Hi Michael: Thanks for the question. That logic vid that ends stating it's an example of Godel's Incompleteness Theorem is precisely the focus of Roger Penrose's emphasis that a calculation is not consciousness. There's two ways of interpreting Godel's Incompleteness Theorem - one is that math is inherently inconsistent or the other is that math is incomplete. The later is what most scientists favor with the Platonic view that technology will continue to disprove hypotheses based on the "proof by contradiction" model of science. Thus with more technological precision therefore a more accurate view of reality arises. Math Professor Luigi Borzacchini points out that this inherent paradox of mathematics (really to do with the discrete and continuous) leads to what he titled his book, "Plato's Computer." The book is in Italian but there's some summary of it - the idea being that mathematics is the structural drive of science but it was based on a "deep pre-established disharmony" (Borzacchini) as the "guiding evoltive principle" of science. This is what is also dubbed by another research group, "the music logarithmic spiral" in an attempt to solve this logical paradox. So Penrose is arguing that ONLY consciousness unifies relativity and quantum physics but not consciousness in the normal sense defined by perceptions or thinking. Rather "protoconsciousness" is this nonlocal process of noncommutativity as "fundamental time" with the future and past overlapping.
The typical physicist (Viktor Toth answering on Quora website is a good example) will argue that the beginning of the Universe can not be defined since it's an "open set" and therefore the mathematics, as per Godel, will always be "incomplete" since reality is defined by the geometric symmetric continuum (this is the unstated Platonic norm of modern Western science). In other words the geometric continuum arises out of an argument of symmetry that Plato claimed was inherent and thus self-evident. His argument was spurious and he was relying on the wrong music theory - secretly - from Archytas and Philolaus. As Luigi Borzacchini points out in his 2007 published article (that he first presented in 1999 on an academic math forum, and I discovered it in 2001) - the secret music origin of the geometric continuum is due to the "negative judgement paradox." So in music theory the "proof" of infinity as this "protoconsciousness" was a NEGATIVE proof and thus required silence as logical inference of the Self - as Socrates taught and practiced. Well Socrates was a kind of vanishing mediator in the creation of Plato - such that the irrational "Alogon" geometry was self-evident. So the symmetric "twoness" exists in the Platonic Realm with time as a static symmetric "image" of eternity.
So there is a book called "From Pythagoras to Einstein" demonstrating how relativity is really just an extension of this logic and so time is converted to a symmetric spatial rest frame. The term "protoconsciousness" as Penrose uses is coined or emphasized by Bernard 'd Espagnat. His book "Veiled Reality" is one of the best books on quantum mechanics - and it won some award also. So basically 'd Espagnat is arguing we can logically infer from the Double Slit experiment, etc. that there is this nonlocal realm of reality that is secretly guiding matter from the future.
Essentially it is the de Broglie-Bohm model that uses this protoconsciousness term but Professor Basil J. Hiley just calls it "active information." So normally we think of "primitive causality" as defined or limited to the "speed of light" but through the double slit experiment we witness a superluminal radiance (that recently was also confirmed in our own microtubules for quantum consciousness!).
The microtubules are resonated as the strongest force or conductance at the ultrasound frequency. It turns out in Tinnitus research it's proven the highest sound we can listen to externally then resonates the brain internally as ultrasound. Ultrasound is a key to both mechanical transformation and electromagnetic transduction via the piezoelectric effect. The most common protein in our body, collagen, is piezoelectric.
So this was the secret to the ancient nonwestern meditation since by LISTENING or logical inference it is proven that musicians can hear up to ten times FASTER than "time-frequency uncertainty" or Fourier Uncertainty. In other words time is no longer limited to being a "linear operator" as in standard quantum physics and in relativity. Fourier Uncertainty arises OUT of noncommutativity as nonlocality.
So Penrose points out that in fact Time is not just cubic but it is also noncommutative and this is what he calls Fundamental Time - a term from Lee Smolin who first studied quantum physics from my same physics Professor Herbert J. Bernstein. If you look up "Eddie Oshins" who coined "quantum psychology" as a physicist working at Stanford Linear Accelerator Center then you discover Oshins solving these "double bind" paradoxes "Cretin's Liar paradox, etc." of Western logic and set theory. Oshins was building off Gregory Bateson's attempt to solve the same Bertrand Russell "error of logical types." I read Bateson's book "Mind and Nature: A necessary unity" on my own in my first year of college - Bateson is struggling in that book.
Oshins basically states the answer is "Neither A nor Not-A." (again the silence as logical inference). Ironically Edgar Allan Poe realized this same truth of logical inference as the basis for creating the modern mystery novel that relies on reversing the future as a noncommutativity of meaning).
Oshins was trying to teach noncommutativity to Karl Pribram (who made his holographic brain model famous). Look up Oshin's paper, (About Models and Muddles, Part I: Why Brown’s Laws of Form and Pribram’s “Hologram Hypothesis” are NOT Relevant to Quantum Physics and Quantum Psychology) - for details. I give the Oshins link on my blog in case his old website gives you problems. He passed on early.
So for example in Richard Feynman's diagrams the photon has both a future and past at the same time and this is necessary to explain the relativistic quantum physics. But Professor Jean Bricmont has pointed out that Bell's Inequality disproved Quantum Field Theory that still assumes a symmetric rest frame for measurement.
The "weak measurements" by Yakir Aharonov's group, along with the Aharonov-Bohm Effect and even the neutron spin experiments (that were first proposed by my own professor in 1967 along with Leonard Susskind and Yakir Aharonov) have proven, as Professor Basil J. Hiley explains, this noncommutative nonlocal process of the future overlapping with the past. Professor Hiley is continuing these experiments of this newly discovered force of the quantum potential. Hiley is also building off Alain Connes' noncommutative lecture to physicists.
Feynman was wrong to dismiss this quantum potential nonlocality into a type of kinetic energy based on the symmetric rest frame of the calculus, as Hiley details! Hiley proves that Feynman's math was wrong.
And so this secret newly discovered force (from the future) was covered up or lost due to the wrong mathematics!
Whether this force can be harnessed for technology - I don't know. I do know that this force is also what Penrose calls the "gravitational potential" as the true dark energy and it originates from noncommutative quantum negative entropy and so it is inherent to life itself from protoconsciousness (Penrose is citing Schroedinger's 1945 book, "What is Life?".
The Hameroff-Penrose model goes into more details on how this evolved into complex life out of this inherent noncommutativity. Hameroff himself just acknowledges this as "negative resonance" from Penrose but Hameroff doesn't understand it is actually noncommutativity! Hameroff does emphasize it is not an "integration" as "calculation" but rather protoconsciousness is literally music. I have delved into the noncommutative music math details in my academia articles and books. See for example "Strange Vibrations" by drew hempel
The concept of noncommutativity is still barely recognized in Western science since as Alain Connes points out most scientists consider noncommutativity to be "strange" and a "nuisance." Most physicists if they do recognize noncommutativity consider it be an "extension" of Riemannian relativistic geometry. But in fact if people study Alain Connes' "Music of Shapes" lecture it's quite clear the noncommutativity was there already in the nonwestern Pythagorean music as an applied body-mind transformation meditation. Eddie Oshins figured this out also since he realized the secret of Neigong or internal martial arts as alchemy is noncommutativity.
And so because math was founded on the wrong music theory then humans developed very precise technology that is not accurate to the foundation of reality: Hence our current ecological crisis due to the entropy of matter being the opposite the entropy of gravity as Roger Penrose explains.
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