The Least Perception

1. Organic MathematicsDistinction as first-order property of the mathematical scienceDoron Shadmi, Moshe Klein

2. The Least Perception Bulge within socket.Socket within bulge.Non of them.

3. The Least Perception Emptiness.Fullness.Non of them.

4. The Least Expression “X is …”“There exists X such that”=“X is…”“Let X be …”=“X is …”“X” represents Element“is” represents RelationFormal expression is at leastRelation \Element Interaction(REI)

5. REI as Least Expression If we use Lisp as an example, then X is the function where y is the parameters, such that (Xy1y2y3 ...).The function is a form of Relation, where the parameter is a form of Element. Some parameter can be a function, for example: (Xy1=X) It does not change REI’s fundamental form, which is (RelationElement1 Element2 Element3…).‘True is WFF’, or‘False is WFF’exactly because it is based onREI, where 'True' OR 'False' areElement(s)and 'is' isRelation. So isP(x,y),wherePisRelationandx,yareElement(s).

6. REI example By carefully research = or ≠ relations, one can conclude that nothing is definable unless some relation is interacted with some element. For example, = relation cannot be but non-local with respect to the researched element, as can be seen by Edge\Node interaction in Graph Theory:The blue edge in the diagram (which is non-local w.r.t the node) is equivalent to =, where the black or red edges are equivalent to ≠. Actually no node is researchable unless it is observed by an edge. In the case of Graph Theory, Relation is called Edge, where Element is called Node. In general, elements are observable only by Relation Element Interaction (REI) (for example only A (where A is an element) is not observable, where A=A is observable).

7. REI Non-locality and Locality The self-reference shown in the diagram, enables Non-locality and Locality to interact as two atomic states that are not derived from each other. By using this notion Non-locality and Locality are mutually independent properties of the same mathematical universe. By following this notion one defines Cardinality as the magnitude of the existence of these properties, such that card(Locality)=0<card(Non-locality)=∞

8. Card(Locality) and card(Non-locality) Bridging is the result of Non-locality\Locality Interaction 0 and ∞ are the weakest and strongest (respectively) magnitudes of existence of this mathematical universe.

9. The intermediate magnitude of existence Interaction (self-reference) Non-local By defining the intermediate magnitude of the existence between 0 and ∞, one defines the concept of Collection, where Non-locality is defined as the Domain aspect of Collection, and Locality is defined as the Member aspect of Collection. Local Local

10. Two kinds of Elements x and y are elements.Definition 1:If only a one relation is used in order to define the relations from x to y, then x is called Local.Example: a point is a local element.Definition 2:If more than one relation is used in order to define the relations from x to y, then x is called Non-local.Example: a line segment can be a non-local element with respect to another element.

11. Relative Observation of Elements(based on REI) x= . x= __ y= __ y= . x is local w.r.t y if: x is local w.r.t y if x<y (example: .__ ) x<y (example: __. ) x=y (example: ( _. , _._ , ._ ) x>y (example: .__ ) x>y (example: __. ) x is non-local w.r.t y if: x< and =y (example: _. )x< and >y (example: _._ ) x= and >y (example: ._ )

12. Absolute Observation of Elements (based on REI) x= point , y= line , z= plane , w= volume If x is observed through w w.r.t z, then x cannot be but on z XOR not on z.By observation w, x is local w.r.t z. If y is observed through w w.r.t z, then y can be on z AND not on z.By observation w, y can be non-local w.r.t z and also y≠x=x≠y(absolute observation through w) .

13. Serial-only Observation "x=y and x>y cannot be simultaneously true by the definition of “>” and inverse element. x-x=0, if x=y then x>y implies x>x implies x≠x. By reduction to absurd, this statement is always false". By using a serial only observation of x,y relations one actually misses the following:1) x=y and x>y is true if y is not x.2) y is not x because x is > and = w.r.t y, where y is not > and = w.r.t x for example: x=__ , y=. , x= and >y(example: ._ )3) By using a serial only observation the claim that "x-x=0, if x=y then x>y implies x>x implies x≠x" is false. "if x=y then x>y" does not imply x>x because x can be simultaneously > and = w.r.t y,where y is not simultaneously > and = w.r.t x. In that case x=y is false and the rest of the argument is wrong.

14. Bridging and Symmetry Bridging is REI’s result and it is measured by the symmetrical states that exist between the Local and the Non-local. 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 ) and serial broken-symmetry (notated by ).

15. Bridging and Modern Math Most modern mathematical frameworks are based only on broken symmetry (marked by white rectangles) as a first-order property. We expand the research to both parallel and serial first-order symmetrical states under a one framework, based on the bridging between the local and the non-local.

16. Organic Numbers Armed with symmetry as a first-order property, we define a bridging that cannot be both cardinal and ordinal (represented by each one of the magenta patterns). The outcomes of the bridging between the Local and the Non-local are called Organic Numbers. Each ON is simultaneously Local and Non-local form of the entire system.

17. Uncertainty and Redundancy Definition 3:xis an elementIdentityis a property of x, which allows distinguishing among it.Definition 4:Copyis a duplication of a single identity.Definition 5:Ifxhas more than a single identity, thenxis calledUncertain.Definition 6:Ifxhas more than a single copy, thenxis calledRedundant .

18. An example of Uncertainty and Redundancy Uncertainty Redundancy Parallel bridging Serial bridging

19. Organic Numbers 1 to 5(Symmetry as first-order property)

20. Some particular case of Fibonacci Series

21. Partition's extension

22. Partition and Distinction , n = 5 Unclear ID Clear ID

23. Symmetry and arithmetic (+1) (1*2) ((+1)+1) (1*3) ((1*2)+1) (((+1)+1)+1)

24. Locality, Non-locality and the Real-line If we define the real-line as a non-local urelement, then no setis a continuum. By studying locality and non-locality along the real line we discover a new kind of numbers, non-local numbers. For example: Local number Non-local number The diagram above is a spatial proof that 0.111… is not a base 2 representation of number 1, but the non-local number 0.111… < 1.The exact location of a non-local number does not exist.

25. Non-local Numbers One asks: “In that case, what number exists between 0.111… [base 2] and 1?”. The answer is “Any given basen>1 (k=n-1) non-local number 0.kkk…”, for example:

26. Organic Fractions(bases 2,3,4)

27. Mixed Organic Fraction

28. Non-locality and Infinity If the real line is a non-local urelement, then Cantor’s second diagonal is proof of the incompleteness of the R set, when it is compared to the real line: { {{},{ },{ },{ },{ },...} {{x},{},{ },{x},{ },...} {{ },{x},{x},{ },{ },...} {{x},{x},{ },{x},{x},...} {{ },{ },{x},{ },{},...} ... } The non-finite complementary multi-set {{x},{x},{},{},{x},…}is added to the non-finite set of non-finite multi-sets, etc., etc. … ad infinitum, and R completeness is not satisfied.

29. A New Non-finite Arithmetic Let @ be a cardinal of a non-finite set such that (Tachyon property): Sqrt(@) = @ @ - x = @ @ / x = @ If |A|=@ and |B|=@ + or * or ^ x , then |B| > |A| by + or * or ^ x Some comparison: By Cantor א 0 = א 0+1 , by the new notion @+1 > @.By Cantor א 0 < 2^א0 , by the new notion @ < 2^@.By Cantor א 0-2^א0 is undefined, by the new notion @-2^@ < @.By Cantor 3^א0 = 2^א0 > א0 and א0-1 is problematic.By the new notion 3^@ > 2^@ > @ > @-1 etc. By using the new notion of the non-finite, both cardinals and ordinals are commutative because of the inherent incompleteness of any non-finite set. In other words, @ is used for both ordered and unordered non-finite sets and x+@ = @+x in both cases.

30. Non-finite and Distinction Dedekind infinite:A set S is infinite if and only if there exists T as a proper subset of S and a bijective map T → S.Since Distinction is a first-order property of Organic Mathematics, any non-finite set has infinitely many superpositions in addition to any cardinal of distinct members. For example, let us research N (the non-finite set of natural numbers) and E (the non-finite set of even numbers): Indeed E is a proper subset of N, but if Distinction is a first-order property of any set, then there is a generalization beyond the particular case of clear distinction, and any non-finite set has infinitely many superpositions that cannot be defined by any collection of clear distinctions (as clearly shown in the diagram above) exactly because no collection of local objects can be a non-local object ( card(collection) < card(Non-locality) ).

31. Further Research Finally, researches done by Dr. Linda Kreger Silverman and more researchers over the last two decades, demonstrates that there are two kinds of learners: auditory-sequential learners (ASL) and visual-spatial learners (VSL). Organic Mathematics is a method that bridges between ASL and VSL, where ASL is serial thinking and VSL is parallel thinking. We believe that using both thinking styles and further research into Relation Element Interaction (measured by Symmetry and based on bridging the Local with the Non-local) is the fruitful way to research and develop the foundations of the mathematical science, where Distinction is a first-order property of it. Thank you