Dear Akinbo,

Thanks so much for reading our essay and for your kind comments. We respect the Platonic view even though it is essentially the opposite of ours. It is that we do not yet see how one could scientifically establish, without appealing to some yet unknown extra-sensory perceptions, the brain making contact with a Platonic `out there' mathematics. Maybe when the field of neurobiology has made further significant advances we will know the answer.

Regarding your comments on the extended point: we readily agree that an extended point can represent a physical reality. But we make a clear distinction between the `thing itself' and `mathematical representation of the thing'. The former is out there and the latter is in our mind, according to our viewpoint.

Unfortunately we could not understand your remarks about velocity of light. We certainly agree that the motion of the earth through space can be detected say via the cosmic microwave background dipole, but you will agree that such a detection does not imply that the speed of light is not a universal invariant independent of the choice of inertial frames.

Thanks for pointing us to your essay - we look forward to reading it in the next few days.

Best regards,

Anshu, Tejinder

While looking forward to your comments on my essay, I wish to clarify my remarks regarding the velocity of light. It is a very common and very crucial misinterpretation what the statement "the speed of light is a universal invariant/ constant" means.

What is implied in special relativity is that the resultant velocity of light is invariant, c v = c or c - v = c, where c is the velocity of light in vacuum and v is the velocity of the observer towards () or away (-) from the observer respectively. Thus, unlike Galilean relativity where resultant velocity can be c v or c - v and so cause earlier or later arrival time of light due to the observer's velocity, v towards or away from light respectively, in Special relativity such observer motion cannot alter light arrival time, hence the SR statement that arrival time over a given distance is independent of the choice of inertial frame (observer frame moving towards or away from incoming light). RESULTANT is the key word. Just as for Sound that has a velocity 340m/s in air, its resultant velocity can be dependent on the choice of inertial frame, i.e the observer's frame. Not so for Special relativity and this was based on the experimental finding of Michelson and Morley, who found no change in light arrival time no matter the direction of earth motion. To buttress let us hear from Einstein himself from his book, The Meaning of Relativity, pg. 27/28.

"But all experiments have shown that electromagnetic and optical phenomena, relatively to the earth as the body of reference, are not influenced by the translational velocity of the earth. The most important of these experiments are those of Michelson and Morley, which I shall assume are known". It is on this that the validity of the principle of special relativity rests.

As you point out in your reply, there are cases where earth motion influences optical phenomena.

That is why there has been a struggle among physicists over the past 100 years. You can also read Herbert Dingle's book, Science at the crossroads when you have the time. I was pointed to this free copy by Pentcho Valev, a contributor on this forum.

Best regards,

Akinbo

Dear Anshu and Tejinder,

I am intrigued to know where mathematics and our language of physics comes from. That this starts with the human brain is a very reasonable working hypothesis and some scientists like Roger Penrose and Stuart Hameroff have even tried to locate our consciousness in brain microtubules. I am curious to learn your opinion about such a controversial subject.

My own mathematical impregnation led me to favour concepts that simultaneously and potentially contain maths and physical aspects like non linear models (and the resulting chaos, fractals, solitons...). For QM and its paradoxes, I was pushed to Grothendieck's "dessins d'enfants" (two-permutation groups possessing cosets, topology, a Riemann surface, algebra over the rationals and geometry). You mention some of these aspects in your paper and you may be interest to read what I have to say in connection to the Monstrous Moonshine.

To conclude, I found your essay stimulating and very well written. I agree with the goal of relating cognition and mathematical physics. I like that you insist that a basic level is that of complex numbers.

Michel

    Dear Michel,

    Thank you for reading our essay and for your kind remarks.

    We have a little bit [though not much] familiarity with the work of Penrose and Hameroff, proposing that quantum coherence and collapse of the wave function in the microtubule environment is the source of consciousness, somehow. With due respect to these great scientists it is our opinion that much more work needs to be done to make this proposal credible. The first issue has to do with the proposal by Penrose, and others, that gravity is responsible for the collapse of the wave-function during a quantum measurement. If you would like to have a look, a recent review can be found here. We find this a very attractive idea but it needs to be developed into a concrete mathematical model first and tested in the laboratory. Only then can we consider applying it to an environment as complex and sophisticated as the brain. Even then, one would have to make a sound mathematical model of the quantum to classical transition in a microtubule. We did not find something like that in the works of Penrose and Hameroff. From what we know, biologists are probably not even agreed on whether microtubules are at all involved in consciousness (open question). Thus while definitely worthy of further study, the idea has a long way to go before becoming believable - so we think.

    We had a first look at your very elegant essay on the maths-physics dialogue, and hope to get back to you soon.

    Best regards,

    Anshu, Tejinder

    4 days later

    Dear Anshu and Tejinder,

    Thank you for the easy to read and thought provoking essay. I like how you start with a definition of math "Mathematics is a precise language in which true statements can be proved starting from a set of axioms, using logic." and a definition of physics "Physics, on the other hand, is an experimental science of the world we observe, where experiments couple with great leaps of conceptual unification." and make a solid argument towards "... it is experiments and concepts first, and then the mathematical formulation."

    I have a small question where you point out for both math and physics, "shape, pattern recognition, counting, space, time, and change." but missing in math is ".. no place in mathematics for matter (material substance), and by extension, for light!". What about the Dirac electron where the mass of the electron is the coupling constant of a pair of spinors, or the more recent work on defining mass through mathematical properties of the Higgs Boson?

    That said, I very much enjoyed how you tied mathematics and physics to cognitive mechanisms in our own human brain. You left lots of things for the reader to ponder, a mark of a well done essay.

    Best of luck in the contest and I hope you get a chance to have a look at my essay here where I take a different approach to our minds understanding of the concepts behind the standard model.

    Thank you for the enjoyable read.

    Regards, Ed Unverricht

      Dear Dr. Anshu Gupta Mujumda

      & Dr.Tejinder Singh

      It's really a pleasure for me to read your nice essay.

      You probably emphasized on a "primordial" logical connection which links both physic and mathematics originated from the "brain perceptions" of "at least one planet full of intelligent beings".

      You also wrote: "The mathematics used in physics comes in only at a later stage, when we seek a precise language to describe the observed physical phenomena..."; and for the mathematics you rightly justified, " Remarkably enough. the primordial roots of mathematics are in the same human perceptions as in physics: shape, pattern recognition, counting, shape, and change. with no significant difference: there is no place in mathematics for matter (material substance), and by extension of light!"

      Whether we can realize such Physics and Mathematics, in broader terms, respectively as 'Hardware' and 'Software' of the nature (including the universe in itself) where those "intelligent beings" are inseparable part in that nature. Is it not true, such biological "intelligent beings" are basically formed by both of those natural hardwares and softwares?

      I also like to add you, is not such an "intelligent" being's centric cognition in this "planet" are fundamentally limited up to any kind of quantized form of message exchanging in-between observers-objects? And any thing, which if exists, conceptually, beyond such messaging limit for that intelligence could be ever rest beyond perceivable limits of that Quantized Cognitive Intelligence (QCI)? Therefore, to such a QCI, up to which it perceives nature through quantized limits of signalling, it's cognition would encourage to believe in some basic axioms of 'casualty' in nature to predict all futures casually within that limit of nature. Beyond that limit as if the nature might appear as a zone of all broken casualties to that QCI.

      Then the same nature looks like having two folds: one as 'Casual' and another as 'non-casual' or 'deterministic' and 'probabilistic'; and the Physics & Mathematics (also being the tools to study that nature) have two similar corresponding folds: 'deterministic' and 'probabilistic'. Therefore, why not, there would be fundamentally two prototype or primordial logics (instead of one) respect to those two folds of nature? And such two prototype logics would connect two such corresponding sets of Physics and Mathematics or Hardware and Software to study as well as fold and unfold of that nature?

      Once again thanks for the essay.

      I invite you also in my submitted essay "A tale of two logics"

      http://fqxi.org/community/forum/topic/2393

      Regrads

      Dipak Kumar Bhunia

        Dear Ed,

        Thanks so much for reading our essay and for your kind remarks.

        Regarding your question, you of course have a good point. However we are making a distinction between physical reality (in this case the mass), and its mathematical representation. It would be like saying that if we hold a ball in our hand and squeeze it, we can feel and appreciate its `materialness' through our senses. On the other hand when we make the statement `We are holding a ball of mass M in our hand' this sort of lingual / mathematical representation of the physical reality is lacking in `materialness' even though we can perceive in our mind what we mean. We do not question the possibility that there can be an elegant mathematical explanation for the origin of mass. We hope (not sure though) this addresses your enquiry.

        We enjoyed your essay and left a post on your page.

        Best regards,

        Anshu, Tejinder

        Dear Dipak,

        Thank you for reading our essay and for your kind remarks.

        You have raised some very nice points. We surely agree that the natural evolution of intelligent beings involves physical hardware (the brain) in which the software (mind, cognition) are operational. For the purpose of the essay, we are compelled to take the brain / mind as given; you will agree perhaps that not enough is known in neurobiology to answer how and why nature evolves hardware and software which then acts back to `understand' nature.

        We agree there maybe fundamental limitations to the efficiency of the observer - observed interaction, but how to explore that scientifically? We also did not quite follow what you meant by `quantised' in this context. Hope to understand this from your essay.

        You also suggest that cognition has a causal / deterministic aspect and a non-causal / probabilistic aspect. Is there a formal construction of this kind in cognitive science? We would have thought that probabilities are attributed to randomness / ignorance of initial conditions, rather than being a limitation of cognition. But you raise an interesting aspect which we need to think more about, and perhaps learn from your essay.

        Best regards,

        Anshu, Tejinder

        "The failure of the Michelson-Morley experiment to detect the motion of the earth through the hypothesised ether led Einstein and others to abandon the ether, and look for a set of mathematical coordinate transformations which allow the speed of light to be the same for all inertial observers."

        That was a dishonest step that eventually ruined physics. In 1887 (prior to FitzGerald and Lorentz advancing the ad hoc length contraction hypothesis), the Michelson-Morley experiment unequivocally confirmed the variable speed of light predicted by Newton's emission theory of light and refuted the constant (independent of the speed of the source) speed of light predicted by the immobile ether theory and later adopted by Einstein as his special relativity's second postulate:

        Alberto Martinez: "In sum, Einstein rejected the emission hypothesis prior to 1905 not because of any direct empirical evidence against it, but because it seemed to involve too many theoretical and mathematical complications. By contrast, Ritz was impressed by the lack of empirical evidence against the emission hypothesis, and he was not deterred by the mathematical difficulties it involved. It seemed to Ritz far more reasonable to assume, in the interest of the "economy" of scientific concepts, that the speed of light depends on the speed of its source, like any other projectile, rather than to assume or believe, with Einstein, that its speed is independent of the motion of its source even though it is not a wave in a medium; that nothing can go faster than light; that the length and mass of any body varies with its velocity; that there exist no rigid bodies; that duration and simultaneity are relative concepts; that the basic parallelogram law for the addition of velocities is not exactly valid; and so forth. Ritz commented that "it is a curious thing, worthy of remark, that only a few years ago one would have thought it sufficient to refute a theory to show that it entails even one or another of these consequences...."

        John Norton: "These efforts were long misled by an exaggeration of the importance of one experiment, the Michelson-Morley experiment, even though Einstein later had trouble recalling if he even knew of the experiment prior to his 1905 paper. This one experiment, in isolation, has little force. Its null result happened to be fully compatible with Newton's own emission theory of light. Located in the context of late 19th century electrodynamics when ether-based, wave theories of light predominated, however, it presented a serious problem that exercised the greatest theoretician of the day."

        John Norton: "In addition to his work as editor of the Einstein papers in finding source material, Stachel assembled the many small clues that reveal Einstein's serious consideration of an emission theory of light; and he gave us the crucial insight that Einstein regarded the Michelson-Morley experiment as evidence for the principle of relativity, whereas later writers almost universally use it as support for the light postulate of special relativity. Even today, this point needs emphasis. The Michelson-Morley experiment is fully compatible with an emission theory of light that CONTRADICTS THE LIGHT POSTULATE."

        Relativity and Its Roots, Banesh Hoffmann, p.92: "There are various remarks to be made about this second principle. For instance, if it is so obvious, how could it turn out to be part of a revolution - especially when the first principle is also a natural one? Moreover, if light consists of particles, as Einstein had suggested in his paper submitted just thirteen weeks before this one, the second principle seems absurd: A stone thrown from a speeding train can do far more damage than one thrown from a train at rest; the speed of the particle is not independent of the motion of the object emitting it. And if we take light to consist of particles and assume that these particles obey Newton's laws, they will conform to Newtonian relativity and thus automatically account for the null result of the Michelson-Morley experiment without recourse to contracting lengths, local time, or Lorentz transformations. Yet, as we have seen, Einstein resisted the temptation to account for the null result in terms of particles of light and simple, familiar Newtonian ideas, and introduced as his second postulate something that was more or less obvious when thought of in terms of waves in an ether. If it was so obvious, though, why did he need to state it as a principle? Because, having taken from the idea of light waves in the ether the one aspect that he needed, he declared early in his paper, to quote his own words, that "the introduction of a 'luminiferous ether' will prove to be superfluous."

        Pentcho Valev

          Dear Pentcho,

          Thank you for your comments. It was my understanding that many many different experiments, carried out independently and using different set-ups, terrestrial as well as astronomical, rule out the emission theory of light to a very high precision.

          With regards,

          Tejinder

          Dear Tejinder,

          Your statement "many many different experiments (...) rule out the emission theory of light to a very high precision" is unfalsifiable - how can I oppose it? We can only discuss the experiments one by one. I hope you agree now that the Michelson-Morley experiment did confirm the variable speed of light predicted by the emission theory, and refuted the constant (independent of the speed of the light source) speed of light predicted by the ether theory and adopted by Einstein as his second postulate. The Pound-Rebka experiment also confirmed the variable (in a gravitational field) speed of light predicted by Newton's emission theory of light:

          Albert Einstein Institute: "One of the three classical tests for general relativity is the gravitational redshift of light or other forms of electromagnetic radiation. However, in contrast to the other two tests - the gravitational deflection of light and the relativistic perihelion shift -, you do not need general relativity to derive the correct prediction for the gravitational redshift. A combination of Newtonian gravity, a particle theory of light, and the weak equivalence principle (gravitating mass equals inertial mass) suffices. (...) The gravitational redshift was first measured on earth in 1960-65 by Pound, Rebka, and Snider at Harvard University..."

          Pentcho Valev

          Dear Pentcho,

          My understanding is that both emission theory and special relativity are consistent with the Michelson Morley experiment, but the former is refuted by subsequent experiments. From what I know, Einstein himself did consider an emission theory of his own, before discarding it in favour of special relativity. As regards the multitude of experiments that refute emission theory, I myself of course do not have the expertise to judge them one by one; I am satisfied that different experiments were reported and published. I would find it extremely hard to believe that a group of experimentalists have over decades indulged in a conspiracy of sorts to deliberately discredit emission theory. I do understand that there are physicists who continue to support the emission theory and probably that is your stance too. I can only say that I respectfully disagree with this stance.

          I wish to add that I am not an all out pro-establishment theorist! :-) I disagree with the establishment view on quantum theory, and I think the theory needs better understanding. However, until a decisive experiment comes along and shows quantum theory to be approximate, the debate is not going to be settled one way or the other. Same holds for the proposal of dark matter: we cannot be sure of its existence until a candidate is found, and I am sympathetic to alternate explanations such as modified gravity and MOND, even though the establishment is strongly pro-dark matter. Another good example I thnk is cosmological inflation: despite all claims to success it is still a hypothesis. I also feel the pro-establishment community is open to considering concrete alternatives, even if these meet pockets of resistance. Thus I am very reluctant to believe that the emission theory is being deliberately suppressed by vested experimentalists and theorists. On the other hand, we do find a healthy response to concrete proposals to modify special relativity.

          Those are my two cents :-)

          Best regards,

          Tejinder

          4 days later

          Dear Anshu and Tejinder,

          I just red your reply following my very positive appreciation of your essay.

          There is another essay about science and cognition by Vincent Douzal that you should not miss.

          Me too I am not yet convinced by Hameroff and Penrose, I met them sometimes ago at a Tucson conference.

          I am now rating your essay highly. I hope it will not be balanced by a stupid 1 as usual. Myself I already got 1 three times.

          Best,

          Michel

            6 days later

            Tejinder and Anshu,

            Congratulations on a weighty discussion. As a modeller in the urbane, offensive and defensive weapons cost and support, and as a teacher of English, I see human mathematics as building on metaphors, not being an entity itself. My background lends that prejudice.

            You do not burden yourself with this question, but do state your feeling: "tempting but erroneous to conclude that the beautiful math description is resident in the physical world."

            I also believe that Cognition draws on the physical world to invent the stable human language of math. Such modelling has led us to discoveries quantum biology, DNA, and LHC through what I see as the connections of math, mind, and physics.

            Thanks for the opportunity to share your views.

            Jim

              Dear Michel,

              Greetings, and thanks for pointing us to Vincent Douzal's paper, which we read and liked. Indeed there are a few papers in the contest emphasising the importance of cognition in the present context (though perhaps too few!). The research works of Lukaff, Nunez, Dehaene, Hestenes, amongst others, are noteworthy.

              Kind regards,

              Anshu, Tejinder

              Dear Anshu,

              Dear Tejinder,

              You are anti-Platonists (or rather non-Platonists), and I am a Platonist aware of all difficulties this option represents. But similar difficulties do undermine ALL options that could be discussed within the framework of this contest. Personally I see (i) this contest as a philosophical one and (ii) philosophy as the choice to focus on issues that do not allow one single answer. Of course sometime an initially philosophical question finally gets a definitive answer, but in this case it ceases to be a philosophical question. Anyway, regarding foundational research about mathematics and/or links between mathematics and physics, we must "empirically" recognize that for n people debating about, there are at least n + 1 opinions. Under these conditions, the BEST we could do is to try a mutually benefit full constructive dialogue, and this primarily with colleagues defending "opposite standpoints." You agree probably that such a constructive dialogue should be for everyone the main motivation to participate in this contest.

              The calm, circumspect, and courteous background tone of your well written, highly interesting essay expressing a wide deepened culture about history of mathematics and physics make me think that you are seeking in turn dialogue.

              Before going further, I would state - even repeating myself - that it has been an enormous pleasure for me to read your really well written elegant essay with its beautiful symmetry between sections I and II. You are absolutely right to emphasize that mathematics and (the interminable stammering of) physics initially knew essentially separate paths, and that the manifest collusion between physics and (a relatively modest part of) mathematics represents at least at first glance a mystery to elucidate; otherwise the subject of this contest would not have any reason to be proposed.

              Well, and now let us discuss some essential points of your essay, with the sole aim to exchange ideas.

              You are wondering how are we ever going to be able to scientifically prove Platonism. Here, my position as a Platonist - I expose it at length in my own essay - is clear. Platonism IS metaphysics and still metaphysics and nothing but metaphysics. Yes, but all competing theories of Platonism are as well metaphysical as Platonism they are trying to "refute". If such a refutation was possible, the competing theory in question would transform Platonism in a no longer metaphysical but scientifically refutable approach being effectively refuted, just like it can happen to other entirely scientific approaches. At first sight it seems certainly strange to postulate that "there is a mathematical universe, which we somehow grasp in an extra-sensory manner". But in my own essay, I quote a reference text of R. Carnap where this categorical anti-metaphysical main representative of logical positivism relegates material realism to metaphysics, making no difference with other ontological theories like idealism. Of course, we receive information on material reality by our sensory channels before cognition processes this information. But it seems to me that the problem is just THERE: To evaluate (i) objectively and (ii) non-metaphysically the degree of adequation between the image we have of reality following its cognitive processing and reality "as such" (??), we should transmute us in pure immaterial spirits able to go out/beyond of our cognition. The fact that all our information about material reality run through our sensory channels, whereas there is nothing equivalent concerning our access to an immaterial mathematical world being objectively given, this only fact does not refute Platonism. First, nothing allows us to predict that humans will "never" discover brain channels able to receiving directly immaterial information. At present we do not know anything about that issue - at least to my knowledge - but anyway, scientists who by definition do not have the vocation to utter metaphysical and /or arbitrary propositions should not play the prophets of what "will or will not be discovered." I would add that the (too) famous "Benacerraf argument against Platonism" - "we have good knowledge about cognition dedicated to the empirical reality, whereas there is no equivalent for an immaterial world" - has not at all "definitely refuted Platonism." Some authors like Penelope Maddy even do not think that Benacerraf would advance an anti-Platonist evidence. Anyway, discussions about "Benacerraf's argument" are continuing for 43 years, and nobody sees the end.

              On the other hand, the material reality which exists - let us admit that it exists - could also not exist. A scientific is generally not interested in the issue "why exist the material reality, instead of not existing." Well, but this issue equivalent to this other: "why should an immaterial objective mathematical world NOT exist instead of existing?" Everywhere we encounter metaphysics, much more metaphysics then we would admit it spontaneously. In other terms: If we deny the existence of material world, we must assume an infinite number of epistemological problems. But we must also ask - without any a priori - if it is not the same for Platonism, and more specifically, if the denial of Platonism does not cause in turn a jolly good number of epistemological problems regarding for example the links between mathematics and physics. In my own essay, I try to raise such problems, leaving it to the reader to appreciate these problems in her/his personal manner.

              In your essay you say that "in their work physicists and mathematicians generally prefer to ignore or 'forget' the brain (...)." Personally, I don't think (i) that it is a question of ignoring or forgetting and (ii) that this point is not a detail. Let me use a metaphor. Imagine you are resolving a mathematical or physical problem, using a computer to facilitate your task. Doing this, you don't "ignore or forget" your computer and its internal functioning. But in this context, the computer is accessory. Perhaps and even probably it is an unavoidable mediator between your work and the given object of your work, may it be mathematics or physical reality. This certainly does not hinder you to be interested about hardware oriented computer science computer science. However, making mathematics or physics as such, you tacitly are bracketing all issues related to computer science or hardware. Now it is clear that your computer and more generally each information processing system must be equipped so that it can carry out its task. Hence the system is supposed to share the logic (and other factors) of (i) the task to be effectuated and (ii) the object that this task concerns. But this does not necessarily mean that information treating system "GENERATE" or "CREATE" the object of its task. Moreover, in the case of a computer or other artificial information processing systems, we can KNOW that the system does not "generate" or "create" the object of its task, since we are outside the circuit linking the system to the object of its task. But it is not the same for our brains, or, more generally, for our cognition.

              Of course, to be able to make mathematics or physics, our brains must be equipped for. Concerning mathematics, it is often said that elementary mathematical structures, principally counting, are innate and do pre-exist in infants to all effectively effectuated operations. Given the complexity of the issue, I am not sure that this point can be categorically asserted, but it is not so important. Not only it is obvious that our brains are disposed for mathematics - otherwise there would be no mathematics, at least no mathematics effectively written on paper - but we can still do some interesting overlaps between the theories of "constructivist" philosophy (Brouwer, Heything ea) reducing any form of mathematics effectively constructable to "intuition of counting", and, on the other hand, modern computational mathematics. Regarding physical reality (in a very vast sense), it is in turn clear that our cognition is able not only to register it, but also to investigate it. Otherwise there would be neither physics, nor other sciences.

              But now complications arise.

              To respond - in cognitive terms - to the following issue "How to explain that (a part of) physical phenomena correspond to (a part of) mathematics being (at least potentially) pre-programmed in / by the human brain?", we need very heavy hypotheses. In a cognitive perspective, if we assume that mathematics is at least potentially rooted in the brain, we must also assume that our brains organize the physical reality in the way that it aligns with given part of mathematics. This lead to a kind of Kantian or neo-Kantian philosophy which inevitably will be the subject of endless controversies. But regardless the option we would adopt in this area, the fundamental problem remains the same: unlike the information processing system mentioned above, where humans being outside the circuit linking the system to the reality can compare the treated information to its original, human cognition CAN NOT "get out from itself" in order to check the degree of adequacy between the reality as such and the reality as it is conditioned by cognition. In other terms, there is CIRCULARITY, and each tentative to break an objectively given circularity leads to metaphysics.

              Nevertheless it is not a deadlock, but the solution can not be classical.

              The position I defend in my own essay is the following: (i) All foundational approaches concerning mathematics and/or links between mathematics and physics are ultimately metaphysical. (ii) Platonism belongs to metaphysics as well as anti-Platonism, whatever it would be. (iii) Even cognitive approaches ultimately DO NOT escape to metaphysics: all investigations of human cognition are conditioned by human cognition. To appreciate this latter "objectively", human cognitive scientists should be able to get out from their cognition, to go "beyond" of all links between cognition and its objects, and this remains a genuine metaphysical idea. (iv) All we can do is to compare several competing metaphysical theories under EPISTEMOLOGICAL criteria such as simplicity in the logical sense of this term, complexity of primary and secondary hypotheses, internal consistency, consistency on the level of consequences and so on. On these bases, I try to show that a necessarily metaphysical Platonism is more plausible than its (also necessarily metaphysical) competing theories. Further details are in my own essay. Here, I try to summarize it as follows: Platonism is based on assumptions neither provable nor refutable. But it's the same for anti-Platonism. On the other hand, anti-Platonism must additionally assume some forms of circularity that Platonism can - at least in purely logical terms - avoid.

              I would be glad to know your counter-propositions about my vision I tried to summarize here above and also to know the criticism that you would not fail to address to my own contribution. It would be precious to of further reformulations on the foundations of a broader vision. Ultimately, this the raison d'être of the present contest.

              With best regards

              Peter

                I did enjoy reading this essay which covers a wide ranging scope of topics in Mathematics, Physics and Cognitive Science.

                I also enjoyed reading your technical end notes on Quantum Theory and completely agree with your assessment that quantum theory might be incomplete. I think you will enjoy reading my essay: 'solving the mystery' which addresses the four oddities that you mention in your essay. The problem is treated from a different conceptual viewpoint (the spacetime wave theory).

                Best wishes

                Richard Lewis