I thought it is pretty clear that when I discuss Hawking's argument, I discuss it under his assumptions, and if I would want to change the assumption I would state it explicitly. Btw, do you think Hawking makes this assumption explicit? Now, assuming anyway that I was not clear under what assumptions I discuss, I think replying to you four times on the same topic should be enough to clarify this :)

Dear Cristinel,

Lots of sweeping statements in this essay and many attempts to refute other arguments without sufficient logic or evidence and more of of attempts to persuade with a general sweeping comment (where is your evidence for "Even if we manage to extend the list with new truths about the universe, there is a mathematical structure which is isomorphic to the universe described by the extended list of propositions" or "Any phenomenon related to time, for example memory, can in principle be described mathemat- ically.").

The main issue I think you should consider is the evolutionary character of mathematics. Mathematics is not a constant where the universe could or could not be isomorphic to it. Mathematics and physics have grown together. For a couple millennia Euclid's fifth postulate was considered to be True, yet a little over 150 years ago it was shown to be one of a few options (it was only true in certain circumstances, so not all circumstances everywhere). Can we say that the universe was isomorphic to Euclildean geometry in 1800, but not now? Our current physics would not be possible without this fundamental change in mathematics beyond Euclid. Since we are all limited to our current time in history, we are limited in what we know of mathematics, which could be, even today, a very small amount.

You also seem to think mathematics is itself consistent and complete, which is impossible to prove (and Godel's proof indicates otherwise for completeness).

Finally, you appear to think that the universe does not have contradictions in it, something that defies all evidence we have at the moment (if reductionism is correct, why is an intelligent being at our scale needed to understand it and to affect the very small as part of the understanding process?) - so this must be an article of faith on your part, which extends as much to mathematics as to the known universe.

Thank you for your essay, but I an unconvinced by your arguments.

Donald

    Dear Donald,

    I read your comments and realized that you did not read with care my essay. Of course our mathematical representations of the universe evolved together with physics. And these representations change, and I even wrote "We know laws that were never contradicted by our observations, and their validity seems to be eternal, but this doesn't mean they will not be contradicted by experiment someday, or that there are no corners of the universe where they don't apply."

    I was talking about the possibility that the physical world is isomorphic with a mathematical structure which we are still searching. So your argument is not against what I said, but what you think I said.

    Also, you call Godel in your defense, but I also discuss this in the essay. Your statement that mathematics is not complete refers to mathematics which include arithmetic, and the incompleteness refers to the incompleteness of a finite set of axioms, in the sense that finite length proofs can't reach any truth. Drawing from this the conclusion that math is some broken toy that can't be fixed is forced. The conclusion, by contrary, is that math is much beyond the limited capacity of humans to express it in finite length proofs based on a finite number of axioms.

    I don't understand your last argument, the one that the universe has contradictions and all evidence show this. You mean logical contradictions?

    Thanks for your comments,

    Cristi

    Now I more closely examined your article "Modern physics, Determinism and free-will", for properly replying to what you actually put there (sorry I was not careful enough last time). Indeed, I confused between your "interpretation" that you support, and your (incorrect) description of the wave function collapse with the role of free will in wavefunction collapse (that you criticize). I see now they were separate sets of concepts.

    About your own "interpretation":

    "even though the states |П€j0 вЊЄ were obtained by a unitary operator from the orthonormal states |П€j1 вЊЄ, they are not necessarily orthonormal, and may even become dependent. In fact, in our case they all become at t0 equal to |П€0 вЊЄ. The operator U is a unitary operator on the total space of |П€вЊЄ|О·вЊЄ, but not on the space of |П€вЊЄ. This shows that it is possible for |П€вЊЄ to satisfy the constraints of both observations, and still have unitary evolution"

    Assuming for a moment that such ideas make sense, then why do you need the evolution to be unitary in the "total space" |П€вЊЄ|О·вЊЄ ? After all, this system is only a subsystem of the Earth. So, if all we need is that the evolution is unitary for the Earth system then the evolution of its subsystems do not need to be unitary. By the way, where is the assumption of unitarity in QM ?

    Now seriously : according to the rules of established QM, what your claim rigorously means is that the first measurement device (that made the first measurement) keeps physically interacting with the system during the time between measurements, so as to progressively align it with the basis of the second observable. Such a claim of interaction is just plain wrong. In reality, after the first measurement, the system no more interacts with the first measurement device, and even if a little bit of influence might subsist, it would be ridiculous to assume that this interaction systematically manages to align the system to one eigenvector of the second observable. You seem to attribute this coincidence to final causes: you seem to argue that there should exist possible initial conditions that make the evolution fit this condition, so that the presence of the second measurement will retroactively arrange the initial conditions to make all things fit so as to achieve this evolution of the system from the eigenstate of the first observable to the eigenstate of the second observable. I consider this as a mere dream since anyway, in good approximation, there is not supposed to be any interaction between the first measurement device and the system after measurement, so that no fine-tuning of the apparatus can make its non-existing interaction with the system succeed to align it with eigenvectors of the second observable.

    "In UIQM I acknowledge that the observed system is in fact entangled with all systems with which it interacted in the past."

    Not entangled, but in continuous physical interaction. You are confusing between (the mathematical expressions of) entanglement and physical interaction.

    In conclusion, your "interpretation" of QM is not an interpretation at all but some fantasy, another theory far from what QM actually says. Now since you did not explicitly claim to reject QM to develop your own theory, but kept giving the illusion that you respect it and only try to interpret it by keeping superficial similarities ("unitary evolution"...), it just shows that you have no serious understanding of quantum physics.

    Now about your discussion of standard QM and the mind makes collapse interpretation:

    "If humans base their choices on random inputs, then this by itself doesn't make them free."

    Of course but only under the assumption that the inputs are actually random in the case under consideration. As I explained in my interpretation of QM, I consider that the same process of wave function collapse usually obeys randomness in non-living systems, but departs from it in the cases of brain processes where free will applies. And I explained in length why I see this sort of "discrepancy" between the kinds of outputs of the same physical process of wavefunction collapse (normal randomness in "purely physical systems", vs. free will in the brain), to be not a discrepancy at all but a very elegant, coherent solution. (I also like the explanation by the last reply of that thread).

    "The configuration after the branching has to evolve, so that the agent can see where it is going"

    Where does that idea come from ? I see no reason for it. My view is that the collapse occurs at some time after decoherence, when the different possibilities are meaningfully distinguished by the observer; there is no need to assume any branching to have occurred before that time, nor to assume that the observer has any explicitly conscious perception of the alternative possibilities among which he is actually (more or less) choosing.

    "in the case of quantum mechanics, where the unitary evolution governed by Schrödinger's equation is interrupted from time to time by the wave function collapse. Assuming that an agent uses this randomness to perform free choices, she must act precisely at the appropriate moment and position where the branching appears."

    "(The discontinuous jump) has never been directly observed. There is no known explicit process leading to the discontinuity. In fact, all interactions we know fit well in the Hamiltonian description, and the measurement devices are made of systems which obey it. So, where does the discontinuity come from?"

    Indeed, if taken separately, these 2 issues would be problems. However, taken together they become zero problem as they resolve each other. Indeed, there is no definition of when the collapse occurs ? It looks like a mystery how the time of free choice may coincide with the time of collapse ? No problem : let the "time of collapse" be conventionally defined as given by the time of free choice. Then, there is no more mystery when and how the collapse occurs, and there is no mystery either how the time of choice may happen to coincide with it.

    "It would violate the conservation laws". This is one of my objections against spontaneous collapse theories. But my interpretation of the collapse is different : I think it occurs not by projection but by selection of a world after decoherence (this is an emergent condition, that fits with my idea of collapse as a non-physical, non-local process) : it is still non-unitary but resolves some defects of the projection postulate. It does not satisfy the conservation laws if taken in a naive sense (expressed by averages) that you wrongly assume to be the necessary form of expression of the conservation laws, but does satisfy a more subtle understanding of the conservation laws that I explained there.

    Generally, my page on the mind makes collapse interpretation replied to all of your objections in your "2.2. The internal tension of quantum mechanics".

    "I find hard to believe that the environment is the cause of selecting the eigen-states, because these depend only on the measurement device. Change the measurement device, and leave the rest of the environment unchanged, and the eigen-states change too."

    You missed that the role of the environment in decoherence happens by interaction with the measurement device. So, change the measurement device, leave the rest of the environment unchanged, then the environment will interact with the new measurement device to do the decoherence according to the new eigenstates given by the new measurement device.

    You speak about the "decoherence interpretation" as if it was an option to accept or reject it. The reality, well-known by all serious participants in QM interpretation issues, is that decoherence is NOT an interpretation but a logically necessary consequence of established QM, completing the picture of what needs to be interpreted.

    I stopped reading your article here. I consider that I found enough fatal flaws in it already. And I see it as a big bug in the academic system, to find such an incompetent "physicist" have any position as physicist there.

      Dear Sylvain,

      Thank you for keeping your promise that you will "read and refute" my article, and to continue our "dialogue". Unfortunately, you are again missing the point and misinterpreting what I write :)

      Let me explain what I am doing in the article you discuss, and in others: I am exploring the consequences of a modification of the standard interpretation of quantum mechanics, under the assumption that the unitary evolution is never broken by a discontinuous collapse. Some reasons why I am doing this: 1) there is no direct evidence of a non-unitarity in quantum mechanics, it was only inferred from the outcomes of the measurement, 2) a discontinuous collapse would break the conservation laws, and this also was never observed, 3) a discontinuous collapse introduces a tension between quantum mechanics and general relativity.

      One of the consequences of maintaining unitary evolution even during the collapse is that the initial conditions of the measurement device and the observed system have to be correlated in advance (as I proven mathematically here). Hence, the initial conditions have to be very special, like in contextuality. This may seem for an observer embedded in time, assuming she would know the complete information about the wavefunctions and not only the outcomes of the measurements, as being retroactive. However, by looking at the complete solution from the "outside", as in the block world view, these correlations appear merely as global consistency conditions, pretty much of the kind of consistency condition which Schrodinger used to find the atomic spectra from his wave equation. I explained this viewpoint more in this talk and this essay.

      Please note that I don't consider that I am the keeper of the absolute truth, I am merely exploring a theory, because I consider it a better alternative for several reasons from which I mentioned some above.

      What I do in that paper is that I explain that, if unitary evolution remains unbroken, then a succession of two incompatible measurements still can be explained without discontinuous collapse, if both of the two measurement devices perturb the observed system in a special way. You misunderstood this, when you said "what your claim rigorously means is that the first measurement device (that made the first measurement) keeps physically interacting with the system during the time between measurements, so as to progressively align it with the basis of the second observable.". No, I don't claim that it keeps interacting with it, only that it perturbed the observed system in such a way I described there. You also said "You are confusing between (the mathematical expressions of) entanglement and physical interaction." You are incorrect, I know very well the difference between interaction and entanglement. What I say is that we can see this as a superposition of composite states (which is entanglement), and each composite state contains a different interaction. So your misunderstanding is clear again.

      You quoted my article that "The configuration after the branching has to evolve, so that the agent can see where it is going", and replied "Where does that idea come from ? I see no reason for it.". Well, as I explained there (where I was talking about free will), to make a choice, you need to know what that choice means, some of its consequences. Otherwise what kind of a choice is that? The alternative is letting the randomness choose for you.

      I want to point out that my unitary interpretation is compatible with a version of decoherence, and also with relative state interpretation. If each of the decohered branches is unitary and not plagued with the discontinuous collapse. But usually the density matrix is given a meaning at the beginning (as representing the state), and another one after is diagonalized (as representing a statistical ensemble representing the branches after decoherence), and this amounts for each branch to a discontinuous collapse. Regarding your comment to my statement "I find hard to believe that the environment is the cause of selecting the eigenstates", I suggest you to take a look at the delayed-choice experiment and see if your explanation still holds. Since you called me incompetent for not agreeing that decoherence solves the measurement problem, let me remind you that also Roger Penrose and Anthony Leggett criticized it. Anyway, although I find your personal attacks not conformal to the rules of the contest, I prefer that your comments will not be removed.

      Cristi

      "to make a choice, you need to know what that choice means, some of its consequences. Otherwise what kind of a choice is that? The alternative is letting the randomness choose for you."

      You are talking about fantasy, not about real facts. Of course it would be nice if we were always correctly informed about the consequences of our choices before making them, but this is not how things usually happen in the real world, or anyway the laws of physics do not provide this. I never claimed that freedom never turns out to be a poisoned gift by lack of proper available means to anticipate the consequences of choices (and personally I do feel it as a very absurd poisoned gift in many cases). To anticipate the consequences of our actions we only have our imagination, which is is fallible, and possibly diverse social structures, but such considerations lead us quite far away from physics.

      I'm not sure what are all the exact details of your errors, since there are hundreds of possible ways to make mistakes, but I know that your interpretation is incoherent (and if I misinterpret things, then it seems at least that your presentation is very unclear). First, if there is entanglement between the first measurement apparatus and the system after measurement, i.e. the system after measurement is not clearly in one of the eigenstates of the first measurement, then it means that the first measurement never happened (in the sense of its given observable that must be defined independently of the second observation : this first measurement is classified as an interaction and not a measurement). Second, again if there is an entanglement, then the observed system does most surely not evolve into any unique eigenspace of the second observable, so that the second measurement gives random results according to some discontinuous projection. Indeed in the absence of physical interaction between subsystems (as is the case after the first measurement, between the measurement apparatus and the system), the evolution of an entangled state will always lead to another entangled state, i.e. its components cannot evolve into pure states (they keep their shape : dimension of the space of possible values, entropy...).

      "Since you called me incompetent for not agreeing that decoherence solves the measurement problem"

      Now you are obliging me to call you illiterate, because you could not even read correctly what I wrote : I never wrote that decoherence solves the measurement problem, only that it is part of the picture that needs to be interpreted. So now I'm done with you, I'll rather comment other essays. Bye.

      "You are talking about fantasy, not about real facts.". Then free will is a fantasy, and that's all. Note that I was not talking about knowing that your choice of lottery numbers is winning or that of you choose to save someone's life, this will not turn out to be a new Hitler. I was talking about being able to perform elementary actions. For example, choosing between going left or right. If the choice is made at the branching, is made by the branching. But randomness is not freedom. If this would be so, then any Geiger counter would have free will.

      Dear Cristinel,

      I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

      All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

      Joe Fisher

      9 days later

      Dear Cristi

      I really enjoyed reading your beautiful essay.

      Here some considerations.

      (1)" A universe in a dot". This is a lovely peace of writing. It reinforces my conviction that the principle of finite information density must hold, as Feynman himself believed (see his 1982 Int. J. Th. Phys.) The continuum is paradoxical: you can write the whole history of the universe on a single atom. This is also related to the following point (3)

      (2) I totally disagree with Smolin on many things, even if I enjoyed reading most of his books, in particular Time Reborn where he already expresses his new ideas that you can also find in his current essay taken from his last book. The two main things I disagree on are: (2a) the physical laws are not constant; (2b) mathematics cannot describe "particularities". I think that both assertions are unscientific according to Popper.

      (2a) Imagine a physical law that changes every minute, and you don't know how it does. It is not a law anymore, right? What about if it is valid for a year, or a billion of year, or not valid e.g. two billions light-years from our planet. What should be the validity time (and space) of the law? If we state a law, it is supposedly eternal. This, however, does not protect the law from possible falsification, which will ask for another law, yet supposedly eternal. In particular, the new law maybe a "meta-law" which regulates the change of the falsified law.

      (2b) A theory cannot describe "reality as it is". It can describe/predict only connections between known conditions/preparations/events and observed facts. Inferring initial conditions in the past needs an increasingly larger knowledge of the current state, the more distant is the past from the now. We can only connects known facts with known facts, or predict facts from known facts: the theory cannot provide the initial conditions. If we want to predict what will happen, we need information about the past. Accounting for the Smolin's "particularities" requires the knowledge of the initial conditions, otherwise it is not science: it is magic.

      I love your ideas about "Is everything isomorphic to a mathematical structure?". I agree with you that the answer is positive, and I enjoyed very much this part of your essay. I think that the point, however, is another one, and again it is about "what is theory". A theory is a reduction of complexity of connections between events/observations. Transforming a set of observations (a peace of "reality") into a list of propositions does not necessarily entail a complexity compression, it is not a theory. Suppose I give you the observation made of the following digits:

      3.1415926535897932384626433832795028841971693993751058209749445923078164062862...

      A good theory would be the Plouffe formula that fits in few bits.

      Also, as Poincaré says, the power of mathematics stays in the induction method, which produces an infinite compression of true statements. This is also the case of a physical law.

      It would be too long to write everything I would like to say ..,

      I hope to meet you again very soon and have a long discussion of philosophy of physics at a dinner table.

      My best regards

      and compliments again

      Mauro

        Dear Mauro,

        Thank you very much for reading the essay and for the interesting comments. I find myself in agreement with what you say. The continuum seems to allow much more information than it actually uses, so seems to be a waste :). So perhaps the continuum is rather an ordering principle for discrete information. With respect to Smolin's recent philosophy, I think you and I agree again, and your points 2a and 2b are right on the spot. And yes, you are definitely right about a theory being an efficient way to compress information. Indeed, we would have a lot to discuss, and I look forward to see you again at a conference, to discuss in more detail.

        Best wishes,

        Cristi

        Daer Cristinel

        I find your essay very interesting, indeed, I consider mine an extension to it. I agree on almost everything.

        I believe, that the universe is "an hyper-equation". "A mathematical structure inside an other, and depends on another external"

        "Mathematics and physics are two sides of the same thing. numbers and physical entities in motion".

        Now, I think we need concrete things from which we can start. For this. I proposed The Paradigm Bi-iterative: a concept that goes beyond quantum mechanics and general relativity.

        Quantum mechanics is not enough, General Relativity is not correct. Maybe, that's why it was impossible to combine the two branches of modern physics. Therefore, it was necessary to introduce the mechanics iterative, as the source, to justify the origin of acceleration, force and time. In addition, the rhythm and the polarization.Two mechanics seemingly separate, connected with a constant, only in this way, we can unify them.

        I am convinced that reality is made up of thin numbers and geometric shapes. For this, I have proposed a Theorem and various mathematical structures (see Annex), which in my opinion, could be the true universe's mathematics.

        The Bi-iterative system's calculation considers reality multi-directional and multi-dimensional. with (1) we mean (1 * 1 * 1). while, the recursive computation and the Fibonacci series uses only the first, the fractal uses the first and the second. So, are wrongly linear, interprets reality on a bi-dimensional sheet. Instead, the system Bi-iterative uses all three.

        Let's take an example:

        (1 + 5 + 7 + 12) =

        聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽= 1 + 5 + 7 + (6 + 8 + 10)

        聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽= 1 + 5 + 7 + (3 + 4 + 5) + 8 +10)

        聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽= 1 + 5 + 7 + (3 +4 +5) + (1 + 6 - 9) + 10

        聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽= 1 + 5 + 7 + (3 +4 +5) + (1 + 6 - 9) + 10

        聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽= 1 + 5 + 7 + (3 +4 +5) + (1 + (3 + 4 + 5) - 9) + 10

        聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽

        聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽= 13

        = 2197

        This structure is compact, inside it could coexist other structures indipendent.

        which can be expressed also in a linear fashion:

        (1 + 5 + 7 + 12) = 13 = 2197

        聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽= (6,5) + (6,5) + (6,5) + (6,5) + (6,5) + (6,5) + (6,5) + (6,5): A straight line.

        "A complex and varied reality , can be described only with a structure capable of transforming".

        This sequence is like a clock going always forward, or rather, reflecting itself, coping itself.

        ( a + b + c + d ) + K

        (1 + 5 + 7 + 12) + K

        In agreement with the concept of light propagation and the distribution of mass and energy, fairly, K is a constant equivalent to (1) ....... + 1, + 1, + 1, + 1, + 1 ........

        "The universe was born so, one bit after another, and continues to do it today."

        Sincerly yours

        BannouriAttachment #1: Theorem_1.jpgAttachment #2: Theorem_345.jpg

          Christi,

          Congratulations on another fine essay, and on hitting the top. Not only do I agree with most of your propositions (I axiomise that maths CAN describe all) but more importantly you set them out and describe them succinctly. I was also pleased to read you agree that a TOE IS possible and that Godel does NOT preclude a mathematical description. Do you agree that recursive higher orders should be able to model that formalism asymptotically?

          One thing perhaps 'missing' was any reminder that a mathematical structure may not necessarily precisely model any physical process for which it's developed, though giving apparently correct outcomes. Do you agree this is possible? I assign this 'false proof' issue to much anomaly and paradox in physics and identify an important example in my own essay, confounding logic in QM. My suggestion is that if we say ALL maths must ALL be right if the bottom line proves right then we can be misguided about how nature actually evolves in time (which I agree is 'physical'). I hope you'll read and give your views.

          You don't suggest a direction for any new formalism pointing towards a TOE, which is not a criticism as very few do, but again I hope you'll analyse the hierarchical 'rules of brackets' which i suggest does so consistently with your analysis (as far as I can see, but your view?)

          I'm not surprised you're in the top group again and hope my score will help keep you there.

          Very best of luck in the final judgements.

          Peter

            Hi Cristinel,

            Your excellent essay makes a compelling case for mapping the universe to mathematical structure, and you presented it in an upbeat and enjoyable narrative. Your demonstration of how pivotal topics from relativity and incompleteness do not limit this isomorphism, provides a very persuasive argument. I like that you explained there is no conflict between mathematics and causality, and your discussion on physical explanation and mathematical description was very pertinent. My essay also discusses the strong connection between physics and mathematics, and for example how changing the mathematical representation affects physical explanation. I also show what was considered a limitation on computation due to (Gödel) incompleteness, can in fact be an expansion when attempting to physically model the mathematical structure of indecidability. This may address or at least support points raised in your essay.

            Also, I appreciated the wide variety of examples encompassed in your essay, and am motivated to check out your work on the Weyl curvature hypothesis, singularities, and free will.

            A very interesting contribution that thoroughly addresses this topic, I give it the highest rating.

            Please take a moment to read my essay and rate it as well. Thanks again,

            Steve Sax

              Dear Bannouri,

              Thank you for the kind and interesting comments. Indeed, in order to advance toward the understanding of the foundations, we have to go beyond what separates quantum mechanics and general relativity.

              Best regards,

              Cristi

              Peter,

              Thank you for the interesting remarks. Let me congratulate you as well for your place. Regarding your question "Do you agree that recursive higher orders should be able to model that formalism asymptotically?", I would say maybe.

              > "One thing perhaps 'missing' was any reminder that a mathematical structure may not necessarily precisely model any physical process for which it's developed, though giving apparently correct outcomes. Do you agree this is possible?"

              I think I addressed the question of models that are not "true" in a section of my essay named "Is mathematics merely a tool in physics?". If you refer to models that give all the correct outcomes, and can't be distinguished by other models by no matter what experiment, I think it is possible. But I think that if these can be related to anomalies and paradoxes, then they can be distinguished and falsified. Regarding "if we say ALL maths must ALL be right if the bottom line proves right then we can be misguided about how nature actually evolves in time", well, correct maths is right, but being right doesn't make it true about our physical world.

              "You don't suggest a direction for any new formalism pointing towards a TOE". I think it is too early for such a suggestion.

              I look forward to read your essay, which is on my todo list.

              Best wishes,

              Cristi

              Hi Steve,

              Thanks for the interesting comments and kind remarks. Indeed, some may qualify my essay as (too) optimistic, and I don't really think there are limitations that should stop us from trying to learn more. By the same criteria, any attempt to understand the universe has to be optimistic. In the same time, I advocate an agnostic and skeptical position regarding the belief in the absolute validity of our theories. And as you said, I indeed see no conflict between mathematics and causality. I think you have a keen eye and understood well my points. Your remarks about how changing the mathematical representation affects physical explanation, and the idea to take advantage from Gödel's incompleteness theorem seem to me very interesting. I look forward to read your essay. Good luck in the contest!

              Best regards,

              Cristi

              Cristi,

              Thanks for being engaged in my essay and for your kind words.

              Jim

              Christi,

              I don't know how I missed your essay.

              Your essay is delightful, like you have the essence of Socrates on your shoulder, distilling with questions and answers his method of inquiry and discussion. I think the potion you create works. It helps to clarify the mind-numbing. You also throw in helpful progressions of studies, GR beyond Newton, singularities.

              Impressive essay.

              Jim

                Thank you Jim, for reading and commenting my essay.

                Best regards,

                Cristi

                Dear Cristi,

                Since this comment relates to both yours and our essays, I am reproducing it here, in your blog, with a slight abbreviation.

                Thank you so much for your compliments; it is a true pleasure to be highly appreciated by one of the experts!

                You underline that your main take away from our essay "is the uniqueness of the laws." Maybe, our laws of nature are not exactly unique, but they definitely belong to a very special and narrow set of mathematical structures, much more narrow than Tegmark's multiverse suggests. In other words, our laws are truly beautiful in that deep meaning of mathematical beauty which was professed by Pythagoreans of all times, from Pythagoras and Euclid to Kepler and Newton and to Einstein and Dirac.

                In that light the statement of your essay that,

                "Mathematics is already there, eternal and unchanging. What we invent is the discovery of mathematics,"

                is revealed as having an even deeper meaning than it may at first seem.

                Cheers and good luck!

                Alexey and Lev