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Stirring the Pot, 2: Godel, the Incompleteness Theorem, Euler’s expression, and the Turing Machine dilemma

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As we continue to stir the mathematics pot, BA 77 has given a link to a video on the significance of Godel’s discovery of incompleteness:

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(Pardon possible embed problems, the links work . . . I am doing this under travel related constraints)

This one, gives a bit more of details on how Turing sharpened the theorem using the Turing machine, that led to the well known algorithm halting problem:

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The issue of the intuitive imagining mind as opposed to an algorithmic machine, is discussed. Worth pondering.

At the same time, we must always bear in mind the famous Euler result:

ei*pi + 1 = 0

 This speaks to astonishing unity in Mathematics, for in one expression, converging from wildly different directions, we find the five most important numbers, three of the most important operations, and work in entire fields of Mathematics.

So, here in the heart of mathematics, we at once confront our limitations, the limitations of logic, algorithm machines, and the issue of the rational, insightful, creative, intuitive power power of mind.

Then, stir in, the apparent pattern that starts with collecting nothing, and successively generating the natural numbers, then extending them via decimals to the continuum, using the rotational view of sqrt_-1 to give a plane, thus space, and the onward ijk vectors to give 3-d space, thence motion and the inertia- force- energy triad that gets us to physics.

So, now, let us further ponder, what is this that mathematics is telling us?

Especially, “the unreasonable effectiveness of mathematics” in helping us understand the physical world? END

KF, The Fine-Tuning of the Universe for Life Just Got Finer - March 15, 2013 Excerpt: In new lattice calculations done at the Juelich Supercomputer Centre [in Germany] the physicists found that just a slight variation in the light quark mass will change the energy of the Hoyle state, and this in turn would affect the production of carbon and oxygen in such a way that life as we know it wouldn't exist. "The Hoyle state of carbon is key," Lee says. "If the Hoyle state energy was at 479 keV [479,000 electron volts] or more above the three alpha particles [helium-4 nuclei], then the amount of carbon produced would be too low for carbon-based life. "The same holds true for oxygen," he adds. "If the Hoyle state energy were instead within 279 keV of the three alphas, then there would be plenty of carbon. But the stars would burn their helium into carbon much earlier in their life cycle. As a consequence, the stars would not be hot enough to produce sufficient oxygen for life. In our lattice simulations, we find that more than a 2 or 3 percent change in the light quark mass would lead to problems with the abundance of either carbon or oxygen in the universe." http://www.evolutionnews.org/2013/03/the_fine-tuning_1070091.html bornagain77
Jerad, to clear up any ambiguity and to more clearly illustrate the centrality of conscious observation and free will in quantum mechanics, I think the following recent experiment does the best in that task: Here’s a recent variation of Wheeler’s Delayed Choice experiment, which highlights the ability of the conscious observer to effect 'spooky action into the past', thus further solidifying consciousness's centrality in reality. Furthermore in the following experiment, the claim that past material states determine future conscious choices (determinism) is falsified by the fact that present conscious choices effect past material states:
Quantum physics mimics spooky action into the past - April 23, 2012 Excerpt: The authors experimentally realized a "Gedankenexperiment" called "delayed-choice entanglement swapping", formulated by Asher Peres in the year 2000. Two pairs of entangled photons are produced, and one photon from each pair is sent to a party called Victor. Of the two remaining photons, one photon is sent to the party Alice and one is sent to the party Bob. Victor can now choose between two kinds of measurements. If he decides to measure his two photons in a way such that they are forced to be in an entangled state, then also Alice's and Bob's photon pair becomes entangled. If Victor chooses to measure his particles individually, Alice's and Bob's photon pair ends up in a separable state. Modern quantum optics technology allowed the team to delay Victor's choice and measurement with respect to the measurements which Alice and Bob perform on their photons. "We found that whether Alice's and Bob's photons are entangled and show quantum correlations or are separable and show classical correlations can be decided after they have been measured", explains Xiao-song Ma, lead author of the study. According to the famous words of Albert Einstein, the effects of quantum entanglement appear as "spooky action at a distance". The recent experiment has gone one remarkable step further. "Within a naïve classical world view, quantum mechanics can even mimic an influence of future actions on past events", says Anton Zeilinger. http://phys.org/news/2012-04-quantum-physics-mimics-spooky-action.html
In other words, if my conscious choices really are just merely the result of whatever state the material particles in my brain happen to be in in the past (deterministic) how in blue blazes are my choices instantaneously effecting the state of material particles into the past?,,, This is simply completely inexplicable to the materialistic/atheistic framework! Of note: since our free will choices figure so prominently in how reality is actually found to be constructed in our understanding of quantum mechanics, I think a Christian perspective on just how important our choices are in this temporal life, in regards to our eternal destiny, is very fitting:
Is God Good? (Free will and the problem of evil) - video http://www.youtube.com/watch?v=Rfd_1UAjeIA
And Jerad, if you don't want to hear the Gospel from a Christian, and the importance of 'freely choosing' Christ in this lifetime (propitiation), perhaps hearing it from the late Christopher Hitchens will bring the point home to you:
The Gospel According To An Atheist Sewell: “Mr. Hitchens, the religion you cite in your book is generally the fundamentalist faith of various kinds. I’m a liberal Christian, and I don’t take the stories from the scripture literally. I don’t believe in the doctrine of atonement — that Jesus died for our sins, for example. Do you make a distinction between fundamentalist faith and liberal religion?” Hitchens’ answer possesses more clarity, precision and accuracy about the Christian Gospel and its implications for mankind than most Christians out there touting the faith can articulate. Hitchens: “Well, I would say that if you don’t believe that Jesus of Nazareth was the Christ and Messiah, and that he rose again from the dead and by his sacrifice our sins are forgiven, you’re really not in any meaningful sense a Christian.” http://www.64fellowship.com/the-64-blog/the-gospel-according-to-an-atheist/
Or perhaps this former militant atheist's Near Death testimony will ring a bell for you Jerad:
video - Howard Storm continues to share his gripping story of his own near death experience. Today, he picks up just as Jesus was rescuing him from the horrors of Hell and carrying him into the glories of Heaven. http://www.daystar.com/ondemand/joni-heaven-howard-storm-j924/#.UKvFrYYsE31
It should also be noted: All foreign, non-Judeo-Christian culture, NDE studies I have looked at have a extreme rarity of encounters with 'The Being Of Light' and tend to be very unpleasant NDE's save for the few pleasant children's NDEs of those cultures that I've seen (It seems there is indeed an 'age of accountability'). The following study was shocking for what was found in some non-Judeo-Christian NDE's:
Near-Death Experiences in Thailand - Todd Murphy: Excerpt:The Light seems to be absent in Thai NDEs. So is the profound positive affect found in so many Western NDEs. The most common affect in our collection is negative. Unlike the negative affect in so many Western NDEs (cf. Greyson & Bush, 1992), that found in Thai NDEs (in all but case #11) has two recognizable causes. The first is fear of 'going'. The second is horror and fear of hell. It is worth noting that although half of our collection include seeing hell (cases 2,6,7,9,10) and being forced to witness horrific tortures, not one includes the NDEer having been subjected to these torments themselves. http://www.shaktitechnology.com/thaindes.htm Near Death Experience Thailand Asia - testimonial video http://www.youtube.com/watch?v=Y8M5J3zWG5g
Verse and music:
Deuteronomy 30:19 This day I call the heavens and the earth as witnesses against you that I have set before you life and death, blessings and curses. Now choose life, so that you and your children may live Third Day - Trust In Jesus http://www.youtube.com/watch?v=2BtaCeJYqZA
The preceding video was made by two Australian University physics professors. As well, as with the scientifically verified tunnel for special relativity to a higher dimension, we also have scientific confirmation of extreme ‘tunnel curvature’, within space-time, to a eternal ‘event horizon’ at black holes;
Space-Time of a Black hole http://www.youtube.com/watch?v=f0VOn9r4dq8
What’s more is that special relativity (and general relativity) also confirm the ‘eternity’ for this higher dimension. i.e. Time, as we understand it temporally, would come to a complete stop at the speed of light. To grasp the whole ‘time coming to a complete stop at the speed of light’ concept a little more easily, imagine moving away from the face of a clock at the speed of light. Would not the hands on the clock stay stationary as you moved away from the face of the clock at the speed of light? Moving away from the face of a clock at the speed of light happens to be the same ‘thought experiment’ that gave Einstein his breakthrough insight into e=mc2.
Albert Einstein – Special Relativity – Insight Into Eternity – ‘thought experiment’ video http://www.metacafe.com/w/6545941/ “I’ve just developed a new theory of eternity.” Albert Einstein – The Einstein Factor – Reader’s Digest “The laws of relativity have changed timeless existence from a theological claim to a physical reality. Light, you see, is outside of time, a fact of nature proven in thousands of experiments at hundreds of universities. I don’t pretend to know how tomorrow can exist simultaneously with today and yesterday. But at the speed of light they actually and rigorously do. Time does not pass.” Richard Swenson – More Than Meets The Eye, Chpt. 12
It is also interesting to point out that this 'eternal' framework for time at the speed of light is also witnessed in Near Death Experience testimonies:
'In the 'spirit world,,, instantly, there was no sense of time. See, everything on earth is related to time. You got up this morning, you are going to go to bed tonight. Something is new, it will get old. Something is born, it's going to die. Everything on the physical plane is relative to time, but everything in the spiritual plane is relative to eternity. Instantly I was in total consciousness and awareness of eternity, and you and I as we live in this earth cannot even comprehend it, because everything that we have here is filled within the veil of the temporal life. In the spirit life that is more real than anything else and it is awesome. Eternity as a concept is awesome. There is no such thing as time. I knew that whatever happened was going to go on and on.' Mickey Robinson - Near Death Experience testimony
‘Time dilation’, i.e. eternity, is confirmed by many lines of scientific evidence but basically the simplest way to understand this 'eternal framework' is to realize that this higher dimensional, ‘eternal’, inference for the time framework of light is warranted because light is not ‘frozen within time’ yet it is also shown that time, as we understand it, does not pass for light. This paradox is only possible for time at the speed of light if temporal time is a lower dimensional time that was created from a higher dimension that ‘contains all temporal time’,,,Yet, even though light has this ‘eternal’ attribute in regards to our temporal framework of time, for us to hypothetically travel at the speed of light, in this universe, will still only get us to first base as far as the eternal framework of quantum entanglement, and/or quantum teleportation, is concerned.
Light and Quantum Entanglement Reflect Some Characteristics Of God – video http://www.metacafe.com/watch/4102182
i.e. Hypothetically traveling at the speed of light in this universe would be, because of time dilation, instantaneous travel for the person going at the speed of light. This is because time does not pass for them at the speed of light, yet, and this is a very big ‘yet’ to take note of, this ‘timeless’ travel is still not instantaneous and transcendent of our temporal framework of time as quantum teleportation and entanglement are, i.e. Speed of light travel, to our temporal frame of reference of time, is still not completely transcendent of our temporal time framework since light appears to take time to travel from our temporal perspective. Yet, in quantum teleportation of information, the ‘time not passing’, i.e. ‘eternal’, framework is not only achieved in the speed of light framework/dimension, but is also ‘instantaneously’ achieved in our lower temporal framework. That is to say, the instantaneous teleportation/travel of quantum information is instantaneous to both the temporal and speed of light frameworks, not just the speed of light framework. Information teleportation/travel is not limited by time, nor space, in any way, shape or form, in any frame of reference, as light is seemingly limited to us in this temporal framework. Thus ‘pure transcendent information’ (in quantum teleportaion experiments) is shown to be timeless (eternal) and completely transcendent of all material frameworks. Moreover, concluding from all lines of evidence we now have (many of which I have not specifically listed here); transcendent, eternal, infinite information is indeed real and the framework in which ‘It’ resides is the primary reality (highest dimension) that can exist, (in so far as our limited perception of a primary reality, highest dimension, can be discerned).
“An illusion can never go faster than the speed limit of reality” Akiane Kramarik – Child Prodigy -
Music and verse:
YOU ARE GOD ALONE, Philips, Craig and Dean http://www.youtube.com/watch?v=OICArFHAa9c Revelation 4:11 “You are worthy, our Lord and God, to receive glory and honor and power, for you created all things, and by your will they were created and have their being.”
Jerad, here are a few more notes on the ‘spirituality of math’: It is interesting to note that ‘higher dimensional’ mathematics had to be developed before Einstein could elucidate General Relativity, or even before Quantum Mechanics could be elucidated;
The Mathematics Of Higher Dimensionality – Gauss and Riemann – video http://www.metacafe.com/watch/6199520/ The Unreasonable Effectiveness of Mathematics in the Natural Sciences – Eugene Wigner – 1960 Excerpt: We now have, in physics, two theories of great power and interest: the theory of quantum phenomena and the theory of relativity.,,, The two theories operate with different mathematical concepts: the four dimensional Riemann space and the infinite dimensional Hilbert space, http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
One peculiar thing to note about the higher dimensional 4-D space time of General Relativity is that it ‘expands equally in all places’:
Where is the centre of the universe?: Excerpt: There is no centre of the universe! According to the standard theories of cosmology, the universe started with a “Big Bang” about 14 thousand million years ago and has been expanding ever since. Yet there is no centre to the expansion; it is the same everywhere. The Big Bang should not be visualized as an ordinary explosion. The universe is not expanding out from a centre into space; rather, the whole universe is expanding and it is doing so equally at all places, as far as we can tell.
Thus from a 3-dimensional (3D) perspective, any particular 3D spot in the universe is to be considered just as ‘center of the universe’ as any other particular spot in the universe is to be considered ‘center of the universe’. This centrality found for any 3D place in the universe is because the universe is a 4D expanding hypersphere, analogous in 3D to the surface of an expanding balloon. All points on the surface are moving away from each other, and every point is central, if that’s where you live.
Centrality of Earth Within The 4-Dimensional Space-Time of General Relativity – video http://www.metacafe.com/w/8421879
And higher (infinite) dimensional quantum mechanics is also very mysterious to look at in that consciousness is found to be the ‘ultimate universal reality’:
“It will remain remarkable, in whatever way our future concepts may develop, that the very study of the external world led to the scientific conclusion that the content of the consciousness is the ultimate universal reality” - Eugene Wigner – (Remarks on the Mind-Body Question, Eugene Wigner, in Wheeler and Zurek, p.169) 1961 – received Nobel Prize in 1963 for ‘Quantum Symmetries’ Four intersecting lines of experimental evidence from quantum mechanics that shows that consciousness precedes the quantum wave collapse of material reality (Wigner’s Quantum Symmetries, Wheeler’s Delayed Choice, Leggett’s Inequalities, Quantum Zeno effect): https://docs.google.com/document/d/1G_Fi50ljF5w_XyJHfmSIZsOcPFhgoAZ3PRc_ktY8cFo/edit
And when one looks at the 4-D space-time of relativity, and the centrality of conscious observation in quantum mechanics, a very interesting ‘anomaly’ pops out:
The Galileo Affair and the true “Center of the Universe” Excerpt: I find it extremely interesting, and strange, that quantum mechanics tells us that instantaneous quantum wave collapse to its ‘uncertain’ 3-D state is centered on each individual observer in the universe, whereas, 4-D space-time cosmology (General Relativity) tells us each 3-D point in the universe is central to the expansion of the universe. These findings of modern science are pretty much exactly what we would expect to see if this universe were indeed created, and sustained, from a higher dimension by a omniscient, omnipotent, omnipresent, eternal Being who knows everything that is happening everywhere in the universe at the same time. These findings certainly seem to go to the very heart of the age old question asked of many parents by their children, “How can God hear everybody’s prayers at the same time?”,,, i.e. Why should the expansion of the universe, or the quantum wave collapse of the entire universe, even care that you or I, or anyone else, should exist? Only Theism offers a rational explanation as to why you or I, or anyone else, should have such undeserved significance in such a vast universe: Psalm 33:13-15 The LORD looks from heaven; He sees all the sons of men. From the place of His dwelling He looks on all the inhabitants of the earth; He fashions their hearts individually; He considers all their works. https://docs.google.com/document/d/1BHAcvrc913SgnPcDohwkPnN4kMJ9EDX-JJSkjc4AXmA/edit
The following is another very ‘spiritual’ finding from mathematics:
The Scale of The Universe – Part 2 – interactive graph (recently updated in 2012 with cool features) http://htwins.net/scale2/?bordercolor=white
The preceding interactive graph points out that the smallest scale visible to the human eye (as well as a human egg) is at 10^-4 meters, which ‘just so happens’ to be directly in the exponential center of all possible sizes of our physical reality (not just ‘nearly’ in the exponential center!). i.e. 10^-4 is, exponentially, right in the middle of 10^-35 meters, which is the smallest possible unit of length, which is Planck length, and 10^27 meters, which is the largest possible unit of ‘observable’ length since space-time was created in the Big Bang, which is the diameter of the universe. This is very interesting for, as far as I can tell, the limits to human vision (as well as the size of the human egg) could have, theoretically, been at very different positions than directly in the exponential middle; There is also found to be a very mysterious ‘higher dimensional’ component in life:
The predominance of quarter-power (4-D) scaling in biology Excerpt: Many fundamental characteristics of organisms scale with body size as power laws of the form: Y = Yo M^b, where Y is some characteristic such as metabolic rate, stride length or life span, Yo is a normalization constant, M is body mass and b is the allometric scaling exponent. A longstanding puzzle in biology is why the exponent b is usually some simple multiple of 1/4 (4-Dimensional scaling) rather than a multiple of 1/3, as would be expected from Euclidean (3-Dimensional) scaling. http://www.nceas.ucsb.edu/~drewa/pubs/savage_v_2004_f18_257.pdf “Although living things occupy a three-dimensional space, their internal physiology and anatomy operate as if they were four-dimensional. Quarter-power scaling laws are perhaps as universal and as uniquely biological as the biochemical pathways of metabolism, the structure and function of the genetic code and the process of natural selection.,,, The conclusion here is inescapable, that the driving force for these invariant scaling laws cannot have been natural selection.” Jerry Fodor and Massimo Piatelli-Palmarini, What Darwin Got Wrong (London: Profile Books, 2010), p. 78-79
Though Jerry Fodor and Massimo Piatelli-Palmarini rightly find it inexplicable for ‘random’ Natural Selection to be the rational explanation for the invariant scaling of the physiology, and anatomy, of living things to four-dimensional parameters, they do not seem to fully realize the implications this ‘four dimensional scaling’ of living things presents. This 4-D scaling is something we should rightly expect from a Intelligent Design perspective. This is because Intelligent Design holds that ‘higher dimensional transcendent information’ is more foundational to life, and even to the universe itself, than either matter or energy are. This higher dimensional ‘expectation’ for life, from a Intelligent Design perspective, is directly opposed to the expectation of the Darwinian framework, which holds that information, and indeed even the essence of life itself, is merely an ‘emergent’ property of the 3-D material realm.
Earth’s crammed with heaven, And every common bush afire with God; But only he who sees, takes off his shoes, The rest sit round it and pluck blackberries. – Elizabeth Barrett Browning
It is important to note that higher dimensions are invisible to our physical 3 Dimensional sight. The reason why ‘higher dimensions’ are invisible to our 3D vision is best illustrated by ‘Flatland’:
Dr. Quantum in Flatland - 3D in a 2D world – video http://www.disclose.tv/action/viewvideo/9395/Dr_Quantum_Flatland_Explanation_3D_in_a_2D_world/
Perhaps some may think that we have no scientific evidence to support the view that higher ‘invisible’ dimensions are above this 3 Dimensional world, but a person would be wrong in that presumption. Higher invisible dimensions are corroborated by Special Relativity when considering the optical effects for traveling at the speed of light. Please note the optical effect, noted at the 3:22 minute mark of the following video, when the 3-Dimensional world ‘folds and collapses’ into a tunnel shape around the direction of travel as a ‘hypothetical’ observer moves towards the ‘higher dimension’ of the speed of light:
Approaching The Speed Of Light – Optical Effects – video http://www.metacafe.com/watch/5733303/
But Jerad, the mystery of the Schroedinger equation goes even deeper to reveal 'the spirituality of mathematics'.
Wheeler's Classic Delayed Choice Experiment: Excerpt: Now, for many billions of years the photon is in transit in region 3. Yet we can choose (many billions of years later) which experimental set up to employ – the single wide-focus, or the two narrowly focused instruments. We have chosen whether to know which side of the galaxy the photon passed by (by choosing whether to use the two-telescope set up or not, which are the instruments that would give us the information about which side of the galaxy the photon passed). We have delayed this choice until a time long after the particles "have passed by one side of the galaxy, or the other side of the galaxy, or both sides of the galaxy," so to speak. Yet, it seems paradoxically that our later choice of whether to obtain this information determines which side of the galaxy the light passed, so to speak, billions of years ago. So it seems that time has nothing to do with effects of quantum mechanics. And, indeed, the original thought experiment was not based on any analysis of how particles evolve and behave over time – it was based on the mathematics. This is what the mathematics predicted for a result, and this is exactly the result obtained in the laboratory. http://www.bottomlayer.com/bottom/basic_delayed_choice.htm "Thus one decides the photon shall have come by one route or by both routes after it has already done its travel" John A. Wheeler Alain Aspect speaks on John Wheeler's Delayed Choice Experiment - video http://vimeo.com/38508798
Wheeler's Delayed Choice Experiment - 2010 Excerpt: The Delayed Choice experiment changes the boundary conditions of the Schrodinger equation after the particle enters the first beamsplitter. http://www.physics.drexel.edu/~bob/TermPapers/WheelerDelayed.pdf
But why should a mathematical equation even care when I decide to implement (freely choose) the boundary conditions for the equation to look at a particle? Mathematical equations can't care about anything! Only God can care if and when and how I decide to look at any particular particle in the universe! In fact, 'the spirituality of mathematics' has now been revealed to an even deeper level through recent quantum entanglement experiments. The foundation of quantum mechanics within science is now so solid that researchers were able to bring forth this following proof from quantum entanglement experiments;
An experimental test of all theories with predictive power beyond quantum theory – May 2011 Excerpt: More precisely, we perform various measurements (conscious observations) on distant entangled photons, and, under the assumption that these measurements (conscious observations) are freely chosen (free will), we give a upper bound on how well any alternative theory could predict their outcomes.,,,, Hence, we can immediately refute any already considered or yet-to-be-proposed alternative model with more predictive power than this (Quantum Theory). http://arxiv.org/pdf/1105.0133.pdf Can quantum theory be improved? - July 23, 2012 Excerpt: However, in the new paper, the physicists have experimentally demonstrated that there cannot exist any alternative theory that increases the predictive probability of quantum theory by more than 0.165, with the only assumption being that measurement (conscious observation) parameters can be chosen independently (free will assumption) of the other parameters of the theory.,,, ,, the experimental results provide the tightest constraints yet on alternatives to quantum theory.,,, http://phys.org/news/2012-07-quantum-theory.html
Now this is completely unheard of in science as far as I know. i.e. That a mathematical description of reality would advance to the point that one can actually perform an experiment showing that your current mathematical theory will not be exceeded in predictive power by another future mathematical theory is simply unprecedented in the history of science! It is, in my unsolicited opinion, a very significant milestone in the history of science! Moreover, the belief that 'maths just is' is simply completely demolished by the fact that our best mathematical description of reality is absolutely dependent on the starting assumptions of conscious observation and free will. Moreover, since our best mathematical description of reality requires conscious observation and free will as starting assumptions, then this necessarily implies that consciousness and free will precede the mathematical equation. Moreover, completely contrary to materialistic thinking, through conscious observation, free will, and operations of logic, a material particle can be reduced to information and instantaneously teleported:
Ions have been teleported successfully for the first time by two independent research groups Excerpt: In fact, copying isn't quite the right word for it. In order to reproduce the quantum state of one atom in a second atom, the original has to be destroyed. This is unavoidable - it is enforced by the laws of quantum mechanics, which stipulate that you can't 'clone' a quantum state. In principle, however, the 'copy' can be indistinguishable from the original (that was destroyed),,, http://www.rsc.org/chemistryworld/Issues/2004/October/beammeup.asp Atom takes a quantum leap - 2009 Excerpt: Ytterbium ions have been 'teleported' over a distance of a metre.,,, "What you're moving is information, not the actual atoms," says Chris Monroe, from the Joint Quantum Institute at the University of Maryland in College Park and an author of the paper. But as two particles of the same type differ only in their quantum states, the transfer of quantum information is equivalent to moving the first particle to the location of the second. http://www.freerepublic.com/focus/news/2171769/posts
The role of each conscious observer, and the (free will) choice(s) of each conscious observer, and the specific operations of logic, used to achieve quantum teleportation in the teleportation experiment are summarized on the following site:
Quantum Teleportation - A summary Excerpt: Assume that Alice and Bob share an entangled qubit ab. That is, Alice has one half, a, and Bob has the other half, b. Let c denote the qubit Alice wishes to transmit to Bob. Alice applies a unitary operation on the qubits ac and measures (i.e. consciously observes) the result to obtain two classical bits. In this process, the two qubits are destroyed. Bob's qubit, b, now contains information about c; however, the information is somewhat randomized. More specifically, Bob's qubit b is in one of four states uniformly chosen at random and Bob cannot obtain any information about c from his qubit. Alice provides her two measured classical bits, which indicate which of the four states Bob possesses. Bob applies a unitary transformation which depends on the classical bits he obtains from Alice, transforming his qubit into an identical re-creation of the qubit c.,,, http://en.wikipedia.org/wiki/Quantum_teleportation#A_summary summary of logical operations http://en.wikipedia.org/wiki/Quantum_teleportation#The_result
Jerad: The 'Spirituality' of Mathematics:
"Nothing in evolution can account for the soul of man. The difference between man and the other animals is unbridgeable. Mathematics is alone sufficient to prove in man the possession of a faculty unexistent in other creatures. Then you have music and the artistic faculty. No, the soul was a separate creation." Alfred Russell Wallace, New Thoughts on Evolution, 1910 (Co-Discoverer of Natural Selection) The Unreasonable Effectiveness of Mathematics in the Natural Sciences - Eugene Wigner - 1960 Excerpt: ,,certainly it is hard to believe that our reasoning power was brought, by Darwin's process of natural selection, to the perfection which it seems to possess.,,, It is difficult to avoid the impression that a miracle confronts us here, quite comparable in its striking nature to the miracle that the human mind can string a thousand arguments together without getting itself into contradictions, or to the two miracles of the existence of laws of nature and of the human mind's capacity to divine them.,,, The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning. http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html Mathematics and Physics – A Happy Coincidence? – William Lane Craig – video http://www.metacafe.com/w/9826382 The Applicability of Mathematics - William Lane Craig - article http://www.reasonablefaith.org/the-applicability-of-mathematics
This applicability of mathematics to describing reality is much more mysterious than Dr. Craig let on in that short video: To give some background, An atheist claimed, in response to my observation that mathematics must ultimately be based in God, that:
"maths just is"
Well, contrary to this commonly held belief that 'maths just is', the belief that 'maths just is' is now demonstrably false. First to be noted, and as Dr. Craig has pointed out, there is a profound epistemological mystery as to why our minds should even be able to grasp and understand reality through the enterprise of mathematics in the first place:
Epistemology – Why Should The Human Mind Even Be Able To Comprehend Reality? – Stephen Meyer - video – (Notes in description) http://vimeo.com/32145998 "You find it strange that I consider the comprehensibility of the world (to the extent that we are authorized to speak of such a comprehensibility) as a miracle or as an eternal mystery. Well, a priori, one should expect a chaotic world, which cannot be grasped by the mind in any way.. the kind of order created by Newton's theory of gravitation, for example, is wholly different. Even if a man proposes the axioms of the theory, the success of such a project presupposes a high degree of ordering of the objective world, and this could not be expected a priori. That is the 'miracle' which is constantly reinforced as our knowledge expands." Albert Einstein - Goldman - Letters to Solovine p 131.
Second, to reiterate, in the last century Godel showed mathematics to be 'incomplete':
Kurt Gödel – Incompleteness Theorem – video http://www.metacafe.com/w/8462821 THE GOD OF THE MATHEMATICIANS - DAVID P. GOLDMAN - August 2010 Excerpt: we cannot construct an ontology that makes God dispensable. Secularists can dismiss this as a mere exercise within predefined rules of the game of mathematical logic, but that is sour grapes, for it was the secular side that hoped to substitute logic for God in the first place. Gödel's critique of the continuum hypothesis has the same implication as his incompleteness theorems: Mathematics never will create the sort of closed system that sorts reality into neat boxes. http://www.firstthings.com/article/2010/07/the-god-of-the-mathematicians Taking God Out of the Equation – Biblical Worldview – by Ron Tagliapietra – January 1, 2012 Excerpt: Kurt Gödel (1906–1978) proved that no logical systems (if they include the counting numbers) can have all three of the following properties. 1. Validity . . . all conclusions are reached by valid reasoning. 2. Consistency . . . no conclusions contradict any other conclusions. 3. Completeness . . . all statements made in the system are either true or false. The details filled a book, but the basic concept was simple and elegant. He summed it up this way: “Anything you can draw a circle around cannot explain itself without referring to something outside the circle—something you have to assume but cannot prove.” For this reason, his proof is also called the Incompleteness Theorem. Kurt Gödel had dropped a bomb on the foundations of mathematics. Math could not play the role of God as infinite and autonomous. It was shocking, though, that logic could prove that mathematics could not be its own ultimate foundation. Christians should not have been surprised. The first two conditions are true about math: it is valid and consistent. But only God fulfills the third condition. Only He is complete and therefore self-dependent (autonomous). God alone is “all in all” (1 Corinthians 15:28), “the beginning and the end” (Revelation 22:13). God is the ultimate authority (Hebrews 6:13), and in Christ are hidden all the treasures of wisdom and knowledge (Colossians 2:3). http://www.answersingenesis.org/articles/am/v7/n1/equation#
In other words, the truthfulness of any given mathematical equation is not found within the equation itself, but the truthfulness of any given mathematical equation must ultimately be derived from a source outside of the equation(s). Moreover, being that mathematical equations are completely transcendent of any space-time constraints, (i.e. mathematical equations are always true no matter what part of the universe you are in and are true regardless of whatever year it happens to be in the universe), then this outside source (cause) that guarantees the truthfulness of any mathematical equation must also be transcendent of any space-time constraints. Also of note, Godel's incompleteness theorem is hardly the only line of argumentation for God in this line of thought:
Not Understanding Nothing – A review of A Universe from Nothing – Edward Feser - June 2012 Excerpt: A critic might reasonably question the arguments for a divine first cause of the cosmos. But to ask “What caused God?” misses the whole reason classical philosophers thought his existence necessary in the first place. So when physicist Lawrence Krauss begins his new book by suggesting that to ask “Who created the creator?” suffices to dispatch traditional philosophical theology, we know it isn’t going to end well. ,,, ,,, But Krauss simply can’t see the “difference between arguing in favor of an eternally existing creator versus an eternally existing universe without one.” The difference, as the reader of Aristotle or Aquinas knows, is that the universe changes while the unmoved mover does not, or, as the Neoplatonist can tell you, that the universe is made up of parts while its source is absolutely one; or, as Leibniz could tell you, that the universe is contingent and God absolutely necessary. There is thus a principled reason for regarding God rather than the universe as the terminus of explanation. http://www.firstthings.com/article/2012/05/not-understanding-nothing
But of more interest as to drawing out the 'spirituality of mathematics', and refuting the 'maths just is' conception of mathematics, it is worthwhile to focus in on the Schroedinger equation:
Finely Tuned Big Bang, Elvis In The Multiverse, and the Schroedinger Equation – Granville Sewell – audio http://www.metacafe.com/watch/4233012
At the 4:00 minute mark of the preceding audio, Dr. Sewell comments on the ‘transcendent’ and ‘constant’ Schroedinger’s Equation;
‘In chapter 2, I talk at some length on the Schroedinger Equation which is called the fundamental equation of chemistry. It’s the equation that governs the behavior of the basic atomic particles subject to the basic forces of physics. This equation is a partial differential equation with a complex valued solution. By complex valued I don’t mean complicated, I mean involving solutions that are complex numbers, a+bi, which is extraordinary that the governing equation, basic equation, of physics, of chemistry, is a partial differential equation with complex valued solutions. There is absolutely no reason why the basic particles should obey such a equation that I can think of except that it results in elements and chemical compounds with extremely rich and useful chemical properties. In fact I don’t think anyone familiar with quantum mechanics would believe that we’re ever going to find a reason why it should obey such an equation, they just do! So we have this basic, really elegant mathematical equation, partial differential equation, which is my field of expertise, that governs the most basic particles of nature and there is absolutely no reason why, anyone knows of, why it does, it just does. British physicist Sir James Jeans said “From the intrinsic evidence of His creation, the great architect of the universe begins to appear as a pure mathematician”, so God is a mathematician to’.
i.e. the Materialist is at a complete loss to explain why this should be so, whereas the Christian Theist presupposes such ‘transcendent’ control of our temporal, material, reality,,,
John 1:1 In the beginning was the Word, and the Word was with God, and the Word was God.
Of note: 'The Word' in Greek is Logos. Logos is the root word from which we derive our modern word 'logic'. bornagain77
Jerad, and how is it that Godel's incompleteness theorem is 'vaguely related' to showing that there must be a mind behind math? If the truthfulness of math does not arise from within itself, as Godel's incompleteness shows, pray tell where this truthfulness for math arises? Shall you posit 'randomness' as the basis of maths???,, You disparaging comment towards the incompleteness theorem reminds me of this cartoon,,, Calvin and Hobbes - cartoon - The Mathematical Atheist http://s3.hubimg.com/u/270622_f520.jpg And Jerad, how will me "really understand(ing) the math" more solve the dilemma that maths cannot form its own basis in reality to explain the truthfulness therein? Mathematics is the language with which God has written the universe. Galileo Galilei Jerad, from what I can gather, you may insist that math itself, or worse yet, randomness, can pull off the ultimate 'pull yourself up from your own bootstraps' program in history of science for explaining the mathematical foundation we find for reality, but I don't have to hold you to be the least bit rational in your belief(s) for holding as such! Notes: The Underlying Mathematical Foundation Of The Universe - Walter Bradley - video http://www.metacafe.com/watch/4491491 Quote from preceding video: “Occasionally I’ll have a bright engineering student who says, “Well you should see the equations we work with in my engineering class. They’re a big mess.”, The problem is not the fundamental laws of nature, the problem is the boundary conditions. If you choose complicated boundary conditions then the solutions to these equations will in fact, in some cases, be quite complicated in form,,, But again the point is still the same, the universe assumes a remarkably simple and elegant mathematical form.” – Dr. Walter Bradley Evidence for an Engineered Universe - Walter Bradley - July 2012 - video http://www.youtube.com/watch?v=nLd_cPfysrE How the Recent Discoveries Support a Designed Universe - Dr. Walter L. Bradley - paper http://www.leaderu.com/real/ri9403/evidence.html The Five Foundational Equations of the Universe and Brief Descriptions of Each: http://docs.google.com/Doc?docid=0AYmaSrBPNEmGZGM4ejY3d3pfNDdnc3E4bmhkZg&hl=en Why Mathematics Works, part 1 - James Nickel - video http://www.youtube.com/watch?v=r1YssV8qi-w bornagain77
BA77: Interesting, though I cannot follow up now, I am in the terror fitted depths of articles and bylaws. KF kairosfocus
Jerad: As you will know, having followed the discussion above and in previous posts, the pivotal question being explored for now is the pervasive pattern of the role of mathematical principles across the physical sciences. That is, we see a pervasiveness of rationality, logic, structure, profound coherence, symmetry and simplicity leading to elegant beauty and intelligibility in the world, a pattern that is positively eerie in cumulative impact and has always excited wonder among many who study mathematics and physical sciences. (Here, I assume that you will agree that logical study of structure, pattern and quantity is an essential feature of mathematics. Indeed, that can serve as a "family resemblance" stand-in for the usual problem that here is no one generally accepted precising definition.) Here is a clip from Dirac on just how significant this pattern can be:
I think that there is a moral to this story, namely that it is more important to have beauty in one's equations that to have them fit experiment. If Schrödinger had been more confident of his work, he could have published it some months earlier, and he could have published a more accurate equation. It seems that if one is working from the point of view of getting beauty in one's equations, and if one has really a sound insight, one is on a sure line of progress. If there is not complete agreement between the results of one's work and experiment, one should not allow oneself to be too discouraged, because the discrepancy may well be due to minor features that are not properly taken into account and that will get cleared up with further development of the theory. Scientific American, May 1963.
In short, one should be led by the principles of order, pattern and elegant but profound, logically sensible simplicity. (An excellent study on this would be the principle of least action, which seems to constrain ever so much in the laws of physics.) Coherence and pervasiveness of organising principles has long been known to be a signature of designing mind at work. So, it is a reasonable question to raise per inference to best explanation at worldviews level, that the mathematical elegance and coherence of the cosmos points to designing mind behind it, especially in a context where one can -- per reasonable principles -- draw out from a set that collects nothing to the natural numbers, thence the continuum, thence space in 3 dimensions, thence change in time through rotation, translation, oscillation etc, thence kinematics, thence vial force, inertia and energy, dynamics, thence a pattern for physical reality. (Which is the step by step process I have taken up.) KF PS: With certain relevant pieces of paper "on the wall" so to speak as well as experience in using the stuff behind such, I have no need whatsoever to prove to you or any other person, that I understand Mathematics. I suggest, rather that you need to think again on the pattern of behaviour that has led you to project such dismissiveness once your apparent commitment to or at least enabling of an ultimately incoherent view of reality has come up. kairosfocus
Uh huh. I tell you what, can you explain to me how the truth (or falsehood) of the Axiom of Choice points to a mind behind the math? Without just posting a lot of links of other people talking about vaguely related topics. Or how about the Goldbach Conjecture? Show me you understand the mathematics and then show me how there has to be a mind back there somewhere. If you just posts comments by others they you're just taking their word for it. Do you really understand the math? Jerad
KF: This is a very interesting developement. Dr. Kirk Durston, in this reply to an old critique of PZ Myers, mentions a method that he, and others, developed to determine if a amino acid site in a protein sequence is interdependent with other sites in the protein sequence.
(A Reply To PZ Myers) Estimating the Probability of Functional Biological Proteins? Kirk Durston , Ph.D. Biophysics - 2012 Excerpt: (Page 4) The Probabilities Get Worse This measure of functional information (for the RecA protein) is good as a first pass estimate, but the situation is actually far worse for an evolutionary search. In the method described above and as noted in our paper, each site in an amino acid protein sequence is assumed to be independent of all other sites in the sequence. In reality, we know that this is not the case. There are numerous sites in the sequence that are mutually interdependent with other sites somewhere else in the sequence. A more recent paper shows how these interdependencies can be located within multiple sequence alignments.[6] These interdependencies greatly reduce the number of possible functional protein sequences by many orders of magnitude which, in turn, reduce the probabilities by many orders of magnitude as well. In other words, the numbers we obtained for RecA above are exceedingly generous; the actual situation is far worse for an evolutionary search. http://powertochange.com/wp-content/uploads/2012/11/Devious-Distortions.pdf
And here is the paper from Durston and company:
Statistical discovery of site inter-dependencies in sub-molecular hierarchical protein structuring - Kirk K Durston, David KY Chiu, Andrew KC Wong and Gary CL Li - 2012 Results The k-modes site clustering algorithm we developed maximizes the intra-group interdependencies based on a normalized mutual information measure. The clusters formed correspond to sub-structural components or binding and interface locations. Applying this data-directed method to the ubiquitin and transthyretin protein family multiple sequence alignments as a test bed, we located numerous interesting associations of interdependent sites. These clusters were then arranged into cluster tree diagrams which revealed four structural sub-domains within the single domain structure of ubiquitin and a single large sub-domain within transthyretin associated with the interface among transthyretin monomers. In addition, several clusters of mutually interdependent sites were discovered for each protein family, each of which appear to play an important role in the molecular structure and/or function. Conclusions Our results demonstrate that the method we present here using a k-modes site clustering algorithm based on interdependency evaluation among sites obtained from a sequence alignment of homologous proteins can provide significant insights into the complex, hierarchical inter-residue structural relationships within the 3D structure of a protein family. http://bsb.eurasipjournals.com/content/2012/1/8
footnotes: Here is the original video from Dr. Durston that PZ Myers had taken a swipe at:
Mathematically Defining Functional Information In Molecular Biology - Kirk Durston - short video http://www.metacafe.com/watch/3995236 Entire video: http://vimeo.com/1775160
also of note:
Measuring the functional sequence complexity of proteins - Kirk K Durston, David KY Chiu, David L Abel and Jack T Trevors - 2007 Excerpt: We have extended Shannon uncertainty by incorporating the data variable with a functionality variable. The resulting measured unit, which we call Functional bit (Fit), is calculated from the sequence data jointly with the defined functionality variable. To demonstrate the relevance to functional bioinformatics, a method to measure functional sequence complexity was developed and applied to 35 protein families.,,, http://www.tbiomed.com/content/4/1/47
isn’t actually very closely equivalent to ‘P and/or not-Q’
Whoops - that should have read 'not-P and/or Q' of course. anonym
ok, which button on my calculator is that again? The pretty blue one? Mung
Leonard Euler, if he is looking down on all this from heaven, probably has a wry little smile and a twinkle in his eye about the euler formula having a helix correlation to DNA!
BA77 I like that picture. The great man suffered horribly from disease and botched operations in his mortal eyes. It's nice to think of him now with eyes shining and twinkling and gifted with heavenly visions that we can't begin to imagine. On another topic, is there a term in logic and argumentation for this tactic that materialists often use? It's a kind of no-big-dealism that attempts to dismiss and trivialize the most magnificent things seemingly for fear their worldview might be undermined. e.g. what's the big deal about the physical constants of the universe. There just a bunch of numbers. Numbers are so trivial - look how easily I can generate them on my pocket calculator. :-) steveO
Semi OT: Is Metaphysical Naturalism Viable? - William Lane Craig - video http://www.youtube.com/watch?v=HzS_CQnmoLQ bornagain77
Jerad as to:
how it points to a mind behind the math escapes. me.
Well Jerad, you are in good (or bad) company.
BRUCE GORDON: Hawking’s irrational arguments – October 2010 Excerpt: Rather, the transcendent reality on which our universe depends must be something that can exhibit agency – a mind that can choose among the infinite variety of mathematical descriptions and bring into existence a reality that corresponds to a consistent subset of them. This is what “breathes fire into the equations and makes a universe for them to describe.” Anything else invokes random miracles as an explanatory principle and spells the end of scientific rationality.,,, Universes do not “spontaneously create” on the basis of abstract mathematical descriptions, nor does the fantasy of a limitless multiverse trump the explanatory power of transcendent intelligent design. What Mr. Hawking’s contrary assertions show is that mathematical savants can sometimes be metaphysical simpletons. Caveat emptor. http://www.washingtontimes.com/news/2010/oct/1/hawking-irrational-arguments/
Kurt Gödel - Incompleteness Theorem - video http://www.metacafe.com/w/8462821
i.e. the 'incompleteness theorem' shows that the 'truthfulness' of any mathematical equation is not held within the equation itself but must be dependent on an 'outside agent' (i.e. on God) to derive its ultimate truthfulness:
Godel and Physics - John D. Barrow Excerpt (page 5-6): "Clearly then no scientific cosmology, which of necessity must be highly mathematical, can have its proof of consistency within itself as far as mathematics go. In absence of such consistency, all mathematical models, all theories of elementary particles, including the theory of quarks and gluons...fall inherently short of being that theory which shows in virtue of its a priori truth that the world can only be what it is and nothing else. This is true even if the theory happened to account for perfect accuracy for all phenomena of the physical world known at a particular time." Stanley Jaki - Cosmos and Creator - 1980, pg. 49 http://arxiv.org/pdf/physics/0612253.pdf
i.e. Logic dictates 'a decision' must have been made in order to purposely create a temporal reality with highly specified, irreducible complex, parameters from a infinite set of mathematical possibilities. Thus the transcendent (timeless, spaceless, massless) reality from which our temporal material reality sprang (and is dependent upon), is shown to be alive by yet another line of evidence besides the necessity for a 'conscious first mover’ to explain quantum wave collapse to each unique point of conscious observation in the universe.
The First Cause Must Be A Personal Being - William Lane Craig - video http://www.metacafe.com/w/4813914 What Necessary Properties Must the Cause of the Universe Have? - William Lane Craig - video http://www.youtube.com/watch?v=1SZWInkDIVI
Supplemental notes:
Alan Turing and Kurt Godel - Incompleteness Theorem and Human Intuition - video (notes in video description) http://www.metacafe.com/watch/8516356/ Are Humans merely Turing Machines? https://docs.google.com/document/d/1cvQeiN7DqBC0Z3PG6wo5N5qbsGGI3YliVBKwf7yJ_RU/edit The 'Spirituality' of Mathematics: https://docs.google.com/document/d/13VBciybSK3D7uJoz6ltldPPSvhL4HJaJAmCmOMkmQxg/edit
Verse and music:
John1:1 In the beginning was the Word, and the Word was with God, and the Word was God. Sara Groves - The Word - music video http://www.youtube.com/watch?v=0ofE-GZ8zTU
You always seem so impressed with Euler's Identity, a special case of e^i*theta. But it just comes out of the Taylor series expansions of e^x, sinx and coax. I think it's 'lovely' as well but how it points to a mind behind the math escapes. me. Jerad
steveO: Leonard Euler, if he is looking down on all this from heaven, probably has a wry little smile and a twinkle in his eye about the euler formula having a helix correlation to DNA! of related note: Alexander Vilenkin, who was the one who dropped a mathematical bomb on Hawking's 70th birthday party, comments on the beauty of mathematics being ideally suited for describing our physical universe (particularly e^ipi+1=0)
Quote: "It appears that the Creator shares the mathematicians' sense of beauty." - Alexander Vilenkin http://rfforum.websitetoolbox.com/post?id=3754268
,,,I find it extremely strange that the enigmatic Euler's identity, which was deduced centuries ago, would find such striking correlation to how reality is actually found to be structured by modern science. In pi we have correlation to the 'sphere of the universe' as revealed by the Cosmic Background radiation, as well pi correlates to the finely-tuned 'geometric flatness' within the 'sphere of the universe' that has now been found. In 'e' we have the fundamental constant that is used for ascertaining exponential growth in math that strongly correlates to the fact that space-time is 'expanding/growing equally' in all places of the universe. In the square root of -1 we have what is termed a 'imaginary number', which was first proposed to help solve equations like x2+ 1 = 0 back in the 17th century, yet now, as Michael Denton pointed out in the preceding video, it is found that the square root of -1 is required to explain the behavior of quantum mechanics in this universe. The correlation of Euler's identity, to the foundational characteristics of how this universe is constructed and operates, points overwhelmingly to a transcendent Intelligence, with a capital I, which created this universe! It should also be noted that these mathematical constants, pi,e, and square root -1, were at first thought by many to be completely transcendent of any material basis, to find that these transcendent constants of Euler's identity in fact 'govern' material reality, in such a foundational way, should be enough to send shivers down any mathematicians spine. footnotes:
in the equation e^pi*i + 1 = 0 ,,,we find that pi is required in; General Relativity (Einstein’s Equation) https://docs.google.com/File?id=dc8z67wz_52c9nxpz2h_b ,,,and we also find that the square root of negative 1 is required in; Quantum Mechanics (Schrödinger’s Equations) https://docs.google.com/File?id=dc8z67wz_51ck47zff3_b ,,and we also find that e is required for; e is required here in wave equations, in finding the distribution of prime numbers, in electrical theory, and is also found to be foundational to trigonometry.,,,this number, e, also appears in banking, because it is the maximum limit for growth of compound interest. Some of the various uses and equations of 'e' are listed at the bottom of the following page: http://www.biblemaths.com/pag03_pie/img0.gif
Notes of Interest
“All the evidence we have says that the universe had a beginning.” - Cosmologist Alexander Vilenkin of Tufts University in Boston - paper delivered at Stephen Hawking's 70th birthday party https://uncommondesc.wpengine.com/intelligent-design/vilenkins-verdict-all-the-evidence-we-have-says-that-the-universe-had-a-beginning/
Of related interest to Vilenkin dropping a 'mathematical bomb' on Hawking's 70th birthday party, it seems that Einstein had a similar experience on his 70th birthday:
THE GOD OF THE MATHEMATICIANS - DAVID P. GOLDMAN - August 2010 Excerpt: Gödel's personal God is under no obligation to behave in a predictable orderly fashion, and Gödel produced what may be the most damaging critique of general relativity. In a Festschrift, (a book honoring Einstein), for Einstein's seventieth birthday in 1949, Gödel demonstrated the possibility of a special case in which, as Palle Yourgrau described the result, "the large-scale geometry of the world is so warped that there exist space-time curves that bend back on themselves so far that they close; that is, they return to their starting point." This means that "a highly accelerated spaceship journey along such a closed path, or world line, could only be described as time travel." In fact, "Gödel worked out the length and time for the journey, as well as the exact speed and fuel requirements." Gödel, of course, did not actually believe in time travel, but he understood his paper to undermine the Einsteinian worldview from within.
And Godel has indeed been vindicated in his 'intuition':
Physicists continue work to abolish time as fourth dimension of space - April 2012 Excerpt: "Our research confirms Gödel's vision: time is not a physical dimension of space through which one could travel into the past or future." http://phys.org/news/2012-04-physicists-abolish-fourth-dimension-space.html
Of interest to this undermining of General Relativity as the complete description of reality, Einstein was asked (by a philosopher):
"Can physics demonstrate the existence of 'the now' in order to make the notion of 'now' into a scientifically valid term?"
Einstein's answer was categorical, he said:
"The experience of 'the now' cannot be turned into an object of physical measurement, it can never be a part of physics."
Quote was taken from the last few minutes of this following video:
Stanley L. Jaki: "The Mind and Its Now" https://vimeo.com/10588094
The preceding statement was an interesting statement for Einstein to make since 'the now of the mind' has, from many recent experiments in quantum mechanics, completely undermined Einstein's General Relativity as to being the absolute/primary frame of reference for reality (even though it is verified to something like 13 decimal places of stunning accuracy; Berlinski). i.e. 'the now of the mind', contrary to what Einstein thought possible for experimental physics, and according to advances in quantum mechanics, takes precedence over past events in time. Moreover, due to advances in quantum mechanics, it would now be much more appropriate to phrase Einstein's answer to the philosopher in this way:
"It is impossible for the experience of 'the now of mind' to be divorced from physical measurement, it will always be a part of physics."
Supplemental note:
LIVING IN A QUANTUM WORLD - Vlatko Vedral - 2011 Excerpt: Thus, the fact that quantum mechanics applies on all scales forces us to confront the theory’s deepest mysteries. We cannot simply write them off as mere details that matter only on the very smallest scales. For instance, space and time are two of the most fundamental classical concepts, but according to quantum mechanics they are secondary. The entanglements are primary. They interconnect quantum systems without reference to space and time. If there were a dividing line between the quantum and the classical worlds, we could use the space and time of the classical world to provide a framework for describing quantum processes. But without such a dividing line—and, indeed, with­out a truly classical world—we lose this framework. We must ex­plain space and time (4D space-time) as somehow emerging from fundamental­ly spaceless and timeless physics. http://phy.ntnu.edu.tw/~chchang/Notes10b/0611038.pdf
Music and verse:
Kari Jobe - Revelation Song - Passion 2013 http://www.youtube.com/watch?v=3dZMBrGGmeE Hebrews 4:13 "And there is no creature hidden from His sight, but all things are naked and open to the eyes of Him to Whom we must give account."
CR, Thanks for expanding so well on what I meant in my first post which was sent " rushed mobile" and resulted in a streak of grammatical and spelling issues :-) KF, I imagine Euler would very much approve of the "Signature of God". He was a man of deep faith who bore many personal difficulties along with (no surprise) the ridicule of some of the high profile atheists of his time. steveO
F/N: Some reading on classical and non classical logic. kairosfocus
CR & Bateman: Rational intuition and creativity coming from an active mind, turn out to be ever so important in reasoning, and it is crucial for it to go beyond algorithmic limitations. Also, the astonishing powers of e and i put together, have transformed analysis of our world. That we see as a drop-out of the e^iwt = cis wt expression, Euler's identity, is a case of getting magic wine out of a chalice, echoing I forget who now. Bang, from a derivation a sixth former or freshman college student can follow, we find an unexpected gift and signature: the world of math, the things we thought we came up with as imaginary constructs and useful values, out of the blue all come together in one. Do we need more of a signature of a rational mind behind reality? I hereby propose that we add to the old Scutum Fidei, Euler's equation as the signature of God! (And, I am only half joking.) KF kairosfocus
Anonym: Material implication captures a key part of implication. Yes, there are paradoxes. What is crucial is, that it turns out that WHY p => q, is important in the world of reasoning. That is what takes skeletal implication logic per true/false inputs and true false outputs, and turns it into a useful tool of insightful inquiry by an active, knowing, reasoning mind that carries background knowledge and rational intuition and insight into the situation. It is not merely plugging in truth values into a table. It is also key to see that ex falso quodlibet -- the principle of explosion -- is routinely employed in modelling theory. Models are generally speaking SIMPLIFICATIONS of reality. That is, they are false, known to be false. But for known, tested zones of validity, they provide accurate results. This is usually put in terms of capturing sufficient of the truth to be useful in that zone. A further implication of this, is that scientific theories, strictly are to be regarded as explanatory models, albeit ones we try to make as accurate as we can. That is also why they are to be held provisional. (Which is of course, why those who wish to pretend that the theory of evolution on the grand scale is practically certain and even fact, are barking up the wrong tree.) But yes, we need to stir it up, stir, stir, stir the pot. KF F/N: This 101 skim on non classical Logics may be helpful. Taking Fuzzy as a starter, what6 happens is that this marks a case where the crucial importance of distinctness of identity is pivotal to the classic laws of thought. Those laws do remain as first principles of right reason, once we have to be distinct regarding identity. Paul's remark on what happens if the trumpet gives an uncertain sound (on the battlefield) turns out to be crucially insightful, one of those little gems in the now too often brushed aside NT. Look a little closer at fuzzy sets and you will see that partial set memberships depending on fuzzy broad borders to zones of membership, depends at a deeper level on the use of these same basic first principles, in order for its reasoning to have stable outcomes. kairosfocus
That's only the beginning of the trouble with formal reasoning. There's also the fun times to be had trying to formalise 'if P then Q' accurately. Turns out that statement isn't actually very closely equivalent to 'P and/or not-Q' at all. Or the semantic paradoxes - Goedel's theorem is not nearly the end of the problems with self-reference, though in fact those paradoxes challenge our intuitions about truth and reasoning themselves, not just our efforts to formalise those intuitions. Graham Priest's Introduction to Non-Classical Logic is a good book, pretty accessible if you're willing to get out the pencil and paper. anonym
e to the i theta It sort of sounds like nerd rap. SteveO, yep Euler's identity is a special case of Euler's formula. e^(ix) = cosx + isinx When x is equal to pi, e^(iπ) = cosπ + isinπ = -1 + 0 e^(iπ) + 1 = 0 The real and imaginary parts of Euler's formula create that double-helix look on the graph. :) BA77, nice link, good info. Thanks for that. Chance Ratcliff
H'mm: the helix works best on x = w*t, i.e. we see time and circular frequency. KF kairosfocus
SteveO that is just plain spooky! The following images show the graph of the complex exponential function, complex exponential function, e^{ix}, by plotting the Taylor series of e^{ix} in the 3D complex space http://www.songho.ca/math/euler/euler.html bornagain77
The narrator mentioned intuition as an important ability that exists outside of mathematics, logic and reason. This reminded me greatly of my short look into quantum physics. Do we know what we think we know? Does this writing sound like gibberish? Good lord, I don't know anymore... Thanks for stirring the pot, kf. Bateman
Another property I find interesting is the close relationship to the Euler formula which when plotted in 3D results in the fundemental geometry of life: a helix ! steveO
F/N: Sorry on embed vid problems. They can be seen at the linked. KF kairosfocus

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