The Larson Research Center

In their books, Lee Smolin and Peter Woit characterize string theory and its accompanying ad hoc inventions, such as its extra dimensions of space and its landscape of solutions, as symptoms of the actual trouble with physics.  As usual, however, the trouble itself is more interesting than the symptoms.

Fundamentally, the trouble with physics comes down to one question: how can nature be both discrete and continuous, at the same time?  Of course, there are many other fundamental physics questions. Such questions as, What is the answer to the foundational problems of quantum mechanics? How can particles and forces be unified? How can general relativity and quantum mechanics be unified? Why are the free constants of the standard model what they are? What is dark matter and dark energy? are all noted by Smolin in his book.  Nevertheless, all of these problems are probably related to the root problem of understanding how nature can be discrete and continuous, simultaneously.

In the mind of Smolin, and that of most theoretical physicists in the LST community, this problem of fundamental unification is understood as the problem of inventing a theory of quantum gravity.  String theorists approach it from the point of view of the invention of particle physics, turning the invention of a dimensionless entity, a point, into an invention of a dimensionful entity, a string, and seeking to integrate the theory of gravity, which appears to come easily from this approach, with a suitable theory of particles, which is much more difficult to attain. 

Non-string theorists, on the other hand, who believe that the string approach is doomed, are also pursuing an integrated theory of quantum gravity, from the point of view of point particles, but, in their inventions, the theory of particles appears to come easily, while the theory of gravity is the much more difficult aspect of the challenge.

Then there is a smaller group of theorists, to which Smolin belongs, who believe that the lesson to be learned from Einstein is that the problem of quantum gravity should be approached from the point of view of the invention of general relativity. For members of this group, it’s not a matter of integrating existing discrete, or continuous, theories, but rather coaxing both theories out of the “ether,” based on Einstein’s invention of spacetime.  The latter approach is admittedly a much more risky and demanding approach, which probably explains why this group is so small.

While these three approaches are very different, they still have a lot in common.  First and foremost, they are all examples of inventive science, not inductive science. Inductive science, as we’ve discussed before, is more rigorous than inventive science, because it must stem from the consequences of hypotheses formed from observation. It does not enjoy the liberty of “free invention.” Nevertheless, the approaches based on inventions are all successful to one degree or another, unifying at least some of the discrete and continuous aspects of nature, and thus their respective adherents eventually become crusaders for their own approach.  That this situation can lead to some very unflattering behavior has become the fodder for physics blogs in the blogosphere that are sometimes not unlike esoteric versions of what used to be called “yellow journalism,” but is now known popularly as the tabloid press, or even just “the tabloids.”

Ironically, however, it’s the blogs that have given the non-professionals greater insight into the nature of the extremely technical issues related to the different approaches to quantum gravity.  Indeed, since even the professionals are non-professionals outside their own speciality, the blogs have greatly enriched the understanding of these issues among the pros, as well. In Woit’s latest post, entitled “Too Much Good Stuff,” he illustrates this new phenomena and identifies an associated dilemma it raises:

I’ve been finding recently that an increasing serious problems with blogs is that there are too many good ones with material worth reading. I’ve learned quite a lot recently from many well-informed blog postings, but the sheer number of these makes it hard to find the time for other things one should be doing.

Of course, because of his stand against the abuses of the string theory community and his book on the subject, his own blog, like several others that are similar, has become a focal point in the interaction between the different camps of theorists, or at least a good reflection of that interaction, with the regular contributors, becoming a lot like the paparazzi of an analogous tabloid, watching and reporting on the moves of the star players in the physics drama.

Thus, the Internet, and the world-wide-web developed on it, is changing the rate of interaction between the scientists allied with the different camps of followers in the three approaches to the theoretical invention of quantum gravity. 

Interestingly enough, however, Peter is not a professional theoretical physicist, but a professional mathematician with a heavy interest in theoretical physics - a virtual physicist, so-to-speak, who is able to say and do a lot of things that a leading real physicist, like Smolin, can’t say or do, like blogging, without causing a lot of professional discontent among his colleagues. 

Without virtual physicists, those of us who are neither professional mathematicians nor professional physicists would have very little access to the insider world of LST physics, but with these guys, the esoteric world that used to be confined to the halls and conference rooms of academia, is now open to the common man on the Internet. 

Perhaps the prototype of the virtual physicist is John Baez, who’s blog is often referred to as the proto-blog for physicists.  John turned from his theoretical physics work to work in mathematics, because he felt that the progress of theoretical physics had stagnated, but then he found that studying mathematics renewed his interest in physics, because of the intriguing connections that were uncovered by the mathematics of obscure, but fascinating patterns in topology, abstract algebra, and group theory.

Like most people who have something to say about these very technical and involved subjects, John is a very smart guy.  Us non-professionals can learn a lot from them about the unification debacle in modern physics, even though many times their “simplified” explanations are still nauseatingly complex.  A very good example of this is the latest entry on John’s blog, pointed to by Peter on his blog.  It’s number 253 in “This Week’s Finds in Mathematical Physics.”

In Week 253, John explains his friend Garrert Lisi’s innovative work on unification that constitutes a new direction:

Garrett is a cool dude who likes to ponder physics while living a low-budget, high-fun lifestyle: hanging out in Hawaii, surfing, and stuff like that. He recently won a Foundational Questions Institute award to think about ways to unify particle physics and gravity. That’s an institute devoted precisely to risky endeavors like this.

Lately he’s been visiting California. So, before giving a talk at Loops ‘07 - a loop quantum gravity conference taking place in Mexico this week - he stopped by Riverside to explain what he’s been up to.

Briefly, he’s been trying to explain the 3 generations of elementary particles using some math called “triality”, which is related to the octonions and the exceptional Lie groups. In fact, he’s trying to use the exceptional Lie group E8 to describe all the particles in the Standard Model, together with gravity.

It’s fascinating to see how these guys jet all over the world to listen to each other’s ideas in conferences and coffee shops.  When I first began to read the above entry on John’s blog, I was excited with anticipation of learning something really insightful.  Unfortunately, however, I was once again disappointed, as I became lost in the complexity of this stuff, which is just mind boggling to me, even with John’s patient and articulate explanations of what they are doing. 

I’ve had this kind of experience so many times before you would think I would get used to it, but, no, it still leaves me with the depressing feeling of exclusion.  Exclusion based on an inability to follow what they are talking about and the inertial mass of it all. It just seems so dense and obtuse, as to be practically impenetrable, at least in a reasonable lifetime. 

However, one wants to believe that the unification of nature simply can’t be based on these elaborate and complex principles that are so esoteric.  Whether it’s the ornate and baroque inventions of the string theorists, the less baroque, but just as abstract inventions of the particle theorists, or the very bizarre inventions of the loop theorists, the conclusion is the same: Isn’t there a simple inductive principle that we can discover that will return us to the good old days of inductive science, which will serve as the basis of a reasonable hypotheses of unification, instead of these learned Rube Goldberg-like contraptions of the LST physicists?

Of course, our firm conviction here at the LRC is that the answer to this important question is definitely yes. The beginning of this understanding is found in Larson’s work, where the principle of reciprocity unveils a whole new foundation upon which the unification of the discrete and continuous aspects of nature are explainable on a grade school level.  As a dramatic illustration of what I mean, I will shortly be sharing the details of a new and exciting development along this line, on The New Math blog.

Stay tuned.

In discussing in this blog Lee Smolin’s new book, The Trouble with Physics: The Rise of String Theory, the Fall of Science, and What Comes Next, we are trying to characterize modern theoretical physics research in general, in what we refer to as the legacy system of physical theory (LST). Smolin’s book, as well as the book, Not Even Wrong: The Failure of String Theory and the Continuing Challenge to Unify the Laws of Physics, by Peter Woit, which is written along the same lines, has stirred up considerable controversy, especially among string theorists, who constitute by far the largest segment of LST theorists in the world today.

However, as these authors point out, the string theory controversy is only symptomatic of the central problem: The crisis in theoretical physics revolves around the fundamental issues that string theory was meant to address, and, thus, its failure, and especially the refusal to recognize its failure on the part of the largest segment of theoretical physicists in the LST community, implies a serious pathology in their ranks.

Unfortunately, most of the string theory community, acting more like a beleaguered government bureaucracy, than an eminent scientific community, have initially tended to defend the implications of the crisis, rather than to take their lumps and to begin to engage in an honest investigation of the probable causes involved. Nevertheless, it seems inevitable that this multi-billion dollar a year enterprise must face the music sooner or later: The truth is that the string theory approach has only managed to exacerbate the already shaky ground of the LST efforts to understand the structure of the physical universe.

Yet, Smolin’s thesis that the crisis has developed into epic proportions from historical roots that hearken back to Newton’s and Leibniz’s debate over the nature of space, which he establishes quite convincingly, only seems to irritate the string theorists whose own point of departure is that the concept of string vibrations solves the fundamental problem that the inverse square law introduces into particle interactions, and, in so doing, leads to a consistent theory of quantum gravity.

Thus, we see that the new definitions of space and matter lie at the heart of the crisis. For Smolin, the major fundamental point to be clarified is that space does not exist a priori, as a fixed structure, to provide the required background for motion, while for string theorists, the major fundamental point to be clarified is that matter does not consist of spaceless point entities, but spatially extended entities. Therefore, the Smolin vs.string theorists controversy actually boils down to an argument of whether it is better, in addressing the fundamental crisis of physics, to redefine space, as a dynamic network of discrete events, satisfying the postulates of geometry, or whether it is better to assume a fixed background and redefine matter instead, as extended objects, removing the infinities caused by the inverse square law.

Smolin points out the challenges and inconsistencies encountered in attempting to redefine matter, as string theorists have done, with their need for extra dimensions of space, and the need to cope with the landscape, etc, and string theorists point out the limited scope of the attempts to redefine the concept of space in order to devise a theory of quantum gravity that doesn’t connect with the force concepts of the standard model. The result is the so-called string wars now raging in the blogosphere and even in the printed media press, where much more heat than light is being generated.

Lately, the controversy has even spread into the field of cosmology, where the LST concept of the hot big bang is being challenged by physicists who are convinced that M-theory, with its membranes of space-time existing in unseen dimensions, allows them to explain the big bang as a recurring event in an endless cycle of brane collisions (see here).  

In the meantime, the pink elephant in the room that everyone is ignoring is that the common understanding of the edifice of LST physics, built on the foundations of relativity and quantum mechanics, which Smolin characterizes as the “unfinished revolution,” is plagued by what Hrvoje Nikoli´ calls “myths.”  Hence, given these problems in our understanding of the foundation of physics, the seemingly endless speculations and inventions of modern physicists, in their attempts to integrate a theory of quantum gravity into the current picture of the standard model and the big bang, and thus finish the spacetime revolution, seem misguided, if not pathetic.

Carlo Rovelli, a colleague of Smolin’s, in a paper entitled “Unfinished revolution,” describes the vision of LST theoretical physics as follows:

At the beginning of the XX century, General Relativity (GR) and Quantum Mechanics (QM) again began reshaping our basic understanding of space and time and, respectively, matter, energy and causality —arguably to a no lesser extent [than the Copernicus/Newton revolution].  But we have not been able to combine these new insights into a novel coherent synthesis, yet. The XX century scientific revolution opened by GR and QM is therefore still wide open. We are in the middle of an unfinished scientific revolution. Quantum Gravity is the tentative name we give to the “synthesis to be found”.

This view, that a synthesis of GR and QM, as a theory of quantum gravity, is now the primary quest of LST physics, implies that these two theories, that have “reshaped our basic understanding” of space and time, and matter, energy and causality, provide a solid foundation to build that theory of gravity upon. Indeed, Rovelli sums up the lesson to be learned as follows:

After a century of empirical successes that have only equals in Newton’s and Maxwell’s theories, it is time to take seriously GR and QM, with their full conceptual implications; to find a way of thinking [about] the world in which what we have learned with QM and what we have learned with GR make sense together —finally bringing the XX century scientific revolution to its end. This is the problem of Quantum Gravity.

But if our common understanding of “what we have learned with QM, and what we have learned with GR,” is plagued with “myths,” what are the chances of making it all “make sense together” by taking it seriously? In the abstract of his paper, “Quantum Mechanics: Myths and Facts,” Nikoli´ explains what he means in using the word myth in this connection:

A common understanding of quantum mechanics (QM) among students and practical users is often plagued by a number of “myths”, that is, widely accepted claims on which there is not really a general consensus among experts in foundations of QM. These myths include wave-particle duality, time-energy uncertainty relation, fundamental randomness, the absence of measurement-independent reality, locality of QM, nonlocality of QM, the existence of well-defined relativistic QM, the claims that quantum field theory (QFT) solves the problems of relativistic QM, or that QFT is a theory of particles, as well as myths on black-hole entropy. The fact is that the existence of various theoretical and interpretational ambiguities underlying these myths does not yet allow us to accept them as proven facts. 

In the introduction of his paper, Nikoli´ carefully explains the difference between the oft-cited success of QM and the problematic nature of its foundational concepts:

On the technical level, quantum mechanics (QM) is a set of mathematically formulated prescriptions that serve for calculations of probabilities of different measurement outcomes. The calculated probabilities agree with experiments. This is the fact! From a pragmatic point of view, this is also enough. Pragmatic physicists are interested only in these pragmatic aspects of QM, which is fine. Nevertheless, many physicists are not only interested in the pragmatic aspects, but also want to understand nature on a deeper conceptual level. Besides, a deeper understanding of nature on the conceptual level may also induce a new development of pragmatic aspects. Thus, the conceptual understanding of physical phenomena is also an important aspect of physics. Unfortunately, the conceptual issues turn out to be particularly difficult in the most fundamental physical theory currently known – quantum theory.

The “particularly difficult” conceptual issues of QM form the number 2 problem of the five, “famously intractable,” problems of theoretical physics, which Smolin articulates in his book.  It is second only to the discrete vs. continuous problem, the number 1 problem of theoretical physics.  As he describes it, this number 2 problem is that, in QM, the structure of the physical universe cannot be described as something that exists independently of the observer.  Smolin writes on page 7 and 8 of his book:

Since quantum theory was first proposed, a debate has raged between those who accept this way of doing science and those who reject it. Many of the founders of quantum mechanics, including Einstein, Erwin Schrodinger, and Louis de Broglie, found this approach to physics repugnant. They were realists.  For them quantum theory, no matter how well it worked, was not a complete theory, because it did not provide a picture of reality absent our interaction with it. On the other side were Niels Bohr, Werner Heisenberg, and many others. Rather than being appalled, they embraced this new way of doing science…

This whole issue goes under the name the foundational problems of quantum mechanics. It is the second great problem of contemporary physics.
    Problem 2: Resolve the problems in the foundations of quantum mechanics, either by making sense of the theory, as it stands, or by inventing a new theory that does make sense.

Unfortunately, however, the schoolmen don’t teach the subject this way, but rather tend to gloss over the problem and this is why Nikoli´ uses the word myth in his paper.  He explains:

Textbooks on QM usually emphasize the pragmatic technical aspects, while the discussions of the conceptual issues are usually avoided or reduced to simple authoritative claims without a detailed discussion. This causes a common (but wrong!) impression among physicists that all conceptual problems of QM are already solved or that the unsolved problems are not really physical (but rather “philosophical”). The purpose of the present paper is to warn students, teachers, and practitioners that some of the authoritative claims on conceptual aspects of QM that they often heard or read may be actually wrong, that a certain number of serious physicists still copes with these foundational aspects of QM, and that there is not yet a general consensus among experts on answers to some of the most fundamental questions. To emphasize that some widely accepted authoritative claims on QM are not really proven, I refer to them as “myths”.

Attempting to build “a theory of everything” from the concept of quantum gravity, which is built on string theory, or from the concept of quantum gravity, which is built on loop theory, given the shaky foundations of quantum mechanics, is just foolish, it seems to me.  We need to know why QM works, even though the concepts of “wave-particle duality, time-energy uncertainty relation, fundamental randomness, the absence of measurement-independent reality, locality of QM, nonlocality of QM, the existence of well-defined relativistic QM, the claims that quantum field theory (QFT) solves the problems of relativistic QM, or that QFT is a theory of particles,” are fundamental concepts that “may be actually wrong.”

If we don’t solve these problems first, how in the world can we think that finding a “synthesis” of GR and QM will ever be possible? And this honesty about the foundations of quantum mechanics is only part of what we need to consider.  What about the foundations of general relativity?  Can we really feel secure in believing that spacetime is something that exists as a dynamic field, fully capable of being warped and dragged about, like a rubber sheet, but not observable in any manner other than in its effects, or could this too be just another myth in our common understanding?

It appears that there is much more to “finishing the revolution” than synthesizing GR and QM into a new theory of quantum gravity. What is really needed is a much more fundamental revolution than we have in GR and QM. In the RST-based theoretical development, both space and matter have been redefined, and the result is that a new vision of the structure of the physical universe emerges, where the paradoxes of particle/wave duality and the existence of a measurement-independent reality not only arise in a natural and intuitive manner, but also provide tantalizing clues to the mystery of gravity.

Thus, it’s not that we have to finish the revolution that was begun in the twentieth century, but that we have to understand it for what it really is: The initial attempt to deal with problem number 1, which has resulted in problem number 2. 

Lee Smolin’s book is doing very well in the UK. One of the reasons is probably the exposure due to the ongoing debates between Smolin and string theorists organized at places like Oxford and the Royal Society of Arts (RSA), where Lee’s conclusions are frankly challenged in a typical English manner. Obviously, it makes for good publicity, but it also reflects the general interest in fundamental questions, I believe, even though the fundamental questions discussed most vigorously in these debates extend to those of a philosophical and a sociological nature, more than those of a physical nature.

In the RSA debate (mp3 audio file), chaired by Chris Isham, two professors, Michael Duff, a string theorist, and Nancy Cartwright, a philosophy professor, address issues raised by the book from the perspectives of offended physicists in the string theory community (Duff) and of a questioned assumption in the philosophical community (Cartwright).

Smolin’s defence is mostly to contend that the offense taken by string theorists, delineated by Duff, was not intended, and that the questioned assumption, raised by Cartwright, regarding the unification of nature, constitutes the gauntlet that nature herself has thrown down. However, he makes it clear that he is convinced that the discovered duality between the one-dimensional string objects and gauge theory is the deep, “core,” idea, as he puts it. Nevertheless, the current formulation of the idea in the context of a background-dependent framework, necessarily leading to nine or ten spatial dimensions and supersymmetry, is the “big bet,” he says. The issue of the unification of nature through higher dimensions and the challenge of how to treat the geometry and stability of the extra dimensions has “been in play” since the early 1920s, he notes, and without success.  Then he asserts something remarkable.  He states that with the

…development of the same idea [i.e the core idea of string theory] in a background-independent context, without the picture of strings moving in a classical geometry, but taking the whole geometry quantum mechanically, as the degrees of freedom that arise out of the dynamics of these extended objects, you get loop quantum gravity and spin foam models, precisely on the nose, that’s what they are.

But this comment, arguably the most valuable one in the whole debate, was his last utterance, at the end; it was not the point of departure at the beginning of the debate, as it should have been.

At one point, near the middle of the debate, Duff insisted that the triumph of string theory in uniting the particles of the standard model with gravity, was its chief justification, since no other theory has done that, but, in his book, Smolin clearly states that this is no longer true, that the other quantum gravity approaches, such as loop quantum gravity, have also succeeded in this. On page 254, he writes:

…one thing is clear. String theory is no longer the only approach to quantum gravity that also unifies the elementary particles…results suggest that many of the background-independent quantum theories of gravity have elementary particles in them as emergent states. And a given theory does not lead to a vast landscape of possible theories. Rather, it shows promise of leading to unique predictions, which will either be in agreement with experiment or not…Science done the old fashioned way is moving ahead.

This was news to me and I’m sure to most others, but because of all the fog raised by the offended string theorists, this point is not being made emphatically enough.  The loop quantum gravity approach and its associated variants, differ from the string theory approach in that they don’t require extra dimensions of space, in a fixed background that can be configured in a gazillion ways.  It is a background-independent approach that redefines space, and the geometry of space, dynamically.

However, the difference is that the particles in loop quantum gravity are emergent and they emerge from the way three 1D objects can be braided, which is something that Smolin and his colleagues learned from an Australian named Sundance Bilson-Thompson.  In his work, Bilson-Thompson shows that, using a toy model, one can represent fundamental objects with helons, which are 1D ribbons with left or right or no twists.  Ribbons with left twists are the inverse of ribbons with right twists, and, together with ribbons that have no twists, they form a group.  Thus, the same three part 1D structure that string theory uses, with its two end points and a vibration between them, analogous to the three part SUDR|TUDR structure of our work here at the LRC, arises once more.

Interestingly, however, unlike the strings of string theory, the structure of these braids is very close to the structure of our SUDR|TUDRs! For example, each twisted helon consists of two, 1 pi, twists, just as each UDR consists of two, 1 unit, progressions of space|time.  Thus, the three types of helons, braided together, constitute four units of positive and negative pi twists, but Bilson-Thompson doesn’t ever braid left-twisted helons with right-twisted helons.  Instead, left-twisted helons are only braided with untwisted helons, or other left-twisted helons, and the same for right-twisted helons.  Only untwisted helons are braided with both types of twisted helons, or with each other. 

On this basis, the three-helon braid of two left-twisted helons, with a neutral helon, has a complemented version of the same thing, with the pattern of the braided helons reversed, and, of course, the three-helon braid of the right-twisted version also has its corresponding complemented braid.  In other words, there is a left and a right-handed braid, consisting of left, neutral, or some combination of these two types of helons, and the inverse of these, the left and right-handed braid, consisting of right, neutral, or some combination of these two types of helons.  Hence, there are chiral versions of braids and anti-braids (positive and negative magnitudes in left and right-handed versions).

Bilson-Thompson’s paper contains the graphic depiction of the braids just described, as shown in Figure 1 below. 

Figure 1.  Bilson-Thompson’s Braids and Antibraids of Positive and Negative Helons 

In the figure above, the left-handed and right-handed versions of each braid of left-twisted (positive) helons are on the top row, while the left and right-handed versions of each antibraid of right-twisted (negative) helons are on the bottom row.  As indicated in the graphic, each type of braid, or antibraid, equates to a particle of the first generation of the standard model, or it’s antiparticle.

As soon as Smolin read Bilson-Thompson’s paper, he knew it was what they had been looking for, because they could use these ideas of braids in loop quantum gravity to represent the emergent particles of the standard model in the theory.  He writes on page 254:

As soon as I read the paper, I knew this was the missing idea, because the braids Bilson-Thompson studied could all occur in loop quantum gravity. This meant that the different ways to braid and knot the edges of the graphs in quantum spacetime must be different kinds of elementary particles.  So loop quantum gravity is not just about quantum spacetime - it already has elementary-particle physics in it.  And if we could discover Bilson-Thompson’s game working precisely in the theory, it would have the right elementary-particle physics.

Eventually they managed to do it, though the achievement certainly isn’t being heralded in the LST community as much as would seem warranted, in part, evidently, because the string theory controversy is eclipsing the real message of Smolin’s thesis: String theory is not the “only game in town.”  He writes:

Plainly, there are different approaches to the five fundamental problems in physics.  The field of fundamental physics beyond string theory is progressing rapidly, and in several directions…While there aren’t as many people (perhaps two hundred, all told) in these [alternative] research programs as there are in string theory, it’s still quite a lot of people to be tackling foundational problems on the frontiers of science.  The big leaps of the twentieth century were made by far fewer.  When it comes to revolutionizing science, what matters is quality of thought, not quantity of true believers.

Of course, at the LRC, there are even fewer people, far fewer people, working on these foundational problems.  However, we are undaunted in our determination to move forward with the new system of physical theory, the RST, exploiting our recent discovery of the Chart of Motion (CM), which is providing phenomenal insight into concepts of numbers and magnitudes of motion and the way to unite them into a single concept that addresses the enigma of the continuum versus the discrete faces of nature. 

However, we are not developing the RSt in a vacuum, understanding the LST community’s challenges is an important part of our work here at the LRC, and there is no greater example of the value of this than what we have been discussing in this post.  To wit: Bilson-Thompson’s work promises to a be a great help to our own efforts in developing the SUDR|TUDR combinations that lead to the standard model’s entities.  Of course, what we bring to the table is an immense advantage over any other approach, because the new system is not only completely background-independent and discrete, but it also is completely free of the conundrum of starting with a definition of the length of space as something with tangible properties.

We don’t need to start with strings; we don’t need to start with ribbons, and we don’t need to start with graphs, in order to discover emergent entities of matter, radiation and energy.  All we need to start with is the definition of motion and the properties of natural numbers.  What a thought!

An effort by Chris Isham and Andreas Doring, of the Imperial College of Science, Technology & Medicine, “to develop a fundamentally new way of constructing theories of physics,” is another topic being discussed by John Baez & company in The n-Category Cafe. In the previous post below, I reproduced a comment I made over there regarding the ad hoc inventions of real and imaginary numbers, but, of course, my comment did not refer to our new system of physical theory. Otherwise, I’m sure it would have been deleted post haste.

However, in writing about their new system of physical theory, Isham and Doring, surprisingly, refer to the role of the invented real and complex (imaginary) numbers, as “critical mathematical ingredients that are taken for granted,” in quantum theory. But it isn’t the numbers per se, that is the focus of their raised eyebrows, but the use of these numbers as a concept of the continuum that gives them pause. They warn:

The a priori imposition of such continuum concepts could be radically incompatible with a quantum gravity formalism in which, say, space-time is fundamentally discrete…

They also make clear that, while they are interested in new interpretations of quantum theory, new insight into the foundations of existing quantum theory isn’t their main goal. Rather, what they are trying to do is to step back even further in search of an entirely new system of physical theory. They write:

However, although we are certainly interested in such conceptual issues, the main motivation for our research programme is not to find a new interpretation of quantum theory. Rather, the goal is to find a novel structural framework within which new types of theory can be constructed, and in which continuum quantities play no fundamental role.

Whoa! “New types of theory…in which continuum quantities play no fundamental role.” Did I read that right? It almost makes one think of rushing to the phone to give them a call. “Hey, fellas,” I would probably blurt out, “we already have just that sort of thing. Come on over and take a look!” But, after a sobering pause, it becomes clear that such a dialog can never take place, not because we couldn’t give them a call, or shoot off an email, but because there is a great gulf betwixt them and us. A gulf of sophisticated mathematics, demarcated by the steep walls of professional sophistication and the associated language of esoteric communications.

Their “basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system.” What is a topos? Well, that’s the gist of the discussion going on at The n-Category Cafe. The authors tell us that “classical physics arises when the topos is Sets, the category of sets,” and that “other types of theory employ a different topos.” However, their particular strategy is more general than this, because it is “based on the intrinsic logical structure that is associated with any topos.” Heavy stuff and, accordingly, they caution the uninitiated physicists: “topos theory is a sophisticated subject and, for theoretical physicists, not always that easy to understand.”

Nevertheless, one need not be a mathematician, or a theoretical physicist, to understand how difficult such a huge undertaking is. Certainly, overhauling the mathematical formalism of current physics is something no one could take lightly, but they are funded by the Foundational Questions Institute to do exactly this sort of thing. They write:

What is needed is a formalism that is (i) free of prima facie prejudices about the nature of the values of physical quantities - in particular, there should be no fundamental use of the real or complex numbers; and (ii) `realist’, in at least the minimal sense that propositions are meaningful, and are assigned `truth values’, not just instrumentalist probabilities. However, finding such a formalism is not easy: it is notoriously difficult to modify the mathematical framework of quantum theory without destroying the entire edifice.

Of course, in our approach, we are even more ambitious, because we are claiming that the foundation of the mathematical structure, underlying quantum theory, is not only the root of the problems plaguing theoretical physics, but, once understood in the proper manner, the new insight that results actually comprises a new framework, or system, for constructing new types of physical theories, in a simple and a straightforward manner, with no need to resort to a mathematical formalism of any kind; that is, we take the position that discovering the properties of numbers and mathematics will naturally lead us to nature’s physical structure. Hence, we too take the realist view in physics, but we take the realist view in mathematics, as well.

I posted the following comment on The n-Category Cafe blog, discussing Smolin’s book:

Victor Grauer writes (in response to a preceding comment of mine):

What is really at stake here is NOT the question of what space and time really really are in the most fundamental sense, but the even more fundamental question of how it is possible to represent them. In other words, we are dealing with not only epistemology but semiotics, the “science” of representation.

Man, I hope this is not considered off-topic. I don’t want to get deleted again. I just want to stress that while Smolin’s thesis, the trouble with physics, is ultimately our inability to unify the discrete and continuous theories, he asserts that there is more than this reflected in the string theory controversy.  It’s as if the string controversy is the center piece of the table, focusing our attention on the state of physics as a whole, not just the latest innovation, which might, or might not, have outlived it’s usefulness.

If string theory is justified on mathematical grounds, it’s not just because it is “beautiful mathematics,” but because it’s beautiful mathematics that, to some extent, unifies the discrete and continuous theories.  The fact that it has no contact with experiment and can’t predict anything right now, is overridden, in the minds of many, by the apparent achievement of a consistent unification of the discrete and continuous, in a very compelling manner.

The details of how the development of string theory has led to the current prospect of “the end of a science” and consideration of the serious question “What comes next?” are not so important at this point.  What is important is gaining a clear understanding of the method of thinking that led us to this point, and without a doubt that thinking is best characterized as the history of developments in the science of mathematics.

String theory must live in a minimum of ten dimensions, but what bothers Glashow, as quoted by Smolin, should bother all of us:

…Worst of all, superstring theory does not follow as a logical consequence of some appealing set of hypotheses about nature.  Why, you may ask, do the string theorists insist that space is nine-dimensional? Simply because string theory doesn’t make sense in other kind of space.

In other words, string theory is not inductive science, it’s inventive science, and the comments of Einstein, regarding the significance of epistemology in science, rise to the top of our thoughts, like the cream in unhomogenized milk.

However, and this question must be asked, if string theory is basically an exercise in mathematics, and it is inventive science, then doesn’t that imply that Glashow’s criticism applies to the science of mathematics represented by string theory?

I believe the conclusion that it does is just inescapable. Writing about this in a historical context, Hestenes sees the development of mathematics as the centuries-long effort to unify discrete numbers with continous physcial magnitudes, which Euclid deliberately kept separate, proving theorems first with line segments and then with numbers.

Clearly, though, this history is a history of inventive science, not inductive science.  String theory mathematics is simply a continuation to unify discrete numbers with continuous magnitudes. Briefly, the three properties of physical magnitudes versus natural numbers are:

1. Continuous vs. discrete quantity
2. Two directions vs. one direction
3. Limited vs. unlimited dimensions

In the development of our inventive mathematical science over centuries, the ad hoc invention of the real numbers addresses number 1; the ad hoc invention of the imaginary numbers addresses number 2; and the ad hoc invention of “compactified dimensions” adresses number 3.

Nevertheless, in the spirit of Glashow’s complaint, wouldn’t we rather have a numerical theory that “follows as a logical consequence of some appealing set of hypotheses about nature?”  Is this even possible in mathematics, or is formalism the only “game in town?”

Of course, I couldn’t give an answer to the question on Baez & company’s blog, but I can here.  The answer is clear: Inventive science is no longer the “only game in town.” Inductive science has returned.  As Larson explained, in his “Principal Address to the Third Annual NSA Conference,” in 1978, inductive science inevitably falls behind the available empirical information of experiment, and thus makes it possible for inventive science to fill the void in the effort to explain natural phenomena.  However, ultimately, the number of ad hoc inventions becomes so great that the human spirit recoils.  We intuitively know that the physical laws of nature are not like the tax laws, designed by government bureaucracy, but most men are not as wise, nor as patient, as Newton, who insisted that basic concepts and laws of nature can only be derived from experience.

The new inductive science is possible, because the new information that Larson discovered, that space is the reciprocal of time, makes all the difference in the world.