Dear Tim:

This is a fantastic essay. And it is very well written.

As we all know, mathematics has been very effective in physics. Its weaknesses to date in modelling physical reality have been twofold:

1. Using open set theory, it does not model that time has an order (whether we interpret this as an order in forward time or an order in backward time);

2. It does not model which directed order (forward order or backward order) corresponds to the observed Arrow of Time.

Your Theory of Linear Structures addresses point 1 - and is therefore important. However, I do not believe it can address the second point. In particular, the initial end point of a line could represent a past instant of time and the final end point of the line could represent a future instant of time - but equally the initial end point of a line could represent a future instant of time and the final end point of the line could represent a past instant of time. The mathematics cannot differentiate between these two interpretations. So, the Theory of Linear Structures, while it can encode an order, will not I believe be able to encode which directed order of a line corresponds to the observed Arrow of Time. Instead, we would have to impose the direction of time from outside the mathematics on the solution - as we do now for open set theory.

What this would mean is that the Theory of Linear Structures - although important - will produce time-symmetric theories, as open set theory does now. We will unfortunately not be able to use it, for example, to have a mathematical theory of evolution. Nevertheless, your work is very good.

If you are interested, in my essay I explain more generally why mathematics cannot, in principle, model the Arrow of Time.

Thank you again for some great research and a clear essay.

Kind regards

Spencer Scoular

    Dear Spencer,

    Thanks for the careful reading. I don't think we disagree on this much: there is a real, physical distinction between the two time directions and it would be useful to have a way to describe geometries that include such a fundamental asymmetry. That is the part my language does. Then we ask a further question: can we give a more profound account of the physical nature of the asymmetry? This could turn out two ways: it could be a fundamental physical structure, and so not admit of further analysis, or there may be a deeper analysis possible. I am open to both possibilities, and it sounds like your work addresses the second.

    My hope is that this mathematical structure can be of use for many different projects. Perhaps yours is one.

    Cheers,

    Tim

    Dear Tim Maudlin,

    I know from your work that you have a strong acquaintance withh Bell's work (B). I arrived at Bell/CHSH inequality from my investigation of Kochen-Specker theorem for multiple qubits mainly through Mermin' treatise (my ref. [19]). At some stage, I observed that the commutation diagram for a set of four observables involved in the violation of the inequality is just a square/quadrangle.

    Hence my attempt to deepen the subject. My work on KS in dimensions 4, 8 and 16 (two, three and four qubits) is in 1204.4275 (quant-ph) [Eur. Phys. J. Plus 127,86 (2012)] where I also mention a paper of P.K. Aravind on BKS.

    My 2013 FQXi essay [also 1310.4267 (quant-ph)] provides the details you ask for. The inequality in p. 4 of my present essay is that of Peres's book [(6.30, p. 164 of Quantum Theory: Concepts and Methods, Kluwer, 1995]. Replacing the dichotomic variables s_i by the appropriate (i.e. commuting like a square) two-qubit operators (or n-qubit operators) that have dichotomic eigenvalues +/-1) as in Peres, p. 174, the norm of C equals 2v2. With two-qubits, there are 90 distinct squares/violations, some involve entangled pairs of operators, others no (as in my example of Fig. 2a). B or KS is not a matter of entanglement but of contexts (compatible observables) as already recognized by many authors. Here I don't refer to an interpretation of QM but to a strict application of its domain of action.

    Of course on can go ahead and try to discover a realm for squares and other finite geometries relevant for BKS as I started to do in the 2010 FQXi essay and my subsequent work. I have found that the application of Grothendieck's dessins d'enfants is very promising in this respect. I have been quite surprised that stabilizing a particular square from the two-generator index 4 free group is an instance of the smallest moonshine group (p. 5) whose structure amounts to that of the Baby Monster group.

    I hope that it clarifies a bit what I wrote. I am currently working at your own ambitious essay and I intend to give you some comments in the coming days.

    All the best,

    Michel

      Dear Michel,

      Maybe it is because you came at this from KS that this seems unfamiliar. If one thinks of the sort of experimental arrangement that Bell had in mind, with observation on the two qubits being made very far apart, then the commutation structure you mention is obvious: any observable on one side must commute with any observable on the other, or else qm would violate no-signalling. And on the same side, the two possible observables cannot commute, or else you do not violate the inequality. But it is not the case that for any set of observables with the commutation structure you show that one can get the maximal qm violation of 2tr(2). so the norm you mention does not follow from the commutation structure you have written down. (Think or what happens as the two possible spin directions on each side approach each other: in the limit, the maximal value becomes 2, not 2rt(2).)

      Regards,

      Tim

      Dear Tim,

      I agree with your point (i): commutation, in my diagram IX commute with XI and ZI but not with IZ, and vice versa.

      I don't agree with point (ii) for qubits. I have checked that for all multiple qubit operators (starting with two qubits) one arrives at the maximal violation 2v2. It is the reason why a finite geometry like the Mermin's square (the 3 by 3 grid) for two qubits has nine proofs of Bell's theorem in it.

      If one makes use of the dessins d'enfants the extension field involved is Q(v2).

      You may have in mind another experimental scheme than the one I am using where the maximum violation does not apply stricto sensu.

      For other type of violations of classical inequalities, there is the paper by Alexis Grunbaum and an optical experiment that I mention in his blog.

      Michel

      Dear Michel,

      Consider the following experimental set-up. On one side, there is a choice between measuring spin in the z-direction and in a direction 5° away from the z-direction, and similarly on the other side. Since the two possible measurement on each side do not commute, and each on one side commute with both on the other, this satisfies your commutation square. But no quantum state gives the maximal value of 2rt(2). If you think one does, maybe you can specify what you think it is.

      Cheers,

      Tim

      Dear Tim,

      More on your comment: "as the two possible spin directions on each side approach each other: in the limit, the maximal value becomes 2": this is classical argument that seems irrelevant in the quantum (not spatial) scheme, either the spins are apart or the same.

      Best,

      Michel

      Dear Tim,

      The two commuting operators on an edge share their states and thus remove the degeneracy occuring in the 4 by 4 observable/matrix. Only these states are involved in the calculation. This is implicit in the norm. If we are talking about a two-qubit experiment I see no other way (and similarly for more qubits).

      Michel

      Dear Tim,

      You write a very attractive essay about a speculated structure of the "physical space-time" that you call a Theory of (Directed) Linear Structures (DLS). " I will argue that standard geometry has been built on the wrong conceptual foundation to apply optimally to space-time". Your essay is based on your recent book at Oxford University Press that unfortunately I could not access yet. But I could find a 2010 paper of you: The Geometry of Space-Time (Tim Maudlin and Cian Dorr). If I am not wrong, your idea is that the DLS have to be structured in terms of open lines with an order of their points reflecting the succession of space-time events.

      A DLS may also be discrete and you write in the aforementioned paper "For example, a space consisting of five points admits of 6,942 distinct topologies and 1,048,576 distinct Directed Linear Structures. These Directed Linear Structures generate 6,942 different topologies: every topology can be recovered from some underlying Directed Linear Structure (DLS), and most can be recovered from many different underlying Directed Linear Structures.", that is the relation between a topology and a DLS is not one to one.

      I have a few remarks that may be are clarified in your book or in the next one to appear.

      * The discrete LS can be seen as point/line incidence geometry and I am curious to see what kind of non-trivial geometries with a few poinst you recover. Is a DLS reminiscent of a Schreier coset graph?

      * As you, I am interested in incidence geometries particularly those that arise from the coset space attached to dessins d'enfants (they are topological objects). I don't see these geometries useful for a space-time physical space time but as an observable space. Not surprisingly (as in QM) they have to do with finite projective spaces. For these geometries having three lines the Veldkamp space (the space of geometric hyperplanes) is isomorphic to a projective space. However, I do not see any reason why it has to with your approach.

      * Can you classifial your geometries in terms of invariants like their genus, or the number of voids or an automorphism group? Could they be finitely represented (in terms of groups)?

      * Are they Cantor sets in the continuous case?

      Thanks in advance for your feedback.

      All the best,

      Michel

        Hello Tim,

        I found a lot to like, in your essay. You articulated well, the problems I've encountered with point set topology, and its limitations for a realistic description of physical form. I have adopted a constructivist and emergentist view toward geometry, in my own research, which reproduces some of the features of your linear structure theory program. And intriguingly; my work linking Cosmology to the Mandelbrot Set necessitated a departure from the standard program, and conclusions similar to yours.

        The Mandelbrot Mapping Conjecture states that the periphery of the Mandelbrot Set encodes the dynamism for the full range of physical processes, from the most to the least energetic. So if it is rotated from the conventional view such that (-2,0i) is on the bottom; it can be viewed as a thermometer. But in cosmological terms; the cusp at (.25,0i) is the moment of the universe's inception. What is clearly described, even when the argument is extended into the quaternion and octonion domains, is that initially spacetime was purely timelike and broken symmetry forced spacetime to evolve relativistically.

        Philosophically speaking; we know that for structures to exist in time, they must have a time-like projection or duration. Accordingly; for spacetime itself to be an enduring feature, it must also exist in time and have a time-like aspect - hence it must possess linear structure. This is something I have attempted to articulate in several papers, but you have summed things up rather elegantly. I will have to take some time to re-read this paper, and fully digest it, before I make a determination. I have some issues to address, I think. But on first look it appears your program has much to recommend it.

        All the Best,

        Jonathan

          Dear Michel,

          In the finite point case, the theory essentially reduces to directed graph theory. I don't see any deep connection to group theory and representations of cosets...the nearer analogs are differential geometry and (a bit) even non-commutative geometry (only the latter with strong restrictions). If you take as a target the space-times that are solutions to the General Relativistic Field equations, then there is not so much reason to focus on automorphisms.

          The continuous case will include all standard Riemannian and semi-Riemannian structures. I can't see any connection to Cantor sets. I think that trying to connect this approach to group theory is not the right way to go.

          Regards,

          Tim

          Dear Jonathan,

          Thanks for the comments, and I hope you find something useful in the program. I would be a bit surprised if it makes contact with what you are doing given that you start from fractals. In fact, my approach tends to make fractals more peripheral than the standard topological approaches, because by the standard topological definition of "continuous" fractal functions are continuous and by my definition they are not. I have a little hope that this might help for a path-integral formulation of the quantum theory, because in the standard approach the measure over path space tends to be dominated by fractals, with makes it something of a mess. But if this is any use for your approach, I would be very pleased,

          Regards.

          Tim

          Meaning no disrespect..

          There has been a lot of heated discussion on various pages of the FQXi forum, regarding Bell's experiments and variations, in the context of Quantum correlations, including their measure and interpretation. Of course; this really stems from concerns raised by EPR, which Bell was hoping to decisively resolve. Unfortunately; there is some ambiguity or inconsistency in the paper by which Bell first articulated this, and successive interpretations have somewhat obscured that.

          There was a comment by Michael Goodband on the thread for one of his previous essays - which I only partially recall - that makes this inconsistency clear, or the ambiguity obvious. One of the key variables is first introduced in Bell's paper as a tensor and thereafter used as a scalar, I think, which restricts the applicability of his conclusions. I see a sensitive dependence on precise definitions and interpretations, surrounding this question, with divergent outcomes for different choices of what principle is most fundamental.

          It appears that you have a settled view of this subject Tim. But for some of the contest participants, at least, there are open issues surrounding Bell's theorem and experiments, how well they address the questions raised by EPR, how well these efforts characterize nature, or what is observed, and so on. The biggest question still remains why we see what we do. Perhaps linear structure theory can help us sort this out. We'll see.

          All the Best,

          Jonathan

          Bruce Lee once said..

          Retain what is useful. I like that advice.

          Regards,

          Jonathan

          Mountains may be hard to define precisely..

          But observers on adjoining peaks can clearly distinguish their positions from each other, and have a distance to travel to be in the same place. I find that pondering questions like 'what is the distance when traveling along the shore in Britain?' make life interesting for me.

          All the Best,

          Jonathan

          Dear Tim,

          The number of topologies (transitive digraphs) on n labeled elements is Sloane sequence 1000798 that is 1, 1, 4, 29, 355, 6942, 209527, ... for n = 1, 2, 3, 4, 5, 6, 7. The number of DLS for n =5 is 2^20 = 1,048,576.

          Do you always have the number of DLS equals to 2^p (for some p) and what is the sequence? I suspect a relation to a finite projective space PG(2,p-1), e.g. for n = 5 the number of DLS is 1|PG(2,19)|. This is reminiscent of a Veldkamp space (set of hyperplanes of a finite geometry with 3 points on a line).

          Regards,

          Michel

          ps/ Today an interesting paper about topologies on a finite set

          http://xxx.lanl.gov/pdf/1503.08359.pdf

          Dear Tim,

          I found your slides of a 2013 talk. The number of DLS is 1,4,64,4096,1048576 (Sloane sequence 1053763). It is the number of simple digraphs (without self-loops) on n labeled nodes. It also corresponds to the number of nilpotent n x n matrices over GF(2).

          May be this property of nilpotency makes sense for a space time as it does for quantum mechanics (Rowlands). Your approach opens many perspectives.

          Michel

          Dear Tim and dear Spencer,

          Are mathematicians in position to at least partially correct mistakes? I was surprised reading in Wikipedia: "once known as the topology of point sets, this usage is now obsolete".

          Of course, while it is possible to attribute a direction to a measure e.g. to a piece of a line, a point doesn't have a direction. Moreover, a set of continuous 1-D pieces can constitute any continuous line while Dedekind just begged to believe that a "dense" amount of 0-D points may constitute a continuous line.

          I agree with Spencer on that the symmetry between past and future cannot be avoided just by means of the otherwise necessary return to Euclid's notion of number. I tried to explain in a discussion with Cristinel Stoica that the direction got inevitably lost due to abstraction from reality to model.

          Serious criticism of my arguments is welcome.

          Regards,

          Eckard

          Dear Eckard,

          It is not clear what it means to apply a direction to an arbitrary measure--think of a volume measure, for example--an in any case we are here talking about geometrical structure that can be defined at a sub-metrical level, without any appeal to measures. Of course, I am not suggesting attributing a direction to a point but only to a line, and every line contains at least 2 points.

          Dedekind was right, of course: the "real line" is just the set of real numbers, and the standard (Lebesgue) measure over the reals gives every individual number measure zero. And the set of reals forms a continuum by any reasonable definition. Since the reals can be put in 1-1 correspondence with the points in a Euclidean line, one can see how a continuum it constituted from 0-dimensional points. I'm not sure what problem you find with Dedekind here.

          Cheers,

          Tim

          Dear Tim,

          If I recall correctly, Euclid spoke of the Unity (1). I asked myself what he meant and perhaps it was my idea to translate it into measure because two units of distance, area, or volume are two according measures. A positive measure like length is naturally directed from smaller to larger. Maybe, I was inspired by one or several of the books and papers on history of mathematics I read mostly in German. I mainly recall O. Becker and W. Gericke but also B. Bolzano, G. Cantor, R. Dedekind, H. Ebbinghaus, A. Fraenkel, F. Hausdorff, J. König, C. Lanczos, D. Laugwitz, Sh. Lavine, W. Mückenheim, D. Spalt, and H. Weyl. I forgot some names, in particular a Spanish sounding one and a Catholic mathematician. Dirichlet, Weierstrass, Heine, and others were more or less involved in the replacement of Euclid's geometric notion of number by the elder and more primitive pebble-like points.

          My essay reminds of the contradiction between something every part of which has parts of non-zero measure (the continuum alias aleph_1) and something that has no parts (a rational, i.e., zero-measure element of aleph_0). Dedekind claimed having filled the gaps by creating irrational numbers. Actually he didn't create a single new irrational number. Dedekind's downward definition by a "cut" proved of no use in contrast to the feasible upward approach by Meray and by Cantor who merely ignored that it is impossible to single out an element from an infinite amount of them. Was it warranted to generalize known limits? Nobody doubts that the limit of 0.999999... is one. However, equivalence is not the same as identity, and the limit pi has no exact numerical correlate. A measure cannot be rational and irrational at a time. Finite and infinite exclude each other.

          By the way, I see my reasoning confirmed in Wikipedia:

          "Any closed interval [a, b] of real numbers and the open interval (a, b) have the same measure b-a".

          I see the academic distinction between open and closed not justified because single points in IR don't matter at all. Their location in IR is not even completely addressable. Isn't this an obstacle for the bijection you are referring to? I see rational numbers as truncated real ones.

          Cantor's transfinite cardinals remind me of his failed attempt to convince cardinal Franzelin of his infinitum creatum. So far, nobody even tried to object when I mentioned that only aleph_0 and aleph_1 proved useful. Cantor's naive point set theory seems to be just a historical burden. If Dedekind did also offer mistakes - and meanwhile I am sure he did - this is much less obvious.

          Cheers,

          Eckard