory you must be able to predict all the experimental facts explained by present theories and more. It is difficult to see how you will do this without setting the whole thing on a rigorous mathematical basis.” Another correspondent asks, “Can you put your theories into a tensor formulation?”

Larson developed the system mathematically, but he did not explicitly develop a mathematical formalism for it. In fact, he regarded the perceived need for a formalism as a symptom of a lack of conceptual clarity, a typical response to conceptual confusion:

These comments reflect a general misconception that has developed in science, particularly in physics, within the present century, in which the “rigor” of the mathematical treatment is judged on the basis of its length and complexity, not on the basis of its adequacy for the task at hand. Following Einstein’s lead in calling upon complex mathematics in an attempt to compensate for conceptual errors, present-day physical theory has become largely a juggling of abstract mathematical relationships, the meaning of which (if any) “we do not ask,” as Eddington says. As so often happens when form is overemphasized, form rather than substance has come to be regarded as the essence. To arrive at a result in the realm of basic theory by plain arithmetic or simple algebra is today unthinkable; unless we can express that result in terms of tensors, or spinors, or matrix algebra, or some other currently fashionable mathematical device, it is automatically unacceptable.

At the same time, however, Larson recognized the intimate and crucial nature of the relationship of mathematics and physics. As he wrote in The Neglected Facts of Science:

A change in [the definition of motion that is] the base of the [new] system naturally necessitates many modifications of the details of physical theory. However, the amount of change that is required is not nearly as great as might appear on first consideration, because the new development calls for very little change in the mathematics of present-day theory. The changes are mainly in the interpretation of the mathematics, in our understanding of what the mathematics mean. Since the case in favor of the currently accepted theories is primarily–often entirely–mathematical, there is little that can be said, in most cases, in favor of current theory that is not equally applicable to the mathematically equivalent conclusions that I have reached. The substantial advantages of a fully integrated general physical theory are thus attained without any violent disruption of the mathematical fabric of the physics of familiar phenomena. All that is necessary in most instances is some alteration in the significance attributed to the mathematical relations, and a corresponding modification of the language that is utilized. These new interpretations, integral parts of a consistent, fully integrated general theory, can then be extended to a resolution of the problems that are currently being encountered in the far-out regions.

This is especially true, when one realizes that the new system doesn't supplant the current system, but rather subsumes it; that is, the base of the current system, vectorial motion, is the motion of objects relative to one another. Of course, this motion is also part of the new system, and, as long as the vectorial motion is not a significant fraction of the speed of light, the new system really has little to add to the current system in its realm of vectorial motion. On the other hand, however, the vectorial motion of the current system cannot be defined without objects. Energy, radiation and matter must be put into the current system in order to apply it to the study of natural phenomena. This is not true under the new system of physical theory.

All forms of energy, radiation, and matter emerge from the RST strictly as a consequence of its fundamental postulates. All the theoretical constituents of the universe of motion and their mutual interactions are either motions, combinations of motions, or relations between motions, including the familiar, one-dimensional, translational, rotational, and vibrational motions, the mechanical (vectorial) motions of matter.

What the RST brings to the table, so-to-speak, is a new kind of motion, hitherto unrecognized as motion, the fundamental scalar motion of the theoretical universe, manifest in the observation of the continuously forward march of time and space. Clearly, the recognition of this new type of motion doesn't change the physical laws that science has already discovered that apply to relatively low speeds of objects, and it doesn't change the usefulness of the existing mathematical language used to express and exploit these laws, but is it really reasonable to think that a new mathematical language, suitable for expressing and exploiting the laws of the new scalar motion, would not actually be helpful?

Given the mysterious nature of the relationship between mathematics and physics, what Wigner characterized as the "unreasonable effectiveness of mathematics in physics," Larson’s view that a new, formal, language is not required for his revolutionary new system, that the mathematical changes required “are mainly in the interpretation of the [existing] mathematics, in our understanding of what the mathematics mean,” seems curiously out of step with the current fascination and emphasis on mathematics. As it turns out, however, this view was extremely insightful, perhaps even prescient.

Mathematics as Science – The Science of Ancient Greece

Larson's conclusion that "the new development calls for very little change in the mathematics of present-day theory" may seem a little like deciding that a new bottle is not needed for the new wine, but if this weren’t the case, just how would one go about developing a new, formal, mathematical language to express and exploit the new system? What are the requirements of the new language? Are we looking for a revolutionary new equation, like say Einstein’s E = mc²? Do we need something that encapsulates the essence of the new system in one, succinct and glorious, algebraic expression of a fundamental new insight into nature?

Of course, such a prospect is exciting, especially to those who have embraced the new system and are anxious to share it with the world, and a few of Larson’s initial followers even made some attempt to do just that. However, it proved more difficult than imagined, because there seemed to be no way to connect the definition of motion in the new system to the definition of motion in the existing system, in any fundamental sense, at least. So, the question came back full circle, back to Larson’s view. It became, “Is there a new interpretation of fundamental ideas,” as Larson asserted, “which will increase our understanding of what the existing mathematics mean?” Although it wasn’t so clear in the beginning, looking back it’s easy to see now that the decision to seek the answers to this question was the beginning of a quest of truly heroic dimensions. It almost seems like returning to the foundations of civilization, to the world of Agamemnon and Odysseus, to look for answers.

For the ancient Greeks, mathematics was science, and there were four fields of that science:

1) Arithmetic - Numbers
2) Harmony – Vibration as ratios of numbers
3) Physics – The relative motion of heavenly bodies
4) Geometry - Physical magnitudes

There was not always consensus on the order of the fields, but it was generally thought that there was a hierarchical relationship between these four fields; that is, that numbers were fundamental, because numbers were required to express ratios, ratios that were not necessarily interpreted in terms of magnitudes of space and time, but as fundamental numbers of harmony. Clearly, geometry too depends upon numbers, but understood in terms of fundamental proportions, not in terms of magnitudes of space and time (motion).

Centuries after the Greeks, Newton clearly understood that the magnitudes of geometry are truly dependent upon the motion of physics (mechanics). Unless magnitudes of motion, with properties specifying dimension, direction, and quantity, can describe the “right lines and circles” required by geometry, geometry would not be able to do its magic. Newton wrote:

The descriptions of right lines and circles, upon which geometry is founded, belongs to mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn...and it is the glory of geometry that from those few principles, brought from without, it is able to produce so many things…

However, this leads us to ask, if harmony depends upon numbers, and if geometry depends upon mechanics, does this mean then that mechanics (physics) depends upon harmony in turn, completing the Greek hierarchy of mathematics (science)?


Figure 1. The Ancient Greek Hierarchy of Science.

Today, we know that harmony is indeed the basis of physics, in the form of simple harmonic motion (SHM). However, not in the sense that completes the hierarchy of the ancient Greek science of mathematics. Something is still missing in the modern concept of these four fundamental disciplines that perhaps prevents us from seeing the mystical relation between them that the ancient Greeks perceived. Harmony was more than the musical vibrations of strings to the Greeks. It was a cosmological concept that is connected with metaphysical meanings of the first four numbers that they called the tetraktys (tetras means four).



Figure 2. The Tetraktys

Thought to have originated with Pythagoras, but more likely to be even more ancient, the tetraktys was so sacred to the Pythagoreans that it formed the basis of their oath:

By that pure, holy, four lettered name on high, nature's eternal fountain and supply, the parent of all souls that living be, by him, with faith find oath, I swear to thee.

The dots represent the numbers 1, 2, 3, and 4, and their descent symbolizes the order of creation of the known universe, and the increasing complexity of its manifestation. The four lower dots represent the four elements; the upper dot represents the first principle. However, the mathematical concept of the tetraktys is harmony. It consists of the four fundamental ratios of vibrating strings: the fundamental ratio, or first principle, which is the vibrating frequency of a string of a given length, the second ratio, double the first, which is the vibrating frequency of a second string, half the length of the first, and two more, the fifth and the fourth, which were harmonics contained within the interval of the first and the second ratios (2/3 & 3/4 lengths).

This fact was highly mysterious to the ancient Greeks, whose observations founded the vast field of today's music of the Western world, but how is the harmony of the cosmos related to numbers? Does it depend on numbers, in a way that is similar to the dependence of geometry on physics; that is, on principles “brought from without?” Is there a glory of harmony that enables it to do many wonderful things based on just a few principles of numbers? And does this glory of harmony relate numbers to physics, through its principles, in a way never suspected in modern times? Actually, there are some tantalizing clues that this is exactly the case.

Recalling that the dots of the tetraktys represent the numbers 1, 2, 3, and 4, and the increasing complexity and order of the universe, it is clear that these numbers are counting numbers that seem to have little in common with the magnitudes of length, area and volume. Geometry deals with magnitudes of points, length, area, and volume, described by vectorial motion (mechanics, i.e. magnitudes of space and time.)

Geometric magnitudes have direction and dimension, and even the ever-forward march of time magnitude has direction in a sense, which is something numbers, as simple quantities, just do not have. Harmony too, as ratios of vibrations, is something with both space and time magnitudes, and these magnitudes of vibration do have direction and dimension, but the counting numbers have neither direction nor dimension. For this reason, Euclid was careful to keep them separate from the magnitudes of geometry, something Hestenes believes was a huge stumbling block to the development of modern civilization.

Intuition Lost – The Ad Hoc Invention of Real Numbers

Yet, there was method to Euclid’s madness, which Hestenes explains:

Today, “to measure’ means to assign a number. But it was not always so. Euclid sharply distinguished “number” from “magnitude.” He associated the notion of number strictly with the operation of counting, so he recognized only integers as numbers; even the notion of fractions of numbers had not yet been invented. For Euclid a magnitude was a line segment. He frequently represented a whole number n by a line segment, which is n times as long as some other line segment chosen to represent the number 1. But he knew that the opposite procedure is impossible, namely, that it is impossible to distinguish all line segments of different lengths by labeling them with numerals representing the counting numbers. He was able to prove this by showing the side and the diagonal of a square cannot be both whole multiples of a single unit.

Of course, it was not like Euclid was the first to discover this, but his fastidiousness, in keeping the two concepts separate, is probably evidence of how universally the “incommensurable” problem of numbers and geometric magnitudes had been recognized by then, more than two centuries after the Pythagoreans had learned it the hard way. Yet, there’s more to it than meets the eye, as Hestenes points out:

The “one way” correspondence of counting numbers with magnitudes shows that the latter concept is the more general of the two. With admirable consistency, Euclid carefully distinguished between the two concepts…This rigid distinction between number and magnitude proved to be an impetus to progress in some directions, but an impediment to progress in others.

This impediment, which resulted from the “one way” correspondence of counting numbers and geometric magnitudes, was serious enough that, eventually, it led to the first revolution in the concept of numbers. A second problem, manifest in what Hestenes’ calls a “breakdown” between counting numbers and geometric magnitudes, was also an impediment to scientific progress, and still is today, in spite of the second revolution in the concept of numbers intended to eliminate it.

The nature of the second problem is, in a sense, the inverse of the first problem. The first problem is that magnitudes appear to be more general than numbers (later we will challenge this conclusion), but the nature of the second problem is that, while numbers can easily be raised to powers greater than three, geometric magnitudes cannot, but are limited to three, or the three dimensions of space that we experience.

This “breakdown” at three-dimensions (four counting zero) is the root cause for the current woes of string theorists too, because they need geometric magnitudes of nine or ten dimensions in order to make their theories work, and while they have numbers with this many dimensions and more, they have to invent a way to find the corresponding geometric magnitudes, and they have invented so many ways to do it that they are bogged down in a landscape (swamp?) of possibilities.

However, as Hestenes points out, this same difficulty, in connection with the ancient Greek science, is not widely recognized: “This ‘breakdown’ impeded mathematical progress from antiquity until the seventeenth century, and its import is seldom recognized even today,” he writes.

Why is this? Hestenes gives us a clue:

Commentators sometimes smugly dismiss Euclid’s practice of turning every algebra problem into an equivalent geometry problem as an inferior alternative to modern algebraic methods. But we shall find good reasons to conclude that, on the contrary, they have failed to grasp a subtlety of far-reaching significance in Euclid’s work.

The subtlety has to do with the important point that Euclid regarded geometric magnitudes as more general than counting numbers, and, evidently, he felt that the differences in their properties must be appropriately taken into account, if the discrete nature of counting numbers is to be unified with the continuous nature of geometric magnitudes. According to Hestenes, the scientific failure of the ancient Greek scientists was that they didn’t develop “a simple symbolic language to express their profound ideas,” but, on the other hand, with the perspective of Larson’s new system, as will be shown below, we can now see that the scientific failure of string theorists is their failure to completely grasp the profound ideas of the Greeks.

The reason for making the distinction between the discrete counting numbers and the continuous magnitudes of line segments, and keeping them separate in calculations and proofs, as Euclid did, is easy to understand, as soon as it’s recognized that measurements of geometric magnitudes include values for which there are no counting numbers. The magnitude of the diagonal of the unit square is the simplest example. However, the other side of the coin is that there are also numbers for which there are no physical magnitudes, numbers with dimensions greater than 3 (four counting zero).

Euclid’s “sidestepping” approach, of re-expressing math problems as geometric problems, worked because he represented numerical products as geometric magnitudes, so that he could represent the product x^., as the “square” with sides of magnitude x, the product xy, as the “rectangle” xy, and the product x³, as the “cube” with sides of magnitude x. However, he couldn’t go beyond these three dimensions, any more than string theorists can, so he encountered this “breakdown” between the correspondence of numbers and physical magnitudes. Again, as Hestenes characterizes it:

…there are no corresponding representations of x ⁴ and higher powers of x in Greek geometry, so the Greek correspondence between algebra and geometry broke down. This “breakdown” impeded mathematical progress from antiquity until the seventeenth century, and its import is seldom recognized today.

But, while Hestenes cites this “geometric” approach of the Greeks as a major factor in the “long period of scientific stagnation” that occurred between the “brilliant flowering of science and mathematics in ancient Greece,” and the “explosion of scientific knowledge in the seventeenth century,” he regards it as inevitable, given that no “comprehensive algebraic system” was available until Rene Descartes was finally able to state explicitly what most, by that time, assumed tacitly. He explains:

Descartes gave the Greek notion of magnitude a happy symbolic form by assuming that every line segment can be uniquely represented by a number. He was the first person to label line segments by letters representing their numerical lengths…the aptness of this procedure resides in the fact that the basic arithmetic operations such as addition and subtraction can be supplied with exact analogs in geometrical operations on line segments.

Of course, the objective of Hestenes’ review of these fundamental developments is to show how his recognition of Clifford’s Geometric Algebra (GA), as he calls his version of it, is actually a completion in the understanding of some fundamental aspects of the modern union of counting numbers and geometric magnitudes that only started with Descartes. He acknowledges the inevitability of Descartes’ explicit articulation of the assumption and notes the independent works of others, such as Fermat, along the same lines, but then he observes:

…Descartes penetrated closer to the heart of the matter. His explicit union of the notion of number, with the Greek geometric notion of magnitude, sparked an intellectual explosion unequaled in all history.

The essence of the change that sparked this explosion is deceptively simple and subtle. At its heart, is the age-old issue of reconciling the discreteness of numerical quantity with the continuousness of physical magnitude, which is the very same issue that continues to plague mankind’s quest for fundamental understanding today. Hestenes writes:

The correspondence between numbers and line segments presumed by Descartes can be most simply expressed as the idea that numbers can be put in a one to one correspondence with the points on a geometric line. The Greeks may have believed it at first, but they firmly rejected it when incommensurables were discovered. Yet Descartes and his contemporaries evidently regarded it as obvious…Of course, such a change was possible only because the notion of number underwent a profound evolution [in the intervening centuries].

In other words, Descartes ignored the catastrophe that proved the downfall of the Pythagorean cult, in that he assumed a correspondence between numbers and geometric magnitudes, in spite of the fact that no ratio of integers can be found to represent the incommensurable magnitudes, such as the square root of 2. Looking back, it would seem like this would have been tantamount to a hue and cry, “Let the philosophical reservations of the Greeks be damned! Let’s move on!” but, according to Hestenes, this was not the case. Evidently, the contemporaries of Descartes were easily persuaded by then that this was the obvious thing to do, although Hestenes offers no explanation of how such a “profound” change came about, except that a symbol for the square root of 2,√, that identified it as a solution to x² = 2, was available by that time, and “once it had been invented, it was hard to deny the reality of the number it names,” especially when it is identified with the diagonal length of the unit square.

Certainly, the most important aspect in the “profound evolution” in the notion of number that Hestenes refers to culminated in the “arithmetization” of the number system in the 19th Century, wherein the so-called “real” numbers were defined in terms of “natural numbers and their arithmetic, without appeal to any geometric intuition of ‘the continuum.’” Regarding this crucial aspect of the “profound evolution in the notion of number,” Hestenes asserts:

Some say that this development separated the notion of number from geometry. Rather, the opposite is true. It consummated the union of number and geometry by establishing at last [the assumption] that the real numbers can be put into one to one correspondence with the points on a geometric line. The arithmetical definition of the “real numbers” gave a precise symbolic expression to the intuitive notion of a continuous line.

Emphasis has been added to the text of the above quote, because it underscores the importance of the fact that it is the universal acceptance of the concept of real numbers, as a successful “consummation” of the union of discrete numbers and the continuum of physical magnitudes, which has been established. Yet, are we safe to say that an actual one-to-one correspondence between the two, by this method of “arithmetization,” has been established, as the correct procedure to follow, given the present trouble with theoretical physics? Certainly, it remains an open question that needs to be emphasized, for, if resorting to the ad hoc invention of real numbers is the correct procedure to follow, one wonders why the issue of uniting these two concepts remains so intractable in the modern theories of physics today. Clearly, however, Hestenes, and almost everyone else in the current mathematics and physics community, regards this reconciliation, based on the arithmetization procedure, as settling the issue once and for all.

Nevertheless, reconciling the discrete quantities of natural numbers with the continuous spectrum of real numbers, addresses only part of the deficiencies in the notion of number vis-à-vis the notion of magnitude. As Hestenes summarizes it:

Descartes began the explicit cultivation of algebra as a symbolic system for representing geometric notions. The idea of number has accordingly been generalized to make this possible. But the evolution of the number concept does not end with the invention of the real number system, because there is more to [geometrical magnitudes] than the linear continuum.

Then, Hestenes identifies the two issues remaining to be addressed, and makes a promise:

In particular, the notions of direction and dimension cry out for a proper symbolic expression. The cry has been heard and answered.

Consequently, Hestenes proceeds to demonstrate, in his work, that the idea of continuous magnitudes, with their properties of direction and dimension, can be symbolically expressed by further generalizing the concept of number. This is accomplished by exploiting the work of Grassmann and Clifford to introduce a concept of “directed” numbers, together with a concept of a new “geometric product” that consists of the combination of the inner product, analogous to the Greek concept of projection, or what we might call the Euclidean inner product, and Grassmann’s outer product, a key innovation that enables the algebraic expression of magnitude with four properties:
1. Quantity
2. Direction
3. Dimension
4. Orientation

The addition of “orientation” to the three universally accepted properties of magnitude, i.e. quantity, direction, and dimension, is recognition of the importance of polarity, as an additional property of magnitude, which will be discussed in more detail later on.

First, however, pausing to analyze what has been discussed to this point, it’s clear that in Hestenes' view, the careful and deliberate separation of the concepts of algebraic numbers and geometric magnitudes, by the ancient Greeks, the separation of discrete and continuous magnitudes, as it were, was overcome through a centuries long effort to “generalize” the concept of discrete counting numbers to accommodate the concept of continuous geometric magnitude, as required to enable the algebraic manipulation of geometric magnitudes.
That is to say, a way was found to directly connect the science of numbers, which formed the foundation of Greek science, to the science of geometry, which constituted the pinnacle of Greek science, without having to resort to the cumbersome procedure of Euclid in which he proved many theorems twice, once for numbers, and once for magnitudes.

But, while this “profound revolution...sparked an intellectual explosion unequaled in all history,” it also destroyed the harmony of the Greek’s scientific hierarchy, illustrated in figure 1 above. In fact, Hestenes notes that the ancient Greek’s would not have let Descartes get away with “his explicit union of the notion of number, with the Greek geometric notion of magnitude….” For example, notes Hestenes, “A careful Greek logician, like Eudoxus, would have demanded some justification for such a far reaching assumption.”
Clearly, the implication is that the Greek notion of the scientific hierarchy was misguided and irrelevant. Hestenes’ view is that, while the compunction of the Greeks, their misgivings over accepting a concept of “unreal” numbers as “real” numbers, so-to-speak, actually impeded scientific progress for centuries, it turned out to be innocuous in the end anyway. This is because the necessary ideas of number in terms of fractions, decimals, and symbolic expression, had to be developed and accepted, before the invention of real numbers could be understood as the means for consummating the union of discrete numbers and continuous magnitudes.

No harm, no foul, we might say today. By the time the “arithmetization” of the continuum was formalized in the 19th Century, ironically crowning Cantor and company’s “real” numbers, the very numbers which the Greeks would have regarded as “unreal” numbers, the long-sought “consummation” of the union of numbers and the continuum was complete, without regard to the cosmological significance of the relation of numbers to harmony, the relation of harmony to physics, or the relation of physics to geometry. It simply relegated the revered science of integers, which the ancient Greeks understood, to the rear car of quaint relics, on the railroad express to modern civilization.
Although not everyone was willing to go along for the ride, without protesting on philosophical grounds, that the arithmetization procedure separates mathematics from geometry, Hestenes will have none of it, as already noted above.

Of course, the “some” to whom Hestenes is mostly referring, is Leopold Kronecker, and those who, with him, believed that “God created the integers, all else is the work of man.” In the MacTutor History of Mathematics article on Kronecker, we read:

Kronecker believed that mathematics should deal only with finite numbers and with a finite number of operations. He was the first to doubt the significance of non-constructive existence proofs. It appears that, from the early 1870s, Kronecker was opposed to the use of irrational numbers, upper and lower limits, and the Bolzano-Weierstrass theorem, because of their non-constructive nature.

According to the article, Kronecker first protested publicly against the prevailing trend to accept an axiomatic “consummation” of integers and the continuum, in 1886. He said that he opposed

... the introduction of various concepts by the help of which it has frequently been attempted in recent times (but first by Heine) to conceive and establish the "irrationals" in general. Even the concept of an infinite series, for example one which increases according to definite powers of variables, is in my opinion only permissible with the reservation that in every special case, on the basis of the arithmetic laws of constructing terms (or coefficients), ... certain assumptions must be shown to hold which are applicable to the series like finite expressions, and which thus make the extension beyond the concept of a finite series really unnecessary.

Needless to say, notwithstanding the protests of Kronecker, and the ancient Greeks, to the contrary, the end seems to justify the means to the minds of most, even in the esoteric field of mathematics and philosophy. The “purist” views of Kronecker were preserved as the branch of logic called Intuitionism, but for the majority of the math and science world, the train had left the station.

Formalism Embraced – Real, Complex & Transcendental Numbers

In his battle with Kronecker, Cantor found an ally in Dedekind, and the work of replacing the intuitive foundations of geometry with the axiomatic foundations of formalisms commenced whole-heartedly. Actually, this is a great oversimplification, but the real story of the development of “the extension beyond the concept of a finite series,” is a “labyrinth of thought,” as explained by Jose Ferreirós Dominguez in his book by the same name. Ferreirós writes:

What was needed was a satisfactory theory of the real numbers, establishing the continuity (completeness) of R …The traditional definition of real numbers relied on the notion of magnitude [as something that may be augmented or diminished]. Number was the ratio or proportion between two homogeneous magnitudes…such an approach had its shortcomings: it did not account for complex numbers or even negative numbers, and, above all, the continuity of R was neither justified, nor explicitly required.

But Ferreirós explains that the axiomatic basis for real numbers, attributed to Cantor and Dedekind, was actually based on a something Hilbert later referred to as “a ‘genetic’ approach, when he proposed the axiomatic approach instead, as more convenient and precise.” The reason why Ferreirós makes this point is telling:

…the fact that the foundations of analysis required an axiom of continuity (or completeness in modern terminology) only became clear after the publications of Cantor and Dedekind in 1872….”

What eventually became an “axiomatic” paradigm of real numbers was initially developed on a “constructive” paradigm of real numbers, because both Cantor and Dedekind initially “regarded arithmetic as a development from the laws of pure thought, as a part of logic…” Therefore, “What they did,” writes Ferreirós, “was in essence to assume the rational numbers as given, and build the system of real numbers on top of the rationals, by means of certain infinitary ‘constructions.’”

The reason that this “genetic” approach was preferred over the axiomatic approach is that axioms were regarded as “true propositions that cannot be proven.” Thus, Ferreirós explains, it “lead them to the idea that arithmetic needs no axioms – everything can be rigorously proven starting from purely logical notions.” He goes on:

Had they simply postulated the basic properties that were needed in order to obtain the desired theorems, in arithmetic and analysis, the question might have arisen, why not talk about points or anything else, instead of numbers? There had to be something that all the different number systems shared, and 19th-century mathematicians looked for a common genealogy, a ‘genetic’ process by which the more complex systems emerge from the simpler. The means by which the genetic process took place was some infinitistic ‘construction’ that in retrospect can be characterized as set-theoretical (although in all cases one had to work with sets of some more or less complex structure).

Of course, the continuing tension between discrete numbers and continuous magnitudes is inherent in all of this. The points of magnitude were intuitively understood to be infinite, as in Zeno’s paradox. However, while numbers, as discrete labels of quantity, could never be divided finely enough to not be discrete, Cantor and Dedekind eventually convinced their contemporaries that, abstractly, a one-to-one correspondence could be established between the infinite points of magnitude and the discrete quantities of numbers, but the inherent paradoxes and doubts of skeptics would never go away.

The important point to understand is that, in the words of Ferreirós, “…the emergence of set theory was the result of a mostly mathematical development, although philosophical ideas played an important motivating role in it.” In other words, even if intuition and the laws of pure thought motivated Cantor and Dedekind, in the end it was a fact that they could not establish the crucial “genetic” connection with philosophy, which they sought in the beginning. Yet, as clear as this is in retrospect, they couldn’t bring themselves to explicitly recognize the ultimate axiomatic basis of their work. Hilbert had to do it for them.

In the beginning, in the late 1850’s, Dedekind “devoted himself exclusively to the question of the foundations of the numbers system for a few months, until he formulated his definition of the real numbers by means of cuts.” This turn in the direction of his thinking was justified on the grounds that the “notion of set” is philosophically acceptable, since it is a “product of our mind.”

Nevertheless, the idea of a product of the human mind, an ad hoc invention, though similar to the logical “development from the laws of pure thought,” has no inductive basis. It is a formulation of the mind that is independent of the necessary consequences that follow from the philosophical foundations of thought. It is always easier to resort to the expediency of such inventions, so that the investigators can continue to progress in the development of ideas that it seems are unyielding to the power of logical deductions, but it comes at a high price, one, perhaps, modern society is still paying.

Ferreirós points out that Dedekind’s earliest writings indicate that his motivation to seek a genetic development of real numbers was probably due to Hamilton’s influence. He read Hamilton’s Lectures on Quaternions and saw there a “key example of the genetic method” as it applied to treating the use of imaginaries in complex numbers as pairs of real numbers, i.e. the use of algebraic couples to define complex numbers. “Thus,” writes Ferreirós, “…Dedekind could regard the problem of the complex numbers as satisfactorily solved, while the reals still posed a difficult problem.”

The rest is history, as they say, because it was Dedekind and Cantor’s work, based on the ad hoc invention of infinite sets, which eventually prevailed as the solution to the difficult problem of the reals. The reason is anybody’s guess, at this point in time, but Ferreirós points out what is now only an obscure footnote in mathematical history:

Hamilton “regards algebra as the science of pure time…. Mathematically, reference to the continuity of intuited time eliminated the problem of the existence of irrational numbers and the need to pin down the difficult idea of the continuum. Time is treated as a given, and [therefore] many basic propositions of arithmetic follow from its properties, as happens with the law of trichotomy (given two temporal instants one has one and only one of the relations a=b, a<b, a>b). Philosophically, Hamilton obtained a beautiful scheme in which geometry emerged from the pure intuition of space, and algebra [emerged from the pure intuition] of time (the two pure forms of intuition, according to Kant).

If this is so, one wonders why Hamilton’s ideas didn’t appeal more to Dedekind and his contemporaries, and especially, later on, to Kronecker, who was so opposed to Cantor and Dedekind’s non-intuitive (actually counter-intuitive) formulations of infinite sets and maps?

The answer to this question probably lies in the fact that Kronecker was the conservative defender of contemporary orthodoxy at the time, while Cantor was the upstart innovator. Kronecker was not looking for an alternative, Cantor was. As is usually the case, however, there was someone to champion the cause of the underdog, and in this case it was Hilbert. Hence, an interesting and absorbing intellectual battle formed among these contemporaries that lasted well into the 20th Century, garnering the riveted attention of the intellectual elite.

Meanwhile, Hamilton’s philosophically rooted work on algebra, as the science of time, was hard to understand, and in any event was eclipsed by the battle over the practical utility of Hamilton’s non-commutative quaternions versus the commutative vector algebra of Heaviside and Gibbs.

Clearly, as discussed earlier, Hestenes discovered the gem of Clifford’s work, in combining the quantitative interpretation of Grassmann’s outer product, and the operational interpretation of Hamilton’s ordered pairs, and used it to pull a powerful new mathematics out of the obscurity of past neglect. Perhaps, this portends that something similar can happen, if we now look back with a renewed appreciation of Hamilton’s algebra, as the science of time, though its luster be ever so encrusted with the “dubious philosophical dressing” with which historians have painted it for more than a century and a half.

Looking back, it’s clear that Cantor’s work on transfinite set theory in the last quarter of the 19th century impacted philosophers and theologians, as well as generations of mathematicians. Kronecker led the opposition to his former protege's work, because he was convinced that only the natural numbers were real mathematical entities. According to his friend and later foe, Weierstrass, “for Kronecker it was an axiom that equations could exist only between whole numbers.” He opposed imaginary numbers, irrational numbers, and the concept of infinite series, due to their lack of reference to natural numbers, regardless of their self-consistency.

However, in all this argument over infinity, the infinity of an unimaginable potential increase of natural numbers versus the infinity of an actualized increase of numbers that could be symbolized and identified, the idea was to deal with magnitude, the infinite parts of a whole. Cantor’s idea was to consider the magnitude of infinity and give it a symbol, and like the symbols identifying irrational numbers, they were found to be consistent and very useful in describing the physical world, but Kronecker objected on grounds that they were not natural.

Yet, what are “natural” numbers? Cantor and company could give many examples where restricting mathematics to the counting numbers limited its usefulness, prompting many to regard Kronecker’s view as too conservative and anti-progressive, relative to the innovative and progressive work of Cantor. The following quote is typical:

The full application of each view on mathematics seems to spell the difference between the mindsets of these two men the clearest. Cantor’s view is far more progressive, allowing mathematics to be freer than Kronecker’s restrictive mindset. While Kronecker’s opinion attempts to suffocate mathematics with arbitrary standards of rigor, Cantor’s is a comparative breath of fresh air. David Hilbert once claimed that “no one shall expel us from the paradise that Cantor has created for us.” Cantor’s mathematics is indeed a paradise in comparison with Kronecker’s.

Today, however, going on a century and half later, there is serious trouble in paradise. It has become clear in the 21st Century that the concept of Cantor’s transfinite numbers leads to Leibniz’s “labyrinth of the continuum,” just as he warned it would, way back in his day. “No one,” he wrote, “will arrive at a truly solid metaphysics who has not passed through that labyrinth.” Today, Gregory Chaitin describes the concept of the continuum in mathematics as “a Swiss cheese.” He writes:

Why should I believe in a real number if I can’t calculate it, if I can’t prove what its bits are, and if I can’t even refer to it? And each of these things happens with probability one! The real line from 0 to 1 looks more and more like a Swiss cheese, more and more like a stunningly black high-mountain sky studded with pin-pricks of light.

Chaitin turns to probabilistic methods to prove that the works of Emile Borel, Alan Turing, and others reveal that a fixed, or formal, axiomatic system, a la Hilbert, cannot uniquely identify a specific real number chosen at random. Such a number cannot be specified, defined, or even referred to. This is tantamount to an “incompleteness theorem” in which it is shown, as Gödel shows, that, contrary to Hilbert’s conviction, no fixed, or formal, axiomatic-based system can be proven. Chaitin observes:

In his book Everything and More: A Compact History of Infinity, David Foster Wallace refers to Gödel as “modern math’s absolute Prince of Darkness” and states that because of him “pure math’s been in mid-air for the last 70 years.” In other words, according to Wallace, since Gödel published his famous paper in 1931, mathematics has been suspended hanging in mid-air without anything like a proper foundation. It is high time these dark thoughts were permanently laid to rest. Hilbert’s century-old vision of a static completely mechanical absolutely rigorous formal mathematics, was a misguided attempt intended to demonstrate the absolute certainty of mathematical reasoning. It is time for us to recover from this disease!

Thus, it is clear that there exists a philosophical crisis in modern mathematics, just as it exists in theoretical physics, and for the same reason: We have become lost in the “labyrinth of the continuum.” Attempts to reconcile discrete numbers with continuous magnitudes, via ad hoc inventions of the human mind, such as irrational numbers, imaginary numbers, complex numbers, transcendental numbers, etc., while successful in many practical ways, as witnessed by the marvels of modern technology, have ultimately led to confusion in modern philosophy, mathematics, and physical theory.

Thus, we are yet compelled, many generations later, to ask ourselves, “What is wrong?” Perhaps numbers were not made to be unified with space alone, and maybe there is something missing in our understanding of the continuum of space magnitudes! It is an interrogative of epic proportions that has grown to a crescendo of controversy in our own time. Unfortunately, the one intellect to perceive its liberating answer failed to persuade generations of brilliant minds to seriously consider it. His name was Sir William Rowan Hamilton.