A Metaphorical Chart of Our Mathematical Ontology by Philip Gibbs

Vladimir, these issues are technically difficult for everybody and we can only move forward by pulling together and throwing in good ideas. I am glad to see you back to do that again. I too find the subjects difficult so I pick out the bits I understand and try to see some of the picture take shape from those.

Thanks for your comments, I will probably be nicking your slime mould idea next time.

Jacel, thanks for your interesting points. We aeem to take different philosophical views. You prefer geometry and I prefer algebra for one thing. Of course I cannot say that I am right and you are wrong. That remains to be seen, or perhaps the truth will be neither or both. All we can do is each take our ideas to their conclusions and see which result fits nature.

"Is there any possibility to get an experimental confirmation of the statement that space and time are emergent? "

That could only be done directly by an experiment so cataclismic that it tears space and time apart. However, indirectly we can find the right TOE and use other more doable experiments to check it. The answer will then come indirectly from that TOE. We will never be totally sure that the TOE is completely correct unless we tests everything it can predict including things like the emergence of space and time, but I hope that once we have the answer it will be something sufficiently convincing.

Ho Jose, I am sorry for not replying earlier. I was busy due to unrelated events and am now back.

I think your assessment of my position is fair and correct. I see mathematics as the realm of logical possibilities and physics as the realisation of those possibilties. From this point of view it obviously makes sense to start with the possibilities and then move on to the realisation.

However, this is just one philosophical position of many and I always recognise that other ways of looking at it can be equally valid. The test of meta-physical ideas is how they lead to more physical theories and from there to experiment. I have described in my essay how I related my philosophy to physics via multiple quantisation, algebraic geometry etc. It will be a long time before we know how it all pans out.

The multiple quantisation idea has been around for a while. I speculatd about the uniqueness of iterated quantisation a few years back at http://arxiv.org/abs/hep-th/9603165 My view of that has not changed but it has developed. I now think that the structure this gives us has to be seen as a system of meta-laws rather than a specific physical theory inside a particular geometry like our spacetime that we can relate to directly through experiment. Physics as we know it is just one solution of these meta-laws. In act you have to go through a cascade of refined solutions to get to the end. So in one sense the end-result of iterated quantisation is unique but the physical theory is not. Perhaps you can realate this to your way of seeing things.

I look forward to reading your essay

Hi Sylvain, it is a nice idea to try and label all the essays with different isms (as on your linked page) so long as you do not take it too seriously.

I think you may also be trying to take the rating too seriously. In my post at the top I say the rating and prizes don't matter to me. What I have found each time is that I start near the top, then later I drop down. You see there are just as many people who hate viXra as people who love it but they arrive later :-)

The reason I keep coming back to the contest is that it is good to formulate thoughts and get some feedback. The FQXi essay contests are very good for that.

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.

Thank you Kimmo, glad to see you here again.

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.

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.