Dear Cristi,

You have written a very attractive introduction to your research while highlighting its solid connections to the contest topic.

A few specific comments:

1. I really enjoyed the playful example at the beginning. While the importance of isomorphisms between seemingly different structures is well recognized by physicists, mathematicians, and philosophers of physics, I believe there is still plenty of room for it to be recognized by people in general, and your example nicely serves that end.

2. Although I had heard of Klein's Erlangen programme, I did not really know what it was about. Knowing that it was based on the recognition of the importance of invariants has caused me to revise one of my beliefs, namely that the general recognition of their importance occurred largely as a result of their importance in relativity (Evidently, Klein anticipated Einstein in this regard by over 30 years). So your essay did change at least one belief, ha!

3. It is interesting to consider whether the notion of fundamentality is purely epistemological or can also be framed ontologically. While I perceive it as a former, I am open to arguments that it can be the latter as well. However, here is a challenge for any ontological fundamentalist: At bottom we might consider the fundamental difference in physics to be between being and non-being. Which is more fundamental? It seems to me, either answer one chooses is open to a counterargument: If one chooses being, then it could be argued that being can be reframed in terms of the absence of non-being, and if one chooses non-being, then it could be argued that it can be reframed in terms of the absence of being. My point is that either choice seems to me to represent a particular worldview, or paradigm, and that makes it inextricably epistemological.

4. The phrase at the end of page 6 "Also they [the GUTs] didn't explain why these particular representations out of infinitely many possible for each group", resonates with me especially. So much of contemporary high-energy theory seems to me like a sophisticated game of pattern-fitting without trying to understand what the patterns really represent. I am often reminded of Feynman's example of the ancient Mayans, who had a sophisticated framework for predicting eclipses and the positions of Venus, but not even a basic concept of planets and the solar system. To me, that is not a secondary but a primary problem with many contemporary approaches to understand nature more deeply. I laud your efforts to try to look behind formalisms and patterns to understand the "why".

5. I admit that I did not follow the math in the latter part of your paper but I would like to raise at least one potential concern: It seems to me not sufficiently broadly appreciated that at different scales, the ratios of different powers of the scale changes, and when considered together with any kind of density, this results in different physical behavior at different scales. In short, to paraphrase Philip Anderson, in my view, bigger is different. Would your Indra's net model be able to account for that?

All in all, a very attractive essay.

    Cristi,

    "This would mean that no region of space can exist, being collections of points."

    This illustrates my point, in that if all those points have no dimensionality, i.e. some modicum of space in the first place, then they are all multiples of zero.

    "Ockham's razor has two edges."

    If I may describe my own view of physical reality, it is a dichotomy of energy and form(information). Energy necessarily manifests all form, or it would collapse into a black hole and form is the definition of all energy. For example, waves are defined by their frequency and amplitude. Try to imagine one otherwise.

    Now I see a good proof of this dichotomy is that after a few billion years of evolution, we developed a central nervous system to process information and digestive, respiratory and circulatory systems to process energy.

    Consider the properties of both. Energy is dynamic and conserved. This means it is always and only present, but necessarily constantly changing configurations, some at faster rates than other changes.

    Form, on the other hand, is stable. If it is unstable, it loses its form. The constituent energy breaks apart, drains away, etc.

    So energy goes from prior to succeeding forms, while the forms coalesce and dissolve. The result is the effect of time. As fairly stable entities, whose consciousness functions as flashes of cognitive forms, we proceed from past forms (and units of time), to future ones, yet these forms go the other direction, future to past. Tomorrow becomes yesterday.

    Think in terms of a factory; The product goes start to finish, being in the future to being in the past. While the production line points the other direction, consuming material and expelling product. Prior to succeeding.

    Life is similar. The individual goes from birth to death, being in the future to being in the past, while the species goes the other direction, onto new generations, shedding old. As our consciousness moves from prior to succeeding thoughts.

    So I would argue time is not an underlaying dimension, measured as duration, but an effect of this activity, much like temperature. Time is the individual frequency, while temperature is masses of frequencies and amplitudes.

    Duration is the state of the present, as events coalesce and dissolve.

    Different clocks can run at different rates because they are separate actions. All being equal, a faster clock uses more energy, like metabolism.

    Time is asymmetric because it is a measure of action and action is inertial. The earth turns one direction, not both.

    The simultaneity of the present was dismissed by arguing different events would be observed in different order from different locations, but this is no more consequential than seeing the moon as it was a moment ago, simultaneous with seeing stars as they were years ago. It is the energy that is conserved, not the information carried by it. That the energy manifesting an event is radiated away is both why we observe it and why it no longer exists.

    So this distinguishes between space and time. We could use ideal gas laws to correlate measures of temperature and volume, but temperature is only foundational to our emotions, bodily functions and environment, not our thought process, so we can presume to be more objective about it.

    Given our nervous system is designed to process the information our environment provides, we do like to study it in detail and math is the most distilled and stable expression of form, but without the energy to manifest it, form does go to zero.

    So, yes, Occam's razor does have two edges and it does cut both ways.

    I would note Edwin Klingman, among others, is also disputing the blocktime aspect of spacetime.

    One more thought, events have to occur, in order to calculate the total input, consequentially the future is not pre-determined, as that would require information traveling faster than the energy carrying it.

    Regards,

    John

    Hello Cristi,

    Much appreciate the number scrabble. We have a 5yo great-grandson in the care of his great-grandmother, who loves games and puzzles, will be delighted with this.

    Glad to see so much interest in Clifford algebras in this year's competition, and particularly the geometric interpretation. Much agree with your statement

    "If there are fundamental geometric structures in physics, we expect them to bring not only a unification of the formalism, but also of principles and of

    entities like particles and fields. If we expect that the holomorphic fundamentalness plays a role in physics, probably the way is by geometric algebras."

    If one takes the vacuum wavefunction to be comprised of the fundamental geometric structures of 3D Pauli algebra - one scalar, three vectors, three bivectors, and one trivector - then wavefunction interactions can be modeled by the geometric product.

    This yields a 4D Dirac algebra of flat Minkowski spacetime, an 8x8 matrix that is the geometric structure of the S-matrix. Time emerges from the interactions in the form of relative phases of the interacting geometric structures that comprise the wavefunctions. It is encoded in the 4D pseudoscalars. Seems like there is no need for complex numbers in this Hestenes formulation of STA. Would much appreciate your opinion on this.

    It seems there is much commonality between what you are doing and the approach Michaele and I have been taking with geometric wavefunctions. What you call Indra's net can be thought analogous to the quantized electromagnetic impedance networks that couple fundamental geometric structures in our model. We are working with Cl(1,3), tho the manner in which we assign quantized EM fields to the 8 component Pauli wavefunction seems to require three or four copies. Our group theory grasp is minimal. Have more questions, would much appreciate some help if you're interested.

    and your hexagrams have me reaching for the I Ching.

    Best regards,

    Peter

      Dear Cristi,

      You get off to a great start showing the isomorphism between 'number scrabble' and 'tic-tac-toe'. You note that "in mathematics, isomorphism's are ubiquitous", mentioning that Euclidian geometry ~ axiomatics ~ symmetries ~ numbers/equations, for example.

      This supports very nicely my thesis that physicists project mathematical structure onto physical reality, and then come to believe that physical reality has that structure. While it is relatively simple for competent mathematicians to 'switch' from one formulation to another isomorphic formulation, the physicist who "freezes" the projected mathematical structure onto physical reality has a tendency to "see" reality is having that structure.

      For example, spins tend to align in fields such that statistically they are aligned or anti-aligned with each other in neighborhood/domains. Based on an over-simplistic interpretation of Stern-Gerlach data, Pauli projected a 'qubit' structure, O|+> = +|+>, O|-> = -|-> onto spin, despite that the SG data is distributed almost exactly as predicted by calculations of 3D spin traversing an inhomogeneous magnetic field. Based on Pauli's 'qubit'-based Hamiltonian, Bell 'believed' the qubit to be real and thus required qubit results: A = +/-1, B = +/-1 rather than variable deflection as seen in the data. The variable data satisfies Bell's relation which he claims is impossible to satisfy.

      In a comment above you state: "Because Bell's theorem is a theorem. Trying to refute it is like trying to find in Euclidean geometry a right triangle which violates Pythagoras's theorem. It is simply impossible." Of course Bell's theorem is a foregone conclusion, from his first equation, in which he forces the only allowed data to be +1 or -1. No physics involved in this, simply an initial condition that is 'projected' onto the reality of spin.

      Thus Bell's 'belief' in Pauli's mathematical projection, causes him to reject 3D spin, which does satisfy ABcos(A,B), and to claim this impossible, leading to "entanglement" as a new mystery, on which thousands of papers can be written. This is compounded by "proofs" of Bell's theorem being conducted with valid two-state experiments, where the states are detection or not of photons.

      Finally, as Bell was forcing 'qubits' on spin, Feynman, who was in love with the two-slit photon experiments, realized that he could apply Pauli's 'qubit wave function' for spin in a manner analogous to the two-slit experiments and he applied this to SG, thus projecting "superposition" on the spin. Although Feynman's gedanken experiments have never been tested, several QM texts now begin with Feynman's two-slit-spin analogy. Thus Feynman and Bell forced a 'mystical' view on spin and Aspect "confirmed" it with photon analogs.

      Once these giants froze the qubit projection onto reality, your isomorphisms go to hell. Isomorphisms are formalisms, qubit spin is (believed to be) physical reality! To seriously question this "reality" can be dangerous to one's career.

      I discuss qubits because the genealogy is so clear cut. I could've discussed iso-spin, in which Heisenberg replaced two real fundamental particles with an imagined particle with 'qubit-like' projections onto reality, etc.

      In my essay I treat another projection onto reality. Einstein, while basing his treatment on Hertz, projected a 4D-coordinate system with a new universal time dimension onto each moving object. The addition of new time dimensions (the physical 'reality' corresponding to the math structure) of course demolished time as universal symmetry and replaced it with "the relativity of simultaneity". This 'freezing' of the 4D-projection on the moving objects has lasted 100 years, despite the fact that the 'energy-time' conjugation in one inertial frame is isomorphic to Einstein's 'space-time symmetry' in two inertial frames, and agrees with all relativistic particle physics data.

      In similar fashion, one can derive Bekenstein's "holographic principle" in terms of energy alone, without ever conceiving of information. But the 'information' projection is now 'believed' by physicists, and the door is closed to isomorphisms.

      In summary, as long as the isomorphisms are mathematical, they are easily seen to morph into one another. But as soon as a mathematical structure is projected onto physical reality, it becomes "frozen" in the mind of the (consensus) physicist, and the fact that other isomorphic interpretations (such as 'classical' versus 'quantum') are equally possible or dismissed or rejected with almost religious fervor.

      You wrote on Jan. 27, 2018 @ 11:32 GMT, that while it is natural to question non-intuitive physics, one has to move on in his career. Nevertheless, you say:

      "But I still think it is necessary to start by questioning everything, and you should never stop."

      I believe that if one projection that leads to non-intuitive 'nonsense' can be replaced by another isomorphism that is compatible with the real data, and yet makes intuitive sense, this change of isomorphisms should be made.

      So thank you, Cristi, for focusing on 'isomorphism' and 'fundamentality' as you have done. Your essay is well written and enlightening. Of course I agree with your proposition that geometric algebra is the tool we should be using. I hope you will read my essay in terms of the above isomorphism's, and I hope you enjoy it.

      My best regards,

      Edwin Eugene Klingman

        John,

        Let's take them one by one. You said "the dimensionless point is the essential geometric concept, yet it is explicitly a multiple of zero, being dimensionless and consequently, mathematically doesn't exist"

        Please start by proving your statement that dimensionless points are multiples of zero. Zero is a number, a multiple of zero means zero times another number, and this is zero. And you equate the number zero with a point, which doesn't seem to be a number.

        Then, assuming that you will be able to prove the previous statement, prove that if something is zero, then "mathematically doesn't exist". Do you mean zero doesn't exist mathematically?

        Here is what I mean by "proof".

        Best regards,

        Cristi

        Dear Armin,

        Thank you for reading the essay and for your relevant comments.

        1,2. Thank you!

        3. Good question. In my essay I argued both for epistemological and ontological fundamentalness. It is not easy to separate the two, because whenever we speak about ontology we do it in a certain epistemological framework, but let's try. I hope section "Quantum holism" makes it clear that there is no way to have "building blocks" in the usual understanding of the word, even when we speak about particles. Particles are described by representations of symmetry groups, in fact they are invariants of such groups. A reducible representation can be decomposed into irreducible representations. This may suggest that the irreducible representations correspond to particles that can be seen as building blocks. This is true, but not in the classical sense, because you can't get a composite particle or system just by putting together elementary particles. For example, even the two electrons in the Helium atom are not just two distinct electrons, they are not separable. This is still consistent with the view of group representations, but not with the usual notion of compositions we have. So if we want to consider something as fundamental, this has to be the whole. Now if we do this, we can proclaim that we just take the entire universe as fundamental and the problem is solved. But we still want to understand the details of the universe. So here the epistemology comes into play, and decomposes the universe in different ways. This doesn't mean that the decomposition is ontological too. But since we can make experiments with separate particles, and since we know we exist even though we can't know the entire state of the universe, there must be something ontological in the way we decompose it. And this is relative in various ways I explained in the essay. For example, epistemologically you can see a general electron as a superposition of plane waves, or a superposition of point electrons, one at each point of space. This simply depends on how we choose the basis in the Hilbert space. But we can also construct states of particles by interference, so this superposition can be used to really make states out of different states. And if we want them to have an ontology, this means that there is also something ontological in their superposition. Things go on like this in various aspects. Now take a quantum field, you can't see it as independent at different places and times, because its values are connected by some equations. Ultimately, if there is a single holomorphic field, what is at each of its points is what is at any other and as the whole. I would expect a fundamental ontology to determine everything, but since this shows that each part determines everything, this makes ontology relative too.

        4. Yes, I agree with you. Is this physics or stamp collecting? :)

        5. I said more about the different scales in The Tablet of the Metalaw. The laws of the fundamental one are always true, but the particular state is determined both downwards and upwards.

        Best regards,

        Cristi

        Hello Peter,

        Thank you for reading the essay and for the insightful comments. Indeed, the Clifford algebras deserve more attention in physics. You mention Hestens, he did great job in formulating parts of physics and geometry in terms of Clifford algebra. His representation of the Dirac electron in terms of the real algebra of spacetime has interesting features, but I think it is not exactly how I would view the electron, because by dropping some of the degrees of freedom it becomes impossible to represent it in a Lorentz invariant way (by this I mean it projects some internal degrees of freedom on spacetime related quantities). The Dirac algebra is complex because it mixes the U(1) symmetry with the spacetime Clifford algebra. I think that complex numbers are in fact rotations in various real spaces, and the usual way physicists use them obfuscates this. For instance, when we take the nonrelativistic limit of the Dirac equation to get the Pauli-Schrodinger equation, different operators which square to -1 all become forgotten, and represented as i, which is the Hodge * operator in 3D. In my complex Clifford 6 model I don't clarify these differences, but I am developing a model in which they become manifest. You made me curious about your approach, I look forward to see it.

        Best regards,

        Cristi

        Dear Edwin,

        Thank you for the interested comments and reading my essay.

        You are right that physicists, like any other humans, project their views onto reality. But I don't think we can use this as argument to simply refute some of the achievements of physics. I would say the opposite is the right way, find where they are wrong and then conclude this was because of a wrong projection. I don't think "they project, so they are wrong" is the right thing to do, because we can say this about anything and we can refute anything like this. So are there places where their projections simply are wrong? I think there are, and the right thing to do is to discuss the arguments.

        When you say "Pauli projected a 'qubit' structure", you make it sound as if Pauli's previous life experience molded his mind to view the world in terms of qubits, and then he started seeing them everywhere, including in the electron's spin. But in fact there was no previous experience of qubits in Pauli's experience. He came with them by reasoning, despite the qubits were not previously present in his experience. So his equation can't be explained as a mere preconception.

        One can argue that Pauli was influenced by the Clifford algebra, his Pauli algebra being nothing but the Clifford algebra of the Euclidean 3D space. There is no sign of this either for Pauli or for Dirac, they both discovered this independently. And I would say unfortunately, since if they knew Clifford algebras some of the confusions in their formulations could be avoided. When I say "confusion" I don't mean they are wrong, their equations turned out to be right and to describe the quantum states and the dynamics quite well in their own domains. What I refer to are some subtleties which involve the geometric interpretation, rather than the empirical adequacy. There is much commitment to historical context in both their theories, which I think would help being deconstructed, but by no means the results are wrong. As you saw in my comment, I was myself opposing when I was very young the conclusions of Quantum Mechanics, but was this because of their projections, or because of my own? I know that it was my projection, because I lived in a classical world, and my intuition was shaped by this and adapted to this. Now I think I know better, but it wouldn't be fair if I would bring my own experiences with this as an argument that you should believe what I say and discard your own views.

        You said "Of course Bell's theorem is a foregone conclusion, from his first equation, in which he forces the only allowed data to be +1 or -1. No physics involved in this, simply an initial condition that is 'projected' onto the reality of spin."

        Here is why I disagree. Bell only assumes that the particle can go up and down, as the Stern-Gerlach experiment shows. He doesn't assume that the Pauli's theory of spin is behind this. He just discusses yes-no measurement. This is very general and with no implicit commitment on what's behind the result. Also, in his theorem he only takes as hypotheses locality (L) and Statistical independence (SI), and he derives a conclusion about the correlations. The experiments proved the conclusion wrong, so either the proofis wrong, or the hypothesis (L and SI). Hence, L or SI or both must be wrong. That's all, no Pauli algebra involved. This works for any kinds of measurements which result in a yes/no outcome, if combined in a similar way. And there are versions in which no spin neither polarization are involved, because two-level systems are everywhere. I remember even a version based on positions and momenta. And the proof was generalized to all sort of quantum states. The reason it always works is because quantum states can live in superposition, and because measurements are represented by operators, and whenever these operators don't commute, things like this happen. And nature stubbornly confirms this.

        > "leading to "entanglement" as a new mystery, on which thousands of papers can be written"

        You can try to make a model of the Helium atom without entanglement. Or reproduce all these predictions of QM which were confirmed by experiments, without entanglement. I agree with you that spin is 3D (well, when more particles are involved the things change). But try to reproduce EPR without forcing Alice and Bob choose the same or opposite directions, but independent ones. To do this you will need either to postulate that something happens nonlocally (thus violating L, like in the Bohmian and GRW interpretations, both endorsed by Bell), or that SI is violated, that is, the initial state of the particles is chosen in a way which depends on what Alice and Bob will choose. My personal position, because I find worse to break Lorentz invariance, is that L is kept (but without rejecting holism), and SI will go away. This is my position, and I know for many is crazier than to give up L. And of course for others it is crazy to drop L. And it is understandable that for others sacrificing L and SI is equally crazy. But to me there is no option to keep both of them except for some very particular cases. And if an explanation works for very particular cases and fails for the general, it must not be the right explanation. No matter how much you qualify the conclusions of QM as "crazy", "mystical", or use quotation marks around words like "confirmed", you still need to prove your point. And before reproducing all we know about QM, try at least to reproduce EPR for spin, for all possible choices made by Alice and Bob, without breaking L and SI. Bell's theorem says you can't. You say Bell was wrong. Prove it. This is the challenge, and I explained I gave up long time ago checking such "proofs" because my time is limited and I have my own crackpot ideas to chase :). But check it for yourself, your model should work for all cases. Then find where Bell was wrong, but in the proof. Write a paper without all this talk about how full of prejudices are Pauli, Feynman, and Bell. Do this if you want after you prove it, but if you want to increase your chances someone from those brainwashed mainstream physicists to read it, make it simple, foolproof mathematically, without handwaving and without psychoanalyzing physicists.

        So let me congratulate you for trying to debunk quantum mechanics and special relativity, perhaps someone has to try this, because everything should be checked, double-checked and so on. I am just a limited being with two jobs and no time to take such attempts seriously enough as they deserve, and from what I am concerned, QM and relativity are correct. But you have my encouragement to dig deeper, good luck!

        Best regards,

        Cristi

        Cristi,

        What I'm saying is that a location, i.e. a point doesn't exist, if it has zero dimension, i.e., spatial extension. Which is why it is a contradiction, according to the principle that any multiple of zero is zero.

        "Please start by proving your statement that dimensionless points are multiples of zero."

        It is common English that the "less" on "dimensionless" means there is no dimension, i.e., zero. Are you saying this is not a correct understanding?

        Numbers are abstractions. The number zero is an abstraction of the empty set. My point is that if a point has no, or zero dimension, it is an empty set. As in there is no point in the set.

        Not to confuse the issue further, but that is why I referred to a "dimensionless apple," as in it would be an empty set, if any apples in it were to be described as dimensionless, i.e., having zero spatial extension.

        Regards,

        John

        John,

        All mathematicians on Earth disagree with the following:

        > a point is a set

        > a point is a number

        > empty set = zero

        > dimensionless = empty

        > dimensionless = not existent

        > a point is multiple of zero

        Here is what mathematicians would say instead of these, and then I wonder if it is the same meaning as you think:

        > a point is not a set, but it can be an element of a set. For example for any point A there is a set {A}, containing only A.

        > a point is not a number, but you can label points by numbers

        > the empty set is not zero, but its cardinal (i.e. the number of its elements) is zero. The cardinal of the set {A} is not zero, it is one.

        > the meaning of dimensionless depends on the context. The empty set is dimensionless, but the set made of a point is dimensionless but not empty.

        > a point can't be multiple of zero, only zero is multiple of zero.

        Are you saying this is not a correct understanding?

        If you mean something with such statements, I think you should define them. Instead, you said "consequently, mathematically doesn't exist", which implies that your statements are mathematical. Maybe they are, but not according to the mathematics all mathematicians on Earth know. Including myself, since I am no better than them.

        Now, as I said, maybe those statements you made mean something to you, but to a mathematician they are either wrong or meaningless. I take it that you had no curiosity about mathematics, which is your right and there is nothing wrong with this. On the other hand I like your adventurous spirit, going in a completely unknown world without even a map.

        Best regards,

        Cristi

        Dear Cristi,

        Thank you for responding to my comment. You have certainly understood my main point, that we project interviews onto reality. However you misinterpret me when you suggest "find where they are wrong and then conclude this was because of a wrong projection."

        That is exactly what I am doing!

        My essay discusses the arguments for one such wrong projection. It is hard to solve other century-old mistakes in a brief comment.

        It does not matter why or how Pauli came up with the wrong projection, only that he did. He was brilliant, and his model was extremely useful. It is only when physicists believe in this model and assume spin is a two-state entity that things go off the track. One can very happily use 'qubits' when it is appropriate. Unfortunately, post-Bell all physicists seem to think it is always appropriate.

        I believe you are wrong about Bell. He does not assume only that the particle can go up or down. He assumes the particle has two states, +1 and -1. This precludes the 3D spin that is deflected in the field by a spin-dependent amount. When one treats 3D spin versus qubit spin, one does obtain the correlation that Bell claims is impossible.

        By projecting qubits onto 3D spin, Bell formulates a false theorem, falsified from his first condition. It is logic past this point, not physics. And the two-state logic ignores the distribution of SG data and is "proved" by two-state experiments on photons, having almost nothing to do with silver atoms in an inhomogeneous field.

        I am surprised and pleased to learn that you do agree with me about 3D spin. That's wonderful!

        You challenge me to make a model of the helium angle without entanglement. I would ask you to try and understand two types of 'entanglement' that physicists do not distinguish between. First, I remind you that I believe in a deBroglie-Bohm-like wave (function) induced by momentum density as discussed in The Nature of Quantum Gravity. The ultra-dense electron induces a gravito-magnetic wave similar to the manner in which a moving speedboat induces a wave. Boat AND wave are physically real. In helium, two electrons interact and their wave states become "entangled". This is a fancy word for simply interacting and influencing each other. It is physically sensible and not surprising in the least.

        This local 'entanglement' is entirely different from Bell type 'entanglement' that exists 'faster-than-light' at any distance. That is the belief derived from Bell's logic based on qubit structure projected onto physics. In short, the entanglement one finds in a helium atom is real and local. It differs from the non-local entanglement of Bell.

        Finally, you say prove Bell wrong. I do so here: Modern Classical Spin Dynamics. I do so by using 3D spins in the magnetic field and calculating the deflections. This maps perfectly over the SG data [see figure 6, page 20]. The model is simply classical spin and the correlation is the same as QM predicts for qubits. As you note, you will not study it, nor will any physicist still active in their careers. So it is a thankless task that yet yields satisfactions, and I thank FQXi for a venue in which we exchange information densely and pleasantly.

        Thanks again for your thoughtful response, and good luck in the contest.

        Best wishes,

        Edwin Eugene Klingman

        Cristi,

        I realize math has rules, but how do they relate to this reality we seem to inhabit?

        Any grouping of people will need some cooperative structure and language in order to communicate and function together, but are these mathematical rules actually handed down from the face of God, or are they arrived at, through the cooperation of human interaction?

        "The empty set is dimensionless, but the set made of a point is dimensionless but not empty."

        Can you point to a dimensionless point, or is it an abstraction, distilled down to its most useful qualities?

        If it is an abstraction, what concept does it represent?

        My impression is that it is an ideal of location. So my view is this point and the coordinate system it locates, is a mapping, or description of the properties of space, relative to an ideal of location, not a set of rules more fundamental than space itself.

        Now the point I'm trying to make, is that in reducing this abstraction to an ideal of location, what is distilled away is any quantity of space, ie. dimensionless, in order to eliminate any conceptual fuzziness.

        The effect though is that it becomes an ideal of location, without any actual location. It doesn't exist in any real spot, because it is dimensionless. There is nothing, not even a Planck unit, to signify its location.

        Then the coordinate system is fixed to it, not arising from it, because the lines and planes could be at any direction and angle. There is no spatial dimensionality to the point, so no structure to give direction to lines and planes. It would seem to be a top down description, not bottom up process, so it is a mapping device, not a fundament to the territory being mapped.

        It seems to me, in their quest for intellectual purity, mathematicians sought to do away with the messiness of reality, but that doesn't mean they really are peering into the face of God.

        Considering we evolved as singular, mobile organisms, on a terrestrial surface, it might give some clue to why points, lines and planes, along with height, are so fundamental to our view of what is most stable.

        Now, I may well be wrong and there are such platonic structures, underlaying the nature of reality, but in the world I inhabit, it is more physical, as well as political, than mathematical. Heck, I can rarely count past about 15, before having to start over, but then again, the sorts of things I have to count are often moving and rarely organized. I live in a world of energies and forms.

        Regards,

        John

        Dear Cristi,

        After responding to you I started looking through 26 Jan 2018 copy of Physical Review Letters I received in the mail today. I was interested to find article 040406 titled

        "Violation of Bell's Inequality Using Continuous Variable Measurements"

        That is essentially the argument I was making above about the continuous variable deflection of silver atoms instead of Bell's constraint of +1 and -1. The current article is based on quantum optics, and therefore does not translate directly into atomic tests, but I hope you can see that it is an isomorphism of the paper I linked to above. The authors [Thearle, e al.] note that for continuum variable quantum optics the Bell test is harder to realize. But, significantly, they state

        "Bell argued that quantum states with positive definite Wigner function would not violate a Bell inequality with respect to continuous variable measurements."

        They claim the first observation of Bell correlations in a continuous variable system. As I said, this does not translate directly into Stern-Gerlach type of atomic tests, but I believe it is isomorphic to the continuous variable deflection measurements that I describe and that I have shown to violate Bell's inequality.

        Best regards,

        Edwin Eugene Klingman

        Dear Cristinel Stoica,

        Your essay is very well written. I can not appreciate it from an expert point of view, because it requires much additional information, but as a simple reader I appreciate it very highly.

        I wish You success with the contest!

        Best wishes,

        Robert Sadykov

          Dear Edwin,

          There are some important parts where I agree with you. I wanted to state this from the beginning of my comment, in order to facilitate reading without feeling that I opposed you too much. There is something where we disagree too, but you will see there is some important part where I tend to agree with you.

          But first let me clarify something. When I say that spin is 3D, I refer to the Block sphere representation (plus the phase), not that spin is a mere 3D rotation. To me the spin is perfectly described by a spinor. I agree with the Pauli spinor as the nonrelativistic limit of the Dirac spinor. If you want to describe the Pauli spin of the electron in a basis, ignoring the position and other degrees of freedom, the basis has two vectors. There are not only two states, there are infinitely many, it seems to be two states because the measurement is done in a particular basis, and by the projection postulate yields two possible outcomes. This description of the spin works perfectly and it is very simple and natural. By "simple" I don't mean is simple to our classical intuition, I mean that it arises naturally when combining special relativity with the requirement of unitarity, see Wigner's theorem. The "3D spin" I mention is the Bloch sphere representation, and the state vector, represented up to a phase factor by a 3D vector, is completely determined by the expectation values of the spin operators along the three axes (which give the components of that vector along the three axes).

          Now here is a bridge over the gap between our views. While I take the spinor seriously, it is not directly observable. The observables are build out of the Dirac spinor by taking various products of Dirac matrices and evaluating the result on the spinor field. You know these are scalar, vector (the electromagnetic four-current), bivector (where the angular momentum is), trivector (or pseudovector), and a pseudoscalar (a tetravector). These quantities are observable, and for a single spin 1/2 particle they behave in many situations quite classically. Now by "classically" I mean a classical spinor field, not a quantized field (as in the so-called second quantization), but the point is that these quantities are differential forms. And in the nonrelativistic limit we can treat an external field as a classical field too, in particular the magnetic field of the Stern-Gerlach device. So I am pretty sure that a quasiclassical analysis of the electron in the magnetic field is useful and relevant. You can even approximate the particle with a ball following a classical trajectory, as long as it is not too localized so that Heisenberg's principle makes the trajectory too fuzzy. This bridge I try to present here to you is something I always found reasonable to be true, and thought that it is important to have such an analysis. My brief glance to your paper gave me the impression that you are doing this in a careful and serious way. I always pictured for myself the electron as interacting continuously with the Stern-Gerlach device and exchanging momentum, energy, and angular momentum with it. I didn't do a careful reading of your paper, but I think you do this, and if I am wrong please let me know.

          As a general approach to quantum mechanics and quantum field theory, I think it is important to understand what happens. I don't belive in magical projectors, and I think measurements are not sharp, they are just interactions. But I don't think there is a description consistent with both L and SI. I will come back to this later. For now, I want to say that I see nothing wrong with the particle passing through a Stern-Gerlach device and landing either in the up region or in the down region, without a collapse or projection. Even though I see this in terms of spinor fields, I think we see this picture similarly. So probably if I will check all your math and physics I expect I will agree with your figure at page 20. If you did this analysis without adding new physics, with the right math, and got that picture at page 20, I think it is an important result.

          Now, I have the feeling that you are not satisfied with this analysis, and want more, namely to disprove Bell. If you are interested in my 0.02$, here is what I would advise you. Take that paper, clean it for claims that Pauli and Bell were wrong (I will explain later why), and try to publish it. If I am right, you can make it be some "mainstream" analysis of the Stern-Gerlach experiment. And I think you can get it published in a journal with ISI IF.

          Now, I promised you I will come back to Bell's theorem. It is completely irrelevant if he labels the two outcomes with +1 and -1, or +1/2 and -1/2, or |up> and |down>, or just "up" and "down". If you think it is relevant, let's consider then another version of Bell's theorem, one which I say is the same, and you may say is a weakened version. Let us refer only to spin being up or down along an axis, not to Pauli matrices, not to two-level systems. By up and down I call the two places where the particle arrives after going through the S-G device, those two regions you reproduce in your picture at page 20. This is also in agreement with my views, because there are no sharp measurements. So we just think in terms of yes/no measurements, answering to questions like "did the particle land on this "lip" of the iconic postcard, when oriented along this particular axis?"

          If you want to prove that Bell was wrong, then your task (for a second paper I would recommend) is to provide an explanation of the EPR experiment based on your theory, in terms of these up and down along diferent axes. So we stick only with what we can see in the experiment, not with the projections you said Pauli made. Maybe you think you already have this proof, but I still suggest you to put it in a second paper, separate from the one-particle paper.

          If I am right, then you are wasting a great opportunity by mixing your one-particle analysis with the idea that this disproves Bell. I think your reasoning is the following sillogism: "(1) I explained the S-G experiment without Pauli matrices and spin operators, (2) Bell assumes Pauli spin, therefore (3) I disproved Bell". I don't think this works, because I don't think you can get the same correlation as QM with your model, unless you add something that breaks either L or SI. If I am right, you can publish the one-particle paper. If you are right, you can publish the one-particle paper, and then make it easier for the reader to accept your model and to read your second paper, where you will explain EPR. So no matter who is right, I think your analysis may result in a paper, which I think will be useful for physics (but I repeat, this is based on a brief glance of your paper, maybe I project my own views on it).

          Best regards,

          Cristi

          Dear Robert Sadykov,

          Thank you for your comments. I tried to say something worth reading for people with different specializations, so I am glad you liked those parts where our expertise overlaps. I wish you success too!

          Best regards,

          Cristi

          Dear John,

          If your position is that you find no use in mathematics as a way to understand the universe, I have nothing against your view. I do find it essential, but I don't care to convince you. You tried to destroy this mathematics (which you maybe see as as a superstition) from the inside, but I hope you realize you are not inside. So let's agree to disagree and stop here.

          If your position is that I, or physicists in general, should no longer use mathematics, or at least no longer use some parts of mathematics, I don't think you can prove it. Maybe you think you are right and I think I am right, so again we can move over.

          You say we "need some cooperative structure and language in order to communicate and function together", but you don't want it to be mathematics. OK, then what language do you propose to use? Is this other language "actually handed down from the face of God, or are they arrived at, through the cooperation of human interaction?"

          You said "Can you point to a dimensionless point, or is it an abstraction, distilled down to its most useful qualities?".

          No, I never claimed that my finger is pointy and that we can see points, so I can't show it to you. But I hope you realize that this doesn't make your previous "mathematical" statements about points right - you never said if you still believe them. You gave me no feedback about my point-by-point reply to your first comment, neither about my discussion of your "unconventional" usage of mathematics. I don't know if we are making progress, or you just hold the same conclusion and only try to change the arguments.

          Maybe there are no points, I explained this possibility 10 years ago, here, where I said that maybe there are no points, but we can still use the mathematics of locales. I realize this is not what you mean, because it is even more abstract mathematics. I consider it too abstract too, so I stick for the moment to sets of points instead of more general locales. What I know is that mathematics, despite using points, works so well, and I may be sad that you refuse to visit this world, but I respect your choice. You claim it doesn't make sense, and you live in a political world. Can politics make better predictions than mathematical formalism in physics? You say you live in a physical world, not a mathematical one. I say they are the same, you don't have to believe me, but you want me to believe that there is no math without bothering to understand what math is.

          I confess that I am very limited, and when I do physics I need to rely as much as possible on mathematics, not on politics or only on the mundane experience. Maybe your statements about points being zero being empty sets and so on make sense to you, but to me, and I claim that to other mathematicians too, they don't. So I realize I will not be a good partner of discussion for you, because I am too trapped in trying to make logical and mathematical sense of your words. I have nothing against your views, but I think there is a reason we are different - we are different so that I can explore the world according to my views, and you according to yours. I am sorry I couldn't borrow your eyes and that I couldn't lend you my eyes for a moment, I believe it would have been an interesting experience for both of us.

          Best regards,

          Cristi

          Dear John,

          Let me express my thoughts in a way more similar to politics and economics.

          We live in a world which gives us few things to rely on. I searched many years some sort of safe or true vision of the world. Maybe it is my fault, but I think that almost everything is impredictable and biased, so almost nothing is trustworthy. Eventually I realized that the best thing that gives me some solid ground and understanding of the world is mathematics. You may think I am wrong, and I am happy if you have other, more solid grounds. But to me, mathematics and logic allow both understanding and prediction of the world, in the best way I found. It is only when I see things mathematically that I feel I really understand. Otherwise, I can keep in my mind various ideas, but to my standards they seem fuzzy and unreliable until I can make them rigorous. But of course the world is more complex to be understood mathematically, so I keep this understanding for physics. My human, mundane side, lives in a world which is still not mathematical, and I don't hope I can make it mathematical, because I know very well the limits of mathematical understanding by limited being as myself. So in this world of complexity and randomness, I take another position, which if I would want to describe is of skepticism and openness. That is, experiences happen, I have no solid proof to qualify them, to judge the intrinsic value of people or events. They happen as in a play, I am spectator and actor, but I gave up trying to make rigorous sense of them. I try to allow people be, because I have no absolute truth about how they should be, and I think it is their right to have their experiences in the best possible way they choose. This makes me able to live in a world which is far from the solid ground provided by mathematics, in a way which makes me happy for my experiences and which allows me allow others have their own experiences. And I am a bit avid in trying to see the world through their eyes too, this makes me realize I have no moral high ground on the others and no way to save them from their problems, since I have my own.

          So I want to thank you for trying to offer me another perspective, and a possible reality check. We both tried, and it didn't work. It seems to me that you propose me to give up mathematics, and I don't know what I will receive instead. Because mathematics is the best I know for understanding and predicting the world, and I know it works. So, if we look at this politically and economically, it seems that I have only to lose from this deal :) Now you may think that you have some special views of the world and some experiences that don't need mathematics. I have too. But is it needed for me to give up what for me is so important, to receive what you think you are offering me? Do you think mathematics prevent people from having experiences like appreciating music or poetry, or love? So let me return your favor and ask you, do you think you will lose yourself if you try to see mathematics for what it is, rather than for what you project on it from very far away?

          Best regards,

          Cristi

          Dear Cristi,

          Thank you for elaborating on your view, which I tink I can understand a bit better now. Your view reminds me a bit of that of the presocratic philosopher Parmenides, who claimed that "All is one" i.e. that the universe and everything in it is like one solid unchanging block, and that any change or smaller parts we perceive is an illusion, a deception by our senses. It seems to me that someone who applies these ideas to fundamentality would not be so far removed from considering it a purely epistemological matter, except for the one ontological truth of the fundamentality of a unified whole.

          The person who in my mind is the best known modern proponent of Parmenidean ideas in physics is Julian Barbour. Perhaps you may also have an affinity for some of his ideas on general relativity.

          All the best,

          Armin

          Dear Cristinel,

          the structure of your essay is quite typical of many others. So allow me a more general criticism at this point. You begin with the human perspective, e.g.: "The universe is rich in complex phenomena and situations of infinite diversity...". From there you jump to isomorphisms, mathematical isomorphisms to be sure. Then, after long discussion and diversion you end up in holomorphic unification. Fine! But then the trouble begins to steer back to the human perspective: "If the program whose first stages were described in the previous section will turn out to work and solve the mentioned open problems....maybe we will find out that this Clifford algebra has a geometric meaning..." So what you say sounds a little bit like: if the squirrel were a horse one could ride up the trees, i.e. you try to solve unsolvable problems. Why? Well, you tried to answer the contest question.

          Mathematical isomorphism is a wonderful tool for mathematicians. However, the only physical 'isomorphism' known today is the 'isomorphism' of Leibniz' relational and explicit three-dimensional Euclidean (Newtonian) space. The problem is that this 'isomorphism' has no positive mapping rule, for relational 'space', the 'space' of human experience, is defined by prepositions, adjectives and adverbs, i.e. possible relations between objects. In other words, relational experience and Euclidean space are incommensurable and, therefore, Absolutely-non-contradictory. Felix Klein was able to capture the entire nature of Euclidean space just because he framed it under a conservation principle (a negation=prohibition!), whereas Hilbert's positive-logical efforts at Euclidean space ended up in a complex and unusable mess.

          So, my general criticism is that your and many other essays set out from the human (anschaulich) perspective (to capture the reader) from where they jump into mathematics (instead of physics) without the slightest chance of ever returning to human experience, hence leaving the scientific reader wanting and the artist sufficiently flabbergasted to turn it into something anschaulich again.

          May the most beautiful castle-in-the-air-win,

          Heinrich