Wednesday, August 7, 2013

Gödel's Incompleteness Theorem is Incomplete

Objective Physics is my attempt to create a completely objective, mathematically complete and consistent 'theory of everything'. Its only premise is "Existence exists", aka the 'axiom of identity', or in Ayn Rand's terms: 'A is A'.

Every mathematical formula objective mathematics generates is ultimately based on the axiom of identity. In other words, objective mathematics is a mathematical system based on objectivism. It provides a formal mathematical basis for objectivist metaphysics.

Logicians and mathematicians may point out that Kurt Gödel's incompleteness theorem proves no mathematical system at or above the complexity of Peano arithmetic can be complete and consistent even relative to itself, let alone complete and consistent in the universal domain.

Consider this: Kurt Gödel's incompleteness theorem proves itself incomplete. ;)

The proof of the incompleteness theorem is based on Gödel numbering, which is a way to use numbers to represent other numbers and mathematical operators 'indirectly'. As such, the proof of Godel's incompleteness theorems uses a mathematical system to prove mathematics incomplete, but in doing so, it proves itself incomplete.

Incomplete means 'not complete'. Nothing can be incomplete unless what it is not, is complete. Every known mathematical system can't be incomplete unless some unknown mathematical system is complete. That means the incompleteness theorem proves a complete mathematical system exists outside the domain of the mathematics covered by the incompleteness theorem.

Objective mathematics is that complete and consistent mathematical system.

Current mathematical systems are based on the semantics of indirect representation. Some symbol, or some number is used to represent some quantity or entity in the system it represents indirectly. All indirect representations are 'not direct'. Nothing can be indirect, unless what it is not, is direct. No indirect representation can exist unless it is based on a pre-existing direct representation. Direct representation is the logical converse of indirect representation. Objective mathematics is based on the direct representation of base one mathematics. In other words, it is based on the direct representational semantics of identity, instead of the indirect representational semantics of equality. It is based on base one numbers instead of being based on numbers with a base >= 2.

In simple terms, objective mathematics falls outside the domain of the types of mathematical systems covered by Kurt Gödel's famous Incompleteness Theorems. Thus it is not subject to the incompleteness theorems.

While the argument above shows that the Incompleteness theorems don't apply, it still doesn't prove that objective mathematics is complete and consistent. As it turns out, proof of the completeness and consistency of objective mathematics is trivial.

Objective mathematics is solely based on the axiom of identity. The axiom of identity states that every existent is itself. Note that I did not say every existent is equal to itself. Identity and equality are not the same thing. Identity is always complete both at the level of every individual existent, at the level of every possible combination of existents, and at the level of the universe as a whole. 1 * 1 * 1 * ... * 1 = 1. (In the foregoing expression, '1' is being used to represent an identity, not simply to represent the number or quantity '1'. The '=' is also used to mean 'is' or 'composes', i.e. I am using it in the complete direct semantic sense of identity, not in the indirect incomplete semantic sense of numeric equality). Equality in indirect mathematics is only defined up to isomorphism. Equality is only defined over some domain and codomain. The domain and codomain are sets. Those sets are subsets of the universal domain. Equality is only consistently definable within some subset of the universal domain. Equality under indirect representation is always incomplete and inconsistent in the universal domain. Conversely, existents that are identities are always complete and consistent over every domain up to and including the universal domain because each identity is itself. Conversely, things that are equal are never complete or consistent in the universal domain because no symbol in an indirect representation can represent itself directly. At best, an equality in indirect representation can never be more than a partial incomplete and inconsistent representation of identity. The problem with equality is it is simply not possible to represent everything in the universal domain completely and consistently. Conversely, using the semantics of direct representation, existents can only be represented completely and consistently because every existent is itself. It is impossible for any mathematical system based on the axiom of identity to be incomplete or inconsistent. In fact, the axiom of identity is even stronger than that. There can only be one complete and consistent direct representation of existence because only one representation of existence can be identical to itself in every possible domain up to and including the universal domain. Of course the converse is true in indirect representation. There can be many different incomplete (partial) and/or inconsistent indirect representations of the same parts of existence.

Since complete equality cannot exist in the universal domain, differences must exist in the universal domain. In particular, potential differences must exist and every potential difference must be itself. The transfinite recursive composition of those potential differences then composes the direct representation of existence. Thus existence itself is a complete and consistent closed base one mathematical system based on the semantics of direct representation and the axiom of identity.

Thursday, August 1, 2013

Objective Physics

1) Despite its ongoing struggle for objectivity, modern physics remains subjective because it is based on consciousness, observation, measurement,  information, logic and mathematics, all of which are subjective. Furthermore, the mathematical tools (tensors) physicists use to try to reduce subjectivity are themselves based on subjective logic, which in turn is based on subjective metaphysics. The root cause of subjectivity in physics lies in philosophical errors in metaphysics.

2) Objective Physics is based on my discovery of the base ONE natural mathematical process the universe uses to autonomously self-generate  all  of physical existence directly from potential differences in energy fields, without any dependence on the a priori existence of any consciousness, measurement, observation, information, reason, thought, logic, mathematics, or metaphysics. Yet in physical form, that self-generating system can generate and evolve the existence of conscious observers that may subsequently use measurement, observation and information to invent their own logic, mathematics, languages, religions, philosophies and sciences to reason about their own existence. This system generates a universe whose mathematics and laws of physics appear to be fully consistent with those discovered by modern physics, science, and mathematics when represented from a conscious observers' reference frame.

 3) In objective physics, existence itself is a kind of self-generating mathematical system. Existence generates itself directly from the transfinite recursive composition of disjoint symmetric potential differences between finite and infinite energy fields.  In other words, existence is a kind of self-generating mathematical , logic, and metaphysical system that generates a mathematically complete and consistent transfinite cumulative hierarchy of automorphism group generators. The resulting direct mathematical system literally is existence. It composes all the energy fields that compose the current quantum field structure and quantum state of existence and causes all ongoing quantum state transitions in the universe. Non-quantized potential differences in the asymmetric, fractal, infinite virtual energy fields that compose the 'quantum vacuum' compose quantized symmetric potential differences in finite symmetric energy fields. The transfinite result causes an infinite series of 'big bangs', and the ongoing generation of the 'quantum vacuum', time, space, all types of energy and energy fields, all particles, all energy quanta, all quantum states, and all quantum state transitions in the universe. None of this requires any consciousness, information, measurement, observation, thought, or decisions. Hence the process is completely objective. It generates pure mathematics, yet requires no mathematician. Since every potential difference, and every generated composition of potential differences is an identity, existence is mathematically complete and consistent in the disjoint union of the finite and the infinite (i.e., in the universe). While our thoughts about existence can be incomplete and/or inconsistent (because we represent thought indirectly), and our current mathematics is incomplete and/or inconsistent (because we represent it indirectly), the objective direct representation and existence of existence itself is always complete and consistent. Existence exists. A is A. No existent, any combination of existents, or the universe can be inconsistent or incomplete.

4) Philosophically speaking, objective physics' metaphysics is based on a blend of Ayn Rand's Objectivism, Monism, and the self-generation of existence as a kind of physical mathematical system. Its only fundamental axiom is the axiom of identity. In other words, existence exists. Equivalently, A is A.

5) The resulting mathematics is instance based, not class based.  In other words, existents themselves always exist as themselves, as existents. Existence does not use classes to represent  classes of existents. Instead multiple existents can appear to belong to the same class or "category of being" from an observer's perspective, simply because those existents are composed from homologous sets of potential differences. Our brain simply composes a neural abstraction that represents those homologous sets of potential differences as a class of existents.

6)  Every symmetric potential difference in energy is an identity.  That means every symmetric potential difference contains it own energy, and every symmetric potential difference is itself. A corollary is that equality does not exist. No two symmetric potential differences in the universe are equal. Whereas current mathematics is based on the axiom of 'equality', objective mathematics is based on the axiom of identity.

7) Every symmetric potential difference also functions as a relation because each one relates two or more symmetric potential differences, or two or more compositions of symmetric potential differences.
     
8) Every symmetric potential difference is also as an entity in its own right. Symmetric potential differences relate other symmetric potential differences, thereby composing the existence of higher-order entities. In turn those higher-order entities themselves can be composed of other entities and the relations between those entities, all of which are ultimately composed of symmetric potential differences.  In other words, an entity is a corollary of the axiom of identity. 

9) An action is something a symmetric potential difference, or a relation, or an entity does. Every action has a cause. There are no causeless actions.  An action is a corollary of the axiom of identity.

10) The cause of every action is a symmetric potential difference between two or more symmetric potential differences, or a symmetric potential difference between higher-order compositions of symmetric potential differences. Thus symmetric potential differences, relations, entities, and actions can all function as the cause of actions.  Cause always precedes action. Cause is a corollary of the axiom of identity.

11) All the so called philosophical "categories of being" , such as qualities, quantities, relationships, actions, causes, states, etc., are merely different partial, incomplete indirect representations (i.e., aspects) of symmetric potential differences. Thus they are all corollaries of the axiom of identity. Symmetric potential differences simply operate on other symmetric potential differences in existence. For nature, the "categories of being" are irrelevant. We must carve existence up into different types of conceptual categories to represent existence indirectly. Existence has no need to do so. It simply represents symmetric potential differences and the all the energy field structures they compose as they interact with each other directly in terms of their very existence.

12) It is important to understand that the causal link relates an entity and its action. The law of causality does not imply that every entity has a cause. Some of the things commonly referred to as "entities" do not come into being or pass away, but are eternal - e.g., the universe as a whole. The concept of "cause" is inapplicable  to the universe; by definition, there is nothing outside the universe that can act as a cause.

13) Causality in existence is a metaphysical fact of reality, independent of consciousness, whether God's or mans. Order, law-fullness, symmetry and regularity are not caused by any cosmic consciousness (as claimed by the religious argument by design). Nor is causality merely a subjective form of thought that happens to govern the human mind (as in the Kantian approach).  On the contrary, causality is a law inherent in being qua being. To be is to be something - and to be something is to act accordingly. Consciousness is the mental faculty of perceiving or thinking about an object, not of creating its existence or changing it. Consciousness is required for an observer to acquire knowledge of existence, not for the existence of existents.

14) Natural law is not a feature superimposed by some agency on an otherwise "chaotic" world; there is no possibility of such chaos. Nor is there any possibility of a "chance" event, if "chance" means an exception to causality. Cause and effect is not a metaphysical afterthought. It is a direct mathematical consequence of the axiom of identity. It is a part of the fabric of reality as such.

15) One may no more ask: who is responsible for natural law (which amounts to asking: who caused causality?) than one may ask: who created the universe? The answer to both questions is the same: existence exists. Existence exists for the same reason energy is conserved. Energy cannot be created or destroyed. It can only change form. Because the total amount of potential difference in the universe is a conserved quantity, it is impossible for existence not to exist, for the simple reason that potential differences in energy act on each other to compose existence. Since it is impossible to destroy any potential difference in the universe, those potential differences can only change form, and those forms can only compose existence. Nonexistence, and zero potential difference are metaphysical impossibilities because both are inconsistent with the axiom of identity.

16) Heisenberg uncertainty is often understood to mean that because we cannot simultaneously measure position and momentum of subatomic particles with complete certainty, their action is not entirely predictable, and that the law of causality therefore breaks down at quantum scales. This is a non-sequiter, a switch from epistemology to metaphysics, or from knowledge to reality.  Even if it were true that owing to a lack of complete information we could never exactly predict a subatomic event, it would not show that "in reality", the event was causeless. Our ability to measure, observe, and predict actions in nature has nothing to do with the cause of those actions, or their existence. Every existent and every action we measure or observe must exist BEFORE we can measure or observe it.  This just goes to show how subjective basing physics on consciousness, observation, measurement, and subjective mathematics, logic and metaphysics can be. Note that this doesn't mean Heisenberg Uncertainty doesn't exist from an observers' perspective. From an observer's perspective Heisenberg Uncertainty is a certainty. However, it is a certainty of the observers indirect representation and knowledge of existence, not a  fundamental property of existence itself.

Sunday, January 13, 2013

The Anthropocentric Foundation of Physics and Mathematics


The Anthropocentric Foundation of Physics and Mathematics

Physics is currently built on an unsound foundation, because it is based on mathematics. Mathematics, as currently defined, is an extended deductive logic system based on a false premise relative to physical existence. Let me explain.

The "unreasonable effectiveness of mathematics in the natural sciences" exists because mathematical structure is a homomorph of the quantum field structure of physical existence. That makes mathematics a very useful tool because we can use it to model parts of many physical systems, and partially predict the outcome of many experiments. The problem is mathematics is only a homomorph of the quantum field structure. It only provides a partial, incomplete, inconsistent, uncertain representation of the observable parts of existence. It is not an isomorph of the totality of physical existence. In less technical terms, that means mathematics does not provide a one-to-one representation of all of physical existence.

Mathematics is not isomorphic to physical existence because the natural numbers are based on the transfinite recursive composition of empty sets. The empty set is an indirect representation of nonexistence. The empty set is an isomorph of, and the logical model for, the number zero (0). In other words, there exists a one-to-one mapping between the empty set and zero, and between zero and nonexistence. The problem is there is no such thing as nonexistence in existence. Nonexistence is the logical complement of existence. It is the complete absence of existence.

Existence cannot represent its own complement, or it would be inconsistent, because it wouldn't be itself. Furthermore, and this is extremely important:

EVERYTHING THAT EXISTS IN THE UNIVERSE IS COMPOSED OF ENERGY

No particle is fundamental. All particles are composed of energy. In this case, I am using the term energy in its broadest possible scope, to include all types of energy, not just the observable kinds. In other words, by the term energy, I mean all types of quantized energy, all types of quantized dark energy, all types of virtual energy, all types of dark virtual energy, the quantum vacuum, and the singularity. Each different type of 'particle' is actually a localized quantum energy field composition with a specific type of quantum field configuration. Its quantum field composition and configuration gives each particle the properties we associate with particles.

The totality of energy in the universe (of all types combined) is a conserved quantity. It is impossible to destroy any energy, let alone all of it. Energy can be transformed from one form to another, including between forms that are observable and unobservable, but none of it can ever be destroyed. In metaphysical terms energy is the arche. In philosophical terms, it is the fundamental 'substance' that composes existence.

Since everything in the universe is composed of energy, that means zero energy, and thus nonexistence, is not part of the universe. Nonexistence is an inconsistent concept relative to physical existence. Even if we remove every particle from a region of space, and cool it down to absolute zero temperature, and shield it from all external energy fields, the resulting vacuum is still full of virtual energy and virtual dark energy. It still contains the quantum vacuum energy. Spacetime itself is composed of energy. There is simply no way to remove all energy from anyplace, let alone destroy all energy in the universe. Where would it go, universe 2? There is no such thing. The universe is everything that exists. There can only be one everything. Since the empty set is an isomorph of nonexistence, the empty set, and by extension the concept of zero, is also not part of physical existence, and is thus inconsistent with physical existence.

It is also easy to see this from the perspective of special relativity. Zero is the  origin of the number system. Yet what is the absolute origin of physical existence? Where is zero in space? When is zero in time? Time and space are both relative. They have no absolute origin. They are relative because they are both composed of energy, and energy is a substance that differences can occur in. In particular, those differences are what causes the existence of potential differences in energy fields, the differences between states, the quantization of energy, and thus the existence of quantized energy fields, quantum states, and finite particles.

Thus right from the start, mathematics is inconsistent because the natural numbers themselves are inductively derived from the transfinite recursive composition of a thing that is not isomorphic to any part of existence.  That means mathematics is a formal system of logical deduction based on a false premise. Relative to existence, that means mathematics is logically unsound. In turn, that means it is possible to use "correct" mathematics to derive false conclusions from a true set of premises. In other words, the mathematics may be correct, but the conclusions relative to existence may not be.

We try to compensate for that by comparing the results predicted by mathematics with the observable results from experiments. We then discard, refine, or modify those theories whose mathematical predictions don't match observable reality. The problem with that is observation is anthropocentric. In fact, the very information we use to represent our measurements and observations in terms of is anthropocentric.  Again, let me explain.

Information is Anthropocentric

For all x in the universe, x must exist BEFORE it can be observed, measured, or represented in terms of information. That means existence cannot be composed of information without reversing the order of cause and effect. That means the "It from Bit" hypothesis in quantum physics, that at its deepest levels, existence is composed of information is false, because it requires a reversal of the order of cause and effect.

Information is created by observation and/or measurement of potential differences in energy. Existence is composed of potential differences in energy fields. Existence does not depend on observation or information. Information depends on the existence of energy, and the existence of observers that can measure potential differences in energy fields, and create information to represent them indirectly. In other words, information is an observers' indirect representation of potential differences in energy fields and the relations between them.

Examine the causality graph between energy, existence, observation and information. Existence is composed of, and thus causally depends on, the existence of energy. Observation depends on energy and existence. Information depends on energy, existence, and the observation of both.

Existence is composed of energy, so it is causally dependent on the existence of energy. Everything that exists is composed of some kind of energy. In reality, existence is energy. Thus the dependence between energy and existence is bidirectional.

Observation depends on existence and energy. If there is no existence, there is nothing to observe, no observer to observe it, and no energy for the observer to use to represent anything with. Energy and existence can exist without observation, but observation cannot exist without energy and existence.

Information is dependent on observation, existence and energy. Energy is a potential difference in an energy field. Those potential differences create charge imbalances, and thus disturbances in the energy fields that compose spacetime, matter, and all forces. Those disturbances form travelling waves that propagate through spacetime. Those waves are signals. We interpret those signals as carrying information. More specifically, observers measure the resulting localized potential differences in the energy field that composes a signal when those potential differences cause changes in the receptive field of a sensory transducer, and use the (usually amplified and signal conditioned) potential differences to construct the representation of  information which they then use to represent the associated part of existence indirectly. In other words, once again, I point out that existence is not composed of information. Only our indirect representation of existence is composed of information.

Heisenberg Uncertainty

In fact, the meaning of information, and its reduction in uncertainty is only a reduction in uncertainty relative to the observer that interprets the meaning of the information. The meaning of information, and its uncertainty is in the mind of its interpreter. Of course at quantum scales, the energy required to take a measurement is a substantial fraction of that which composes the localized field being measured, so it changes that field, making subsequent measurements of the same localized field uncertain. Nevertheless, the uncertainty is part of the information we use to represent existence, not part of the energy that composes existence itself. Physical existence itself has no uncertainty. No representation that is complete and consistent in the universal domain can have any uncertainty. Mathematically speaking, uncertainty can only exist in incomplete and/or inconsistent representations. In simple terms, this means information, observation, and measurement are unreliable tools for representing existence, because they are anthropocentric. They are incomplete, inconsistent, and uncertain. As long as physics relies solely on them, it too will remain incomplete, inconsistent, and uncertain.

Direct representation and direct mathematics is not dependent on observation or measurement. In physical form, it generates itself directly from symmetric potential differences in the singularity. It has no undecidability and no uncertainty. Heisenberg Uncertainty is not a fundamental property of physical existence. It is a fundamental property of information. Since we use information to represent all of our observations, we mistakenly attribute the uncertainty in information about existence to existence itself. This isn't difficult to see. The quantity of information in Shannon information theory is mathematically defined in terms of how much it reduces uncertainty. If there is no reduction in uncertainty, the quantity of information is zero. In other words, if there is no reduction in uncertainty, there is no information.

There is a second reason any fixed formal mathematical system is either incomplete or inconsistent. All formal systems are based on information, and information is a kind of indirect representation, in which some observer x, associates some symbol y with some element z to represent z in terms of y indirectly. In any such representation, the symbol y is not the same thing as that which it represents z. In other words, in all indirect representations, y z. That makes it impossible for any indirect representation to be complete and consistent in the universal domain, because it means that nothing that is represented indirectly is itself. In an absolute sense, all indirect representations are inconsistent because they are like a dog chasing its own tail. No fixed formal system based on indirect representation, or any collection of such systems, can ever catch up to itself to represent all of itself completely. Indirect representations can only represent the universal domain incompletely and inconsistently. In other words, information is at best, only an incomplete, partial, inconsistent, uncertain representation of existence. Since numbers are all indirect representations, and since mathematics is based on numbers, mathematics is necessarily incomplete and inconsistent in the universal domain. Of course we already knew that. Kurt Gödel proved as much in his celebrated incompleteness theorems in his 1931 paper entitled "On Formally Undecidable Propositions of Principia Mathematica and Related Systems."

The only possible way all of existence can be complete and consistent is if, for all x in the universe, x = x. The only way that can happen is in the logical converse of an indirect representation: direct representation. In direct representation, everything that exists is composed of energy, and everything that exists contains its own energy. In other words, for all distinct x and y in the universe, the energy that composes x the energy that composes y. Direct representation is the representation of being, not merely information about being. Energy composes everything that physically exists directly, in terms of the potential differences in the energy fields that compose each thing. Those potential differences are the direct representation of direct mathematical relations. Not only do potential differences represent relations, they are energy differences that can cause changes in the things they compose. Hence, they act like mathematical functors. Those functors operate on themselves, transforming their current state into their next state, thereby constructing the current moment of existence throughout all spacetime in the universe. More precisely, potential differences in energy fields are the direct mathematical automorphism group generator generator of physical existence. (No, that "generator generator" isn't a typo). The direct mathematical equations represented by potential differences in energy fulfill all the properties required by the definition of a mathematical group. (I've derived the equations that prove it). Since everything in direct representation represents itself, it is trivially isomorphic to itself. Since there is only one universe, that group can only act on itself, so it has to be an automorphism group.

Physical existence is a self-generated direct mathematical system that generates all of itself directly from the transfinite recursive composition of symmetric potential differences in the singularity. In other words, instead of representing numbers from the transfinite recursive composition of empty-sets, nature represents existence from the transfinite recursive composition of symmetric potential differences in the singularity. In effect, physical existence is a faithful isomorph of the complex number system, except there is no zero. Zero is replaced by the infinite singularity. The infinite singularity acts like an inaccessible cardinal. In addition, nature's equivalent of the complex numbers starts with very large potential differences at each big bang (there are an infinite series of big bangs...), and the magnitude of the potential differences decreases over time as spacetime cools and expands and energy and dark energy compose higher and higher order forms of matter and energy. Eventually, the energy in the quantum vacuum will be exhausted and the expansion of spacetime will slow and then reverse. More and more black holes will form, and space, time and the quantized energy forms will be converted back into their infinite forms, where they will remain trapped in the singularity, until the omega black hole consumes all remaining spacetime in the universe, causing a collapse of its own gravitational field, and the subsequent initiation of the next big bang. Hence, in this model, the universe is an infinite series of big bangs. Between each big bang spacetime undergoes a short period of exponential inflation, followed by a long period of expansion, then an exponential increase in the rate of black hole production near the limit of spacetime expansion resulting in a short period of exponential deflation, followed by a gradually decelerating asymptotic deflation until the omega black hole finally consumes all remaining spacetime in the universe and causes the next big bang.

The result is a symmetrically self-limiting four dimensional pentachoron simplicial complex spacetime lattice composed from a cumulative hierarchy of complex tensor valued quantum energy fields with Gaussian integer coefficients.  (The space between the quantized lattice points is filled by complex and imaginary fractal dimensional non-quantized complex tensor fields with complex real non-integral coefficients that represent the composition of virtual energy and virtual dark energy fractal strings in the quantum vacuum fields). Thus we get time, space, all forms of energy, matter, and complex Hilbert Space tensor hierarchies of quantum field compositions. Time and space both exist physically. Since they both exist, they both must be composed of energy. Spacetime energy field compositions of energy and dark energy compose photons and the elementary particles, and those and their relations compose larger particles and particle systems. We get whole part hierarchies of composition at all spacetime scales above the quantum scale, but because the symmetry of pentachoron simplices is topologically self-dual, the dimension of spacetime never increases beyond the fourth dimension. Instead, as quantum bubbles of temporal energy and anti-temporal dark energy are created via ongoing spontaneous particle pair creation out of the quantum vacuum, it causes the ongoing expansion of four dimensional spacetime. We get time and space, the arrow of time, and the fact that cause precedes effect at all spatial scales above the quantum scale.

Subspace - A New Model of the Quantum Vacuum


Below the quantum scale, spacetime and order break down because virtual energy and virtual dark energy have an imaginary dimensional fractal structure below the quantized unitary imaginary dimension of time, so it has no fixed measure, and time, space, quantum states, and order do not exist. I call the fields that result subspace. Photons and the speed of light are meaningless in subspace. Subspace is filled by the infinite forms of energy. It is infinite because it lacks a measure, not because it is infinite in quantity. Strictly speaking, without a measure, the concept of quantity is undefined and does not exist. Changes can propagate through subspace instantaneously because time and distance have no meaning there. This is what enables the 'spooky action at a distance' observed in entangled quantum systems. It also explains why quantum state changes occur in quantum leaps in zero time.

Physical Existence is Complete and Consistent


Since everything that is itself is consistent with itself, the transfinite recursive composition of everything is also consistent with itself. Thus the universe, and everything that physically exists in it is consistent with itself.

Since the universe is everything that exists, it also has to be complete. That means the universe and everything that physically exists in it is complete and consistent. That means the universe is the union of the finite and the infinite. The quantized forms of energy and dark energy represent the existence of the finite, and the non-quantized forms of energy represent the quantum vacuum, virtual energy, virtual dark energy, and the singularity. In effect, there is a one-to-one isomorphic mapping between the complex field (with the singularity replacing zero), and physical existence. In essence, physical existence itself is a kind of direct mathematical complex number system. The universe as a whole is thus both complete and consistent in the universal domain. The universe is the union of the finite and the infinite, and all relations between them. It is only our indirect representation of existence, based on information, that is incomplete, inconsistent, and uncertain.

The Universe is Finite And Infinite


The universe as a whole is both finite and infinite. In other words, it has finite components and infinite components. Energy comes in finite and infinite forms. Only half of its finite forms are observable. Energy quanta are observable. Dark energy is finite, but unobservable because its arrow of time is reversed. Time and space are finite. All quantized energy forms are finite.

The Solution to the Quantum Vacuum Catastrophe

Virtual energy, virtual dark energy, and the singularity compose the quantum vacuum. Those forms of energy are not quantized, and they are unobservable and unmeasurable. Energy in its infinite forms has no mass or gravitational field. The infinite forms of energy exist beneath and between the quantized dimensions of spacetime. Mass and gravity cannot exist without spacetime. Thus while the singularity and quantum vacuum contain an immense amount of energy, that energy exists in a form that has very little effect on the observable forms of energy. In particular, it has no effect on the cosmological constant. This resolves the so called quantum vacuum catastrophe in quantum physics.

Spontaneous Particle Pair Creation and Annihilation

It is also important to understand that spontaneous particle pair creation and annihilation is a misnomer. In reality, in particle pair creation, virtual energy and virtual dark energy is being transformed from its infinite form into its finite form as energy and dark energy quanta. Thus, those quanta appear spontaneously from the quantum vacuum. The same process works in reverse in particle pair annihilation. In that case, finite energy and dark energy quanta are converted back into their infinite virtual energy and dark virtual energy non-quantized forms. No creation or destruction of energy is involved in this process. It is just another kind of energy transformation. The only thing that makes it appear special is that the energy is being transformed between observable and unobservable forms.

Time and Space are Quantized Energy Forms

Since energy composes everything that exists, and since time and space exist in the universe, time and space must both be composed of energy. Time and space are finite, and they must exist because we can measure them and travel through them. In addition, gravity creates spacetime curvature. It is not possible to create curvature in something that doesn't exist.

The Future of Mathematics and Physics

The only way to represent physics completely and consistently is to use a mathematics that itself is complete and consistent. The only way to do that is to redefine mathematics, and the concept of number in terms of direct representation. Science will never be isomorphic to existence as long it is based on mathematics that isn't.

I am deriving a new foundation for mathematics based on an isomorph of direct representation. The resulting mathematics is both complete and consistent in the universal domain. When completed, in theory it could simulate the generation of the totality of existence, starting from the singularity, completely and consistently. Of course, in reality, resource constraints will make that impossible, but it will still be able to represent many parts of existence that are inaccessible using mathematics represented in terms of indirect representation. For example, we will be able to consistently represent the singularity, We will be able to consistently represent what happens inside black holes. We will be able to consistently represent the complete quantum state of existence consistently, with no uncertainty. We will be able to represent all relations between energy and dark energy. We will be able to represent the quantum vacuum, and all possible relations between it and the finite, observable parts of existence. We will be able to obtain a complete and consistent understanding of the quantum field structure of existence. At least we will have the equations and be able solve them. Admittedly, some of the equations become extremely complex, and it will take quite some time to interpret them and understand all their implications.

Tuesday, April 3, 2012

The Observer Paradox


THE OBSERVER PARADOX 


For all x in the universe, the existence of x PRECEDES measurement and observation of x. Without the existence of x there is no x to measure or observe. That means the existence of x cannot depend on measurement or observation of x because the latter occurs AFTER x already exists. In other words the belief that existence depends on measurement, and observation violates the order of cause and effect. Since all information about existence is based on measurement and observation of x, existence cannot be composed of information. This means 'It from Bit' is incorrect because it violates causality.

Existence does not have to measure or observe itself because it already is itself. 
Only OUR INFORMATION about existence depends on measurement and observation. 

The observer paradox is very important. All of logic, mathematics, and science are currently based on measurement, observation, and information. The existence of existence cannot depend on any of those things. All of those things depend on the apriori existence of existence. 

WE are not the center of the universe. The existence of the universe does not depend on our personal existence, or even the existence of our entire species. Existence does not depend on what we think of it, what information we learn about it, or what equations we derive to represent it. All of those things are anthropocentric.


SOLVING THE OBSERVER PARADOX 


The entire universe is a naturally self-defining, self-organizing, self-modifying quantum computer. That quantum computer operates on the current quantum state of existence to produce its next quantum state. It naturally generates a complete and consistent direct mathematical system that generates all ongoing changes in the current quantum state of existence. It requires no creator, no programmer and no program. 

The direct representation of existence solves the observer paradox completely and consistently in the universal domain without the need for any measurement, observation, information, or consciousness. It completely and consistently represents the existence of the infinite, the finite, and all relations between them. It represents everything that exists in terms of gravitationally induced potential differences between the infinite singularity's grand unified field and the transfinite recursive composition of a cumulative hierarchy of orthogonal energy and dark energy topological quantum field simplices. Those quantum field interactions then compose the ongoing changes in the current quantum state of existence. The resulting direct mathematical quantum system completely and consistently represents the existence of the infinite singularity, time, space, all types of energy and dark energy, matter and dark matter, and every other force, subatomic particle, atom, molecule, object, process, and thing that exists in the universe. That includes the existence of all observers, observation, measurement, information, neurons, and the neural representation and computation of perception, thought, meaning and consciousness.

In theory, it should be possible to create a computer simulation of this quantum computer, allow it to compute a complete and consistent isomorph of the direct representation of part of existence, and then observe its results to learn about the energy and dark energy quantum fields, and the quantum field dynamics of the simulated part of existence. Since the simulation is complete and consistent, it has to be isomorphic to existence because there can only be one complete and consistent representation of the universe. Since we would only be observing a complete and consistent simulation of existence and not the energy and dark energy quanta that compose physical existence itself, we can gain a complete understanding of the quantum field structure and dynamics of the simulated part of existence with no Heisenberg Uncertainty. This should also allow us to observe the dark energy and dark matter components of existence, and their quantum field interactions with observable and unobservable energy and matter. It will also allow us to observe what happens inside black holes. In other words, this model represents all of existence completely and consistently; not just the observable and measurable parts.



Sunday, March 11, 2012

Beyond Information - A Mathematically Complete and Consistent Quantum Field Theory of Everything

Abstract

This paper introduces the natural 'mathematical' foundation the universe itself constructs and represents all of existence in terms of.  It is based on an alternative foundation for the representation of logic, mathematics, and information that provides a closed, complete, and consistent representation of the current quantum state and all quantum field interactions and processes in the universe. This paper also explains what the infinite singularity is, and how all time, space, energy, dark energy, matter and dark matter are naturally generated from it via the transfinite hierarchical composition of orthogonal symmetric differences in the infinite singularity. This mathematical foundation is also background independent. The natural construction of existence is not causally dependent on human measurement, observation, or the indirect representation of information in any way whatsoever. Instead of being designed to represent things relative to human observation, measurement, decisions, and computations, this representation is designed from first principles to naturally generate itself, organize itself, modify itself, define itself, and compute itself via the ongoing composition of symmetric differences in the infinite singularity. Those symmetric differences (and the ongoing expansion of spacetime) are generated by ongoing spontaneous symmetry breaking in the infinite singularity. The ongoing symmetry breaking is caused by the ongoing decay of the black and white hole gravitational fields that compress, form, and contain the infinite singularity.    

All of existence is based on the transfinite recursive composition of symmetric differences in the infinite singularity, aka the grand unified field. Those symmetric differences compose virtual energy and virtual dark energy open strings, and if those differences reach an integer multiple of a Planck length, they can compose energy and dark energy quanta. Those differences represent all charges, all energy and dark energy quanta, all quantum states, all quantum field interactions and all relations and processes in the universe. Since nature represents all energy and dark energy in terms of the transfinite composition of differences in the infinite singularity, it does not need to decide how to represent each thing. It does not need to decide whether to represent something as a quantum state, as a quantum field interaction, as a process, or as a relation.

This allows nature to construct existence without any need to make decisions. Decisions are anthropocentric. We make decisions. Most of nature does not. Since most of nature has no need to make decisions, there is no need to decide on a representational domain, co-domain, or range. Consequently, this representation has no domain limitations. There is also no need to decide what to represent. Nature simply represents everything that exists using the same self-defining, self-organizing, self-modifying direct mathematical system. It uses the same ontology, the same representation, and the same computational process to represent all of existence. In fact, in direct representation, the ontology, its representation, and its computational process are all the same thing. They are all represented by the transfinite recursive composition of symmetric differences in the infinite singularity. That makes the representation complete in the universal domain. Since everything represents its own existence in direct representation, everything is itself. In turn, that ensures the representation is consistent in the universal domain. Thus direct representation is both  complete and consistent in the universal domain. That explains how nature gets around Kurt  Gödel’s  Incompleteness theorems. By avoiding the need for the indirect representation of information, direct representation also ensures the completeness and consistency of the totality of existence.

All differences are finite. Since the infinite is not finite, infinity must be the complete absence of differences. Thus, the infinite is the logical complement of the finite. The infinite singularity functions as a direct representational monad, and the null space of existence. Symmetric differences in the singularity compose the quantum field structure of existence via the transfinite recursive composition of an orthogonal  cumulative hierarchy of energy and dark energy fields via the direct representation of virtual energy and virtual dark energy open strings, and the direct representation of energy and dark energy quanta (closed strings). Those symmetric differences in the singularity are caused by the ongoing decay of the black (and white) hole gravitational fields that form, compress, and contain the infinite singularity. The decay of the gravitational field allows the energy and dark energy in the singularity to expand, thereby creating differences in the singularity. Those differences represent a change of state and a potential difference in charge. Hence they simultaneously represent states, relations, and processes. Because of this, nature does not have to 'decide' whether to represent something as a state or a relation or a process. In turn, this causes particle wave duality. All differences are finite. Thus all virtual energy and virtual dark energy strings and all energy and dark energy quanta are finite. Everything that exists is ultimately composed of energy and/or dark energy, including the infinite singularity. The singularity itself is just another form of energy. It is the superposition of energy and dark energy when all differences are gravitationally compressed to zero. When all differences are reduced to zero, there are no differences, no dimensions, no states, no boundaries, no surfaces, no relations and no processes within the infinite topological space of the grand unified field. Hence a sufficiently strong gravity field converts energy from its finite form as energy and dark energy quanta and virtual energy and virtual dark energy open strings into its infinite form in the infinite singularity. In other words, energy is always conserved. In its infinite form, energy is eternal. It has no beginning and no end. Ongoing changes in gravitational fields convert energy back and forth between its infinite and finite forms. The union of the infinite singularity's topological space and the finite topological space, compose the entire universe. Gravity then defines and causally relates the finite and the infinite.

Nature's representational equivalent of the integers is the unitary direct representation of energy and dark energy quanta. However, instead of being represented as a transfinite cumulative hierarchy of differences between empty sets by reference, energy quanta are represented as a transfinite cumulative hierarchy of differences in the infinite singularity by value. Since all quanta are represented by value, every energy and dark energy quantum is its own unique instance and has its own unique identity. In addition, since energy quanta compose by value, and time is a kind of scalar energy field (see discussion on time below), all of their temporal relations are represented directly via their structural composition. Thus existence is composed from the energy and dark energy quanta that represent the quantum states that compose the current moment of time (potential energy) and the differences between those quantum states that compose all quantum field relations and cause all quantum state changes (kinetic energy).

At the level of energy quantum instances, every quantum is unique. In other words, no two energy quantum instances are equal. Every energy quantum exists in its own context. In fact, each energy quantum is composed of the energy quanta and energy quanta differences (virtual energy strings) that compose it. All energy quantum composition is instance based. It is all based on value semantics, not reference semantics. 

Topologically, energy and dark energy quanta and their differences compose simplicial complexes up to dimension four. At each dimension, the simplicial complex forms an n-dimensional convex hull that minimizes potential energy and dark energy differences within and between every type of energy and dark energy field within the cumulative hierarchy of its subspaces. The singularity functions as the null space of existence. Each successive field is orthogonal to the cumulative hierarchy of fields that compose it. For example the infinite singularity is an orthogonal subspace of the temporal and anti-temporal fields. The temporal field is an orthogonal subspace of the electromagnetic field. In turn, the EM field is an orthogonal subspace of the color field. In turn, the color field is an orthogonal subspace of the weak field. In turn, the weak field is an orthogonal subspace of the spacetime field.

Spacetime itself is a dual pentachoron simplicial complex composed of equal parts energy and dark energy. Energy and dark energy fields compose a dual pentachoron simplex that forms a stable nilpotent cubic spacetime lattice. It is no coincidence that the permutation group of a pentachoron simplex is self dual. That means when one of the tetrahedrons that compose a pentachoron spacetime simplex absorbs a virtual energy string, or an energy quantum, it simply subdivides that tetrahedron, converting it into another pentachoron, and causing the ongoing expansion of spacetime. As a result of this process, all quantum field relations within the quantum energy and dark energy fields that compose spacetime are conserved under spacetime expansion. That also explains why spacetime is four dimensional. All fermions and baryons are naturally represented in terms of how they relate to the four dimensional energy and dark energy fields that compose the quantum vacuum of the spacetime they occupy. As a result, fermions and baryons all have mass and they all create curvature in spacetime. The end result is everything in the universe exists in four or less dimensions in direct representation.

Since all four dimensional polynomials have closed form solutions,  the entire quantum state of existence can be solved in direct representation. 

Time is the first dimension. It is composed of a scalar energy field composed from virtual energy open strings and energy quanta (i.e., closed strings). As the gravitational field that contains the singularity decays, the singularity expands, thereby creating a potential difference in the singularity. That potential difference creates virtual energy and virtual dark energy strings. As the singularity continues to expand, the length of those virtual energy and dark energy strings increases, until they reach an integral multiple of a Planck length, at which time they can minimize their potential difference by forming closed loops without self interference. Those closed string loops are temporal energy quanta and anti-temporal dark energy quanta. The circular symmetry of those closed loops then represent a field of stable, unitary 'points' that persist through time. Those fields persist through time because their circular symmetry provides the degree of freedom required for the potential difference they represent to exist in stable form.  Physically, each temporal energy quantum is the event horizon of a black hole quantum microsingularity. Each anti-temporal dark energy quantum is the event horizon of a white hole quantum microsingularity. In aggregate, those microsingularities form a scalar field. Those event horizons separate the finite from the infinite. They represent the scalar temporal field. They have to because they are the first difference in the infinite singularity. They compose the first dimension of the finite. The first dimension of the finite is the current moment of time. All other forms of energy and dark energy are composed from symmetric differences in the temporal and anti-temporal fields, so all other forms of energy exist in time and/or anti-time. Differences in the the temporal and anti-temporal fields create a change in state, thereby representing  the emergent composition of higher order / higher dimensional energy and/or dark energy fields. We call the energy that causes the change in quantum state kinetic energy. We call the energy that composes the states before and after change potential energy. Transfinite potential differences between energy and dark energy quanta create all other  types of energy, dark energy, space, matter and dark matter. As a consequence, all other types of energy, dark energy, space, matter, and dark matter exist over time. They exist over time because they have a temporal and/or anti-temporal scalar field component. 


We are unaware of temporal charge because we can't observe or measure it. It is not observable or measurable because its energy (and dark energy) is the dominant component of the energy and dark energy that composes the quantum vacuum of spacetime that we measure all other energy relative to. Thus from the perspective of indirect representation, we assume the energy and dark energy that composes the quantum vacuum is zero, even though it is actually 15,569 times more powerful than the strong nuclear force. The extreme energy density of this field relative to all other types of energy is the underlying cause of increasing entropy, the arrow of time, and the order of cause and effect. 


The transfinite recursive composition of differences between the energy and dark energy quanta that compose the quantum vacuum of spacetime compose all fermions and baryons. That explains why all fermions and baryons occupy spacetime, cause curvature in spacetime, obey Fermi-Dirac particle statistics, obey the Pauli exclusion principle, and have mass. Of course, the energy and dark energy fields that exist beneath spacetime (and that compose the quantum vacuum of spacetime itself) do not curve space time, so they represent all the massless bosons and obey Bose-Einstein particle statistics. The net result is the transfinite recursive composition of symmetric differences in the infinite singularity generates all time, all space, all energy, all dark energy, all matter and all dark matter in the universe. Thus direct representation composes all of existence.

In turn this means all higher order, higher dimensional types of energy and dark energy fields exist in time because they have a temporal field component. For example, it means the electromagnetic field has a temporal field component. The anti-electromagnetic dark energy field has an anti-temporal field component. This means a south magnetic pole is simply a time reversed north magnetic pole and vice-versa. In turn, the color field that carries the strong force that binds the quarks that compose nucleons has electromagnetic and temporal field components. Similarly, the weak force responsible for radiation has color field, electromagnetic field and temporal field components. In turn, spacetime has weak field, color field, electromagnetic field, and temporal field components, along with their dark energy dual field counterparts. 

The composition of the quantum field structure of existence is based on selection and variation. I call this the quantum evolution process. Charles Darwin's evolution of species by means of natural selection is a special case of quantum evolution. In the case of quantum evolution, selection is based on stability. Only those energy configurations that are stable enough to persist through time can participate in the subsequent composition of higher order fields. The unstable field configurations emit quanta or virtual energy strings and decay back into simpler forms until their component fields reach stable configurations. Stability can be static (a balance of forces over one or more dimensions) or dynamic (via the formation of an attractor, a system of attractors, or adaptation to a changing fitness landscape of some kind). This means selection is self-organizing. Since everything in direct representation represents itself, and direct representation is complete and consistent in the universal domain, the selection process itself is part of the representation of existence, so it is also self modifying. Variation occurs via emission or absorption of energy and dark energy quanta, and emission or absorption of virtual energy and virtual dark energy open strings. Variation occurs when it can reduce the magnitude of the potential energy  and potential dark energy differences in the current quantum field configuration, and still comply with the simplicial complex topological and orthogonal subspace constraints. On average, variation tends to increase stability over time because it reduces local potential differences. The end result is the composition of larger and larger statically and dynamically stable quantum field structures along with a  general increase in entropy over time. The order of cause and effect is ensured because the temporal field energy density dominates that of all other fields, and because higher order structures can only be composed of components that already exist. Of course, the general increase in entropy over time outside the event horizon of a black hole causes the second law of thermodynamics. Entropy always increases or stays the same outside the event horizon of a black hole. Entropy always decreases inside the event horizon of a black hole, until it becomes nonexistent in the singularity.

The resulting logic and mathematics can be used to compute and represent all of existence - including the mathematical representation and computation of the infinite singularity, energy, dark energy, time, space, matter, dark matter, abstraction, thought, and consciousness. In addition, this new philosophical foundation for logic, mathematics, and science is not causally dependent on observation, measurement, human decisions, or the representation of information. It naturally generates itself, all the laws of physics, all forces, all time, all space, all energy, all dark energy, all matter, and all dark matter directly from the transfinite recursive composition of symmetric differences in the infinite singularity, aka the grand unified field. 


By creating a self-generating, self-organizing formal representation of mathematics that is not dependent on measurement, observation, or the indirect representation of information, we can create a representation of logic and mathematics that is complete and consistent in the universal domain. We can overcome the incompleteness and inconsistency limitations imposed on formal systems by  Gödel’s  Incompleteness theorems. We can also eliminate the need for Heisenberg Uncertainty in the formal representation of quantum mechanics - so long as we are not using QM to represent the results of observation. (Observation is a form of indirect representation, so it necessarily requires incompleteness, and Heisenberg Uncertainty).

This paper argues that humanity has currently only explored one of three possible forms of representation. Logic, set theory, mathematics, information, and human communication are all 'indirect' forms of representation. All indirect representations use one thing to represent another. For example, all symbolic representations are indirect representations. Each symbol in a symbolic representation represents whatever we decide to allow it to represent. No symbol represents its own existence. Instead, each symbol represents the existence of its referent. All indirect representations depend on observation because something or someone has to interpret the meaning of the representation. Information itself is a kind of indirect representation. 


The existence of indirect representation implies the existence of its logical converse, 'direct' representation. A direct representation represents particulars from the first person direct perspective of each particular itself, instead of from the third person indirect perspective of an observer. Existence is a direct representation. Direct representations are not dependent on observation. They represent existence directly via composition. For example, all energy and dark energy quanta are direct representations. All of existence (including all indirect representation) is composed of energy and dark energy. Thus all of existence is a direct representation. If direct and indirect representations exist, then to complete the powerset of representation, a third form of representation should exist that is universal; i.e., both direct and indirect[1]. Universal representation arises as a natural extension of direct and indirect representation. Universal representation represents neural abstractions, concepts, thought, and consciousness.

This paper argues that the universe itself is a closed, consistent, and complete direct representation. It argues that the representation of thought is a closed, consistent, and complete universal representation. It argues that information cannot be the correct foundation for the representation of existence because it would violate causality.

This paper also identifies the immaterial bivalence responsible for the direct representation of existence, and in doing so, identifies the first cause of symmetry, the first cause of all forms of energy, and a new conservation law more fundamental than the law of conservation of energy. It also identifies the universal bivalence responsible for the representation of thought. It identifies the neural representational basis for the first person direct relation between meaning and existence at all levels of abstraction in all contexts. It identifies a single universal of computation responsible for the direct neural processing and representation of all perception, awareness, understanding, meaning, and consciousness. It also explains how to create formal representations that can represent everything in the universe and avoid the adverse consequences of Gödel’s Incompleteness theorems. It concludes by recommending the creation of very high priority research programs to create new axiomatic foundations for the direct representation of existence and the universal representation of thought.

Introduction

Our species uses information as the basis for the representation of all communication. Humans have spent about 2,400 years developing logic, mathematics and science based on information and it has served us well. We have been able to develop theories and scientific laws that allow us to predict the outcome of experiments, develop useful technologies, and understand quite a bit about the composition and function of the universe. Our successes have led most to believe that information is the only possible basis for representation. In fact, the philosophy of information goes so far as to posit that at the very deepest levels, existence itself is derived from bits and based on the representation of information. [1] This paper provides strong arguments to the contrary. It presents a convergent argument that the representation of existence is direct. It argues that the incompleteness of mathematics arises precisely because mathematics is an indirect representation. It argues that mathematics is not isomorphic to the direct representation of existence. Moreover, it argues that it is impossible for mathematics to represent existence directly because mathematics itself is based on the indirect representation of set theory. Representing the direct representation of existence using an indirect representation is incomplete and excessively complex. This paper proposes a direct representation of existence as an alternative to its indirect representation using information. It also identifies the first cause of symmetry and proposes a new conservation law that is more fundamental than the law of conservation of energy.

This paper also argues that the representation of thought is both direct and indirect, and that the brain has no need to use, nor does it use, information to represent or encode thought. We think directly, from the first person perspective in context as in Cogito Ergo Sum. It is not possible to think from the first person direct perspective in context using a third person indirect context free representation. It would be combinatorially too complex, and there would be no way to ground semantic meaning. It would also make it impossible to avoid Ryle's regress; i.e., it would make it impossible to avoid the need for an infinite series of observers or interpreters to interpret the meaning of an indirect representation inside the brain. Both of these problems are avoided in universal representation. A brief introduction to the representation of thought is presented.

The paper concludes by recommending the creation of high priority research programs to formulate new axiomatic set theories for the direct representation of existence and the universal representation of thought. The former should allow us to accelerate development of theoretical physics exponentially. The latter leads directly to the creation of sentient computers, improved methods for teaching, improvements in treating brain injuries and mental illness, and eventually, a substantial increase in human intelligence.

Keeping Things in Perspective

Humanity would do well to keep things in perspective. Human beings are only one species among millions on a single planet circling one star in a very large universe. According to the latest scientific estimates, the universe is between 13.60 and 13.84 billion years old.[2] Anatomically modern humans first appear in the fossil record in Africa about 130,000 years ago, although studies of molecular biology give evidence that the approximate time of divergence of homo sapiens sapiens from the common ancestor of all modern human populations was about 200,000 years ago.[3][4][5] Even if we use the earlier date, our species appeared on earth approximately 13.7 billion years after the beginning of the universe. Our entire species has existed for less than 0.0015% of the age of the universe.

Existence cannot represent itself indirectly from the perspective of an observer. Even if it could, existence has no need to use a context free, fixed symbolic encoding to provide a shared basis for the communication of information between particulars in existence. Why should the requirements for the representation of human communication be the same as those for the representation of existence? What is the probability the representation of information our species uses for communication, logic, mathematics, and science just happens to be the same as the representation the entire universe uses to represent itself?

Information is an Indirect Representation

The representation of information enables communication between observers. It describes things from the third person indirect perspective of an observer. Therein is the problem. The representation of existence is direct. Existence cannot represent itself indirectly from the third person perspective of an observer. Existence is logically, physically, and causally prior to observation. Something has to exist before it can be observed or described using information.

Particulars in existence can only represent themselves directly from their own first person direct perspective. In addition, because information must describe things from the third person indirect perspective of an observer, it must use a fixed context free encoding to provide a shared basis for the communication of meaning between observers using a shared communication protocol. The purpose of the representation of existence is the direct physical representation of existence, not the indirect communication of information about existence to an external observer. Consequently, the representation of existence does not need to use a fixed context free encoding, and it categorically does not need to represent itself abstractly, symbolically, or indirectly.

Mathematics is proven incomplete by Gödel’s Incompleteness theorems.[8,9,10] Mathematics is incomplete because it is an indirect representation. Indirect representations are incomplete because they cannot represent anything directly. That means mathematics cannot even represent itself directly. It is impossible for mathematics to represent things directly because it is based on axiomatic set theory. Axiomatic set theory is an indirect representation. The most commonly accepted theory for the foundation of mathematics is the Zermello- Fraenkel, with Axiom of Choice, or ‘ZFC’ set theory. [7] There are many alternative set theories, but they all have one thing in common. They are all indirect representations.

Set Theory is an Indirect Representation

Axiomatic set theories represent the universe of mathematics from the third person indirect perspective of an observer. Set theory is an indirect representation. The most fundamental concepts of set theory reflect this. For example, set members can be atoms or other sets. Atoms are references for things in the real world, or references for abstract concepts like numbers. The references can represent anything we like, but they are indirect. They typically take the form of a label or a name. For example, the set {barry} contains the name ‘barry’. ‘barry’ is a reference for the person named barry. It is not the human being named barry or a direct representation of barry as a human being because it does not have to include the representation of all barry’s components; i.e., barry’s arms, legs, skin, teeth, hair, muscles, molecules, and all their relationships and interactions.

The most fundamental relations of set theory reflect the fact that it is an indirect representation. The set membership operator is not transitive.[6] For example:

2 is a member of {1,2}
And {1,2} is a member of {{1,2},{3,4}}
but 2 is not a member of {{1,2},{3,4}}.

This means set membership does not represent the ‘is part of’ relation. If the representation of set theory were direct, then the set membership operator would be transitive because transitive whole-part relationships are fundamental to the ontology of existence.

Set theory’s equality relation ‘=’ also reflects the indirect representation of sets.  In set theory {1, 2, 3} = {1, 2, 1, 3} because identity is by reference, not by value. In set theory, the two occurrences of ‘1’ in {1, 2, 1, 3} are considered to be the same object because they refer to the same object. Again, this could not happen in a direct representation. In a direct representation, representation = existence. In direct representation, everything represents itself, and the representation of every particular in existence is a singleton.

Direct representations cannot represent things indirectly, but they can represent everything that exists in the direct representation completely and consistently. The logical complement of an incomplete, indirect representation is a complete direct representation.

Set theory represents the set with no members as { }, the empty set. It must do so because set theory is an indirect representation founded on the transfinite recursive composition of sets of empty sets. Set theory does not represent existence directly; it represents it indirectly using sets so it must represent empty sets.

In a direct representation, representation = existence. Therefore, the empty set does not exist; i.e., the representation of nonexistence is nonexistent, just as it is in the actual universe. Nonexistence is physically nonexistent because it is impossible to destroy any energy, let alone all of it. Since all energy always exists in some form, it is logically and physically impossible for nonexistence to exist in the universe. In effect, that means current mathematics is an extended closed deductive logic system based on a false premise: the existence of nonexistence. While that system can be self-consistent, it is inconsistent with parts of physical existence.

From the foregoing discussion, it should be clear that set theory is poorly suited for the representation of phenomena whose existence is based on direct representation. Set theory can only represent direct representation indirectly. That makes it very complex to represent existence.

It also makes it impossible to directly represent thought from the first person direct perspective. There is no direct basis for semantic grounding using an indirect representation. First person direct context dependent representation and understanding of meaning cannot be based on a third person indirect context free representation.

In principle, all of mathematics is based on axiomatic set theory. That means all of mathematics is indirect. The representation of the universe itself is direct. That means we are trying to represent existence using a representation whose most fundamental elements, relations, and ontology are not isomorphic to that of existence. The universe of mathematics is not isomorphic to the universe of existence. The universe of mathematics is more flexible and more general than the direct representation of existence. While that indirection increases generality, it is not without cost. The cost of that indirection is incompleteness and a combinatorial increase in complexity. The cost of that incompleteness and increased complexity is incredible. It is the reason the mathematics used to describe physics is so complex. It is the reason it has taken humanity more than 2000 years to reach our present understanding of physics and indeed, essentially all of science.

First Order Logic is an Indirect Representation

First order propositional logic represents everything from the third person indirect perspective of an observer. Sentence letters represent particulars indirectly. They are labels for abstract concepts, or labels for objects in the real world. The same sentence letters may have different meanings in different contexts. This could not happen in a direct representation. The concepts of ‘True’ and ‘False’ are themselves labels for abstract concepts.

The representation of the universe is direct and physical. It is concrete. It is not abstract, and it is not indirect. First order logic fails to distinguish between the indirect, abstract representation of thought about reality, and the direct, concrete representation of reality. It fails to distinguish the difference between an indirect representation of existence and the direct physical representation of existence itself. It is also too static. It fails to model time. It fails to represent how things and relations between things change over time. In hindsight, this was probably unavoidable. We experience and think about the world indirectly and abstractly. Because thought seems to be indirect[1], we attempted to represent everything indirectly. Lacking an understanding of the representation of thought, we did not understand where to draw the line between thought and reality.

Propositional calculus depends on propositional logic. Predicate logic depends on propositional logic. Predicate calculus depends on propositional calculus. Axiomatic set theory depends on predicate calculus. Mathematics depends on axiomatic set theory. 'Bits’ represent particulars indirectly. A ‘bit’ is an indirect representation or label for an abstract concept, or for an object in the real world. The same bit may have different meanings in different contexts. Information is composed of and represented in terms of bits, so it too is an indirect representation.

Information Blindness

The fact that our species uses information as its exclusive basis for communication makes our species blind to the possibility that other bases of representation exist. The widespread presumption that information is the only available basis for representation is species centric. In hindsight, our exclusive reliance on indirect representation will prove to be no better than the Ptolemaic geocentric astronomy European and Arabic astronomers mistakenly labored under for 1,393 years prior to the advent of Copernican heliocentric cosmology and the start of the scientific revolution.

The representation of existence is context dependent, not context free. Particulars in existence always exist in some context. Existence uses a relative relational encoding, not a fixed context free encoding. Most importantly, the representation of existence must be consistent and complete. The entire universe must be represented by a single universe of discourse. There can be no domain limitations. There can only be one ontology and one direct representation of existence for the entire universe. All other alternatives increase complexity combinatorially in the number of representations by making it combinatorially more complex to maintain the consistency and completeness of multiple overlapping representations of existence.

The fact that logic, mathematics, and science have succeeded in representing many different limited fixed domains of discourse using many different formal systems each with its own representation, its own ontology and its own ontological consistency rules is not a logically sufficient basis for assuming that information is the basis for the representation of the entirety of existence itself. The ability to represent limited domains of existence is not the same as the ability to represent all of it at once. Representations based on information are incomplete. They are domain limited. They are complex. They are brittle and fail easily in the face of unexpected input. They are inefficient. Most significantly, they require a priori knowledge of what is to be represented before a suitable representation can be formulated. Existence is logically, physically, and causally prior to observation. Therefore, the use of information as the basis for the representation of existence violates causality. Continuing to base all representation on information despite these facts is illogical and wasteful in the extreme. The only logical alternative is to move beyond the representation of information to overcome these problems.

Viewing the Universe through the Lens of Information

Physics has had many successes. However, it has been unable to answer many of the most basic questions about the universe using information. For example, what force causes a photon to travel through space at the speed of light? What causes that force? How large is that force? How can a photon carry electromagnetic charge when it has no charge? Why are energy and matter quantized? What causes the quantization of energy and matter? What causes like charges to repel and opposite charges to attract? What is time and what causes it? Why does energy exist? What is the first cause of energy? What created the Big Bang? What came before the Big Bang? What created space? What created the dimensions of space? What causes symmetry? Why is symmetry so prevalent in the universe? What ensures the consistency of the Universe? How could an information-based representation ensure the global consistency of existence, given all the different domains of discourse, representations, ontologies, and ontological consistency rules it would seem to require? The fact that we have been unable to answer these most basic questions is a sure sign that we are missing something very fundamental. It is as if we have been trying to analyze and understand the entire universe by looking through the lenses of millions of microscopes, each viewing the universe in a limited spectrum and each having a limited, isolated field of view, each described using its own specialized symbols, models, and languages. Looking thru the incomplete, domain limited lens of information, we cannot see or reach all the squares on the chessboard of reality because the physical representation of the universe itself is not based on information. Information only provides an incomplete, partial representation of reality. We need to go beyond the limitations and constraints of information if we want to understand Physics completely. We need to be able to model and represent all of reality as a complete, consistent, integrated whole in all of its context dependent splendor using a single complete and consistent representation that is isomorphic to the full representation of existence. The same is true of all physical sciences.

Thought and Information

The fact that we communicate using information is also not a logically sufficient basis to assume that our brains use information as their internal neural basis for the representation of thought. People must communicate with each other using information with fixed encodings to establish a shared basis for understanding via communication using a common alphabet and language. However, the neurons in our brain do not communicate directly with neurons in other people’s brains. Our neurons do not communicate with anything other than the other neurons inside their own nervous system. The nervous system is a closed representational system. Neurons have no need to establish or maintain a public shared basis for the internal communication of information. They are free to use their own private language and their own private encoding. In fact, by removing the fixed encoding constraints required for external communication, neurons can vary their encoding as a function of that which they represent to minimize code length and storage space. They can use a relative relational encoding unique to the current state of knowledge stored in each individual’s brain. They can use a representation that is direct and indirect, instead of one that is only indirect. In fact, neurons must use a representation that is both direct and indirect. Without a basis in direct representation, there is no basis for the first person direct representation and understanding of meaning. Meaning cannot be grounded indirectly.

Neurons exist physically. Existence is a direct representation. Our neurons operate from the first person direct perspective of the direct representation of existence, but because they represent and implement the ontology of abstraction, they also allow us to represent things abstractly and indirectly- thereby allowing us to communicate abstractions indirectly using information. Neurons convert the indirect external representation of information into the direct representation of thought for internal processing. They convert the internal direct representation of thought back into the indirect representation of information for external communication. While this conversion may seem complex or difficult when viewed from the perspective of information, it is a simple matter for the representation of thought[2].

The brain's internal knowledge representation operates much faster and much more efficiently when we do not make ourselves think in terms of information. I would like you to try a quick little thought experiment. Look out your window. See how fast you can recognize all the objects, all their relationships, all the textures, all the colors and understand what you are seeing? Now try to describe the same scene in words and see how many words it takes to describe it to the same level of detail you could perceive, recognize and understand in less than a second. Now give that description to somebody else and see how long it takes him or her to understand the contents of the scene. See how much information was lost in the conversion to information?

Now try to describe the same scene using mathematical equations. See how long it takes somebody to understand that, see how much could not be represented using mathematics, and see how much information was lost in the process. That will give you a good feel for the relative efficiency of the brains internal knowledge representation vs. the representation of information. The brain uses the same knowledge representation and computational model for seeing and understanding that scene out your window as it does to think and reason using symbolic information. The difference in efficiency is almost entirely due to the inefficiency of the representation of symbolic information. When we try to represent and understand the universe in terms of symbolic information, we force our brain to continuously translate back and forth between the indirect representation of information and the brains direct native representation it uses internally to reason and think. That slows the brains native thought process tremendously. It also loses just as much information as the difference between looking out your window and understanding the scene in less than a second vs. trying to describe the scene in words or equations and understand it. Humans have a huge untapped potential to increase the speed and depth of comprehension of abstract knowledge and increase intelligence. To unlock this potential, we need to learn the brains’ native representation of thought and teach ourselves to use it directly. Until we do that, we will continue degrading our innate mental capacity by forcing our brain to think indirectly in terms of what for it is a terribly inefficient, complex, symbolic, foreign representation of information.

We Must Move Beyond the Representation of Information

The assumption that information is the basis for the representation of existence is incorrect. The assumption that information is the basis for the representation of thought is also incorrect. On what basis do I make these claims?
  • Existence is logically, chronologically, physically, and causally prior to the observation of existence. 
  • Set theory and the representation of information are both indirect representations. 
  • Everything that exists in the universe is ultimately composed of energy and dark energy. Energy and dark energy are direct representations. They exist directly and interact with each other directly, not indirectly as a consequence of some observer measuring them, thinking about them or representing them indirectly. 
  • Energy and dark energy quanta compose the entire universe. They compose all time, all space, all kinds of energy and dark energy, and all forms of matter, and dark matter. In the final analysis, everything that exists for more than a Planck time is composed of energy quanta, and its existential dual, (unobservable) dark energy quanta. 
  • Every quantum state, and every quantum field relation is composed of energy or dark energy quanta, or differences between them. 
  • In fact, at the lowest level of existence, temporal field energy quanta themselves are a gravitationally induced difference in the infinite singularity, aka the grand unified field. 
  • The infinite singularity itself is the complete absence of differences. Think about that carefully. 
    • Every difference is finite. Every finite difference in existence is physically represented by a difference in potential (a charge of some kind) in some kind of energy or dark energy field. 
    • When energy is compressed into singularity in a black hole, the dimension of the spacetime it exists in is gravitationally compressed to zero. That means all differences in its quantum energy field and quantum state must be reduced to zero
    • Without any differences, there are no differences between quantum states, and no relations. 
    • Without any difference, it is impossible to distinguish between two quantum states, or between states of any kind. The same is true of relations. 
    • That means the infinite singularity can have no charge, no state, no properties, no relations, no boundary, and no surface. It can have no gravity, and no entropy. (The gravity is a property of the spacetime curvature surrounding the black hole's event horizon, not a property of the singularity). 
    •  Thus all differences are finite, and the complete absence of differences is infinite. 
    • Infinite literally means 'not finite'. Since all differences are finite, the absence of all differences is infinite.
  • Energy is the transfinite recursive composition of differences between the finite and the infinite. All of existence is composed from energy and dark energy, and thus all of existence is composed from the transfinite recursive composition of differences between the finite (energy and dark energy fields) and the infinite singularity. 
    • At the smallest scales, this means every energy quantum contains a quantum microsingularity at its core. 
    • All charges are ultimately differences between quantum microsingularities. 
    • Black hole quantum microsingularities are energy sinks and dark energy sources, whereas white hole quantum microsingularities (dark energy) are energy sources and dark energy sinks. 
    • In turn, this means the origin of existence is the infinite singularity, not nonexistence, or zero. 
      • The universe expanded from the infinite singularity in the big bang, not from zero, or nonexistence, or an empty set. 
      • In turn, that means the origin of the number system is not isomorphic to that of existence. 
      • It also means we can compose a number system that is isomorphic to existence from the transfinite recursive composition of differences in infinity, instead of from the transfinite recursive composition of differences in empty sets. 
      • Such a number system is based on direct representation and value semantics, instead of indirect representation and reference semantics. 
      • Direct representation is the logical converse of indirect representation. 
      • The result is the creation of a number system that is closed, complete and consistent in the universal domain, where the universal domain represents the entire universe, including both the finite and the infinite and all relations between them. 
      • In turn the ongoing generation and expansion of that direct mathematical system ensures the consistency and completeness of the totality of physical existence.
 (Don't worry if some of this goes over your head. It is only meant to be a high level summary. Space precludes presenting this theory or the mathematics it is based on in detail here. You can see my presentation on 'Direct Representation' http://www.slideshare.net/bkumnick/direct-representation-second-draft for further details. Even that presentation leaves out over a thousand pages of notes - again in the interest of brevity).


2)    I have discovered two entirely new classes of representation that are not based on the representation of information. One is combinatorially less complex than the representation of information. It appears to be the direct representation of existence. It provides direct intuitive interpretations for the foundations of quantum physics with minimal complexity. It provides direct answers for many of the deepest unsolved mysteries in Physics. For example, it explains the first cause of symmetry and the cause of the quantization of energy. It explains the cause of the universal consistency and completeness of existence and proves it mathematically. From the axiomatic definition of existence, it derives the meaning of the finite, the meaning of infinity, the meaning of universe and their relationships mathematically.

     The second representation is geometrically less complex than the direct representation of existence. Therefore, it is geometrically combinatorially less complex than the representation of information. It is exponentially more powerful than the indirect representation of information, logic, and mathematics. It is also complete and consistent in the universal domain. It is the representation of thought. The representation of thought is direct and indirect. Its ontology is isomorphic to the ontology of abstractions and concepts and isomorphic to the branching topology of individual neurons. It is also isomorphic to the overall branching pattern of neural connectivity in the human neocortex. It explains the neural basis for the representation of abstract thought, concepts, meaning, perception, awareness and consciousness. Unlike the representation of information, the representations of existence and thought are both provably complete and consistent. The representation of thought is based on the direct representation of the ontology of abstraction. The ontology of abstraction is direct and indirect, and unlike information or logic, it is both intensional and extensional. It consists of a single universal of computation, a single representational primitive, and a single ontology, all of which are represented by the same thing, a living, functioning neuron.

     The branching structure of a neuron is isomorphic to the ontology of abstraction. From a high- level perspective, neural connectivity in the human neocortex is logically organized as a top down hierarchy of concepts where each node in the hierarchy contains a bottom up hierarchy of abstractions. The hippocampus is located at the top of the concept hierarchy and the sensory receptors and nerves that control the muscles are located at the bottom. The association cortices are located in the interior. This allows us to think abstractly, it allows us to think conceptually, and it allows us to think in context. It also allows us to represent and understand meaning from the first person direct perspective in context at multiple levels of abstraction simultaneously. The derivation and formal definition of 'abstraction' and 'concept', their ontology, and their relation to neural topology is presented in http://www.slideshare.net/bkumnick/direct-representation-second-draft

3)    Gödel’s Incompleteness Theorems [8,9,10] prove information based formal systems incomplete. The complement of an indirect, incomplete representation is a direct, complete representation.

4)    Indirect formal systems are only complete in domains of discourse of limited size. To avoid incompleteness we must limit and fix the size of the domain of discourse. This causes domain limitations. Domain limitations lead to a combinatorial increase in complexity because they force us to resort to the use of multiple domains, multiple representations, multiple ontologies, and multiple sets of ontological consistency constraints to cover the representation of the universe as a whole. In such a system, there is no known way to ensure the global consistency or completeness of the representation of the universe as a whole.

      Think about this. To create an equation to represent anything, the first thing we must do is decide what thing(s) to represent, and what relation(s) to represent between those thing(s). The result is always a partial, and thus incomplete representation of existence. The problem is nature as a whole cannot 'decide' which parts of existence and which relations to represent. Heck, most of nature can't even decide whether to represent a thing as a state or a relation. All decisions are anthropocentric. People make decisions using their brain. Most parts of nature cannot make decisions. Most parts of nature simply exist and interact directly via the direct interaction of the energy and dark energy quanta that compose them. Furthermore, nature must represent all of existence completely and consistently, or else the universe itself would be incomplete and inconsistent. By the definition of 'universe', it is impossible for the universe to be incomplete or inconsistent. The cause of incompleteness and inconsistency is indirect representation itself. It is impossible for any mathematical system based on indirect representation to represent the universe completely or consistently.

5)    Information is represented from the perspective of an observer. The universe cannot represent itself from the perspective of an observer. Existence is logically, chronologically, physically, and causally prior to observation. From what observers’ perspective could the first thing in the universe have been represented? We know there had to be a first thing. The universe has a lower size limit[3]. Therefore, it is finite. Therefore, time had a beginning. Therefore, there had to be a first thing. If there was a first thing, it could not have had an observer. Furthermore, there could not have been any perspective to view it from because the perspective itself (i.e., spacetime) would have had to preexist. Furthermore, without an observer, there would have been nothing to ask yes-or-no questions to decide how to interpret and encode information, no basis for the formulation of the questions, nothing to measure the results with, and nothing to record the binary answers on or in. Since existence is logically and chronologically prior to observation, existence has to exist before it can be observed. Since information is dependent on an observer, either the representation of existence is not based on information, or existence does not exist. Since the universe exists, we must conclude existence cannot be based on the representation of information.

6)    ON.2 It from bit.[1] Otherwise put, every “it” — every particle, every field of force, even the space-time continuum itself — derives its function, its meaning, its very existence (even if in some contexts indirectly) from the apparatus-elicited answers to yes-or-no questions, binary choices, bits. “It from bit” symbolizes the idea that every item of the physical world has at bottom — a very deep bottom, in most instances — an immaterial source and explanation; that which we call reality arises in the last analysis from the posing of yes-no questions and the registering of equipment-evoked responses; in short, that all things physical are information-theoretic in origin and that this is a participatory universe. (Wheeler [1990], 5);

While this argument seems plausible on the surface, it is deeply flawed. First, the premise is ambiguous because it fails to distinguish between the direct physical existence of particles, fields of force and the space- time continuum, and their indirect representation as information. While an indirect, set theoretic, information based description of the function, meaning and existence of particles, fields of force and the space time continuum may be derived from the answers to yes-or-no questions and represented as bits, that representation is only an indirect description, or model, of reality. It is information intended to describe reality indirectly; it is not reality or a direct representation of reality itself. It cannot be a direct representation of reality. The axiomatic definition of ‘set’ makes it impossible for a set to represent reality directly. In addition, the meaning of the information is only in the mind of an observer. The direct representation of the actual physical existence of particles, fields of force and the space-time continuum itself is not abstract. It is not symbolic. It is not based on bits. It is not indirect. It is direct. It is physical. Reality itself is composed of the direct physical existence of energy and dark energy quanta and the fermions, bosons, and the quantum field interactions they compose, not bits. Reality is a direct representation, not an indirect representation.

Second, relative to the first thing in existence, who or what is going to formulate the yes-or-no questions?

Third, relative to the first thing in existence, who or what is going to measure, interpret and record the answers as bits?

Fourth, relative to the first thing in existence, where are the bits going to be recorded and stored?

Fifth, relative to the first thing in existence, what is going to interpret their meaning?

Sixth, yes-or-no questions are abstract, but the representation and process of abstraction are not immaterial. The representation and process of abstraction are carried out in the mind of an observer who thinks. Neurons represent thought. Neurons are physical, not immaterial. Therefore, the representation of yes-or-no questions is not immaterial, nor is the representation of the bits by the neurons that represent their answers.

Furthermore, if ON.2 argues for the immateriality of the representation of thought, it contradicts ON.1.

ON.2, “It from bit” is almost correct. The fundamental direct physical representation of the universe is bivalent, and at the deepest level, it is immaterial, but the immateriality and bivalence are not based on the answer to yes-or-no questions. The basis for the bivalence is not one or zero or true or false. There is no apparatus for formulating yes-or-no questions. There are no questions. There are no answers. There is no observer. The meaning of the immaterial basis for the bivalence of the universe has been misinterpreted. Therefore, the meaning of the bit has been misinterpreted. As currently defined, the bit is isomorphic to the foundations of mathematics and logic. It is isomorphic to the representation of information and human communication. It is not fully isomorphic to the direct physical representation of existence.

7)    The representation and encoding of information is context free. The representation and encoding of existence are context dependent. Representing direct context dependent systems using indirect context free representations increases complexity combinatorially in the number of contexts (and the number of representations) used to represent them.
   
8)    The representation and encoding of information is indirect in that bits represent that which they encode indirectly; i.e., they are a substitute or label for that which they represent, they are not the thing they represent. The representation and encoding of existence are direct. In direct representation, Representation = Existence. The physical representation of existence is existence; it is not information about existence.

9)    The equation that represents thermodynamic entropy and the equation that represents information entropy are the same up to a constant not because existence is composed of information, but because thermodynamic entropy and information in the mathematical theory of communication are both modeled by the same kind of probability model. Obviously, if you use the same type of mathematical model to represent thermodynamic entropy and information, you will end up with the same equation, up to a constant.

Existence Beyond Information

The immaterial basis for the representation of existence is the infinite singularity, aka the grand unified field - not nonexistence. There is no such thing as nonexistence. The concept of nonexistence is an abuse of abstraction. Nonexistence is an ungrounded abstract concept. Do you want proof? Consider the law of the conservation of energy. Energy cannot be created or destroyed. It can only change form. That means no energy can be destroyed, let alone all of it. Since no energy can be destroyed, energy is eternal. That also means energy outlasts the existence of time itself. Time begins and ends with each big bang. Energy has no beginning or end. It simple transforms between its finite forms and its infinite form. The main point is that since energy always exists in some form, nonexistence cannot exist anywhere at any time in the universe.

The key to understanding infinity and the infinite singularity is to first understand the finite. Consider this fact: every difference is finite. Now ask yourself what happens if all differences are removed? Without any differences, there is no way to distinguish between quantum states. In fact, without any differences, there is no way to distinguish between any kind of states. The very concept of 'state' becomes indefinable. By the same token, without differences, there can be no relations, no boundaries, no surfaces and no dimensions. The result is infinite, i.e., infinite literally means not finite.

Now consider what happens to the quantum energy fields that compose the spacetime inside the event horizon of a gravitational black hole as they get gravitationally compressed into the black hole's singularity. Spacetime and energy (and dark energy) get compressed more and more as they get closer to the singularity. All differences in energy and dark energy wavelengths must converge to zero as the spacetime they exist in itself is compressed. In other words, all energy and dark energy is compressed to the same wavelength in the singularity. By the same reasoning, since there can be no differences in the singularity, and no dimensions, all potential difference must be compressed to zero. In other words, all energy and dark energy in the singularity must get compressed to the same absolute potential, thereby removing all potential differences, and unifying all types of charge into the grand unified field, aka the infinite singularity.

The same reasoning means the singularity can have no distinguishable properties. That means it can have no entropy, no mass, and no gravitational field. In other words, the gravitational field is carried by the curvature in the spacetime that surrounds the black hole's event horizon. The gravitational field is a property of the spacetime surrounding the black hole, not a property of the singularity itself. That means the singularity itself has no mass and no gravity. It also has no entropy. There can be no mass or entropy without dimensions! Hence, black holes recycle the entropy in the universe. They convert high entropy energy and dark energy at the end of time in each instance of existence back into the zero entropy singularity from which they expanded. (It has to be the same singularity because there can only be one infinity, and thus by definition, there can only be one infinite singularity).

Eventually, as it consumes more and more of the energy and dark energy in the universe, the omega black hole starts to consume the energy and dark energy that composes the spacetime whose curvature composes its own gravitational field. As a result, the omega black hole's gravitational field starts to decay, so the gravitational forces that compress and contain the singularity decrease. As a result, the singularity undergoes spontaneous symmetry breaking, and the first potential differences occur in the singularity. This creates the first dimension. It creates a scalar field of virtual energy and virtual dark energy open strings. As the external gravitational field continues to weaken, those difference increase in wavelength until they reach a Planck length. When the wavelength reaches a Planck length, the open strings can close, forming closed loops with circular symmetry. Those closed loops are temporal field energy quanta (and anti-temporal field dark energy quanta pairs). The temporal and anti-temporal scalar fields form the first dimension of existence. Physically, the closed temporal energy loop is the event horizon of a black hole quantum microsingularity. Similarly, the closed anti-temporal loop is the event horizon of a white hole quantum microsingularity. (A white hole is just a time reversed black hole). The black and white hole quantum microsingularities are stable, and persist through time because of the circular symmetry of their event horizons. The event horizon's circular symmetry provides the degrees of freedom necessary for the potential difference that composes temporal and anti-temporal energy quanta to persist in stable form over time.

Each temporal quantum microsingularity functions as a point in time. Because those energy quanta persist through time, differences between them represent the first dimension of existence. They represent time. That explains what time is, where dimensions come from, and how the finite is related to the infinite.

From this, we can see that time is a scalar potential energy field. Everything else that exists in the universe exists in time because it is composed of differences in the temporal energy field. In other words, energy and dark energy quanta are nature's only direct representational primitives. They are the natural equivalent of integers, except they are based on differences in the infinite singularity, instead of differences in nonexistence, or differences between empty sets. (The empty set represents nonexistence which is an existential fallacy; nonexistence and the empty set do not, and cannot exist in the universe). Their quantization makes energy and dark energy quanta unitary. This allows nature to form the direct representation of existence via the transfinite recursive composition of finite symmetric differences between the temporal energy and anti-temporal dark energy fields.

This leads directly to a new kind of mathematics that is both complete, consistent, and closed in the universal domain that is the universe itself. It is closed over both the finite AND the infinite. Instead of being based on indirect representation, this mathematics is based on its logical converse - direct representation. Instead of being incomplete and/or inconsistent in a local domain, it is both complete and consistent in the universal domain. Instead of being based on reference semantics, it is based on value semantics. Instead of being context independent it is context dependent. Instead of being observer and measurement dependent, it is observer and measurement independent. Despite these converse relations, the ontological structure of direct and indirect representation are similar. It is just that current mathematics can only represent parts of existence incompletely, whereas direct representation can represent all of existence completely and consistently.

Consider this: the first thing we do when we want to represent something mathematically is decide what parts of existence we want to represent, and which relations we want to represent between those things. The problem with that is nature as a whole cannot represent part of existence. It must represent all of existence, and it must represent existence completely and consistently. Nature has to represent how everything that exists in the universe is related, whether it is finite or infinite. Nature cannot decide to just represent part of existence. If it did, existence itself would be incomplete and inconsistent, and the 'universe' could not exist. For the same reason, nature cannot distinguish between states and relations. States and relations are both partial representations of existence. Hence, nature can only use a single representational primitive, and that primitive must represent both states and relations. In fact, that representational fact explains the fundamental cause of particle wave duality in nature. This also explains why everything that exists for more than a Planck time is composed of energy (and dark energy) quanta. The quantum is nature's monad. It is the single primitive unit of representation that all the finite parts of existence are composed of.

Energy exists in finite form in the form of energy quanta and virtual energy strings, and dark energy quanta and dark virtual energy strings, and in infinite form in the form of the infinite singularity. Literally everything that exists in the universe is composed of energy and/or dark energy of some kind. (Dark energy just the time reversed dual of energy).

It is interesting to note that both current mathematics and direct mathematics are constructed from the transfinite recursive composition of differences in something. Whereas current number systems and mathematics are composed from the transfinite recursive composition of differences between sets of empty sets, direct representational mathematics is composed from transfinite recursive differences in the infinite singularity. The difference is the infinite singularity physically exists, whereas the empty set is a fallacy because it is impossible for it to exist physically. In other words, relative to physical existence, current mathematics is an extended deductive logic system based on a false premise; that false premise being the existence of nonexistence, and thus the existence of the empty set. That makes current mathematics logically unsound relative to existence. In other words, current mathematics is self-consistent, but it is inconsistent and incomplete relative to the totality of existence.

Anyway, the net result of this is that time is by far the most powerful form of energy. However, we cannot directly measure or detect that energy because its energy composes the quantum vacuum that we call spacetime. We measure all energy relative to the background energy of spacetime itself, which we arbitrarily call 'zero'. Of course by ignoring the energy (and dark energy) that compose the quantum vacuum, we blind ourselves to knowledge of its existence, and make it impossible for us to understand things like what is energy, where does charge come from, what happens inside a black hole's event horizon, what is dark energy, what is time, and what is space. This ignorance also prevents us from leaning how to control the interaction of the quantum energy fields that compose spacetime itself, thereby severely limiting our ability to bend or fold the spacetime field, and limiting our ability to control energy field interactions at subatomic scales. In turn, that limits our nano-engineering and nano-materials capabilities.

There is nothing more immaterial than the infinite singularity. The infinite singularity has no constraints, no boundaries, no dependencies on the existence of numbers, quantity, or dimension, and no dependence on an observer. It has absolutely no domain limitations. It cannot be destroyed. It can only be transformed. Instead of making mathematics isomorphic to the indirect representation of information and human communication, we need to make it isomorphic to the direct physical representation of existence. Instead of building a Von Neumann Universe from the transfinite recursion of empty sets via nested powersets to create the indirect mathematical universe of ordinals, we need to build a direct set theoretic representation of the universe itself based on transfinite recursion of symmetric differences in the infinite singularity. This requires that we redefine the axiomatic concept of set using an ontology and semantics that are isomorphic to the direct representation of existence.

The fundamental unit of existence is a quantized nilpotent symmetric difference in the infinite singularity. At the quantum scale, all matter and energy are quantized. All physical quanta exist fully or not at all. All quanta are created and destroyed instantaneously[4]. At no time are they partly in existence and partly nonexistent. Quanta are the indivisible atomic units of existence. They cannot be subdivided. Quanta are universally, physically, existentially, and completely bivalent. They exist fully, or they do not exist at all. If they are destroyed, they don't cease to exist. They get converted into virtual energy open strings, or they become part of the infinite singularity. Quanta are also much more general than the answers to yes-or-no questions. As far as we know, everything in the Universe is composed of quanta and their relationships at the deepest level of existence. Thus, a representation based on the transfinite recursion of quanta and their relationships is well founded, complete, and not domain limited. Furthermore, quanta and their direct relationships avoid the need for an observer, they avoid the need to formulate or ask yes or no questions, they avoid the need to measure and interpret the answers to those questions, and they avoid the need to store the answers as bits of information. Therefore, they avoid the need for the universe to describe itself from the indirect perspective of an observer, and they avoid the consequent violation of causality that entails. Given all the facts above, it is clear that quanta and their relationships are the true basis for the bivalence of existence, not the indirect representation of bits of information.

Existence is not based on the answers to true-or-false questions. It is based on the direct physical existence or nonexistence of quanta and their relationships. The quanta and the relationships between quanta are based on the transfinite recursive composition of symmetric differences in the infinite singularity. The existence of symmetry, space-time, energy, all forces, and the evolution of the universe itself, are a consequence of the conservation of the infinite singularity. The infinite singularity must be conserved because infinity has no beginning and no end. The conservation of energy is a special case of the conservation of infinity. The conservation of infinity is responsible for the existence of symmetry, all quanta, all forms of energy and dark energy, all forms of matter and dark matter, time, space, and all relationships between quanta, energy, dark energy, time, space, matter, and dark matter.

Thought Beyond Information

The key idea behind the representation of thought is to represent one and only one thing directly, but that one thing then represents everything else indirectly. The one thing that represents everything indirectly is abstraction. Abstraction itself is a kind of direct representation. This then provides a direct representation of indirect representation. It directly represents everything indirectly. With indirect representations like logic, set theory and mathematics, we attempt to represent everything represented by direct representation indirectly. Logic, set theory and mathematics do just the opposite of what the brain does. Instead of directly representing everything indirectly, logic, set theory and mathematics attempt to indirectly represent everything, directly. It is impossible to indirectly represent everything directly because the indirect representation of everything is too complex and it is inconsistent or incomplete or both. Doing things the other way around, the representation only has to represent one thing completely and consistently. If there is only one thing to represent in a domain of discourse, the only way for it to be incomplete or inconsistent is for it to be incomplete or inconsistent relative to itself. It is impossible for a direct representation based on relative relational encoding to be inconsistent or incomplete[5]. This then allows us to avoid the adverse consequences of Gödel’s incompleteness theorems.

Fortunately, it is possible to represent one thing completely and consistently using information - provided the complexity of that one thing is not too great. Therefore, we can use a computer to indirectly represent the direct representation of abstraction, and then use multiple instances of that one simulated direct representation of abstraction to ‘directly’ represent everything else indirectly. We use the same strategy used by nature in the brain, but it is a little less efficient due to the additional level of indirection. Nevertheless, it still provides the means to represent everything indirectly completely and consistently. It also still has all the same benefits in terms of the geometric combinatoric reduction in complexity and storage size. It allows us to create sentient computers that represent and understand the meaning of information from the first person direct perspective in context.

Summary

There are three main branches in the tree of knowledge.

· Direct Representation
· Universal Representation
· Indirect Representation

Logic, set theory, mathematics, information, and human communication are all forms of indirect representation.

Formal systems are incomplete because they are indirect representations. Indirect representations cannot represent themselves or anything else directly.

Direct representation is complete and consistent. Direct representations can represent themselves and all that they represent directly.

Everything that physically exists in the universe is represented by a direct representation. This includes the physical existence of neurons and the process of abstract thought, including the thought process that led to the human development of indirect representation. Direct representation led to universal representation which led humanity to the development of indirect representation and the representation of information.

Many of the “unsolved” mysteries and complexities encountered in the physical sciences are due to our attempts to represent complete, direct, context-dependent phenomena using incomplete, indirect, context- free representations. Things are a lot simpler if viewed from the correct perspective using the correct representation.

Existence is a direct representation based on symmetric differences in the infinite singularity. Symmetric differences in the singularity represent all energy and dark energy quanta, all bosons and fermions and all energy relations between bosons and fermions. Composition of those difference creates all quantum field energy and dark energy compositions, and all quantum state compositions. They compose the existence of the entire universe.

It is impossible to destroy the infinite singularity. This suggests the existence of a new fundamental physical law for the conservation of the singularity. The conservation of the singularity is the first cause of symmetry, the cause of energy and dark energy, the cause of matter and dark matter, the cause of the conservation of energy, the cause of all forces, and the cause of the principle constraints that govern the ongoing evolution of existence.

Universal representation is the most powerful and most compact of the three classes of representation. It is direct and indirect, intensional and extensional, context dependent and context free, and complete and consistent. It is based on the direct representation of the ontology and process of abstraction. The direct representation and process of abstraction represents abstractions and concepts directly and indirectly. It also represents the relation between intensional meaning and extensional existence, and does so in context across all levels of abstraction. It converts the external indirect representation of information to and from the direct internal representation of thought and knowledge.

Recommendations

Based on the foregoing arguments, we should place very high priority on research in the following areas:

1) We should develop a new axiomatic set theory based on the direct representation of existence, instead of the indirect representation of information. This will complement existing set theories, existing mathematical logic, and existing mathematics and create a complete and consistent mathematics for the direct representation of existence and Physics. It will allow us to understand the nature of existence with combinatorially less complexity than we can by using an indirect, incomplete representation to represent the direct, complete representation of existence. It should accelerate the development of theoretical physics exponentially.

2) We should develop a new axiomatic set theory based on the universal representation of thought. That is, we need to develop an axiomatic set theory that is both direct and indirect. The resulting logic is geometrically, combinatorially less complex then the indirect representation of current set theories and it is complete and consistent. It is the representation of thought. This leads directly to the development of an extension of information theory that provides the basis for the development of sentient computers that will be able to amplify human intelligence in a manner analogous to the way our machines amplify our muscles. In turn, an understanding of the representation of thought will allow us to improve the methods we use to educate our children. It will also improve our ability to treat brain injuries and mental illness. Most importantly, it will substantially increase human intelligence by teaching us how to think directly in terms of our brain’s native knowledge representation, instead of trying to think indirectly in terms of information.

3) We should develop sentient computers and true machine intelligence. This should be done in two steps. First, we should develop sentient computer software simulations. The author of this paper has already developed the mathematical equations required for this advance. This will teach us a lot about how we think. It will also allow us to create sentient computers that can solve problems that are currently beyond the capacity of human intellect. The results of software simulations can also be used as the basis for the development of sentient computer neuromorphic processors.

4) We can embed sentient software, and/or neuromorphic processors in machines and robots to create autonomous machine intelligence.


Authors Email: barry.kumnick@gmail.com


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[1] The representation of thought is actually both direct and indirect. This is explained later in this paper.
[2] A paper that describes this transformation in detail is in preparation.
[3] The lower size limit is the Planck length.
[4] Creation and destruction occur within a Planck time.
[5] In a relative relational encoding, the representation and the encoding are fully encapsulated. The encoding itself is a function of that which it encodes. The representation of existence is defined relative to symmetric differences in nonexistence. Nonexistence is the only thing ‘inside’ and ‘outside’ the universe that has no dependencies.

[1] To complete the Powerset of representation, a null representation would also exist, but it is uninteresting.