GENESIS OF A PYTHAGOREAN UNIVERSE by Alexey and Lev Burov

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

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.

Dear Alexey,

I was hoping you and Lev find explicit mistakes in my system in the spirit of the contest. But of course I do understand if you do not find the time or the inclination.

Thanks and good luck to you.

Dear Alexey,

I am an engineer by trade with interest in physics, not a philosopher or anything fancy. I Just took a good guess which was very similar to other people's guesses like Wolfram(NKS), Conway and others, However my guess was just very successful in my opinion. This is exactly the lesson people could draw from the "effectiveness of math". It is for other people to evaluate my system and consider it as useful or not as it stand. There is also the possibility of improving it so it maybe more convincing, or somehow I would discover that the system is trivial and I would be happy to kill it and use my time to make more money :-) Sorry for wasting your time.

Thanks and good luck.

Dear Alexy and Lev,

Thank you for an interesting essay.

You write that there is no reason for the rules to be selected. In my essay I explain why some of the rules related to some of the phenomena are, in fact, selected.

Thanks,

Noson

Dear Alexy and Lev, I keep reviewing other essays, expanding my list of the best essays I found, and now I would like to recommend you the essay Was there a choice? by William Nelson, which I found to have interesting common topics with your own, even if their reasoning is different.

Dear Alexey,

Dear Lev,

I have carefully read your essay and I agree completely, with your chain of reasoning and all details.

First of all, we never underline sufficiently that the mere presence of laws of natures taking mathematical forms does not explain anything. Within papers or discussions belonging to this contest, we can often read that mathematical truths are deducted from physical experience. But this, as you notice it rightly, does not explain how/why natural phenomena "behave" in the way that their "behavior" matches given laws. Hence there must be "something" doing that physical phenomena "behave" "law like" and not anyhow.

I particularly appreciate your proof framework: On the one hand, the configuration of natural constants compatible with the emergence of life characterizes already itself by a probability infinitely close to zero. On the other hand, the finesse of fine the tuned universe is not fine enough to allow any experimental justification of phenomena satisfying WAP conditions. So, instead of formulating far-fetched hypotheses, it is better to see in more-than-fine-tuned-universe related phenomena the confirmation of this Pythagorean/Platonic faith which in turn had guided the approach of many physicists being at the origin of modern and contemporary physics.

Your essay is courageous, very clear, well written and documented. I have just given it the highest rating.

I would like to stay in contact with you even after the end of the contest to continue exchanges of ideas, knowing that there is still a lot of work in this domain.

Good luck, best regards,

Peter

Dear Alexey and Lev,

Thank you for your thoughtful comments on my essay. As you indicated, I agree with your view that chaosogenesis (to use your term) must be rejected. Let me try to advance the discussion by mentioning three points.

First, about the argument (on page 7 left column): "Because the logical structure of our universe cannot be explained by chaos, and because it cannot explain itself, we are left with only one possible explanation remaining, that it was conceived and realized by a mind." What about another alternative, namely, that the logical structure and indeed the existence of our universe cannot be explained at all? On this alternative, the universe simply is, and nothing more can be said. In recent years a number of writers have advanced theories which amount to chaosogenesis in one form or another. As the weaknesses of such theories become more and more obvious, many of these people will probably retreat to this position of fundamental inexplicability. The conclusion that you present (i.e., "that it was conceived and realized by a mind") is, I think, preferable to the alternative of no explanation at all, but the interesting question is how to formulate the arguments.

Second, the argument on page 3, left column, is important, and I believe it deserves to be elaborated. I refer to the argument from the forms to their unity and then to absolute mind. As you point out, Tegmark does not bring up the matter, and when I read some of Tegmark's writings, I never thought of anything like what you say here. I think you are saying something like this: The mathematical forms are real. If a form is real, then it must be self-consistent. But self-consistency of each individual form is insufficient to guarantee the reality of the total realm of forms. If there is a mathematical universe, then all the forms in that universe must be mutually consistent. But what constitutes that over-all consistency of the mathematical universe, and for what reasons do we believe the mathematical universe to be consistent? (In my view, these are two separate questions, although related.) The ground of the self-consistency of the mathematical universe cannot itself be a formal proof or other mathematical structure. We can see why that will not work. So, the ground and guarantee of mathematical consistency has to be something outside the network of mathematical structures. Considerable additional argument is required to show that the external something is a mind or at least mind-like. You state that the logical terminus of the argument is absolute mind. As I said, your ideas on this point are new to me, but I am inclined to agree with you. I am sure that we should not simply take mathematical structures as the unproblematic starting point.

Third, in this same place on page 3 you assert that Tegmark's mathematical absolute has "total indifference to the forms it contains." This contrasts with your position that Ultimate Mind (or Absolute Mind) grants to each of its constituents "its own special significance", as you say in a comment above. I think this contrast is important. I believe you would also maintain that, from the actual contents and order of nature, we can infer something about the primal valuations which are intrinsic to what you call "the inexpressible potentiality of being". Here I am inclined to agree more with you than with Tegmark. In any event, you have brought forward an issue that has been too much neglected. Perhaps in the future you will say more about it.

Best wishes,

Laurence Hitterdale

Dear Alexey and Lev,

Very good essay. To repeat my reactions for the sake of readers here: The writing was excellent, in the sense of being both technically apt as well as readable to a general literate audience (it reminds me of Penrose's writings, and in outlook as well.) I think you have an acute grasp of conceptual foundations and issues (like, the problem of existential asymmetry for specially-selected possible worlds.) Well put. First, I agree with you that physics is more than math, and that our world is not a math structure. Math by itself cannot tell us more than about its own contents (like, why there "are" five Platonic solids in that sense). However, as you well argue, the math we find in the universe can tell us much more. You correctly note the flaw in the argument that observed fine-tuning can be adequately explained (in Bayesian terms) as no more than a self-selection effect. True, if that were so, then the precision and elegance of the world would probably be less. (However, let's all admit that with continua we do have a measure problem. Still, even without enumerable sets to compare, the relative "areas" of numerical ranges give us a rough idea of what we should expect.)

Actually I think the problem is even worse. If we really consider the full range of math structures, then we have to include inconsistent ones like e.g. the splicing together of y = x2 with y = x4. In that case, rules would not even be consistent over time etc. There are many more possible messy worlds than orderly ones, a problem noted about David Lewis' modal realism.

These foundational arguments are fascinating and important, but I am particularly proud of my novel (in its broad execution at least) argument for why space had to be three-dimensional. It constrains possible worlds more than previously realized, although as I noted: only to the extent that we expect lawful consistency in "worlds" in the first place. And what really makes "worlds" different from mere structures of math? I basically agree with the sentiments pleaded by Roger Penrose (whose diagram is borrowed for your essay). Quote:

"One can argue that a universe governed by laws that do not allow consciousness is no universe at all. I would even say that all the mathematical descriptions of a universe that have been given so far must fail this criterion. It is only the phenomenon of consciousness that can conjure a putative 'theoretical' universe into actual existence! ... Yet beneath all this technicality is the feeling that it is indeed 'obvious' that the conscious mind cannot work like a computer, even though much of what is actually involved in mental activity might do so. This is the kind of obviousness that a child can see--though that child may, in later life, become browbeaten into believing that the obvious problems are 'non-problems', to be argued into non-existence ... ."

- Roger Penrose, in The Emperor's New Mind (1988), pp. 447-448.

Finally, I invite readers to look at my own essay. It is one of a few that directly presents specific new physical insights (in this case, for why space has three large dimensions.)

Dear Alexey and Lev,

It was a great feeling when I discovered that you share the same ideas on metaphysics. I enjoyed reading your essay and gave it a high grade. I have a couple of comments, which can be thought of as suggestions for a further research.

The Godel theorems in logic imply that there is no a TOE, which has a profound implications for a platonic metaphysics. This fact is also consistent with the fact that beside the mathematical ideas (concepts and structures) exist non-mathematical ideas (also called by Tegmark as non-computable structures). As I described in my first FQXi essay, "Temporal Platonic Metaphysics", one needs the idea of passage of time, as well as of consciousness (observer) in order to obtain a more realistic metaphysics than the one proposed by Tegmark. In your essay you raise a very interesting question concerning the multiverse, i.e. which universe is realised and why is it realised. From my point of view, an abstract Universe is realised if it is temporal. It is clear that our universe is temporal and it contains cosmic observers, i.e. us. You also raise a deeper question, which is which abstract universe is realised, and you use the arguments related with fine tuning to justify the existence of universes with observers. I think that the deeper question is why our universe is temporal, i.e. real, since there is an infinity of abstract mathematical universes. You correctly identify that there are two logical possibilities: it is God who decides or all mathematical universes are temporal.

I also liked that you mentioned the 3 worlds: platonic, mental and material. I would like to point out that in my aforementioned essay, I described a relationship between them: the mental and the material worlds are subsets of the platonic world of ideas. The material world consists of mathematical ideas immersed in time, while the mental world consists of nonmathematical and mathematical ideas bundled together in our brain.

Best regards,

Aleksandar

Dear Alexey,

I cannot find your email adress in your author bio; perhaps it had been suppressed by organisators.

Here is my own personnal adress:

peter.punin@wanadoo.fr

Please could you just return a blanc mail to this adress to establish contact?

On the other hand I have consulted the home page of the Fermi Society of Philosophy. Is there a possibility to join it not only on forum discussions but as an active member?

Best regards

Peter