A Metaphorical Chart of Our Mathematical Ontology by Philip Gibbs

Interesting points. I dont claim to have all the answers. These are the Marianas Trench of philosphical questions and the bottom is still a long way down. However, I am guided by the idea that there cant be any miracles. There must eventually be some way to understand our experience without harking back to something more fundamental, even something like simplicity is too much of a human construction.

I am able to accept that the range of logical possibilities is an acceptable starting point which does not require any miracle. Mathematics is just the analysis of these logical possibilities. We can start from there and try to find a bridge that takes us to physics without putting in any miracles. Universality is just that bridge. It must be there because we got here somehow.

I dont know exaclty how the bridge is constructed but I use metaphors from complexity theory to try and get some ideas. Most of the things we know about universality in complexity theory exist within some bigger context. Everything has to be intrinsic rather than extrinsic. We have to avoid the need for some measure on the moduli space of possible theories because that measure would need an explanation itself. I think some forms of universality such as the universality of computability do not need such a measure. I think there is a similar universality principle at work in category theory but it is harder to find.

I dont think social constructs are involved in forming these meta-laws. I do think that social constructs such as anthropomorphic selection is relevant in selecting the solution to the meta-laws that forms our experience of reality. So there can be no fine-tuning in the meta-laws. They must be perfectly natural.

Dear Philip Gibbs,

In your essay you wrote, "Today the more progressive physicists take a different view. Space and time are seen as emergent from a yet unknown new way of looking at the universe that must go beyond the bounds of standard quantum field theory, but that much is widely accepted and therefore is not the defining feature of the new paradigm. What is harder to accept is the multiplicity of the vacuum - the idea that there may be more than one stable solution for cold empty space and that the one we know is nothing special or unique." If spacetime needs to be replaced, there might be 2 basic possibilities: (1) there exists a continuous, non-commutative geometry for the string landscape with different Lagrangian formulations of quantum field theory; (2) quantum field theory is an approximation generated by Wolfram's automaton via approximations to string vibrations on a finite lattice. Could there be a third alternative? According to Jacob Bekenstein, "The present paper introduces TeVeS, a new relativistic gravitational theory devoid of a priori fields, whose nonrelativistic weak acceleration limit accords with MOND while its nonrelativistic strong acceleration regime is Newtonian. TeVeS is based on a metric, and dynamic scalar and 4-vector fields (one each); it naturally involves one free function, a length scale, and two positive dimensionless parameters, k and K. TeVeS passes the usual solar system tests of GR, predicts gravitational lensing in agreement with the observations (without requiring dark matter), does not exhibit superluminal propagation, and provides a specific formalism for constructing cosmological models." -- J. Bekenstein, "Relativistic gravitation for the MOND paradigm", 2004, arxiv.org

What is my fundamental objection to TeVeS? I doubt that it makes sense in terms of string vibrations. Digital physics might, or might not, be empirically valid. In any case, MOND is empirically valid -- on the basis of the work of Milgrom, McGaugh, Kroupa, and Pawlowski. MOND might suggest 2 basic possibilities: (1) the equivalence principle is 100% correct but the concepts of gravitational mass and inertial mass need to be slightly changed by a complicated modification of Einstein's field equations; (2) the equivalence principle is 100% correct for particles that are measured but fails for dark matter. Why do I think that the first alternative in the preceding statement is wrong? If you modify GR by adding two or more new mathematical mechanisms then you need to explain the modifications in terms that make sense to most physicists. You also need to explain what went wrong in Einstein's original derivation. I say that the -1/2 in the standard form of Einstein's field equations should be replaced by -1/2 + dark-matter-compensation-constant. This means that the multiverse has a boundary and an interior and that gravitational energy is lost from the boundary; the process eventually results in an instantaneous collapse of each matter universe and each antimatter universe into a synchronized big bang that occurs every 81.6 ± 1.7 billion year. Let us suppose that the fundamental string domain really is 10-dimensional in empirical reality. If nature is infinite, the 10 dimensions somehow curl-up into 4 spacetime dimensions. If nature is finite, the 10 dimensions DO NOT curl-up. There are 4 dimensions of spacetime in a universe composed primarily of matter, 1 dimension of Wolframian time that determines the nonmeasurable clock speed of Wolfram's automaton in a matter universe, 4 dimensions of spacetime in a universe composed primarily of antimatter, and 1 dimension of Wolframian time that determines the nonmeasurable clock speed of Wolfram's automaton in an antimatter universe. The Wolframian time is needed to explain the discrepancy between astronomical time and atomic time. Here "astronomical time" and "atomic time" are as described by Fernández-Rañada and Tiemblo-Ramos.

Phil,

Count me among your admirers. Though I haven't always understood what you're saying, I find this essay crisp, clear and abundantly meaningful.

"What then would happen if we treat the whole of mathematics as a statistical physics system or as a path integral over the moduli space of all possible theories [4]? Would some universal behaviour emerge that could describe the meta-laws of physics?"

I think so! Taking your reference to chess-playing aliens, one finds that that every chess game, as complex as the game is, has a critical point where the game can go either way, or to the equilibrium state of a draw. I agree also with your view of vacua -- my conclusion is that nature cannot respect any vacuum state without respecting all vacuum states.

Years ago I suggested that even mathematics (at the level of analysis) is a self organized system. You get my highest score, and I hope you can read my essay when you get a chance.

All best,

Tom

Hi Philip,

I am just curious how you think an AI could invent the concept of real numbers. If everything a computer can deal with is computable, how could a computer handle uncomputable real numbers (as opposed to computable reals such as pi or e) which have an infinite amount of information and cannot be referred to in any unambiguous way? The information content of most "real" numbers cannot be compressed. How would a computer define operations such as addition on these types of numbers?

Please check out my Digital Physics movie essay if you get the chance. I'd be interested to hear your thoughts.

Thanks,

Jon

Dear Philip,

Thank you for the comments you left on my essay's page. About a week ago, following a reference at the end of Jonathan Dickau's essay, I came upon your almost twenty-year-old essay Theory of Theories, and I found your extension of the idea behind Feynman's path integral to the space of all possible theories absolutely fascinating. Quoting from that essay:

"We might well ask if the same can be applied to mathematical systems in general to reveal the laws of physics as a universal behavior which dominates the space of all possible theories and which transcends details of the construction of individual theories."

I then reread your entry in this year's contest, where you expand upon this idea, whose significance I had missed on first reading, and followed your reference to the recent paper by Seth Lloyd and Olaf Dreyer, The universal path integral.

If I were to rewrite my essay today, I would certainly mention these ideas. I totally agree with you that, if all possible mathematical/physical universes have potentially the same existence as ours, the anthropic principle is not enough by itself to explain why we find ourselves living in a universe that is so regular and relatively simple. Something like your Theory of Theories could "collapse" the chaotic ensemble of all mathematical possibilities, via something like a path integral, to a reduced set of relatively well behaved "coherent" scenarios, on which the anthropic principle would then act. The principle of stationary action has always been my favorite idea in all of physics, and to think that something similar could play a role in "regularizing" the "smorgasbord" of the multi/Maxiverse is very appealing to me!

I agree with you that a future FQXi contest on the relationship between consciousness and physics would be absolutely fascinating! In this year's contest, we have splits between mathematical platonists and anti-platonists, as well as the usual split between the "let's evolve physics from the current accepted theories" crowd and the "bring back local realism and/or absolute space-time" crowd. Imagine if we add a split between "consciousness-first" and "matter-first" views, and between the "free-willers" and the "free will is an illusion" camp... Oh what a wonderful, delicious and mad cacophony this would be! :)

Marc

P.S. I have also posted this on my essay's page, and I will be back with a proper review of your essay, hopefully within the next few days!

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?

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

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.

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

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

Philip,

I am uploading an Excel spreadsheet with the contest results. This might be useful if you decide to do a comparison between viXra authors and FQXI authors in general.

Thanks again,

Gary SimpsonAttachment #1: FQXI_2015_Results.xlsx

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