Organic Mathematics Distinction as first-order property of the mathematical science Doron Shadmi, Moshe Klein.
Download ReportTranscript Organic Mathematics Distinction as first-order property of the mathematical science Doron Shadmi, Moshe Klein.
Slide 1
Organic Mathematics
Distinction as first-order property
of the mathematical science
Doron Shadmi, Moshe Klein
1
Slide 2
The Least Perception
Bulge within socket.
Socket within bulge.
Non of them.
2
Slide 3
The Least Perception
Emptiness.
Fullness.
Non of them.
3
Slide 4
The Least Expression
“X is …”
“There exists X such that” = “X is …”
“Let X be …” = “X is …”
“X” represents Element
“is” represents Relation
Formal expression is at least
Relation \ Element Interaction (REI)
4
Slide 5
REI as Least Expression
If we use Lisp as an example, then X is the function
where y is the parameters, such that (X y1 y2 y3 ...).
The function is a form of Relation, where the parameter
is a form of Element. Some parameter can be a function,
for example: (X y1=X)
It does not change REI’s fundamental form, which is
(Relation Element1 Element2 Element3 …).
‘True is WFF’, or ‘False is WFF’ exactly because it is
based on REI, where 'True' OR 'False' are Element(s)
and 'is' is Relation. So is P(x,y), where P is Relation and
x,y are Element(s).
5
Slide 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
6
observable).
Slide 7
REI Non-locality and Locality
The self-reference shown in the diagram, enables Nonlocality 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
7
card(Locality)=0 < card(Non-locality)=∞
Slide 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.
8
Slide 9
The intermediate magnitude of existence
Interaction
(self-reference)
Non-local
Local
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.
9
Slide 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.
10
Slide 11
Relative Observation of Elements
(based on REI)
x=.
y = __
x = __
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: ._ )
11
Slide 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) .
12
Slide 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.13
Slide 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 ).
14
Slide 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.
15
Slide 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.
16
Slide 17
Uncertainty and Redundancy
Definition 3:
x is an element
Identity is a property of x, which allows distinguishing
among it.
Definition 4:
Copy is a duplication of a single identity.
Definition 5:
If x has more than a single identity, then x is called
Uncertain .
Definition 6:
If x has more than a single copy, then x is called
Redundant .
17
Slide 18
An example of Uncertainty and Redundancy
Uncertainty
Redundancy
Parallel bridging
Serial bridging
18
Slide 19
Organic Numbers 1 to 5
(Symmetry as first-order property)
19
Slide 20
Some particular case
of Fibonacci Series
20
Slide 21
Partition's extension
21
Slide 22
Partition and Distinction , n = 5
Unclear
ID
Clear
ID
22
Slide 23
Symmetry and arithmetic
(+1)
(1*2)
((+1)+1)
(1*3)
((1*2)+1) (((+1)+1)+1)
23
Slide 24
Locality, Non-locality and the Real-line
If we define the real-line as a non-local urelement, then no set is 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.
24
Slide 25
Non-local Numbers
One asks: “In that case, what number exists between 0.111…
[base 2] and 1?”. The answer is “Any given base n>1 (k=n-1)
non-local number 0.kkk…”, for example:
25
Slide 26
Organic Fractions
(bases 2,3,4)
26
Slide 27
Mixed Organic
Fraction
27
Slide 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 nonfinite multi-sets, etc., etc. … ad infinitum, and R completeness
is not satisfied .
28
Slide 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 א0 = א0+1 , by the new notion @+1 > @.
By Cantor א0 < 2^א0 , by the new notion @ < 2^@.
By Cantor א0-2^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
29
sets and x+@ = @+x in both cases.
Slide 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 30
nonlocal object ( card(collection) < card(Non-locality) ).
Slide 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 visualspatial 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
31
Organic Mathematics
Distinction as first-order property
of the mathematical science
Doron Shadmi, Moshe Klein
1
Slide 2
The Least Perception
Bulge within socket.
Socket within bulge.
Non of them.
2
Slide 3
The Least Perception
Emptiness.
Fullness.
Non of them.
3
Slide 4
The Least Expression
“X is …”
“There exists X such that” = “X is …”
“Let X be …” = “X is …”
“X” represents Element
“is” represents Relation
Formal expression is at least
Relation \ Element Interaction (REI)
4
Slide 5
REI as Least Expression
If we use Lisp as an example, then X is the function
where y is the parameters, such that (X y1 y2 y3 ...).
The function is a form of Relation, where the parameter
is a form of Element. Some parameter can be a function,
for example: (X y1=X)
It does not change REI’s fundamental form, which is
(Relation Element1 Element2 Element3 …).
‘True is WFF’, or ‘False is WFF’ exactly because it is
based on REI, where 'True' OR 'False' are Element(s)
and 'is' is Relation. So is P(x,y), where P is Relation and
x,y are Element(s).
5
Slide 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
6
observable).
Slide 7
REI Non-locality and Locality
The self-reference shown in the diagram, enables Nonlocality 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
7
card(Locality)=0 < card(Non-locality)=∞
Slide 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.
8
Slide 9
The intermediate magnitude of existence
Interaction
(self-reference)
Non-local
Local
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.
9
Slide 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.
10
Slide 11
Relative Observation of Elements
(based on REI)
x=.
y = __
x = __
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: ._ )
11
Slide 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) .
12
Slide 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.13
Slide 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 ).
14
Slide 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.
15
Slide 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.
16
Slide 17
Uncertainty and Redundancy
Definition 3:
x is an element
Identity is a property of x, which allows distinguishing
among it.
Definition 4:
Copy is a duplication of a single identity.
Definition 5:
If x has more than a single identity, then x is called
Uncertain .
Definition 6:
If x has more than a single copy, then x is called
Redundant .
17
Slide 18
An example of Uncertainty and Redundancy
Uncertainty
Redundancy
Parallel bridging
Serial bridging
18
Slide 19
Organic Numbers 1 to 5
(Symmetry as first-order property)
19
Slide 20
Some particular case
of Fibonacci Series
20
Slide 21
Partition's extension
21
Slide 22
Partition and Distinction , n = 5
Unclear
ID
Clear
ID
22
Slide 23
Symmetry and arithmetic
(+1)
(1*2)
((+1)+1)
(1*3)
((1*2)+1) (((+1)+1)+1)
23
Slide 24
Locality, Non-locality and the Real-line
If we define the real-line as a non-local urelement, then no set is 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.
24
Slide 25
Non-local Numbers
One asks: “In that case, what number exists between 0.111…
[base 2] and 1?”. The answer is “Any given base n>1 (k=n-1)
non-local number 0.kkk…”, for example:
25
Slide 26
Organic Fractions
(bases 2,3,4)
26
Slide 27
Mixed Organic
Fraction
27
Slide 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 nonfinite multi-sets, etc., etc. … ad infinitum, and R completeness
is not satisfied .
28
Slide 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 א0 = א0+1 , by the new notion @+1 > @.
By Cantor א0 < 2^א0 , by the new notion @ < 2^@.
By Cantor א0-2^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
29
sets and x+@ = @+x in both cases.
Slide 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 30
nonlocal object ( card(collection) < card(Non-locality) ).
Slide 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 visualspatial 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
31