Complementary Complementary MathematicsMathematics

Various Degrees of the Number’s Various Degrees of the Number’s DistinctionDistinction

Doron Shadmi, Moshe KleinDoron Shadmi, Moshe Klein

Gan-Adam Ltd.Gan-Adam Ltd.

Hilbert’s 24Hilbert’s 24thth challenge challenge::

In the end of his famous ICM 1900 lecture Prof. In the end of his famous ICM 1900 lecture Prof. David Hilbert was concern about the possibility David Hilbert was concern about the possibility that Mathematics will be split up into separate that Mathematics will be split up into separate

branchesbranches that do not communicate with each that do not communicate with each other, and he said: other, and he said:

““I do not believe this nor wish it.I do not believe this nor wish it. Mathematical Mathematical science is in my opinion an indivisible whole, science is in my opinion an indivisible whole, an an organismorganism whose vitality is conditioned upon the whose vitality is conditioned upon the

connection of its parts”connection of its parts”

Complementary MathematicsComplementary Mathematics::

This work is motivated by Hilbert’s organic This work is motivated by Hilbert’s organic paradigm, and our solution is based on the paradigm, and our solution is based on the

relations between the local and the non-local, as relations between the local and the non-local, as observed among 5 years old children that were observed among 5 years old children that were

asked to describe the relations between asked to describe the relations between a point and a line.a point and a line.

55 years old children observation 1years old children observation 1::

NevoNevo::"We imagine some shape in our mind, and then we draw it. "We imagine some shape in our mind, and then we draw it. But we can draw shapes without lines and points and use them in But we can draw shapes without lines and points and use them in order to create new shapes; all these shapes are in our imaginationorder to create new shapes; all these shapes are in our imagination..""

Nevo positions his source of creativity in his mind; he discovers that Nevo positions his source of creativity in his mind; he discovers that an abstract idea is independent on any particular representation.an abstract idea is independent on any particular representation.

Sivan:Sivan: "The shape is in our imagination; we can think about it, but it "The shape is in our imagination; we can think about it, but it does not exist. So, we draw it, because we can see it within our does not exist. So, we draw it, because we can see it within our hearts."hearts."

SivanSivan is aware of the difference between potential and existing is aware of the difference between potential and existing things. In his experience, the innovation emerges from the heart.things. In his experience, the innovation emerges from the heart.

55 years old children observation 2years old children observation 2::

Tohar:Tohar:" " If a point and a line are not friends and they are not If a point and a line are not friends and they are not reproduce, then they will remain few and nothing will be bornreproduce, then they will remain few and nothing will be born."."

ToharTohar is aware of the distinction between a point and a line. He is aware of the distinction between a point and a line. He realizes that new drawings can be created by associating them.realizes that new drawings can be created by associating them.

OfriOfri: "The point tries to catch the line, but it cannot catch it : "The point tries to catch the line, but it cannot catch it because the line is too highbecause the line is too high."."

Ofri Ofri understands that a line has a property (height) unreachable understands that a line has a property (height) unreachable by a point.by a point.

Membership, two optionsMembership, two options:: Element Element isis eithereither a seta set oror an urelement.an urelement.

Sub-element Sub-element isis an element that defines another an element that defines another element.element.

Option 1: Option 1: A membership between an element and A membership between an element and

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

elements (elements (notated bynotated by €€, where , where ₡ ₡ is “is “not a not a membermember””). ).

Bridging Bridging = An = An option 2option 2 membership = membership = €€

Locality and non-localityLocality and non-locality::

xx and and AA are placeholders of an element.are placeholders of an element.

Definition 1:Definition 1:

IfIf xx €€ AA xorxor xx ₡₡ AA then then xx is is locallocal..

Definition 2:Definition 2:

If If xx €€ AA andand xx ₡₡ AA then then xx is is non-localnon-local..

is a is a locallocal member (if member (if xx= = andand AA==____ then then ____ xorxor ____ ) )

A set’s member is a A set’s member is a locallocal member ( member (xx €€ AA xorxor xx ₡₡ AA))

____ is a is a non-localnon-local member (if member (if xx== ____ andand AA== then then ____ andand ____ ) ) .. ..

.. ..

..

....

Bridging and SymmetryBridging and Symmetry::A bridging A bridging is measured by symmetrical states that exist between is measured by symmetrical states that exist between

local elements and a non-local urelement.local elements and a non-local urelement.

No bridging (nothing to be measured)No bridging (nothing to be measured)

A single bridging (a broken-symmetry, A single bridging (a broken-symmetry, notated by notated by ))

More than a single bridging that is measured by several symmetrical More than a single bridging that is measured by several symmetrical states, which exist between parallel symmetry (notated by states, which exist between parallel symmetry (notated by ) and serial ) and serial broken-symmetry (notated by broken-symmetry (notated by ).).

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

Organic Natural NumbersOrganic Natural Numbers::Armed with symmetry as a first-order property, we define a bridging that Armed with symmetry as a first-order property, we define a bridging that cannot be both a cardinal and an ordinal (represented by each one of the cannot be both a cardinal and an ordinal (represented by each one of the magentamagenta patterns). The products of the bridging between the local and patterns). The products of the bridging between the local and the non-local are called the non-local are called Organic Natural NumbersOrganic Natural Numbers..

Organic Natural Numbers 4 and 5Organic Natural Numbers 4 and 5::

Complementary relations between Complementary relations between multiplication and additionmultiplication and addition

+(1)

(1*2)

(1*3) ((1*2+)1) +(((1+)1+)1)

+((1+)1)

Locality, non-locality and the Real-lineLocality, non-locality and the Real-line::If we define the Real-line as a non-local urelement, then no setIf we define the Real-line as a non-local urelement, then no set is a is a continuum. By studying locality and non-locality along the real-line continuum. By studying locality and non-locality along the real-line we discovered a new kind of numbers, the we discovered a new kind of numbers, the non-local numbersnon-local numbers..

For exampleFor example::

The diagram above is a proof without words that 0.111… is not a base The diagram above is a proof without words that 0.111… is not a base 2 representation of number 2 representation of number 11, but the non-local number , but the non-local number 0.111…0.111… < < 11..

The exact location of aThe exact location of a non-local number non-local number does not exit.does not exit.

Non-local numbersNon-local numbers::One can ask: "In that case, what number exists between One can ask: "In that case, what number exists between

0.111…0.111… [base 2] and [base 2] and 11?". The answer is "Any given base?". The answer is "Any given base nn>2 >2 ((kk==nn-1) non-local number -1) non-local number 0.kkk…0.kkk… ", for example", for example::

Non-locality and InfinityNon-locality and Infinity::If the real-line is a non-local urelement, then Cantor’s second If the real-line is a non-local urelement, then Cantor’s second diagonal is a proof of the incompleteness of diagonal is a proof of the incompleteness of RR set, when it is set, when it is compared to the real-line:compared to the real-line:

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

The non-finite complemntary multiset The non-finite complemntary multiset {{{x},{x},{},{},{x},…{x},{x},{},{},{x},…}} is is added to the non-finite set of non-finite multisets, etc., etc. … ad added to the non-finite set of non-finite multisets, etc., etc. … ad infinitum, and infinitum, and RRcompleteness is not satisfiedcompleteness is not satisfied . .

Incompleteness and proportionIncompleteness and proportionAA={1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12, …}={1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12, …}

BB={2, 4, 6, 8,10,12, …}={2, 4, 6, 8,10,12, …}

If some member is in If some member is in BB set, then it also in set, then it also in AA set and since 12 is in set and since 12 is in BB set, in this particular example, then it is also in set, in this particular example, then it is also in AA set, and the set, and the accurate 1-1 mapping is:accurate 1-1 mapping is:

AA={1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12, …}={1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12, …} ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ BB={2, 4, 6, 8,10,12, …} etc. , … etc. ad infinitum={2, 4, 6, 8,10,12, …} etc. , … etc. ad infinitum.

Sets Sets AA and and BB are non-finite sets; applying the proportion concept, are non-finite sets; applying the proportion concept, any non-finite set is no more than a potential-infinity, while any non-finite set is no more than a potential-infinity, while distinguishing between locality and non-locality.distinguishing between locality and non-locality. ..

A new non-finite arithmeticA new non-finite arithmeticLet @ be a cardinal of a non-finite set, and let Let @ be a cardinal of a non-finite set, and let CC and and DD be non-finite be non-finite sets.sets.

If |If |CC| = @ and || = @ and |DD| = @-2^@, then || = @-2^@, then |CC| > || > |DD| by 2^@. By using the new | by 2^@. By using the new notion of the Non-finite, we have both non-finite sets and an arithmetic notion of the Non-finite, we have both non-finite sets and an arithmetic between non-finite sets, which its results are non-finite sets, for example:between non-finite sets, which its results are non-finite sets, for example:

By Cantor By Cantor 00+1+1א א = = 00א א , by the new notion @+1 > @. , by the new notion @+1 > @.By Cantor By Cantor 00אא̂^22 > > 00א א , by the new notion @ < 2^@. , by the new notion @ < 2^@.By Cantor By Cantor 00אא̂^00-2-2א א is undefined, by the new notion @-2^@ < @. is undefined, by the new notion @-2^@ < @.By Cantor 3^By Cantor 3^ 00אא < < 00אא̂^22 = = 00אא and and 00-1-1אא is problematic. is problematic.By the new notion 3^@ > 2^@ > @ > @-1 etc.By the new notion 3^@ > 2^@ > @ > @-1 etc.

By using the new notion of the Non-finite, both cardinals and ordinals By using the new notion of the Non-finite, both cardinals and ordinals are commutative because of the inherent incompleteness of any non-are commutative because of the inherent incompleteness of any non-finite set. In other words, @ is used for both ordered/unordered non-finite set. In other words, @ is used for both ordered/unordered non-finite sets and finite sets and xx+@ = @++@ = @+xx in both cases. in both cases.

A further researchA further researchWe believe that further research into various degrees of the number's We believe that further research into various degrees of the number's

distinction (measured by symmetry and based on bridging distinction (measured by symmetry and based on bridging between locality and non locality) is the right way to fulfill between locality and non locality) is the right way to fulfill

Hilbert's organic paradigm of the mathematical languageHilbert's organic paradigm of the mathematical language..

Finally, research by Finally, research by Dr. Linda Kreger SilvermanDr. Linda Kreger Silverman over the last two over the last two decades demonstrates that there are two kinds of learners: decades demonstrates that there are two kinds of learners:

Auditory-Sequential Learners (ASL) and Visual-Spatial-Learners Auditory-Sequential Learners (ASL) and Visual-Spatial-Learners (VSL). (VSL). Complementary mathematicsComplementary mathematics is a model that bridges is a model that bridges

between ASL and VSLbetween ASL and VSL..

Thank youThank you