Dear Alexey,

Thank you for posting an interesting essay. You presented many questions I believe we should further consider in the discussion of discovering a unified theory describing nature.



The fact the universal laws were well constituted, established, and in practice billions of years before humans were cognitively/consciously capable of developing symbols to describe...

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Dear Alexey,

Thank you for posting an interesting essay. You presented many questions I believe we should further consider in the discussion of discovering a unified theory describing nature.



The fact the universal laws were well constituted, established, and in practice billions of years before humans were cognitively/consciously capable of developing symbols to describe and interpret them suggests our involvement in the theoretical/calculable discovery of these laws plays an insignificant (if not minimal) role in their fundamental purpose, function, and prevailing existence. This clearly suggests we are only capable of deriving various tools and methods of communicating physical phenomenon by observation for contemplation among our own human kind using a preferred but limited practical method of describing the logic of reality - mathematics and physics. In other words, the moon and other planets existed before we were capable of observing, discovering, and offering them their descriptive names for identification and definition as planet and moon. The universe however, observes a celestial body, humans, during our limited methods of understanding, tend to classify these observations for identification which leads to interpretation and confusion regarding the fine tuning of an equitable definition of the observation itself. We are fundamentally at fault for our own chaotic misconceptions in our attempts to understand quantitative observations.

Chaos is not without structure, it is however a calculable distinct derivation of a “defined” ordered state. We may never discover or possess the experimental capacity to confirm definitely the existence of conscious intellectual beings residing in the opposite sides of our universe (as they may simply assume we do not exist), but we may speculate their existence without confirmation just as we may speculate the undiscovered laws governing nature must exist, but is currently undiscovered or recognized and therefore appears unfathomably possible from our limited progress of exploratory pursuits. Perhaps, it is likely, we have been looking in the wrong direction while utilizing the many available tools at our disposal. This does not suggest the tools are useless from being derived by our own conception, the impossibility of a unified theory describing nature is nonexistent, or complication from chaos plays a pivotal role in its evasiveness. It simply means we have not looked in the appropriate direction when attempting a discovery of what we assumed cannot be found. Furthermore, humans are often flawed by our own distractions of self, we may only need to pursue this discovery with our eyes and minds open to alternative methods of discovery not yet considered and contemplated by conventional means. But regarding a unified theory if it exists, we will find it to be self-evident, self-explanatory, and independent of any other supportive methods conceived by our human applied interpretation, therefore it will remain unbounded yet simple.

Best Regards,

D.C. Adams

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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...

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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^∞ 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

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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...

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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

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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...

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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.)

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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...

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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

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As I said I appreciate the general ideas of your essay, as a logical argument against some widespread conceptions, but still I have remarks. That is, what does the landscape of logically possible laws of physics, look like. Of course as you explained we can abstractly consider any ugly arbitrary law as a "possible law", and in this vast chaotic landscape of "possible laws", find many possibilities...

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As I said I appreciate the general ideas of your essay, as a logical argument against some widespread conceptions, but still I have remarks. That is, what does the landscape of logically possible laws of physics, look like. Of course as you explained we can abstractly consider any ugly arbitrary law as a "possible law", and in this vast chaotic landscape of "possible laws", find many possibilities of biological evolution similar to what happened on Earth, with the difference that the fundamental laws would not be intelligible. However it seems to me (and maybe I read from other sources) that, among all "possible laws" able of biological evolution, our particular laws (quantum field theory, the standard model) differ from the rest of this landscape not just by their intelligibility, but also in other ways : something like, apart from modifying the values of physical constants, they are not so much modifiable in the small details of their consequences without destroying all their mathematical coherence.

Namely, while we can conceive of possible variants (such as was hypothesized as alternatives to the Higgs boson, some supersymmetric models now refuted, etc), they are not so many (if we want to restrict the study to simple ones, thought the motivation for this restriction may be questioned, e.g. for its links to intelligibility). For example, there are not so many conceivable kinds of particles that are mathematically coherent in quantum field theory. The Higgs boson was predicted because there are not many logical possibilities of what may give a mass to particles. We may see it as a beauty of mathematics (a restriction on the number of possibilities, and some of the most elegant ones are actually realized).

More importantly, there are not many logically conceivable alternatives to quantum physics that behave in a roughly similar manner. It is even doubtful whether there is such a logical possibility at all. An important search for such things is the spontaneous collapse theories, however usual attempts of such theories violate both relativistic invariance and the conservation laws, even if very slightly. But a violation of conservation laws, even a very slight one, is something that (when formulated in proper terms) General Relativity dismisses as absolutely impossible. Thus, when trying to conceive logical alternatives that would also lead to a biological evolution, they may have to either be very different from how things go in our universe (very different chemistry, etc), or be of the form "These laws do not always apply, they are sometimes broken", but still in ways that do not result in breaking the whole Universe apart (as "breaking conservation laws" would logically imply), that would be... very odd kinds of laws.

But what I see remarkable about the trouble in trying to locate our laws of physics in a landscape of logical possibilities, is not only the lack of phenomenologically similar possibilities (beyond variations of the values of physical constants) but also that it is questionable whether the laws we found of our universe (quantum physics) may meaningfully be considered as a logical possibility at all. Because if we strictly look at them as a logical possibility, then it logically drives us to the Many-worlds interpretation. But, how can a many-worlds universe be considered as a real universe at all ? It does not look like one ; it is questionable (and a metaphysical question) whether we can make sense of "probabilities" in it, unlike the effective role of probabilities (the Born's rule) in our physical reality.

And, in connection with what I explained in my essay, I see all this as not a trouble, but as the simple consequence of the fact that the whole discussion, as followed by Tedmark and other scientific circles, is usually expressed in the wrong terms. Because if the question is about the possibilities of real universes, then it has to involve an ingredient of "reality" beyond mathematical existence, i.e. beyond pure logic, and this ingredient is consciousness. Since, due to its non-mathematical nature, this ingredient cannot be located in any mathematical landscape of possibilities, the very expression of "logical possibility" is not applicable to it.

For more details, see my notes on spontaneous collapse and the Many-worlds.

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