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:

"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).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.

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:

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.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!

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:

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: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.

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!”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.

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:

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

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 remarkable thing about observables is that they often correspond to

symmetries. This connection is much more direct in quantum mechanics

than in classical mechanics.

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:

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.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”.

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:

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.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).

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:

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

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.