Copyright © May 2010 by Tienzen (Jeh-Tween) Gong

In the "Linguistics Manifesto," I stated that linguistics is roaming in a space which is composed of three tier hierarchy of axiomatic systems --- from the formal system, Godel system to Life system. Many readers might take for granted that the Life System connotes a system similar to the biological life system. In a sense, it is correct. However, in the "Linguistics Manifesto," the "Life System" is an extension of the formal - Godel systems which are both mathematics. Yet, there is no such system known in the current mathematics. Thus, I must provide an explanation for what the Life System is all about in terms of linguistics and mathematics.

I also stated that the necessary condition for a language is that it must encompass, at least, one formal system. Why? I have chosen not to define language but to discuss its attributes.

- It contains information.
- It processes information.
- It expresses intelligence.
- etc..

In a sense, a formal system is an attempt to organize a chaotic system with two steps.

- Finding some orders in a chaotic system.
- Finding some rules (definitions or axioms) to derive those orders.

In the past two thousand years, Aristotle formalized syllogisms and Euclid formalized geometry. Then, there is a barren period for almost 2,000 years until Giuseppe Peano formalized arithmetic in the late 19th century. By the earlier 20th century, the "formal system" was well-established by the work of Russell and Whitehead, the "Principia Mathematica." Then, many paradoxes popped out from those "formal systems," such as, the Russell's paradox, the Grelling's paradox, etc.. In general, those paradoxes arise from the "self-referential loop" of the system. Thus, Russell and Whitehead invented the "theory of Types" to expel all self-referential loops. Nonetheless, the development of mathematics gave two verdicts on formal systems.

- The life of formal system is guaranteed -- for any chaotic system, it can always be formalized or partial formalized. The following theorems provide the support on this.
- Ramsey's large number theorem -- system B is chaotic with a "large number" of members, then, some orders can be found in B.
- Shadow theorem of Fractal -- every chaotic system is a shadow of an ordered system.
- Two code theorem -- a computable system can always be represented with two codes, such as (0, 1), (yin, yang), etc..
- etc..

- The power (knowledge) of formal system is limited -- For a powerful enough (or complex enough) chaotic system B, no formal system can encompass it. There is always some information (knowledge) of B is unknowable by its corresponding formal system. The following theorems provide the support on this.
- Church's (Alonzo Church) undecidability theorem of formal system.
- Tarski's (Alfred Tarski) indefinability theorem.
- Two incompleteness theorems of Godel.
- etc..

All these limitation theorems state a fact that every formal system is a shadow of a chaotic system which is larger than that formal system. Nonetheless, that chaotic system can be almost wholly represented by a progressively enlarging formal systems.

Thus, if a given nature language is a chaotic system, then it can be almost wholly represented by a formal system schema. And, for constructing a comprehensive linguistics theory, it will be a good starting point to revisit the theory of formal system first.

A formal system is a system which is described with a set of inference rules from a set of axioms which are the rules for determining the membership for the system. In fact, there are two types of axioms, the universal axioms and the membership axioms. A universal axiom states a universal truth. A membership axiom has no true-false value but defines the membership of a domain (or a club). Nonetheless, both types of axioms cannot be derived by principles of deduction nor by mathematical proofs. They are starting points for constructing a formal system. Every formal system consists of the following parts.

- Some members (in finite number or in infinity) -- they can be called as "symbols."
- Some undefined terms.
- Some definitions (including operations, function, etc.).
- Some axioms (including inference rules, derivation procedure, etc.)

From the above, something can be produced.

- String or sentence -- the composite of symbols via some operations (or functions).
- Theorem -- a sentence which is derived from definitions and/or axioms.

Now, some issues arise.

- Issue -- Is a given sentence a theorem?

Answer: We can construct a "decision procedure" to find it out.- Issue -- Does there always have a decision procedure for any given sentence?

This is called a halting problem.

Answer: If a procedure halts, a decision can be known. If it does not halt, there is no decision procedure.

- Issue -- Can one theorem contradict another theorem in a given formal system?

Answer: This is the issue of "consistency."

If one contradiction arises, the formal system is "internally" inconsistent.

- Issue -- Is a theorem of a system always true (in the sense of the pre-formal system period)?

Answer: In the pre-formal system period, a sentence is true if and only if it states a fact about the real world (the reality). That is, a theorem of a formal system needs not to be true. If all theorems of a formal system are true, then it (the formal system) has the "external" consistency.

- Issue -- A sentence B of a formal system is true. Must B be a theorem?

Answer: If B is not a theorem, then that formal system is incomplete.

Now, we have discussed some very important concepts.

- decision procedure
- halting problem
- consistency
- truth
- completeness

There are only two types of formal system.

- Type 1 -- it is consistent and complete. That is, no contradiction among theorems, and all truths are theorems.
- Type 2 -- it can never be consistent and complete at the same time. If it is consistent, then it must be incomplete.

The type 1 system is very weak without much complexity. On the contrary, the type 2 system can grow to an infinite complexity. Thus, I am not interested in the type 1 system. The type 2 systems are described by the Godel's incompleteness theorems (the first and the second). As the Godel's theorems are well-established and well-known, I will not repeat them here. Yet, I will use those theorems to develop a new system, the Life System.

I will begin with --

- an arbitrary system F which is a type 2 formal system. And, the governing rules for F are:
- the "principle of noncontradiction"
- the "complementary principle"

- as a type 2 system, F is incomplete. Thus, there is a sentence C of F which is undecidable in F. That is, C or -C (negative C) can both be true in F.
- in order to eradicate that incompleteness, I simply add "both C and -C" (not just C or -C) as two new axioms of F. With these new axioms, F becomes F(1) = {F, C, -C}.
- Yet, the Godel's theorems guarantee that F(1) is still incomplete. There is a sentence C1 ...
- Not willing to be defeated, I repeat the step 3 over and over. Soon, I have constructed the system F(m).

By now, the F(m) is fundamentally different from F. When m is large, the most of axioms of F(m) are contradictory statements. The F(m) is dramatically different from any known mathematic system. I call it a Life System. The Life System is constructed with a genuine formal system as the seed, and it grows with the Godel process. The figure A depicts a Life System.

What is the governing principle for a Life System? Obviously, the contradiction is no longer prohibited but is a norm for the system. As the founder of the modern "formal system theory," Alfred North Whitehead recognized that the contradiction is the norm for many biologic systems. That is, the opposites (the contradictions) are the roots of each other. Yet, he was unable to incorporate it into his formal system theory. Nonetheless, it became a part of his philosophy, the mutual immanence. Yet, Whitehead did not see that mutual immanence coexists simultaneously. He wrote, "Any set of actual occasions are united by the mutual immanence of occasions, each in the other. To the extent that they are united they mutually constrain each other. Evidently this mutual immanence and constraint of a pair of occasions is not in general a symmetric relation. For, apart from contemporaries, one occasion will be in the future of the other. Thus the earlier will be immanent in the later according to the mode of efficient causality, and the later in the earlier according to the mode of anticipation, ..."

On the contrary, that the opposites are parts of each other is an innate nature of the Life System. They coexist simultaneously at all time. In fact, they are permanently confined in each other. That is, they can never be pulled apart. The figure B below depicts the concept of mutual immanence. Many of those bubbles (except a few) can be viewed as both a bubble and a counter-sink. In fact, no decision procedure of any kind can distinguish it as one or the other definitely. Thus, the governing rules for the Life System are:

- Mutual Immanence Principle.
- Permanent confinement.

Up to now, the formal system theory was discussed purely as mathematics. Its sentences are forms without any context. Yet, even without context, they do carry some meanings, such as, theoremhood, truth, consistency or as a sentence (the composite of symbols and operations). And, these are their intrinsic meanings, the innate meanings. In fact, all meanings of a sentence of a formal system must be springing out from its innate meanings via some manifestation pathways.

- via interpretation --

For the form a + b = c,

if 1 + 9 = 10, it represents a decimal system.

if 1 + 1 = 10, it represents a binary system, etc..

- via assignment --

The composite of four symbols "book" is artificially assigned to represent "a bound printed pages."

The combination of some symbols and operations (I read book) is assigned via some assignments and some assigned rules to represent "going over the printed pages while trying to understand some messages on those pages."

Obviously, the interpretations and the assignments are done in many levels. The meaning of one level can only be "read" in a different level.

- First order manifestation --- Of course, those interpretations and assignments cannot arise without the innate meanings. They are the first order manifestation (particle).

- Second order manifestation --- Then, these first order manifestations can be interpreted and assigned the second time. Linguistically, the first order manifestation is normally done by an author. The second order is done by a reader.

While the first order manifestation has very little leeway away from the innate meaning, there can be a large canyon between the first and the second manifestations. The second manifestation can be very much context laden. That is, the meaning of the first order particle can be changed by the "interaction" between it and the surrounding particles.

- The higher order manifestation ---Then, there are higher (third and up) order manifestations. At this point, the innate meaning, the first order manifestation, the second order manifestation, the context and the whatnot are all entangled. Very often, the original meaning (the innate and/or the first order) is buried at this level.

Linguistics is about information, information processing, meaning manifestation, etc.. Thus, the "life process" can be a good example of it.

All life processes are about two objectives.

- Reproduction (replication)
- Maintaining the life system (metabolism)

- an preexisting state:
- There is DNA (DeoxyriboNucleic Acid). It is the information warehouse for a life.
- It can be read as a sequence of amino acids
- It always comes in two identical (in mirror images) pairs.

- There is cytoplasm which is the life soup.

- There is DNA (DeoxyriboNucleic Acid). It is the information warehouse for a life.
- The replication and the metabolism processes.
- Step one -- getting the blueprints with three DNA-enzymes.
- DNA-endonuclease --- unzipping the DNA
- DNA- polymerase --- copy the information and move away
- DNA-ligase --- restore the unzipped DNA

The requirement for this step: preexisting- DNA
- three DNA enzymes

- step two -- producing an engineer to read the blueprints
- to produce ribosome --- with rRNA (ribosomal RNA) + some proteins

- rRNA
- cytoplasm
- some proteins

- step three -- building a prototype
- to produce amino acids --- with Ribosomes (as interpreter) reading and translating mRNA (the blueprints)

- step four -- building a factory
- to produce proteins, including all enzymes --- with Amino acids + tRNA (transfer RNA)

- tRNA

- step five -- large production
- With enzymes, all amino acids which are parts for the replication of DNA can be produced in a large quantity. The parts for all proteins for metabolism can also be produced in a large quantity.

Result --- the tasks of replication of DNA and of metabolism can be carried out. - step six -- producing a production manager
- Some enzymes act as inhibitor or repressor to control and to fine tune the large production.

- step seven -- producing a body
- via morphogenesis

- Step one -- getting the blueprints with three DNA-enzymes.

- has information
- processes information
- reaching the information (unzipping DNA)
- copying the information
- reading and interpreting the information
- reproducing the information in large quantity
- etc.

- has meaning manifestations
- first order manifestation
- producing a messenger
- producing an engineer
- producing a manager

- second order manifestation
- replicating DNA
- maintaining metabolism

- higher order manifestation
- morphogenesis -- from genotype to phenotype, producing the organism, the body.

- first order manifestation

The above example is, in fact, the simplest linguistic system. That is, it must contain all the essentials for any linguistic system. There are some "necessary" features in this system.

- DNA exists in pairs (mirror images of each other)
- the same information is processed in different ways (in tiers)
- the entire process is a repeated self-reflection --- The mRNA copies a gene or genes of DNA. A gene is the portion of the DNA strand which codes for a single enzyme. The above bio-process is to reproduce the gene via producing ribosome, enzymes, amino acids which are the different expressions of the same information, the gene. The entire process is simply a repeated self-reflection; the same information is processed over and over in different ways and in different tiers.
- every process is a part of self-referential loop (the prerequisite of an earlier process is that the later process was done already)
- it is a well-defined Godel system

What do these features mean? Are they unique to the biological language? Or, are they universal? As it is a well-defined Godel system, we should revisit the formal system theory again.

For Russell and Whitehead, the self-reference problem of the formal system was viewed as a demon which must be eradicated. After the discovery of the Godel theorems, this self-referential demon was recognized as an essential part of any formal system in mathematics. In fact, this bio-system above is all about self-reference in one form or the other, the self-replication, the self-reflection, entangled self-referential loops, etc.. Thus, we must answer the following questions.

- Question one -- Why is DNA in pairs? For a reason of redundancy? Or, it is essential and universal!
- For every type 2 formal system F, it has three expressions.
- The nutshell (the four part) expression, F.
- The Godel expression -- with theorems and incompleteness. The incompleteness can be viewed as an umbilical cord (um), and we make this um as a new axiom to express F more fully. Thus,

G1 = F + um(1)

Gm = F + um(1) + ... + um(m) = G(m-1) + um(m)

Gm is consistent and ordered. - The chaotic expression G(T) -- T means total.

G(T) = Gm + ... + ... to infinity. - high level programming language with machine language
- word tokens with their pronunciations.
- etc.

As a universe of itself, these three expressions of system F (F, Gm, G(T)) are identical. They are self-reflections of one another. In fact, they are permanently confined with one another. This permanent confinement is expressed in figure C.

- Question two -- Is the self-referential loop unique to the bio-system? Or, it is essential and universal!
- The vocabulary of all languages are recursively defined. The figure D depicts a self-referential loop. The right hand is drawn by the left while the left is drawn by the right.

- Question three -- Must the same information be processed in many different tiers? Is this just a unique feature for the bio-system? Or, it is essential and universal!
- The symbol "b" is processed in a string "book" and again processed in a sentence "I read book." The tier structure is essential and universal to all formal systems, including all linguistic systems. The figure E depicts a tier structure.

That is, all (each and everyone) features of linguistics are features of FGL. Then, what are the FGL features? They must be discussed in different levels.

- The structure and the governing rules
- Type 2 formal system, F -- consistency (principle of noncontradiction and complementary principle) and incompleteness.
- Godel system, G -- self-referential loops (constructed with a type 2 formal system plus its self-referential sentence as a new axiom)

G(1) = F + um(1), um(1) is F's self-referential sentence

G(m) = F + um(1) + ... + um(m)

G(T) = G(m) + um(m+1) + ... + um(n) + ... to ad infinitum, T means total.

G(T) cannot be reached with this self-referential process. - Life System, L -- mutual immanence and permanent confinement (constructed with the union of G(T) and -G(T) ).

- The features and the properties
- F is computable and meaningful -- "every" F encompasses "all" recursive functions which are computable. This feature guarantees that a mapping of F to the real world does exist. Thus, F can be interpreted and be meaningful.

- There are a few very important features for G (the Godel systems)
- G(m) is unconscious of itself -- the self-referential sentence of G(m) is "always" outside of G(m). While G(m) is conscious of G(m-1), it is unconscious of itself. This fact is vividly depicted in the figure C.

- The white crab (G, the foreground) cannot encompass the black crab (the background, the self-referential space).
- The white crab cannot be separated from the black crab. This is a permanent confinement and the first order of mutual immanence.

- G(T) is unreachable in G itself -- again, this fact is vividly depicted in the figure C. The edges of figure C is chaotic, and they cannot be eradicated by expanding the whole sheet.

Although this chaoticness cannot be removed in G, it can be encompassed with a very special process, the "renormalization" process. In physics, when a particle is not seen, it is not a particle. In a seeing process, a particle must interact with the seeing agent. This "interaction" will make a particle "visible." And, this interaction is called the renormalization for the particle. On the same token, G(T) becomes visible only after it interacts with a particle not of itself.

That is, after it is renormalized, G will become a "self" which is conscious of itself. Yet, G(T) cannot be renormalized by itself.

- G(m) is unconscious of itself -- the self-referential sentence of G(m) is "always" outside of G(m). While G(m) is conscious of G(m-1), it is unconscious of itself. This fact is vividly depicted in the figure C.
- L (the Life System) is conscious and intelligent [see thesis(III) below] -- in L, G(T) and -G(T) are renormalizing each other. They are permanently confined and mutually immanent between each other. At this point, both G(T) and -G(T) are conscious of themselves, so is the L(T) = {G(T), -G(T)}. See "Linguistics Space (II)-- the Intelligence."

- F is computable and meaningful -- "every" F encompasses "all" recursive functions which are computable. This feature guarantees that a mapping of F to the real world does exist. Thus, F can be interpreted and be meaningful.
- Bottom out and top out
- Every FGL system has a bottom. In fact, the F is the bottom of FGL.
- Every FGL system has a top. In fact, the L(T) is the top of FGL. Yet, it is interesting to know that how to calculate the top.

G(T) = G(m) + um(m+1) + ... + um(n) + ... to ad infinitum

= G(n) + um(*), [um(*) = um(n+1) + ... to ad infinitum]

Now, what we are interested in is about the complexity of G(n), the C(G(n)).

At one n, when C(G(T)) = C(G(n)), then n is G's top out number.

Is such a "n" always existing for all G's? The answer is Yes. This top-out process is called renormalization (see "Linguistics Space (III) -- the New Mathematics" at http://www.prebabel.info/newmath.htm). Often, a top-out can become a bottom of a higher system, and this recursion goes ad infinitum. But, this recursion will also be topped out finally (FTO). For religious people, FTO can be called with an over used term "God". In science, this FTO can be called TOE (Theory of Everything). Nonetheless, we do know a few real examples to get a sense of this topping-out process.- For a language, n could be equal to 6.
- word roots
- words
- word phrases
- sentences
- essays
- language (top-out)

- For a higher level system
- language
- books
- ...
- cultures
- humanity (top-out)

- For the bio-system above, the n = 7.
- Higher level bio-system
- single cell
- ...
- ...
- humans
- ecosystem (top-out)

- For a language, n could be equal to 6.

- Every chaotic system can always be formalized.
- Not so fast! Every powerful formal system (Godel system) has a chaotic boundary which cannot be eradicated by "itself".
- Don't worry! Every chaotic boundary can always be renormalized.

These three points can be vividly described with one good example -- the interaction between a bacterium and an invading virus.

- For a bacterium G(bact, T), it is consistent. That is, all sentences (amino acids, proteins, etc.) in G(bact, T) are theorems of it. This is point 1.
- Thus, the DNA of a virus should not be a theorem of G(bact, T). Yet, it can still be a truth in G(bact. T). So, the non-theorem can write its own theorems (amino acids, etc.) in G(bact. T). That is, G(bact, T) is incomplete, the point 2.
- Both bacterium and virus are different from the bio-system above in one sense. While the above-stated system is written on paper, both the bacterium and the virus have interacted with the environment, and they can be distinguished from the environment and between each other. These interactions are the renormalization processes, the point 3.

Now, I have described that what a Life System is. It is an extension of the formal-Godel systems but goes far beyond them. This is the same difference between a factory and its product. For example, cars are produced by a car factory. Not only the cars are different from their factory but their product can reach another level, producing the human activities, the economics, etc.. Another example is about the brain and human intelligence. The intelligence is produced by the brain but is in a higher level to the brain. With this understanding, I am introducing two theses.

That is, anything which can be handled by human intelligence can be described by linguistics. The details of this will be discussed in the paper ["Linguistics Space (II)-- the Intelligence," at http://www.prebabel.info/aintel.htm].

That is, the Godel theorems are not applicable on this L(T), Life System (T). L(T) can always be renormalized. The details of this will be discussed in the paper [Linguistics Space (III) -- the New Mathematics].

With these three new theses,

- the nature
- the human intelligence
- the linguistics

Now, a detailed framework and its applications of linguistics can be outlined.

- The framework
- Tertiary -- The tertiary feature on word form, word sound and word meaning is not an happenstance. It is the intrinsic essence of every formal system which has three expressions [the nutshell expression, the theorems (orderly expanded) expression and the chaotic expression].
- Logic -- It can describe "all logic" as it encompasses "all" formal systems.
- Hierarchical -- It has many levels of hierarchies (words, word phrases, sentence, essays, etc.) which is the result of self-referential loops, guaranteed by the Godel's theorems.
- Mutual immanence -- It encompasses all contradictions with mutual immanence, the union of G(T) and -G(T).
- Renormalization -- It encompasses all infinities and infinite chaos with renormalization (having a bottom and a top). Thus, the following is guaranteed.
- the bottom -- the PreBabel word root set. There is only one bottom.
- the top -- the universal language. There is only one top.

Between the bottom and the top, a few theorem (expanded) systems can be developed. The PreBabel (Chinese) is one actual example, which is available at http://www.chinese-word-roots.org. - the chaotic expression -- all different languages.

- The consequences
- The framework of linguistics is universal -- the "Large Complex System Principle (LCSP)" will govern "all" large complex systems. Thus, linguistics is "isomorphic to"
- human intelligence system. The four features of linguistics are four pillars of intelligence
- logic -- formal system
- hierarchical -- self-referential loops (Godel systems) and self-similarity transformation of fractal.
- mutual immanence -- encompassing all contradictions, the union of G(T) and -G(T)
- renormalization -- reigning in infinities and infinite chaos

- bio-systems
- physics
- mathematics
- political sciences (visit the page http://www.chinese-word-roots.org/cwr016.htm)
- economics (visit the page http://www.chinese-word-roots.org/econom01.htm)
- etc.

- human intelligence system. The four features of linguistics are four pillars of intelligence
- One real example of its application

Seemingly, physics is far removed from linguistics. On March 30, 2010, the Large Hadron Collider (LHC) at CERN saw its first high-energy proton collisions. Its objective is to find a Higgs boson which is the foundation for the Standard Model of elementary particle physics. The graph below is the "current" bottom for the Standard Model.

- Be a part of this bottom. Then, this additional particle will destroy this 4 x 4 matrix. It makes this simple bottom becoming more complicated. In a sense, it violates the bottoming principle of linguistics. Thus, if such a Higgs boson were discovered, it cannot form a true bottom. There must be a bottom lower than the Higgs boson.
- Be a new bottom. If Higgs boson is a single particle, then this new bottom has only "1" of something. From (4 x 4) to 1, it is seemingly a too big of a drop.

- The framework of linguistics is universal -- the "Large Complex System Principle (LCSP)" will govern "all" large complex systems. Thus, linguistics is "isomorphic to"