Wednesday, January 28, 2009

A Major Paradigm Shift Is Coming

A Major Paradigm Shift Is Coming

I am working on research that will result in a major paradigm shift that will turn out to be as significant to science and society as that of the shift from the Ptolemaic world view to that of the Copernican world view. My research has resulted in the discovery of two entirely new classes of representation that are both mathematically consistent and complete.

Why is this important? All of logic, mathematics, computation, and science are currently based on the theory of information. Godel's Incompleteness theorems proved that all formal systems above the complexity required to represent Peano arithmetic are incomplete, or inconsistent, or both.

The cause of incompleteness and inconsistency is not the concept of a formal system itself. The cause is the underlying representation of information formal systems are represented in terms of. Information is an indirect representation. It can only represent things INDIRECTLY. Information can not represent ANYTHING DIRECTLY. All indirect representations (of any complexity) are necessarily incomplete because they rely on something outside themselves, the mind of an intelligent observer, to decide the meaning of the bits and symbols represesented by information. The information does not represent meaning. The meaning of information is inferred by an external system, namely the observers brain. The internal operation of the observer's brain is not based on the representation of information, any more than the physical existence of the universe is. Just because our brain can interpret the meaning of information, it does not mean that our brain's operation itself is based on the representation of information. The belief that the brain's internal operation is based on the representation of information is a fallacy. I have proven this.

The problem for science is, the physical universe is complete and consistent, but our logical and mathematical representation of it IS NOT. Information is the wrong basis for the representation of physical existence. Information is incapable of representing the universe completely or consistently. For example, physical existence does not have any representation for the number zero. Physical existence can not and does not have a physical representation of nonexistence. Nonexistence does not exist as long as spacetime and dimensionality exist. Zero is a placeholder in the representation of information for the representation of nonexistence, or the representation of a possible existence. Nature does not represent the possible. It only represents that which actually exists. In terms of information, it is as if Nature only represents the ones.

Because nature only represents what actually exists phyically, nature does not have to make any decisions to represent existence or the physical configuration of the state space of existence. Nature does not have to interpret the meaning of information. The laws of physics create and enforce themselves because they are based on a complete consistent representation of existence that can ONLY represent that which exists. The laws of physics exist because they are the only possible laws that nature can represent using the direct representation of existence. There are no decisions to make. Only man wastes bits, energy and time representing the nonexistent. Nature is exponentially less complex. Since existence exists, and its representation cannot be based on information, its representation must be based on another form of representation. All indirect representations are incomplete and or inconsistent as Godel proved. The logical converse of an indirect representation is a direct representation. The converse of an incomplete, inconsistent, indirect representation is a complete, consistent direct representation. The representation of existence is based on a complete and consistent direct representation.

The universe is the transfinite recursion of the direct upper ontology of existence. Its recursive operator is based on the composition of symmetric differences between existence and incomplete (aka partial) nonexistence.

Using this representation one can generate new forms of logic and mathematics exponentially more powerful than those based on information. These mathematical forms provide a complete and consistent description of the universe. This description literally generates and enforces all the laws of physics. It can describe the creation of the universe and the evolution of everything in it starting from the bosonic singularity of complete nonexistence, where complete nonexistence is defined as the absence of all spacetime, matter and dimensionality. It describes what causes the quantization of energy. It describes what causes symmetry. It describes what space and time are and how they are formed. It describes the cause of the big bang and the lifecycle of the universe. It describes what happens inside the event horizon of a black hole. It describes the cause of the zero point field. It describes the cause of mass and the cause of gravity. It describes the cause of zitterbewegung. It describes the cause of subatomic structure and ultimately, the cause of all higher order forms of matter and energy in the universe. It provides the key to a complete understanding of physics.

My two main discoveries include:

1) The direct upper ontology of the representation of physical existence. I have discovered an upper ontology that is one-to-one isomorphic to the direct representation of physical existence. In other words, I have discovered a method that will allow us to create a direct one-to-one mathematical representation of the physical existence of the universe. Physical existence itself is a kind of representation, but it's representation is not based on information. Its representation is based on, and is, the transfinite recursion of the direct upper ontology of existence. By using that representation in a computer, it will allow us to directly model the creation of the universe, the creation of everything in it, and the relations between everything in it in context. Unlike conventional upper ontologies that are designed to be a foundation for lower level domain specific ontologies, the upper ontology of existence eliminates the need to create any domain specific ontologies. This should reduce the complexity of the mathematical representation of complex systems combinatorially. It will also allow us to conveniently represent the existence of phenomena whose behaviors vary depending on the context they exist in.

Black Holes

Given upcoming events at the LHC we better make sure we really understand black holes before we create them. The current theory of black hole formation and operation is based on information theory. It is incomplete.

Current theories as to what goes on inside the event horizon of a black hole are incorrect. Black holes are not created, nor do they operate as current Physics predicts. Specifically, their power source and the source of their gravity field is not the energy in the particle stream that creates them. A black holes power source is derived primarily from the collapse of the fermionic field that composes spacetime inside the event horizon. Exceeding the speed of light destroys the consistency of the representation of existence and causes the collapse of the fermionic field that composes the zero point field that composes the dimensions of spacetime. When it collapses, the fermionic field transforms to its inner representation, a bosonic field. Bosons do not occupy spacetime. That is why bosons can all occupy the same non-dimensional "point". That is why black holes cause singularities. The vast majority of the energy contained in a black hole comes from the conversion of the fermionic field to a bosonic field.

Inside the event horizon the zero point field that composes the structure of spacetime has collapsed. Dimensionality ceases to exist. Energy cannot escape the black hole because it has no spacetime to travel thru, not because it can't travel faster than the speed of light.

The main power source for a black hole is the difference between the zero point field energy outside the event horizon and the localized nonexistence inside the event horizon. A black hole is the only phenomena in the universe that frees up all the energy in the zeropoint field and makes it available to do work. The almost limitless zero point field energy outside the event horizon flows to "ground". It causes the black hole to ingest spacetime and collapse its fermionic field, converting it to a bosonic field and allowing it to collapse into a singularity.

The conservation of energy can not be used to calculate the energy in a black hole becuase it does not consider the difference in the zero point field energy that composes spacetime outside the event horizon and the absolute zero energy density inside the event horizon. Evaporation of the black hole (if it evaporates) exposes the singularity at its core, and allows the naked singularity to convert most of its energy back into its outer representation - a fermionic field and the subsequent creation of the zero point field and space time. The explosive expansion in space time that results transports massive amounts of high energy gamma rays as it expands and the zero point field reforms itself. This is the cause of gamma ray bursts. In the extreme case of the primordial black hole, it causes the big bang.

This paper also identifies a new mechanism for the creation of black holes that makes it unlikely they would be created by cosmic rays, but increases the likelihood they may be created in a particle accelerator. Be advised, the energy in the particle stream used to create a black hole is only the energy needed to trigger the formation of the black hole. It does not account for the energy the black hole will ingest and convert from the spacetime surrounding the event horizon. Creation of an artificial black hole could be catastrophic. Due to variability in the way a black hole could be created in a particle accelerator, it is not possible to quantitatively ascertain the amount of energy that would be released if it evaporates and exposes its singularity. In addition, the mechanism thought to account for black hole evaporation was based on the representation of information. It is not reliable.

2) The universal representation of thought. This appears to be the basis for the biological neural knowledge representation and upper ontology responsible for all human thought, perception, awareness and consciousness. It solves the unitary binding problem in neural science. It provides a completely new model of computation and a completely new coding theory that will allow us to develop sentient computer systems that can perceive, think and understand the meaning of information and knowledge from their own first person direct perspective in context. Just as the universe is the transfinite recursion of the direct upper ontology of existence, the mind results from the transfinite recursion of the direct (and indirect) upper ontology of abstraction. Both the representation of thought and the representation of existence form the basis for mathematical systems exponentially more powerful, and more compact than the representation of information. In fact the representation of thought provides exponential compression relative to the direct representation of existence. That is why we can store so much knowledge within the limited volume of our craniums.

Philosophical Basis

Some time ago, I discovered an inconsistency in the Philosophy of information, in the principles of ontological neutrality that has been lurking there since the early 1960's. Its significance was apparently overlooked.

ON.2 (It from Bit) is inconsistent with ON.1, ON.3, and ON.4. See http://plato.stanford.edu/entries/information-semantic/#1.6 for the definitions of these philosophical principles. Proof of the inconsistencies in ON.2 relative to the interpretation of physical existence can be found in earlier entries in this blog.

I was able to resolve the inconsistencies by separating the ontological representation of existence from the ontological representation of information. This preserves ON.2 but restricts its domain to that of communication and computation.

By creating a separate ontology for existence, it allows each thing in existence to represent itself from its own first person direct perspective in context. It also accounts for the ubiquitous fact that each thing that exists is composed of other things. Its existence is defined in terms of how it relates to the things it is composed of and how it relates to the things in its external environment. The complexity of the representation and ontology of existence is constant, and independent of the number of types of things in existence. By contrast, the complexity of the representation of information grows combinatorially in the number of types of things it represents.

The ontology and direct representation of existence eliminates the problem of the observer in Physics and in physical existence, and it provides a mechanism that automatically accounts for the consistency and completeness of the totality of existence itself. It also provides a physical existential ontology that can solve the horizon and flatness problems in cosmology without violating the speed of light as inflation does.

Information is Incomplete and too Complex to Represent the Totality of Existence

Imagine you are a proton. A proton can't represent itself from the 3rd person indirect perspective of an observer. Which observers' perspective would it choose to represent itself from? Which observers' frame of reference would it represent itself from? If you are a proton, you must experience forces from the perspective of your own existence, relative to the spacetime context and frame of reference you exist in. You can't experience forces as seen from the third person perspective or reference frame of any observer.

While we can model protons or most other things indirectly using information and current mathematics, it quickly becomes very complex to model large systems of interacting fields and particles mathematically, especially if their behavior varies based on the context they exist in. For example this makes it extremely difficult or impossible to solve many body problems in quantum mechanics for systems larger than a hydrogen atom without simplifying them into collections of simpler problems, or ignoring parts of the problem to create a problem simple enough to solve. Using the ontology and direct representation of existence, it should be possible to model and solve very large many body problems directly because the ontology and direct representation of existence model context dependent relations directly. Using the ontology and direct representation of existence, the encoding and representation of many body problems is dynamic and the representation alters itself dynamically based on the changing contextual relationships between the bodies in the problem. The result should be a combinatoric reduction in the complexity of the equations required to solve many body problems, and a combinatoric increase in the size of many body problems we can solve. Consequently, further development of this theory could lead to rapid advances in our knowledge of quantum physics and relativistic quantum field theory among other things.

Seen from this perspective, the idea of representing the physical existence of the universe in terms of information is ludicrous. Protons aren't physically composed of bits or information. In addition representing the physical existence of protons using information would violate cause and effect because physical existence itself is a representation. Physical existence is logically and physically prior to observation. The universe existed long before conditions in it could support life. Nobody could have been around to represent existence from the third person indirect perspective of an observer. Therefore, the physical representation of existence cannot be based on information. It cannot be based on any representation that requires an observer, other than each particle itself, because if we go back in time to the existence of the first photon in existence, only the first person perspective of the existence of the first photon could have been available.

Each thing that exists has to represent itself from its own first person direct perspective. This is true whether we are talking about subatomic particles, energy quanta, atoms, molecules, proteins, neurons, people, rocks, trees, stars, galaxies or anything else that exists. The only representation that is the same across all of existence is the first person direct representation of each thing that exists. The representation of physical existence emerges from the composition of the first person direct representation of each thing that exists and the relationships among those things. All of those relationships and all of those representations are based on and constrained by the upper ontology of existence.

The representation of information was designed to support human communication. It wasn't designed to represent the physical existence of fields of force and subatomic particles. This has been known since the 1960's in Philosophy, but it looks like nobody noticed what the implications of those inconsistencies implied. Specifically:

1) By separating the ontology of information and the ontology of existence, we can create a direct one-to-one representation of the existence of the physical universe. We can generate a direct set theoretic representation of physical existence from the transfinite recursion of complete nonexistence and a nilpotent symmetric difference in nonexistence in a manner analogous to the way the Von Neumann Universe of mathematics is generated from the transfinite recursion of an empty set and the set that contains the empty set. This will provide the basis for a new kind of direct set theory and direct mathematics that is isomorphic to physical existence itself. It will also provide the ontological foundation required to further the development of quantum computation.

2) Human communication and computation are based on the transfer or communication of information between computers and individuals, but the requirements necessary to support communication are not the same as those needed for the computation of meaning from the first person direct perspective in context. Why should the human brain be based on the requirements of information? Most parts of the brain evolved long before the development of speech in our species. Why should we base computation on the requirements of a mathematical theory of communication? Why not base it on the requirements of abstract computation, and then translate the results to and from information for external communication? That is what the brain does. It turns out evolution was a lot smarter than we were. Evolution figured out a way to avoid the limitations of Goedel's Incompleteness Theorems, and a way to compute everything using a single universal computational algorithm, and a single knowledge representation with absolutely no domain limitations. What's more it does so with constant computational complexity, while computational power scales combinatorially for each stage of neural processing, and storage is compressed as a combinatorial of combinatorals for each subsequent stage of neural processing.

I developed a separate ontology for the direct representation of existence that looks like it will be able to explain all of Physics, even the cause of the Big Bang itself. I.e., where did all that energy come from? It also explains the first cause of symmetry, it explains what energy is, not just what it can do. It explains why existence is quantized, it explains what created spacetime and what it is, it provides an alternative explanation for black holes, it provides a much more intuitive explanation for quantum mechanics, and it looks like it may explain the cause of mass. It also predicts the Higgs mechanism is incorrect. It disproves the many worlds interpretation of quantum mechanics. With some further development, we should be able to use it to solve very large many body problems.

It looks like my thesis on the direct representation of existence is consistent with most of the standard model except:

1) It includes the physical existence of spacetime in the form of a fermionic field with spin 1/2 that assembles the zero point field and creates the four dimensions of spacetime.

2) It provides an alternative explanation for the expansion of the universe.

3) It provides an alternative explanation for the cause of black holes and the source of their gravity and power.

4) It provides an additional law of nature more fundamental than the conservation of energy - that causes the conservation of energy.

5) It provides an alternative explanation for the cause of mass.

6) It predicts the Higgs mechanism is incorrect and that the Higgs' Boson does not exist.

7) It identifies the root cause of the quantization of energy.

8) Instead of representing existence using a fixed number of dimensions, it represents it in the minimal combination of dimensions required to represent each thing in existence with maximal entropy. The mathematical system this forms allows one to solve systems of equations independent of the dimensionality of that which the system represents. It also represents everything in context so the representation automatically accounts for all contextual dependencies.

9) It eliminates the need for observer relative or observer dependent reprsentation.

10) It ensures the consistency and completeness of the universe.

11) It eliminates the need for a "decider" to determine which possibilities exist and which are only potential.

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The Universal Representation and Ontology of Thought

I have also discovered what appears to be the ontology and neural knowledge representation responsible for the representation and computation of all abstract human thought, perception, and consciousness. It provides a new model of computation and a new coding theory that will allow computers to represent the meaning of information from the first person direct perspective in context, and understand its meaning. It will also allow computers to perceive the world around them and form their own subjective experiences. The representation of thought will allow the development of sentient computers that compute directly in terms of abstractions and concepts from the first person direct perspective in context.

Like the ontology of existence, the ontology of thought is an upper ontology that eliminates the need to create any domain specific ontologies. It is based on the ontology of abstraction itself. The ontology of thought is one-to-one isomorphic to the spatial topology of biological neurons. Everything we can think, experience, feel, perceive, or do can be represented as an abstraction. Neurons in the brain represent everything we think, experience, feel, perceive, and do as abstractions. Each neuron is a living expemplar of a whole collection of related abstractions.

This is a major advance in computation. Not only will we be able to create sentient computers that can think and understand meaning using the same knowledge representation used by our brains; it provides a model of computation that has constant complexity, independent of the complexity of the problem being solved. Processing capacity scales geometrically. If we assume each neuron in the network can compute the result of a 100 term abstract equation, then:

- First level neurons can each compute a 100 term equation.
- Second level neurons can each compute a 10,000 term equation.
- Third level neurons can each compute a 1,000,000 term equation.
- Fourth level neurons can each compute a 100,000,000 term equation.
- Fifth level neurons can each compute a 10,000,000,000 term equation.
Etc.

In addition, computation time is constant at each level. If each level takes 5 mS to compute its result, we could compute the results of n 1.0 x 10^10 term abstract equations in 25 ms where n is the number of neurons in each layer of the network.

Doing this in realtime would require development of a new type of massively parallel hybrid neuromorphic CPU, but even with conventional hardware, a lot can be done, even with current computers. The computational model is very, very efficient. It uses a universal of computation. It is based on a single function that can compute anything a sentient system can percieve, feel, think, learn, understand, or do from the first person direct perspective of the system itself.

Storage is also extremely compact. Storage is compressed combinatorially at each layer in the network. Compression efficiencies scale in direct proportion to computational capacity so compression is geometric combinatorial in the number of levels in the network. Hence the amount of knowledge the network can store is astronomical.

Even better, the direct representation of existence and the universal representation of thought are both complete and consistent and have no domain limitations. I'm not just talking truth functionally complete and consistent. I am talking universally complete and consistent. As in the ability to compute anything in the universe with no domain limitations.

These representations both get around the limitations of Goedels Incompleteness Theorems. This will allow us to develop new set theories and new forms of mathematics that are universally consistent and complete. We'll be able to work through logical problems and compute anything in the universe, or anything we are capable of thinking using a single ontology and a single knowledge representation with absolutely no domain limitations.

That's pretty powerful stuff.

Big changes are coming.

Stay tuned for further developments.

Friday, January 23, 2009

Ontology

Ontology

Since the direct representation of existence and the universal representation of thought are both based on ontologies, a brief review of ontology is in order.

According to Tom Gruber, an ontology is an explicit specification of a conceptualization (Gruber, 1993). That is, an ontology is a description (like a formal specification of a program) of the concepts and concept relationships that are of interest in a domain of discourse. The term “ontology” is borrowed from philosophy, where an ontology is a systematic account of existence.
For knowledge-based systems, what “exists” is exactly that which can be represented.

Formally, an ontology is the statement of a logical theory; i.e, an ontology is the (representation / description / encoding) of a logical theory. Most ontologies defined to date have been based on indirect representations. Thought is represented in terms of and relative to each agents’ universal ontolology.

When we represent the knowledge of a domain in a declarative formalism, we call the set of objects that can be represented the universe of discourse. This set of objects, and the describable relationships among them, are reflected in the representational vocabulary with which a knowledge-based program represents knowledge. Thus, we can describe the ontology of a program by defining a set of representational terms. In such an ontology, definitions associate the names of entities in the universe of discourse (e.g., classes, relations, functions, or other objects) with human-readable text describing what the names are meant to denote, and formal axioms that constrain the interpretation and well-formed use of these terms.

We use common (shared) ontologies to describe ontological commitments for a set of agents so that they can communicate about a domain of discourse without necessarily operating on a globally shared theory. We say that an agent commits to an ontology if its observable actions are consistent with the definitions in the ontology. Pragmatically, a common ontology defines the vocabulary with which queries and assertions are exchanged among agents. Ontological commitments are agreements to use the shared vocabulary in a coherent and consistent manner. The agents sharing a vocabulary need not share a knowledge base; each knows things the other does not, and an agent that commits to an ontology is not required to answer all queries that can be formulated in the shared vocabulary.

In short, a commitment to a common ontology is a guarantee of consistency, but not completeness, with respect to queries and assertions using the vocabulary defined in the ontology.

A domain ontology (or domain-specific ontology) models a specific domain, or part of the world. It represents the particular meanings of terms as they apply to that domain. For example, the word ‘card’ has many different meanings. An ontology about the domain of poker would model the ‘playing card’ meaning of the word, while an ontology about the domain of computer hardware would model the ‘punch card’ and ‘video card’ meanings.

In information science, an upper ontology (top-level ontology, or foundation ontology) is an attempt to create an ontology which describes very general concepts that are the same across all domains. The aim is to have a large number of ontologies accessible under this upper ontology. It is usually a taxonomy of entities, relationships, and axioms that attempts to describe the representation of those general entities that do not belong to a specific problem domain.

Representational Encodings

Representational Encodings

The type of encoding used by a representation confers upon it unique properties and abilities enabling each type of representation to serve its distinct purpose. Without their distinct encodings, the three types of representation would not possess the properties and abilities required to represent existence, thought and information respectively. Only by understanding the advantages and disadvantages of the different types of encodings can we know how best to utilize each encoding and each representation that uses it.
There are two basic types of representational encodings: Fixed, and Relative.


Fixed Encodings
Fixed encodings represent each symbol with a constant fixed code or fixed numeric value, or fixed pattern. Information uses fixed encodings. For example in a computer, the ASCII code for the letter ‘A’ is always decimal 65 or binary 01000001. Every computer that uses the ASCII encoding represents an upper case A as the decimal number 65. In printed text using the Latin alphabet, the uppercase letter ‘A’ is always represented by a character that looks like ‘A’. Fixed encodings are typically based on standards, conventions or agreements. Fixed encodings are context free. The value of the code used to represent each symbol is fixed. It does not change as a function of the context it is used in. Information uses fixed encodings. Fixed encodings are well suited for communication. Their weakness is they are not very compact, and they do not scale well when representing complex, context dependent information.


Relative Encodings
In contrast to fixed encodings, relative encodings have no public fixed “symbols”, fixed codes, or fixed values. Relative encodings are private and context dependent. Relative encodings encode the representation of particulars in terms of how they ‘relate’ to ‘other particulars’, where the ‘relations’ and ‘other particulars’ are defined relative to the context in which they are used, or they are represented by an instance of a previously defined relative encoding within the context of definition of that which they participate in the definition of.

Both fixed and relative encodings can represent how things relate to each other, but they do so differently. Fixed encodings represent relationships external to the encoding of an entity; that is the relationships are used to define the intension of an entity, but the encoding of the relationships themselves is independent of the context they are used in. For example, in a fixed encoding the representation and meaning of an addition operator is defined outside of the context in which it is used. Its’ encoding and meaning are not affected by the context it is used in. In a fixed encoding, the relationships are represented by the encoding, but they are defined external to it. Hence, in a fixed encoding, the relationships have an existence and a coding independent of their use in the intensional representation of an entity.

Relative encodings embed the encoding and representation of the relationships between entities as part of the representation of the entity itself. The entity being defined encodes the representation of the relationship relative to, and in terms of the entities own existence or direct representation. The relationships are internal to the encoding of the representation and are defined by, in terms of, and relative to it. The encoding of the relationships is inseparable from the encoding of that which they relate. In a relative encoding, the relationships do not have independent context free definitions. They are only defined by and only have meaning relative to the context they are used in. This is a critical distinction. With a relative encoding, either all the parts of the representation of a particular are embedded inside the particular, or the particular does not exist.

Relative encodings encapsulate the representation of their component parts. Fixed encodings do not. With a fixed encoding, each part of the representation is separable and context independent. Fixed encodings allow partial representations of particulars. Relative encodings do not. The use of relative encoding to encode the direct representation of existence is the cause of the bivalence of existence. At the level of quanta in Physics, quanta exist fully or not at all. Empty space has physical existence because it has dimension. Distances can be measured in space. If space did not exist, there would be no measurable distance between objects in space. Space can be curved. Space also contains vacuum energy fields, also known as the zero point field, or dark energy, thought to be responsible for the cosmological constant and the accelerating expansion of space. Additional physical evidence for considering the physical existence of space will be covered in more detail later. Fundamental particles (i.e., fermions) and bosons also exist fully or not at all. Nothing physical partially exists at the fundamental level of physical quanta.

The relative encoding of existence is also the cause of the Pauli Exclusion Principle in Physics. It is the reason matter cannot pass through matter, even though according to the standard model of particle physics, matter is known to be 99.999999999999% empty space. Matter is composed of curved or knotted zero point energy vacuum fields. The space-time encapsulated in matter is part of the representation of matter. It is not possible to separate matter from the space-time that composes it because the direct relative encoding of matter encapsulates the representation of the space-time from which the matter is composed. Removing the space-time from the representation of matter, or changing the space used in the representation of matter would be the same as removing part of the representation of matter. It would break the encapsulation of the representation of existence, which would make it inconsistent and the matter would cease to exist. It is not possible to break the encapsulation of the representation of existence without destroying it. The encapsulation of the representation of existence is a fundamental property of existence.

The encapsulation of the representation of existence is responsible for the bivalence of existence. If the existential representation were not encapsulated, existence would not be bivalent; i.e. quanta could partially exist, and quantum states would not exist, and the universe would not exhibit quantum behavior or operate according to the laws of Quantum Mechanics . The encapsulation of the representation of existence is the cause of all quantum states in Physics. If the fundamental building blocks of existence did not have quantum states, then the conservation of nonexistence would be violated (we'll cover this later after we formally define the representation of existence), it would be possible to destroy nothing, and the laws of Physics would be inconsistent.

Without quantum phenomena, the fundamental building blocks of existence could partially exist and existence would be continuous, incomplete and inconsistent instead of quantized, complete and consistent. Partial existence does not occur at the quantum level. Nothing that exists can exist half inside and half outside the universe. An individual quantum property must be in one quantum state or another. It cannot exist at a level partly between two different quantum states. Quantum behavior is a fundamental property of existence precisely because it is dependent on the representation of existence. It is a necessary fundamental property of existence because of the bivalence between existence and non-existence and the encapsulation of the representation of existence.

Wednesday, January 21, 2009

Representation

Representation

The Merriam Webster’s Collegiate dictionary definition of representation is something that serves as a specimen, example, or instance of something[1].

Before we dive into the detailed representation of thought, existence, and information, we need to take a step back and examine the fundamental nature of representation in general. Although representation is one of the most fundamental phenomena in existence, the most fundamental question: what is it? – has rarely been answered directly.

The overwhelming majority of work on representation has been based on the symbolic representation of information. This includes the development of logic, mathematics, and information systems.

Information and everything based on it only address one of the three fundamental types of representation. It is important to view the entire landscape of representation so we can see how logic, mathematics, and information relate to the other fundamental types of representation. It is also important to see if some other fundamental type of representation is better suited to the representation of thought than a representation based on symbolic logic, mathematics, or information. We need to look beyond the symbolic representation of information. If we confine ourselves to only one of the three fundamental types of representation, we limit our ability to reason to that which can be represented by that single fundamental type. We should not limit our ability to reason needlessly. We certainly should not do so blindly.

An Orthogonal Classification of Representation
In the most general sense, representations can be classified along two orthogonal principal axes: The first axis of representation is the direct - indirect axis. The second is the intensional - extensional axis. These axes form an abstract two dimensional concept space within which we will map and analyze the different fundamental types of representation.

The Direct - Indirect Axis of Representation
There are three fundamental classes of representation along the direct – indirect axis; direct representations, indirect representations, and universal representations.

[1] Merriam Websters Collegiate Dictionary, 11th Edition, Meriam-Webster, Inc., 2003

Direct Representation

Direct Representation

A direct representation is the thing itself; i.e., everything that exists in the universe is a direct representation of itself. For example, the direct representation of a particular proton is that particular proton. In fact, there is, and can only be, one direct representation of each particular thing that exists in the universe. Hence, the referent (i.e., extension) of a direct representation IS directly composed of the representations intension, where the extensional representation is the only instance of the thing represented. Composition in the sense used here is equivalent to containment; i.e., in a direct representation, the extension is composed of its intension because it contains its intension. The existence of the intension is equivalent to the existence of the extension. Everything that exists represents itself. In short, existence represents itself. This can be represented symbolically by the equation: Existence = Representation. Mathematically, a direct representation is endomorphic. Each particular or “thing” that exists in the universe is the only direct representation of its own existence. Direct representations are “first-person” representations. They represent things from the first person “inside-out” perspective of the things themselves instead of from the third person indirect “outside-in” perspective of an external “observer”. In a direct representation, any calculations, computations, or processes occur directly on, and in terms of the actual thing itself, and hence directly on its own representation. The importance of direct representation has been seriously underestimated. Development and application of direct representations is the key to solving many of the deepest unsolved problems in Philosophy, Physics, Cognitive Science, Artificial Intelligence (AI), and the theory of representation. One of the most difficult unsolved problems in logic, AI, and cognitive science is how to create a knowledge representation that can perform computation from a “first-person” perspective. How do we make a computer self-aware? How do we give a computer the ability to represent and understand meaning from a first person perspective? How do we give it an intrinsic sense of self; e.g., cogito ergo sum? I think, therefore I exist. How do we create a conscious machine? Direct representation is the key to the solution to ALL of these problems.

Direct representation is also the key to the creation, representation and ongoing construction and operation of existence itself. Everything that exists must have some representation. Without representation, there can be no existence. The direct representation of existence IS existence. Furthermore, things must be able to exist independent of the existence of an observer. Existence cannot represent itself from the perspective of an observer. Existence is logically prior to observation. Thus, the representation of each thing that exists must exist independent of the existence of an observer. Just because nobody is around to observe a thing does not mean that thing does not exist. Space-time, matter and energy were all present in the universe long before there were any observers to experience them. This means the representation of existence cannot depend on an observer in any way whatsoever. The representation of existence must be entirely observer independent. The only way to eliminate the observer in representation is to define a type of representation in which everything that is represented is its own observer. We must use an endomorphic representation in which the referent of the representation of a thing refers to the things own representation. Everything in existence is then represented from its own “first person” perspective. In existence, things are not represented relative to an observer. The representation of everything in existence is represented relative to itself. Existence represents itself. Understanding this is critical. If you do not understand this, you will never fully understand existence. Do not despair if you don’t fully understand this immediately. Direct representation is actually very simple, yet it can be a difficult concept to grasp because we are so used to representing things indirectly to express them as information.

Direct representations are also context dependent. Direct representations always represent things in context. Their representation is defined in terms of and relative to the context in which the representation exists. Everything that exists exists in, and is represented relative to some context. By contrast, indirect representations are context free. In an indirect representation, the representation does not vary as a function of the context the object is contained in or used in. For example, the representation of the letter ‘e’ does not vary as a function of the word it is contained in or as a function of the sentences it is contained in. The same word can represent different meanings in different contexts, but it is spelled the same way in every context. In an indirect representation, the meaning is not encoded as part of the representation so the fact that the meaning of a word may be interpreted differently in different contexts is independent of the representation of the word.

Direct representations are encapsulated because they are defined and encoded relative to, and in terms of, the context they are represented in, and because each thing represents itself. Consequently, each particular requires its own representation. The representation of each particular can only be used once in the context it is part of. There is always a one to one relationship between the representation and its only instance. By contrast, indirect representations are unencapsulated. In an indirect representation, the representation of a particular is indirect. The extension is a substitute for the thing represented by its intension, not its intension itself. In addition, in an indirect representation, the encoding of the relations that define the intension are independent of the encoding of the intension, not defined relative to, and in terms of it. Hence, in an indirect representation, the ontology does not constrain the completeness of the intensional representation, so it can be incomplete unless constrained by domain specific ontological consistency rules outside the ontology itself. In an indirect representation there can be a one to many relationship between its representation and its instances. Many instances of the same thing can occur in many different contexts. In a direct representation the extension is an identity for its intension.

Direct representation can seem very odd because it is counter to the way we normally represent information. We must use information to communicate. The meaning of information is always defined and understood relative to an observer. A book does not understand the words it contains. The letters and words in a book have no meaning, in and of themselves. The meaning is only in the mind of the book’s reader. Each observer interprets and understands the meaning of information relative to their individual state of knowledge when they read the book. The point is, the meaning of ALL information is inherently relative to an observer. Yet we know logically that the representation of existence cannot depend on ANY observer. Therefore, the representation of existence must not be based on information. If the representation of existence is not based on the representation of information, but we only use information to communicate and reason about existence, we constrain our ability to reason to that which can be represented and communicated using information. If existence itself is not represented using information, then how can we hope to fully understand it? If we base our understanding solely on information, our understanding will always be constrained by the limits of the representation of the information, logic and mathematics we use to reason and communicate it.

To understand existence fully, we must create and use a system of representation that has the same, or fewer limits and constraints than the representation of existence itself. We must create a logic and mathematics based on the first person direct representation of existence instead of the third person indirect representation of information. After we do this, things that are extremely complex and difficult to understand, represent and compute using information will be simple and optimally efficient. Once we use the proper representation, we will be able to represent how anything relates to anything else in any combination of any number of dimensions in any context and perform logical and mathematical operations irrespective of the dimensionality of the representation. Put in less abstract terms, we will be able to perform arithmetic directly on systems of any combination of dimensions. We will be able to represent everything in its most efficient number of dimensions and perform all calculations the same way regardless of the dimensionality of the representation or the complexity of the computation. For a less abstract example, imagine being able to directly add vectors of any combination of different dimensions together, or imagine being able to directly take the dot product of vectors of any combination of different dimensions. Imagine a single operation that is the universal of computation in the same sense that an entity is the universal of information. In terms of the representation of thought, imagine being able to calculate directly in terms of concepts and abstractions at the speed current computers calculate in bits. That is the magnitude and import of what I am talking about here. The potential gains in human understanding through application of this knowledge boggle the mind.

Our ability to understand the representation and operation of existence is not as hopeless as it may seem from the discussion above. We are all born with an internal knowledge representation that can transcend and surpass the constraints and limitations of logic and information. The human brain’s knowledge representation is actually less constrained, and more capable than the representation of existence. It is not our innate ability to reason that is fundamentally limited. It is our inability to fully communicate the results of our reasoning via the translation to and from information that limits our understanding.

We think directly at the level of concepts and abstractions. We just cannot communicate and transfer that knowledge directly to others. Instead, we have to convert it to information first. It is not that we cannot represent anything we want to with information. Subject to language limitations, we can. It is just impractically complex, lengthy, and time consuming to do so.

Information does not encode the meaning of knowledge. It can only encode information about knowledge. Normally, pragmatic time, space, and complexity constraints only permit us to encode a minuscule fraction of a small portion of selected aspects of our knowledge for communication. Even then, the information transmitted is subject to misinterpretation and may be misunderstood by its recipients if their preexisting knowledge of the topic of communication is not sufficiently similar to that of the sender. Even worse, if we only think in terms of symbolic information, we hobble our intellect. We limit our thinking to that which can be represented by information, and we slow our thought by making things combinatorially more complex than they really are if we allowed ourselves to think and reason directly in our brain’s direct internal representation.

Have you ever wondered why you can think much faster than you can reason using symbolic logic, or perform mathematical calculations? Have you ever wondered why a picture is worth a thousand words? Have you ever wondered why we can grasp complex relationships almost instantly with the help of a good illustration or recognize an image in a picture almost immediately, yet trying to understand the same content if it is described in words or mathematical equations is slow and error prone if it can be expressed in words or equations at all? The same is true of listening to music, tasting a good wine, or smelling a flower. Representing these things using symbolic information is complex, slow, and often difficult or impossible. Sometimes we can teach ourselves specialized languages or specialized notations and train ourselves to do it but it really slows things down.

What if there was a way to think about, analyze and understand highly abstract concepts as easily as you can understand a picture? By understanding and internalizing the representation of thought, and the representation of existence, you will be able to do so. It will not happen overnight. At first, the change will be very slow, almost imperceptible. It will not seem like anything is different. Then you will catch yourself understanding how you thought about something right after you thought about it. Gradually, you will notice an increased ability to understand abstract topics like mathematics and quantum physics. The rate at which you can understand abstract topics will continue to accelerate. Learning the representation of existence and the representation of thought is the gift that keeps on giving. The only thing that will slow you down is the necessity to convert your understanding into words to communicate it and teach it to others. Alas, that cannot be avoided. Communication is only possible using the representation of information.

A block diagram that shows how things are represented using direct representation is shown below. In this diagram Thing1 is related to Thing2 by relation R1. It is also related to Thing3 and Thing4 by relation R2. The intension of the representation of each thing is composed of the representations of the things that compose it. Therefore the representation is fully encapsulated. All items are represented "by value", and each item is a singleton. In other words, each item has a unique identity and there is only one instance of each item. The consistency and completeness relations are part of the ontology of direct representation. No intelligent observer is required to define them, and no extra representation is required to represent them. In the representation of existence, representation = existence. We can use a variant of a direct representation in a computer to represent things directly, as long as we maintain a one-to-one relationship between each thing in existence and its representation in the computer. In other words, a direct representation is characterized by a one-to-one relation between the existence of each thing and its representation.

Indirect Representation

Indirect Representation

Indirect representations are surrogates for something else. Indirect representations are “third-person” representations. They represent things from the third person “outside-in” perspective of an external observer. The referent of an indirect representation is whatever the representation represents. Indirect representations take many forms. Among the most developed are first order predicate logic and mathematics. Other types of indirect representations include computer programs, and various types of knowledge representations. A knowledge representation (KR) is most fundamentally a surrogate, a substitute for the thing itself, used to enable an entity to determine consequences by thinking rather than acting, i.e., by reasoning about the world rather than taking action in it [Davis et al, 1993]. As far as we know, no other species creates indirect representations at the high level of abstraction of homo sapiens. As far as we know, no other species creates persistent indirect representations, i.e., writing, or some equivalent. However, other species do communicate and the act of communication entails the use of indirect representations, so other species are capable of creating transient indirect representations. Human beings often encode indirect representations symbolically as information. However, indirect representation does not have to be symbolic. Indirect representations can be encoded in a wide variety of forms for communication.


A block diagram that shows how indirect representation represents things is shown below. This diagram represents the same relationships that were shown in the diagram of direct representation. Thing1 is related to Thing2 by relation R1. Thing1 is also related to Thing3 and Thing4 by relation R2. However, in indirect representation, all the representation is indirect. Thing1, Thing2, Thing3, Thing4, R1 and R2 are all represented indirectly. The actual representation of the thing that Thing1 refers to is located outside the representation of Thing1 itself. Thing1 and its representation are two different things. The same is true of every other thing and every relation that is represented. The advantage of this type of representation is that the definitions of things only have to be stored once, and then multiple instances of those things can refer to the same definition. This saves storage. However, the disadvantage is the representation is unencapsulated, and extra representation needs to be added if we want to make sure the representation is complete and consistent. In order to define the consistency and completeness conditions, an external intelligence must know what is to be represented and it must decide how to represent it. Contrast this with direct representation. For a direct representation, no intelligent observer is required to define consistency or completeness constraints nor do the constraints need to be represented explicitly. Instead they are represented implicitly by the ontology of the representation.

Universal Representation

Universal Representation

It turns out there is also a third fundamental class of representations that has heretofore been overlooked. I call this a universal representation. Universal representations combine direct and indirect representation. They can represent themselves and everything else directly and indirectly. Universal representation is based on the representation of abstraction. Anything can be represented by an abstraction indirectly, including the representation of other abstractions, but abstractions themselves are represented and processed directly. This provides complete and consistent direct representation of everything indirectly while avoiding the inconsistency and incompleteness of pure indirect representations and avoiding the limited representational power of direct representations.

The key idea 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 the ontology of abstraction. We use a direct representation of an instance of the ontology of abstraction to represent each thing we want to represent indirectly. That allows us to represent anything directly as an abstraction, yet the abstraction represents things indirectly. This provides a direct representation of indirect representation. It directly represents everything indirectly. With indirect representations like logic, set theory and mathematics, we 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 try 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. If a representation only has to represent one thing, the representation is simple enough that we can make it complete and consistent. 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 one thing, and then use that one simulated direct thing 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.

The representation of thought is a universal representation. While we have not defined the terms and laid the prerequisite groundwork needed to understand the representation of thought, a brief introduction is possible.

In a universal representation, the intension and extension are direct, but the direct use of the extension within the representation of the intension is indirect. Consequently the intension defines meaning in context from the direct first person perspective, but the extension can be used in multiple contexts and be included as part of the representation of multiple intensions. In this case, the representation of the intension and extension are represented directly by value, but the use of the extension is represented indirectly by reference.

A neuron is a universal representation because its dendritic trees directly represent the intension of a concept and the intension of the set of abstractions that constitute the intensional representation of the concept. Detection of the direct satisfaction of the intensional conditions via the process of dendritic integration triggers the firing of the neuron’s axon which signals the existence of the concept extension and the existence of the particular abstraction extension it represents in all intensional contexts it participates in the definition of. All neural processing is direct, but the representation is simultaneously direct and indirect. The direct processing and direct representation entail self-awareness and first-person understanding of the meaning of concepts and abstractions from the first person direct perspective in context, while the indirect representation of the use of the extension entails third-person reasoning about external things at multiple levels of abstraction in terms of how those things relate to other things in context. If you do not understand this right now, do not worry. We will cover all of this later in much more detail after we present the prerequisite definitions and concepts required to do so.

The diagram below shows how the same system that was represented for direct and indirect representation is represented using a universal representation. From this illustration, you can immediately see how the ontology of a universal representation resembles neural topology. In fact, there is a one-to-one mapping between the ontology of abstraction and the spatial topology of neurons. In the figure below, the solid lines with arrows leading away from each box represent the extension of an abstraction (and the concept it partially represents). They also represent a neurons axon. The dotted lines with arrows pointing toward each box represent the intension of an abstraction (and the intension of all the abstractions that represent the concept represented by the box). They also represent a neurons dendritic trees. Just like the direct and indirect representations, this diagram represents Thing1 as being related to Thing2 by relation R1, and Thing1 being related to Thing3 and Thing4 by Relation R2. When R1's axon fires, it causes synapse R1 to fire at time t. Then Thing2's axon fires at such a time that synapse T2 fires just as the electrotonic potential from synapse R1 reaches the dendritic location of T2. The electrotonic potentials then superimpose and sum and their sum flows down the dendritic tree to the axon hillock of Thing1 and arrives at some time ta. Meanwhile R2's axon fires which causes synapse R2 to fire and send electrotonic potential down the dendritic tree towards Thing1. Synapse R2 fires at just the right time so that its electrotonic potential will reach the axon hillock at time ta. Sometime after synapse R2 fires, Thing3 and Thing4 fire their axons at such a time that the electrotonic potential from their synapses also reaches Thing1s axon hillock at time ta and superimposes with the other electrotonic potentials. Because the electrotonic potentials from synapses R2, T3, and T4 are all spatiotemporally correlated, they superimpose at the position they intersect their common dendritic path and the integrated sum of the superimposed electrotonic potentials travel down to the Thing1s axon hillock together. Of course if the relative firing times are not correct, then Thing1 doesn't reach its activation threshold and doesn't fire its axon. If the relative synaptic firing times are correct, then the electrotonic potential does reach Thing1's activation threshold and it fires its axon. The firing of thing1's axon means the intensional conditions that define how Thing1 is related to Thing2 by R1, and how Thing3 and Thing4 are related to Thing1 by R2 were met. Therefore the firing of Thing1's axon represents the existence of the abstraction that represents Thing1. It signals the satisfaction of Thing1's intensional conditions and thus represents the relation between the intensional meaning of Thing1's definition and its existence. The firing of the axon doesn't encode any information directly. However, when fired, it signals the occurence of an instance of the abstraction represented by Thing1. Therefore it can transmit an arbitrary amount of meaning without the need to encode any information. That means the size of the representation is constant, irrespective of the complexity of the set of abstract relations that it represents. The size of the representation is independent of the complexity of whatever it represents in univeral representations. In addition, this means there is no neural code. There are too many other representational and computational advantages to cover here and we haven't discussed the prerequisites needed to understand them. I will cover them when I discuss the neural knowledge representation in more detail in a future blog.

We have now introduced the direct indirect axis of representation. We learned that there are three fundamental types of representation along the direct indirect axis; direct representations, universal representations, and indirect representations. The representation of existence is a direct representation. The representation of thought is a universal representation, giving thought the power to represent things directly and indirectly simultaneously. The representation of information is an indirect representation. The representation of information provides the basis for natural language, communication, symbolic logic, computer programs, and mathematics among other things.

While these descriptions do not cover the direct - indirect axis of representation in exhaustive detail, they do provide enough of an introduction to make it worthwhile to move on to describe the second principal axis of representation – the intensional – extensional axis.


The Intensional - Extensional Axis of Representation

The Intensional – Extensional Axis Of Representation

The second principal axis of representation is the intensional – extensional axis. All representations have intensional and extensional aspects. The intension of a representation can be stated equivalently as that set of conditions which must be satisfied by any object, within the given universe of discourse, for the representation to exemplify the object. If an object satisfies the intensional conditions, then it is an exemplar of the object.

In indirect representations, the intension of a concept represents the syntactic definition of that which it represents. In indirect representations, the epistemic meaning of a concept can be inferred or interpreted from the representation of the concept’s intension by an intelligent observer, but not by the object itself. For example, a dictionary definition cannot read itself. It does not understand the meaning of the words it contains. A computer does not, and cannot, understand the meaning of its data. Indirect intensions represented using information contain syntax but not semantics. Consequently, the representation of indirect intensions using information is incomplete.

In direct representations, Representation = Existence. Direct representations represent things in terms of how they relate to the things that compose them and in terms of how they relate to those things that affect them in their external environment. The intension is directly composed of the extensions of the set of objects and object relationships that the object being represented is composed of and related to. The existence of the set of objects and object relationships that represent the objects that compose the object being represented and the existence of the relationships between its component objects and those objects it is related to in its external environment directly represent the existence of the object being represented. In a direct representation, the extension IS the representation and existence of the intension. Symbolically we have: Existence(Extension) = Existence(Intension). This is equivalent to Existence = Representation. In a direct representation, the representation of an objects’ intension encapsulates the representation and thus the existence of the extensions of the objects and object relationships that the object being represented is composed of and related to. (Intensional relations exist between the objects that compose the object being represented and between the object being represented and those it is related to in its external environment). Direct representations represent existence as an nth order relational hierarchy of composition. Objects can be composed of objects that are composed of objects that are composed of objects, to any required degree. Any object can be composed of zero or more objects, each of which may be composed of zero or more objects. The same is true of relations. There can be first order relations between objects: O R O, second order relations between relations: O R(R) O, third order relations: O R(R(R)) O, etc to any required degree. At the most fundamental level of the representation of existence, both the objects and the relations are composed of the same thing and represented the same way, by differences (i.e., asymmetries) in incomplete, inconsistent nonexistence. We’ll have a lot more to say about this when we cover the detailed representation of existence later.

The extension enumerates all members of the set that satisfy the relational conditions specified by the intension.

In a direct representation, the extension represents, and is, the only instance of the object represented by the intension. The existence of the intension IS the existence of the extension. In a direct representation, the cardinality of the extensional set is always 1. The extension represents the existence of the intension. In a direct representation, the extension can be represented by one bit of information, irrespective of the complexity or level of composition of the intension. However, the bit has no meaning in and of itself. Its existence simply signals the existence of the extension, and thus the satisfaction of the intensional conditions represented by the extension. If the intensional conditions are not met, the extension does not exist. This explains why fundamental particles cannot be split and why energy is quantized. It is impossible to divide the fundamental quanta of existence. All quanta exist completely, or not at all. Their representation is always complete and consistent. It is impossible for a quanta to have a partial state of existence. It cannot be part in the universe and part outside it. Creation and destruction of quanta are instantaneous, indivisible events. If the representation of a quanta is made inconsistent or incomplete, it ceases to exist. For this reason, the representation of existence is complete and consistent. It is impossible for it to become incomplete or inconsistent, so there is no need to represent or enforce ontological consistency.

In an indirect representation, the extension enumerates all members of the set that satisfy the relational conditions specified in the intension and that represent exemplars or instances of the object or concept being represented. The members of the extension represent the instances of that which is represented by satisfaction of the intensional conditions, they are not the instances themselves.

A universal representation combines intensional and extensional representation. In a universal representation, the intension is direct, but its extension is indirect. Consequently the intension defines meaning in context from the direct first person perspective. The intension of a direct universal representation defines and understands its own meaning from the first person perspective in context. The indirect extension can be used in multiple contexts and be included as part of the representation of multiple intensions. In this case, the representation of the intension is represented directly by value, but the extension is represented indirectly by reference. This will be covered in much more detail when we derive the representation of thought.

Table 1 provides an overview of the three classes of representation, their relationships, and some examples of things represented by each class.

The representations in column 1 are direct. Direct representations represent the physical existence of everything in the universe. We will refer to representation at the level of physical existence as direct representation, level 1 representation, type 1 representation or existential representation.

The representations in column 2 are universal; thus, they are both direct and indirect. Universal representations are used to represent thought. In the rest of this book, universal representations will be called type 2 representations. We will also refer to representation at the level of thought as level 2 representation.

The representations in column 3 are indirect. Indirect representation is the type of representation used to represent information. Most human generated representations are of this type. This type of representation is required for communication. It is used for logic, mathematics, writing, speaking, drawing, and computation. Indirect, or information based representations will be called type 3 representations. Representation at the level of information will also be called level 3 representation.




1-Direct


(Existence)



2 –Universal


(Thought)



3 –Indirect


(Information)



A -Intensional



A1
:Direct*


Intensional


Representation
of:


Conservation
of nonexistenceà


Symmetry,
Conservation of Energy, Quantum Mechanics, General Relativity, Physics



A2:Universal*


Intensional


Representation
of:


Meaning
(Direct),


Neurons’
Dendritic Trees, PostSynaptic Terminals



A3:Indirect*


Intensional


Representation
of:


Logical
Predicates,


Mathematical
Axioms, Laws of Physics



B- Universal



B1:Direct*


Universal


Representation
of:


Evolution of the universe.
Existence=


Representation



B2:Universal*


Universal


Representation
of: Thought: Neurons, Abstractions, Concepts, Qualia, Consciousness,


Cogito
Ergo Sum



B3:Indirect*


Universal


Representation
of:


First
Order Predicate Logic, Mathematics, Knowledge Representations



C -Extensional



C1:Direct*


Extensional


Representation
of:


Existence
of the universe



C2:Universal*


Extensional


Representation
of:


Existence
(Indirect),


Neurons’
Axonal Tree, Presynaptic Terminals



C3:Indirect*


Extensional


Representation
of:


Mathematical
Sets




Table 1: Types of Representation

Intensional Representation

Intensional Representation

The intension of a representation often takes the form of a definition. For example, in the domain of mathematics, an intensional definition is a function. For example, we can define a successor function S that takes an integer argument and returns the integer + 1.
int S(int i)
{
return i + 1;
}

By using this simple function, and a single input of 1, we can inductively define the infinite set of Natural numbers by repeatedly calling the successor function using the output from its previous evaluation as the input to its next evaluation. If called an infinite number of times, the successor function will function as a generator and it will generate the infinite set of Natural numbers.

int i = 1;
for (;;) // repeat forever
{
i = S(i);
}

In this example, the intensional definition would include the definition of the successor function S, the initial input value of 1, and the rule specifying that the successor function be called repeatedly using its output as its next input. The extension would be the entire infinite set of Natural numbers. We can generalize this example to allow intensional definitions to include more than one function, and/or to include generalized functions. For example, we can create intensional definitions that include operators; i.e., functions that take functions and other types of mathematical objects as arguments. For example, an operator could take tensors, or matrices, or even matrices of operators as arguments. Our operators could take any number of any type of mathematical arguments and return any number of mathematical results of any type. Integrals and derivatives are simple examples of mathematical operators.
Intensional definition also applies to rules or sets of axioms that generate all members of the set being defined. For example, an intensional definition of "square number" can be "any number that can be expressed as some integer multiplied by itself." The rule -- "take an integer and multiply it by itself" -- always generates members of the set of square numbers, no matter which integer one chooses, and for any square number, there is an integer that was multiplied by itself to get it.

Similarly, an intensional definition of a game, such as chess, would be the rules of the game; any game played by those rules must be a game of chess, and any game properly called a game of chess must have been played by those rules.

Intensional definitions can take many forms. They need not be logical or mathematical. A dictionary definition of a word is an intensional definition. A set of rules is an intensional definition. For example, an intensional definition of physics consists of the scientific laws of Physics.

If a set contains all possible instances of a logical predicate, that set represents the extension of the predicate. The predicate represents the intensional definition of the set. An intensional definition defines the necessary and sufficient conditions for belonging to the set being defined.
For example, an intensional definition of "bachelor" is "unmarried man." Being an unmarried man is an essential property of something referred to as a bachelor. It is a necessary condition: one cannot be a bachelor without being an unmarried man. It is also a sufficient condition: any unmarried man is a bachelor.