Wednesday, August 7, 2013

Gödel's Incompleteness Theorem is Incomplete

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

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

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

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

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

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

Objective mathematics is that complete and consistent mathematical system.

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

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

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

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

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

Thursday, August 1, 2013

Objective Physics

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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