the organic unity of mathematics - various degrees of the numbers’ distinction

8/14/2019 The Organic Unity of Mathematics - Various Degrees of the Numbers Distinction

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The Organic Unity of MathematicsVarious Degrees of the Numbers Distinction

Doron Shadmi, Moshe Klein

Keywords: Symmetry, Locality, Non-locality, Bridging, Organic Natural Numbers, Infinity

Urelement, NXOR (Not XOR), Non-Local Numbers

0. Abstract

Consistency is the most important property of a strong and interesting mathematical framework,

because in an inconsistent framework we can prove X and its negation. In that case, the whole ideaof proof is meaningless. Classical Logic's consistency is based on XOR connective, where concepts

like True and False prevent each other (If X is True, then it cannot be also False and vice versa).It enables the existence of a strong and interesting framework, which is sufficient enough to

prove or disprove strong assumptions. In addition there are cases which are undefined or cannot be

proved or disproved within Classical Mathematics.

We show that in order to avoid a framework that can prove X and its negation, Classical Logic

uses a hidden assumption, which is based on the ability to compare between X and its negation in a

one framework in order to conclude that we prove, disprove, or cannot prove or disprove someassumption.

We think that the ability to compare between X and its negation must not be undefined logically,

or in other words, the logical connective of the concept of Framework must be defined. We

generalized it by using the minimal truth table of 2-valued logic, and noticed that the answer to this

question is based on a NXOR (Not XOR) connective.

NXOR connective, if used, enables us:

1. To define the concept of Framework (or Context, if you wish) logically (we avoid the hiddenassumption).

2. As a result we invent/discover a consistent and strong mathematical universe that exists between

X and its negation, where the XOR connective Classical Logic's consistency is the particular casewhere the middle (the universe between X and its negation) is excluded.

3. From a NXOR connective point of view both NXOR\XOR connective complement each otherand Complementary Logic (where Excluded-middle is a particular case of it) is a generalization ofClassical Logic.

4. XOR is not a hidden assumption from a NXOR point of view.

5. NXOR is a hidden assumption from a XOR point of view.

6. NXOR and XOR contradict each other form a XOR point of view.

7. NXOR and XOR complement each other form a NXOR point of view.


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LetZbe NXOR\XOR Logic.

The truth table ofZis:

0 0 T (NXOR)

0 1 T (XOR)

1 0 T (XOR)

1 1 T (NXOR)

By Z we may fulfill Hilbert's organic paradigm of the mathematical language. Quoting Hilberts

famous Paris 1900 lecture:

The problems mentioned are merely samples of problems, yet they will suffice to show how rich,

how manifold and how extensive the mathematical science of today is, and the question is urged

upon us whether mathematics is doomed to the fate of those other sciences that have split up into

separate branches, whose representatives scarcely understand one another and whose connection

becomes ever more loose. I do not believe this nor wish it. Mathematical science is in my opinion an

indivisible whole, anorganism whose vitality is conditioned upon the connection of its parts.

1. Introduction

The Membership concept needs logical foundations in order to be defined rigorously.

Let in be "a member of ..."

Let out be "not a member of ..."

Definition 1:A system is any framework which at least enables to research the logical connectivesbetween in ,out.

Let a thing be nothing orsomething.

Letx be a placeholder of athing.

Definition 2:x is called localif for anysystemA,x is inAxorx isoutA returns true.


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The truth table oflocality is:

inout0 0 F

0 1 T (in ,out are not the same) = { }_

1 0 T (in ,out are not the same) = {_}1 1 F

Letx be nothing.

Definition 3:x is callednon-localif for anysystemA,x is inAnorx isoutA returns true.

The truth table ofnon-locality whenx is nothing:

inout0 0 T (in ,out are the same) = { }

0 1

F1 0 F1 1 F

Letx be something.

Definition 4:x is callednon-localif for anysystemA,x is inAandx isoutA returns true.

The truth table ofnon-locality whenx is something:

inout0 0 F

0 1 F

1 0 F1 1 T (in ,out are the same) = {_}_

LetsystemZ be the complementation betweennon-locality and locality.

The truth table ofZ is:

inout

0 0 T (in ,out are the same) = { }

0 1 T (in ,out are not the same) = { }_

1 0 T (in ,out are not the same) = {_}

1 1 T (in ,out are the same) = {_}_


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2. Locality and non-locality, basic terms and definitions

The set is a fundamental concept used in many branches of mathematics. Although not

rigorously defined, a set can be thought of as a collection of distinct members whose order is notimportant. If we expand the membership concept beyond the set/member relation, then new

mathematical frameworks emerge. For example:

or is a logical OR connective.

xor is a logical EXCLUSIVE OR connective.

and is a logical AND connective.

= is "Equal to ".

is "Not equal to ".

Urelement is not a set, but can be a member of a set.

Element is either a set oranurelement.

Sub-element is an element that defines another element.

is a member of an element, but not necessarily a sub-element of an element.

is the negation of.

x and A are placeholders of an element.

Definition 5: IfxA xorxA, thenx is local.

Definition 6: IfxA andx A, thenx is non-local.

x (represented by ) is a local urelement ofA (represented by ), that is,

x AxorxA. Also x is not a sub-element ofA, because A is an urelement (an example of

locality: xor ).

x(represented by ) is a non-local urelement ofA (represented by ), that is,

xAandxA. Alsox is not a sub-element ofA, because A is an urelement (an example ofnon-

locality: and ).

The Membership concept is expanded beyond the set/member relation (In a set/member relation

each x member is local and a sub-element of set A) and we logically define the relations between

mutually independent urelements (as can be seen by the true tables in pages 2 and 3).


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The Membership concept has at least two different options:

Option 1: Membership between an element and its sub-elements (notated by ).

Option 2: Membership between an element and other elements, which are not necessarily its sub-elements (notated by ).

The first option is the standard set/member relation, while the second is called bridging. From a

meta view on the mathematical language, each context-dependent axiomatic system is a localmember that can be related to a non-local urelement (please see systemZ in page 1). The bridging

(represented by | ) between non locality (represented by __ ) and locality (represented by one ormore ) is measured by symmetry, for example:

Figure 1

No bridging (nothing to be measured)

A single bridging (a broken symmetry, notated by )

More than a single bridging that is measured by several symmetrical

states, which exist between parallel symmetry (notated by ) andserial broken symmetry (notated by).

Most modern mathematicalframeworks are based on only

broken symmetry, marked by

orange rectangles, as a first-

order property. We expand the

research to both parallel and

serial first-order symmetrical

states in one organic meta-framework, based on bridgingthe local and non local.


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Here is an example of a parallel/serial bridging between locality and non locality (bridging spaces 4and 5):

Figure 2


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Axiomatic systems are based on mutually independent, self-evident true statements calledaxioms. Since the contextconcept is equivalent to a non-local urelement, then each distinct axiom

is equivalent to a local member of a broken-symmetry context. However, if the symmetry concept is

used as a first-order property of the axiomatic approach, we provide a new mathematical framework

that is not necessarily based on broken symmetry. In that case, uncertainty and redundancy are usedas first-order concepts of a symmetry-based mathematical approach. Figures 1 and 2 define the non

distinct (based on bridging between local and non local) as a first-order mathematical property.

Definition 7: Identity is a property ofxorA, which allows distinguishing among them.

Definition 8: Copyis a duplication of a single identity.

Definition 9: IfxorA have more than a single identity, thenxorA are called uncertain.

Definition 10: IfxorA have more than a single copy, thenxorA are called redundant.

Here is an example of uncertainty and redundancy:

Figure 3

Parallel Bridging

Serial Bridging


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4. Organic Natural Numbers

First, here is the standard definition of the natural numbers.

Definition 11: The set of all natural numbers is the set N = {x |xIfor every inductive setI}.

Thus, a setx is a natural number iff it belongs to every inductive set. Each member of an inductive

set is both a cardinal and an ordinal, because it is based on a broken-symmetry bridging,represented by each blue pattern in figure 4. Armed with symmetry as a first-order property, we

define a bridging that cannot be both a cardinal and an ordinal, represented by each magenta patternin figure 4. The outcomes of the bridging between the local and the non-local are called organic

natural numbers.

Figure 4

Each pattern in Figure 4 is both local/global state of the organic natural numbers system, as can beseen in the case of the blue broken-symmetry patterns.


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5. Locality, non-locality and the real-line

A sequence is a collection of elements ordered by some rule. A continuum is a property of a non-

local urelement. If we define the real line as a non-local urelement, then no sequence is acontinuum. By studying locality and non locality along the real line, we discovered a new type of

numbers, the non-local numbers. For example,

Figure 5

Figure 5 illustrates a proof that 0.111 is not a representation of the number 1 in base 2, but thenon-local number0.111 < 1. Are there any numbers between 0.111 and 1? Yes, for any given

base n>1 and any 0.kkk[base n] (where k=n-1) there is 0.nnn[base n+1] such that

0.kkk[basen] < 0.nnn[basen+1] < 1, for example:

Figure 6


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Figure 6 demonstrates that between any given pair ofR members, which are local numbers, thereexists a non-local number, whose exact location on the real line does not exist. A local number isnot a limit of any non-local number, because local and non-local numbers are mutually independent.

Furthermore, no set of infinitely many elements can reach the completeness of a non-local

urelement. Thus, any non-finite set is an incomplete mathematical object, when compared to a non-local urelement.

A finite set has an accurate cardinal because its size is based on a whole local number along the

non-local urelement real-line; however, a non-finite set does not have an accurate cardinal because

it cannot get the completeness of the non-local urelement real-line. From this new notion of the non

finite, Cantor's second diagonal argument is understood as a proof of the incompleteness of a non-

finite set. For example, assume a non-finite set composed of unique non-finite multisets, whereeach non-finite multiset has a unique order of empty and non-empty sets:

{

{{ },{ },{ },{ },{ },...}

{{#},{ },{ },{#},{ },...}

{{ },{#},{#},{ },{ },...}{{#},{#},{ },{#},{#},...}

{{ },{ },{#},{ },{ },...}

...

}

We can then define another unique, non-finite multiset, which is the non-finite, diagonal

complementary multiset {{#},{#},{ },{ },{#},...} that is added to our non-finite set of

non finite multisets, etc., etc. ad infinitum. From this point of view, the identity map of a non-

finite set, composed of unique non-finite multisets, is incomplete; furthermore, its cardinality isunsatisfied.

We know that the Cantorean transfinite universe is an actual infinity, where the limiting

"process" is a potential infinity. However, by the new notion of the non finite, we realize that sinceno non-finite set can reach the completeness of a non-local urelement, then only a non-local

urelement is considered as an actual infinity. Here any non-finite set is no more than a potential

infinity. These sets can have infinitely many potential infinities, but none reaches the completeness

of a non-local urelement.

Let @ be a cardinal of a non-finite set such that:

Sqrt(@) = @

@ -x = @@ /x = @

If |A|=@ and |B|=@ + or * or ^x , then |B| > |A| by + or * or ^x

Some example:

By Cantor 0 = 0+1 , by the new notion @+1 > @.By Cantor 0 < 2^0 , by the new notion @ < 2^@.

By Cantor 0-2^0 is undefined, by the new notion @-2^@ = @.

By Cantor 3^0 = 2^0 > 0 and 0-1 is problematic.

By the new notion 3^@ > 2^@ > @ = @-1 etc.


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This new approach to the non-finite is not counter intuitive as the Cantorean transfinite universe,because it clearly separates between the continuum (which is not less than a non-local urelement)

and the discrete (which is no more than a set of finitely/infinitely many elements).

From a Cantorean point of view, cardinals are commutative (1+0 = 0+1) and ordinals are not

(1+

+1). By using the new notion of the non-finite, both cardinals and ordinals arecommutative because of the inherent incompleteness of any non-finite set. In other words, @ is

used for both ordered and unordered non-finite sets; moreover, the equalityx+@ = @+x holds in

both cases.

6. Summary

Armed with bridging, we use symmetry as a measurement tool leading to a new notion of the

natural number, based on its internal distinction degree.By clearly distinguishing between the actual infinity (which is not less than a non-local uelement)

and the potential infinity (which is no more than a set ofinfinitely many elements), we state thatany given non-finite set is an incomplete mathematical object. We believe that further research into

various degrees of the number's distinction (measured by symmetry and based on bridging betweenlocality and non locality) is the right way to fulfill Hilbert's organic paradigm of the mathematical

language. Quoting Hilberts famous Paris 1900 lecture once more:"The organic unity ofmathematics is inherent in the nature of this science, for mathematics is the foundation of all exact

knowledge of natural phenomena. That it may completely fulfill this high mission, may the new

century bring it gifted masters and many zealous and enthusiastic disciples!"

Acknowledgements:

Ofir Ben-Tov,thank you for your important suggestion to add Distinction as the third property of

the Organic Natural Numbers, in addition to Cardinality and Ordinality.

Dr. Ruben Michel,thank you for your excellent editing.


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References

Hilbert David: Mathematical Problems, Bulletin of The American Mathematical Society, Volume37. Number 4, Pages 407-436, S 0273-0979(00)00881-8.

Tall David: Natural and Formal Infinities, Mathematics Education Research Centre, Institute of

Education, University of Warwick.

Tall David, Tirosh Dina: Infinity - the never ending struggle, published in Educational Studies inMathematics 48 (2&3), 199-238.

Joseph W. Dauben:George Cantor and the battle for transfinite set theory , Department of history,

University of New-York.

Ian Stewart: Nature's Numbers, Orion Publishing 1995.

Wittgenstein: Lectures on the foundation of mathematics, Cambridge 1939.

the organic unity of mathematics - various degrees of the numbers’ distinction

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