Think of an indeterministic random walk. It is both the case that the particular details of any walk admit of a mathematical description (2 steps right, then one left, then three right, then four left...) and, in many cases, that some statistical characteristics of the walk can be predicted with high reliability. So indeterminism is not incompatible with mathematical description. Quantum theory is generally considered to be indeterministic, but still amenable to precise mathematical description.

5 days later

Dear Prof Maudlin,

From Wignar "How do we know that, if we made a theory which focuses its attention on phenomena we disregard and disregards some of the phenomena now commanding our attention, that we could not build another theory which has little in common with the present one but which, nevertheless, explains just as many phenomena as the present theory?"

His answer: It has to be admitted that we have no definite evidence that there is no such theory.

I am not sure of the detail regarding "Theory of Linear Structures", but I think there is room for multiple models, where details will either pass of fail the test of real physical experiments.

Regards and wondering if you have a internet link for more detail?

Ed

    Hi Matt,

    I have not tried to get the quantum-mechanical aspect of it, but I have done some work on using this to describe discrete Relativistic structures. So think of this as in the spirit of causal sets. I can get a simple discrete approximation to a 2-D Minkowski space-time and to a 3-D inflating space-time with horizons, and this is just from trying a few simple constructive rules for the Linear Structure and then analyzing the results. I have an idea for a general scheme for writing down constructive rules (both deterministic and stochastic) for generating Relativistic discrete Linear Structures, but there is a lot of work to do.

    Just to give a taste of how this differs from causal sets, using the usual way that causal sets are generated no pair of events will be null related. But doing it my way, the entire space-time structure is built from null related events: it is all light-like in the foundations, as it were. I can also easily put in place constraints on the constructive rules that avoid some of the issues that come up for causal sets, which basically arise from the fact that the kind of graph they want to get is very much not a random graph.

    The analytical advantage of a discrete space is that it comes already equipped with a natural measure--counting measure--but in the Relativistic case you have to be careful about what to count. I know that sounds cryptic, but it would take to long to explain properly...maybe we can talk about it sometime.

    It may be that just being able to generate good discrete approximations to classical solutions in GR would yield clues about how to implement a fully quantum treatment, but that prospect is too far away now.

    Cheers,

    Tim

    This was a good essay with some interesting ideas such as the temporalization of sapce but ity is an idea that cannto be tested in a laboratory and abstract as it is it is pure speculation and increases the complexity of physical models and introduces more ambiguity. I think a theory like that presented in usenet years ago would be fircly attacked and the creator woudl be called names. What are any new predictions this temporalization ofefrs? Therefore, although I through the essay was good I think it makes undjustifuable claims that should not be made at the level of professional physics.

    "Whereas it is often said that Relativity spatializes time, from the perspective of the Theory of Linear Structures we can see instead that Relativity temporalizes space"

    Most theoreticians now believe that the spatialization of time (a consequence of Einstein's 1905 false constant-speed-of-light postulate) was wrong:

    "And by making the clock's tick relative - what happens simultaneously for one observer might seem sequential to another - Einstein's theory of special relativity not only destroyed any notion of absolute time but made time equivalent to a dimension in space: the future is already out there waiting for us; we just can't see it until we get there. This view is a logical and metaphysical dead end, says Smolin."

    "Was Einstein wrong? At least in his understanding of time, Smolin argues, the great theorist of relativity was dead wrong. What is worse, by firmly enshrining his error in scientific orthodoxy, Einstein trapped his successors in insoluble dilemmas..."

    WHAT SCIENTIFIC IDEA IS READY FOR RETIREMENT? Steve Giddings: "Spacetime. Physics has always been regarded as playing out on an underlying stage of space and time. Special relativity joined these into spacetime... (...) The apparent need to retire classical spacetime as a fundamental concept is profound..."

    Nima Arkani-Hamed 06:11 : "Almost all of us believe that space-time doesn't really exist, space-time is doomed and has to be replaced by some more primitive building blocks."

    Pentcho Valev

      Dear Prof Maudlin,

      For your subquestion 1) „Which mathematical concepts seem naturally suited to describe features of the physical world, and what does their suitability Imply about the physical world?"

      I suggest three main candidates for the mathematical concept:

      bit (it was the subject of the competition FQXi 2013);

      exp(x) (You know the unique features of this function);

      Euler's identity. There are other useful functions, but less importance.

      Suitable use of pervious can to describe features of the physical World.

      What are your main candidates? If you agree with me, part of the solution can be found in my essay.

      Best Regards,

      Branko Zivlak

      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.