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I'm an outsider so forgive me if my comments seem naive. It is clear you all know details that I do not know, yet great thinkers often ignore details because it can lead you astray (e.g. "Mathematical Discovery: Hadamard Resurected). For example, Einstein didn't know more details than his peers, but he did question their assumptions.

Did relativity and quantum theory evolve from classical...

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I'm an outsider so forgive me if my comments seem naive. It is clear you all know details that I do not know, yet great thinkers often ignore details because it can lead you astray (e.g. "Mathematical Discovery: Hadamard Resurected). For example, Einstein didn't know more details than his peers, but he did question their assumptions.

Did relativity and quantum theory evolve from classical physics? That would imply that modern physics is a special case of classical physics, starting with the same axioms but more refined. This is obviously false. It is clear that classical physics is actually a special case of modern physics. Put another way, for any of you who have any interest in evolutionary theory, classical physics was a thought "dinosaur". And just like real dinosaurs that were unlikely to evolve into the current ecosystem of Earth, modern physics needed to evolve from new axioms, not directly from classical physics. So I question the approach of trying to extend relativity and quantum theory to explain the as yet unexplained. In other words, I'm not sure dinosaurs will ever evolve into humans.

So while I can't point to an answer, I can point to some assumptions that might be wrong. First of all, the problem of consciousness is obviously one we've inherited from culture and we can't solve it because it isn't really a problem. Many philosophers have gotten rich calling it a problem, but there is no evidence of any problem beyond the fact that unless you and I are the same consciousness, we won't ever know the the other is conscious. It is wreckless to mix in the notion of consciousness in a forum discussing quantum physics, for one will never help the other just as you can't distinguish two entangled particles until they are unentangled.

Most importantly, there is no evidence of this thing called time. Until we can all admit this simple fact, we will continue to be trapped by our culture and human form into concepts like "evolve", "measure", etc. Comparing one state to another requires memory, which requires observing an observation of an observation. As Bruno nicely pointed out "we don't know and cannot know which computational history support us." Making a measurement requires change. Change itself cannot be observed. Therefore when I say something changed then I mean that what I most recently observed is different from a memory of what that thing was like. We have no way of knowing (with our current understanding) if in fact all possible things happen from one instant to the next. Or if our Universe was just created from a computer simulation in its current state, including my own memories of the "past". This assumption that what we observe as time is actually real is a major assumption, and you may not want to question it. Many did not want to question absolute time when Einstein proposed relativity. It complicated the Universe a whole lot. But the Universe made more sense for it. So is anyone willing to question this assumption and see if any predictable features of our observed Universe can be made?

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Garrett, I guess this is why I have become so disappointed in scientists, because they never want to discuss anything off the beaten path. They are always so afraid of looking unprofessional or unscientific or unintelligent. It is as if we live in a brutal dictatorship and everyone is afraid to discuss freedom. Consequently you can’t discuss anything unique with scientists, no matter how...

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Garrett, I guess this is why I have become so disappointed in scientists, because they never want to discuss anything off the beaten path. They are always so afraid of looking unprofessional or unscientific or unintelligent. It is as if we live in a brutal dictatorship and everyone is afraid to discuss freedom. Consequently you can’t discuss anything unique with scientists, no matter how “clever”. Here we are at a website founded on fundamental questions in cosmology. You yourself asked a question looking to get feedback on an unexplored issue but the talk immediately reverts to the safety of the familiar. There needs to be a revolution from within.

Lisi wrote:

“a classical observer making an observation of a system that may inhabit one of a number of possibilities.”

Isn’t a classical system deterministic, so there is no such thing as a classical system that may inhabit one of a number of possibilities. The classical observer only assigns probabilities to events he/she has limited information about. Knowing every property of an actual coin flip, the classical observer knows the precise outcome. In principle the information is available. Contrastingly the quantum observer cannot know all the properties of a quantum event. The information isn’t localized. So the roles are not the same. They are only similar if the classical observer lacks information and then the similarity is that they both lack information. In lacking information they both can calculate the chance of say winning the lottery, but in doing so the classical observer assumes a measure of uncertainty that doesn’t exist in his/her classical world.

If we had complete information of a classical universe would we still question if other universes were possible, since complete information would predict only the one universe. Perhaps there is a fundamental flaw in the notion that complete information can predict a single outcome (Godel?).

However, I do realize you are pointing to something. I have wondered if quantum mechanics is somehow tied to the quantum observer’s lack of information, a mistaken theory resulting from our role in reality as observers. Of course we can’t say where the particle is if we don’t measure or observe it. But beyond this surface thought I sense you've brought up something deeper, something basic concerning predictability and uncertainty.

The classical observer’s ability to calculate probabilities based on limited information is due to how limited information predicts more than one possible outcome. Any limited information about a system naturally predicts a set of possible outcomes, many of which are equally possible (coin flip). Likewise in a quantum event, the lack of local information (the measurement problem) predicts a set of possible outcomes, many of which are equally possible. I have had this insight before but never clear enough to record it, but I think this is the fundamental reason why uncertainty exists in nature, or in time, because there are equally probable outcomes, in the atomic world, as well as for the observer, (viewing Schrodinger’s cat), and equally probable outcomes interfere with one another (heads and tails), in human calculations and quantum events.

Imagine walking down a stairs but instead of one step below there are many. Why? Because there are many different steps that could proceed from the step you are standing on. In the range of possibilities, there are many next steps that are as equally probable or possible. This is how nature works, and uncertainty exists in time, because a past state cannot lead to one single future state. Time cannot be linear because the information in past states is by nature unable to determine a single specific future state. The present configuration of the universe cannot ignore all the equally possible futures and dictate a single future, because the equally possible futures are equally real (and our universe is a sampling of that reality).

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There are three fundamental and interrelated relationships of reference for perspective. The three fundamental relationships of reference for perspective are the relationship of the dimensions of Cartesian coordinates with the dimensions of polar coordinates, the relationship of finite with the infinite, and the relationship of 'still' with motion (or constant motion with accelerated motion). Each...

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There are three fundamental and interrelated relationships of reference for perspective. The three fundamental relationships of reference for perspective are the relationship of the dimensions of Cartesian coordinates with the dimensions of polar coordinates, the relationship of finite with the infinite, and the relationship of 'still' with motion (or constant motion with accelerated motion). Each of these three fundamental relationships are not, in-and-of themselves, a 'given' (that is, not an independent and necessary foundation of physics), but rather, the result of the ‘overlapping’, or exchange, of the designated parameters of the other two. And so, it is the relationships of fundamental references that define tiered perspective (that is, a perspective co-existing within an ‘Other’ overriding perspective) and perspective exchange (that is, consideration, or ‘thought’) as the foundation for all perception (physical observations and symbolic considerations). That is, there is no 'point' of origin, but instead, a relationship determines existence and experience through tiered interrelated combinations of fundamental references into perception ternaries. In other words, various tiers, or ternary combinations, of the constituent references of the three fundamental relationships represent all perspectives and perspective exchanges. The three fundamental relationships are not completely independent of each other, but rather, are interrelated and overlapping (like interacting ‘whirlpools’), in one or two of the three constituent references comprising the involved respective perception ternaries. So, every ‘object’, number, and experience, every physical and mathematical observation and consideration, is represented by overlapping ternary combinations of the constituent references within the three fundamental relationships of reference. And the relationship of tiered combinations forms the perception ternary—the perceived, the perceiver, and the reference of measurement. The model/metaphor is a self-proving totality that is entirely perspective-based, and the necessitated foundation for existence that results from the tiered-ternary relationships of fundamental references, providing its own inter-exchanging context and its own reciprocating cause.

This presents a physics/mathematics continuum, or overlapping exchange, that is based upon the binary relationship of ‘within’ and ‘without’, representing the one-or-two-of-three of the constituent references in a perception ternary. That is, the overlapping of one, or the overlapping of two of the constituent references in the various constituent tiers, or ternary combinations, determines the relationship as ‘without’ or ‘within’, respectively. Thus, in physics (or ‘without’—reference paired with perceived in perception ternary), the three fundamental relationships of reference form, by perspective, the three fundamental overlapping and interrelated concepts of distance, time/speed, and mass/acceleration; and in mathematics (or ‘within’—the symbolic pairs the reference with the perceiver in the perception ternary), the three fundamental relationships of reference form, by perspective, the three overlapping and interrelated fundamental ‘constants’ which divide, or provide the exchange for, the three fundamental concepts of physics—namely, epsilon, pi, and phi (the golden ratio). That is, epsilon represents mathematically the divide, or reference of exchange, between distance and speed, pi represents mathematically the divide, or reference of exchange, between linear distance and rotational distance, and phi represents mathematically the divide, or reference of exchange, between linear distance and area (dimensional shift). Further, every ‘physical’ observation is symbolic in one-of-three of the perception ternary; and conversely, every ‘mathematical’ consideration is physical (more truly, of other tiered consideration) in one-of-three of its analogous perception ternary. So, mathematics, whether as numerical values, operations, or their corresponding geometry, is the result of tiered co-perception (or, tiered perspective and perspective exchange), just as with physics, with overlapping and exchanging references ‘within’ and ‘without’, so that each completes the other.

In this way, all values can therefore be defined completely only through the multiple and interrelated contexts (references) of tiered perspectives, thereby representing a ‘number’ (or perception ternary) not as an exact, or ‘given’, absolute, but rather in relation to another perception ternary which is separated by one-or-two of their constituent fundamental references—for instance, relating, as a perceived proportion, a fundamental concept like that of time or dimensions to distance. So, ‘value’ and ‘path’ (or, act of calculation) ‘trade places’ (exchange) and are defined in the same way that ‘observer’ and ‘object’ do in their physical correlation. That is, the ‘constants’ (constant only by co-perspective), pi, phi, and epsilon, as well as the exchanges between them, represented by the polar and Cartesian forms of complex numbers (‘i’), frame the foundation for the mathematical correlate of perspective and perspective exchange. And, as with the three fundamental concepts of physics, these values are overlapping and interrelated.

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"an aura of mystical speculation surrounding quantum mechanics and the role of consciousness."

I think the mystical interpretation sends a cold shiver down the spine of most skeptical scientists. For all intents and purposes Dieter Zeh and decoherence have resolved the measurement problem. I'm content to leave the ontology a mystery, but I think entropy may provide a reasonable explanation...

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"an aura of mystical speculation surrounding quantum mechanics and the role of consciousness."

I think the mystical interpretation sends a cold shiver down the spine of most skeptical scientists. For all intents and purposes Dieter Zeh and decoherence have resolved the measurement problem. I'm content to leave the ontology a mystery, but I think entropy may provide a reasonable explanation rather than the many worlds interpretation or the role of “consciousness”.

"The role of the observer making an observation in quantum mechanics is the same as that of a classical observer making an observation of a system that may inhabit one of a number of possibilities. You look at it, and then you know what happened -- until you look at it, you don't know and can only assign probabilities to what may have happened."

"The weirdness of quantum mechanics is that the possibilities interfere with one another, whereas classically they don't."

Classically probabilities do interfere. For instance, in a system of Einstein solids the probability of the system's overall macrostate depends on the microstates of each individual solid. In fact, when they are combined the macrostate they form is overwhelmingly probable even for a small amount of energy and small number of oscillators. The probability distributions are wavelike Gaussians that do interfere with each other.

Dieter Zeh describes the Copenhagen interpretation as a non-interpretation. My belief is that the imaginary number in the exponential time evolution of the wavefunction

exp(iEt/hbar)is why it is considered a non-interpretation. However, I'm not convinced that the imaginary number belongs in the wavefunction's time dependent evolution.

I agree with Dieter Zeh that most of those who dislike QM or are confused by it believe that the probabilities represent incomplete knowledge. The incomplete knowledge is wrong quantum mechanics is fundamentally probabilistic there are no hidden variables that will change it's probabilistic nature. However, I do not agree that, "Heisenberg's original hope that the quantum system was disturbed during the measurement is not tenable." I think entanglement and Heisenberg's original hope are perfectly compatible.

This is the problem, the cat is alive or dead when measured, but quantum tells us it is a superposition of both at the same time until measured.

In other words flip a coin and the possibilities are H or T and quantum says it is both heads and tails at the same time.

A possible solution is that what is observed after the collapse is the systems macrostate. For instance with three coins the possibilities are:

TTT = 0 Heads macrostate | probability 1/8 | multiplicity 1

TTH = 1 Heads macrostate | probability 3/8 | multiplicity 3

THH = 2 Heads macrostate | probability 3/8 | multiplicity 3

HHH = 3 Heads macrostate | probability 1/8 | multiplicity 1

If a measurement were to change the microstates, introducing one thousand coins into the system, then the overall macrostate is overwhelmingly likely to be a 1/2 heads macrostate and the no heads macrostate has a probability of 1/2^1000.

I used these concepts from entropy to give a possible alternative to the MWI and consciousness explanations for the collapse. I used the derangement from combinatorics to produce a real wavefunction time dependence as well as real probability amplitudes and eigenstates for any state |PSI>. I did not have to use the Hilbert space to get to the probability space i.e. without requiring the inner product first . Wavefunctions can still entangle and the exp(-a) dependence still "drops out" in a way analogous to using the complex conjugate during a "measurement". I would be interested in getting a physics beating by the experts or settle for their encouragement.

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Dear Brian,

let me briefly expose what i meant with what you call a big idea (it isn't so much a big idea in my opinion, merely a discrepancy between two very sucessful theories):

The evolutionary theory in its neo-darwinian form assumes the arising of consciousness as an "epiphenomenon", not as a phenomenon "in its own right". Evolutionary theory, as it is widely accepted, explains...

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Dear Brian,

let me briefly expose what i meant with what you call a big idea (it isn't so much a big idea in my opinion, merely a discrepancy between two very sucessful theories):

The evolutionary theory in its neo-darwinian form assumes the arising of consciousness as an "epiphenomenon", not as a phenomenon "in its own right". Evolutionary theory, as it is widely accepted, explains all biological properties to be selected due to their impact for the sucessul survival and reproduction. It is assumed that the emerging of the biological property of consciousness has an impact of being able to decrease exact these reproduction chances. Not by intelligence being a sexually selected trait (what is also true but not the whole story concerning the explanation of consciousness in evolutionary theory), but by intelligence as being capable of avoiding certain situations that could lead to the organism's dead. The more frequently an organism can avoid its own dead by grasping a dangerous situation in its mind and reacting sucessfully to that effect, it also has more options to reproduce itself (be it by sex or by another mechanism) and therefore entail his intelligence to its offspring. Would the most of his spezies including hiself die out (for whatever reasons), they probably could not create sufficient much offspring that can survive accidents and on their part create further offspring.

Let us now assume that this widespread explanation of the emergence and resistence of a biological property named "intelligence" is true - and -complete.

Now, let us assume that there exists a - yet to be discovered - fundamental physical theory, named here as TOE, that is strictly deterministic in its framework.

What follows, is, that both theories, namely evolutionary theory and this deterministic TOE (label the latter as Bohmian quantum mechanics or whatever you want) cannot both be true - and - complete at the same time. Because a strictly deterministic TOE asserts that everything is only and only determined by its perceding event(s) and evolutionary theory states that some organisms must be partially independent of such a strict determinism for the aim of changing the tide of events codified in the fundamental equations & initial conditions of this TOE (for example to be not eaten by a tiger by seing him coming towards myself with the aim of eating me i jump on a tree and climb as fast as i can on the top of this very tall tree and wait untill the tiger disappears). The discrepancy is, that the latter behaviour of an organism is, at least for myself, hardly to imagine as having been established in a universal wavefunction (or something like that at the beginning of all times) where all events are predetermined.

"How is this tying into the wavefunction? Is PSI(x,t) = exp[(iEt/-h-)] the determinism you're referring to? What causes the wavefunction to collapse?"

I have no clue on this. Rather i assume that there cannot be in principle a path from abstract knowledge to ultimate reality. Because abstract knowledge cannot distinguish between truth and assumption. Therefore i cannot decide what ontological status the wavefunction really has. The determinism i refer to is independent of a specific physical theory, it could be every TOE that is thinkable.

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Hi Brian, thanks for the reply. I wanted to start a new topic on this subject so as not to diverge too much from the scope of the original thread. Unfortunately, I cannot create a new post. My apologies to the OP for hijacking his thread.

Too often you find someone employing Ocaam's Razor as if it is a measuring tool for scientific truth or validity. Simplicity does not imply validity and...

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Hi Brian, thanks for the reply. I wanted to start a new topic on this subject so as not to diverge too much from the scope of the original thread. Unfortunately, I cannot create a new post. My apologies to the OP for hijacking his thread.

Too often you find someone employing Ocaam's Razor as if it is a measuring tool for scientific truth or validity. Simplicity does not imply validity and it is not a fundamental litmus test for scientific truth. The litmus test for scientific truth is empirical observation. If it looks like a duck, acts like a duck, and quacks like a duck, then it's a a duck; even if the equations infer it must be a cow.

Someone can come out of the woodwork and put forth the18'th century hypothesis that' the Sun is simply a burning lump of coal and attempt to justify the theory by asserting that the Coal theory is certainly much more simple and elegant than the explanation provided by nuclear physics. Right out of the gate, however, the lack of dominance of carbon emission lines throws a monkey- wrench into this theory. The empirical observations of the emission lines force a much higher level of complexity on any theory that attempts to to explain this observation.

Also, I think it is important to separate methodological simplicity from causal simplicity when discussing concepts like Ocaam's Razor. A scientist does not start with the goal of constructing a theory which is simple and elegant. A scientist sets out to construct, in the most general and fundamental terms possible, an explanation for the phenomenon under scrutiny. This explanation and the associated methodology may or may not be more simple than any arbitrary theory which precede it or competes with it. If anyone doubts this, simply read a paper on the theory of vortex flow or non-linear dynamics.

In terms of methodology, relative simplicity results because we choose a frame of reference that often, but not always, makes it possible to intuitively conceptualize the problem under study. As a very simple example, I can choose to utilize polar coordinates to calculate the trajectory of an object undergoing rectilinear motion even though it is much easier and more intuitive to utilize Cartesian 3-space or some arbitrarily simply Vector space. Both reference frames will produce the same results and allow me to predict and explain. To me, that is the gist of Ocaam's Razor -- there is no need to trouble yourself with unneeded complexity when a simpler explanation will do. Ocaam's Razor does not imply that something is very wrong when a theory is not relatively simple, elegant, and beautiful.

I also believe that it is important to separate heuristic simplicity from causal simplicity when discussing a subject such as Ocaam's Razor. We generalize concepts to make them more subtle and compact. This flows out of our operational requirements. We formulate more fundamental generalizations to eliminate higher levels of relative complexity in our understanding of phenomenon. In terms of causal simplicity, a more fundamental theory is naturally going to be much more simple and elegant than those that precede it because it provides a more fundamental and generalized way of viewing the phenomenon.

....

As far as interpretations of QM, interpretations are only worthwhile for a scientist if they lead to a framework that allows the scientist to create novel and testable predictions that the current theory cannot. Otherwise, different camps would simply be doing the same things using different metaphysical labels. When a novel conceptual understanding of fundamental ontology leads to a new theory that can explain and predict where others cannot, the result is a paradigm shift that revolutionize how one thinks about nature.

IMO, Physics has been in a metaphysical funk for quite some time -- since the 20's, actually -- and it is a community that is not really sure of itself or where it wants to go. If any period in the history of Physics as in need of a paradigm shift, this is it. As I stated in an earlier post, we are not in Kansas anymore and pretending that we are is likely only going to result in more time spent in this funk.

Modern Theoretical Physics has become String Theory vs the Standard Model vs Loop Gravity vs [insert theory here]. Very few have stopped to consider that perhaps everyone is digging for oil in the wrong place. Perhaps the community needs to find a totally different conceptual landscape in which to probe for oil. Unfortunately, budgets and manpower are all tied up and little of anything is left for any undertaking which challenges the status quo or heads in a novel direction. Physics is no longer 19'th century Natural Philosophy where scientists work largely autonomously and can head out into new ventures on their own. The current politics, sociology, and budgetary considerations of 21'st the century scientific enterprise preclude the possibility of a paradigm shift happening any time soon.

IMO, a Reductionist blind-spot exists in our attempt to understand nature and formally relate lower levels of complexity to higher levels, and vice-versa. One thing that is apparent from our current understanding of nature is that there often exists no clear and systematic causal delineation between different levels of structure in the physical world. I would liken this to the notion of the 'so-called' missing links in evolution that don't permit one to empirically map one level of organizational structure to the next. For example, our distinction between atomic and molecular structures is methodological and heuristic--nobody has managed to come up with a complete and internally consistent physical model that applies, across the board, as to how molecular structure arises from atomic structure.

We can explain the structure of the water molecule by appealing to the concepts of energy levels,stability, valence, polarity, and all these other useful concepts employed in chemistry;however, we would be hard-pressed to deductively predict the existence of the water molecule and it's properties based only on the the laws governing physical structure at the atomic level. In chemistry, scientists still have trouble explaining certain unique molecular properties such as chirality. In fact, scientists are still debating the number of bonds each water molecule makes with its neighbors. In our study of nature, a reductive blind spot always seems to exist when we go from one level of structure to another.

In short, knowing and understanding the properties of lower level structures does not always allow us to predict or account the properties of higher level structures based only on the properties of those that exist at a lower level -- in any formal and systematic way. There will always be gaps, special cases and oddities along with the blind-spots in our reductive schemes. To abstract even further, would an assumed TOE which accounts for the lowest level structures be able, in theory or practice, to explain the migratory patterns of humpback whales or explain the existence and content of Shakespeare's plays? I am withholding my opinion as to whether or not I believe emergence is due to ontological or methodological concerns.

Historically, physicists have been working with the metaphysical presupposition that discreet and autonomous bits of substance represent the most fundamental property that can be attributed to natural phenomenon. The autonomous parts that comprise themselves are all that is needed to account for complexity -- at least that's the current paradigm.

This ontological and methodological framework is what I question.

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Back to the subject of interpretations.

I think it's worthwhile to take a quick survey of ideas and concepts throughout history that presented similar conceptual and ontological difficulties. It certainly won't offer any solutions to our problem but will give us an idea of how such issues eventually were resolved.

If you every get a chance, browse through Newton's Principia and...

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Back to the subject of interpretations.

I think it's worthwhile to take a quick survey of ideas and concepts throughout history that presented similar conceptual and ontological difficulties. It certainly won't offer any solutions to our problem but will give us an idea of how such issues eventually were resolved.

If you every get a chance, browse through Newton's Principia and Optiks. Compare and contrast the tones present in the works. In Optiks, you will find Newton offering many hypothesis and explanations for the phenomenon he is investigating. In Principia, he explicitly refrains from offering any opinions whatsoever concerning the nature or of the gravitational force.

Action at a distance is so abhorrent to a mechanistic world-view that it is nonsensical. Forces are a local phenomenon and force is transmitted through direct contact. In the everyday world of the 18th century, a force is something that only occurs between two bodies that are in direct contact. It was impossible to fathom how two bodies separated by great distances could influence one another.

Unlike the corpuscular theory of light he put forth, Newton pretty much dealt with the subject of gravitation in a pedantic way and left it at that. The subject was never revisited. Action at a distance was a hot potato and the subject was usually dropped from discussions. It was something that most everyone eventually accepted but nobody ever really spent a whole lot of time trying to think about. There were no Copenhagen or Many-Worlds interpretations for this spooky behavior. Just as we do today, Newton and his contemporaries simply acknowledged that they did not understand the manner in which such a phenomenon manifest itself. Like today, they simply knew that the mathematics worked, could make accurate predictions, and offered a complete accounting for the observations. Life goes on and the work continues. One does not need to understand the complete picture in order to continue to make use of the knowledge one has gained.

In the 19th century, the development of analytical mechanics changed the focus away from force and onto the concept of the energy of a system. In the formulations of LaGrange and Hamilton, force is never an explicit part of the picture. One does not need to visualize the system as individual particles exerting forces on one another. The goal of nature is to minimize the integral of kinetic and potential energy differences. This makes it much easier and efficient, as one can deal with scalar quantities and generalized coordinates. It also means that the notion of action at a distance subtlety disappears from the scene. The problem of action at a distance basically gets tucked away in the attic. It isn't until Einstein that anyone seriously attempts to revisit the subject and attempt to make any sense of the ontological nature of gravity.

A similar justification was used for accepting the field concept after it's introduction by Maxwell and Faraday. With Maxwell, a non-local force between distance charges never explicitly enters the picture. The electromagnetic field permeates space and is the causal agent responsible for the interactions of parts of the 'electromagnetic fluid.' As the field permeates space, matter is influenced by it's direct interaction with the field at the point at the point on the field at which it finds itself. The field is the causal mechanism of the electromagnetic force and action at a distance becomes superfluous.

Later developments obviously model the forces of nature as being mediated by the exchange of particles or energy borrowed from the vacuum.

The spooky notion of action at a distance completely disappeared from the picture. Or so we thought. After a short hiatus, it seems to be back again, only in a slightly different form. Entangled particles seem to be working some strange magic and we once again find ourselves in the unenviable position of trying to play poker with half a deck. It took almost 200 years and a radical shift in methodology and theory before we were able to form a coherent picture of gravitation that let go of the notion of action at a distance. How long it will take for us to make sense of the present conundrum remains to be seen. At least we know how Newton and his contemporaries felt when trying to make sense of the nonsensical. The intrigue and wonder is what draws most people to science. If we had the answers to everything, it would be a very boring world.

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Continuing with the discussion(or monologue in this case). I can never resist discussing this subject, even on Internet message boards. It's a good way to just sit back and unwind -- some might say a very geeky way, but a good way nonetheless. :)

I am not trying to sustain a running monologue but am trying to add a different dimension to the subject. Jump in y'all !

There are lots of...

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Continuing with the discussion(or monologue in this case). I can never resist discussing this subject, even on Internet message boards. It's a good way to just sit back and unwind -- some might say a very geeky way, but a good way nonetheless. :)

I am not trying to sustain a running monologue but am trying to add a different dimension to the subject. Jump in y'all !

There are lots of things we don't know.

What do we know?

-- QM in all its incarnations works. It represents the most successfully theory to-date in terms of it's power to predict and account for observed phenomenon on the scales in which we apply it.

-- For nearly a century, the most brilliant minds in the scientific community have been trying to come to terms with the implications of this theory.

-- These brilliant minds have put forth a myriad of interpretations but there is no consensus and currently there exists no methodological or empirical way of establishing the validity of one interpretation over another.

Unfortunately, that's pretty much all we can really say at the moment. It's fun to speculate but we are all basically shooting in the dark when offering our interpretations.

If someone were to ask me my opinion, I might just say, "I haven't the slightest idea. If Feynman or Wheeler couldn't figure it out, I sure as hell can't." Then I would put on my philosophers hat and offer some metaphysical speculations, because that's really all that can be done without some new discovery or an entirely new reformulation of the subject. Obviously, our classical deterministic tendencies that we have gleaned from the macro world have failed us on this most fundamental level.

We did not build our current paradigm from the ground up. We occupy a very peculiar corner of the Universe. The majority of the visible matter in the universe exists in the peculiar high-energy state of matter that we call plasma. Our tiny little corner of the cosmos is a Universal freakish oddity characterized by stable low-energy states of matter known as solids. If we were plasma-beings living in the chaotic soup of high-energy substance, our physics textbooks would look very different. Our world would be dominated by forces so powerful that it would probably take quite some time before we ever get around to the idea of gravitation, if we ever did at all. We would likely never catch onto the notion that a relatively stable substance called solids was something real. We would probably never speculate on the existence of relatively stable entities such as molecules,solids, or polypeptide chains.

We find ourselves on the opposite end of the spectrum. Our world is characterized by relatively low-energy states where the effects of forces such as gravity are the most discernable features of our existence. It is in this environment that we formulated the rules of the game. We had to start big and work our way backwards down the causal chain into the relatively high-energy world of particle interactions. There is no reason not to believe that our intuition gleaned from our oddball existence will fail us in this environment. This strange and tiny quantum world looks and behaves nothing like the one we live in here in our corner of the cosmos. If a plasma-being happened onto our little corner of the world, he would bey as equally perplexed and dumbfounded.

All the plasma-scientists would be running amok shouting, "Nothing behaves like it should ! What the hell is going on?!!" We would simply retort, "Of course nothing behaves like it should -- you live in a plasma! What did you expect? What made you think that your world should behave like ours?"

We always assumed that the micro-world must behave like our oddball world full of colliding pool balls and masses sliding down frictionless planes. We were wrong. Just one look around tells us that we ain't in Kansas anymore, Toto.

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Continue on with this subject..

Violation of Bell's Inequalities appear to have put us in the position of either rejecting the idea of hidden variables or accepting the notion of hidden variables as they apply to non-locality.

Some theorists who do not like the experimental results will say, "I will concede the existence of non-locality as long as I get to retain the ability to rely...

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Violation of Bell's Inequalities appear to have put us in the position of either rejecting the idea of hidden variables or accepting the notion of hidden variables as they apply to non-locality.

Some theorists who do not like the experimental results will say, "I will concede the existence of non-locality as long as I get to retain the ability to rely on hidden variables." Others will simply reject the experimental results entirely and argue the fine points of the experimental setups. They will concentrate on finding ways to show how the results are invalid. Again, going back to my points earlier, this sounds more like an attempt to hold onto a particular worldview rather than taking an objective look at the evidence and trying to reach the most logical conclusion that represents the most plausible interpretation. If the inequalities had not been violated and the results of experiment had told them what they wanted to hear, would they be pressing the issue of validity? I would tend to believe not.

The experimental results have been replicated many times. If someone were to demonstrate and empirically prove that hidden variables indeed exist, I would have no choice but to accept their existence. That evidence, however, does not exist and nobody has put forth any consistent theory that would account for or explain such hidden variables. This does not mean I am right. It simply means that I have formed my opinion in as objective a way possible, irrespective of my own prejudices and metaphysical bias. It doesn't really matter to me if it turns out that hidden variables do indeed exist anymore than it would matter to me if it turned out that we discovered that the Universe really was the back of a Tortoise floating in a gigantic pool of water. I simply see no reason to currently accept either proposition. They appear to be conjectures based on criteria that appeals to an individuals metaphysical desires and inclinations about how the world should operate.

In the absence of evidence for hidden variables, I can only objectively infer that classical reductionism likely died a very slow death as we discovered more and more about the workings of nature on scales that are far removed from our senses. Many have refused to attend the funeral and hold out hope that it simply went AWOL and will one day reappear or manage to resurrect itself. The theoretical community appears to be going through the progressive states of Kubler-Ross -- Denial, Anger, Bargaining, Depression, and Acceptance. Currently, many seem to be somewhere between the Bargaining-Depression state.

I believe that any resolution of the conceptual and ontological issues we face are likely going to result not from a reformulation of Quantum Mechanics or a discovery of hidden variables, but will involve a major paradigm shift that reorganizes our intuitive and causal notions of how nature is structured on a phenomenological level, not only microscopically but macroscopically as well.

To give you an idea of what I mean, we should consider some examples from History. In the history of our quest to understand the structure and properties of the phenomenon that exist in the natural world, there have been two fundamental reasons for shifts in thought that lead us to reformulate current principles. Such shifts either result from observations that force this change upon, as is the case with QM, or they result from a radical ideological shift in perspective. Both will lead to a reformulation of fundamental notions or concepts about the world but the latter proves more powerful as it forces one to reformulate ideas on a global scale. It forces a new approach to an old problem using entirely novel ontological conceptions about the nature of the Universe itself, not just a specific phenomenon associated with an arbitrary scale of influence.

I would present two examples here that explain each type of methodological shifts .

Contrary to popular opinion and many textbooks, Copernicus did not discover that the Earth orbited the Sun. Copernicus presented a Hypothesis--one that was neither logically or empirically necessary, given the information available at the time. The Ptolemaic system could account for the observations of the behavior of heavenly bodies with equal precision. Unlike our venture into the Quantum world, the Heliocentric hypothesis was not one which was formed as a result of some new observation that could not be accounted for within the confines of the accepted theory of the time. Copernicus did not come out and say to the world, "Hey guys, we’ve got this all wrong. I was just observing the sky last night and discovered that the Earth is actually circling the Sun. We've got this whole thing wrong !"

The Heliocentric hypothesis was one that had fermented in the cellar for centuries. You can find the hypothesis present in some schools of thought all the way back to the Greeks. Copernicus framed the heliocentric hypothesis on a much more refined and accurate scale, using the precise data available at the time. He was able to do something the Greeks could not--he could match empirical data to a hypothesis and give it some life. Still, I must emphasize that the Heliocentric hypothesis was not necessary to be true in order to account for the data -- the Ptolemaic model did so with equal precision. Since displacing the Earth from it's primal and privileged position was abhorrent to the common sensibilities of the time, why, then, did it gradually gain acceptance? It offered simplicity and elegance that the Ptolemaic system could not. It accounted for the astronomical observations in a much more straightforward and rational way. In short, it made more intuitive sense; whereas the Ptolemaic system, with its bizarre system of epicycles and deferent, did not. One could more easily explain and account for observed phenomenon.

The period that followed represents a time when the ideology of the world was turned totally upside down as a result of the inferences we derived from the theory. It marks the beginning of the scientific revolution and allowed us to proceed in ways previously undreamed of. We have never looked back since. Obviously, the idea was slow to be accepted. As with the Theory of Evolution today, it infers things about our existence that many people may not want to hear. If our position in the Universe is not special then perhaps we are not as special as we thought. In the case of Copernicus, the idea started to take shape that perhaps the world does not revolve around us, either literally or figuratively. IMHO, the Copernican Revolution represents the most radical shift in naturalistic and rational thought in the History of the Western world.

Compare this Paradigm shift to the changes in reasoning that occur when nature forces upon us, by means of direct observation, certain tenets or principles which cannot be avoided but do conflict with the current paradigm.

A misconception in the history of physics as it is portrayed in most texts is that Planck was a willing accomplice in the development of Quantum Theory. Planck's motivation for introducing the notion of quantization was entirely methodological and not thematic. He was simply using quantization as a neat mathematical trick. His goal was to create a mathematical model which could be used to reproduce the spectral observations gleaned from the study of blackbody radiation.

Another misconception is that he was motivated to do so due to the Ultraviolet Catastrophe. This is not the case -- in any way, shape, or form. Planck held great reservations surrounding the methodological application of Boltzmann’s statistic to observed phenomenon. He never trusted such an approach as a internally consistent method of relating thermodynamic properties of macroscopic systems to micro-phenomenon. He saw the application of M-B statistics as rather arbitrary and contrived. He set out to create his own methodology that would not rely on assumptions about the mathematical forms of the distributions themselves. In other words, he was asking, 'What mathematical form of the distribution equations would allow one to successful model the spectral phenomenon?' His goal was to model the behavior of the system, not start with assumptions about the statistical behavior of an ideal ensemble.

Using Classic Electrodynamics, Planck tackled the issue by imagining that all of the component structures contained in a body that gave rise to the EM waves could be imagined as charges on the end of springs, vibrating back and forth like a simple harmonic oscillator. He worked out an equation to model the energy distribution among the components that would give the exact results measured in experiment. He gradually found out that he could never apply a continuous distribution of energy among the oscillators and model the empirical results. The only way he could make the mathematics work is if he assigned each oscillator only discreet values of energy which were proportional to an arbitrary constant. We know the rest of the story.

The gist here is that Planck thought of quantization only as a mathematical trick that would allow him to model the phenomenon. His goal was not to explain the nature of the underlying entities which made up the system. He, and others, refused to even entertain the idea that the quantization of energy states had anything to do with the entities themselves. When he published his results, he presented the equations as a mathematical tool to model the observed phenomenon. He was not making a statement about the nature of the underlying entities themselves.

It wasn't until Einstein took up the study of the Photoelectric effect and direct experimental proof of quantization of light was shown to exist that he finally came to grips with the idea that quantization indeed occurs on a fundamental level. He admits he only grudgingly accepted the notion. It is for this reason, btw, that many consider Einstein the founder of Quantum Theory -- or at least co-founder. It is also ironic that both men eventually came to reject the theory which arose from their work. Neither would ever accept Quantum Theory as a complete and accurate representation of nature. Two of the most brilliant men in the history of physics let their prejudices concerning the ontological nature of reality interfere with the acceptance of a theory.

The point here is not to bloviate but to offer a different perspective on the subject of interpretations. There are two different ways in which we come about altering our perspectives.

Shifts in understanding may come from areas of research that are largely ignored and still in their infancy. In our current case, I believe that such areas of study include chaos theory, complexity, and emergence. These are realms where processes take on a more fundamental role than discreet substance and represent different ways of tackling an issue. I believe that work in these areas could possibly allow us to make sense of things that we may be missing in transitions as we go from one level of structure to the next.

We have made a lot of ontological assumptions about the nature of physical reality and how systems are constructed. Many of them are hard to hold onto in the face of current observations of the world beyond our senses. I think we need to ask the question, what is fundamental to nature? The assumption has always been that discreet substance(whatever that means on the subatomic level) is the most fundamental property of nature that is responsible for all observed systems. The notion that substance may not be fundamental may seem absurd and nonsensical but on close inspection it is not as absurd as one may think -- can it be more absurd than believing that a photon possesses certain ontological properties that allow it to simultaneously be both a wave and a particle?

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Hi

When people explore the nature of consciousness in an attempt to resolve the weirdness of QM, they are making an implicit assumption about the scope and structure of the theory we have built -- namely, that the theory itself represents something more than an elaborate Turing machine which we have constructed to allow us to model phenomenon on the level of detail beyond our senses....

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Hi

When people explore the nature of consciousness in an attempt to resolve the weirdness of QM, they are making an implicit assumption about the scope and structure of the theory we have built -- namely, that the theory itself represents something more than an elaborate Turing machine which we have constructed to allow us to model phenomenon on the level of detail beyond our senses.

As I mentioned in my prior post, the mathematical structure and formalism of QM was something forced upon us by the nature of our observations. Such a construct is logically necessary. The variants of QM(e.g. Canonical, Path Integral, Field Theory) represent the only systematic ways one can construct a formal mathematical structure that takes into account the quantization of observables. We have carried over some of the the basic principles and structures from Classical Mechanics, such as the implicit notions of least action present in the Lagrangian and Hamiltonian dyanmics. We have also carried with it the ontological notions about casualty that have arisen from our intuitive understanding of the world of our senses. In other words,we have taken along with us the philosophical baggage we have intuitively derived from our sense experiences and this the reason for our cognitive dissonance.

The fundamental question is not what QM is describing as it relates to our intuitive understanding of the world of our senses, as this picture will always be incomplete given that we are mixing categories. In my opinion it makes no sense whatsoever to assign properties like 'partlecness' or 'waveess' to things like electrons or photons. Intuitive concepts like 'particleness' and 'waveeness' have arisen from the intuitive concepts we have formed from our sense experiences of the world in which we live.

In other words, from my perspective, we cannot retain any logical consistency in our world-view if we employ intuitive constructs such as an electron being either a particle or a wave -- or anything resembling the structures we assign to things in our world. These only represent neat and abstract mathematical constructs that allow us to explain in a stochastic fashion, the behavior of the systems under scrutiny.

I liken this to being at a party and playing charades with your friends. Your form solutions to the problems of identification by trying to relate abstract clues in a game of association. You rely on your memory and intuitive understanding of relations, causal structures, and imagery. You form a set of al possible things which could account for the clues and then narrow them down to form an identification by selecting one member from the set. You come up with an answer -- you are a cow ! Woohoo -- pass me a beer !

When we speak of substance on the level in which we apply QM, what are we really talking about? When trying to answer this question, we can only form the answer using, as a basis, our intuitive understanding of phenomenon gleaned from our classical sense-experiences of the world around us, on the scale in which we live. Like charades, we form the set of all possible things which can account for the clues. The problem is, there are no members of the set which can account for the things we are describing in QM. We have no intuitive or sense-experience information in our thought processes for an entity called a 'wavicle'. A wavicle is not a member of the set of things formed by our experience and intuition. We would never be able to intuit an object with such properties from the sense-impressions we have formed through our existence.

Take, for example, a physical system whereby we might model an electron conconfined to a potential. First of all, we are assuming that there is an entity which corresponds, in a classical sense, to something being confined -- like confining a pool ball to the confines of a table. IMO, this notion is FUBAR to begin with. We are mixing ontological categories of being and structure. There is no direct map to the set of all our sense to the set of all properties and behaviors observed on the QM level. We just try to play the game assuming there are. That's when things go haywire.

It is much more logically sound and consistent to speak instead not of substance as something fundamental to a phenomenon, but as a fundamental process that determines the changes any arbitrary system will undergo when we think of in the classical sense. In a classical sense, strip a particle of it's properties then what is it? What is left of an electron if you take away its properties of charge, mass, and energy? Nothing. You could not identify such a thing --there is no substance left. The properties themselves ARE the substance. They define the substance in a classical intuitive sense, not the QM sense. In other words, there is nothing there in the classical sense of a point-like particle running amok like pool balls flying about on a table.

It is more consistent and sane to think of something like an electron not as substance but as an emergent feature of some fundamental underlying process that is beyond our ability to comprehend or intuit. Thing such as the appearance of an electron, photon, etc represent the possible degrees of freedoms of such a fundamental system. Interactions,forces, fields, in the classic sense, represent the possible degrees of freedom.

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Hello Folks,

I've given this topic a lot of thought, so I figured I should weigh in. I gave a presentation at the 10th Frontiers in Fundamental Physics conference last Fall, about a common basis for non-locality and entropy, which focused on the role played by decoherence. I have to admit to incomplete knowledge of the subject, however. My proceedings paper here did not pass peer-review,...

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Hello Folks,

I've given this topic a lot of thought, so I figured I should weigh in. I gave a presentation at the 10th Frontiers in Fundamental Physics conference last Fall, about a common basis for non-locality and entropy, which focused on the role played by decoherence. I have to admit to incomplete knowledge of the subject, however. My proceedings paper here did not pass peer-review, but subsequent correspondence with H.D. Zeh, Erich Joos, and others, has given me a pretty clear notion of what was wrong with it (I will revise), and a clearer idea of what decoherence really means. I'll share what I'm fairly certain of here.

First off; the wavefunction does not collapse. The global wavefunction remains a unified and coherent entity which evolves according to Schrödinger's equation. However; its nature is essentially non-local. It is field-like, wave-like, and pervasive. When we introduce a local frame of reference; this reference frame is automatically identified with the material or particle-like nature, by virtue of its locality. Any local observer exists in contrast with the non-local reference frame of the global wavefunction, and induces (or observes) components of that wavefunction to decouple whenever there is an interaction or observation taking place.

Non-local components of the wavefunction that had been global become associated with one or another discrete or observable entity. In the process; what happens is exactly as Dieter says - "various systems (the observed one, the apparatus, the observer, and the environment) get entangled." In Bell's and GHZ experiments, sequential weak measurements are made that effectively break off one component at a time, so that wavefunction components of the particle under study (corresponding to energetic degrees of freedom) are transferred from that entity to the measurement apparatus (until all the degrees of freedom are linked to it). Thus; components preserving the entanglement of one particle with its distant counterpart become dedicated instead to entangling the particle with the measurement system.

So; there is no collapse as such. There is always an evolving wavefunction. But wavefunction components which had been associated with one system become linked with another, leaving the two systems entangled. One of the chief observations of decoherence theory, therefore, is that there are no isolated systems. Instead, entanglement is universal, and the manner in which various sub-systems are entangled evolves over time.

I shall have lots more comments on this topic, especially if someone responds to these. I want to talk about the observer's role as participant. I just skimmed professor Leiter's Journal of Cosmology paper and feel that it bears some commentary relating to this subject - especially as it seems to offer some confirmation of my non-locality and entropy linkage. I thought the new paper explained his ideas much more lucidly than his FQXi contest essay, and I commend him on his clarity.

All the Best,

Jonathan

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Logically it makes sense why they act like particles when observed and waves when not observed. Quantum physics is common sense to anyone who understands the statistics of 2 coin flips and how those statistics are affected by observing the coins. I will explain why the double-slit experiment is the same experiment as something you can do with 2 coins.

In the double-slit experiment, an...

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Logically it makes sense why they act like particles when observed and waves when not observed. Quantum physics is common sense to anyone who understands the statistics of 2 coin flips and how those statistics are affected by observing the coins. I will explain why the double-slit experiment is the same experiment as something you can do with 2 coins.

In the double-slit experiment, an electron (or other particle/wave) has 2 holes it can go through and then is detected hitting somewhere on the back wall. If it goes through the left hole, statistically it will paint a pattern on the back wall. If it goes through the right hole, it paints a different pattern. It goes through each hole equally often as the other. Most peoples' common sense tells them that statistically it doesn't matter if you know which hole it went through because you can simply average the 2 patterns to get the pattern on the back wall for when it could go through either hole. But its a very different pattern from the average of the left-hole-pattern and right-hole-pattern. Its the same pattern as waves interfering with each other.

Sometimes electrons act like particles and sometimes like waves, but why? I'm going to explain why that happens using common sense instead of equations. The problem is most people don't have all the parts of common sense that they think they have. If you understand the following about 2 coin flips, and you see the patterns created in the double-slit experiments, then you can put them together and understand why electrons (and other particles/waves) sometimes act like particles and sometimes act like waves. Logically, without considering the specific equations of physics, we can know there has to be something like that in physics somewhere. Here's the 2 coin question:

If I flipped 2 coins and at least 1 coin landed heads, then whats the chance both landed heads?

Its 1/3, not 1/4 or 1/2 like most people think, because there are 4 ways 2 coins can land and I only excluded "both tails" when I said "at least 1 coin landed heads" so that leaves 3 possibilities and I asked what is the chance of 1 of those 3 things which happen equally often. Its 1/3. If you still don't believe it, flip 2 coins many times and only ask the question when at least 1 of them lands heads and you will see that 1/3 of the time you ask the question they both land heads. The flaw in Human minds is the need to choose 1 of the coins and say it certainly landed heads, but I did not tell you any specific coin landed heads, and it does change the answer if you take that shortcut.

Most peoples' common sense tells them that since its a symmetric question (between the 2 coins), it can't matter if they start with 1 of the 1-or-2 coins that landed heads, and they think it will get the same answer as not knowing if a specific coin is heads or not. How could it matter? We know at least 1 of the 2 coins landed heads, so I'll just define a variable called coinX=heads and figure out if coinY=heads or coinY=tails. Since coinY was randomly flipped and coins have a 1/2 chance of heads, then the chance both are heads must be 1/2. But then they think about the extra information I told them: at least 1 coin landed heads. That has to change something, so how could coinY be 1/2 chance of heads by itself and with coinX? CoinX and coinY are symmetric. You can trade them in this question and not change the answer. So whatever is true of coinY has to also be true of coinX on average. So maybe the chance both are heads is 1/4. Most people go back and forth between 1/2 and 1/4, but the answer is 1/3 as I explained above.

How is the 2-coin experiment related to the double-slit experiment?

The patterns of the 2 coins (how often they land heads) individually can not always be averaged to get the pattern of both coins together. If at least 1 coin landed heads and you observe a specific coin being heads, then the chance they are both heads is 1/2. If at least 1 coin landed heads but you don't observe any coin, then the chance both are heads is 1/3.

Logically, observing a specific case of something you know has to be true in general, about the 2 coins, produces a different outcome than only knowing its true in general.

The analogy to quantum physics is that when you observe a heads or tails, you collapse the wavefunction (including the other coin you didn't observe) to a particle and the other becomes a different wavefunction, but if you do not observe any heads or tails then its a symmetric wavefunction between the 2 coins.

I can say the same thing about the 2 holes in the double-slit experiment. If I put an electron detector past the left hole, and shoot an electron that could go through either hole, and the detector observes or does not observe an electron, then I get a different pattern (statistically on the back wall of where the electrons hit) than if the detector was not observing the space between the left hole and the back wall. If any part of the possible paths are observed (as containing or not containing an electron), then the other possible paths are affected even though they were not observed. The electron could have gone through both slits or neither or left or right, but still the path on the right is affected by observing the path on the left.

Most quantum physics scientists explain it as the electron going through the left hole, the right hole, both holes simultaneously, or bouncing off the thing containing the 2 holes without going through either hole. If the electron does not go through either hole, they do not count that in any of the patterns on the back wall.

In the 2-coin experiment and double-slit experiment, there are 4 possibilities, and 1 is excluded. I need to label the 2 coins for this, like the left and right holes/slits are labeled "left" and "right". One coin is a nickel and the other is a dime. This is not the only way to pair the 4 possibilities. Its just a way to explain that they are the same problem:

(1) Nickel heads. Dime tails. Electron left slit. Electron not right slit.

(2) Nickel tails. Dime heads. Electron not left slit. Electron right slit.

(3) Both coins heads. Electron goes through both slits.

(4) Both coins tails. Electron bounces off the thing containing the slits and does not go through either. It is not true that "at least 1 coin landed heads" so I don't ask the question or keep statistics of it. The electron didn't hit the back wall so its not part of the statistical patterns.

In the design of both experiments (2-coin and double-slit), cases (3) and (4) are opposites and cases (1) and (2) are symmetric. Exactly 1 of (3) and (4) is not counted in the statistics, but the chance is equally balanced between (1) and (2). It works the same way if you swap the left and right slits or swap the nickel and dime or swap heads and tails or swap going through a slit with not going through a slit. Its practical to test it going through the slit but not practical to test it after it bounces because bouncing is an observation by the thing it bounced on.

Quantum physics is a kind of statistics. So is the 2-coin experiment. In the double-slit experiment and the 2-coin experiment, observing any part changes the outcome statistically. I'm not saying the math of the double-slit experiment is exactly the math of a bayesian-network (which is the kind of statistics used for the 2-coin experiment), but I explained enough similarities that quantum physics scientists should take this seriously.

The double-slit experiment is a variation of the 2-coin experiment that uses continuous angles instead of only heads/tails.

That is why observing things changes the outcome and why electrons/photons/etc act like particles when observed and act like waves when not observed.

If I flipped 2 coins and at least 1 coin landed heads, then whats the chance both landed heads? The most important thing to remember is the question is symmetric between the 2 coins, but observing either of those coins changes the outcome, like observing what goes through either slit changes the outcome.

Quantum physics is common sense to anyone who understands the statistics of 2 coin flips.

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I'm posting this here to see how it looks online. It is my first instance of my paper online incase I wish to show it to someone instead of using email. I hope you don't mind. I basically worked like 5 years on this. If you wish to contact me for further discussion then contact the Fields Institute or Clay Math Institute they have my paper and my...

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I'm posting this here to see how it looks online. It is my first instance of my paper online incase I wish to show it to someone instead of using email. I hope you don't mind. I basically worked like 5 years on this. If you wish to contact me for further discussion then contact the Fields Institute or Clay Math Institute they have my paper and my email.

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Proving the Relative Consistency of Both Hilberts 1st Problem and the Polynomial Time/Non-Polynomial Time Problem

　



　

Paper Outline:



　

Part 1 - Introduction

Part 2 - History

Part 3 - Proving the Relative Consistency of Hilberts 1st Problem

A) First Method of Proving the Relative Consistency of Hilberts 1st Problem

B)Second Method of Proving the Relative Consistency of Hilberts 1st Problem

Part 4 - Proving the Relative Consistency of the P-NP Problem

Part 5 - Relative Consistency Principles Similiar To Hilberts 1st and P-NP Problem in Other Areas of Math

Part 6 - Conclusion



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Part 1: Introduction

　

　

"A disciple asked, "What is the difference between the enlightened and the unenlightened man?"　

The Master replied, "The unenlightened man sees a difference, but the enlightened man does not."" - old monist teaching

　

　

　

It all comes back to Schrodingers Cat. The ultimate duality of circumstance. Did the cat live or die? Until we open the box, we will always have 2 possible outcomes to consider but we have forever known the principle of the uncertainty would leave us with this dual perspective encountered as one tries to foresee the outcome. This paper deals with Hiberts 1st Problem and the P-NP Problem and I will be attempting to "prove their relative consistency by finitistic methods". Ironically, I could have wrote up a near infinite long paper (hehe) considering the most fundamental and basic principles and foundations of math are required with this paper where I inevitably found that most of these relative types of math and algebra that I came across always applied in some way. I tried to sift through and only use what was relevant or important in any quotes or examples I use. I know what empirical equations are and I know how they can be used. My greatest challenge is teaching the laymen what "Relativity" means ... or that many principles are relative. Any ideas based on measure, comparing, ratios, or probabilitys etc. would be relative. My other key word is "Axiom". Once I focused on what an axiom is and how to use it to show what relativity was, was when I found I could maybe write up some fundamentals regarding Hilberts 1st Problem and the P-NP Problem and try to present a paper on them. I'll be honest, I have a good idea that these are "Relative Problems" and relative problems most likely occur in most of todays top listed math problems not as of yet solved ... or perhaps half of them but I think many of them are relative problems although I'll admit I have only read up on these 2 problems in this paper and have not looked at the other problems!



"The First If"



Without axioms math wouldn't exist. Axioms simply ARE what mathematics and measurement is about. As you read my quotes from information websites, watch to see how axioms determine the "System" that is being used as the tool for solving. Remember that when we encounter an"If", we have obviously used an axiom. I also liked the following word and will quote it since it's a great start for what you are about to read in my paper;

　



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

　

"The term syntax is also used to refer to the rules governing the behavior of mathematical systems, such as formal languages used in logic—see syntax (logic)—and computer programming languages—see syntax (programming languages)"

　

　

　　

Every word of that definition will be relevant to this paper. I will discuss relative axioms showing relative qualitys in the relative principles. Inevitably, we will see how computer language/logic applys as well and I feel is just as relevant to proving these relative principles. I basically consider Hilberts 1st Problem and the P-NP Problem to be the same problems but only in a different form based on their obvious relative nature ... and have already guessed that many more unsolved problems fall into this "Relative Problem" category. I am going to base most of the paper on 1 diagram found on Wikipedia ( http://en.wikipedia.org/wiki/File:P_np_np-complete_np-hard.s
vg ) because its the most fundamental diagram showing the principles discussed in both problems I talk about in this paper.　

　

I apologize for not linking any math equation.jpg or chart.jpg mentioned in any quoted links I use because it would take a long time to upload every symbol or chart and then insert its link into the information I quoted in this paper. If you go to the link I give it will be there to better see but yes you will haveto sift through the page to find the quote I used for it sorry. This is actually my first real math paper. I've written a couple other math/physics papers before this but its just an amateur hobby of mine really and I just write these up as best I can hoping its understandable. My earlier papers were bold and eager but I'm going to try to be more straight forward and convincing in this one. Thank-you for your time and I hope this paper makes sense to you.



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Part 2: History

　

　

　

I became interested in String Theory around the late 90's in my twentys. I had drawn up an image of what I only just recently discovered to be the fractal Koch Snowflake and visited a math professor at the local university with a simple equation that read "x=1/13y". I had with me a 12-sided die (dodecahedron) and tried to explain that when adding up all 13 spheres that make up a dodecahedron (12 spheres around 1 sphere in the middle) that if all 13 spheres were numbered 1-13 that the total is 91 and so thus the average (or Mean) of 91 is 91 divided by 13 so 7 is the Mean. I had heard that " 7" was the biblical "number of completion" and thought that interesting but , anyways, I felt that this geometric truth must be significant since the dodecahedron is very sphere-like and therefore common and also many consider the 12 houses of the zodiac, 12 apostles, and 12 hours of a clock etc. to be more than just coincidences (article depicting our universe as a dodecahedron;> http://www.world-science.net/exclusives/041005_univshapefrm.
htm ) ... So, the math teacher I was talking with said my diagram was fractal and that I was doing fractals. I didn't have any philosophy to why I liked "x=1/13y". I said it was empirical to the dodecahedron and so must be of some value to other structures but I really did not have any real clue as to what I was trying to say. I left somewhat deflated but still enthused and considered it just one step in my journey. I took a 3 or 4 year break and then got back into String Theory maybe around 2004. I started reading on many topics and learned a great deal but it was only maybe around 2006 or 2007 that I finally stumbled upon something called Sample Space. It was when reading Sample Space that I saw literally the step I used to add all 13 spheres of the dodecahedron as simply being a Sample Space of 13 numbers! I soon realized that now I could not only find a Mean average of the 13 digit string/spheres of the dodecahedron, but I now could apply an equation that always added any length string/spheres which could be summed and then divided to find its Mean and so this equation now empirically handles ANY size string and so I right away assumed I had a String Theory or a type of String Theory. I was excited and did not have the easiest time putting into an equation this function of finding a Mean for any length of string. I finally came up with "Energy= (sum Sample Space)/Length". Also, and still surprising to me, I looked at this equation for quite awhile and wrote down its solved values. I was always looking for patterns and so I also thought I'd like to see if theres any pattern in looking at the percentages between the numbers of the various lengths of strings I was calculating ... and so I found a very handy percentage calculator on the internet and it had both "percentage of" and "percentage from" functions so I could look for even more patterns. Thus, by fortune, I saw some pattern showing that inevitably helped me eliminate one of the variables in my first formula. I now had an equation that read "(sum Sample Space)/Length=(Length+1)/2". Both these equations performed the same function and both empirically worked for any length string and so I was even more excited and thought I must have a String Theory equation here somewhere. I wrote up a physics paper covering many topics and I also tried very specifically to factor in the various pivotal fundamentals involved with important topics like String Theory such as Bosons, Tachyons, and also Dimensions, and Blackholes, and Superfluids etc. etc. and I made philosophy an absolute necessity for being the foundation and logic to my theorys and ideas in regards to celestial mechanics and Blackholes and Photons and String Theory etc. etc. I'm bringing this up because I know how deep many of these particles and concepts are and I know how tedius and similiar many particles and concepts can be! I will just get to the point here and say that many, many concepts and particles tend to share many of the same qualitys and often many concepts are discussed across various fields but have many different names. Ideas like Convergance, Wave Collapse, Focal Points, or Locality, Wilson Loops, and things like Feedback, or Rings, or Solitons all can be found in many different forms but still be basically the same concept or action. When I was trying to pinpoint what my 2 equations were, I felt that the empirical geometric and symmetrical nature of my equations surely meant we were discussing quantum propertys or possibly Blackholes and even Tachyons. Still, to this day, I have learned that Mathematical Induction (discussed by Plato and Socrates) is empirically a quality of the geometric and symmetrical realm that we may very well call Blackholes, or Tachyons, or even the Higgs Boson. I am explaining this for those that read my first ever physics paper where I tried to distinguish the more important qualitys of my 2 equations that I felt could be describing Blackholes, Tachyons, Holonomy, Superfluids, or Zero-Point etc. etc. These equations may very well describe those qualitys since they added empirically and we do not yet understand completely just what Blackholes are or what Zero-Point is or how the Higgs Boson functions etc. I am bringing this up really because my first physics paper tried to sort into 2 basic groups the property of Scalars versus Vectors and so I had 1 equation as Scalar and 1 equation as Vector and I tried very hard to sort what category Zero-Point would be in or where the Tachyon should go or if I had a "Blackhole Equation" or a "Higgs Equation". My ambitiousness resulted in a paper with tons of assumptions and many things left open for debate. To start, I didn't actually have String Theory equations since only now recently is when I found out my first equation was actually a form of a Pythagorea Mean and my second equation was my rediscovery of a process called Mathematical Induction. That glaring error of me thinking I had done String Theory really deflated the sails of anyone with the slightest bit of doubt about my paper. Right from the start, in my paper, I had stated my first equation looked very Pythagorean and I also was absolutely certain both equations were empirical, since they were, and that both equations must be significant to the physics world because I added a "proofs" section where I tried to post as many Wikipedia articles I could find that showed any equation using a pattern simply known as "an Infinite Set". I even found that the pattern of my equations that were finding the Mean could be found in whats known as "Magic Squares" that interestingly were used to teach Chinese creation myths with the I-Ching and is also used as the international symbol for Feng Shui (hehe). The pattern showed itself easily in odd-numbered Magic Squares with my Mean number solved from the equations being the centre number of the Magic Square when you added the total number of squares and then divided by that number. You see, I had noticed all along that the function I was using Sample Space for was the very exact and same function of the Infinite Set. I was finding examples of Infinite Sets in many, many classic and empirical equations and therefore I was using this to show that if my equations use Sample Space and its function is seen in the use of Infinite Sets in many other formulas, then surely my equations must be verifiable and legitimate! Well, despite eventually discovering that my equations were legitimate once I found them online, after finding about 4 or 5 university and physics and math proffessors to email my paper around to , I still did not get one person to respond back to me except for the local university math teacher whom I emailed and insisted he give me some feedback. He did recognize my first equation and said it already existed but he never did actually identify it as a form of a Pythagorean Mean equation as I later figured out by myself (alltough to be honest, my version of a Mean equation was original and only the Quadratic Mean equation even comes closest to looking like it, but I digress.). He did not even recognize my second equation as to be a form of Induction but, either way, he seemed to think I knew nothing about math. He even called me crazy. I've been called crazy twice infact (the other person knows who he is). Basically, I sent out a very deep and thought provoking paper (my first ever) to about 10 people asking for any kind of feedback but only a couple responded with unkind words yet not one of those people ever replied back to say, "Hey, your first equation is a form of Pythagorean Mean equation and your second equation is whats known as Mathematical Induction". Had even one person emailed back I would have felt a little better about myself but instead it wasn't until only recently, like a year after my first paper, that it would be myself that would find out I had rediscovered legitimate empirical equations. I actually had called up an old highschool math teacher named Mr. Dajka to help me solve and reduce down the Induction equation that had only 2 variables. By luck, while reading Wikipedia, right after Mr Dajka solved my equation down farther did I by chance happen to be reading the article about Summation and it gave a classic example equation of Mathematical Induction. I was gazing upon literally my reduced second equation. So, there's where I was. I was proud that my equations did mean something but dissatisfied that they weren't original or ground-breaking. I let this stage of my journey chill for awhile and then maybe 5 months later I got back to reading and delving deep into various concepts and philosophys ... re-reading about Pythagoreans 3 Means and then I'll guess around February 2011 where I stumbled upon Transfinite Numbers. The write-up for Transfinite Numbers right away seemed to imply or hint at a "Mean" or "middle-point" paradox that could be conceptualized to exist somewhere between the rational and the infinite. I had realized that the function of summing the Sample Space gave the empirical value of what was the outcome of the Induction equation that used Summation. Sample Space as a single variable was giving the same answer that a summed Infinite Set was giving and so I thought there must be some new equation that exists where I find the Mean between an Infinte Set(or summed Sample Space) and another set that wasn't infinte. I felt the answer to that equation would be the Mean or Transfinite Number since that would philosophically and logically be what the Mean should be. My theory was that if I can find the Mean between an Infinite Set of Natural numbers with another set that is anything but Infinite, it should beTransfinite and maybe a Theory of Everything could be an equation that computed a value between the rational and the irrational or the Quantum and Classical realms. I looked very long at the classic Mathematical Induction equation I stumbled upon and realized the Summation notation of that equation could be used to create a larger more complex Induction formula but that now included another set to be added to this classic Induction pattern and then divided by 2 to get its Mean average. By looking for the pattern I somehow came up with the equation that I am now finally about to explain as my Extended Diophantine Mean equation. I also consider it luck that I wandered into Set Theory and learned about Alephnaught and Alephone as perfect "book-ends" to be used as variables for finding the Mean between the two. I want to repeat, as good as the equation looks with the variables named Alephnaught and Alephone, still, all that matters is the overall pattern. Any real knowledgable mathematician knows that many variables have the same qualitys but just a different name or letter. Even the "constants" like e, pi, the speed of light c, or phi, all can be found in many, many equations and from similiar structured formulas. Changing the letters around does not invalidate the structure. Anyways, the next section explains my handy examples of Tautological Ontological Empirical equations I call TOE's (hehe) and eventually why I think I can show how that the Continuum Hypothesis and the P-NP Problems are relative and that I have equations to solve #1 of Hilberts 23 Problems where he asks if Alephone and Bethone are the same cardinality. If anything in this paper, I show that the Pythagorean Means and Mathematical Induction are related in function and I show that Infinite Sets and Sample Space also share similiar functions. I decided only far more recently (May 2011) as I write up this paper now that it seems the Polynomial Time/Non-Polynomial Time Problem may well also be one and the same type of relative consistency problem as the Continuum Hypothesis. I also recently had noticed that an article on Divergent Series explained how the late mathematician Srinivasa Ramanujan had claimed that the sum of an infinite Divergent Series equaled "-1/12" ... which I found intrigueing since that is only 1 sphere off of a dodecahedron (13 total spheres) of which my first ever empirical equation I created for the dodecahedron was "x=1/13y". Could the difference of just that 1 centre sphere be the difference between Convergence from Divergence or a plane from a point? Also interestingly, I decided to compute the percentage difference of just 1 sphere of the dodecahedron to another ... or 12/13 from 13/13. I found that 12/13ths was 92% and therefore the Mean between 92% and 100% was 96% ... is 96% significant? Well, intrigueingly, yes it may be significant. The ratio of all dark "stuff" in our universe is roughly 96% dark to 4% visible. Also, it was surprising to do an internet search and find out that humans' nearest animal relation, the chimpanzees, are 96% related!

　

*Final note,I would love to find the person who created this diagram (http://en.wikipedia.org/wiki/File:P_np_np-complete_np-hard.
svg ) That diagram IS my entire paper and whoemever created that diagram should enjoy this work and I consider that person a peer to whom I owe much gratitude.

　　　

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Part 3: Proving The Relative Consistency Of Hilberts 1st Problem



　

Comment: I will show 2 different forms of determining the cardinality(size) of the sets a and b. The first form will be an axiom of choice applied to the diagram used to explain the P-NP Problem which uses 1 Extention applied on one side of the diagram from the other side. The second form to determine the cardinality(size) of the sets a and b will be an axiom of choice applied to an Extended form of a Diophantine Mean equation. First, I will link to a basic tutorial with basics I want to make sure are understood;

　

http://s134.photobucket.com/albums/q107/m
rgf/?action=view¤t=MathematicalInductionFoundInExample
s.jpg



http://s134.photobucket.com/albums/q107/mrgf/?action=view&cu
rrent=MathematicalInductionin1DimensionalStrings.jpg



http://s134.photobucket.com/albums/q107/mrgf/?action=view&cu
rrent=MathematicalInductionin2DimensionalSurfaces.jpg



http://s134.photobucket.com/albums/q107/mrgf/?action=view&cu
rrent=MathematicalInductionin3DimensionalSolids

　

http:
//s134.photobucket.com/albums/q107/mrgf/?action=view¤t
=OhmsLaw.jpg

　

http://s134.photobucket.com/albums/q107/
mrgf/?action=view¤t=fig6y-MeanEquations.jpg　



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

　

htt
p://en.wikipedia.org/wiki/Thurston_model_geometry

　

htt
p://en.wikipedia.org/wiki/Venn_diagram

　

　

ӌ
8;

　　

　

A) First Method Of Proving The Relative Consitency of Hilberts 1st Problem:

　

　

Comment: First, lets create an axiom of choice that we could use to solve the problem of the cardinality of sets a and b. Let's create a "Circle Axiom" that says that "All circles are the same cardinality" and apply it to the P-NP Problem diagram which is found at the website link (http://en.wikipedia.org/wiki/File:P_np_np-complete_np-hard.
svg ). Remember to consider, ironicallly, that pi is the same size for any circle. ;

　

　

http://s134.photobucket.com/albums/q107/mrg
f/?action=view¤t=VenndiagramofContinuumHypothesisusing
1AxiomCircleAxiomand1Extension.jpg



　

Comment: Notice only 1 more circle has been added to the left diagram than on the right diagram. This is key since its merely an expansion (OR EXTENSION) of 1 more circle to the left diagram which thus now represents more "integers/sets/choices" or "colors/dimensions" to the universe.

　

On the right, we see that since a does not equal b, then obviously b must be greater than a (the integers) and certainly not greater than c (the infinite reals), and so therefore b exists between them and thus the Continuum Hypothesis would not hold since it states that "no set exists between the integers and the reals". On the left, we see that a=b (by the Circle Axiom since both a and b are circles) and therefore we could assume by syllogism (a 3-step logic) then that ,"If all circles are the same cardinality, and all sets are circles, then all sets are the same cardinality" and therefore no set exists between the integers and the reals and therefore the Continuum Hypothesis does hold.

In again some ironic (yet logical) way, we see that this diagram tells us that only in the "expanded by 1 circle" left side diagram can it even be possible to solve and confirm the cardinality of sets using a "Circle Axiom" (we see 2 full circles a and b on the left diagram and therefore only in that expanded side of the diagram can we say that yes a = b). I am saying this is ironic since its this very notion of "expansion" that is the very same technique called "Extention" used to find other empirical functions such as Bézout's identity. Also, ironically, I had called an equation I used to show the relative nature of Hilberts 1st Problem an "Expanded" Diophantine equation only to later learn that it was the same technique used to find Bézout's identity which was called "Extention"!

　

　

　　

　

　


B)Second Method of Proving The Relative Consistency Of Hilberts 1st Problem:

　

　

　

Comment: First, let's revisit Aritys and Induction;

　

　

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


　

"Arities greater than 2 are seldom encountered in mathematics, except in specialized areas, and arities greater than 3 are seldom encountered in theoretical computer science. In computer programming there is often a syntactical distinction between operators and functions; syntactical operators usually have arity 0, 1 or 2."

　

　　

　

http://en.wikipedia.org/w
iki/Mathematical_induction

　

"Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers (non-negative integers). It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one."

　　　

　

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

　

"In mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +∞ and −∞ (read as positive infinity and negative infinity respectively)."

　

"The affinely extended real number system turns into a totally ordered set by defining −∞ ≤ a ≤ +∞ for all a. This order has the desirable property that every subset has a supremum and an infimum: it is a complete lattice."

　

　　

　

　

Comment: Next, let's learn what Extension is and where it's used;

　

　

　　

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

　

"The extended Euclidean algorithm is an extension to the Euclidean algorithm. Besides finding the greatest common divisor of integers a and b, as the Euclidean algorithm does, it also finds integers x and y (one of which is typically negative) that satisfy Bézout's identity

ax + by = gcd(a,b).

The extended Euclidean algorithm is particularly useful when a and b are coprime, since x is the multiplicative inverse of a modulo b, and y is the multiplicative inverse of b modulo a."



　



　　

Comment: We see here that by "extending" or "expanding" Euclid's algorithm by 2 additional sets of variables (g,c and a,b), it now can be used to find further algebraic properties called "the Bézout coefficients for (a,b)". I am just an amateur mathematician but I am going to presume the "Surgery" that Grigori Perelman used for proof of the Poincare Conjecture was most certainly a form of Extension as given above in my couple of examples;

　

　



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

　

"Accord
ing to Perelman, a modification of the standard Ricci flow, called Ricci flow with surgery, can systematically excise singular regions as they develop, in a controlled way."　

　

　

　

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

　

"Therefore, given a manifold M of dimension n = p + q and an embedding , define another n-dimensional manifold M' to be





One says that the manifold M' is produced by a surgery cutting out and glueing in , or by a p-surgery if one wants to specify the number p."　

　

"Surgery is symmetric in the sense that the manifold M can be re-obtained from M' by a (q − 1)-surgery, the trace of which coincides with the trace of the original surgery, up to orientation.

In most applications, the manifold M comes with additional geometric structure, such as a map to some reference space, or additional bundle data. One then wants the surgery process to endow M' with the same kind of additional structure. For instance, a standard tool in surgery theory is surgery on normal maps: such a process changes a normal map to another normal map within the same bordism class."

　

*KEY* "In the classical approach, as developed by Browder,Novikov,Sullivan, and Wall, surgery is done on normal maps of degree one. Using surgery, the question "Is the normal map of degree one cobordant to a homotopy equivalence?" can be translated (in dimensions greater than four) to an algebraic statement about some element in an L-group of the group ring Z[π1(X)]. More precisely, the question has a positive answer if and only if the surgery obstruction is zero, where n is the dimension of M."

　

　

　

http://en.wikipedia.org/wiki/Forc
ing_%28mathematics%29　



"Intuitions:

Forcing is equivalent to the method of Boolean-valued models, which some feel is conceptually more natural and intuitive, but usually much more difficult to apply.

Intuitively, forcing consists of expanding the set theoretical universe V to a larger universe V*. In this bigger universe, for example, one might have lots of new subsets of w = {0,1,2,…} that were not there in the old universe, and thereby violate the continuum hypothesis. While impossible on the face of it, this is just another version of Cantor's paradox about infinity. In principle, one could consider

,

identify with (x,0), and then introduce an expanded membership relation involving the "new" sets of the form (x,1). Forcing is a more elaborate version of this idea, reducing the expansion to the existence of one new set, and allowing for fine control over the properties of the expanded universe."

　

　

　

　　

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

　

*KEY* "Forcing(mathematics):

Paul Cohen invented the method of forcing while searching for a model of ZFC in which the axiom of choice or the continuum hypothesis fails. Forcing adjoins to some given model of set theory additional sets in order to create a larger model with properties determined (i.e. "forced") by the construction and the original model. For example, Cohen's construction adjoins additional subsets of the natural numbers without changing any of the cardinal numbers of the original model. Forcing is also one of two methods for proving relative consistency by finitistic methods, the other method being Boolean-valued models."

　

　　

　

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

　

*KEY* "The cardinality of R is often called the cardinality of the continuum and denoted by c, or , or (beth-one).　



"A more abstract example of an uncountable set is the set of all countable ordinal numbers, denoted by Ω (omega) or ω1. The cardinality of Ω is denoted (aleph-one). It can be shown, using the axiom of choice, that is the smallest uncountable cardinal number. Thus either , the cardinality of the reals, is equal to or it is strictly larger. Georg Cantor was the first to propose the question of whether is equal to . In 1900, David Hilbert posed this question as the first of his 23 problems. The statement that is now called the continuum hypothesis and is known to be independent of the Zermelo-Fraenkel axioms for set theory (including the axiom of choice).

　

Without the axiom of choice:

　

Without the axiom of choice, there might exist cardinalities incomparable to (namely, the cardinalities of Dedekind-finite infinite sets). Sets of these cardinalities satisfy the first three characterizations above but not the fourth characterization. Because these sets are not larger than the natural numbers in the sense of cardinality, some may not want to call them uncountable.

If the axiom of choice holds, the following conditions on a cardinal are equivalent:



and

, where and is least initial ordinal greater than

However, these may all be different if the axiom of choice fails. So it is not obvious which one is the appropriate generalization of "uncountability" when the axiom fails. It may be best to avoid using the word in this case and specify which of these one means."





　

　　

　

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

　

"Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals."

　

"Some basic sets of central importance are the empty set (the unique set containing no elements), the set of natural numbers, and the set of real numbers."

　

　

"Some ontology:

　"A set is pure if all of its members are sets, all members of its members are sets, and so on. For example, the set {{}} containing only the empty set is a nonempty pure set. In modern set theory, it is common to restrict attention to the von Neumann universe of pure sets, and many systems of axiomatic set theory are designed to axiomatize the pure sets only. There are many technical advantages to this restriction, and little generality is lost, since essentially all mathematical concepts can be modeled by pure sets. Sets in the von Neumann universe are organized into a cumulative hierarchy, based on how deeply their members, members of members, etc. are nested. Each set in this hierarchy is assigned (by transfinite recursion) an ordinal number α, known as its rank. The rank of a pure set X is defined to be one more than the least upper bound of the ranks of all members of X. For example, the empty set is assigned rank 0, while the set {{}} containing only the empty set is assigned rank 1. For each ordinal α, the set Vα is defined to consist of all pure sets with rank less than α. The entire von Neumann universe is denoted V."　

　



"Determinacy:

Determinacy refers to the fact that, under appropriate assumptions, certain two-player games of perfect information are determined from the start in the sense that one player must have a winning strategy. The existence of these strategies has important consequences in descriptive set theory, as the assumption that a broader class of games is determined often implies that a broader class of sets will have a topological property. The axiom of determinacy (AD) is an important object of study; although incompatible with the axiom of choice, AD implies that all subsets of the real line are well behaved (in particular, measurable and with the perfect set property). AD can be used to prove that the Wadge degrees have an elegant structure."

　

　

　

　　

Comment: Let's now see where axioms of choice are used;

　

　

　

http://en.wikipedia.org/wiki/Ty
chonoff%27s_theorem

　

"All of the above proofs use the axiom of choice (AC) in some way. For instance, the third proof uses that every filter is contained in an ultrafilter (i.e., a maximal filter), and this is seen by invoking Zorn's Lemma. Zorn's Lemma is also used to prove Kelley's theorem, that every net has a universal subnet. In fact these uses of AC are essential: in 1950 Kelley proved that Tychonoff's theorem implies the axiom of choice. Note that one formulation of AC is that the Cartesian product of a family of nonempty sets is nonempty; but since the empty set is most certainly compact, the proof cannot proceed along such straightforward lines. Thus Tychonoff's theorem joins several other basic theorems (e.g. that every nonzero vector space has a basis) in being equivalent to AC."

　

　

　　

　

Comment: Axioms are quintessential to analyizing any non-singular system. The best forms to analyze would haveto be the geometric and symmetrical forms. Geometric and symmetric are empirical qualities utalized by algebraic equations to help us understand the fundamental properties of more complex systems. Diophantine equations are first order empirical structures to be used to solve and comprehend such geometric and symmetric qualities.;

　

　



　

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

&
#12288;

"Another angle comes from Arakelov geometry, where Diophantine equations are studied using tools from complex geometry. The theory involves complicated comparisons between finite fields and the complex numbers. Here the existence of F1 is useful for technical reasons."

　

　

　

　

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

　

"In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. "

"In more technical language, they define an algebraic curve, algebraic surface, or more general object, and ask about the lattice points on it."

"A linear Diophantine equation is an equation between two sums of monomials of degree zero or one.

While individual equations present a kind of puzzle and have been considered throughout history, the formulation of general theories of Diophantine equations (beyond the theory of quadratic forms) was an achievement of the twentieth century."

*KEY* "The point of view of Diophantine geometry, which is the application of techniques from algebraic geometry in this field, has continued to grow as a result; since treating arbitrary equations is a dead end, attention turns to equations that also have a geometric meaning.The central idea of Diophantine geometry is that of a rational point, namely a solution to a polynomial equation or system of simultaneous equations, which is a vector in a prescribed field K, when K is not algebraically closed"

　



　

Comment: I noticed that Diophantine equations not only reflect geometric properties but they also display an important concept known as Recursion which is simply another anologue of Induction;

　

　



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

　

"Many mathematical axioms are based upon recursive rules. For example, the formal definition of the natural numbers in set theory follows: 1 is a natural number, and each natural number has a successor, which is also a natural number. By this base case and recursive rule, one can generate the set of all natural numbers."

　



　

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

　


"Matiyasevich's theorem says:

Every computably enumerable set is Diophantine. "

　

　

　

　

http://en.wikipedia.org/wik
i/Hilbert%27s_tenth_problem



"Diophantine sets:



Sets of natural numbers, of pairs of natural numbers (or even of n-tuples of natural numbers) that have Diophantine definitions are called Diophantine sets. Diophantine definitions can be provided by simultaneous systems of equations as well as by individual equations because the system

is equivalent to the single equation

A recursively enumerable set can be characterized as one for which there exists an algorithm that will ultimately halt when a member of the set is provided as input, but may continue indefinitely when the input is a non member. It was the development of computability theory (also known as recursion theory) that provided a precise explication of the intutitive notion of algorithmic computability, thus making the notion of recursive enumerability perfectly rigorous. It is evident that Diophantine sets are recursively enumerable. This is because one can arrange all possible tuples of values of the unknowns in a sequence and then, for a given value of the parameter(s), test these tuples, one after another, to see whether they are solutions of the corresponding equation. The unsolvability of Hilbert's tenth problem is a consequence of the surprising fact that the converse is true:

Every recursively enumerable set is Diophantine."

　

　

　　

　

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

　


"Equivalent formulations:

　

The following are all equivalent properties of a set S of natural numbers:

Semidecidability:

The set S is recursively enumerable. That is, S is the domain (co-range) of a partial recursive function.

There is a partial recursive function f such that:

Enumerability:

The set S is the range of a partial recursive function.

The set S is the range of a total recursive function or empty. If S is infinite, the function can be chosen to be injective.

The set S is the range of a primitive recursive function or empty. Even if S is infinite, repetition of values may be necessary in this case.

Diophantine:

There is a polynomial p with integer coefficients and variables x, a, b, c, d, e, f, g, h, i ranging over the natural numbers such that

There is a polynomial from the integers to the integers such that the set S contains exactly the non-negative numbers in its range.

The equivalence of semidecidability and enumerability can be obtained by the technique of dovetailing.

The Diophantine characterizations of a recursively enumerable set, while not as straightforward or intuitive as the first definitions, were found by Yuri Matiyasevich as part of the negative solution to Hilbert's Tenth Problem. Diophantine sets predate recursion theory and are therefore historically the first way to describe these sets (although this equivalence was only remarked more than three decades after the introduction of recursively enumerable sets). The number of bound variables in the above definition of the Diophantine set is the best known so far; it might be that a lower number can be used to define all diophantine sets."





　

　

Comment: In the History part of this paper I discovered that Induction was a form of a Mean equation ... and the last line in that last quote I linked asks if there is a form of Diophantine Set with a lower number of bound variables than in the defintion of a Diophantine Set that the article gives. Well, we see that yes my Extended Diophantine Mean equation shows there is a Diophantine Set equation with a lower number of bound variables. The point is is that Diophantine equations have not only historically been used to solve previously unsolvable math problems, but they are quite fundamentally the perfect vehicles to be used for such a purpose. I explained in the History section how I came to chose an Extended Diophantine Mean equation to be used to solve Hilberts 1st Problem, and now I am going to also link the articles describing the sets that I decided upon to be used as variables in my Extended Diophantine Mean equation which can be used to solve Hilberts 1st Problem. Notice above it said Alephone was reresented by an Omega and Sample Space can also use the same symbol Omega of which I said a summed Sample Sapce represented the same function of an infinite set that I had eventually found out through reading and my 2 equations( Diohantine and Induction equations)? The sets I am chosing for the equations to prove these relative consistency problems are Alephnaught and Alephone;

　

　

　

http://en.wikipedia.org/wik
i/Aleph-naught#Aleph-naught



" is the cardinality of the set of all natural numbers, and is the first transfinite cardinal. A set has cardinality if and only if it is countably infinite, which is the case if and only if it can be put into a direct bijection, or "one-to-one correspondence", with the natural numbers. Such sets include the set of all prime numbers, the set of all integers, the set of all rational numbers, the set of algebraic numbers, the set of binary strings of all finite lengths, and the set of all finite subsets of any countably infinite set.

If the axiom of countable choice (a weaker version of the axiom of choice) holds, then is smaller than any other infinite cardinal."

　

　

　

http://en.wikipedia.org/wi
ki/Aleph_number#Aleph-one

　

" is the cardinality of the set of all countable ordinal numbers, called ω1 or (sometimes) Ω. Note that this ω1 is itself an ordinal number larger than all countable ones, so it is an uncountable set. Therefore is distinct from . The definition of implies (in ZF, Zermelo-Fraenkel set theory without the axiom of choice) that no cardinal number is between and . If the axiom of choice (AC) is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus is the second-smallest infinite cardinal number."

　

　

　



Comment: Therefore, I have my type of equation determined (Extended Diophantine Mean equation) and now have my variables specified (as opposed to simply using a and b) to create an Extended Diophantine Mean equation from which an axiom of choice can be applied to show the relative nature of Hilberts 1st Problem. I am proposing that in Computability Theory, that if an axiom of choice is applied to an Extended Diophantine Mean equation that results in an unchanged outcome from the equation, then the Continuum Hypothesis will hold. I am proposing that it is the wording of the axiom of choice that determines the result of the changed or unchanged outcome of the equation that is being used to test the Continuum Hypothesis (or Hilberts 1st Problem)! *Please note, calling any equation a "Theory of Everything" today draws extreme scrutiny and originally I felt all Induction/Recursion/Diophantine equations WERE what many conjectured would be a Theory of Everything and so in my early first draft of my class of Extended Diophantine Mean equations I had labelled them TOE's since I knew they were empirical and foundational but I have now since decided to label this handy group of fundamental equations simply "Tautological Ontological Empirical" equations or "TOE" for short *grin. *Please also note, that the first example of my TOE equations is just the Summation example of the Extended Diophantine Mean equation and its only the latter 2 example equations where theres more than one Alephnaught and Alephone in the equation that can be used via an axiom of choice to show the relative nature of the Continuum Hypothesis;

　

　



http://s134.photobucket.com/albums/q107/mrgf/?action=view&cu
rrent=SummationInductionandTautologicalOntologicalEmpiricale
xamples.jpg

　

　

　

Comment: The Extended Diophantine Mean equations are at the bottom - only the last 2 are for use in solving Hilberts 1st Problem because they have 2 sets of Alephnaught's and Alephone's - the first equation at the bottom is the Summation version which still equals the other Extended Diophantine Mean equations but has only 1 Alephnaught and 1 Alephone and therefore cannot be used to solve Hilberts 1st Problem.

This is where we create an "Axiom of Choice" that states, "Change all Alephone's to Bethone's" to an Extended Diophantine Mean equation . If, in Computational Theory, the resulted outcome still computes the same size ratio/mean value - meaning: although the variables may be different (Bethone instead of Alephone), the size of the value of the final outcome is still the same as before the variables were changed from Alephone to Bethone and therefore the Continuum Hypothesis still holds because obviously Alephone must equal Bethone in cardinality. Conversely, if the axiom of choice is reworded so that it instead states, "Change only one Alephone to Bethone" in an Extended Diophantine Mean equation, we will see that now the final outcome computed no longer shows the same size ratio/mean value as it did before only one Alephone was changed to Bethone and therefore the Continuum Hypothesis does not hold and we would more convincingly see that the axiom of choice is what determines the relative result of the Continuum Hypothesis of whether Alephone equals Bethone (Hilberts 1st Problem)! This is simply a logic gate that computes a different size value determinant on how the axiom of choice is worded.

-----------------------------------------------------
------------------------------------------------------------
--------------------------------------

　

　

Part 4: Proving The Relative Consistency of the P-NP Problem

　





　

http://en.wikipedia.org/wiki/File:P_np_np-complete_n
p-hard.svg

　

　

Comment: I am going to simply state that if I have proven through 1 axiom and an Extended Diophantine Mean equation the relativite consistency of the Continuum Hypothesis (does set a = set b?) and then have shown in Venn diagram form the relative consistency of the Contiuum Hypothesis is the same seen in the exact same Venn diagram used to show the relative consistency of the P-NP problem, then I'd think intuitively that both problems must be equivalent or similiar.

　

I believe the difference between the sets of integers (Alephnaught) and the sets of reals (Bethone) in the Continuum Hypothesis must be the same difference between the group of Polynomial Time type problems with the group of Non-Polynomial Time type problems.

　

The following I would consider a relevant point about the relative consistency observation that Potential Theory in two dimensions is different from Potential Theory in more dimensions. Again, I am saying that Potential Theory may also be another anologue of the Continuum Hypothesis or the P-NP Problem;

　

　

http://en.wikipedia.org/wiki/Potenti
al_theory

　

"Two dimensions:

　

From the fact that the group of conformal transforms is infinite dimensional in two dimensions and finite dimensional for more than two dimensions, one can surmise that potential theory in two dimensions is different from potential theory in other dimensions. This is correct and, in fact, when one realizes that any two-dimensional harmonic function is the real part of a complex analytic function, one sees that the subject of two-dimensional potential theory is substantially the same as that of complex analysis. For this reason, when speaking of potential theory, one focuses attention on theorems which hold in three or more dimensions. In this connection, a surprising fact is that many results and concepts originally discovered in complex analysis (such as Schwartz's theorem, Morera's theorem, the Weierstrass-Casorati theorem, Laurent series, and the classification of singularities as removable, poles and essential singularities) generalize to results on harmonic functions in any dimension. By considering which theorems of complex analysis are special cases of theorems of potential theory in any dimension, one can obtain a feel for exactly what is special about complex analysis in two dimensions and what is simply the two-dimensional instance of more general results."

　

　



http://en.wikipedia.org/wiki/P_%28complexity%29

　

"P can also be viewed as a uniform family of boolean circuits"　

　

"P is known to contain many natural problems, including the decision versions of linear programming, calculating the greatest common divisor, and finding a maximum matching. In 2002, it was shown that the problem of determining if a number is prime is in P."

　

　

Comment: Here we see that the prime numbers are in P and we know that Set Theory deals with natural and prime numbers. It is simply a case of apples and oranges when it comes to differentiating between Hilberts 1st Problem and the P-NP Problem. From the following quote I want to say I was very pleased to know there is other people out there pondering the common relativity of these various famous math problems;

　

　



http://rjlipton.wordpress.com/conventional-wisdom-and-pnp/

&#
12288;

"Miklos/December 30, 2010 6:44 pm

　

I keep wondering if P=NP could be independent of the ZFC axioms, just like the continuum hypothesis. This would mean that a practical implementation of P = NP could not be found, but it would not be possible to prove that they are different either. Very intriguing…"



------------------------------------------------------------
------------------------------------------------------------
-------------------------------

　



　

Part 5: Relative Consistency Principles Similiar To Hilberts 1st and P-NP Problem in Other Areas of Math

　

　

Comment: Pay attention to all the instances of the set of pairs a and b or multiple sets. It may seem trivial or arbitrary to list many examples of the set of pairs a and b but I say it is anything but arbitrary and quite obviously the essential foundation of empirical formulations or axioms found in mathematics and thus relevant to my paper discussing the relativity of set a with set b (Hilberts 1st Problem) or the relativity of the group of Polynomial Time problems with the group of Non-Polynomial Time problems (the P-NP Problem);

　

　

http://s134.photobucket.com/albums/
q107/mrgf/?action=view¤t=ANDorOR.gif

http://s134.photob
ucket.com/albums/q107/mrgf/?action=view¤t=XOR.gif



　

　　

http://itee.uq.edu.au/~cogs2010/cmc/
chapters/BackProp/index2.html

　

"From a geometrical perspective, the perceptron attempts to solve the AND, OR, and XOR problems by using a straight line to separate the two classes: inputs labelled "0" are on one side of the line and inputs labelled "1" are on the other side. For the AND and OR functions, this is easy to do whereas for XOR it is not. The line separating the two classes is determined by net = theta. For two-diminensional problems such as AND, OR, and XOR, the line corresponds to

　

(I1 w1) + (I2 w2) = theta

　

Solving for I2 yields

　

I2 = -(w1/w2)I1 + (theta/w2).

　

In higher dimensions the boundary separating the two classes is a hyperplane

　

net = (sum)iwiIi.

　

All problems which can be solved by separating the two classes with a hyperplane are called linearly separable. The XOR problem (as presented) is not a linearly separable problem.

　

However, we can recode the XOR problem in three dimensions so that it becomes linearly separable "

　

　

　

Comment: I consider the difference between the set of integers (Alephnaught) and the set of reals (Bethone) to not only represent the difference between the group of Polynomial Time problems and the group of Non-Polynomial Time problems but also another anologue of the difference between the basic AND/OR logic gates with the XOR logic gate. I also believe the dual nature of the relative unprovable contradicting states found in both the Continuum Hypothesis and the P-NP Problem is also seen not only in Hardys Paradox, but also in the "quantum pairing" of Cooper Pairs. I'd simply refer to the decisions by the 2 "persons" involved in the Hardys Paradox example to represent both sides of the axiom of choice and I would also simply refer to the gate capacitor in the Cooper-pair box to represent an on/off axiom of choice ;

　

　

　

http://www.sciencedaily.com/release
s/2009/03/090304091231.htm

　

"Hardy's Paradox, the axiom that we cannot make inferences about past events that haven't been directly observed while also acknowledging that the very act of observation affects the reality we seek to unearth, poses a conundrum that quantum physicists have sought to overcome for decades."



　

　

　

http://granades.com/2009/03/30/hardy
s-paradox-or-the-economist-is-dismal-at-science/

　

http
://granades.com/wordpress/wp-content/uploads/2009/03/paths3.
jpg

　

"It turns out weak-valued probabilities don’t have to be positive definite, but what does that mean? In their paper, Lundeen and Steinberg say,　

"Recall that the joint values are extracted by studying the polarization rotation of both photons in conicidence…. As in all weak measurement experiments, a negative weak value implies that the shift of a physical "pointer" (in this case, photon polarization) has the opposite sign from the one expected from the measurement interaction itself." "

　

　

　

http://en.wikipedia.org/wiki/File:C
ooper_pair_box_circuit.png

　

　



　

http://en.wikipedia.org/wiki/Weierstrass_approximati
on_theorem

　

"In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on an interval [a,b] can be uniformly approximated as closely as desired by a polynomial function. Because polynomials are among the simplest functions, and because computers can directly evaluate polynomials, this theorem has both practical and theoretical relevance, especially in polynomial interpolation. The original version of this result was established by Karl Weierstrass in 1885."

　



　

　

http://en.wikipedia.org/wiki/B%C3%A9zout%27s
_identity

　

"In number theory, Bézout's identity for two integers a, b is an expression

　

ax+by=d \,

　

where x and y are integers (called Bézout coefficients for (a,b)), which is such that d is a common divisor of a and b. Bézout's lemma states that such coefficients exist for every pair of nonzero integers (a,b), with in addition d > 0, which is then in fact their greatest common divisor and also the smallest positive integer that can be written in this form for any integers x,y. This value of d is therefore uniquely determined by a and b, but the Bézout coefficients are not unique. A pair of Bézout coefficients (in fact the ones that are minimal in absolute value) can be computed by the extended Euclidean algorithm. "

　

　



　

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

2288;

　

"It follows that there are also infinite solutions if c is a multiple of the greatest common divisor of a and b. If c is not a multiple of the greatest common divisor of a and b, then the Diophantine equation ax + by = c has no solutions."

　

　

　

　

http://en.wikiped
ia.org/wiki/Diophantine_set

　

"Examples:

　

The well known Pell equation

x2 − d(y + 1)2 = 1

is an example of a Diophantine equation with a parameter. As has long been known, the equation has a solution in the unknowns x,y precisely when the parameter d is 0 or not a perfect square. In the present context, one says that this equation provides a Diophantine definition of the set

{0,2,3,5,6,7,8,10,...}

consisting of 0 and the natural numbers that are not perfect squares. Other examples of Diophantine definitions are as follows:

The equation a = (2x + 3)y defines the set of numbers that are not powers of 2.

The equation a = (x + 2)(y + 2) defines the set of numbers greater than 1 that are not prime numbers.

The equation a + x = b defines the set of pairs such that "

　



　

　

http://en.wikipedia.org/wiki/Kernel_extensio
n_theorem　

　

"Formally, the dimension theorem for vector spaces states that

Given a vector space V, any two linearly independent generating sets (in other words, any two bases) have the same cardinality.

If V is finitely generated, then it has a finite basis, and the result says that any two bases have the same number of elements.

The proof of the existence of a basis for any vector space in the general case requires Zorn's lemma and is in fact equivalent to the axiom of choice, and the proof given below assumes trichotomy (all cardinal numbers are comparable), which is also equivalent to the axiom of choice. The theorem can be generalized to arbitrary R-modules for rings R having invariant basis number."

　

　

　

　

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

　

"If a is algebraic over K, then K[a], the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. In the special case where K = Q is the field of rational numbers, Q[a] is an example of an algebraic number field.

A field with no proper algebraic extensions is called algebraically closed. An example is the field of complex numbers. Every field has an algebraic extension which is algebraically closed (called its algebraic closure), but proving this in general requires some form of the axiom of choice.

An extension L/K is algebraic if and only if every sub K-algebra of L is a field."

　

　



　

http://en.wikipedia.org/wiki/Euler%27s_formula

ӌ
8;

"The original proof is based on the Taylor series expansions of the exponential function ez (where z is a complex number) and of sin x and cos x for real numbers x (see below). In fact, the same proof shows that Euler's formula is even valid for all complex numbers z."

　

"Now, taking this derived formula, we can use Euler's formula to define the logarithm of a complex number. To do this, we also use the definition of the logarithm (as the inverse operator of exponentiation) that

and that

both valid for any complex numbers a and b.

Therefore, one can write:

for any z ≠ 0. Taking the logarithm of both sides shows that:

and in fact this can be used as the definition for the complex logarithm. The logarithm of a complex number is thus a multi-valued function, because φ is multi-valued.

Finally, the other exponential law

which can be seen to hold for all integers k, together with Euler's formula, implies several trigonometric identities as well as de Moivre's formula."

　

　

　

　　

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

　

"The first proof of the existence of irrational numbers is usually attributed to a Pythagorean (possibly Hippasus of Metapontum),[7] who probably discovered them while identifying sides of the pentagram.[8] The then-current Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as the other. However, Hippasus, in the 5th century BC, was able to deduce that there was in fact no common unit of measure, and that the assertion of such an existence was in fact a contradiction. He did this by demonstrating that if the hypotenuse of an isosceles right triangle was indeed commensurable with an arm, then that unit of measure must be both odd and even, which is impossible. His reasoning is as follows:

　

* The ratio of the hypotenuse to an arm of an isosceles right triangle is c:b expressed in the smallest units possible.

* By the Pythagorean theorem: c2 = a2+b2 = 2b2. (Since the triangle is isosceles, a = b.)

* Since c2 is even, c must be even.

* Since c:b is in its lowest terms, b must be odd.

* Since c is even, let c = 2y.

* Then c2 = 4y2 = 2b2

* b2 = 2y2 so b2 must be even, therefore b is even.

* However we asserted b must be odd. Here is the contradiction."

　

　

　



http://en.wikipedia.org/wiki/%CE%98_%28set_theory%29

　


" In set theory, Θ (pronounced like the letter theta) is the least nonzero ordinal α such that there is no surjection from the reals onto α.

If the axiom of choice (AC) holds (or even if the reals can be wellordered) then Θ is simply , the cardinal successor of the cardinality of the continuum. However, Θ is often studied in contexts where the axiom of choice fails, such as models of the axiom of determinacy.

Θ is also the supermum of the lengths of all prewellorderings of the reals.



Proof of existance:



It may not be obvious that it can be proved, without using AC, that there even exists a nonzero ordinal onto which there is no surjection from the reals (if there is such an ordinal, then there must be a least one because the ordinals are wellordered). However, suppose there were no such ordinal. Then to every ordinal α we could associate the set of all prewellorderings of the reals having length α. This would give an injection from the class of all ordinals into the set of all sets of orderings on the reals (which can to be seen to be a set via repeated application of the powerset axiom). Now the axiom of replacement shows that the class of all ordinals is in fact a set. But that is impossible, by the Burali-Forti paradox."

　

　

　　

　

http://en.w
ikipedia.org/wiki/Intersection_theory

　

"This is a symmetric form for n even, in which case the signature of M is defined to be the signature of the form, and an alternating form for n odd"

　　

"By Poincaré duality, it turns out that there is a way to think of this geometrically. If possible, choose representative n-dimensional submanifolds A, B for the Poincaré duals of a and b. Then λM(a, b) is the oriented intersection number of A and B, which is well-defined because of the dimensions of A and B. This explains the terminology intersection form."

　

"Intersection theory in algebraic geometry:

　

William Fulton in Intersection Theory (1984) writes

　

... if A and B are subvarieties of a non-singular variety X, the intersection product A.B should be an equivalence class of algebraic cycles closely related to the geometr

view post as summary

reply to this thread

cont'd:

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

z
88;

"This is a symmetric form for n even, in which case the signature of M is defined to be the signature of the form, and an alternating form for n odd"

　　

"By Poincaré duality, it turns out that there is a way to think of this geometrically. If possible, choose representative...

view entire post

cont'd:

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

z
88;

"This is a symmetric form for n even, in which case the signature of M is defined to be the signature of the form, and an alternating form for n odd"

　　

"By Poincaré duality, it turns out that there is a way to think of this geometrically. If possible, choose representative n-dimensional submanifolds A, B for the Poincaré duals of a and b. Then λM(a, b) is the oriented intersection number of A and B, which is well-defined because of the dimensions of A and B. This explains the terminology intersection form."

　

"Intersection theory in algebraic geometry:

　

William Fulton in Intersection Theory (1984) writes

　

... if A and B are subvarieties of a non-singular variety X, the intersection product A.B should be an equivalence class of algebraic cycles closely related to the geometry of how A∩B, A and B are situated in X. Two extreme cases have been most familiar. If the intersection is proper, i.e. dim(A∩B) = dim A + dim B − dim X, then A.B is a linear combination of the irreducible components of A∩B, with coefficients the intersection multiplicities. At the other extreme, if A = B is a non-singular subvariety, the self-intersection formula says that A.B is represented by the top Chern class of the normal bundle of A in X. "

　

　

　



http://en.wikipedia.org/wiki/Many-one_reduction



"Formal languages

　

Suppose A and B are formal languages over the alphabets Σ and Γ, respectively. A many-one reduction from A to B is a total computable function f : Σ* → Γ* that has the property that each word w is in A if and only if f(w) is in B (that is, A = f − 1(B)).

If such a function f exists, we say that A is many-one reducible or m-reducible to B and write

If there is an injective many-one reduction function then we say A is 1 reducible or one-one reducible to B and write

"



　

　

　

　

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

　

"In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function. (The terminology comes from the Ancient Greek meros (μέρος), meaning part, as opposed to holos (ὅλος), meaning whole.)

　

Every meromorphic function on D can be expressed as the ratio between two holomorphic functions (with the denominator not constant 0) defined on D: any pole must coincide with a zero of the denominator.

The Gamma function is meromorphic in the whole complex plane.

　

Intuitively then, a meromorphic function is a ratio of two well-behaved (holomorphic) functions. Such a function will still be well-behaved, except possibly at the points where the denominator of the fraction is zero. (If the denominator has a zero at z and the numerator does not, then the value of the function will be infinite; if both parts have a zero at z, then one must compare the multiplicities of these zeros.)

　

*KEY* From an algebraic point of view, if D is connected, then the set of meromorphic functions is the field of fractions of the integral domain of the set of holomorphic functions. This is analogous to the relationship between , the rational numbers, and , the integers."　



　



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

　

"IIn field theory, the expression is only shorthand for the formal expression ab−1, where b−1 is the multiplicative inverse of b. Since the field axioms only guarantee the existence of such inverses for nonzero elements, this expression has no meaning when b is zero. Modern texts include the axiom 0 ≠ 1 to avoid having to consider the trivial ring or a field with one element, where the multiplicative identity coincides with the additive identity.."

　

　



　

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



2288;

"Definitions



G is a connected, reductive algebraic group over an algebraically closed field.

B is a Borel subgroup of G

W is a Weyl group of G corresponding to a maximal torus of B.

The Bruhat decomposition of G is the decomposition

of G as a disjoint union of double cosets of B parameterized by the elements of the Weyl group W. (Note that although W is not in general a subgroup of G, the coset wB is still well defined.)"

　

　

　



　

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

　

"I
n algebraic geometry, a Schubert variety is a certain subvariety of a Grassmannian, usually with singular points. Described by means of linear algebra, a typical example consists of the k-dimensional subspaces V of an n dimensional vector space W, such that

for j = 1, 2, ..., k, where

is a certain flag of subspaces in W and 0 < a1 < ... < ak ≤ n. More generally, given a semisimple algebraic group G with a Borel subgroup B and a standard parabolic subgroup P, it is known that the homogeneous space X = G/P, which is an example of a flag variety, consists of finitely many B-orbits that may be parametrized by certain elements of the Weyl group W. The closure of the B-orbit associated to an element w of the Weyl group is denoted by Xw and is called a Schubert variety in G/P. The classical case corresponds to G = SLn and P being the kth maximal parabolic subgroup of G."



　



　

http://en.wikipedia.org/wiki/Dirichlet_L-function



2288;

"These functions are named after Johann Peter Gustav Lejeune Dirichlet who introduced them in (Dirichlet 1837) to prove the theorem on primes in arithmetic progressions that also bears his name. In the course of the proof, Dirichlet shows that L(s, χ) is non-zero at s = 1. Moreover, if χ is principal, then the corresponding Dirichlet L-function has a simple pole at s = 1."

　

"Just as the Riemann zeta function is conjectured to obey the Riemann hypothesis, so the Dirichlet L-functions are conjectured to obey the generalized Riemann hypothesis."

　

　





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

　


"In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a recursively presented group G is the algorithmic problem of deciding whether two words represent the same element. Although it is common to speak of the word problem for the group G strictly speaking it is a presentation of the group that does or does not have solvable word problem. Given two finite presentations P and Q of a group G, P has solvable word problem if and only if Q does.[citation needed] So in this case no confusion is caused by speaking of the word problem for G. When the group is recursively presented but not finitely presented, the distinction becomes important.

　

The related but different uniform word problem for a class K of recursively presented groups is the algorithmic problem of deciding, given as input a presentation P for a group G in the class K and two words in the generators of G, whether the words represent the same element of G. Some authors require the class K to be definable by a recursively enumerable set of presentations."

　





　

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



"A cobordism is a manifold W with boundary whose boundary is partitioned in two, .

Cobordisms are studied both for the equivalence relation that they generate, and as objects in their own right. Cobordism is a much coarser equivalence relation than diffeomorphism or homeomorphism of manifolds, and is significantly easier to study and compute. It is not possible to classify manifolds up to diffeomorphism or homeomorphism in dimensions because the word problem for groups cannot be solved, but it is possible to classify manifolds up to cobordism. Cobordisms are central objects of study in geometric topology and algebraic topology. In geometric topology, cobordisms are intimately connected with Morse Theory, and h-cobordisms are fundamental in the study of high dimensional manifolds, namely surgery theory."



　

　

　



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

&#
12288;

"Definition:　

Let Z*(X) := Z[X] be the free abelian group on the algebraic cycles of X. Then an adequate equivalence relation is a family of equivalence relations, ∼X on Z*(X), one for each smooth projective variety X, satisfying the following three conditions:

(Linearity) The equivalence relation is compatible with addition of cycles.

(Moving lemma) If are cycles on X, then there exists a cycle such that α ~X α' and α' intersects β properly.

(Push-forwards) Let and be cycles such that β intersects properly. If α ~X 0, then ~Y 0, where is the projection.

The push-forward cycle in the last axiom is often denoted

If β is the graph of a function, then this reduces to the push-forward of the function. The generalizations of functions from X to Y to cycles on X × Y are known as correspondences. The last axiom allows us to push forward cycles by a correspondence."



　



　

http://en.wikipedia.org/wiki/Motive_%28algebraic_geo
metry%29　

　

"In algebraic geometry, a motive (or sometimes motif, following French usage) denotes 'some essential part of an algebraic variety'. To date, pure motives have been defined, while conjectural mixed motives have not. Pure motives are triples (X, p, m), where X is a smooth projective variety, p : X ⊢ X is an idempotent correspondence, and m an integer. A morphism from (X, p, m) to (Y, q, n) is given by a correspondence of degree n - m."

　

"The theory of motives was originally conjectured as an attempt to unify a rapidly multiplying array of cohomology theories, including Betti cohomology, de Rham cohomology, l-adic cohomology, and crystalline cohomology. The general hope is that equations like

　

* [point]

* [projective line] = [line] + [point]

* [projective plane] = [plane] + [line] + [point]

　

can be put on increasingly solid mathematical footing with a deep meaning. Of course, the above equations are already known to be true in many senses, such as in the sense of CW-complex where "+" corresponds to attaching cells, and in the sense of various cohomology theories, where "+" corresponds to the direct sum.

　

From another viewpoint, motives continue the sequence of generalizations from rational functions on varieties to divisors on varieties to Chow groups of varieties. The generalization happens in more than one direction, since motives can be considered with respect to more types of equivalence than rational equivalence. The admissiable equivalences are given by the definition of an adequate equivalence relation."

　

　

　

http://en.wikipedia.org/wi
ki/Zorn%27s_lemma

　

"Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory that states:

　

Suppose a partially ordered set has the property that every chain (i.e. totally ordered subset) has an upper bound. Then the set contains at least one maximal element.

　

It is named after the mathematicians Max Zorn and Kazimierz Kuratowski.

　

The terms are defined as follows. Suppose (P,≤) is a partially ordered set. A subset T is totally ordered if for any s, t in T we have s ≤ t or t ≤ s. Such a set T has an upper bound u in P if t ≤ u for all t in T. Note that u is an element of P but need not be an element of T. A maximal element of P is an element m in P such that for no element x in P, m < x.

　

Zorn's lemma is equivalent to the well-ordering theorem and the axiom of choice, in the sense that any one of them, together with the Zermelo–Fraenkel axioms of set theory, is sufficient to prove the others. It occurs in the proofs of several theorems of crucial importance, for instance the Hahn–Banach theorem in functional analysis, the theorem that every vector space has a basis, Tychonoff's theorem in topology stating that every product of compact spaces is compact, and the theorems in abstract algebra that every nonzero ring has a maximal ideal and that every field has an algebraic closure."

　

　

　

　

　

Comment: Again, I am stating to consider here the Bruhat decomposition, the Schubert cell, the Dirichlet L-function, the Word Problem for groups, Cobordisms, adequate equivalence relations, Motives, and Zorn's Lemma can all be said to display a relative consistency using finitistic methods and all are more analogues of the sets a and b of the Contniuum Hypothesis and the groups of the P-NP Problem. Here are just a few more areas of math that may be relevant to the ideas of this paper;

　

　

　

　

http://en.wikipedia.or
g/wiki/PR_%28complexity%29

　

"The Ackermann function is an example of a function that is not primitive recursive, showing that PR is strictly contained in R."

　

　

"PR functions can be explicitly enumerated, whereas not all functions R can be. This shows that 'PR' has a syntactic definition, whereas R lacks one.

　

On the other hand, we can "enumerate" any recursively enumerable set (see also its complexity class RE) by a primitive-recursive function in the following sense: given an input (M, k), where M is a Turing machine and k is an integer, if M halts within k steps then output M; otherwise output nothing. Then the union of the outputs, over all possible inputs (M, k), is exactly the set of M that halt."

　

　

　

http://en.wikipedia.org/wiki/A
lgebraically_closed_field

　

""If a proposition which can be expressed in the language of first-order logic is true for an algebraically closed field, then it is true for every algebraically closed field with the same characteristic. Furthermore, if such a proposition is valid for an algebraically closed field with characteristic 0, then not only is it valid for all other algebraically closed fields with characteristic 0, but there is some natural number N such that the proposition is valid for every algebraically closed field with characteristic p when p > N"

　

　

http://en.wikipedia.org/wiki/Euler's_phi_f
unction



"He introduced a replacement for the prime numbers in the cyclotomic field Q(ζp), expressed the failure of unique factorization quantitatively via the class number hp and proved that if hp is not divisible by p (such numbers p are called regular primes) then Fermat's theorem is true for the exponent n = p."

　

　



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

　

"Quantum theory also indicates that molecular orbitals (MO) of identical symmetry actually mix. As a practical consequence of this mixing of diatomic molecules, the wavefunctions s+s and pz+pz molecular orbitals become blended. The extent of this mixing (or blending) depends on the relative energies of the like-symmetry MO's."

　

　



http://physics.aps.org/articles/v2/32

　

""But Aharonov, together with David Albert and Lev Vaidman [3] understood where the problem was. They realized that measuring Sπ/4 was tantamount to simultaneously measuring the spin along the x and z directions, an impossible task since they would disturb each other. But one could in fact limit the mutual disturbance if one is willing to pay the penalty of allowing the measurements be less precise. They called such measurements “weak” because of the reduced disturbance and these outcomes are called “weak values.” When such measurements are performed, Sπ/4 does indeed take the impossible value of √2/2."

　

　





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

　

"The complexity of the best known algorithms is exponential in the smaller of the two parameters N and P. Thus, the problem is most difficult if N and P are of the same order. It only becomes easy if either N or P becomes very small.

　

If N (the number of variables) is small, then an exhaustive search for the solution is practical. If P (the number of place values) is a small fixed number, then there are dynamic programming algorithms that can solve it exactly.

　

What is happening is that the problem becomes seemingly non-exponential when it is practical to count the entire solution space. There are two ways to count the solution space in the subset sum problem. One is to count the number of ways the variables can be combined. There are 2N possible ways to combine the variables. However, with N = 10, there are only 1024 possible combinations to check. These can be counted easily with a branching search. The other way is to count all possible numerical values that the combinations can take. There are 2P possible numerical sums. However, with P = 5 there are only 32 possible numerical values that the combinations can take. These can be counted easily with a dynamic programming algorithm. When N = P and both are large, then there is no aspect of the solution space that can be counted easily."

　



　

　

http://en.wikipedia.org/wiki/G%C3%B6del_numb
er

　

" To encode an entire formula, which is a sequence of symbols, Gödel used the following system. Given a sequence x1x2x3...xn of positive integers, the Gödel encoding of the sequence is the product of the first n primes raised to their corresponding values in the sequence:

"



"Once a Gödel numbering for a formal theory is established, each inference rule of the theory can be expressed as a function on the natural numbers. If f is the Gödel mapping and if formula C can be derived from formulas A and B through an inference rule r; i.e.

then there should be some arithmetical function gr of natural numbers such that

This is true for the numbering Gödel used, and for any other numbering where the encoded formula can be arithmetically recovered from its Gödel number.



*KEY*Thus, in a formal theory such as Peano arithmetic in which one can make statements about numbers and their arithmetical relationships to each other, one can use a Gödel numbering to indirectly make statements about the theory itself. This technique allowed Gödel to prove results about the consistency and completeness properties of formal systems.



Generalizations:



In computability theory, the term "Gödel numbering" is used in settings more general than the one described above. It can refer to:



Any assignment of the elements of a formal language to natural numbers in such a way that the numbers can be manipulated by an algorithm to simulate manipulation of elements of the formal language.

More generally, an assignment of elements from a countable mathematical object, such as a countable group, to natural numbers to allow algorithmic manipulation of the mathematical object.



Also, the term Gödel numbering is sometimes used when the assigned "numbers" are actually strings, which is necessary when considering models of computation such as Turing machines that manipulate strings rather than numbers."

　

-------------------------------------------
------------------------------------------------------------
------------------------------------------------





Part 6: Conclusion

　

　

In attempting to prove relative consistency of the Continuum Hypothesis and the P-NP Problem by finitistic methods, I have shown many, many, relative empirical math principles usually involving sets or pairs of sets with an axiom applied to show the relative qualities observed which are relevant to the objective of this paper (to show the relative duality of the Contiuum Hypothesis and the P-NP Problem). If, in the very least, you agree I have used an Extended Diophantine Mean equation to prove the relative consistency of the Continuum Hypothesis, then I feel you should understand the 2 Venn diagrams (one addressing the Continuum Hypothesis and the other addressing the P-NP Problem) I showed are both the very same chart but only the variables renamed. If you agree they are the same chart then you would haveto agree the problems they address must be the same.

　

　

I hope you enjoyed the paper and I look forward to, if possible, future discoverys or solutions to come from it.

　

　

Regards,

GF

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All form is the result of resonances of space or D, which are products of velocity and time. Particles are merely the interference patterns of those resonances and can be associated with various matrices associated with resonances. In certain conditions it is clear that certain particles are associated with resonances, but then again when at scales that subdivide those resonances, those...

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All form is the result of resonances of space or D, which are products of velocity and time. Particles are merely the interference patterns of those resonances and can be associated with various matrices associated with resonances. In certain conditions it is clear that certain particles are associated with resonances, but then again when at scales that subdivide those resonances, those measurements fail since the subdivisions and associated particles are less than the products that resulted in said resonances. But all form is the result of matrices of velocity and time acting in various dimension of resonance D of the ether space.



The concept of ether field means that there would be propagation of effects and action at a distance, but also since due to the inductive nature of fields there also has to be immediate action since the existence of 1 is tied automatically unto another in v and t. The differences between the nature of related fields is merely functions related to time and velocity. The difference between magnetic field and current is time and velocity respectively, and the similarity is D or resonance space. The force is a product of space or D. so maybe in evaluation of those differences, some are instantaneous and others a propagating, and thus from my point what has value in assigning which as special, related to time, and related to velocity, has meaning through propagation and induction. And then perhaps a propagating resonant field does both, act at a distance and those that are instantaneous are related to time and velocity.

All things are products of time and velocity. Equivalency means that all things through all dimensions are products of time and velocity held together by a math or matrix and related as scalar, (additive), proper (multiplicative) within various dimensions connected through states of resonance and induction. Space is made from loops of a velocity and time matrix, woven together with properties almost like a fluid. The geometry, compression, and configuration of this through multiple dimensions of inductive states through dipole action result in a continuous continuum that is multidimensional. So therefore the atom is the perfect Trans dimensional model, exhibiting all states and transitions of the matrix or space, so there is no distinguishing between the cores or the space. The particle is merely the overlapping of the matrix in nodes of interference. The centres of all particles are the singularities, of the boundaries that go from the interval to infinity. One of the controls is the number pi. The motion of these various dimensional or resonant states act together one relating to another. So therefore the entire system of space and particle is an inertial field or matrix. All these relationship at the smallest level are products of time and velocity. So then the idea of mass, particularity used to define force and energy, are arbitrary assignments. Since the smallest denominator has nothing to do with mass directly, and the defining of systems of ether and atomic states in terms of a larger configuration or "energy" in joules is not periodic, meaning not in system of the assignment of parts and properties as we have in a periodic table of chemistry. If we redefine all properties as arising from three units, to include v and t, then the associated properties are an adjunct to the real structure, and that v or t is the energy (new word) that is in all things, and particles associated with various states. The Higgs is more a test of the system of units, the defining of mass, than the finding of a particle.

The evaluation of all particles in terms of mass and energy is linear and not periodic. Mass is a higher periodic state than velocity and time which are at the bottom. v and t as the basics and through the periodic development of v and t we finally get the effect of mass. Therefore to evaluate those things which arise out of more simple relationships of vt, such as mass less particles, with something much more evolved, such as mass, is not providing the clarity to the structure of the real universe.

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