Dear Peter,

You're absolutely right... I did neglect to post my remarks on your thread. Things got a bit chaotic there for a while, and my own thread became rather busy, though not quite a busy as yours! In any case, I see you are near the top for the second year in a row, so I congratulate you.

I read your essay a couple of times before, but I should read it again before attempting to address you seriously. I do recall having a couple of questions about the scenarios involving multiple media in your scene 2, and the exact implications of the optical axis rotation on page 7. Also, I'm not quite sure in what context you refer to the "invalidity of motion in geometry," both in your essay and in your remark here. For instance, timelike curves in Minkowski space certainly "represent" motion in an obvious sense. I happen to agree that motion is derivative, but then of course I view geometry itself as derivative!

Anyway, thanks for the gentle reminder... I will certainly take another look at your work. Take care,

Ben

Ben

Re 1: Yes, but that answer would need an awful lot of caveats.

Re 2: Because the common assumption is that time is a characteristic of reality (specifically, a dimension), so one can end up using the same incorrect phraseology in refuting that!

The underlying point here being that, in terms of existence, there are only physically existent states (or realities) of whatever constitutes our reality. When compared, difference is revealed, which indicates alteration/change. Difference is not physically existent, neither is alteration/change. But these concepts reflect physicality, they are not an illusion. What they refer to are the characteristics of the difference between realities, not features of any given reality. So timing can only be an extrinsic measuring system, as its purpose is to calibrate the relative rates at which changes occur (by comparing numbers thereof within different sequences against a duration reference, ie time).

Re 3: It is not an assumption. By definition, ie given how our reality must occur, the identification of any given physically existent state (ie a reality) involves establishing what existed at any given time. There is no change in a physically existent state, otherwise it cannot exist, and then alter, because the incidence of change means there is more than one state involved. The choice of what time is irrelevant in that it does not determine the state, just what happened to exist then. But, as with any such form of comparison, having selected a reference this must be maintained in order to ensure comparability of different measurements. In other words, that same point in time would have to be used to discern what other physically existent states occurred at that time in other sequences. They do not have their 'own time'. Only different times if they existed at different stages in the sequence.

[Just for the record! This misconception lies in Poincare's flawed concept of simultaneity, and became the surrogate variable for dimension alteration (usually referred to as length contraction), which Lorentz, et al, were uneasy with, ie time variance seemed more rational. So the tail got pinned on the wrong donkey, so to speak. In repeating simultaneity in 1905 (the A-B example, 1st section), Einstein went a step further, thereby compounding a felony, by using light speed to assess distance, but this was effected in terms of A to B, and then (ie subsequently) back (B to A), thereby, again, reifying time as a feature (a variance) within a reality. Apart from inadvertently attributing light speed with some relationships which are non-existent. In the meantime(!) Minkowski reified time with his spacetime model, following through the flawed concept of simultaneity.

t = x/v is an expression of what timing is. The number always equals the sum on the other side, and the property is the same on both the top and bottom of that side. So it is an expression that timing is rate of change (1/t, per time). Which it is. Having understood this, one then has to be very careful how alternative variables are introduced into this 'equation'.]

Re 4 To which I would say, does the concept of superposition have any existential substance? Sequence is different from time, and neither is this to do with a 'classical picture'. The differentiation of classic/quantum is a false dichotomy, there is only one form of our reality, with distinct generic features. Sequence is how change occurs, ie one at a time. It is like a film. Time is about the speed of change. At the existential level, it might be that all change is a function of the same feature, or different features which function at the same speed. So however change eventually manifests, it is, at that level, occurring at the same speed, which then makes the concept redundant, at that level. But this needs to be proven first. And anyway, deconstructing physical existence to this level is probably impossible to achieve, and certainly does not help in understanding most situations. Try describing one physically existent state of the 'object' known as The White House. Trillions of dollars worth of equipment and a million years later, we would probably still be at it!

Paul

Ben

In scrolling down I noticed the post above on symmetry. After our first exchange I did have a think about those 6 key principles you refer to, and came to the immediate conclusion that symmetry (amongst others) was a meaningless concept, given the fundamental nature of our reality, but did not want to 'overload' the exchange. However, since you indicate above that this is important, I looked at your post (19/9 17.29) to David Alves. In it you say:

"Shape dynamics deals principally with symmetric relations, since the separation between two points has nothing to do with their order; X is a distance D from Y if and only if Y is a distance D from X"

Indeed, but this is just a truism determined by how existence occurs. There is no symmetry, or relations, involved. [Incidentally, this happens to relates back to my point about the AB example, which I threw into my last post]. Distance is a function of the relative spatial footprint, and the relative spatial position, of the physically existent states involved. It is an artefact and has no physical presence. One has to presume, until proven otherwise, that as at each subsequent point of time, a change will occur to one or more of the physically existent states involved which will impact on distance. But, in reality, the distance has not altered, the physically existent states have. In other words, it is no longer X &/or Y, it is a distance being different states. So, yes, by definition, X to Y must be the same as Y to X, but it is not a case of "only if", because what must be under consideration here is an existent state which involves no change. The distance is just the distance is just the distance, as are the physically existent states which determine it.

Then you say:

"Spacelike separation in relativity is similar. However, two events may also be causally related, and in this case the relation between them is generally asymmetric because the order matters; X is in the causal past of Y if and only if Y is in the causal future of X. In all but extreme cases, causal relations correspond to timelike separation"

Indeed, but again this is just a truism determined by how existence occurs. There is no asymmetry, just a sequential order of different states. Neither is there asymmetry in the sense that the other circumstance had symmetry, because there was none. In the first circumstance there was no sequential order, only one state within such an order. Now the circumstance involves more than one state of the order.

And, by definition, for any given physically existent state to have had a physical influence on any subsequent one, there must have been a specific relationship between the two in terms of both sequential order and relative spatial position. So, in terms of sequential order, a potential causal state must have been one of the immediate predecessors, because physical influence cannot 'jump' sequential order. Then in terms of relative spatial position, for any given state to be a potential causal factor, it must have been adjacent to, or in, the spatial position of the successor, because, again, physical influence cannot 'jump' spatial position. Given these criteria for potentiality of being a causal factor, in any given circumstance the causal factor could have stemmed from a) a previously existent physically existent state altering of itself, b) any combination of influence from all the other previously existent states, c) any combination of a & b.

Paul

Dear Paul,

I think most of our mutual difficulties come from different conceptions and uses of the terms "existence" and "reality." In particular, this leads you to insist on one hand that a particular entity "does not exist," or "is not real," and at the same time insist that such an entity has particular properties and must be understood in a particular way.

For instance, you say, "Difference is not physically existent, neither is alteration/change. But these concepts reflect physicality, they are not an illusion." Similarly, you insist that time is "not real," yet insist that it is an "extrinsic measuring system!"

What I am trying to do is propose a model that may be useful in describing the physical world. Trying to argue about whether or not certain aspects of this model are a priori "physically real" on philosophical grounds is a waste of time. Ultimately, nature is the judge of whether or not a theory is useful. For instance, it's useless to argue against the superposition principle on the philosophical basis of whether or not it has any "existential substance." The fact is, it seems to describe how the world works. Similarly, the classical/quantum "dichotomy" is not really a dichotomy, but just reflection of archaic language. Again, "E pluribus unum" seems to describe how the world works.

Symmetry and asymmetry are not meaningless; they have very well defined meanings. The extent to which they are physically important is another matter. I argue that symmetry is slightly less physically fundamental than most physicists believe. To argue that it is physically meaningless, however, seems again to ignore the evidence of nature on philosophical grounds. Take care,

Ben

Dear all,

Some time ago, Jonathan Dickau raised the question of what the correct formulation of path summation is in various contexts. He attached a well-written paper by Steven Kauffman advocating a Hamiltonian phase space path integral approach and deprecating the Lagrangian approach. I would like to make a few remarks about this.

1. In causal theory, the natural version of path summation involves paths defined by order morphisms from linearly ordered sets into the causal order. Assuming local finiteness and acyclicity, these may be identified with sequences of relations. In terms of acyclic directed graphs, such paths are just sequences of directed edges.

2. The natural object to sum over in this context is a "relation function;" i.e., a map from the set of relations to a ring, which conventionally is the complex numbers, but may be some other algebraic object. Finite fields, quaternions, octonions, etc. are interesting in this regard.

3. There is a technical information-theoretic reason why an "element function" won't work: "antichains of elements are permeable."

4. This relation function doesn't correspond directly to a Lagrangian, but to a phase, which in Feynman's Lagrangian approach is the exponential of the classical action. The action comes from integrating the Lagrangian with respect to time, but in the locally finite case, the integral is a sum, and a single relation represents the smallest subdivision of a path. In this sense, the relation function is an "infinitesimal path functional," which is what a Lagrangian is in abstract mechanics. This is the sense in which the approach is "Lagrangian;" it need not have anything to do with conventional Lagrangians or actions.

5. Path summation over a single classical universe violates background independence, so the path sums I have in mind are over causal configuration space.

6. Conventional phase space involves "position" and "momentum" coordinates, and the Hamiltonian is an "energy function" on 2n-dimensional phase space. However, "momentum" is emergent in causal theory. As of now, I don't know how one would could carry out background independent "phase space" path integration in the context of causal theory, or if this even makes sense.

So further thoughts on this would be appreciated! Take care,

Ben

    Ben,

    Very wise to read again, and slowly. There are a whole related set of quite new assumptions to assimilate. The scene 2 (Act 1) media do have more different scenario's than we've assumed. I really needed a dozen more pages! Perhaps read my recent posts to Eckard & Pentcho.

    But the concept is simple and physical. Six buses and six clouds of electrons flying round in space all represent different and mutually exclusive 'inertial frames', and domains for Maxwell's (near field) equations. Each then has spatial limits, with a boundary state and mechanism that absorbs and re-scatters to the local c precisely like a fluid dynamic coupling (as an auto gearbox torque converter). It is variously, magnetohydrodynamic shocks, surface charge/fine structure, the far field transition zone, the heliosheath, galactic halo, etc. It contains Boscivitch's 'sphere of influence' Einstien's 'spatially extended mass' and represents Galileo's states of motion, so, yes, derives discrete space time geometries, and at ALL scales, from a particle to a universe.

    The structure is precisely that of Truth Functional Logic. Substitute 'frame' for 'proposition' and quantum relativity emerges. It's also continuous spontaneous localisation generalised to matter and systems in common states of motion at all scales.

    The whole 'concept' of motion is of course invalid in geometry. Descartes tried a solution, but space and matter are 3D. It is the 'whole geometry' that moves, as a frame, not some unreal 'wire frame' where the wires have to cross. 'Getting wires crossed' may assume a new meaning!

    Optical axis rotation was the last big piece that fell into place to complete the puzzle. If a plane wave in a medium hits ('charges') a beach ball or gyroscope (ion) at rest in the medium, which absorbs and re-emits it, the re-emission may be expected to be symmetrical (an ion/plasma is 'self focussing') and at c wrt the ion.

    But if the ion is in motion through the medium fings appen! (effects from causes). Motion on the x (source) axis causes change to lambda L and, inversely, frequency f, (new but entirely obvious) which conserves c (c = fL).

    Lateral motion v means the charge cannot be symmetrical. Ergo the re-emission is likely to be rotated, and in proportion to v. All of a sudden we find this maps atomic scattering direct to some of the most poorly understood physics we have; Stellar Aberration, Kinetic Reverse Refraction (the same thing). lensing, a whole host of related phenomena in optics and astronomy, and, yes, when we include harmonic resonance subject to f, and Huygens construction, apparently quantized refraction and 'curved space time.' Scary indeed. But it all needs clearer translation from English and logic into into physics and maths. It just needs someone who speaks both and also understands it.

    That's almost another essay! but just scraping the surface. Do just ask every question you think of. I'll then link you to more if you can take it.

    Thanks

    Peter

    Ben

    I am not aware of any contradiction, although at times it is difficult to find an exact expression, because our very language incorporates a particular view as to how our reality functions. The differentiation is between something that is (or was) physically existent, and an artefact which reflects that, but cannot, of itself, be existent. Though your repeated reference to time is different from that, it being just a human devised measuring system, and therefore, obviously, has no physical existence.

    And whilst that might appear to be somewhat pedantic or philosophical, it is not because it highlights the crux of the problem. And this is contrary to your assertion: "Trying to argue about whether or not certain aspects of this model are a priori "physically real" on philosophical grounds is a waste of time. Ultimately, nature is the judge of whether or not a theory is useful".

    The argument is not philosophical, but physical. Because, before any analysis can get underway as to what (ie content) is happening, we must first establish how existence is detected, and then how that (ie our reality) must occur generically (ie form). Form comes before content. And our reality has a definitive, independent, physical manifestation of its own, and there are physical processes which enable its detection. All of which is discernable, and thereby invokes a set of pre-conditions which must not be contravened in the analysis, construction of representational devices, etc. For example, given the fundamental characteristics of our reality ("nature"), the question then becomes, does the concept of superposition have validity? And the answer is: no, because it contravenes how our reality must function.

    In respect of symmetry, again in the context of the form of our reality, I was pointing out that the first circumstance had no symmetry in it, there was nothing to be symmetrical with. And the second circumstance was not asymmetrical, it was just existential sequential order. Any concept of 'a direction of progress' (ie there is more than one possibility) is a fallacy. Which leaves me wondering, again, whether the concept of symmetry has any meaning. The more interesting point was about distance. This is another artefact. And there can only be distance between existent states which occurred at the same time. There cannot be a distance between an existent state and one that has ceased to exist, or indeed has not yet existed. All of which points to the necessity to understand the sequential order of the sequences and their relative status, otherwise we will have a jumble of hypothesised physical influences which actually could not have occurred. And since we depict a reality at a higher level than that at which it exists, and we have deemed that change (aka time) can be a characteristic of it, then it is highly likely we have that jumble.

    Paul

    ahahah learn french and read my poems ahahah uncle spherical gives you a lesson.and you know it.Don't try with the faith and the spherization. You are not really relevant in fact.a simple sockpuppet player is your name,in fact you are only good for the computing, the rest oh my god but what is your foundamentals. ahahah in fact you are only good for these kind of play.not for the gnerality.You repeat always the same in fact. ahahah frustrating no? me I have faith, you no.It is there your error sockpuppet inventing false names. Even your maths are always the same. In fact several people fear for their funds. They fear that their institute looses a little of their credibility.So they utilize the competition like excuse, but in fact they are just pseudos. They prefer the publicity instead of generalities. They are simply people lost in the meanders of the society. You confound a little the things in fact.You do not evolve, you just make a pub because you fear for your jobs perhaps I don't know.Or it is just the taste of monney who interests you.In all case, it is not a good parameter that you utilize. You have just simply profitted about a discovery, revolutionary and you try simply to prove to yourself that your hormons are ok. Have you probelms in the bed with women or What ? if you want I can give courses. I am understanding your hate.You can evolve, buy a bibble I don't know me. I can also give courses of maths, piano, physics, theaters,poesy,guitar,taxonomy of evolution,indeed I class all.You know animals and vegetals I suppose. And you try with what ? the spherization, ahahah let me laugh.I have quantized the mass me, you no. I am arrogant I agree but it is the reality. The story begions dude, you know the evolution and its optimization spherization, we are young, we are aged of 14 billions years dude, I have time to explain you the spherization.I am going to explain you what are the maths of spheres of light and its uniqueness giving the singularities. You think really that my soul of spheres of light is going to accept your play.You can even kill me you know.We are eternal my friend, you shall be head in the soil in front of the infinite light poor thinker. Uncle what? make surf dude, and pray for your soul. All is said in fact. Jesus Christ and Siddartha Gottama Buddah are with me.Ahahah I have the syndrom of the elected. and what ? what is the probelm, put me also on a cross also if you want, I will laugh in front of your face, I will resist with universal love and faith.you can even burn me if you want,I will resist poor dude.

    You are already in the bad boat.

    Spherically yours band of comics.

    Ben

    At the risk of causing a riot, a problem with that Smolin/Magueijo paper is that it presumes SR includes gravitation in the first place. Which it does not. So in that sense, there is no issue to resolve. Einstein defined SR several times, and it is not 1905.

    In 1905, the two postulates are "apparently irreconcilable". Which is a bizarre statement when juxtaposed against the assertion that "These two postulates suffice for the attainment...theory...electrodynamics of moving bodies". The irreconcilabilty is because light is presumed to be in vacuo, whilst everything else is not, because they are subject to dimension and momentum variance. And the cause of that is subsequently revealed to be gravitational forces. In other words, light and matter cannot be co-existent and so 1905 is not a singular, cohesive, theory. So in expounding GR, where everything (ie including light) is subject to gravitation, Einstein refers to a special/entirely theoretical circumstance where there was no gravitation, ie SR. This is why SR only involves uniform rectilinear and non-rotary motion, light that travels in straight lines at a constant speed, and rigid bodies.

    Einstein: Relativity 1916 section 18: "...the special principle of relativity, i.e. the principle of the physical relativity of all uniform motion..."

    Einstein: Foundation of GR 1916 section 3: "...the case of special relativity appearing as a limiting case when there is no gravitation"

    Einstein: Relativity 1916 section 28: "The special theory of relativity has reference to Galileian domains, ie to those in which no gravitational field exists...In gravitational fields there are no such things as rigid bodies with Euclidean properties; thus the fictitious rigid body of reference is of no avail in the general theory of relativity"

    Einstein: Relativity 1916, section 22: "Let us further investigate the path of light-rays in a statical gravitational field... Let us find out the curvature which a light-ray suffers. [equation (74) refers] ...a curvature of rays of light can only take place when the velocity of propagation of light varies with position...

    Paul

    The relationship between position configuration variables and momentum can first be seen by looking at the diagram I attach. This illustrates a scattering process with 5 input momenta and 5 output momenta. The "blob" is the region with virtual or off shell processes, which can be realized by on shell processes by BCFW recursion. I will ignore that matter for the moment. The momenta labeled with number 1, 2, 3, ..., 10 must all add to zero if we reverse the sign of the outgoing momenta. This is a trick used in working out the S, T, U channels. These momenta can then form a polygon. It is tempting of course to think of the configuration variables as defining the momenta by p1 = x1 - x2, p2 = x2 - x3, and so forth. However, this polygon has a dual polygon, which may be constructed by drawing a line through the mid points of the momentum edgelinks and then finding where these lines intersect. This will be the dual polygon. This dual polygon is then the vectors which represent the position variables of these particles. This is the more sophisticated way of such a representation for it does not rely upon an explicit reference to either set of variables to derive the other, but rather depends upon a duality principle.

    In three dimensional space the diagram is more complex and the polygon is replaced with a polytope in three dimensions. Further, since the diagram on the left is really a spacetime diagram, with time running to the right, the polygon is really replaced with a polytope in four dimensions. The fundamental polytopes in four dimensions are the 24-cell, which is self-dual and the 120-cell that is dual to the 600-cell. In order to construct a one to one self duality between momentum and position configuration variables the 24-cell is the obvious model. For systems with more particles than can be represented by a single 24-cell tessellations of 24-cells may then be considered.

    The 24-cell is a representation in Hurwitz quaternions of the F_4 exceptional group. The F_4 group shares a relationship with the B_4 = SO(9)

    F_4/B_4:1 --- > spin(9) --- > F_{52/16} --- OP^2 --- > 1

    And of course spin groups have a double cover to orthogonal groups

    1 -- > Z_2 --- > Spin(n) --- > SO(n) --- > 1

    The group SO(9) plays an important role with string theory or holography. Physics on an infinite momentum frame, physics observed boosted enormously, reduces the relevant physics to one dimension less, and the observed physics by time dilation is effectively nonrelativistic by time dilation. I illustrate this briefly below. This reduces the 10 dimensions of supergravity to 9, and the symmetry group of this spacetime is then SO(8,1) ~ SO(9).

    It is easy to how an extremely boosted system appears nonrelativitstic. We consider the invariant momentum

    m^2 = E^2 - p^2,

    here with c = 1. The energy is then

    E = sqrt{p_x^2 p_y^2 p_z^2 m^2}

    We then consider the momentum p_z as enormous, and p_z >> p_x,y. We factor this out with

    E = p_z sqrt{1 (p_x^2 p_y^2)/p_z^2 m^2/p_z^2}

    Where binomial theorem gives us

    E = p_z (p_x^2 p_y^2)/p_z m^2/p_z

    The momentum p_z then acts as a time dilation factor approximating a Lorentz factor. We may group all of this to define a new energy or Hamiltonian

    H = (E - p_z)p_z = p_x^2 p_y^2 m^2,

    where the right hand side of this equation is a nice classical nonrelativistic Hamiltonian. The mass squared then plays the role of a potential energy.

    Cheers LCAttachment #1: scattering_polygon.GIF

    Lawrence/Ben

    Bear with me while I put up what may appear like a simplistic comment, but sometimes a person without all the background sees the 'wood for the trees'. I have no idea how this could be represented as a model, let alone analysed in practice, but the logic of our reality must be:

    In establishing what constitutes dimension, distance and space in our reality, it must be recognised that we are, in effect, conceiving of any given physical reality (ie physically existent state) as if it was being divided into a grid of spatial positions. And in order to deconstruct it to its existential level, the 'mesh' size of this grid would have to be equivalent to the smallest physical substance in our reality. [Note: there is no form of change within any given state of physical existence within our reality, only spatial characteristics, because it can only occur in one such state at a time].

    Only physically existent states exist, being comprised of physical substance. That is, concepts either reflect that physicality, or are an artefact of it. By definition (ie what constitutes physical existence within our reality), any given physically existent state must have a definitive dimension/size/shape (ie spatial footprint), this being a function of its constituent physical substance. That, with reference to the conceptual grid, can be defined as spatial positions 'occupied'.

    It could be argued that a direct comparison between states is possible, and therefore there is no need for the concept of a grid. But this is a fallacy, because logically the two circumstances are the same. The physically existent state used as a reference is just a surrogate grid. Indeed, in order to ensure compatibility with other comparisons, that state would have to be maintained as the reference (ie a de facto grid).

    'Mapping' other states that were existent at the same given time, would reveal not only, obviously, both the spatial footprint of those states and their comparability with each other, but also, distance. That is an artefact, a function of the physicality of the particular existent states involved. It is a difference, defined by comparison. So, there cannot be a distance between physically existent states which existed at other times, because there cannot be a distance to a physically existent state which does not exist. Distance is usually measured between the two nearest dimensions of the existent states, but could include any combination of dimensions. And depending on the spatial relationship of the states involved, distance could involve a relationship in terms of separation of the states, or one within another, that again being with respect to specified dimensions.

    Dimension is a specific aspect of spatial footprint (ie spatial positions 'occupied' when existent). It relates to the distance along any possible axis. So, three is the absolute minimum number of spatial dimensions that is still ontologically correct at the highest level of conceptualisation of any given physical reality. But is not what is physically existent. At that existential level, the number of possible dimensions that any given physical reality has is half the number of possible directions that the smallest substance in our reality could travel from any given spatial point.

    Paul

    Largely what Dribus connects with is causal net, where physica can be established within a minimal set of postulates and is based on the causal succession of events. There is the shape dynamics which involves polyhedra in space and their dynamics. One involves time (causal nets) and the other involves space (shape dynamics), where I think there is some possible duality here between these pictures. I have suggested this might have some categorical relationship to supersymmetry.

    Cheers LC

    In respect of any given physically existent state, ie the existential state of our reality (or constituent parts thereof) at any given time, there are only spatial characteristics. These being a function of the physical substances which comprise our reality. Space, as in the meaning of not-substance (ie rather than spatial footprint), which is not to be confused with just a different form of substance, needs to be proven. By definition, since a physically existent state involves no change, ie it is one state in the sequence, it only has one set of spatial characteristics.

    Any circumstance where there is 'time' involves change, ie sequence, sequence order and the rate of turnover thereof. That is, the characteristics revolving around what and how a physically existent state is altering, which is derived from comparison of more than one of them in the sequence. Alteration must function in accord with specific rules relating to sequence order and spatial position, ie any given physically existent state can only be the potential cause of another if it is physically possible. Causality only relating to direct physical influence, since, by definition, all physically existent states which comprise our reality at any given time are ultimately physically interrelated.

    The issue is what constitutes a physically existent state. And given the logic of our reality, that must be associated with being in any one of the possible conditions that the properties of the elementary substances could attain. The problem is that we do not conceptualise our reality at its existential level, and therefore fail to identify the entirety of a sequence and confuse the relative timings at which any given state in a sequence occurred.

    Paul

    Dear All,

    Another idea I'd like to inject regarding the possibility of shape/causal duality occurred to me while trying to understand the "top-down causation" philosophy presented by George Ellis and others. I struggled with this a bit after first reading about it; in particular I seemed to run into trouble thinking about how one might make it precise. I recall Lawrence expressed some agnosticism about this on his thread too.

    Anyway, it occurred to me that, very generically, it might be problematic to expect duality between a theory involving classical holism (e.g. shape dynamics) and a theory with complete reductionism at the classical level (e.g. causal sets). Ellis' top-down causation, suitably represented in a pure causal theory, might incorporate a degree of classical holism into causal theory and make the idea of duality more feasible.

    (continued below)

    (continued from previous post)

    Whether this idea would work or not, one might still wonder how top-down structure could be incorporated into pure causal theory in a sensible way, and what the actual quantitative differences would be. The answer, I think, is that top-down causation elevates the binary relation generating the causal order to a "power-space relation." The causal metric hypothesis still applies here, but the vertices of causal graphs no longer represent spacetime events. Below, I copy the relevant material from the post I made about this on George Ellis' thread.

    (continued below)

    (continued from previous post)

    After initially struggling with the idea, I've been thinking a bit about

    how your [George's] top-down causation idea might look from the

    perspective of nonmanifold models of fundamental spacetime structure that

    emphasize the role of causality. It seems that top-down causation might

    provide an interesting new perspective on such models. For definiteness

    and simplicity, I use Rafael Sorkin's causal sets approach as an example.

    Causal sets, as currently conceived, are by definition purely bottom-up at

    the classical level. Causality is modeled as an irreflexive, acyclic,

    interval-finite binary relation on a set, whose elements are explicitly

    identified as "events." Since causal structure alone is not sufficient to

    recover a metric, each element is assigned one fundamental volume unit.

    Sorkin abbreviates this with the phrase, "order plus number equals

    geometry." This is a special case of what I call the causal metric

    hypothesis.

    In the context of classical spacetime, top-down causation might be

    summarized by the statement, "causal relationships among subsets of

    spacetime are not completely reducible to causal relations among their

    constituent events." In this context, the abstract causal structure

    exists at the level of the power set of classical spacetime, i.e., the set

    whose elements are subsets of spacetime. Discrete models very similar to

    causal sets could be employed, with the exception that the elements would

    correspond not to events, but to families of events. Two-way

    relationships would also come into play.

    Initially this idea bothered me because of locality issues, but such a

    model need not violate conventional classical locality, provided that

    appropriate constraints involving high-level and low-level relations are

    satisfied.

    This idea is interesting to me for the following reasons.

    1. The arguments for top-down causation presented by you [George] and

    others are rather convincing, and one would like to incorporate such

    considerations into approaches to "fundamental theories," particularly

    those emphasizing causality.

    2. One of the principal difficulties for "pure causal theories" is their

    parsimony; it is not clear that they contain enough structure to recover

    established physics. Top-down causation employed as I described (i.e.

    power-set relations) provides "extra structure" without "extra hypotheses"

    in the sense that one is still working with the same (or similar) abstract

    mathematical objects. It is the interpretation of the "elements" and

    "relations" that becomes more general. In particular, the causal metric

    hypothesis still applies, although not in the form "order plus number

    equals geometry."

    3. There is considerable precedent, at least in mathematics, for this type

    of generalization. For example, Grothendieck's approach to algebraic

    geometry involves "higher-dimensional points" corresponding to

    subvarieties of algebraic varieties, and the explicit consideration of

    these points gives the scheme structure, which has considerable

    advantages. In particular, the scheme structure is consistent with the

    variety structure but brings to light "hidden information." This may be

    viewed as an analogy to the manner in which higher-level causal structure

    is consistent with lower-level structure (e.g. does not violate locality),

    but includes important information that might be essential in recovering

    established physics.

    4. As far as I know, this approach has not yet been explicitly developed.