Dear Alexy and Lev.

Your paper is well argued. I will admit to being very agnostic about these sorts of ideas. In particular I am very agnostic about Tegmark's hypothesis, which seems not mathematically provable, nor scientifically testable. Even string theory is only at best indirectly testable, but Tegmark's Mathematical Universe Hypothesis seems impossible to test.

A couple of points I mention first. The WAP as I understand it is the statement that the universe observed must be of sufficient complexity and structure to permit such observers. It does not mean that any cosmology that exists must admit observers. I think that is the strong AP (SAP). The other point is that chaos, at least within the meaning of Hamiltonian chaos or strange attractor physics, means that a system can execute a vast number of complex dynamics, all of them separated by very small initial conditions. This means that phase space is specified to a very small fine grained detail. Given this is cut into N boxes or pieces, and in each is one of the possible states (0, 1), the degree of complexity is 2^N = e^{S/k}. This is the dimension of the Hilbert space corresponding to this classical setting and the entropy S = k ln(2)N = k ln(dim H), H = Hilbert space. Chaos then in fact implies a high level of complexity.

I did not make much mention of this in my essay. It could be said that mathematics has a body and soul. The body concerns things that are numerically computed and can in fact be computed on a computer. The soul involves things that involve infinitesimals and continua. These tend to be at the foundations of calculus with limits and related arguments. Even though my essay discusses homotopy, this is argued on the basis of continuous diffeomorphisms of loops or paths. However, in the end this is not what we directly compute in mathematics. We are interested in numbers, such as indices or topological numbers, and in physics that is much the same.

If you crack open a book on differential geometry or related mathematics you read in the introduction something like, "The set of all possible manifolds that are C^в€ћ with an atlas of charts with a G(n,C) group action ... ." The thing is that you are faced with ideas here that seem compelling, but from a practical calculation perspective this is infinite and in its entirety unknowable. This along with infinitesimals, or even the Peano theory result for an infinite number of natural numbers, all appears "true," but much of it is completely uncomputable. This is because the soul of mathematics touches on infinity, or infinitesimals.

The soul also involves things that quantum mechanically are not strictly ontological. These are wave functions or paths in a Feynman path integral. The existential status of these is not known, for the standard idea of epistemic interpretation is now found to be not complete. This differs from classical physics, where the physics is continuous, with perfectly sharply defined paths and energy values and so forth.

I am somewhat agnostic about the existential status of the soul of mathematics. In some sense it seems compelling to say it exists, but on the other hand this leads one into something mystical that takes one away from science. So it is not possible as I see it now to make any hard statement about this. We seem to be a bit like Garrison Keillor's Guy Noir, "At the tenth floor of the Atlas building on a dark night in a city that knows how to keep its secrets, one man searches for answers to life's persistent questions, Guy Noir private eye."

I will give your essay a vote in the 7 to 10 range. I have to ponder this for a while.

Cheers LC

There is a profound principal in the universe that says there is no central entity or notion anywhere, and that everything has no special significance than any other things in physics terms. This principal dispelled 'earth-centric' idea and later the Newtonian absolute time-space concept. It is a universally accepted principal in modern science. If math and physics are intertwined inextricably then it seems natural numbers ought to have an equal standing as any other numbers, irrational, complex, or even numbers yet to be invented.

Is there any physical underlying reason for natural numbers' special status? Or are the natural numbers just a convenient way for people to count and were invented by macro intelligent beings like us?

Since all natural numbers are mere derivatives of the number '1', so let's look closely at what this number one really means. There are two broad meaning of the number one. First it registers a definitive state of some physical attribute, such as 'presence' or 'non-presence'. We can find its application in information theory, statistical physics, counting and etc. The second interpretation of number one is that it denotes the 'wholeness' of an entity.

In physics, natural numbers virtually have no sacred places prior to the establishment of quantum mechanics. After all, we don't need any natural numbers in our gravity functions or the Maxwell electro-magnetic wave functions. Some sharp observers would argue that the 'R squared' contains a natural number 2. However on close examination the number 2 is merely a mathematical notation for a number multiplying by itself, and it has no actual physical corresponding object or attribute. The fact that there is no natural number in the formulae represents the idea that time-space is fundamentally smooth. For instance there is no such law in physics that requires 7 bodies (non quantum mechanical) to form a system in equilibrium.

Had we obtained calculus capability before we can count our fingers, we probably would have been more familiar with the number e than 1-2-3. We might have used e/2.718 to represent the mundane singletons. There is no logical requirement that we couldn't or shouldn't do it. It is all due to the accident that people happened to need to count their fingers earlier than the invention of calculus. There is no physical evidence that the number '2' is more significant than the any other numbers in the natural world.

However with the standard model of quantum mechanics, energy is quantized, that is, it can only take natural numbers. This idea profoundly altered the status of natural numbers in physics and is a direct contradictory of the notion of 'no center in the universe' principal. In this sense it is far more unorthodox than the two relativity theories combined because the latter in fact enhance the 'no center in the universe' law. Why does the quantum have to be integer times of a certain energy level, and not an irrational number like square root of 17, or the quantity e? Does it really mean there are aristocrats in the number world, where some are nobler than others? Were the ancient Greek mathematicians right after all, who worshiped the sacredness of natural numbers and even threw the irrational number discoverer into the sea?

From this standpoint we can almost say that quantum theory has some bad taste among all branches of natural science.

Before the quantum theory got its germination, actually people should have noticed the unusual role natural numbers play in rudimentary chemistry. For instance, why two hydrogen atoms and not five, are supposed to combine with one oxygen atom to form a water molecule? If scientists are sharp enough back then they ought to be able to be alarmed by the oddity underlying the strange status of natural numbers. It could almost be an indirect way to deduce the quantized nature of electrons.

Fundamentally if natural numbers indeed play a very unusual role in nature, then nature resembles a codebook not just from a coarse analogy standpoint. It is the ultimate codebook filled with rules for a limited number of building block codes. The DNA code is an excellent example.

If it's a codebook, inevitably it takes us to surmise if information itself is the ultimate being in the universe. It is probably not electrons, strings, quarks or whatever 'entities' people have claimed. It is the information that is the only tangible and verifiable entity out there. Everything else is a mirage or manifestation of some underlying information, the codebook.

In this sense physics has somewhat gone awry by focusing on the wrong things, the 'attributes' such as momentum, position and etc. Instead, information is what contemporary physicists talk about and experiment with. Otherwise, the physicists would have no right to laugh at the medieval scholars who based their intellectual work on the measurement of the distance between a subject and God's throne.

The nature has revealed her latest hand of cards to us. It looks like it's the final hand but no one can be sure of course.

    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