Friday 26 May 2023

Language and Spacetime Shift of Time From SAPIR Edward to KAWAMATA Yujiro. 2007

 

Language and Spacetime Shift of Time From SAPIR Edward to KAWAMATA Yujiro. 2007


 
1 Language is expressed as structure of spacetime.
2 Spacetime is expressed as manifold.
3 Affine algebraic variety is selected for description of spacetime.
Algebraic variety is pasted together from affine algebraic variety.
Affine algebraic variety is irreducible affine algebraic set.
Affine algebraic set is the set that consists of common zero point of finite polynominal
Polynominal is in n-dimensional complex affine space.
4 Now n-dimensional projective space P n is presented.
C n+1 \ {O} / ~
O is the coordinate’s origin of complex affine space.
~ is mathematical equivalence on elements of set.
5 Projective space P n is covered by n+1 affine space.
6 Now abelian category and derived category are presented.
Abelian category and algebraic variety is placed together.
Derived category is abelian category’s coherent sheaf’s complex that is composition of successive arrows becomes 0.
7 From derived category, distinguished triangle is presented.
8 Here time conjecture of language is presented.
(1)Distinguished triangle makes the model for shift of time on language.
(2)Time on language is closed, successive and circular in word.
Circulation is worked between starting point and ending point of word.
The origin of shift of time is derived from the following.
The concept of time in language is taught from the following.
SAPIR Edward   LANGUAGE  An Introduction to the Study of Speech   Harcout, Brace & Co. Inc
Special thanks to KAWAMATA Yujiro for the mathematical approach on language research, especially from the following.
KAWAMATA Yujiro   Daisukikagaku to doraiken   Sugaku 58-1, January 2006
 
Tokyo April 20, 2007



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Wednesday 17 May 2023

Meaning minimum 2017 Edition On Roman Jakobson, Sergej Karcevskij and CHINO Eiichi

 


Meaning minimum 2017 Edition
On Roman Jakobson, Sergej Karcevskij and CHINO Eiichi


TANAKA Akio


1.
Meaning minimum is one of the kernel concepts for the model of language universals on my study. The concept was at first thought from Roman Jakobson's semantic minimum on which I first read at his book, ESSAIS DE LINGUISTIGUE GENERALE, 1973.　His concept was yet intuitive at the linguistic study  history in the latter half of the 20th century.Comparison with his concept, my definition of meaning minimum was a certain basis prepared in the learning of mathematics, especially on algebraic geometry, that is the most fantastic approach to the construction of the language model. But the contents of meaning minimum is vacant. This concept shows the minimum unit of one big constructive meaning of word. meaning minimum seems to be correspondent with element of set theory, which theory and foundations of mathematics had been my favourite mathematical basis in 1970s, my youth time. Bourbaki was always echoed around us. Grothendieck was a highest star in this world.

2.
Time went vast from at that time. Set theory became one of the premise field of mathematics. But in my part, set theory's agenda was put at a another point. Because language has a certain expanded world that seems to be continuous. Set theory's atomic discreteness does not match in my primary learning level.  So, in my  age 30s, I had sank in the philosophically intuitive thinking often referring the tradition of 1920s, especially of the Linguistic Circle of Prague. On the circle my teacher CHINO Eiichi had taught me from time to time on the campus of university or coffee shop near the station we used. CHINO had gone to the Czechoslovakia Republic from 1959 to 1967. I first met him in 1969 at his Russian class at my  third year of university student. I was the age 21 and he was 37.

3.
At the age 23's 1971 spring,  I graduated university and  once became a high school's teacher and again returned back to university in 1979 after 8 year job of the school. At that time I thought of characters' distinctive features on Written  Chinese classics. I mainly read WANG Guowei, ZHANG Binglin, DUAN Yucai, WANGYyinzhi being guided by Japanese modern scholar KANO Naoki. Expressly I had attracted to  WANG Guowei and his book Guantangjilin. Besides reading these China's Qing dynasty's linguistic peaks, I had always thought on Ludwig Wittgenstein for his endless pursuit on language. So I resigned school and came back to the campus where I again met with CHINO. I was age 30s and he was 50s. He was already the big scholar at the linguistic field  but I was a poor return student. But I dare to say we were colleagues for language study. He taught me the detailed and strict tradition of the linguistic Circle of Prague. He frequently talked on Sergej Karcevskij and his eminent discernment on language. In the later year's masterpiece, Janua Linguisticae Reserata, 1994, he wrote only Karcevskij as genius in the great linguists.

4.
Being led by CHINO, I again started linguistic learning on meaning that I had been interested in from my 20s but too hard to approach by my talent. This time I had Karcevskij's fine insight to meaning enough absorbing the fertile tradition of Prague, where also exist Jakobson and Mathesius. Through the learning I gradually lean to desire to write clear definitions on language. I again remembered the little learning of my 20s age's mathematics. Bourbaki, Godel, TAKEUCHI and their set theory, foundations of mathematics and that Incompleteness theorem. I had learnt mathematics little by little, inch by inch.

5.
CHINO Died in 2002 at age 70 and I became 55. The next year 2003, I wrote a short paper titled "Quantum Theory for Language". This paper was showed at a international symposium on Silk road for dealing with language from Chinese characters on linguistic viewpoint. I knew  that Asian civilisation and history had great concern not only from Asia but also European continents. At the symposium some 400 researchers gathered in the various scholarly fields. It was a awesome encounter for my study, namely, East meet West. Probably Chinese character's agenda will be written by Europe oriented mathematics. WANG Guowei will meet with karcevskij mediated through mathematics' description. The target confronted at that time was time inherent in characters, or time in word.  In Chinese, particularly in classical written Chinese, all the characters show enough independent meaning in one character probably including even time. It was my first conjecture taught from Karcevskij and CHINO. Meaning minimum is on the boat going across to the opposite shore. This metaphor was derived from WANG Guowei's famous paper, "Yin- bu zhong suojian xiangong xianwang kao"

6.
The concept of time inevitably led to the concept of distance. In 2004, I wrote a paper titled "Distance Theory". But the paper was yet intuitive and not clear for descriptive definition. So hereinafter I learnt algebra inch by inch being assisted with the rich heritage of geometry. In the centre of learning, always exist time that connotes finiteness and infinity. But infinity is not easily obtained without probably only loop space at the present. And again returns back to meaning minimum as the passenger of the boat named time property inherent. This time the passenger on the boat  is called operad or algebraic language.

7.
After all I came back to the very dream that I had embraced since the high school days. It was  a fundamental ask on language related with mathematics and physics. The root of language would be able to describe by mathematics and physics. In my mind language is always put at the centre of the pursuit that was what anyone can clearly understand. Description by mathematics, but physics why (Note 1)?. Physics treats with substance that constructs the world in which I had desire to let language enter. It started at Substantiality of language (References 2. 2). It was my dream and probably will be so, hereafter.

References
1.

Cell Theory Continuation of Quantum Theory for Language / From Cell to Manifold For LEIBNIZ and JAKOBSON / Tokyo June 2, 2007

The Time of Language Ode to The Early Bourbaki To Grothendieck / January 10, 2012
2.

Fortuitous Meeting What CHINO Eiichi Taught Me in the CLass of Linguistics / Tokyo December 5, 2004

Linguistic Circle of Prague / Tokyo 13 July 2012, 19 July 2012 Added
3.

The Complete Works of WANG Guowei / Tokyo 24 May 2012

The Time of Wittgenstein / Tokyo January 20, 2012

Notes for KARCEVSKIJ Sergej / Note for KARCEVSKIJ Sergej's "Du dualisme asymetrique du signe linguistique" / Tokyo September 8, 2011
4.

Quantum Linguistics / Growth of Word Dedicated to TAKEUCHI Gaishi / Tokyo January 30, 2006
5.

Quantum Theory for Language Synopsis / Tokyo January 15, 2004


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Sunday 14 May 2023

Distance Theory Algebraically Supplemented / Brane Simplified Model / Distance / 26 October 2007

 


Distance Theory Algebraically Supplemented
Brane Simplified Model <Continuation of Escalator Language Theory>

2
Distance
Direct Succession of Distance Theory

TANAKA Akio

1
Metric model of 5-dimensional spacetime is expressed below from Randall and Sundrum (1999). <RS model>
ds2 = e2U(y)ηmndxmdxn + dy2
Branes exist at y = 0 and y = d.
Our world is regarded as brane y = 0.
U(y) is called <warp factor>.
2
Using <orientifold> of circle ( radius R )’s identification, y is expressed by <line segment> that scales from 0 to ±πR.   
Distance is defined in <line segment>.
According to <line segment>, <warp factor> of <RS model> is measured in bulk spacetime of 5 dimensional Anti-deSitter space.
3
Distance in <line segment> is expressed from <y = –πR> to <y = 0> and from <y = 0> to <y = +πR>.
Now “from <y = –πR> to <y = 0>” is called <minus side> and “from <y = 0> to <y = +πR>” is called <plus side>.
Values of <warp factor> are same at <minus side> and <plus side>.
4
In <Distance Theory Algebraically Supplemented> (abbreviation; DTAS), word is regarded by <warp factor>’s value.

BraneDistance.jpg

5
Word has distance at <minus side> and <plus side>.
6
Now distance at <plus side> is called distance of <real language> and distance at <minus side> is called distance of <mirror language>.

[References]
<Quantum Theory for Language Map>
2.1.3 Distance Theory
2.1.4 Reversion Theory
2.2.1 Prague Theory
2.3  Warp Theory
2.4  Time Theory
4.3.1 Guarantee of Language
4.3.2 Place where Quantum of Language Exists
4.3.3 Actual Language and Imaginary Language
4.5.1 Mirror Language
4.5.2 Mirror Theory
<Aurora Theory>
[AT11] Distance and Time  Language Multi-Time Conjecture 3  5th Time for KARCEVSKIJ
<Aurora Time Theory>
[ATT3] Enlarged Distance Theory
[ATT4] Opened Time and Closed Time
<Algebraic Linguistics Linguistic Result>
1 Deep Fissure between Word and Sentence
2 From Cut and Glue Dimension to Krull Dimension  8th For KARCEVSKIJ Sergej

Tokyo October 26, 2007

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Farewell to Language Universals Revised 2021

 
Farewell to Language Universals Revised 2021
 
Farewell to Language Universals
18/03/2020 19:40 ;  

First uploaded date to Sekinan View I ever thought of language from the vie point of the core elements usually assumed as one of language universals.
Every element has been important for language phenomena, but selecting these elements from the infinite language world, they are seemed to be arbitrary pointing-out act for me now.
So the below quoted  facts at paper 1 and paper 2 are, at the end, considered to be one of the important aspects on vast and boundless language for me now.

Tokyo
18 March 2020 First uploaded
27 June 2021 Text revised
Sekinan Library
Paper 1
Mathematical description for three elements of language universals


Energy, dimension and distance can be described by mathematical writing.
Energy in language is now preparatory description till now.

vide:

  1. Energy of Language / Stochastic Meaning Theory
  2. Energy and Distance / Energy Distance Theory
  3. Energy and Functional / Energy Distance Theory
  4. Potential of Language / Floer Homology Language

For dimension, definite results are presented being aided by arithmetic geometry.

vide:

  1. Three Conjectures for Dimension, synthesis and Reversion with Root and Supplement

For distance, its vast and vagueness of the concept can not be grasped up. But related papers of mine are probably the most in number.

vide:

  1. Distance / Direct Succession of Distance Theory / Distance Theory Algebraically Supplemented
  2. Distance of Word / Complex Manifold Deformation Theory

This paper is not finished.Tokyo

27 February 2015

SIL 


Read more: https://srfl-essay.webnode.com/news/at-least-three-elements-for-language-universals/

The upper paper, titled as Mathematical description for three elements of language universals, is now one of the most fundamental studies of language at present outlook.


Tokyo

28 February 2020

SRFL Paper

[Note]

28 February 2020

I ever wrote a essay titled Half Farewell to Sergej Karcevskij and the Linguistic Circle of Prague

Read more: https://srfl-paper.webnode.com/news/half-farewell-to-sergej-karcevskij-and-the-linguistic-circle-of-prague-with-references/

Paper 2

Half Farewell to Sergej Karcevskij and the Linguistic Circle of Prague with References

TANAKA Akio

I have thought on language through the rich results of the linguistic Circle of Prague and its important member Sergej Karcevskij.

But now my recent thinking has inclined towards algebraic or arithmetic geometrical method and description. Probably it is the time of half farewell to those milestones which led me to the standing place here with rather sufficient results in my ability.
Great thanks to all that always encouraged me for hard and vague target on language especially meaning and its surroundings. And also to CHINO Eiichi with love and respect who taught me all the bases of language study.

For recent results see the following papers group named AGL Arithmetic Geometry Language and related essays.


Tokyo

23 October 2013

Sekinan Research Field of Language


[References]


Papers

Arithmetic Geometry Language (AGL)

  1. Dimension of Language (AGL 1)
  2. Synthesis of Meaning and Transition of Dimension (AGL 2)
  3. Birth of Word, Synthesis of Meaning and Dimension of New Word (AGL 3)
  4. Dimension Conjecture at Synthesis of Meaning (AGL 4)

Essays

  1. Parts and Whole
  2. Edward Sapir’s Language, 1921
  3. Macro Time and Micro Time
  4. Meaning Minimum
  5. Disposition of Language

24 September 2014, references added

Sekinan Research Field of Language

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von Neumann Algebra 3 Note 2 Purely Infinite

 


von Neumann Algebra 3

Note 2
Purely Infinite  

TANAKA Akio


[Theorem]
The necessary and sufficient condition for what von Neumann algebra N is purely infinite ( Ⅲtype) is what semi-finite normal trace that is not 0 does not exist over N.

[Explanation]
<1 Trace>
<1-1>
Trace over von Neumann algebra N          τ : N+ → [0, ∞]  0∞ := 0
τ is the map that has next condition.
(i) τ ( A+B ) =τA +τB,   ∀A,B∈N
(ii) τ (λA ) = λτ ( A )      ∀A∈N+,   ∀λ∈[0, ∞)
(iii) τ ( A*A ) = τ ( AA* )   ∀A∈N
<1-2>
Trace over von Neumann algebra N          τ
(1) τ is faithful.     A∈N, τ (A) = 0 → A = 0
(2) τ is normal.     Increase net {An} ⊂N+   τ (supα Aα) = supα τ (Aα)
(3) τ is definite.    τ (I ) < ∞
(4) τ is semi-definite.     When A(≠0)∈N+,    there exist B(≠0) ∈N+  while B≦A and τ (B) ≠0.

To be continued
Tokyo May 1, 2008
Sekinan Research Field of Language
www.sekinan.org
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von Neumann Algebra 3 Note 1 Properly Infinite

 


von Neumann Algebra 3

Note 1
Properly Infinite  

TANAKA Akio


[Theorem]
On von Neumann algebra N, next are equivalent.
(i) N is properly infinite.
(ii) There exist {En : n∈N}⊂P(N) and En~I, ∑nEn = I.
(iii)There exist E∈P(N) and E~E⊥~I.

[Explanation]
<1 Objection Operator>
<1-1>
Hilbert space     H
Linear subspace of H     Subspace
Subspace that is closed by norm || ・|| of H    Closed subspace
Arbitrary subspace of H         K
K⊥: = {x∈H ; <x, y> = 0, ∀y ∈K}     Orthogonal complement of K
Subspaces of H     K, L
<x, y> = 0 ∀x∈K  ∀y∈L     It is called that x and y are orthogonal each other. Notation is K⊥L.
Direct sum K⊕L : = {x+y ; x∈K, y∈L}
<1-2>
x∈H
d = dist(x, K) : = inf{||x-y|| ; y∈K}
z∈K
d = ||x-z||
z : = PKx
PK is called objection operator from H to K.
<1-3>
von Neumann algebra     N
All of objection operators that belong to N     P (N)
All of unitary operators that belong to N     U (N)

<2 Bounded operator>
<2-1>
Hilbert space      H, K
Subspace of H     D
Map    A
A(λx+μy) = λAx+μAy, ∀x, y∈D, ∀λ, μ∈C
A is called linear operator from H to K.
D     domain of A    Notation is dom A.
Set {Ax ; x∈D}     range of A    Notation is ran A.
<2-2>
dom A = H
Constant M>0
||Ax|| ≦M||x||  (∀x∈H)
A is called bounded operator from H to K
All of As     B(H, K)
H = K
B(H) := B(H, H)
<2-3>
A∈B(H)
A*∈B(H)
<x, Ay> = <A*x, y>
A* is called adjoint operator of A.
A = A*
A is called self-adjoint.
A*A = AA*
A is called normal operator.
A = A* = A2
A is called objection operator.
||Ax|| = ||x|| (∀x∈H)
A is called isometric operator.
A*A = AA* = I   ( I is identity operator.)
A is called unitary operator.
Ker A := {x∈H, Ax = 0}
A that is isometric over (Ker A)⊥ is called partial isometric operator.
<2-4>
von Neumann algebra     N
Commutant of N     N ‘
Center of N     Z := N∩N ‘      
Z = CI
N is called factor.
E∈P(N)
Central projection     E that belongs to Z    
All of central projections     P(Z)
<2-5>
Projection operator     E, F∈P(N)
Partial isometric operator     W∈N
F1∈P(N)
F1≦F
E ~ F1
Situation is expressed by E ≺ F.
≺ gives P(N) partial order relation.

<3 Comparison theorem>
<3-1>
[Theorem]
For E, F∈P(N), there exists P∈P(Z) , while EP≺FP and FP⊥≺EP⊥.

<4 Cardinality>
<4-1 Relation>
Sets     A, B
x∈A, y∈B
All of pairs <x, y> between x and y are set that is called product set between a and b.
Subset of product set A×B     R
R is called relation.
x∈A, y∈B, <x, y>∈R     Expression is xRy. 
When A =B, relation R is called binary relation over A.     
<4-2 Ordinal number>
Set     a
∀x∀y[x∈a∧y∈x→y∈a]
a is called transitive.
x, y∈a
x∈y is binary relation.
When relation < satisfies next condition, < is called total order in strict sense.
∀x∈A∀y∈A[x<y∨x=y∨y<x]
When a satisfies next condition, a is called ordinal number.
(i) a is transitive.
(ii) Binary relation over a is total order in strict sense.
<4-3 Cardinal number>
Ordinal number    α
α that is not equipotent to arbitrary β<α is called cardinal number.
<4-4 Cardinality>
Arbitrary set A is equipotent at least one ordinal number by well-ordering theorem and order isomorphism theorem.
The smallest ordinal number that is equipotent each other is cardinal number that is called cardinality over set A. Notation is |A|.
When |A| is infinite cardinal number, A is called infinite set.
<4-5 Countable set>
Set that is equipotent to N     countable infinite set
Set of which cardinarity is natural number     finite set
Addition of countable infinite set and finite set is called countable set.
<4-6 Zermelo’s well-ordering theorem>
If there exist Axiom of Choice, there exists well-ordering over arbitrary set.
<4-7 Order isomorphism theorem>
Arbitrary well-ordered set is order isomorphic to only one ordinal number.
<4-8 Axiom of choice>
∀x∃f [ f ∈Map(x, ∪x)∧∀y[y∈x∧y≠0 → f(y)∈y]]


To be continued
Tokyo May 1, 2008
Sekinan Research Field of Language
www.sekinan.org

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