Part 2

Fortunately, we have one important point of agreement. You acknowledge that a theory is valuable due to its utility to solve the problems under consideration. And you also agree that if a theory assuming the PSR can explain most experimental observations, it would be worth of consideration. Indeed, this is why I support the PSR and the fluid assumption.

If we are looking for parsimonious theory, I think I have it because it is based on only one single postulate, that is, that space is a massive fluid. This postulate alone suffices to explain most of the problems of modern physics. But this requires a radical change in our present conceptions of the physical world. By this I mean, for instance, that we have to abandon the assumption that fields do not need a medium. Certainly, this is the opposite view that you and current physics hold and this is a big problem. So, I am not going to convince you or anybody else only by epistemological or ontological arguments but also by providing a new theoretical and consistent framework where the facts could be satisfactorily explained. Actually, I am working on this part.

Among the problems that my proposal has the potential to explain (or eliminate) are: the dark matter and dark energy problems, the expansion of the universe, the flatness problem, the vacuum catastrophe and the horizon problem (which from the perspective of my proposal are all illusory problems in the sense that they are the result of the choice of the fundamental assumptions in the prevailing theories such as there is no aether, fields are not states of the aether, etc.), the fly-by and pioneers anomalies, the baryon asymmetry, the GZK limit, the cosmological constant problem and so forth. Some other problems of theoretical character such as the Hawking paradox, the arrow of time, the reality of the wave function, the wave-particle duality and the nonlocality of QM can also be solved. In fact, my proposal has the potential to make QM to be utterly local. In addition to all of this, the theory I support not only will solve most of the problems but also will make the physics very intuitive. Modern theories lack this virtue and so they lose credibility. Their lack of an intuitive picture is the result of too much mathematical formulation and very few philosophical reasoning. Fortunately, physicists have realized this and they are going back to the old way of doing physics (e.g., check the Trouble with Physics of Lee Smolin or look up at the Herman Minkowski Institute on the web), actually this contest is a recognition that the old way of doing physics should be recovered.

to be continued...

Israel

Part 3

Finally, you said: guy has filmed light moving! Well, if your idea is to present this experiment as a test that the one-way speed of light or any other physical entity can be measured, I invite you to analyze it in great detail so you can convince yourself that measuring speeds is not just defining a distance and measuring the time it takes for a physical entity to travel it. This will be useful as an exercise for you to realize that the one-way speed of light can not be measured. It will also help you to understand that one has to assume a special system of reference where the one-way speed of light is isotropic (see my reference 17 to follow the approach).

You: My above comments are related to this paragraph. What I can figure out is that you are confusing the experimental implications of the theory (i.e., the predictions of the theory) with the internal consistency of the theory. From the experimental point of view relativistic effects are real, they do occur (and they are real because the PSR must exist)''

Can´t see why they are real because the PSR exist.

According to the LT objects in motion really contract and clocks in motion really dilate because to accelerate an object from absolute rest one has to apply forces. Being the aether a fluid, the aether opposes to the applied force and the object deforms (contracts). Clocks in motion relative to the aether dilate because electromagnetic fields are states of the aether and so fields have to travel longer distances in the aether as the clock moves through it. Again, imagine the light clock placed in a perpendicular direction to the motion. Light has to travel in a diagonal in the forward and backward journeys, covering a longer distance than when the clock is at rest on the aether. The faster the clock moves the slower the ticks because light traveling trough the aether will take a longer absolute time to reach the upper mirror and go backwards. So, as measured by an observer at rest in the aether, the process of completing a tick lasts more while in motion than at absolute rest.

If you have understood this picture, then you will probably understand why it is said that in SR relativistic effects are only apparent and not real. The word "REAL" implies an absolute system of reference in the sense of LT. Since in SR there are no PSRs, the effects cannot be real or absolute. To understand this problem further you have to study it deeper. My suggestion is that you should do the exercise above and consider my reference 17 and references there in. As long as a relativist does not spend some time studying this problem he will never understand it (and what is worst he will be denying it for the rest of his life).

I hope you have grasped a little the great importance of reintroducing the aether and the PSR. So, if you are interested we could be in touch far beyond this forum. In any case, I see in you someone with great potential for physics, and it would be a shame if you focus your efforts in something fruitless. It took me many years of hard work to realize where the problem was and I'll move on with this approach because I am 99.9% that I am right. If you would like to join my endeavor now or in the future you will be more than welcomed.

Good luck in your life and the contest

Best regards

Israel

Dear Israel

We have definitely come to a point of agreement. Thanks for your comments and for inviting me to participate in your research. I´m having a little bit of trouble now, because studying (enginnering, not physics) in my city in Brazil does not give me academic status enough to be admitted in the top physics universities of the world, so I´m desperately trying to convince people my independent study was effective. After a quick research I found lots of papers about one-way speed of light issue, so there´s a lot of interesting material I haven´t heard of before. Certainly, if you can show more concretely how the PSR assumption might lead to fruitful achievements I will be convinced, as the rest of the physics community around the world. I think our main conclusion was that of identifying the strengths and weakness of some different conceptions of motion... and to find a criterion for judgment: utility.

Thanks again for an awesome discussion!

Daniel

Dear Daniel

I have commented above all the problems that the reintroduction of a medium for light solves. I do not understand what you mean by "if you can show more concretely how the PSR assumption might lead to fruitful achievements I will be convinced". Do you want to see the references? If you do, check my reference 19, and look for C. Christov papers. He develops a theory assuming space as a continuous massive fluid and propose some experiments to detect it. By the way that paper also analyses some misconceptions about the michelson-morley experiment. There will also find my email, in case you would like to keep in contact.

On the other hand, If you are interested in understanding paradoxes of SR and how relativists see other people who denied the veracity of SR, I recommend you to read the essay of Chris Kennedy which is related to the clock paradox.

Best regards

Israel

Dear Israel

By ''if you can show more concretely how the PSR assumption might lead to fruitful achievements I will be convinced'' I intended to mention the problems in the big list of problems you said the PSR might help to solve. as you said:

''the dark matter and dark energy problems, the expansion of the universe, the flatness problem, the vacuum catastrophe and the horizon problem (which from the perspective of my proposal are all illusory problems in the sense that they are the result of the choice of the fundamental assumptions in the prevailing theories such as there is no aether, fields are not states of the aether, etc.), the fly-by and pioneers anomalies, the baryon asymmetry, the GZK limit, the cosmological constant problem and so forth. Some other problems of theoretical character such as the Hawking paradox, the arrow of time, the reality of the wave function, the wave-particle duality and the nonlocality of QM can also be solved. In fact, my proposal has the potential to make QM to be utterly local.''

I remain skpetic about the PSR assumption (aren´t there other ways to attack the issues that motivate it according to you?). But we shouldn´t discuss that here right now...maybe in the future. The relational approach of motion has also advantages (no unobservable statements for instance), and also has potential for explaining a lot of things, including a novel path for quantum gravity. Again I feel ''utility'' is the main word. But are we restricted to absolute or relational motion? My essay was about how distinct conceptions of motion may lead to distinct physics.

Best Regards

Daniel

Dear Daniel

Sounds good.

With respect to your question, I can only conceive two ways of modelling physics, absolute or relative; although I uphold the opinion that when something undergoes forces, it really moves.

If you have realized, contemporary physics assumes that space is only geometry (epitomized by the GR) and filled with fields (fermion, gauge, Higgs, EM). I hold the opposite view. Space is a massive fluid (not empty geometry) and the fields are states of this fluid. I found this view more useful than the other.

I also thank you for this stimulating and interesting discussion and I hope we could keep in contact in the future.

Best regards

Israel

Daniel

As per your request on my blog

Forget the theories, and just consider the fundamental nature of physical reality as is manifest to us. There being no other, unless one resorts to belief/assertion.

Motion is the incremental alteration in relative spatial position. As with any such attribute, it can only be calibrated by using a reference. Any reference can be selected, but having done so, that must be used consistently so that results are comparable. So, by definition, motion can only be known in relative terms. We are unable to transcend our existence.

To establish motion (and other features such as size, etc) we conceptualise reality as being in a spatial grid. The distance from one point to another on this grid being the equivalent of the smallest 'thing' in reality. Thus at any given point in time, A is deemed to occupy X configuration of spatial points, whilst B occupies Y. As at each subsequent point in time any alteration can be identified. That is the easy bit to say! In practice I doubt if this could be achieved. But our failure to be able to discern physical reality at its properly differentiated level does not mean we can then just embark on a false approach.

There is no existent phenomenon which corresponds with space, it is only 'things' which exist. Neither is there any time in physical reality, because the concept of timing relates to the rate at which change occurs. That is, it is associated with a feature of alteration between realities (ie speed of alteration), not of any given one.

Paul

    Daniel,

    You wrote: "...quantum mechanics uses a mathematical and conceptual formalism suitable for an absolutist picture while general relativity is almost perfect relational."

    The opposite is true: Quantum mechanics uses Newtonian concepts according to which any motion (speed), even that of light, is relative. In Einstein's relativity the motion (speed) of light is absolute. If Walther Ritz had lived longer, the name "Einstein" would be unknown nowadays:

    Walther Ritz 1908: "The only conclusion which, from then on, seems possible to me, is that ether doesn't exist, or more exactly, that we should renounce use of this representation, that the motion of light is a relative motion like all the others, that only relative velocities play a role in the laws of nature; and finally that we should renounce use of partial differential equations and the notion of field, in the measure that this notion introduces absolute motion."

    Alberto Martinez: "Two months after Ritz's death, in September 1909, his exchange with Einstein barely echoed at a meeting of the Deutsche Naturforscher und Ärtze in Salzburg, where Einstein delivered a lecture elaborating his views on the radiation problem but made no explicit reference to Ritz's views. Two years later, however, in November 1911, Paul Ehrenfest wrote a paper comparing Einstein's views on light propagation with those of Ritz. Ehrenfest noted that although both approaches involved a particulate description of light, Ritz's theory constituted a "real" emission theory (in the Newtonian sense), while Einstein's was more akin to the ether conception since it postulated that the velocity of light is independent of the velocity of its source. (...) Ritz's emission theory garnered hardly any supporters, at least none who would develop it or express support for it in print. As noted above, in 1911, two years after Ritz's death, Ehrenfest wrote a paper contrasting Ritz's and Einstein's theories, to which Einstein responded in several letters, trying in vain to convince him that the emission hypothesis should be rejected. Then Ehrenfest became Lorentz's successor at Leiden, and in his inaugural lecture in December 1912, he argued dramatically for the need to decide between Lorentz's and Einstein's theories, on the one hand, and Ritz's on the other. After 1913, however, Ehrenfest no longer advocated Ritz's theory. Ehrenfest and Ritz had been close friends since their student days, Ehrenfest having admired Ritz immensely as his superior in physics and mathematics; but following Ritz's death, Einstein came to play that role, as he and Ehrenfest became close friends."

    Pentcho Valev

    5 days later

    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

      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

        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