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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. 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.

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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.

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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?

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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.

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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.

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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

, 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.

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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.

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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.

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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

.

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

. 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

. 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

. 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.

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8. Godel, Kurt. “Some metamathematical results on completeness and consistency (1930b)”. __From Frege to Godel, A Source Book in Mathematical Logic, 1879-1931__. Ed. Jean van Heijenoort. Cambridge, MA: Harvard University Press, 1967. 595-6.

9. Godel, Kurt. “On formally undecidable propositions of Principia mathematica and related systems I (1931)”. From Frege to Godel, A Source Book in Mathematical Logic, 1879-1931. Ed. Jean van Heijenoort. Cambridge, MA: Harvard University Press, 1967. 596-616.

10. Godel, Kurt. “On completeness and consistency (1931a)”. From Frege to Godel, A Source Book in Mathematical Logic, 1879-1931. Ed. Jean van Heijenoort. Cambridge, MA: Harvard University Press, 1967. 616-7.