I think the world around us gives us all type of CLUEs to our intellectual pursuit. Great thinkers are great observers of hints and clues. Ultimately the physical world (not the bio-sphere or human society) is far simpler than anyone can think of, because of the equilibrium status of the universe. Or read differently the static nature of the universe. The universe is a dead body waiting for eons to be dissected. Complexity is largely suppressed because of the cancelling effect of forces, as well the elimination of complexity by evolution of a closed system. The complexity is decreasing as a function of increasing approximation to an equilibrium.

The trick is to change our human centric mind set and think out our own box. I pointed the natural number conundrum. It is an example of homo sapiens obsession. New type of understanding is needed and can be achieved by shedding unconscious constraints posed upon ourselves.

    Dear Lawrence, I am following your good idea to double the comment.

    *****

    Dear Lawrence,

    Thank you so much for your generous compliments to our essay. As you see, we are showing there how Tegmark's MUH is refuted on the scientific ground. Yes, it goes against the dominating opinion of the community of cosmologists (and your own), that the full-blown MUH is unfalsifiable, but our refutation looks very solid for me.

    About your 'couple of points'. First, your distinction of WAP and SAP fully agree with the conventional one, as I may judge. It isn't clear to me what point you were trying to make about them. Second, we use the word "chaos" in its ancient meaning, as we stress it when this word is used the first time, pointing there to Platonic philosophy. This meaning sometimes is expressed by such words as "nothingness" or "nothing". This formless entity, chaos/nothingness, is a source of pure accidental, random, causeless factors. It has little to do with the mathematical concept of "dynamical chaos" you mention, which assumes certain mathematical forms already given.

    Your ideas about "the soul of mathematics" sound very interesting to me, and I would very much wish to discuss them with you in much more detail than this specific place and occasion allows. You know how to find my email. Please be assured that I would highly value communication with you on these and other questions.

    All the best,

    Alexey.

    Math is all about mapping from one artificial domain to another artificial domain. The mapping works well but does not give rise to the legit or such mappings.

    I propose that quantum mechanics needs an entirely new type of math. The current approach to quantum mechanics is bewildering and confusing. The fundamental rules of math need change to adapt to the quantum mechanics world.

      Hi Lawrence,

      I have read your essay and many of your posts and appreciate what you are trying to do. But please take a look at my essay if you haven't already and give it some effort since you are a good programmer, it should not take much of your time. You will see that physics can be done based on the most elementary mathematics known(addition, comparison ...etc). So fundamentally no need for exotic math or to worry about infinity, compute-able ... etc. The system can be put on more formal level(fundamentally it is a geometric probability problem), but through simulation the origin of the design of reality is so clear and makes a lot of sense. Then MUH is established(confirmed) with minimum effort, we start by postulating it and end by confirming it. Our reality's existence is the proof of the Platonic realm of mathematics.

      Essay

      Thanks and good luck

      P.S. But please read some of the first post in my thread about running the programs if you like to delve in them more.

        Congratulations Lawrence on your high standing..

        I'm just finishing reading your excellent paper, and I'll have some comments after I catch some sleep. This is one of your best essays so far, and it appears that high marks are well deserved.

        All the Best,

        Jonathan

          "I don't remember if I read your paper or not.=

          Isn't this one more unnecessary insult?

          I recall joy Christian and Einstein's anus mirabilis.

          The meaning of chaos might in that sense also mean void or nothing. Of course the trick is that there can exist vacua at different energy. Since this is quantum mechanical there can be quantum transitions or tunneling between two vacua.

          Your essay did propose an alternative to the MUH. My main point with untestable is empirical. My main issue with MUH has been that it tries to "prove too much." The idea is a sort of monism, which tries to reject duality between forms and substance, but in the end it runs into difficulty I think.

          Cheers LC

          Actually equilibrium in general relativity is not well defined. Suppose you have a black hole that has the same horizon temperature T ~ 1/M as the cosmic background temperature. If the black hole emits quanta it becomes hotter and the probability that it will then emit more photons to the universe increases. Conversely, if the black hole absorbs a photon from the background universe it becomes colder which makes it more probable that it will absorb more photons than emit them. This is a bit odd with respect to semi-classical or quantum physics in spacetime.

          LC

          There have been some ideas along these lines. There was axiomatic quantum field theory, but this never really seemed to catch on. There is also quantum logic, which has lead to some interesting developments. There is though Bohr's statement that quantum physics is best described by a system that permits discussion in ordinary language.

          LC

          Dear Adel,

          I suddenly see that I got a fair number of posts and have risen considerably up now to the top. It is interesting to respond to two people, Alexey and Lev Burov above, who refute the MUH, and then within the same half hour discuss somebody who embraces it. Of course I will have to read your essay first. I will try to get to that as soon as I can.

          Cheers LC

          Jonathan,

          Thanks for the positive assessment. Indeed, I seem to have popped up considerably in the last day or two. I will have to take a look at your essay as well to refresh my memory on it. I can't recall if I scored it as yet.

          Cheers LC

          I will have to respond in detail later. It is the morning before heading off to work. In many ways physics is coming around to the idea that information is at the foundation of what is important with respect to phenomenology.

          LC

          Excellent job Lawrence,

          This essay is well-written and presents a Tour de Force of interesting Maths relevant to Physics. You have managed to work in a lot of topics that are very interesting to me, and about which I have much to learn, such as Homotopy Type Theory. The HoTT program is especially interesting to me, as it has a constructive geometric basis on the one hand, and a rigorous analytic procedure on the other. I also like that you wove in the Bott periodicity, which I was trying to find a way to fit into my own essay, because it is one of those invariant structures that one seems to bump into - as though it was there before you found it.

          Being a constructivist, I think that perhaps numbers and counting are not the first Maths to arise, however. Having a Set of objects requires preexisting elements of geometric topology, so that objects with surfaces and containers to hold them are well-defined. Also, it is seen in young children that a sense of greater and lesser quantity is a kind of numeracy that exists apart from counting itself, and develops sooner. I would think that just as ontogeny recapitulates phylogeny, for developing organisms; so individual patterns of learning are reflected in the development of cultures. Perhaps counting is merely the earliest form of mathematical reckoning that could be written down.

          I think you come out on the side of the formalists and logicists in the Brouwer Hilbert debate, while I am firmly in the intuitionist camp - and while this is sometimes termed anti-realist, I believe it is more realistic yo imagine that everything should be constructable for Physics. But at least you mention that there is a debate about this among mathematicians, which some might miss otherwise. It was a great effort overall, and you get high marks from me.

          Regards,

          Jonathan

            Jonathan聽

            I am rather agnostic on any of these ideas about mathematical foundations. I don't hold to any of them to much degree. For one thing these things are a bit removed from physical theory, which I am more interested in than pure mathematics. The other reason is there seems to be no way we can decide whether one is better than the other. In some sense maybe it is best to consider them as metaphysical tools that can be used or not depending on the situation.

            The homotopy and Bott periodicity involves my observation that groups involved with quantum information appear to have this period 8 structure to their topology. This seems to extend into the exceptional and sporadic groups as well. This means the quantum bits associated with a black hole event horizon have a type of degeneracy. This is the main reason why I think it is possible that this homotopy based mathematics with a correspondence to quantum bits might form the foundations of mathematical physics through this century. It would be curious to see what mathematical physics looks like in 75 years.

            Really I don't pretend to know the relationship between physics and mathematics. It is a completely mystery really. It may just come down to an instrumentalist argument that because physical science involves measuring things according to numbers that the subject must necessarily involve mathematical consistency.

            LC

            Dear Lawrence,

            Since the question of this contest is about universe theoretizability (using the word of our essay), the answer apparently cannot refer to such specific terms as 'vacua', 'energy', or 'quantum mechanics'. Such references are logical flaws, aren't they?

            As to Tegmark's MUH, we are refuting it on the factual ground, namely, on the grounds of the logical simplicity, large scale and high precision of the already discovered laws of nature.

            Cheers,

            Alexey Burov.

            Thanks for the thoughtful reply Lawrence..

            In his (non-contest) essay 'A View of Mathematics', Alain Connes speaks about Math as a single corpus, almost like a biological organism, that makes it hard to separate the parts or say what came first. I am of the opinion, however, that there must be some set of most elementary rudiments, from which that entire body of knowledge flows. Perhaps this does not mean it can be constructed deterministically, but there have to be some bones to hang the meat on somewhere.

            You might find interesting Andrei Rodin's book 'Axiomatic Methods and Category Theory' which relates strongly to the HoTT program, and is available as a download from arXiv. But I think it is wonderful that mathematicians are pursuing a better understanding of the underpinnings of Math, to expose its underlying simplicity, or inherent congruency, as well as probing the complexities and the details of that knowledge.

            All the Best,

            Jonathan

            I think you made reference to how mathematics by itself is not a minimal guide for physics. Many mathematical areas are vastly complex systems, which might not serve as an effective foundation to reality.

            Maybe a part of the problem is that we do not have a "unified theory of mathematics," assuming such a thing is possible. I think the foundation of the universe involves zeta functions, 8-fold periodicity related to Bott periodicity, homotopy indices, Langland number theoretic correspondences and so forth. This may involve some unification of some subjects in mathematics. I am not sure if this is comprehensive though. Physics in one sense involves working on a similar type of problem on deeper levels, while mathematics often involves pursuing the study of entirely different sorts of structures. These new structures can come into play with physical problems, but the method of thought is often very different between how a mathematician works out the consistency of some type of structure, and how a physicist frames a type of theory or solves phenomenology.

            I would agree that Tegmark's MUH does not appear to satisfy certain minimal conditions we would prefer. In that I would tend to agree with you. I am not sure if this is exactly a proof though. For all we know our requirement for simplicity could be a bias that is wrong. Maybe the universe is vastly complex at is foundation. There are areas of mathematics that have physical implications which involve a huge level of complexity. The universe might in fact have some extremal Kolmogoroff complexity condition, which means the foundations are not only bewilderingly complex, but unknowable. I am not saying I think this is the case, but on the other hand I do not know that this is not the case.

            Cheers LC

            Dear Lawrence,

            Following your last post, this is the type of application we can discuss. Until now, I focused on dessins due to their relationship to quantum geometries and contextuality as in my [12] and [17], now I mentioned in the essay the link to most sporadic groups, there are plenty of other applications, some have to be discovered. Cheers.

            Michel

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            • Post #114

            There are several statements in your comment, dear Lawrence, to which I'd like to respond.

            1. "I would agree that Tegmark's MUH does not appear to satisfy certain minimal conditions we would prefer." I cannot accept the verb "prefer" here. We are refuting the MUH not on the ground of preferences, but on the ground of facts.

            2. "For all we know our requirement for simplicity could be a bias that is wrong." In a sense, I share this caution. As Einstein used to say, "Subtle is the Lord but malicious He is not". We, humans, should not underestimate how subtle He might be. However, without a strong belief in the human ability to comprehend the yet unknown, this unknown would be unknown forever.

            Cheers,

            Alexey.

            The Rodin book looks to be a very long read. It could have some interesting insights into things. There are connections through groupoids to category theory and Grothendieck type of theory and cohomology.

            It is hard to know what the totality of mathematics is. It could be infinite in extent, which of course makes it difficult to know how this applies directly to physics. That would be difficult with respect to the Tegmark MUH conjecture. The one thing that is apparent is how many areas of mathematics are mapped into each other according to functors and categories.

            Cheers LC