Dear Philip,

Following our previous conversations, I just reread your essay. Indeed, we share a lot of the same views.

I like the way you begin your essay by considering the shortcomings of the "univacuum assumption". I googled "univacuum" and I think you are the first to use the term in this context. I would say that I am clearly in the camp of the "multivacuers"... but I think there are many "univacuers" out there! You say that the idea of the multiplicity of the vacuum "bruises the egos of particle physicists who thought that the laws of physics they were unveiling were special in a very fundamental sense." This echoes the final sequence in the third of my "This Is Physics" videos (submitted to the recent FQXi video contest) where I claim that "Most physicists don't like the idea of the Multiverse, because if it is true, it means that they have devoted their lives to master only ONE physics of ONE universe instead of THE physics of THE universe."

I agree with you when you explain that it is more "parsimonious" to accept that "all solutions exist in some higher sense, whether inside or outside our universe." I claim essentially the same thing in my essay when I discuss the issue of Occam's razor (the "law of parsimony") in the context of the multi/Maxiverse.

Like I said in the most recent reply to my conversation with you and "En Passant" on my essay's page, maybe we are obscuring the issues by insisting in labelling as "mathematical" or "physical" the fundamental structures that make up reality. In your essay, you take a safe and wise approach when you simply talk about the "timeless and spaceless" ensemble of "all things that are logically possible".

For me, the highlight of your essay is when you say that "Universality brings together all the logical possibilities of mathematics under one metaphorical path integral". Indeed, emergent universal behavior, as expressed by the Meta-Laws of physics, is what we need to understand better if we are to explain why, in the space of all possible worlds, we find ourselves living in a universe that obeys stable and relatively simple laws. I like how you use the concepts of path integrals and critical points in the context of finding out the Meta-Laws of physics.

Finally, you nicely address the subject of this year's essay directly when you explain that "mathematicians and physicists are attracted towards the same critical point of universality".

Great job!

Marc

P.S. As you say, the fact that the Monstrous Moonshine Conjectures have been studied by using the methods of string theory is an astounding demonstration that "there are deep relations between ideas from physics and mathematics". I have to confess that I'm having some difficulty understanding clearly the complicated concepts behind the Monstrous Moonshine Conjectures and Borcherds' approach. Do you know of any (relatively) accessible sources that deal with this fascinating topic?

My use of the word "metaphorical" when talking about the path integral is deliberate. We get an intuitive idea feel for universality when we think of it as a sector of the path integral that dominates or where phases do not cancel out, as in maximum entropy or the classical limit. But even in quantum mechanics that is not the whole story. The path integral for fermions works differently through some less intuitive algebraic mechanism.

Universality of the concept of computability is another example that is less intuitive. Some people do not even recognise it as the same use of the word universality, but I think it is. The universailty that I am looking for in this essay may also be more subtle than just a path integral dominated by some kind of comonality, but it is a good metaphor and possibly a source of toy models that might be instructive.

The monstrous moonshine conjectures are quite deep and a bit beyond my expertise, but you can get an idea by learning a little about the Golay code and the Leech Lattice and the related groups. There are a lot of mathematical ideas mysteriously connected by the number 24 which are also connected to the number of dimensions in bosonic string theory. John Baez wrote some easily understandable articles about that.

6 days later

Dear Phillip,

There are opinions that Euler's identity is of particular importance for physics. In this competition is very rarely mentioned. I have no idea about it, but I see the possibility of complementarily exp(i*pi), (from Euler's identity) and exp(2*pi), which I use in my essay, under the name cycle. I ask: what you think about the possible progress in using Euler identity in physics?

Regards,

Branko

    Dear Brabko,

    Euler's identity was an early example of the surprising connectedness of ideas in mathematics and physics. It linked [math]e[/math] and [math]\pi[/math] in a surprisingly direct way where previously they seemed unconnected. Now we take it for granted because it is part of complex analysis that is common place in maths and physics, for example it is used in Fourier analysis. However it is a good idea to use it in this contest.

    Hi Phil,

    It is a pleasure to meet you again in FQXi Essay Contest. Even this year, you made an excellent work as I found your Essay very interesting and enjoyable. I must confess that I have read your Essay after reading a nice sentence of yours reported in the Essay of our friend Jonathan J. Dickau which claims that "the laws of physics are a universal behaviour to be found in the class of all possible mathematical systems". In any case, here is a couple of comments on your nice Essay:

    1) Some years ago I discussed in San Marino with two great physicists, i.e. Hagen Kleinert and Alexander Burinski, on the new paradigm that is emerging in fundamental physics concerning the new way of looking at the universe. All of us three agreed that gravity is the key. We think that gravity goes beyond string theory and quantum field theory.

    2) I disagree on the issue that the holographic principle is required to resolve the black hole information loss puzzle. This assumption by Susskind is based on the Maldacena conjecture. But I agree with Mathur's criticisms, see here and here. The key point is that the AdS/CFT duality works only for low energy processes, where black holes do not form. In other words, if we force the gravity theory to be the dual of a given CFT, then we cannot assume that black holes will form in the theory. Thus, the duality seems to break down in presence of black holes.

    In any case, I still remark that I find your Essay very intriguing. Thus, I give you a deserved highest score.

    I hope you will have a chance to read my Essay .

    I wish you best luck in the Contest.

    Cheers, Ch.

      Christian, It is good to see you here again.

      I agree that gravity and spacetime is an important key. For me it is about how to understand that the geometry of gravity emerges from somthing more algebraic and abstract, but there are other ways to see it.

      I dont think the holographic principle is really based on AdS/CFT. The principle was formulated by 't Hooft and Susskind before the Maldacena conjecture and is independent of that or string theory. The problem is to know the general context in which the conjecture could be true. For that I have suggested that "complete symmetry" is the required element. Of course it is not unreasonable to be skeptical of the principle. It is speculative, as is string theory. I am sure it will be many years and there will be many interesting discussions and revelations before any consensus is formed.

      Philip,

      Heavy stuff. As some politicians would say, "I am not a scientist specializing in quantum gravity, so ..." Well, what they say is simpler.

      "All is possible in the quantum realm but there is a hierarchy of classical limits ... these limits define worlds in which math rules are played out according to the law of quantum averages."

      Can we simply start with trying to explain a mystery in the classical world, for example, how European robins navigate N and S seasonally. A theoretical physicist and a molecular genetics professor did this. They combined a receptor for the avian chemical compass with a protein capable of generating entangled electrons that interacted with the Earth's magnetic field. Does their study impose a classical limit which utilizes what they know about quantum mechanics?

      I know I am cherry-picking your quotes but I am struggling to understand.

      Incidentally what does a successful BICEP2 discovery of a primordial B-mode cosmic inflation caused by gravitational waves do for quantum gravity?

      Like to see your thoughts on my essay: http://fqxi.org/community/forum/topic/2345.

      Jim

        Phil,

        Well up to you usual high standard and with signs of greater maturity of view. I hope my score helps keep you at the top. I particularly like your identification that the question can be just as well reversed (why is it that physics...). I do however have a couple of questions;

        1) Do you really believe there must be a greater or 'super' symmetry hiding away behind the 'dipole' symmetry considered 'breaking' by Green and most particle physicists? ..and what would it bring. Do you think of the concept that the dipole may perhaps be the very quiescence of 'matter' (inc anti) as opposed to (maybe dark?) 'energy' alone?

        2) This may be semantic, but suggesting a 'map of all things logically possible" would seem to many to be excluding QM and non-locality. Do you suggest QM; a) CAN have a 'logical' explanation. Or b) ??

        Re 2; I hypothesise a 'quasi classical' mechanism that seems to reproduce it and reveal the mathematical 'sock switch' trick that hides it in my essay, so prefer a).

        I hope you get to read mine. I feel that in 'stabbing in the dark' I've felt something, but how can we then expose what it really is when each of us has a different vision? Perhaps the value of reading these a essays is in converging those vision.

        Well done, and very best of luck with the definitive judging.

        Peter

          Dear Philip,

          Because universality is the central concept in your ontology, I wonder whether it would be appropriate to put logic at the point of universality. We usually think of universal concepts as the concepts which apply to everything. Similarly, universal principles are defined as those principles which apply to everything. The common view seems to be that the concepts and principles of logic are universal in this sense. Your chart includes all logical possibilities. If we try to organize and understand the realm of all logical possibilities, then I think we would begin by using the concepts and principles of logic. In a sense a very basic part of the meta-laws are the laws of logic. Of course, there are more specific meta-laws as well, including meta-laws for physics. But maybe I am misunderstanding the role of universality here. In any event, thank you for a stimulating essay.

          Best wishes,

          Laurence Hitterdale

            James, the world seems classical but that is only superficial. Quantum mechanics plays many roles in life.

            If the BICEP2 discovery had stood up it could indeed have allowed us to explore the effects of quantum gravity in the early universe. Sadly that seems not to be unless we can find a clear enough window through the dust. Other opportunities to see similar effects may come from direct observation of primordial gravitational waves or low frequency radio waves.

            Peter, supersymmetry was first conceived as a component of quantum gravity where it may be very hard to detect. If is important in making perturbative quantum gravity more consistent. In my idea of "complete symmetry" where there is a degree of symmetry for every degree of freedom supersymmetry is essential simply because there are fermions. All this is specualtive of course. I dont think there is any good evidence even in dipole measurements.

            So called quantum logic can be described using ordinary ideas in logic. We can pretend that it is something more general but I dont see it that way. Of course there are some mysteries in quamtum mechanics but I dont see them as questions beyond logic. Perhaps I am wrong.

            Laurence, I dont think you have misunderstood it. You have expressed it very nicely.

            Philip,

            Shark time when some are pulled under, so I am revisiting essays I've read to assure I've rated them. I find that I rated yours on 4/20, rating it as one I could immediately relate to. I hope you get a chance to look at mine: http://fqxi.org/community/forum/topic/2345 as the hours tick down.

            Thanks,

            Jim

            11 days later

            Dear Philip,

            This is very well written essay, and an enjoyable read.

            I read quite a long time ago, and I think I am now on my third read. I guess that I really have a different perspective.

            You have a logically-ordered ontology of mathematics which you present metaphorically. And central to this is the idea that there is some sort of universality around which the mathematics converges rather like a critical point.

            While on the surface, there is something attractive about this idea. However, there are some seemingly paradoxical aspects that arise the more that I think about it. First, you don't seem to differentiate between types or classes of mathematical theories or descriptions. And because of this, it then seems odd to develop such an ontology, since in doing so, the act of creating an ontology (or ordering) would be inherently mathematical. So this then begs the question "What type of mathematics allows you to develop an ontology of mathematics and where does this fit into the resulting ontology?" There is something circular about this that is unsettling.

            On a different point entirely, I think that it is very telling that you (even in your title) are relying on a metaphor. This creation of models via metaphor is a critical aspect of science. David Hestenes' essay takes this stance, which leads to mathematics as being an analogy-based tool for thinking.

            I agree strongly with his approach. Symmetries are particular cases of analogies, and in my essay, I show explicitly how the symmetries of associativity and commutativity (along with closure and ordering) result necessarily in additivity (up to invertible transform). Thus, any description of a system that has those symmetries must result in an additive theory. This suggests that the universality lies in fundamental symmetries (such as commutativity, associativity, distributivity---which are not the same as physics-based (higher-order) symmetries such as isotropy of space, gauge invariances, etc).

            Now, you actually make some comments about symmetry and note that some people see symmetries as being emergent. I believe that some are. They are emergent from the chosen description. But they still could be the source of the laws. Another objection that you point out is the fact that some theories known to be dual to one another are based on different symmetry groups. However, this is not an argument against the universality of symmetry. Instead it highlights consistency in/and among the chosen description/s.

            In the post above from Laurence Hitterdale, he points to logic as being the universal principle. In your response, you seem to agree with this. However, it is not specified which logic you two are discussing. But either way, logic is a particular example of symmetry/order, which again places those concepts at center stage.

            To me it seems that your exercise in constructing a metaphor for an ontology of mathematics highlights the critical nature of metaphor and analogy in science, which supports symmetries as being central as Hestenes and I discussed in our essays.

            I think that there are some deep ideas/insights here that can be extracted. I would like to know your thoughts if you have a chance.

            Again, thank you for a very enjoyable and thought-provoking essay.

            Kevin Knuth

              Kevin

              Thank you for your insightful comments

              I don't think there are really different types of mathematical theories. There is just one self-referential logical whole. We see it from the inside as participants and like children who rep4eatedly follow every answer with the question "why?" we are never satisfied with a final answer. Yet I think that the class of logical possibilities is as far back as you can go. We have to accept that it is consistent because we know that cannot be proven from within the system other than by the fact of our own being.

              I may overstate the metaphorical aspects of my ideas but I do so to try and keep things separate from the physical models and ideas that our minds are programmed to look for. Sometimes we seek explanations for things that are not there. Inspired by the words of Marc Seguin I would put it like this , consciousness is biology plus noithing else, biology is physics plus nothing else, physics is maths plus nothing else, maths is logical plus nothing else and logical is just nothing else. We look for more in our minds by trying to decide what exists and why and where it came from. We can only express these questions by analogies from our physical experiences. This is a good way to gain some philosophical understanding but we should not lose sight of the fact that they are just metaphors

              I agree on te importance of symmetries is algebra but I think that gauge symmetries are the same thing. That is where they come from. In fact the meta-laws have much more algebraic symmetry and the symmetry we know of in physics must be part of a much greater whole. This is the only way to explain the holographic principle for example. The algebraic symmetries are more fundamental but the physical gauge symmetries are what remains of them when the solutions of the algebraic equations are mapped to emergent space and time. I hope one day people will understand in detail how this works using the principles of category theory, algebraic geometry and the like.

              I think computers would understand real numbers the same way we do, through symbolic logic. You dont have to be able to understand every individual real number to be able to explore the properties of real numbers as a whole. This is the same for humans as it is for an AI. Mathematica and other symbolic logic programs can already handle real numbers in this sense.

              I think that once we know the rules for constructing the possible vacuum states it is going to be a huge challenge to work out exactly what it is.

              It may turn out not to be so hard or it may be so hard that we can never work it out completely, either because the necessary experiments are out of reach or because the computation is too complex.

              The most interesting scenario would be that it is possible but only after some very clever experimentation and computation, but we will have to accept whatever nature has in store for us.

              It was the most abstract, complete and general algebraic object I could think of.

              Science works both ways. Sometimes theorists make predictions that are then verified by scinece, but sometimes experienters find things that were not predicted and theorists have to find the right theory for them. There are plenty of examples of the latter e.g. the muon, law of black body radiation, constancy of the speed of light etc.

              Particle accelerators search for predicted particles but they can also find things that were not predicted. This is done by continually measuring every quatity at higher and higher energies and checking to see whether they match predictions from accepted theory.

              While accelerators are limited in energy scale, the other experiments I mentioned are not. proton decay could give us information about the GUT scale for example.

              You dont need a formal framework to find something new. As soon as something does not match the existing framwork you can start looking for what might explain it.