Hi Cristi,

I didn't interpret Tegmark's falsifying criterion into existence. He states it explicitly in his book (p.356): "Looking toward the future, thee are two possibilities: If I'm wrong and the MUH is false, then physics will eventually hit an insurmountable roadblock beyond which no further progress is possible; there would be no further mathematical regularities to discover even though we still lacked a complete description of our physical reality. For example, a convincing demonstration that there's such a thing as fundamental randomness in the laws of nature (as opposed to deterministic observer cloning that merely *feels* random subjectively) would therefore refute the MUH. If I'm right, on the other hand, then there'll be no roadblock in our quest to understand reality, and we're limited only by our imagination!"

I explore this falsifying criterion in my essay.

The question of computability enters because Max approaches the hypothesis the same way that Wigner describes the role of mathematics in physics: discovery of physical regularities leads to discovery of mathematical regularities. I argue that if MUH is true, it has to work both backward and forward -- i.e., discovery of mathematical regularities leads to discovery of physical regularities. MUH is true if, and only if, the hypothetical final theory is mathematically complete.

(By student of Bar-Yam, I didn't mean a classroom student but rather a follower. He is a dozen years younger than I -- showing once again that there's little correlation between age and wisdom.)

Best,

Tom

Hi Cristi,

As I told you in my FQXi page, I have read your pretty Essay. Here are some comments:

1) I find the title very intriguing.

2) I think the most brilliant text ever written is Einstein's 1915 paper on general relativity!

3) The idea that all human creation and knowledge is a point in the interval [0, 1] is upsetting!

4) I disagree with Smolin's statement that "there is no mathematical object which is isomorphic to the universe as a whole". I think we merely do not know if such a mathematical object exists or not.

5) Congrats for generalizing the Weyl curvature hypothesis, you know that I am an estimator of your research work.

6) I do not think that you are too reductionist in claiming that there is a mathematical structure which is isomorphic to the universe described by the extended list of propositions. In fact, I agree with you.

7) I am not completely sure of the real existence of a a theory of everything, but it should be a great pity if it does not exist! The same for black holes!

8) Your statement that "if we can understand the universe, it is because this immense complexity can be reduced to a small number of laws" is very intriguing. It I think should have been appreciated by Einstein who claimed that "The most unintelligible thing about the universe is that it is intelligible at all."

In any case, the reading of your intriguing Essay was interesting and enjoyable. It surely deserves the highest score that I am going to give you.

I wish you best luck in the Contest.

Cheers, Ch.

    Hi Christian,

    Thank you for the comments and for the kind words! With 2), I think I very much agree with you, but I wanted to connect with most readers, and perhaps upset others. With 3), as well as 1), indeed I wanted to upset the reader. So connect and upset, to make sure what you say will be remembered one way or another :)). About the other points, I think you are right too. I was happy to read your excellent essay, and I hope this edition you will get a well-deserved prize!

    Best wishes,

    Cristi

    Dear Cristi,

    The introduction to your essay is very clever. Most people do not appreciate that when going from the rationals to the real numbers, one enters a whole new ballgame, so to say. I did not appreciate it until I began to study some set theory, but your example brings this home very nicely.

    I essentially agree with almost everything in the essay, but would like to add the following comments:

    1. I think that at least in part, some of the difficulties pertaining to the metaphysical status of mathematics are linguistic. Specifically, I think "exist" is simply not a good word to apply to mathematical objects (unless one subscribes to the MUH) because it has too much baggage associated with the physical world. There should be expressions which capture subtle ontological nuances, similar to how Eskimos purportedly have over a dozen different words for snow. These nuances might allow us to refer to things like physical objects, feelings, thoughts and mathematical objects in such a way that we can keep separate what it is that we are talking about from everything else, and aid in investigating how they relate to each, if they do. I do agree with you that in the specific sense that you described it, every book, piece of music, etc. is already there.

    2. Taking the ability to name something in the real world to entail being able to give a list of propositions that apply to it seems intuitively pleasing, but I still have a lingering feeling that there might be something above and beyond a complete list of propositions that characterizes physical objects. One could of course simply define this relation to be just so, but that seems like a cheap way out to me.

    3. I am personally of the opinion that there is no "theory of everything". In fact, I consider the idea that there should be one as one of the last remaining vestiges of anthropocentrism in physics in the following sense: presupposing the existence of such a theory also presupposes that all processes in nature can be described using a small set of laws by one kind of observer of which we are a special example. Who is to say that there are not other kinds of observers whose observations of nature could also be described by a small set of laws but which would be *in principle* inaccessible to us? To give a specific example, who is to say that it is not possible to formulate a theory of nature from the frame of reference of an observer associated with null geodesics in spacetime? I often see the argument that it makes no sense to imagine such frames because no spacetime observer can transform to them, but that exactly makes my point: We are observers in spacetime, and to suppose that speaking of such frames makes no sense reflects a very subtle form of anthropocentrism.

    Again, yours was a very thought-provoking and well-written essay.

    Best wishes,

    Armin

      Dear Armin,

      Thank you for reading my essay and for the comments. You raised interesting points, and I will address them.

      1. As you know, in mathematics the term "exists" is the same as in logic. For example, "there exists a field which extends the field of real numbers and is algebraically closed". This doesn't have the meaning of physical existence, but rather of logical consistency. Whenever I used "exists", the sense of mathematical existence or physical existence follow from the context, otherwise, I specified that I was talking about mathematical existence or physical existence. To change the terminology would be unnecessary and would introduce confusion. You are right that they can be identified in the context of MUH. You are, of course, right, that the standard terminology is not the most fortunate, but I hoped that my precautions were enough.

      2. Probably the reason why you feel that there is more to existence than relations is that one considers more existent the things with which we have a relation, directly or indirectly. But this is relation too. If you could be more specific about that property that escapes the relational viewpoint, that would be helpful. Otherwise, I think my statement is not merely a way out, but is the only way which avoids reference to things which don't have observable effects by themselves.

      3. I agree with you about the necessity to avoid anthropocentrism. This is why I wrote "Being able to guess them and then test them would mean either that we are that lucky, or that the universe wants to be completely understood by us, who are just tiny waves on its surface." As you could see, I did not claim that this theory must exist with certainty, and certainly didn't claim that, even if it exists, we can find it. But I think that it is likely that it exists, and even that we find it, simply because we are so close. Most of the physical laws are contained already in general relativity, quantum theory, and the standard model, which really are a small set of laws accessible to us. Of course, this doesn't ensure us that TOE exists and can be found. You also said "To give a specific example, who is to say that it is not possible to formulate a theory of nature from the frame of reference of an observer associated with null geodesics in spacetime?" Well, you are right, physical laws can be described very well in coordinates whose constant surfaces are lightcones. Also, in Finkelstein coordinates and Kruskal-Szekeres coordinates, and also in Penrose-Newman formalism and Penrose-Carter diagrams, null coordinates are used. There is no need for an observer to travel at the speed of light, this is simply the diffeomorphism invariance of laws in general relativity. So the problem of antrhopocentrism introduced by reference frames was solved in general relativity, for other reasons. I think this was a good point you raised, because it could lead to the diffeomorphism invariance, or at least is another good reason to use them.

      Thank you for the excellent points you raised, they allowed me to clarify some perhaps unclear elements, and to see some things I knew in a different light.

      Best wishes,

      Cristi

      Dear Cristi,

      Thanks for your kind words, which honour me. I am happy to see that we are in perfect accord.

      I wish you will get a well-deserved prize too! Let us cross our fingers!

      Cheers, Ch.

      Dear Christinel,

      "When physicists describe the laws governing the physical world, mathematics is always involved." This sentence of your abstract is the only one without question mark.

      Let me ask you to provide something indispensable but possibly neglected: What about causality and about the border between past and future?

      Regards,

      Eckard

        Dear Cristi,

        Thank you for your comments on my essay. I discussed your comment about the paradoxical nature of immortality in the context of a multi/maxiverse in a reply on my page.

        Congratulations once again on an outstanding entry to this FQXi contest! I found myself highlighting a lot of your statements that I wholeheartedly agree with. In particular, in the section "Is there something that can't be described by mathematics", you nicely explain that, despite what mathematical universe critcs often affirm, consciousness and the flow of time could very well, in principle, "emerge" from mathematics , even if we don't understand all of the details yet. I also agree that "emergence", whatever it is, does not need a non-mathematical explanation (whatever that would mean), even if we, once again, do not understand all the details. There is an interesting parallel between some arguments against a mathematical universe and God-of the-gaps-type arguments : in the same way that some religious believers zero-in on the unexplained details of our scientific theories (for instance, the origin of the first life form on Earth) to see in them evidence that some sort of God is needed for that step, I think that many critics of the mathematical universe hypothesis zero-in on the parts of our understanding of the world that are not quite satisfactory (what is consciousness? why does time appear to flow?) to see in them evidence that the world cannot be fundamentally mathematical.

        I agree with your conclusion that Gödel incompleteness and indecidability do not act as "show-stoppers" when one considers the fundamental relationships between mathematics and physical laws, because "to obtain an inconsistency, we should make the physical laws assert their own indecidability, but how could this be done?"

        I also agree with you that the statement that our world is isomorphic to a mathematical structure is a "plain truth that doesn't make predictions at all, and doesn't explain anything", and that any universe complex enough for universal Turing machines to exist could sustain our existence: that's why, in my essay, I argue that we live simultaneously in an infinite number of larger contexts.

        As I said in my reply to your comment on my page, I have some questions concerning your affirmation "at least we know that there is room for free will, whatever this may be". While formulating them, I followed the trail from your reference pages and fallen into a rabbit hole of cross-linked articles... I will continue to think about this and come back to you soon, whether I have free will to do so or not...

        Good luck in the contest... I hope you make it to the top this time!

        Marc

          Dear Eckard,

          Thank you for the questions, but I would like to ask you to be more specific. Could you please describe the specific features or instances of causality and present which you refer to, and why you consider that math is not involved in them? Or if this wasn't what you meant, could you please explain me the question?

          Best regards,

          Cristi

          Dear Marc,

          Thank you for the kind and interesting comments. You said "I found myself highlighting a lot of your statements that I wholeheartedly agree with." I had the same feeling while reading yours! There is a perfect harmony and complementarity between our essays :)

          You wrote "I have some questions concerning your affirmation 'at least we know that there is room for free will, whatever this may be'." Yes, I gave some citations to previous essays and other works. I think a place to start are these two, Flowing with a Frozen River (pages 4-7) and Modern physics, determinism and free-will, both used in Aaronson's The Ghost in the Quantum Turing Machine. My arguments come from quantum mechanics, and the conclusion about free-will is very close to Hoefer's Freedom from the Inside Out. Also in "The Tao of It from Bit" I discuss a bit the issue. Please let me know what you think, or if you have questions. The bottom line is that I think free will is compatible with both determinism and indeterminism (indeterminism alone anyway doesn't guarantee it, because If quantum randomness would equal free will, then any Geiger counter would have free will.. But I only say there is room from free will, I don't know what it is :)

          Best wishes,

          Cristi

          You're right, Cristi. I invite you to see my analysis of the philosophical foundations of mathematics and physics, the method of ontological constructing of the primordial generating structure, "La Structure mère" as the ontological framework, carcass and foundation of knowledge, the core of which - the ontological (structural, cosmic) memory, and information - polyvalent phenomenon of the ontological (structural) memory of Universum as a whole. I believe that the scientific picture of the world should be the same rich senses of the "LifeWorld» (E.Husserl), as a picture of the world lyricists , poets and philosophers.

          Kind regards,

          Vladimir

          Dear Christinel,

          You are convinced that maths and physics are much related, as in Tegmark's thesis. I suggest you read Leifer's essay and in an another direction the multiverse essay of Laura Mersini-Houghton. As you worked in cosmology and QM, I would be glad to have your view about the multiverse as a possible way to connect these two separate fields. Myself I am quite innnocent on this subject. I am working at this essay by Laura.

          I am also rating your essay now.

          Thanks in advance.

          Michel

            Dear Michel,

            Thank you for the comments. Indeed, I tried to bring some arguments supporting/explaining the connection between math and physics. The essays by Matt and Laura are on my to do list, I hope to get there soon. Regarding the connection between cosmology and QM, if you refer to the connection between inflation and QM, as advocated by Sean Carroll, I am not sure what to say about this. If you refer to Quantum Gravity and Quantum Cosmology, I think that it is premature the standard view that perturbative methods fail for Quantum Gravity. I have a paper on the connection between singularities and dimensional reduction in perturbative quantum gravity (Metric dimensional reduction at singularities with implications to Quantum Gravity, Annals of Physics 347C (2014), pp. 74-91). Many researchers found that various dimensional reduction effects may help quantum gravity. I argue that we don't need to put by hand these various dimensional reductions, since they occur already due to singularities in GR. But I also hope there is a better, nonperturbative way to quantize gravity, yet to be found.

            Best wishes,

            Cristi

            Hi Christi,

            This is a very interesting essay indeed! I have to object on one point though. Just because you can list certain properties about a system doesn't mean you know that there must exist a mathematical structure describing what the system does. It is trivially true that if you collect certain quantities that describe some system, then that is some kind of table, which you can call a "mathematical structure" if you wish, but this is just data collecting. The point of science is to come up with a useful mathematical model for this. In my essay I explain what I mean with this in more detail. I enjoyed reading your essay, also because it is well structured.

            -- Sophia

              Dear Sophia,

              "Just because you can list certain properties about a system doesn't mean you know that there must exist a mathematical structure describing what the system does. It is trivially true that if you collect certain quantities that describe some system, then that is some kind of table, which you can call a "mathematical structure" if you wish, but this is just data collecting."

              Well, I didn't refer to data collecting. I meant that if you have all the propositions that are true for a system, then there is a mathematical model which makes these propositions true. You have an axiomatic system (the collection of true statements), and there is a mathematical model of it (in the sense of model theory). It would be ridiculous to think that collecting some numbers that follow from a series of measurements would give a mathematical model. So I agree with you that "The point of science is to come up with a useful mathematical model for this.". I was not talking about collecting data from experiments, but about the existence of a mathematical model of a system which can be described by propositions and is free of logical inconsistencies.

              Best wishes,

              Cristi

              Dear Christi,

              Thank you for the references. I am interested by the quantum gravity subject but it may take a while before I am able to produce a good idea. These days I was exploring F-theory also because it involves modular group concepts.

              Michel

              Dear Michel,

              I too tend to focus more where the previous research leads me. I was not interested in perturbative Quantum Gravity, but the singularities which I studied turned out to have dimensional reduction effects, so this is how I become interested. To make a parallel, I move from one field of interest to another rather by analytical continuation, than by jumps :) So I guess for you is more natural to explore F-theory, given that it involves modular group concepts.

              Best wishes,

              Cristi

              Dear Cristi,

              Being a fan of the good old Budeanu, I hope you as a new voice of Bucuresti might forgive me my misspelled Cristinel and decide without prejudice which variant is most appealing to you:

              1) The late Einstein called the distinction between past and future an illusion. Nonetheless admitting that the now worries him seriously he considered it something outside science. In principle, this is the accepted "spacetime" view of modern physics.

              2) Spencer Scoular instead argues for a qualitative theory of physics.

              3) Tim Maudlin suggests a notion of number that provides the arrow of time. Spencer and Tim further discussed their views in Maudlin's thread.

              4) I gave already in Fig. 1 of an earlier essay of mine an alternative explanation: Only elapsed time is measurable. Only future processes can be influenced. I agree with Tim Maudlin on the good old Euclidean notion of number as a measure, not a pebble.

              5) You might have your own idea.

              With best hopes and wishes,

              Eckard