From Cell to Manifold

 

Cell Theory　

Continuation of Quantum Theory for Language

 

From Cell to Manifold

 

For LEIBNIZ and JAKOBSON

 

 

TANAKA Akio

 

1 Cell is defined by the following.

 n-dimensional ball Dⁿhas interior that consists of cells. Cell is expressed by Dⁿ – δDⁿ and notated to eⁿ that has no boundary.

δis boundary operator.　

Homomorphism of Dⁿ is notated to ēⁿ.

ēⁿ  – δēⁿ = eⁿ

2 Set of no- boundary-cells becomes cell complex.

3 Some figures are expressed by cell. h_n is attaching map.

n-dimensional sphere      Sⁿ =  ē⁰ ∪h_n  ēⁿ   

n-dimensional ball          Dⁿ = ( ē⁰ ∪h_n-1  ēⁿ ) ∪ ēⁿ

Torus                              T² = ( ē⁰  ∪h₁  ( ē⁰ ∪ēⁿ ) )∪h₂ ē²

3 Grassmann manifold is defined by the following.

Grassmann manifold G^R(m, n) is all of n-dimensional linear subspaces in m-dimensional real vector space.

                                        S¹ = G^R( 2, 1 )

4 Canonical vector bundle γ is defined by the following. E is all space. π is projection.

γ= ( E, π, G^R(m, n) )

5 Here from JAKOBSON Roman ESSAIS DE LINGUISTIQUE GÉNÉRALE, <semantic minimum> is presented.

Now <semantic minimum> is expressed by cell ē³.

6 <Word> is expressed by D².

7 <Sentence> is expressed by Grassmann manifold’s canonical vector bundle γ¹ ( G^R(3, 1) ).

 

Tokyo June 2, 2007

Sekinan Research Field of Language

www.sekinan.org

 

[Reference note / December 23, 2008]

 

Time of Word / sekinan.wiki.zoho.com

Road to Language Universals

TANAKA Akio 

Sekinan Research Field of Language was established in 2003 for the study on language universals, that was the core theme since my youth time some 40 years ago in the 1970s. But real start was much later in the late 1980s.In 1986 I established the Sekinan Library for the bace of study at Tachikawa, Tokyo. At that time my main theme was declined to the study of old Chinese character’s classical semantics using the traditional study of the form of characters reading Qing Dynasty’s WANG Guowei. The study from the direction had not given me so rich results. I at the same time read some mathematical books influenced from Bourbaki that was the big stream in 1970s in Japan. Set theory represented by Godel was also the favourite at the time.

In 1990s I determined the direction of study only to language universals taught from CHINO Eiichi, one of the modern typical philologists of Japan. He taught me almost alls WANG Guowei the aspects of European philology that were unkown for me. In the various themes he gave me, the most impressive was the existence of the Linguistic Circle of Prague. Above all Sergej Karcevskij was overwhelmingly deep and moving. But Karcevskij’s  approach was seemed to be intuitive and far from clear descriptiveness.

I began to learn the theme hinted by Karcevskij while learning Godel, Bourbaki and Wittgenstein that was also gave me the big influence form the age 20s. I had freuently read Wittgenstein’s TRACTATUS. But my base of math was so poor and never became the descriptive use for language’s delicate phases. So I again restarted math from geomatry that gave the most familiar and clear image for me. My intuitive image and basic math description were seemed to be able offer in nearly milleium days.

In spring 2003 I knew the application paper for the international symposium held  by a certain institute. I arranged the short paper for it till early summer and sent it. The paper was happily adopted  as one of the language and literature section of the symposium. The paper’s title was Quantum Theory for Language. It was again arranged next year 2004 as the title Quantum Theory for Language synopsis.

The establish date of Sekinan Research Field of Language  was 23 August 2003, that was the date of completion of proto-conceptive paper of Quantum Theory for Language at the hutte in Hakuba, Nagano. Now the time passed rapidly 10 years. The home site, sekinan, has become full by the papers accumulation. So the new site has become indispensible.

Tokyo
31 December 2012