Dear Tim Mauldin

An enjoyable essay to read. I agree that "one could easily write a companion paper to Wigner's called "The Unreasonable Relevance of Some Branches of Mathematics to Other Branches"". For me there are 2 places where this particular unreasonableness of maths transfers over to physics in a sort of pincer movement that constrains physics and potentially your proposed descriptive language.

The first is causation for any continuous physics over a space with a metrical structure: these conditions specify the maths description must be in terms of the norm-division algebras. Both General Relativity and Standard Model are in terms of NDA valued fields, which constrains every attempt to unify them to be able to reproduce this NDA description. But the physics conditions also constrains all alternative descriptions to be capable of reproducing the NDA based description.

The second side of the pincer comes from requiring any theory to reproduce quantum theory results. My hidden propagator dynamics analysis came from having a particular theory in mind: one with discrete topological defects in a space with compactified dimensions. This is a discrete theory with a potentially discrete space. However, I was surprised to discover that my HPD analysis revealed that in order to connect with experimental results the details of the discrete nature of the theory must necessarily be erased, leaving only the same NDA-based description of experimental results as quantum theory. This is a general result for any theory with discrete particles: to connect with experimental results all discrete elements of the theory are erased, leaving only an NDA based description. This means that the entire class of HPD theories are experimentally indistinguishable from each other, as they must all reproduce the same descriptive form for experimental results.

This pincer movement would seem to include your new descriptive language of the Theory of Linear Structures. Any new descriptive language must reproduce the descriptions of existing theory. In this bigger picture context of connecting with experimental results that are already successfully described by NDA valued fields - won't the new description be lost in the process of experimental prediction?

Michael Goodband

    Dear Professor Maudlin,

    this is a very interesting essay and a very interesting program of research. It reminds me very much of the causal set research program of Rafael Sorkin. In particular, like causal sets it is a beautiful and austere way of looking at spacetime structure, but perhaps a bit too austere for my taste. Below I append the sort of questions that I usually ask about the causal set program which I think also apply to your program. If you have time to reply to any of these questions, that would be very much appreciated.

    --David Garfinkle

    (1) Your method gives a conformal structure, which in the case of a spacetime is equivalent to the conformal class of the metric. There is then a standard result that also giving the volume element will determine the metric. Is there a simple and natural way of specifying a volume element within your formalism?

    (2) Presumably a model for a spacetime within your formalism would be a set with your structure which also (exactly or approximately) admits a differential structure (thus making it a manifold) and a metric. But most of the sets with your structure will not be manifolds, even in an approximate sense. Is their a way within your formalism of figuring out which sets are manifolds? (or approximately manifolds).

    (3) I admire the austere beauty of your approach. But I'm pretty happy with the standard notion of spacetime as a manifold with a metric and with the usual definition of a manifold as a topological space with an atlas. What do I lose by not adopting your approach?

      Dear Michael,

      I do not feel confident that I am in command of the technical details of your analysis, but let me say at least this. The sense in which any new theory must "reproduce the descriptions of existing theory" is obviously a matter of approximation, not exact derivation. That is, all experimental data come with error bars, and so recovering the predictive success of present theories (such as quantum theory or the General Theory of Relativity) requires matching their predictions where they have actually been tested to within the tolerance of the errors. If one were to demand a higher degree of match than this, then the new description would of course be "lost". But the expectation is that the new description will deviate in its predictions from the present theory, but only to a small degree and only in certain circumstances. What that degree is, and which circumstances are relevant depends on the theory itself.

      Regards,

      Tim Maudlin

      Dear Professor Garfinkle,

      I completely understand seeing a similarity between this approach and causal sets, but at the level of implementation and exact detail they are quite different. So, for example, in the causal set approach there is essentially zero chance for any pair of events to be null related, and in this approach to a discrete structure all the fundamental space-time geometry is null. Also, as you note, in the causal set approach the idea is to get the conformal structure from the causal graph and then fill out the rest of the geometry by a volume measure. But in this approach to a discrete Relativistic space-time (which is not described here) one does not add a volume measure but rather a measure corresponding to the Interval measure on continuous paths. In a word, the basic metrical notion is length rather than volume.

      Let me answer your questions.

      Well, actually what I have said above answers 1) for the discrete case. The discrete case is nice because (as Riemann observed) the discrete case comes equipped with a measure: counting measure. But you have to be careful about what you count! In this case, one does not count nodes (which is what the causal set people do to recover a volume measure), and what you are quantifying is not volume but length. This may be a bit cryptic, but in short in the discrete case one can define a "corner" in a continuous path through the space-time as opposed to an "unbent segment", and when quantifying the length of a path one counts corners rather than nodes in the path. The result is that the number of corners on a path that lies on the light cone is zero (even though the number of nodes may be unboundedly large). The connection to the Interval should be obvious.

      In a continuum, the measure must be imported from outside (just as Riemann said). I have not tried yet to somehow analyze that measure in terms of anything else in the continuum case: it is just an additional piece of space-time geometry supplementing the conformal structure, just as in GTR.

      To the second question:

      Investing a space with a Linear Structure automatically invests it with a standard topology, via the definitions I give in the appendix. So it is easy to tell if you have a manifold. But in fact, even familiar continuous space-times (e.g. Minkowski) turn out not to be manifolds once one takes account of the directionality of time. One of my points is that the whole idea of a manifold arose in the context of purely spatial (Riemannian) geometry, and the use of those mathematical tools to deal with space-time (Lorentzian) structures is a mistake. So I am not even aiming at recovering manifolds: I am aiming to recover geometries with an intrinsic light-cone structure, which a manifold does not have.

      To the third:

      One of the main advantages of my approach is that one can deal with contiinua and discrete spaces using the very same analytical tools and definitions. No one knows whether at (say) Planck scale space-time is discrete or not. So it would be nice to articulate theories in a mathematical form that can be adapted to each possibility. As things are, one uses manifolds and differential geometry for continua and graph theory for discrete spaces. Graph theory (including infinite graphs) is a special case of the Theory of Linear Structures, but the Theory of Linear Structures can be used to analyze continua as well. It also allows you to take ideas developed in the context of continua and see how they play out in a discrete setting. So I think that is one thing that is an advantage.

      One other advantage, which I do touch on, is that the Theory of Linear Structures allows for a distinction between intrinsically directed and intrinsically undirected geometries, which is relevant to the description of time. Indeed (as I mention) it even allows one to make a distinction between intrinsically directed and intrinsically undirected topologies, which (to my knowledge) no one working in standard topology has made, and maybe cannot even be drawn without using the resources of the Theory of Linear Structures. So if space-time is discrete (which it may be) and if time is intrinsically directional (which I think is obvious!) then the analytical tools available are better than standard topology and hence manifolds.

      Cheers,

      Tim Maudlin

      the central point here is :(Q) what properties mus physical reality have for math to be applicable? As the author knows the brings in some traditional problems of scientific realism. In the simple cases treated first we know reality and then apply integers or geometric forms. In the difficult cases we know physical reality through the way successful physical theories represent reality. Here Q i still applicable but more difficult to answer. You eventually sharpen Q to: What physical features space or space-time must have to be represented by the topology of an open set? This leads into your development of a theory of linear structures. I don't feel qualified to comment on that, though it looks good. I tend t think of open sets as a mathematical trick with no direct physical significance. Maybe the development of your theory will chang that.

      Ed MacKinnon

      Dear Tim Maudlin,

      You have mentioned that for mathematics to be used as the language of physics, physical world has that sort of structure to be represented mathematically? That depends on the mathematical language being used Physical characteristics are required for mathematical structures to describe a physical situation.

      Yes I agree with you and thats why I have propounded Mathematical Structure Hypothesis to explain their origin in the same line.

      Question is - Who decides the symphonic structure of that language? For any mathematical structure to be compatible to explain the physical structure, we need to match their intrinsic "laws of invariance" otherwise their applications would be wrong.

      This is why in context of Skolem's paradox: "A particular model fails to accurately capture every feature of the reality of which it is a model. A mathematical model of a physical theory, for instance, may contain only real numbers and sets of real numbers, even though the theory itself concerns, say, subatomic particles and regions of space-time. Similarly, a tabletop model of the solar system will get some things right about the solar system while getting other things quite wrong."

      You have classified the mathematical structures into two categories based on Wigner's essay

      1) One which are naturally suited to physical world e.g. Integers and what does their suitability imply about the physical world?

      2) Others which are not e.g. advanced concepts e.g. complex numbers should have use in physics.

      I have explained on the basis of Mathematical Structure Hypothesis that whether it falls in any category, its basically physical characteristics behind the development mathematical language which describes the physical characteristics of the physical world i.e. whether Integer or Complex numbers.

      Wigner talks about Complex numbers as advanced concept but what decides the structure of complex numbers and why they are so effective in Quantum Mechanics.

      Eugene Merzbacher in his book on QM has explained by deriving that for certain physical characteristic to be satisfied( for quantum waves,any displacement in the space & time dimension should not alter the physical characteristics of waves) and to satisfy these criteria, the mathematical parameters turns out to be "i"(complex number).

      Here is the reason the structure of mathematical language has been matched/molded to suit the physical characteristic of quantum waves(physical world).Infact, its not the mathematics describing physics here rather their corresponding law of invariance. So, what is the law of invariance behind complex number. Its answer lies in the definition of why negative multiplied by negative turns out to be positive? Why not positive multiplied by positive also become negative? Here is hidden laws of physics behind the definition of mathematical operators structure and vice versa.

      This is because mathematical structures abstractness and physical reality both are creations of the same thing Vibration, which my Mathematical Structure Hypothesis has propounded.

      Anyway, your essay is indeed great.

      Thanks & Regards,

      Pankaj Mani

        Dear Pankaj Mani,

        Thank you for the comments. The use of complex numbers in quantum theory is a very interesting case, which needs a lot of discussion. My own work here is just on space-time structure, so does not touch on quantum theory directly. But I think it may help to recall that time-revesal is implemented in quantum formalism by taking the complex conjugate of the wave function. This immediately suggests a connection between the use of complex numbers and the temporal structure, indeed a connection with the direction of time.

        It is harder to deal with quantum theory because there is no agreement at all about just what physical entities the theory is committed to, particularly what Bell called the "local beables" of the theory. The observable behavior of laboratory apparatus should be determined by the behavior of these local beables at microscopic scale. If you don't even know what these are, then interpreting the significance of the mathematical apparatus becomes essentially impossible. Pure space-time theory is a bit more straightforward.

        Regards,

        Tim Maudlin

        Dear Tim,

        Concerning you criticisms in my Essay page, some clarifications could be needed. For rotating frame in my Essay I mean the frame in which the observer sees the detector at rest (the absorber orbits around the source). Clearly, in that frame photons propagate in the radial direction. You are of course correct in highlighting that Equivalence Principle has local behavior. On the other hand, rotating frames generate the centrifuge acceleration in the radial direction cited above, which, in turn, defines a locally accelerated frame. Thus, it seems to me that the application of Equivalence Principle is completely legitimate. I also stress that the use of the Equivalence Principle in rotating frames in general and in the Mössbauer rotor experiment in particular has a long, more than fifty-year-old, history. In the paper of Kündig, i.e. ref. [3] in my Essay, which is dated 1963, one reads verbatim: "when the experiment is analyzed in a reference frame K attached to the accelerate observer, the problem could be treated [7] by the principle of equivalence of the general theory of relativity". Reference [7] in the paper of Kündig is the historical book of Pauli on the theory of relativity dated 1958. Thus, it seems that you were wrong in those criticisms. Here the key point is not the viability of the Equivalence Principle in treating this problem, but the issue that previous literature did not take into due account clock synchronization.

        I will read, comment and score your Essay soon. I wish you best luck in the Contest.

        Cheers, Ch.

        Dear Tim,

        I was reading your excellent essay with growing interest as you were getting closer and closer to the geometry of spacetime. You argue that standard geometry has been built on the wrong conceptual foundation to apply optimally to spacetime and you propose an alternative geometrical language. That is really promising as I am just looking for such languages. You finally claim that the Theory of Linear Structures (TLS) is capable of describing the geometry of continua [...]. Is TLS designed exclusively for a specific spacetime description or is it possible to describe e.g. Thurston geometries (the geometrization conjecture, proved by Perelman)? This is double-dealing question because the Thurston geometries, in my opinion, we can treat as a space-like, totally geodesic submanifolds of a 3+1 dimensional spacetime. In three dimensions, it is not always possible to assign a single geometry to a whole space. Instead, the conjecture states that every closed 3-manifold can be decomposed into pieces that each have one of eight types of geometric structure, resulting in an emergence of some attributes that we can observe. 3+1, in turn, means that the constant curvature geometries (S3, H3, E3) can arise as steady states of the Ricci flow, the other five homogeneous geometries arise naturally where the dynamics of the Ricci flow is more complicated and where topological changes (neck pinching or surgery) happen. That way the space flows with time and becomes a dynamical medium - spacetime. Then if TLS can really suit, what is the profit from that approach in comparison to the standard?

        I agree that "Physicists seeking such a mesh between mathematics and physics can only alter one side of the equation." However if we use a proper correspondence rule with the empirical domain than it is really sufficient to discover the set of geometric structures isomorphic to physical reality. Details in my essay.

        I would appreciate your comments.

        Best regards,

        Jacek

          Dear Jacek,

          As I mentioned in the appendix, there are topologies that cannot be recovered from any underlying Directed Linear Structure, but I am morally certain that these have no bearing at all on Thurston's conjecture, which deals with manifolds. I see no reason at all to doubt that every one of the manifolds that the conjecture deals with is generated by at least one (and in fact infinitely many) distinct Directed Linear Structures. (Indeed, by infinitely many Linear Structures, with no direction). And the way to implement the analog of surgery theory in the Theory of Linear Structures is completely straightforward: one specifies how two Linear Structures are to be combined by specifying which lines in one are continuations of lines in the other.

          As for the profit: the Theory of Linear Structures describes geometrical structure in finer detail than standard topology: that is why many distinct Linear Structures typically can generate the same standard topology.

          I'm not sure what you mean when you say that one cannot always assign a geometry to the whole space. I think you must be using that term in a non-obvious way. As you say, the theorem is about decomposing closed 3-manifolds into a set of pieces, each with a specified geometry. But there is obviously a geometry (in the sense of a topological and differentiable structure) assigned to all of these objects: that's why they count as 3-manifolds.

          Regards,

          Tim

          Thank you Tim for inspiration and clarifying as the TLS is new for me. I do not feel it yet and I have to catch up.

          When I say that one cannot always assign a (single) geometry to the whole space I mean this is not possible in three dimensions. This is specific feature of three dimensions pointed out in the geometrization conjecture. Differently, for two-dimensional surfaces you can freely assign a single geometry to a whole space. It was really not clearly stated.

          Best regards,

          Jacek

          Nicely argued essay, Tim!

          I believe your idea that time (motion) imposes linear ordering on space is fundamental. I suppose you know that the idea was fundamental to Newton's fluxions. And you have convinced me to look at your book on "The Theory of Linear Structures."

          In regard to geometry, I submit that your arguments may benefit from the Geometric Algebra mentioned in my essay. It may help you with the notion of areas defined by linear ordering of line segments, and volumes defined by linear ordering of areas.

          Respectfully....David Hestenes

            Thanks David! I have always been an admirer of Geometric Algebra...one of my students (Doug Kutach) became much more adept at it than I did. I hope to be able to understand it more deeply. And on a different note, I have recently been looking into Zitter theory. It looks like if one uses Linear Structures to model discrete Relativistic space-times something like Zitter theory must be the right picture of matter. (The fundamental structure of the discrete models is all light-like, so all particles have to follow light-like paths at micro scale.)

            Cheers,

            Tim

            Dear Tim,

            just a quick first reaction to your enjoyable text. We are all very pleased to live in a physical world not completely described by fluid mechanics, but in one where chairs, tables, even living bodies are possible. The step from the physics of this type of world to the mathematics of natural numbers is short and reasonable. But in this reasoning (which I am not questioning) one is actually going form physics to math, from object to description, from territory to map. The reversed scenario - from math to physics - is also interesting, and more challenging. For example, the Mathematical Universe Hypothesis (MUH) does that: it puts math at the roots of the physical world, which would neatly explain why math is also good at describing the physics: math--(is)-->physics--(described-by)-->math. However, MUH does not specifically address the key question of why there are objects (thus natural numbers) rather than just fluids, a circumstance that scientists should also put in the box of 'unreasonable' facts about the universe (nice topic for next year Contest...).

            In my opinion, the most convincing explanation for the emergence of distinguishable 'objects' in our universe, thus of natural numbers for their mathematical description, is, currently, the one that attributes a fundamentally algorithmic nature to the dynamics of spacetime, or of whichever discrete structure one figures sitting at the bottom of the universal architecture. We have today plenty of experimental evidence for the 'miraculous' emergence of distinguishable-denumerable 'objects' - object/background patterns - from the computations of even the simplest programs.

            Best regards

            Tommaso

              Hi Tim,

              After reading Tommaso's comment (but before I've read your essay), I wanted to chime in to say that the piece he describes as missing is precisely what I found almost 30 years ago, and touch on in my essay. The attached JPEG shows explicitly the quantum hydrodynamic analog within the Mandelbrot Set, showing where objectified forms appear just past the primary Misiurewicz points. This image is my Mandelbrot Butterfly figure, everted about (-1, 0i) such that concentric circles are laid flat.

              When I sent this image to John Bush, he replied that he found it 'quite interesting,' which I suspect is because the analogy with quantum hydrodynamics is pretty obvious. This research is still a work in progress. But if it turns out there is a robust connection, where M reveals the process of pinch-off and nucleation by which fluids form droplets - this raises the issue that Tommaso raises above. How could this be, unless the physical reality flows directly from the Math - rather than the other way around?

              All the Best,

              JonathanAttachment #1: Plateau.jpg

              ... and another point.

              You are against attributing an abstract mathematical status to democritean atoms: for being mathematically tractable (e.g. counted), they need some physical properties, that european mountains to not possess. You conclude that attributing a mathematical structure to physical items is not the same as postulating that they are mathematical entities.

              I do not know which precise physical properties can be attributed to democritean atoms (probably not color, volume, spin...), but let us consider an atomic event, or atom of spacetime, as conceived in a causal set, a model mentioned higher up in this blog. These are points, with no other attributes than those you assign to the ideal mathematical/geometrical point. They are countable - crucially, for recovering volume information - and yet totally featureless. I would say that their nature (their ontological status?) is mathematical, not physical. And yet, when collected in a causal set, or a superposition of these, they are conjectured to originate, by emergence, just about Everything - I mean the physical Everything.

              I find irresistible the argument that the deeper you go in magnifying the 'fabric of the cosmos' the more the familiar physical properties we are used to recognize tend to vanish. Brought to the limit, this means that physical reality pulverises into mathematics, or 'baggage-free', purely abstract objects.

              I guess you disagree with this, If I understand your points correctly...

              Sorry for the length

              Tommaso

              Dear Tommaso,

              I am a Platonist about mathematics, in the sense that I think there are objective facts about abstract mathematical objects, including facts that lie beyond our abilities to prove. (For example, Goldbach's conjecture is certainly either true or false, and if it is true we may never be able to prove it.) So there are all of these non-material mathematical structures, whose existence is independent of the physical world. (The physical world could not come out one way that makes Goldbach's conjecture true and another way that makes it false.) As I understand Tegmark's hypothesis, every single one of these abstract mathematical structures is a concrete physical world. I think that there are very, very severe problems of different sorts with that hypothesis. One is that the vast majority of possible mathematical structures are not regular enough to be described simply (think of all possible sequences of integers: for most there is no compact way to specify it). So if all mathematical structures are physical, most physical worlds are not compactly describable. And it would be almost a miracle that ours is.

              That is a completely different matter than the one about algorithmic dynamics. Here I think we agree: indeed, relatively few structures can be generated by a compact algorithm, just a relatively few sequences can be generated by simple rules. The search for such an algorithm is a form of the search for simple laws. And the simplicity of the laws should explain the comprehensibility and predictability of the physical world.

              Regards,

              Tim

              Dear Dr. Maudlin:

              I have downloaded a fair number of FQXi-2015 Contest Essays, and tried to read through as many as I can manage. Needless to say that my understanding of the essays is based on the framework I used to view them, and that framework is described in my essay http://fqxi.org/community/forum/topic/2456 .

              Simply put, I view the world "analogically," as contextually sensitive set of duals: i.e. I frame Wigner's Refrain of mathematics and physics as freedom and determinism (among others I can choose) and then try to understand Dr. Maudlin's:

              (1) "Wigner's question is this: why is the language of mathematics so well suited to describe the physical world? A proper answer to this question must approach it from both directions: the direction of the mathematical language and the direction of the structure of the physical world being represented. In order for the language to fit the object in a useful way the two sides have to mesh."

              (2) "Physicists seeking such a mesh between mathematics and physics can only alter one side of the equation. The physical world is as it is, and will not change at our command. But we can change the mathematical language used to formulate physics, and we can even seek to construct new mathematical languages that are better suited to represent the physical structure of the world."

              Given my formulation of Wigner's Thesis, there is nothing that I can disagree with Dr. Maudlin's views as expressed in the two paragraphs above, but I like to know how the "meshing" might be accomplished in the project.

              Regards,

              Than Tin

              Dear Tommaso,

              I missed the second post, so a comment. I am perfectly happy with the sort of ideas in causal sets, or with point-particles or point-events. Mathematically, an infinitude of different causal graphs exist as abstract objects. If one of these accurately describes the physical universe, it is because the physical universe is composed of physical point-like entities. If one denies the difference between the abstract mathematical objects and the physical objects, then I suppose one ends up with Tegmark's view, which has insuperable difficulties. There are all sorts of mathematical objects (e.g. operators) that can be used to describe physical things (e.g. time evolutions) but are not themselves physical things.

              Physical point-event are certainly stripped down: they have little in the way of intrinsic physical characteristics. But that does not make them abstract in the sense that mathematical objects are. It just makes them have few physical characteristics.

              Regards,

              Tim