Tim,

I read your essay with a mixture of exhilaration and misgiving. Exhilaration for the sheer audacious brilliance of it, and misgiving that I have not yet introduced myself to your work. I plan to start correcting the latter in short order.

One comment: "This (metric - ed.) distance is just the minimal 1 length of a continuous path between the points. It can have an affine structure, which sorts continuous paths into straight and curved. It can have a differentiable structure, which distinguishes smooth curves from bent curves. But beneath all these, already presupposed by all of these, is the most basic geometrical structure: topological structure."

The straight line being a special case for the curve, an analytical "twoity" (LEJ Brouwer's word) guarantees curved structure of metric properties. In a 2006 conference paper I identified the complex plane structure that guarantees a counting function without appealing to the axiom of choice, with a physical definition of "time: n-dimension infinitely orientable metric on random, self-avoiding walk."

Looking forward to immersing myself in your research.

Tom

    Dear Tom,

    Thanks for the comments. I was not aware of the Brouwer, and a quick look at some discussions shows that it will not be an easy thing to really understand. It is, of course, possible to describe the geometry of a space with enough structure to define the affine structure but not enough for a full metric. The so-called "Galilean" or "Neo-Newtonian" space-time is like this (if you try to use a standard full metric is it degenerate). This is particularly nice if one is trying to translate physical laws into a purely geometrical vocabulary. Newton's First Law, for example, becomes "The trajectory of a body is a straight line through space-time unless a force is put on it". The fundamental distinction between the affine and metrical structure also shows up when one demands, in General Relativity, that the metric be compatible with the connection on the tangent bundle.

    Regards,

    Tim

    Hi Tim,

    I think a full metric description is, just as you imply -- native to point set topology, and not to affine space. There is an arithmetic theorem that any point maps simultaneously to any set of points provided it is far enough away. In reverse, this gives us the degenerate result of Galilean or Newtonian space. There is no time parameter.

    Einstein, by introducing time by way of Minkowski space, may have hoped the point "far enough away" would avoid the singularity and instead found that expanding 4 dimension spacetime (by Hubble's result) places the singularity at every arbitrarily chosen point of 3-space. There will be singularities in general relativity. No point is far enough away to overcome the Poincare-Hopf theorem.

    Best,

    Tom

    Dear Tim,

    I was very happy to find your essay here. I read it with pleasure and I like it so much. I am a mathematical physicist working in general relativity (singularities). Also, I started recently teaching a master class at the Faculty of Philosophy, together with the philosopher Iulian Toader, and we are using as main resource your book Philosophy of Physics. Space and Time, which we both consider great.

    In addition to the part containing the general discussion about the effectiveness of mathematics in physics, I enjoyed very much the part about your theory of Linear Structures. I like the idea that, once you have the linear structure, with directed (causal) lines, you can recover not only the topological and the causal structures of relativistic spacetime, but also the conformal structure, that is, the metric up to a scaling factor. I have some comments and questions.

    You said somewhere "Whether those axioms could be modified in a natural way to treat of pointless spaces is a question best left for another time." I think the answer to this is positive, and that in the same way linear structures reconstruct topology, a sort of linear structures can reconstruct the generalization of topology which may or may not have points, named "locales".

    The linear structure able to lead to the recovery of an Euclidean or relativistic spacetime has to be very special. In other words, it has to be subject of some constraints, which lead to the topology, the affine structure, and the metric of the Euclidean space.

    In the case of relativity, without an underlying structure similar to the usual topological structure of spacetime, the directed lines can be distributed in so many ways. Consider first that we have a point and a local homeomorphism to R4 around that point. The directions of the directed lines in R4 at that point can be any subset of the sphere from R4 centered around the origin of the lines. This means that, in order to get the causal cones in relativity, one needs to ensure that at least the topology of the directions is that of a cone. Otherwise, we can obtain various kinds of spacetimes, in particular the Galilei spacetime is of this form.

    But how to recover the topology of R4 from directed lines in relativity? The problem seems to me to be that future oriented timelike vectors of length smaller than the unit form a noncompact set, and they can't be used to reconstruct the topology. In addition, the spacelike directions, which have to be undirected if we want to talk about them, are disconnected from the timelike ones, so it is difficult to use them even together to reconstruct open sets. I have some ideas how to do this, but I am still not sure if this reconstruction can be made simpler than the standard one.

    The condition of homeomorphism with R4 is quite natural in the standard notion of topology, but you can object that this is because we are in the old paradigm. However, the condition that the directed lines give causal cones is more natural assuming a (4D) tangent bundle (which can only be defined if we assume a 4 dimensional topological manifold endowed with a differential structure), on which a Lorentz metric is defined, or at least the corresponding conformal structure. Both these conditions seem to me to make the case for 4D open sets rather than lines, because it seems difficult to recover the differential structure from lines rather than open sets.

    Did you find additional axioms to those of a directed linear structure, which would make it in a 4D manifold with differential structure and causal cones just as in general relativity, in a more natural way? Because at this point it seems to me that adding such axioms would lead to a much more complicated definition of relativistic spacetime, and by this would make the advantages of the simplicity and concision of the linear structure vanish.

    Maybe some of my questions are already answered in your second volume or other works. Or in your future results, since it is natural to think that such a theory takes some time to answer to the most important questions.

    Anyway, thinking at this led me to some ideas of simple constraints to supplement your causal structures to recover relativistic spacetime. If you are interested, I can try to detail them.

    Reconstructing relativistic spacetime in a natural way, based on more intuitive and physical principles, can be a good starting point for generalizing the structure to include matter and quantization.

    I want to make clear that the fact that in order to recover relativity one has to add to the linear structure less natural axioms than the standard is due to the richness and generality of the linear structures, and it happens the same even if we start from open sets topology. And is not necessarily a disadvantage. By contrary, it may be an advantage, because we have the freedom to use other axioms, that give something different than general relativity. For instance, consider the possibility that the linear structure behaves differently at different scales, maybe this would lead to the dimensional reduction which would be useful to perturbatively quantize gravity (my own approach to quantum gravity is based on singularities, which are naturally accompanied by dimensional reduction).

    Another advantage of linear structures approach over standard general relativity is that it is much richer. Maybe this richness can be used somehow to describe matter on spacetime, although I don't have a clue how to do this. Also, I think that Sorkin's causal sets can be seen as a particular case of your linear structure approach to general relativity.

    Another feature I liked at your theory of Linear Structures is that it equally works for discrete geometries. This made me think at the following. In 2008 I proposed a mathematical structure, based on sheaf theory, which allows to construct all sorts of spacetimes and fields on them - a general framework which contains as a particular case any theory in physics (but it is not a TOE). This sheaf theoretical approach works equally for discrete approaches like causal sets, spin networks, CDT etc., but I did not develop it beyond what is in that paper, for lack of time. My sheaf theory approach works with both discrete and continuum theories and captures some essential features in a simpler and more general structure, otherwise there is no parallel with your theory.

    Best wishes,

    Cristi Stoica (link to my essay)

      Dear Crista,

      Thanks so much for the comments. I am familiar with some work on pointless spaces (some of it older that the things coming from category theory) and I can only say that there seemed to be no obvious way to adapt this approach there. The basic problem is that the points in a 1-D open manifold are linearly ordered automatically, but while regions in a pointless space can include one another, it is hard to define the same sort of linear order if the basic structures are not 1-D (and hence constructed from things are are 0-D, i.e. points). This is not so say it is impossible, but that it is at least not obvious.

      The language of Linear Structures is, as you appreciate, very flexible. That is good in one way (lots of possible geometries) and bad in another (you need strong constraints to narrow down to what you want). One interesting place to look is discrete spaces, when one can consider various constructive rules for generating a geometry and then analyze the character of the geometries that result. I have quickly found rules that give good discrete approximations to a 2-D flat space-time and the 3-D expanding space-time with a horizon structure. It will take more research to figure out exactly what features of the constructing rule control the outcome.

      The idea is a bit like Causal sets, but the actual implementation is quite different. The most fundamental structure is light-like rather than time-like, and the Causal sets it is time-like.

      Cheers,

      Tim

      Dear Tim,

      I am happy to announce you that I found a natural way to supplement linear structures (directed null lines) with simple constraints to recover relativistic spacetime in n dimensions, without directly imposing to be locally IRn. If you think it would be interesting, I can post it here, I don't think it will take me more than a couple of pages to explain it in detail.

      Best wishes,

      Cristi Stoica

        Tim Maudlin: "The Relativistic account of space-time geometry makes the light-cone structure of space-time a fundamental part of its geometry. This, rather than the "constancy of the speed of light" lies at the heart of the theory."

        This is wrong, Tim Maudlin - the constancy of the speed of light does lie at the heart of the theory, even if you want to hide it behind the light-cone structure of space-time. And since the speed of light is not constant (you know that don't you?), you will have to join Steve Giddings, Nima Arkani-Hamed and Lee Smolin in their rejection of Einstein's space-time.

        Pentcho Valev

        Dear Tim Maudlin,

        I really enjoyed the thread of your argument: simple arithmetic and geometry, based on our acquaintance with the physical world, leading to complex mathematical concepts, and these in-turn having some role in physics, and thus the Wigner's puzzle being solved. The whole thing would have been a smooth curve, but for your sudden jump: in explaining geometry, you jumped from the 'geometry of the bodies' to the 'geometry of the space'.

        "And as soon as items are countable, other mathematical concepts can be brought to bear: ratios and proportions for example." 'Countable' is a factor; but, imagine the universe has a finite number of atoms, but does not change with time, what structure will it have? For any change, these countable entities should move, or 'motion' is the most fundamental factor. Motion is a space- time relation that obeys mathematical laws, and so the changing world obeys the laws of mathematics. That provides the simplest solution for Wigner's puzzle. Please go through my essay: A physicalist interpretation of the relation between Physics and Mathematics.

        "We can even seek to construct new mathematical languages that are better suited to represent the physical structure of the world". As you have pointed out, any new mathematics depends on the basic mathematics, and cannot be 'genuinely new'. The theory of linear structures, in fact, is not new mathematics; the 'newness' is in the 'directional aspect' which is a property, a property which you assume. You have assumed the conceptual primitive, regarding space, to be lines instead of points. The mathematical rules remain the same, but the resultant structures are different; so there is justification in calling it a new mathematical language used to describe space-time.

        The physicalist point that I propose is that the properties we assume, whether about space or about atoms, should be in conformity with what we observe directly. As pointed out by you, everything started from the simple observations in the nature. The concept can become more and more complex, but should not go against that simple facts. The essence of your arguments in the first part of the essay is also the same.

          Dear Jose Koshy,

          I agree that the motion of matter is described by some mathematical law, but the formulation of that law is beyond the scope of my present project. If we start with the general idea that physics is the theory of matter in motion, this presents two targets for an exact theory: matter and motion. The "motion" of a localized object seems to be best understood (and this is not how Newton would have conceptualized it) as a trajectory through space-time. So that leaves us with the problem of describing the geometry of the space-time, and the problem of understanding how the trajectories of objects are produced. You have to solve the first in order to even approach the second. That is, you need to understand the geometry of the space-time before you can begin to write down a dynamics. This project is then even more basic: not yet "what is the geometry of space-time?" but rather "what is the best mathematical language to use to describe the geometry of space-time?". If the fundamentals of Relativity are correct, this looks like an especially promising sort of language. The language is new, in the sense that (as far as I know) no one has written it down before. But at a basic level, I am a Platonist about mathematics, and in that sense nothing is really new.

          Regards,

          Tim

          Dear Tim Maudlin,

          Your stand is clear: if the fundamentals of (General) Relativity are correct, then there is four-dimensional space-time, and you are proposing a new language to describe the geometry of the space-time. I have just downloaded "New foundations for physical geometry" to know exactly what you referred as the project.

          My argument is that physicists have not explored all the possibilities classical three-dimensional space offers. For example, the model proposed by me (refer finitenesstheory.com) views that motion at speed 'c' is the fundamental property of matter, and the reaction to this motion creates gravity. Consequently, the path of a body, in a classical 3-dimensional space, is bent by its own gravity. This can give similar results like that of GR.

          From a cursory glance at the material downloaded, I get the impression that you consider going back to classical three-dimensional space is a retrograde step. However, I think that you will ultimately arrive at that conclusion.

          Dear Jose Koshy,

          I would not at all consider quite different space-tmes from GR as "retrograde": I am quite open to all sorts of possibilities. Here are two considerations. One is that if one rejects the idea that space-time is a continuum and uses a discrete space-time instead, the most natural way to implement that idea using this formalism results in the basic geometrical structure of space-time being all light like (null) paths. It would then follow automatically that any matter in continuous motion (i.e. following a continuous path) is always "traveling at the speed of light", and massive particles, such as electrons, must really engage in Zitterbewegung: rapid vibration. The other consequence of going to a discrete space-time is that the geometrical structure has a natural unique foliation, which can be considered a "backward" step to a more classical structure. (Not full Newtonian picture, with a single space persisting through time, but a structure with an intrinsic "simultaneity" built in, together with a light-cone structure.) Is that bad? It is actually the most straightforward way to be able to implement the non-locality implied by violations of Bell's inequality. So I am very open to these possibilities. As I said, in the first place this is just a new mathematical language. There are many things that can be described using it.

          Reagrds,

          Tim Maudlin

          Dear Tim Maudlin,

          One last question: Devoid of physics, can the Theory of Linear Structures be regarded as a new branch of mathematics related to topology? Or is there already any related branch in mathematics, and you are only trying to use that in physics?

          The aim of the Theory of Linear Structures is to provide a precise a formalized mathematical language (axioms for the basic structure and strict definitions) that make precise the same informal notions addressed by standard topology (continuity, connectedness, closure, limit points, etc.). These new definitions are different from those of standard topology...for example some functions that are continuous according to the standard definition are not continuous according to my definitions. Conceptually, I think my definitions are better: they match better what our informal judgments are. But more importantly, in the case of physical space-time, one can see from a physical perspective why the geometry should be well described in the this language, and it is not clear why it should be well described in the standard language. So: it is a new branch related to topology and, in some sense, competing with standard topology, which I am trying to use to do physics. It also provides a common mathematical language for describing both continua and discrete spaces. Standard topology does not provide this. So insofar as we are unsure whether physical space-time is continuous or discrete, we can can still use the basic conceptual tools provided here knowing they will work in either case. This also provides clues about how to formulate discrete approximations to continuous structures.

          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