Dear Akimbo,

First thank you for your kind interest. This post is a tentative response to your question having in mind your very pedagogical essay about monads.

You: Monad - a fundamental unit of geometry; that of which there is no part;...

i. extended objects, not further extensible or compressible.

ii. they are fundamental and not a composite of other 'its'.

iii. they are the fundamental units of geometry, both body and space.

Me: The points of the geometries I am dealing with could perhaps be seen as monads. (e.g. the 7 points of the Fano plane in Fig. 1a. Then in Fig 1b the same points are extended as edges).

You: monads are 'it' and their change between two alternate states is the 'bit'.

Me: Agree. One edge in Fig. 2b is either black (bit 1) or white (bit 0).

You: the two-valued attribute

denoted by 0 and 1 must really occupy the deepest part of the basement!

Me: Agree, but as two elements of a triple {0,1, \infty}.

Stephen Anastasi: (above) "not only does the universe collapse to a single minimally simple omnet, all of mathematics went down the tube with it.",

Me: The translation of this sentence would be the Belyi theorem (see the step 3 in my Sec. 2 giving the definition of a child's drawing) and the property that the child's drawing D itself is the preimage of the segment [0,1], that is D=f^-1([0,1]), where the Belyi function f corresponding to D is a rational function. All black vertices of D are the roots of the equation f(x)=0, the multiplicity of each root being equal to the degree of the corresponding vertex. Similarly, all white vertices are the roots of the quation f(x)=1. Inside each face, there exits a single pole, that is a root of the equation f(x)=\infty. Besides 0, 1 and \infty, there are no other critical value of f.

Sorry about the technicalities.

You: But what about the space then?

Me: Although the model of dessins d'enfants may be applied differently, practically, in my essay, it corresponds to the (Heisenberg) space of quantum observables such as the Pauli spin matrices, or tensorial agregates of them. You would say that they cannot be monads in such a case! But they cannot be divided in the sense that the parties (let's say Alice, Bob and Charlie for the three-partite case, I used the Fano plane for this case) are linked once for all, whatever state they share, entangled or not. I don't know about Mach, I have to think more.

I am sure that it does not dissolve your question, at least it gives you a hint, hopefully, of what this kind of maths may do.

Please rate my essay if you like it.

Best wishes,

Michel

Dear Michael your essay is very good

The symmetry, the groups and their intimate relationship with the information; whose culmination, in regard to the observation process is the Bell theorem. His essay is technical for the average of the overall level of this competition. I especially liked your exposure on the geometric and topological aspects, which without doubt are directly connected to the concept of the information and its mathematical quantification.

I think you'll agree with me that only by pure numbers generated by the ratios of the masses, fundamental constants, etc., only in this way will be possible to advance the unification of physics. Physicists have before their eyes a theory of strings that is already developed, so basic, in the foundations of quantum theory. I refer to model a rope in a box. In my work I have shown that a string compactified on seven dimensions, finding the probability for a dimension, a single string, it is the ratio of the Higgs boson mass in relation to the value of the Higgs vacuum. It is no coincidence that the geometry of the tetrahedron this closely related to the spins and the electric charges, because: no tetrahedral angle 1 = cos (spin 2), 1 tetrahedral angle cos = cos (spin 1/2), and so angle GUT unification = cos (spin 3/2). The sum of the cosines of all spins has, among others, this property: [SUM (cos (all spins)) / 2] ^ 2 x 246.221202 = ~ 127.2 GeV (Higgs boson mass). I am Going to more carefully read your essay, rate it certainly high. Thank you very much. Regards

    Dear Angel,

    Thank you for your interest. I agree that the tetrahedron may be a basic piece of

    an unification model. I suspect that Klein's theory of invariants is related to your calculations. For the tetrahedron, the Belyi function is the cube of the ratio between the two invariants as given in Klein's book about the icosahedron (Dover, 1956, [5], p. 104). But we can discuss this by email when the competition ends. The tetrahedron may be seen as the 3-simplex, it can be driven in 6 distinct ways by a dessin d'enfant arising from the cartographic group (as I answer above to Stephen Anastasi), I wonder if one can attach some physical significance to these facts.

    In what regards your essay, I find it extremely attractive because you are producing numbers that seem to correspond to mass ration in particle physics.

    It would have to be organized in a more academic style but I don't worry at this stage. Your essay is also relevant to the topic of observer participancy. You know the sentence requoted in Wikipedia article about Preintuitionism

    In fact Kronecker might be the most famous of the Pre-Intuitionists for his singular and often quoted phrase, "God made the natural numbers; all else is the work of man."

    I am a fan of number theory and produced several papers on this topic.

    I give you an extremely high rate to promote your research. I would like to understand the details of your calculations. My email is

    michel.planat@femto-st.fr

    Good luck,

    Michel

    Michel,

    Thanks for your kind comments on my work, and helpful links, though I could make little sense of your arXiv papers (I'm sure my fault not yours).

    I've now re-read your essay and have found some connections I didn't previously notice. I support PBR, but as it's consistent with the DFM's realist ontology, and still find a valid re-interpretation of Copenhagen. I understand you do too but on different grounds.

    We think differently as I've eschewed the conventional 'shut up and calculate' era approach. I've found the belief in the power of manipulating symbols which are supposed to precisely represent nature's evolution is almost pagan mysticism. Can we formulate Kolmogorov complexity? I suggest a new 'stop and think' era is overdue. The McHarris essay on chaos is then more important than most realise.

    Can a child's drawing of a curve represent the non-linear correspondence of a circle to a line? or explaining Borns' Rule by the DFM; a sphere to a plane? so a Bayesian cosine distribution which contains natures truths hidden BETWEEN the integers 0 and 1 inaccessible to our present mathematics?

    You stand more chance than me of rationalising this in the language most with influence will understand so I hope you can. I also think your essay deserves to finish in the final placings so am pleased to assist.

    Best wishes

    Peter

    Dear Peter,

    Thank you for helping me to float. Apart from several unfair votes this game is quite democratic at the level of exchanging deep thoughts and knowledge.

    I still don't know what can ultimely be reached with this Grothendieck's dessins d'enfants. I try to extract them from the treasure trove of mathematics in the context of quantum paradoxes but I also had interesting feedbacks with a few philosophically oriented FQXi competitors, and you can find the tracks of these discussion above. The secret is in the understanding of the so-called Belyi theorem. I would have to tell more on this and display many examples to convey the beauty of the concept that has fascinated Grothendieck in his

    http://en.wikipedia.org/wiki/Esquisse_d'un_Programme

    Peter, I wish you not be overlooked this time.

    Michel

    ps: I realize that we are both born in 1951.

    Michel,

    I was October, and we could see France from our house in Royal Road. I have a yacht in Ramsgate Harbour and we often race over to the Dunes, Cote D'Opal, Normandy and Brittany.

    Can you get any contact with Alain Aspect? He's ignored my letter advising of the theoretical explanation of the 'orbital asymmetries' he found. He discarded all that data (the vast majority) as no theory existed at the time!

    If he is involved with the explanation he will look very good, but I think maybe not if he ignores it! What do you think?

    Peter

    PS. Yes, democracy is good, unfortunately power often corrupts and can ultimately prevail.

    Dear Peter,

    May be Alain Aspect just forgot to answer. After his breakthrough experiment he was interested by other topics in quantum optics (Bose-Einstein condensates...). He is not a theoretician. He got the CNRS gold medal in 2005, I suspect that he will also win the Nobel Prize after Cohen Tannoudji and Serge Haroche.

    Of course I met the three guys at conferences but they don't know me. I am not in the main stream and I don't try to be within it.

    If you like, we can continue our exchange through the email

    michel.planat@femto-st.fr

    Best regards,

    Michel

    Do you speak some French?

    Dear Michel,

    Please see below statements and their implication in mathematics as you have posted a question in my thread about zero = I = infinity. I am using the symbol "~" to represent infinity.

    If 0 x 0 = 0 is true, then 0 / 0 = 0 is also true

    If 0 x 1 = 0 is true, then 0 / 0 = 1 is also true

    If 0 x 2 = 0 is true, then 0 / 0 = 2 is also true

    .

    .

    .

    If 0 x i = 0 is true, then 0 / 0 = i is also true

    .

    .

    .

    If 0 x ~ = 0 is true, then 0 / 0 = ~ is also true

    It seems that mathematics, the universal language, is also pointing to the absolute truth that 0 = 1 = 2 = i = ~, where "i" can be any number from zero to infinity. Any number on its own means absolutely nothing (zero) or itself (infinite or undefined). Only when compared to numbers before it or after it does it have a relative meaning. Theory of everything is that there is absolutely nothing but the self or i.

    I have also explained that the universe is an iSphere and we humans are capable of interpreting it as a 4 dimensional 3Sphere manifold.

    Love,

    Sridattadev.

    Hi Michel,

    Thank you for your excellent essay, which advances the discussion of the topics of quantum non-locality and contextuality, and thanks for your response with the references, which I will follow up shortly. Your work is highly relevant to my own and will be one of the most important of the competition for me. On Friday, July 26, I gave your essay a very high rating.

    Sincerely,

    Charles Card

    Greetings Michel,

    I have just given you a well deserved boost, after reading (or reviewing) your wonderful paper. As I understand it; the Dessins are contextual maps, showing the connections involved and the object-observer relationships. You state that the Fano plane is the smallest projective plane possible. I guess that means the octonions are the smallest irreducible representation of object and observer context. Anyhow; it was a very well written and fun introduction to a subject which could have been a lot less engaging. Your enthusiasm for your subject is infectious Michel, and you definitely portray the child-like playful explorer well.

    Have Fun!

    Jonathan

      Dear Jonathan,

      My best acknowledgements for the boost. I appreciate that the technicalities do not discourage you. I hope that the infection will propagate!

      You are right that the octonions lurk around the examples I selected. Not all dessins are feature quantum contextuality. The simplest case given here is Mermin's square and can be seen as an archetype (in the language you use in your essay). The next case is Mermin's pentagram, there are 12096 of them with three qubits and 12096 is also the size of the automorphism group of G2(2) (related to the octonions as John Baez explains in his famous note). This is discussed in our recent papers on ArXiv.

      My kind regards,

      Michel

      Thank you Michel!

      I am pleased to help your wonderful essay rise higher. I expect that many more wonderful insights await, in the collection of your papers (or where you are an author) I have downloaded from arXiv. The overlooked importance of something small like the fundamental nature of the (0, 1, /infty) triple is seldom made known. I noticed you commented to this effect on Akinbo's essay site as well as in your own.

      People are unaware that in an ab initio formulation, if we actually start at the very beginning and move forward from first principles, one can only know there is an extent; we can call it 1 but it could also be infinite as there is only nothing to compare it to. That is the rule for constructive geometers. In Ian Durham's essay; he makes the point that even knowing something is a unit, we still don't know how 'big' it is (e.g. - a bit or a trit). Perhaps ternary digits are more useful, after all.

      But I like imagining that 1 is a nice balance point between 0 and infinity.

      All the Best,

      Jonathan

      Dear Michel,

      Thank you so much for your essay. As far as I understand, in your essay, the conventional logic is used. When we use the different type logic, for example, topos http://arxiv.org/abs/quant-ph/0703060 , how to relate your approach? Also, the problem seems to be completely solved.

      Best wishes,

      Yutaka

        Dear Yukata,

        My essay don't refer to logic at all (although Grothendieck's topos may be hidden in it at some level).

        Should I remind that Grothendieck introduced the concept of a topos.

        It seems that you did not understand what my essay is about, but still scored it low, this is unfair.

        http://en.wikipedia.org/wiki/Topos

        I don't uderstand your sentence "the problem seems to be completely solved"!

        Michel

        Dear Jonathan,

        As far as dessins d'enfants are concerned, the members of the triple (0, 1, /infty) have well defined meaning. Sorry that I just copy my earlier post:

        The Belyi theorem (see the step 3 in my Sec. 2 giving the definition of a child's drawing) and the property that the child's drawing D itself is the preimage of the segment [0,1], that is D=f^-1([0,1]), where the Belyi function f corresponding to D is a rational function. All black vertices of D are the roots of the equation f(x)=0, the multiplicity of each root being equal to the degree of the corresponding vertex. Similarly, all white vertices are the roots of the quation f(x)=1. Inside each face, there exits a single pole, that is a root of the equation f(x)=\infty. Besides 0, 1 and \infty, there are no other critical value of f.

        In experiments you will have 0 or 1 as the result of the experiment (in the single or multiple qubit context) but the unobserved \infty is needed in the explanation. The way the black points (bit 0) and white points (bit 1) ly on the dessin (a graph on the oriented surface such as the sphere S2, or a Riemann surface with holes) is such that sigma(0)*sigma(1)*sigma(infty)=id, where

        sigma (0) is the permutation group attached to the black points 0 (how the edges incident on the black points rotate) and sigma (1) is the permutation group attached to the white point 1 (how the edges incident on the white points rotate).

        It is still binary logic but in a more clever way (may be this has to do with Grothendieck's topos, I have not thought about this aspect).

        Thanks again for your interest.

        I intend to write you again about the Hopf fibrations.

        My kind regards.

        Michel

        Dear Michel,

        For completeness, I'm posting the following (slightly modified) response from my FQXi blog. Sorry to see that you are also the victim of some "unfair" voting. Regrettably, I scored many "ones" without the accompanying "critiques (the very reason that I entered, as spelt out in my essay).

        Gordon

        ......

        Dear Michel,

        Many thanks for continuing the discussion. I hope we will get to do more of it in the future.

        As for my acceptance (or otherwise) of COUNTERFACTUAL REASONING, let me offer the following proposition:

        Perhaps the related problems are due to COUNTERFACTUAL TESTING!?

        For example: In my Essay, referring to the CHSH Inequality -- page 7, equations (21)-(22) -- you will see that the inequality is based on a TRUISM (21). But we do NOT test the truism; rather, we test the best approximation that we can (22).

        So, by this view, it is not counterfactual reasoning that's at fault. It's the failure, even the impossibility, of testing it.

        Hence the question: Why should that impossibility be regarded as a valid strike against a rational local-realism?

        Especially when QM fails to deliver in the same impossible context?

        WHILE both theories deliver the same experimental outcomes!

        Thus the need for further discussions continues.

        With best regards; Gordon

          Dear Michel,

          yes, I read your essay but was on the vacation before I had the chance to write you. I like your geometric model very much (I rated your essay long ago with maximum score).

          Now after a second reading I have some questions:

          - You used the dessin d'enfants to visualize the contextuality. I understood the Mermin square but how did I see it in dessin d'enfant (Fig. 3b). Is it the number of half-edges (odd number) which produces the contradiction?

          - Why is the transitive action so important? In case of a non-trivial orbit, you can check every point seperately.

          One remark about the triple 0,1,infty: In the projective geometry, you always have the invariance w.r.t. the inversion operation. In the context of your model it means you have the operator and its inverse operator. Then 0 is related to infty and 1 is related to itself via inversion.

          Thanks in advance for the answers.

          Torsten

          PS: Did you saw my post (June, 17) above?

            Dear Gordon,

            Thank you for your continuing interest. I really appreciate your feedback at this time of the competition. We can certainly learn more from each other after the end of the contest. I will rate your essay highly, as it deserves, I would like to see you in the finalists, hopefully I will be too.

            Concerning counterfactuality, as soon as a good theory of quantum observability is written, one will be able to check it as others assumpions in science. I claim that Grothendieck's approach with dessins d'enfants is an excellent starting point because it has all attributes of an archetype (read Dickau's essay) or a monad (read Ojo's essay) and other good ontological properties which I don't list here. Topos theory is not too far.

            There are important essays here that pushed me to see the dessins d'enfants as "explicate imprints" of a more general (possibly spatio-temporal) algebraic geometry. I have in mind the Hopf fibrations as an excellent tool. For example you can lift S2 (the Riemann sphere) to S3 (the 3-sphere, i.e. the space of a single qubit (Jackson's intelligent qubit?), also the conformally compactified Minkowski space (see Matlock' essay and in relation to Bell's theorem Joy Christian 'realistic' approach).

            Local/nonlocal arguments are insufficient, I think, mathematics should help in revealing the hidden machinary of the physical and ontological universe. May be this is Einstein's dream, not contradicting Wheeler, at the end of the day because we are, more or less, their children in knowledge.

            Yes our discussion should live.

            All the best,

            Michel

            • [deleted]

            • Post #89

            Dear Torsten,

            Thanks for your careful reading.

            1) It is not straight to see the contradiction in the dessin of Fig. 3b, I failed to see it in general (for other contexts). Also there is not a single dessin leading to Mermin's square but many, why is it so? More work is necessary. This non-bijection is general for most geometries I have tried to reconstruct from the n-simplices to projective configurations such as Desargues, Cremona-Richmund (i.e. the doily W(2) of two-qubit commutatitivity) and others.

            2) You are right that transitive action may not be a necessary condition. The geometry is constructed by having recourse to the stabilizer of each point in the permutation group relevant to the dessin.

            3) Last remark, the geometry is of the projective type not the dessin. Here you have to refer to the theory that is well explained, for example in Lando and Zvonkin (my ref. [6]).

            Torsten, please check that you vote was recorded.

            Thanks Michel for your message on my blog. My arguments are from a philosophical and classical perspective. It is possible that when viewed from your perspective we may well be saying something similar. I am not expert on the math involved in quantum theory.

            Following additional insights gained from interacting with FQXi community members, perhaps you will find the the judgement in the case of Atomistic Enterprises Inc. vs. Plato & Ors, delivered on Jul. 28, 2013 @ 11:39 GMT easier to understand my thinking.

            Best regards,

            Akinbo

            *I have already rated your essay so you may do likewise.