Dear Daniel,

Excellent writeup! A few remarks come to mind. First, regarding relational theories:

1. I would like to point out that there are at least two very different types of relations that play crucial roles in fundamental physics. 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. 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.

2. Different approaches to fundamental physics place different emphasis on these two types of relations. For example, shape dynamics takes the symmetric relations as fundamental, and causal set theory takes the asymmetric ones as fundamental. The theory of causal dynamical triangulations takes both to be fundamental.

3. My own belief is that there is one fundamental type of relation, but I am not sure that it is strictly symmetric or strictly asymmetric. I think it is "mostly asymmetric," and hence I refer to it as the relation generating the causal order, but one should bear in mind that this is a definition of what causality means and not a hypothesis. The hypothesis I make is that this relation is sufficient to describe both metric and causal structure. If you are interested, you might look at my essay here On the Foundational Assumptions of Modern Physics.

4. Another possibility is that there is some type of duality in which spacelike relations can be recovered from timelike relations and vice versa. I mention some metric recovery theorems in my essay (not my theorems, you understand) that allow for recovering most of the metric structure from causal relations.

5. You remark that "relativity is almost completely relational." True, but the "almost" is important. Time-travel paradoxes and various other problems arise from the fact that it is not completely relational.

Regarding relationships among time, space, objects, and motion, and category theory.

1. I think you are right on target by suggesting that perhaps any one of these concepts can be given meaning in terms of the others (page 6 of your essay).

2. I agree that category theory (and more generally, graph theory) is a very promising language in which to develop this view.

3. My viewpoint on this is somewhat different from that of Baez. I think that in certain important ways elements (events), relations, objects, and morphisms can all be viewed interchangeably. Viewing morphisms as objects arises in multicategory theory. Viewing elements as objects arises in the theory of categorification. I mention both of these in my essay, but only briefly.

In conclusion, you have touched on a lot of points that I have thought about too, and I think you have done so in a very promising way. I would appreciate any remarks you might have on my submission if you get a chance to read it. Take care,

Ben Dribus

    Dear Bejamin

    Thanks for the comments, I´m really glad to see your post and I think we can have a very interesting discussion. I will adress each of your points.

    ''1. I would like to point out that there are at least two very different types of relations that play crucial roles in fundamental physics. 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. 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.

    2. Different approaches to fundamental physics place different emphasis on these two types of relations. For example, shape dynamics takes the symmetric relations as fundamental, and causal set theory takes the asymmetric ones as fundamental. The theory of causal dynamical triangulations takes both to be fundamental.''

    It´s a good thing that you remarked these two different types of relations. Shape dynamics recovers GR, including its whole causal structure, from relational first principles (which as you put, is based on symmetric relations). I don´t know very much about causal dynamical triangulation and causal set theory and about the main motivations for them to be dealing with assymetric or symmetric (or both) relations. But an interesting point about shape dynamics is exactly its motivation: it comes from a definition of motion at the classical level that cuts off (in a sense) the unobservable structure of absolute space and time, which was historically introduced in part due to the ''accident'' that we live in a nearly perfect stable enviroment (rotation of earth substitutes unobservable time parameter, distant stars substitute absolute space). Had humanity appeared in a different enviroment, would we ''need'' to introduce absolute structures? Barbour´s theory say we wouldn´t. When applied to a 3-D metric field theory, the result is GR. There are many more interesting results.

    As regards to your points 3 and 4, I will take a careful look at your essay.

    ''5. You remark that "relativity is almost completely relational." True, but the "almost" is important. Time-travel paradoxes and various other problems arise from the fact that it is not completely relational. ''

    That is true and interesting. The relation between GR and shape dynamics (which claims to be completely relational) can be found here.

    ''1. I think you are right on target by suggesting that perhaps any one of these concepts can be given meaning in terms of the others (page 6 of your essay).

    2. I agree that category theory (and more generally, graph theory) is a very promising language in which to develop this view. ''

    Good to know we have similar views. One thing I would like remark about that is: by seeking semantic completness (giving time and ''position'' a meaning) at the classical level Barbour gets to GR via relational conceptions of motion. But the relational programme is not complete in this semantic endeavour: the notion of object, or ''state'' remains primary. What would happen if we attached a meaning to it? If Barbour got to GR, could a semantic complete formalism for dynamics bring us any closer to quantum mechanics? One interesting point is that if we use the categorical approach and definitions I presented, the semantic functor has tight relations with the concept of ''observation''. In the traditional Machian picture, two configurations ''mean'' the same if they are the same upon observation. That is, two configurations of the universe related by an absolute translation are the same because this translation cannot be detected/observed. This may point a direction on why observation is so special in QM: it could be the basic ingredient of the semantic functor. It could be a reason for why ''observation'' is so different from all other physical phenomena.

    If you have any thoughts or if you would like to investigate this more deeply, please let me know.

    ''3. My viewpoint on this is somewhat different from that of Baez. I think that in certain important ways elements (events), relations, objects, and morphisms can all be viewed interchangeably. Viewing morphisms as objects arises in multicategory theory. Viewing elements as objects arises in the theory of categorification. I mention both of these in my essay, but only briefly.''

    I feel N-category theory seems the perfect language for studying the fundamentals of motion. I also feel it is perfect for extendind machian thoughts as I explained above.

    Best Regards,

    Daniel

    Dear Lawrence

    Thank you very much for the comments. Barbour and his collaborators are trying to bring the relational conception of motion to quantum mechanics to create a full machian quantum gravity. I don´t know their work in details, but you may like to take a look at Barbour´s and Sean Gryb´s essays and ask them directly.

    Best regards,

    Daniel

    Dear Paul

    ''As with any such attribute, it can only be calibrated by using a reference.''

    I feel motion can only be defined using a reference. When we say something has moved, it must have moved in relation to some other thing, even if it is in relation to the invisible absolute space.

    ''To establish motion (and other features such as size, etc) we conceptualise reality as being in a spatial grid.''

    This grid is unobservable and the question immediatly comes: can we make physics without that grid? Barbour´s research argues that we can.

    ''As at each subsequent point in time any alteration can be identified.''

    Maybe we don´t need the grid to identify alterations, as Barbour´s research has argued. The question then becomes ''what is the most fruitful conception of motion?''

    Daniel

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    • Post #51

    I just read Barbour's paper. It along with his video on FQXi makes reference to similar work you have. I suggested a way to think about this using null rays on his blog site.

    Cheers LC

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    • Post #52

    One can look at this according to how one axiomatizes a space. If you have a space in n dimensions one can represent the positive tensor dimension as ||| ...|•ε = 0, where | represents an element such as a vector or spinor and the set |||...| means an exterior product of these. The ε means a Levi-Civita symbol and this is a skew product. This can be seen equivalently as a skew symmetrization of the |||...| in a higher dimensional space. If this is zero, then the space of tensors is symmetric. This system however requires there to be the |||...|•g, where g is a symmetric tensor. This way of thinking is what might be called Penrose-ology. Again this is equivalent to a symmetric trace in a higher dimensional space. The "dimension of these tensors" are n and -n respectively. They correspond to the symmetric and antisymmetric sets of tensors, which have a duality.

    This duality between symmetric and skew symmetric elements, or for two tensors products of the sort

    {ψ^a, ψ^b} = g^{ab}

    [φ^a, φ^b] = ω^{ab}

    involves supersymmetry. In the case of spacetime the generators of supersymmetry Q_a and \bar-Q_b construct Lorentz boosts

    [Q_a,\bar-Q_b] = iσ^μ_{ab}∂_μ.

    The relationship between the symmetry and antisymmetric approaches, say shape dynamics and causal set theory, might then have functors to Fermi-Dirac fields and boson fields, and a system which includes both might then have a graded Lie algebra with Grassmann generators that connect the two.

    Cheers LC

    Daniel,

    I will have to think carefully about what you said and wrote about semantic completeness and observation. One of the weak points in my knowledge that has come up over and over again on this forum is the original ideas of Mach, although it seems that much of this involves covariance/equivalence principles, which I have thought about. Part of my task is just understanding the terminology, since I come mostly from the math side. Take care,

    Ben

    Lawrence,

    Again, thanks for the insight. I have been having a somewhat similar discussion about possible duality/complementarity with Sean Gryb over on my thread. I won't remark further about this particular idea here without Daniel's permission, since it's somewhat tangential to the focus of his essay, but such discussion is always welcome on my thread. Take care,

    Ben

    Dear Lawrence and Ben

    Unfortunately, I must admit I still don´t have enough technical language to participate in the discussion you have proposed, but please, feel free to discuss those matters as you wish.

    Ben, sorry, I think I was not clear enough, but tomorrow I will prepare a more elaborate reply to you to explain those machian and extensions of machian thoughts I mentioned.

    Daniel

    Dear Ben

    I will now try to explain the point I was trying to make. Let´s place ourselves in the 17th century and try to build physics from the scratch, that is, for the sake of the argument, let´s ignore any complications due to modern physics. I must firts say that Mach´s thoughts are philosophical. For someone coming from a math background this may seem extremely vague. But Mach´s philosophy has tight relations with GR, and this makes it very important.

    Due to our stable enviroment, we were led to think that there was an invisible space and time background. The rotation of the earth provided a parameter to which all motion should be labeled and the distant stars provided an infinite grid to which all distances should be measured. It was then very natural to take the mathematical gadgets such as R³ to model the physical world. Then you can build equations between 'stuff' defined on R³ such as ma=-grad(V) but this theory cannot be tested because this R³ spatial grid cannot be seen... a closer look reveals that the distant stars also seem to move! There is no epistemological way to identify the grid. The best that can be done is to find a visible object, measure distances upon it and check if ma=-grad(V) would hold (such an object would provide an inertial frame of reference). But even so we could not identify a grid defined this way with absolute space, because the inertial frame object can be moving with constant velocity in relation to absolute space.

    So, was absolute space a historical mistake? Could we formulate a physical theory without unobservable structure, in a purely relational (because all we see and measure are relations) way? This is what Mach intended, though he never wrote down a complete physical theory. But then, Barbour has shown that, in a sense, it suffices to impose a relational framework to a 3-metric field to RECOVER general relativity from first principles.

    If one recovers GR from relational first principles, it becomes compelling to study the origin of these first principles... what´s the origin of Mach´s thoughts? What I have been thinking is that maybe it is possible to see Mach´s procedure as a part of something bigger.

    Mach´s unease with classical mechanics can be seen as coming from the following: what do we mean when we say an object´s position at a time t is (x,y,z)? Mach´s criterion of meaning is OBSERVATION. To say that the position is (x,y,z) at a time t can only MEAN that the relative distance between the object and a reference is (x,y,z) and that a clock (which is a physical object) has marked t. For Mach, all statements of classical mechanics should MEAN only what can be observed upon then. Unobservable statements should be cut off from the start, because they don´t MEAN anything. What Mach was ultimately searching was MEANING, using OBSERVATION as criterion. And remarkably, this leads to GR via Barbour´s argument!

    But the relational philosophy is not complete: it gives time and position a meaning upon the concept of physical object. But the concept of physical object is still MEANINGLESS! This is why I proposed to try to build a dynamics where time, space, motion and objects all gain their meaning upon each other, to see what would be the correct mathematical structure amd what result we would achieve. I call a completely meaningful description of the universe ''semantically complete''. Now one of the mysteries of QM is the nature of observation: why is observation so different from other physical phenomena? What causes the wave function to collapse? How can we classify a physical process as an ''observation''? Category theory now comes into the game: imagine a category where objects are semantically complete descriptions of the universe and functors that send such descriptions to descrptions that MEAN the same (let´s call them semantic functors). Now it seems that the concept of ''observation'' could be given a precise mathematical meaning: it is ''that thing that is used to build the semantic functor''! And relational physics can be simply stated as ''the square commutes'' as I put in my essay! This is the outline of what I was thinking.

    I think the first step should be to cast barbour´s relational physics in a category theoretic framework. Barbour´s procedure of eliminating absolute structures is his method of best-matching, as I explained in the essay. I was thinking of trying to find the origin of best matching via category theoretical considerations. I don´t know if this is possible, but my intuition tell me it is. Once we do that, then I think everything would be easier to understand. Maybe you will find that interesting. I´ve seen that Derek Wise was trying to find relations between Barbour´s theory and Cartan geometry, which has some relations to category theory, we could investigate that. Anyway, I will be waiting for your feedback.

    Best Regards,

    Daniel

      Dear Daniel,

      I have been thinking a bit more about the last few sections of your paper. I am in the process of trying to learn several new things at once (and also write a dissertation about something completely unrelated!) so you'll have to forgive my delayed response.

      First, I appreciate your explanation in the previous post; I think I have a better idea now of how you are using certain terminology. In particular, I realize now that a large part of what you are presenting is your own ideas, so it's not surprising I haven't heard of this view before. Now let me itemize a few remarks.

      1. Regarding the concepts "time, space, object, motion," it seems that you want to define each in terms of the others. Now it seems clear that any logical or mathematical system (at least any system satisfying a suitable finiteness assumption) will have either undefined concepts at its lowest level (in terms of which the remaining concepts are defined), or will have redundancy at its lowest level (where the fundamental concepts define each other). It seems that these two possibilities are interchangeable: if you have redundancy, you can eliminate concepts one by one until the redundancy disappears and the remaining concepts are undefined. Conversely, you can define new concepts in terms of the fundamental (undefined) concepts and take these to be "equally fundamental." I suppose this is analogous to finding a basis of a vector space from a spanning set, or augmenting a basis to a larger spanning set that is no longer linearly independent. There are plenty of situations in which a larger redundant set of concepts is useful, so parsimony is not the only consideration here.

      2. It seems that "semantic completeness" as you define it requires redundancy, because if every concept can be defined in terms of others, then some of these concepts can be eliminated (at least, if there are a finite number of fundamental concepts).

      3. We must be very careful about the use of the word "object," because it has more than one meaning. It has a precise, axiomatic, but very flexible meaning in the context of category theory; as you point out, categorical objects could be Hilbert spaces, or logical propositions, or whatever. It seems to have a vaguer but more specific meaning in the sense of "physical object." When you mention defining time in terms of objects, space, and motion, the objects you are talking about in this case must mean "physical objects," such as "particles" or "fields," and to define "time," they must somehow be indentifiable as "the same object" after undergoing the "change" that defines time. In other words, I don't think a pair of different structures by itself can define time in a Machian sense; rather, it is necessary to be able to identify the "second" structure as being the "result" of "changing" the "first" structure. I use quotation marks to indicate that I am not attempting to be precise at this point! What I am trying to get at is that the concepts of "change" or "motion" require that the "initial and final states" be identified as different states or configurations of the "same object" rather than two totally unrelated structures.

      4. I am glad you pointed out the work of Derek Wise. I have not read these notes yet, and they seem very relevant to what we have been discussing.

      5. There is much more to discuss, but no time at the present to do so. I see you have an email address on your paper, and I also have one on mine... that way we can keep in touch after the essay contest is over.

      Take care,

      Ben

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      • Post #58

      Dear Ben,

      Thanks for your reply. I understand your lack of time. A few remarks to your remarks:

      ''It seems that these two possibilities are interchangeable: if you have redundancy, you can eliminate concepts one by one until the redundancy disappears and the remaining concepts are undefined. Conversely, you can define new concepts in terms of the fundamental (undefined) concepts and take these to be "equally fundamental."''

      That´s the point. I was thinking that maybe we can define ''motion'' using ''time'', but then times becomes undefined... or we can define ''time'' using motion, but then motion becomes undefined... and so on. In the end, maybe all fundamental terms: space, time, motion, physical objects could have this ''duality''. The reason that points me for thinking this is that machian philosophy (which leads to GR) can be seen as a PART of this duality!

      ''2.It seems that "semantic completeness" as you define it requires redundancy, because if every concept can be defined in terms of others, then some of these concepts can be eliminated (at least, if there are a finite number of fundamental concepts).''

      I don´t know if they could be eliminated... but the CHOICE of fundamental terms would not be unique! Again, we could consider that motion gains meaning from time, space and objects, OR that time gains meaning from motion, space and objects. Then, we could postulate that no matter how we choose to represent the universe, there should be no physical change upon a different choice of fundamental terms. Absolute and relational views of motion would be a part of the same structure... and there would also be the ''something else''!

      Ultimately, what I propose is that by investigating the ''meaning'' of classical statements about motion, we can find new ways to conceive motion, and then build physics using this conceptions.

      ''What I am trying to get at is that the concepts of "change" or "motion" require that the "initial and final states" be identified as different states or configurations of the "same object" rather than two totally unrelated structures.''

      We can also define motion without an absolute structure by introducing a ficticious ''background'' structure for the final and initial state and then eliminating it using barbour´s best matching: take for instance two configurations of point particles defined in cartesian frames, say (xi,yi,zi) and (x'i,y'i,z'i), representing distinct instants of time. Now hold the first frame fixed and perform rotations and translations of the second until some incongruence measure such as SQRT((xi-x'i)2+(zi-z'i)2+(yi-y'i)2) gets minimized. This is defined as the best-matched distance between these two configurations, and this value can be calculated using only information meaningful in the relational view of motion.

      I´ll keep in touch for further discussion.

      Best regards

      Daniel

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      • Post #59

      and one redundance and one for the sortings of datas and informations, and now you are going to make some logarythms for the sortings, we know we know.

      and after a mthematical universe proof of course of course.

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      • Post #60

      Dear Daniel,

      I completely agree that "if the appearance of observation in the semantic functor could bring us any closer to quantum mechanics" since I see "observation" as a mapping from physical states to recorded experimental outcomes, i.e. to descriptions encoded in some memory medium using classical information. So observation is itself a functor, from a category in which the objects are quantum states and the morphisms are unitary transformations to a category in which the objects are descriptions encoded in classical information and the morphisms are formal operations defined on those descriptions. The criterion for descriptive coherence is precisely diagram commutativity. But this is not your "semantic functor" which you have defined as a category automorphism. It is the function that tests whether two physical configurations "mean" the same thing from some observer's point of view.

      Such "observation" functors are very familiar: they define the semantics that we associate with computer hardware. We pretend that "classical" computers are classical. This is of course nonsense; they are quantum systems just like everything else. Nonetheless, when we look at them, we assign a semantics under which their physical dynamics is mapped to formally-specified execution traces of classical algorithms. In my view, this is what ALL observation is.

      My PhD advisor, Rob Cummins, used to tell us all to imagine that our PCs grew up overnight in our back yards. I agree: this forces us to think about the semantics we assign to physical events in a coherent way.

      Good luck with your research,

      Cheers,

      Chris

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        • Post #61

        Hi Daniel,

        To be honest, I doubt that it is reasonable to follow Barbour and somehow replace time. Your age refers to the time of your birth which is certainly not likely to be chosen as reference point for an absolute time. May we conclude that there is no absolute zero of time? I do not suggest referring to the hypothetical moment of a Big Bang. Being an old engineer, I see only the actual border between past and future a natural fix point suited to refer to in an non-arbitrary manner. You might try and find some flaw in my criticism .

        As for mathematics, it would not be unreasonable to completely avoid non-zero integration constants by agreeing on a definition of integration that always refers to the lower border zero.

        Eckard

          Dear Chris

          Thanks for your comment. I´ve read your essay and, as I said, I deeply impressed!

          ''So observation is itself a functor, from a category in which the objects are quantum states and the morphisms are unitary transformations to a category in which the objects are descriptions encoded in classical information and the morphisms are formal operations defined on those descriptions.''

          I see we have slightly different views. To describe a ''quantum system'' or a ''classical system'' we need to use structures like space and time. However, these structures may come with a large degree of redundancy, depending on how we conceive motion in the first place, as I have explained in my essay. So there should be a functor connecting all those semantically ambiguous states, and the outcome of physical process should not depend on how we describe it. This is where the diagram commutes in my view. And the functor that connects all the semantically ambiguous states should be built by using a criterion of ''meaning upon observation'' to fulfill the principle that ''empirical indiscernibles are physical indiscernibles'' (as Robert Spekkens put). My hope is that this would have relational physics and GR as a sub-product. But I see the process I have in mind could be greatly enhanced by first characterizing observation as a functor in the first place, as you said. I will have to think more about that.

          Best regards,

          Daniel

          Dear Eckards

          Barbour´s research on time deals exactly with the following redundancy: if eveything speed up in the universe, including clocks, we could never tell the difference. So, why not consider time an abstraction from motion instead of an invisible parameter?

          This is the relational view of time, and by combining with a relational view of space, one gets a completely different mathematical and physical structure than those of absolute view of motion. So the point is that distinct conceptions of motion may lead to distinct physics, and I have argued in the essay that maybe we´re not limited to absolute or relational conceptions.

          Best regards

          Daniel

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          • Post #64

          Dear Daniel,

          The contests solicited to reveal wrong assumptions. As an engineer I know how to describe an invention. Accordingly my essay begins with a list of obvious shortcomings. Will your suggestions provide solutions to such enigma and suspected flaws?

          I do not deny that Barbour's approach might be reasonable to some extent. I merely doubt that it will help for instance to overcome Einstein's tense-less view. I rather see it a step in the wrong direction. See 1364.

          Best regards,

          Eckard

          ''The contests solicited to reveal wrong assumptions. ''

          As I say in the abstract, ''It is argued that there may be other ways to conceive motion and that a systematic investigation of these different conceptions may produce new physics. ''

          ''Will your suggestions provide solutions to such enigma and suspected flaws?''

          Who knows. Everybody is speculating to some extent here.

          ''I rather see it a step in the wrong direction.''

          Even if it is a step in the wrong direction, my conclusion remains: different conceptions of motion at the classical level may lead to distinct physics.

          Best regards

          Daniel

          Dear Daniel

          Reading your essay I could see you have the talent, mental stamina and imagination to discover new physics. The problem of understanding motion is a very interesting one. You promote Barbour's theory which I value for relegating time to an emergant status, but think the SS idea unnecessary theoretical complication. Mach's ideas are very interesting, but there are many ways to interpret them.

          Starting from very different premises developed in my 2005 Beautiful Universe Theory on which I based my fqxi essay Fix Physics! . I concluded that motion is the translation of a pattern (a concept you use) in the energy and orientation of dipolar nodes making up a universal lattice. In my theory general relativity becomes very simple, but motion involvs a self-convolution of the pattern not only of the object involved but (a la Mach) the nodes making up the surrounding gravitational field, on to those of the entire universe. For example a force on matter causes the pattern to compress first, and then start moving with its length contracted, 'pushing' the external gravitational node pattern ahead as well. This combines Newtonian SR and GR notions. I attach a figure from the BU paper to explain this qualitatively.

          I wish you all success, and urge you to stick close to simple physical concepts - one can easily get too abstract using clever mathematics.

          VladimirAttachment #1: 2_BUFIG26.jpg