I want to respond to some of the unanswered questions here as follows.

I see mathematics as something that comes before physics but I don't see mathematics as a platonic realm. Mathematics is just the study of logical possibilities and our physical experience is just a stream of those logical possibilities being played out. We don't need to explain existence and reality any more than that. Our intuition may demand a causal and structural explanation for why it all happens but that is just part of our psychological makeup and has no answer. However, we do need to explain why the laws of physics follow certain mathematical rules.

When people started doing mathematics they were interested in counting and measurement. There was no mystery about why the mathematics was effective in physics, because it was derived from it. But then mathematicians saw interesting logical structures that did not have obvious applications, such as prime numbers. Mathematics took on a life of its own.

Logical possibilities include stuff that is very interesting to mathematicians and stuff that is less interesting. The interesting stuff is characterised by its universality. It is applicable to a range of problems. Mathematicians are delighted when something they formulated for one problem turns out to be useful for another. They get a sense that those logical structures are discovered while others are merely invented. This is what distinguishes pure mathematics from other intellectual endeavours such as art and literature where we consider things to be created rather than discovered. All these things are logical possibilities but the mathematically interesting structures are more universal. They would probably be discovered by an alien race of mathematicians no matter what point they started from.

Already there seems to be some mysteriousness about this universality. Why is it there? Some people are not convinced. They see no mystery yet, so let's look further.

As pure mathematicians continued to study these objects for their own sake without any remaining interest in physics they went beyond the naturally occurring logical structures. They discovered the mathematics of complex numbers, non-Euclidean geometry and higher dimensional spaces. They did not expect these things to be useful to physicists but later they were. Already the unreasonable effectiveness of mathematics seemed mysterious to Wigner, but some people still shrugged their shoulders. They can say that these things were still inspired by physical ideas originally or that the universe is obviously going to be mathematical so of course these things are going to be useful.

For me the real clincher came when string theory was used to prove the monstrous moonshine conjectures. These were mysterious problems connecting areas of number theory and algebraic geometry that nobody would have expected to be connected to the real world. String theory has not yet been shown to be real physics but it was certainly discovered by physicists generalising the framework of quantum field theory with the goal of forming new physical theories. The distance across mathematics spanned by string theory and monstrous moonshine could not have been greater, and yet they turned out to be deeply connected in an unavoidable way.

This is no longer just a simple matter of the unreasonable effectiveness of mathematics in physics. It is also about the unreasonable effectiveness of physics in mathematics. The mystery is deep and cannot be shrugged off. It demands an explanation.

In my opinion, that explanation will come from a better understanding of universality and how it emerges in complex systems. It must be a self-organised structure embedded in the complex system of logical possibilities and their interrelations. It may be characterised by scale-free networks, self-similar fractal structures, path integrals over the grand ensemble of algorithms, iterated quantisation, n-category theory, symmetries etc. Our experience then unfolds according to laws of physics that form as a hierarchy of solutions derived from these universal systems. That is how our universe is put together and it explains the mysterious links that bind mathematics and physics together because this universality is ultimately what both mathematicians and physicists are drawn towards for different reasons.

Hi Philip,

I agree with Ken Seto about the need for concentrating more on physical models. I have witnessed their potential in my own research and theory (TOEBI). Also, don't be too judgemental towards people who think that the contemporary physics has gone into woods for a long time ;-) I'm one of those people.

Anyway, you are a solid performer as usually, thanks for your essay! Check out my essay, it's a good example of a more physical theory.

    Do you think that there is an underlying physical, concrete, entities which are the building blocks of our universe? For example photons, electrons... concrete or not? Lets forget our contemporary theories about them, clean table view and your opinion.

    • [deleted]

    • Post #82

    Kimmo, that's an interesting question from the point of view of universality which I did not have space to cover in the essay, so thanks for asking it.

    The idea of universality is that the laws of physics emerge from an ensemble of different complex systems. Because of this they dont tend to have unique best descriptions. Take as an example the concept of universal computing as defined by Turing. He used a Turing machine but he could equally well have used something else like any programming language. You can show that the different possible definitions are equivalent. The concept of universal computing is unique and would be classed as something that mathematicians discovered, yet the Turing machine is invented and not special or unique.

    If the laws of physics emerge from a principle of universality I expect the same thing to happen. I dont think reductionism will lead to a unique end point with a fundamental set of building blocks such as parrticles or strings. Instead there will be multiple ways of ariving at the laws of physics via definitions, none of which will be obviously the best or simplest or right way to go.

    We already see this in some (still speculative) theories of physics such as elecromagnetic duality. In this case you can regard electrically particles particles and fields as fundamental and particles with magnetic charges are derived or composite, but you can also start from a descritpion where the magentic fields are fundamental and the electric fields are derived

    Perhaps you are trying to amke a distinction between theories based on abstract mathematical ideas and theories based on concrete physical ideas? I dont see how there can be such a disinction. How do you define concrete?

    How do you define concrete?

    Good question... for example something having the boundary for an impenetrable volume.

    I dont think any mainstream physicist would be looking at theories of concrete objects in that sense, but thank goodness we have people outside the mainstream to fill the void.

    Hi Philip,

    Sorry for my last post ,it was cryptic and done in a hurry. In this one I will elaborate .

    I was trying to dig up some concrete material on your random graphs and random matrices and how they lead to Necklace algebra and symmetries. That is because as I explained very very briefly that my system which is based on simulation is indeed nothing but a Buffon's needle type, something like two needles which are larger than the gap(the space between the needles).

    As you well know Buffon's needle is a geometric probability problem which is well connected to integral geometry which is the theory of measures on a geometrical space invariant under the symmetry group of that space.

    So my theory and your theory and the theory of particles in standard physics are connected.

    Can you post some links to your materials, and what do you think about my way of thinking of linking all these ideas. I know I am asking for too much effort on your part, but I need some clear help in the direction which I will be taking to put my system in a more formal incarnation. But of course I understand if you cannot oblige.

    Essay

    P.S. Some info about the setup of distance in the program

    Thanks and good luck.Attachment #1: 1_dist.png

      Just a note

      I can't find any mathematical write ups on you material , they are too general, so DO you have more concrete material. Thanks

      Dear Philip Gibbs,

      In your essay you wrote, "If there is indeed a class of many possible solutions for the vacuum, is only one of these real? I think it is more parsimonious to accept that all solutions exist in some higher sense, whether inside or outside our universe. Some physicists have speculated that there is an eternal process of inflation with vacua decaying to different solutions so that our own universe is just one bubble inside a larger arena. Others have looked at evolving universes where the laws of physics evolve in leaps where new universes are born from old. We can learn a lot from thinking about such possibilities whether they are eventually testable or not but we should not get carried away by thinking they are less speculative or more testable than they really are."

      If the foundations of physics are mathematical equations that restrict energy and spacetime then there might two basic possibilities: (1) the restrictions cause spacetime to curl up according to energy-density based upon generalized quantum information, or (2) the restrictions cause approximations to energy and spacetime to build up from Fredkin-Wolfram information below the Planck scale. How many fundamental particles need to be added to the roster of the Standard Model of particle physics? My guess is that there 2 basic possibilities: (1) if the Heisenberg uncertainty principle should be generalized to use both hbar and alpha-prime, then some form of supersymmetry is empirically valid, or (2) if Einstein's equivalence principle fails for dark matter then the finite nature hypothesis is empirically valid (because the multiverse has a boundary and an interior). Google "witten milgrom" for more information. What is your opinion of the space roar and the photon underproduction crisis?

        Adel, thank you for the questions. Since you ask I will give you a potted history of how my work developed and you can compare with your own path.

        As a PhD student I worked on Lattice Gauge Theories and wrote programs to do Monte Carlo calculations, much like the Buffon's needle trick except there are many more variables in the calculation.

        I left academia but was still interested in doing some monte carlo simulations on my home computer (a Commodore Amiga) Full blown lattice gauge computations were out of the question but some people were looking at random triangulation models for quantum gravity and I wondered what would happen if it was simplified to just a random graph. I was conditioned to think about symmetries so I thought the permutation symmetry might be spontaneously broken to form an emergent spacetime.

        I found that this was possible but only in contrived ways so I wondered in the adjacency matrix for the random graph could be generalised to a full random matrix so that the permutation symmetry $S_N$ becomes a matrix group like $SO(N)$ or $SU(N)$ which would allow the spacetime symmetry to be unified with gauge symmetry.

        This was in 1987-1990. I had no internet or other way to look at other peoples research so I did not even know that there was a mathematical literature on random graphs and random matrices.

        In 1992 I was working in France and had access to the internet so I found out about arXiv (as we now call it). I did a catch up on string theory and realised that my ideas of emergent spacetime could be relevant to what people were asking about spacetime in string theory and what happened to it in the "topological phase", so I worked on it some more and put some papers on arXiv about random graphs and random matrices with generalisations to include sypersymmetry.

        While random matrices were interesting I saw that they were also limited. I felt that the ultimate model should have complete symmetry so that the field variables themselves are in one-to-one correspondence with the generators of the symmetry. For a matric model this would mean using a single matrix, but single matrix models do not have a rich enough structure, so I started to look at generalisations involving tensors in addition to the matrices. I wanted to produce a random model inspired by string field theory.

        I thought I had done it in 1995 when I heard that an old friend Richard Borcherds had succeeded in using symmetry structures from string theory to prove the Moonshine conjectures so I showed him my string inspired symmetry algebras to ask if there could be a connection. He pointed out with a counterexample that my symmetry did not close. Luckily the counter-example made me realise the way to correct the problem and I published this on arXiv and the Int J Theor Phys.

        The Lie algebras I had constructed for discrete strings were a form of necklace Lie algebra, but these were not known to me at the time so I did not use that term until later. That is probably why you cant find them in my work. see http://arxiv.org/abs/hep-th/9510042 http://arxiv.org/abs/hep-th/9609118

        These papers generated a little interest at the time from people like Leonard Suskind who wrote to me to say that he was also looking at discrete strings while trying to solve the black hole information paradox. Soon after he published him Matrix Theory. Where I spoke of spacetime events he spoke of instatons so nobody noted the connection with my event-symmetry. Another group did play on the connection between the permutation symmetries and diffeomorphism invariance but there was never any mention or citation of my work so nobody follows the idea through to the necklace lie algebras.

        Much later in 2006 the idea of quantum graphity arrived which reinvented the idea of random graphs using permutation symmetry. Again there was no reference back to my work so nobody followed to where I had taken the idea (Later they did give me a citation)

        So for twenty years I have been sitting on this idea of Necklace Lie Algebras. The maths is very tidy. It falls into place naturally and can be generalised through an iterative process that I think is related to multiple quantisation. The complete symmetry is just what would be needed to formulate a holographic theory that everybody is puzzling over. I learnt that necklace lie algebras similar to mine are of interest to mathematicians. Even that the free lie algebra can be arranged into the form of a necklace lie algebra and there are ways of mapping this through iterated integration to spacetime. The amplituhedron also uses Yangian symmetries with a linear structure and they scratch their heads wondering how these might be extended to string theory.

        So each year I write an FQXi essay and try to promote my ideas in different ways, but always everyone knocks it down and the winners are safer ideas with nothing very new or radical. I dont mind because I would rather write about what is meaningful to me than something safe and accepted that other people already agree with.

        I think eventually people will see that necklace lie algebras, multiple quantisation, complete symmetry and all that fit in, perhaps in another twenty years time it will happen. From my experience so far I expect that when they do they will use a different language and a different interpretation and so they will not recognize the connection to my work even then. That is what happen when you work independently outside the system. I dont mind that. I am happy that I have known stuff for over twenty years that other people are still confused about and I may have another twenty years of it before they finally get it.

        So my advise to you is keep working on your stuff yourself is nobody else will listen. Make sure your results are out there somewhere permanent and dont be disappointed if nobody joins in to work on it with you. Just enjoy the pleasure of having a different way of looking at things that others cant see yet.

        Allow me to just point out that Geometric Algebra \neq Algebraic Geometry. I got confused about that at one point when I was a Ph.D. student and started learning some Geometric Algebra when what I really needed was Algebraic Geometry. Still, it was a very interesting diversion to see the mathematics of spinors presented in terms of GA.

        Hi Philip,

        Your central thesis that mathematics and physics converge due to universality is very intriguing to me. However, it still leaves open the question of why the meta-laws should exhibit such universality. We still have a "miracle" to deal with, but just at one higher level than Wigner's miracle of the effectiveness of mathematics. Of course, all answers to Wigner's question will open their own new questions, so I do not see this as a defect of your view. However, I think that maybe if we take into account that human knowledge is developed by a social network of individuals, that might help to explain why universality should inevitably emerge in our fundamental laws. It is perhaps a bit controversial to bring social factors into the analysis of fundamental physics, but I think we can safely admit that knowledge is partly shaped by the society that generates it without descending into full-scale social constructivism.

          David, thanks for these points. I feel they go beyond the subject of my essay which does not have much to say about details in cosmology. The anomalies (dark roar and photon underproduction) you mention are very interesting and not ones that I am familiar with. I fear that the resolutions will be rather mundane but I hope there will turn out to be something fundamental at work and I will keep an open mind on it until more compelling data is available either way.

          I follow the digital physic and finite nature ideas which are very interesting to someone like me who sees information and qubits as fundamental but I dont agree with some of the stronger claims or theories about cellular automata. This is because I steer away from the idea that simplicity has nay part in determining the laws of physics and use universality instead. Nevertheless digital physics is also interesting from the point of view of universality.

          • [deleted]

          • Post #93

          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.

          8 days later

          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!