STATE[-][MENT[-]]VERSE i.e. STATE[MENT]VERSE (Incomplete)

The physical dynamism/dynamics/mechanics of reality and the dynamics of nature itself and the laws of the dynamics independent of and anything other than as information of state, it is argued, is questionable and even has no significance and is redundant, as there exists no actual dynamics at the fundamental level, much like which is indicative in classical Zeno’s paradox, independent of state.

Putting forward the argument that the only entities are the states, since, and that, they are the only realizable entity, state(s) is presented and taken to be the fundamental entity and defined to be a set of information(s), which eventually is a piece of information itself, with information defined as a mathematical statement. Hence, asserting that any physical entity/property, including any dynamism, is nothing but an information/state or a part of that information/state. Mathematical statement here refers to any random statement. Any state is regarded as a “unique” set of, or say, strand of information(s), which, again, can be inferred as a piece of information.  

Further, it is proposed that these informations/states constitute a state (information) – space, which again is/can be represented as a piece of information. State-space, defined, is a good old quantum mechanical state-space, which is a complex Hilbert space in which the possible instantaneous states of the system may be described by unit vectors, however, with some alterations and (via which) generalizations to this Dirac formalism and to this definition of state and of state-space. It is argued that the definitions of state and state-space in Dirac formalism deduces to the, as argued, more general definitions presented here.

Using these definitions of information, state and state-space, which as argued is more general and that Dirac formalism deduces to it, a new world-view to the physical reality is presented and how various physical entity/property in this picture gets displayed.

Also, an idea/argument that classical-physical and classical-mathematical contradictions/mutual exclusiveness reduce to truth statements, i.e. all mutually exclusive entities are realized, and possibly the contradicting/mutually exclusive entities are all identical, though different, i.e. mutually exclusive though not. This follows from the adaptation of definitions of mutual exclusivity/orthogonality/contradictory, which in turn is deduced from the quantum state-space definition presented here.

INTRODUCTION

Explaining the physical world involves interpreting the “physicality“ in the definitive paradigm of the brain (at least of human brain) —mathematics, more precisely, classical mathematics/logic. The what and why queries related to the physical world involves and leads to the how questions—the mechanics and dynamics, of the physical world. Arguably, the “how-questions” apparently can be regarded as the primary, the only significant queries and queries that are equivalent to and, possibly, identical to what and why questions.

However, the how questions involving the physical dynamism and the physical dynamics itself and the laws associated with it (i.e. the dynamical models), though quite vivid and convincing, if brought into detailed consideration and examined, cannot be realized but only as the information of state. By state, quantum-state is implied and quantum-state is taken, here, as the fundamental and general definition of state. But now we may ask what information is and what is quantum-state? Consider state-space:

Conventional quantum state-space is a complex Hilbert space in which the possible instantaneous states of the system may be described by unit vectors. (Cohen-Tannoudji, 1977)

Now, the dictionary definition of information is that: “Information is the resolution of uncertainty; it is that which answers the question of “what an entity is” and thus defines both its essence and nature of its characteristics. Information relates to both data and knowledge, as data represents values attributed to parameters, and knowledge signifies understanding of a concept.” (Merriam-Webster)  

Information according to classical information theory, specifically Shannon information (i.e. information in communication theory), relates not so much to what is being communicated, as to what could be communicated. “That is, information is a measure of one’s freedom of choice when one selects a message.” In communication theory, in more definite terms, “the amount of information is defined, in the simplest cases, to be measured by the logarithm of the number of available choices.”

Consider  number of choices/possibilities, with respective probability of being chosen being . Information in this set (say)  of possibilities  is then given by:

                                                                                      (E.Shannon & Weaver, 1964)

In Shannon’s communication theory, information is about the degree of freedom that any set/system has, i.e. the information (contained) in the set/system is about how many possible states the system can (potentially) be in.

For example, consider a set of six possibilities (say) a die: , each with equal probability of 1/6. And another set of two possibilities (say) a coin or a binary bit: , each with equal probability ½. 

Now, if we take any set, say , then the only Universe/possibility is this set. Therefore, by definition, the only thing that exists is this set. Hence, the only thing that “is” is this set of six possibilities. Now, this set, is defined and is about the six possibilities: . Hence the only existence that “is” are these six possibilities, hence the probability of set  is 1, which simply means that the only entity that “is” is set .

Now, since, by definition, each possibility has equal probability and, also by definition, the probability of  is 1. So, the probability of each possibility, i.e. the share/contribution of existence (truth value T/1) and being, of each possibility must produce a total existence value/probability of being, of , equal to 1.

Hence the two definitions:

1. Only   exists, i.e. probability of  is 1. And
2. The existence of each possibility of  has equal share on existence of .

Along with the definition that states: , where  are the respective probabilities of possibilities , and definition,   , gives .

Similarly for set , .

[NOTE: It is important to note that this all is due to the definitions of set along with some other definitions which can be put together as a set of these definitions.]

Now, since set , compared to set , has more possibilities, i.e. since  can be in more possible states (by definition) it requires more information to describe/define this set. It is more difficult to guess its state.

Any particular guess made, about any set/system, the existence of any guessed state/possibility contributes less to the existence of set compared to the contribution due to existence of states that is exclusive of the guess, provided that the set has number of possibilities more than two.

For set of only two possibilities, the contribution is equal. For set of only one possibility, the only contribution to the set’s existence is due to that possibility, hence the existence of its possibility is its existence. Hence, there is no difference between the set and the set element.   

Now any set can be reduced to the set of elements/possibilities, call it guess set  with possibilities ,  for guessed state, and , for state(s) exclusive of  guessed state .  can be written as , for guessed state being true and  by , for guessed state being false. i.e. .  States  may contribute to  by any amount, i.e. these states may share any value of probability from total unit probability.

Now, since set  is a set of six possibilities, the guessed state has contribution 1/6 and the complement set of states that is exclusive of the guessed possibility has contribution 5/6 in D’s existence. Hence for ,   is such that probability of , , and probability of ,  . Hence any particular guess made, has the highest probability of being FALSE/wrong (i.e. ) of and lowest probability of being TRUE/correct (i.e. ), where  is a state that the set (here set ) actually assumes.

Hence, for set , more pieces of information is required to completely define/describe it, compared to set . Hence, more pieces of information is required to be absolutely confident about, i.e. to absolutely specify, set/set’s actual state of being , as the possibilities/states/elements of the set increases. The more the possible states, the more is the information-pieces required to obtain information about set’s actual state of being. It is required, with the increase in the possibilities of a set, to know more about the set’s current status to know set’s realized state.

For example: In case of set , i.e. say on rolling the die, suppose we ask a question, “is it in state 1?” and the answer is “NO”. Now we know/have gained information that set  hasn’t assumed state “1”. And we also have an information, by definition of , that “set  must then be in one of the states other than state “1”.”. This information is due to, or can be considered as, two informations, one that states, “set  can only assume one state when it assumes/chooses a state.” And the other that states, “ set  has possibilities {1,2,3,4,5,6}.”, which both is present/exists as information that is set ’s definition.

So, we have two pieces of information: “NO-1” and “definition”. However, we often take information-“definition” for granted, hence we have one piece of information: “NO-1”. Set ’s state of choice isn’t known yet. So, we proceed with next question, “ is it in state 2?” and say the answer is “NO”. So we have two pieces of information: “NO-1” and “NO-2”. But the actual state of set  isn’t known yet. Again we proceed and ask, “is it state 3?” and the answer is “NO”. So we have information: “NO-1”, “NO-2”, “NO-3”. We proceed similarly and acquire information: “NO-1”, “NO-2”, “NO-3”, “NO-4”. Finally we accumulate information: “NO-1”, “NO-2”, “NO-3”, “NO-4”, “NO-5”, which is equivalent  to information set: “NO-1”, “NO-2”, “NO-3”, “NO-4”, “NO-5”, “YES-6”. This is due to the information that states, “when all possibilities but some are exhausted, the only possibilities are the remaining possibilities.”, which is involved in the definition, specifically, due to the information that states, ” when all possibilities but one is exhausted, the only possibility is that one remaining possibility.”—- “[IN-1]”, which is involved in the definition.

This set of information, [“NO-1”, “NO-2”, “NO-3”, “NO-4”, “NO-5”] or [“NO-1”, “NO-2”, “NO-3”, “NO-4”, “NO-5”, “YES-6”], or information  [“NO-1” & “NO-2”& “NO-3”& “NO-4”& “NO-5”& “YES-6”] or [“NO-1” & “NO-2”& “NO-3”& “NO-4”& “NO-5”& “[IN-1]”], completely specifies the current state of set , which, here, is 6. In general, information [“NO- ” & “NO- ”& “NO- ”& “NO- ”& “NO- ”& “[IN-1]”] completely specifies any current state of set .

We could argue that the set of information, for example, , [“NO-1”, “YES-2”] or , [“NO-1”, “NO-2”, “YES-5”] could specify the state of set , namely state-2 and state-5 respectively. However, this does not represent the maximum possible information content of set  i.e. maximum possible information required to specify any realized state of set , i.e. maximum potential of set  to keep its state undefined.

Information, corresponding to a set, relates to the number of non-redundant steps that is to be taken (questions to be asked) from a maximal uncertainty (minimal information) to minimal uncertainty (maximal information) through gradual decrease in uncertainty and increase in information such that the path is the longest (extremum) i.e. number of steps taken is maximum i.e. involves all the possibilities of the set.

Information content in a set/system relates to how much the set’s assumed state is uncertain i.e. to the probability of state exclusive of the guessed state  i.e. to how much information, that specifies the system’s state, is missing.

In case of set , the maximal uncertainty/missing information on guessing any possibility is or is related to , and minimal information that is available is or is related to .         

In case of set , the maximal uncertainty on guessing any possibility is or is related to , and minimal information is or is related to .        

Set  can be expressed as a 3-bit binary set . Thus the information content in set , i.e. possibilities of set  can be expressed in binary terms by a subset  of set .

OR, by a subset  of set . OR any binary set.

Here  is for bit-a unit of piece of information, and  is for 3 bits. Here  is a set of two possibilities 0 and 1.

Now to specify or obtain information that completely defines the assumed state of set , only information regarding each bit values of the 3-bits  is required. Hence, we can proceed and ask, “is the 1st bit 1?” and say the answer is no, then we have information: . We further ask question, “is the 2nd bit 0?” and say the answer is yes, then we have information: . We further ask question, “is the 3rd bit 1?’ and say the answer is yes, then we have a complete information: , which completely specifies the realized state of set . Hence the information corresponding to set  is 3-bits .

Set  can efficiently be expressed as set (say) , which has three possibilities only, unlike set  which has six. Each three possibilities is a bit because each possibility has two possibility of either being 0 or 1, i.e. of being True or False, i.e. , where  is for True and  is for False. For example, consider . Here, possibility  and  and . So, asking whether possibilities are True or False and suppose get answer , we get , which specifies the state of set . So, acquiring information about each bit defines state of set .

Now, we cannot reduce set  any further to set with smaller number of possibilities, given that we agree on what number is and how one number relates to the another, i.e. on the definition of number, and on definitions of set, and every mathematical rules/definitions used here in this context.

We can call this set a generalized set, which shares its property/concept with generalized co-ordinate (system).

Generalized co-ordinate: Generalized co-ordinates are a set of convenient co-ordinates used to describe the configuration of a particular system, such that the number of independent generalized co-ordinates defines the number of degrees of freedom of the system with dependent generalized co-ordinates having some dependency with each other as a result of additional constraints. With  generalized co-ordinates and  constraints , the difference  between the number of generalized co-ordinates used and the number of constraints/constraint equations is equal to the total degrees of freedom of the system.

(Amirouche, 2005) 

Now, since set  cannot further be reduced to set of lesser possibilities than set , which defines set , set  can be regarded as generalized set for describing set of possibilities such as set , with not only independent possibilities but also with least independent possibilities/information that describe sets like .  

In Shannon’s theory, however, each possibility is about the potential gain of information. For example, consider coin, which is a set of two possibilities: . The information that can be gained or the information that is required, i.e. information that is to be generated, in order to describe this set is maximum (i.e. ), provided that the coin is unbiased. However, it is less than for, for example, set die which has 6 possibilities: : . Hence the maximum potential gain in information corresponding to set .

Now consider a coin that is biased for head such that it has head on both its sides. So, the only information available is that: “It has both sides heads”. Hence this set has only one possibility, i.e. . Hence the uncertainty for this set is zero and certainty that the outcome is head is 1. Hence, according to Shannon, there is no any gain in information from this set. There is only one information (working as a constraint), which is its definition that describes this set, and no any other information is required. Hence there is no gain/generation in/of information: . However, this set has a piece of information that defines it, which is taken for granted, and is responsible for its existence like it is with any other set and its definition. Therefore, it is argued that the information corresponding to this set is non-zero, and this holds true for any set.

Now, putting forward following arguments:  

In case of set , for which one single information is required/is to be gained in order to define the set’s acquired state completely, the information that’s gained is about the state, which is either  or . So, information gained is 1. However, this also involves information that defines set , which involves information that states the mutual exclusiveness/independence of the two possibilities  & , which we take for granted and so on.

Now, in case of set , suppose that the set is not defined, similar to realized state  incase of set . Then say we define this set as . On defining this set, information, which is set’s definition, is acquired, realizing the set and bringing it to existence. Thus the information gained in defining this set is 1. Now, putting this in the context of Shannon’s theory, we could say that set can have any infinite  i.e the set can be any set, with for example set  as one of the possibilities. We will come to this later.  

Information, it is defined, is not just about the degree of freedom that any set/system has, i.e. the information (contained) in the set/system is not just about how many possible states the system can (potentially) be in. Information relates to the very definition of the set/system, that brings it to existence, definitions of the states and every definitions/statements being considered, that the definition of/statement about the set incorporates. What is implied here is that with every bit of information we can associate a statement that defines that bit of information i.e. that brings into existence that bit of information.      

But what is the number of pieces of information

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However, it is not necessary that a statement correspond to a conventional bit (i.e. a bit defined by Shannon). For example, consider statement:

“Set .”

This statement defines a certain set and therefore corresponds to a conventional bit.

On the other hand, consider statement:

“Set .”

This statement defines a certain set of two possibilities and also defines what possibility the set has assumed/realized. Hence, this statement involves two conventional bits, one that defines the set and the other that defines the realized possibility.

If a (single) bit/bit of information is defined to denote a (single) statement, then the correspondence between a (single) statement and a (single) bit is mutual and both ways.

It can be noticed here that, what Shannon defines as “gain in a bit of information”, i.e. gain in a conventional bit of information, corresponds to the difference between the two statements, in this case the above two statements.

This gives a clue how Shannon’s (conventional) definition of bit and the statement-definition of bit is different (NOTE), and also gives a clue about how statement(s) is viewed in conventional way of thinking about them, or in conventional logic (NOTE, Find reference if possible), and how that differs from idea of statement presented here.

The idea of statement/state presented here is that the only entity (fundamental) is the state/statement. Any entity previously/conventionally conceived as the “combination” of statements, is also a single statement-piece and not a “combination” and/or “non-combination”. It is also a quanta like previous/conventional individual statements which is a non-combination of conventional individual statements. Conventionally there are combined and non-combined statements. In the idea of statement here, there are statements and that’s it.

Further analyzing this:

Conventionally: Consider statements “A” and statement “B”. Then statement “A & B” is the combination of statement “A” and statement “B”, i.e. statement “A & B” is related to statement “A” and statement “B”.

In statement theory presented here: Consider statement “A” and statement “B”. Then, there is statement “A & B” and that’s it. Combination, no-combination, relation, no relation, etc isn’t considered – st.(statement) 1. Even st.1 is not considered – st.2. Even st.2 is not – st.3. Even st.3 is not – st.4. Even st.4 is not – st.5. Even st.5 isn’t –st.6. And so on… There is no combination, no-combination, relation, no relation, – st.(statement) 1. Not even st.1 – st.2. Not even st.2 – st.3. Not even st.3 – st.4. Not even st.4 – st.5. Not even st.5 –st.6. And so on… There is statement “A & B” and that’s it. Even mentioning the phrase “and that’s it” is a statement in itself. Therefore, to be precise, there is statement “A & B”. What is meant and emphasized here is that, because statement is the fundamental entity in the statement theory presented here, any further than any statement is just a statement in itself. To avoid any confusion and for convenience we will identify any statement of concern within “ ”.

[There are three ways of, call it, statement logic.

1. In which: not every statement is the not of statement A, and only one statement is the not of A, which is called not A. So, not all statements are mutually exclusive of every other. This is conventional logic. Within this lies the concept of a number and it’s negative.
2. In which: every statement is a not of every other statement. So, A is not of B, not C, not D… and C is not A, not B, not E… and K is not 0, not R, not A… and so on… So, each statement is mutually exclusive of the other statement. Each statement is the complement of the other.
3. In which: If there is statement A then there is statement A and that’s it. For statement B, there is statement B and that’s it. For statement C there is statement C and that’s it. And so on…
4.  

[Nothing, which we now days refer to absolute nothing, is Everything. Here’s how:]

 [Probab. Duty cycle of ]

                                           FIG: Probability duty cycle

Now, examining this definition of state-space:

First defining what a quantum state is:

State-Instantaneity: