2019年6月1日土曜日

最近聴いている曲リスト

【中世音楽/教会音楽】

Graduel D'Alienor De Bretagne - Plainchant et polyphonies des XIII & XIV siecles



Fontevraud修道院の尼僧院長,BretagneのAliénorが所有していたgradual(ミサの楽曲を集めた本)にある典礼歌を,中世音楽の演奏で有名なEnsemble Organumが再現したもの.冒頭の"Natus est hodie Dominus",二曲目のKyrie,四曲目の"Per te Dei genitrix",10曲目の"Hodie donum datur gracie",11曲目の"Viderunt omnes fines terre"が特に好き.

Anon. 1225 - Miri it is while sumer ilast



13世紀前半の歌で,現存する英語の歌で最古のものらしい.中英語だけど,何となく意味を拾える.中世の音楽で残っているのは宗教音楽が多いと思うけど,これは冬の厳しさを歌った世俗の歌で,本の中に楽譜の書かれた紙片がはさまれていたことで現代に伝わっているらしい(https://earlymusicmuse.com/mirie-it-is-while-sumer-ilast/

Мария, Дево чистая, Пресвятая Богородице



Агни Парфене (Agni Parthene)というロシア正教の聖歌.もともとはΑγνή Παρθένεというギリシア語の歌で,19世紀にエギナの聖ネクタリオスが作曲したもの.ギリシア語版しか知らなかったけど,ロシア語版の方が好きかもしれない.ヴァラーム修道院の合唱団が歌うバージョンも良い ↓

Valaam Brethren Choir - Agni Parthene



【Future Garage】

Vacant & Sorrow - Requiem



最近聴いた曲の中で一番好きかもしれない.純粋なる恍惚としか言いようがない.

Vacant - Overnight



Vacantの曲は全部良いので聴いて欲しい.

Phelian - With You



Phelian - Lost



Victoriya - For You



【Electronic/Chillstep】

Phaeleh - Feel You Fade (ft. Augustus Ghost)



歌詞が良い."Would you take me intravenously? / Feel me dancing in your bloodstream / Could you love more easily / If I were dancing in your bloodstream?"

Synthetic Epiphany - This Metal Skin



Synthetic Epiphanyが作る複雑で奥行きのある音は昔から大好き.

Synthetic Epiphany - Motion



Synthetic Epiphany - The Dying of the Light



【Dubstep】

Vacant x Aesthetic Kid - Knøw



【Ambient】

Grandyzer - Foreign Body



Oleg Byonic - 8848 Meters Above Sea Level



【Trance】

The Blizzard With Gåte - Iselilja

 

この曲自体は昔から知ってたけどこのバージョンは最近初めて聴いた.後半の盛り上がり方が良い.

2019年3月30日土曜日

Natus est hodie Dominus (Lyrics)

Natus est hodie Dominus

Source: Ensemble Organum, Graduel de Aliénor de Bretagne: Plain-chant et polyphonies des XIIIe & XIVe siècles



Natus est, natus est, natus est hodie Dominus
qui mundi, qui mundi, qui mundi diluit facinus
quem pater factor omnium
in hoc misit exilium
ut facturam redimeret
et paradiso redderet.

Hec minuit quod erat
assumens quod non erat
sed carnis sumpto pallio
in virginis palatio
ut sponsus de thalamo o
processit ex utero o

flos de iesse virgula a
fructus replet secula a
hunc predixit prophetia
nasciturum ex maria
quando flos iste nascitur
diabolus confunditur
et moritur mors et moritur mors et moritur mors.

Igitur, igitur, igitur mundana fabrica
iam nova, iam nova, iam nova concrepent cantica
pax pax pax est in terris reddita
per prothoplaustum perdita
orta prole summi patris
sacro sancta carnem matris
cipressus ex platano
veniens a libano o
est inclinata deitas
ut assumeret humanitas

o quanta leticia a o et quanta gracia a
tanta rei gaudia se ineffabilia a
o nativitas miranda o et dies veneranda
o stella maris inclita
eternem solem rogita
ut adiuvet nos ut adiuvet nos ut adiuvet nos.

Lux anni, lux anni, lux anni reducta ferculo
festiva, festiva, festiva refulget seculo
quam lux hominis filius
carnem suscepit virginis
in huius valle iherito
de germine Davitico
o magnus misterium !
o quale solatium !
deum qui cuncta condidit
puelle mater edidit.

Cuius puerperium
mundo fert remedium
missa pascit hostia
de superna gratia a.

Ergo fratres exultemus
et hec festa celebremus
ut hec nobis festivitas
sit omnibus prosperitas
Diex amen amen Diex amen !



He is born, He is born, He is born this day, the Lord
Who from the world, Who from the world, Who from the world hath washed our crime,
Whom the Father, the Creator of all things,
Hath sent into this exile,
That He redeem what hath been committed
And restore us to paradise.

He hath diminished what He was
Assuming that which He was not,
But having clothed Himself in flesh,
In the virginal palace,
As a bridegroom from the bridal chamber,
He came from the womb.

Flower of the staff of Jesse,
Fruit that replenisheth the ages,
This was foretold by prophecy,
That He would be born unto Mary.
When this flower was born,
The devil was confounded,
And death died, and death died, and death died.

Accordingly, accordingly, accordingly throughout the created world,
let new songs resound,
Peace, peace, peace hath returned to the world,
Which was lost by the first man.

Thanks to the offspring of the Almighty Father,
Issued from the most holy flesh of the Mother,
Cypress issued from the plane-tree
Come from Lebanon,
His Divinity deigned to lower itself
In order to assume human form.

Oh, what joy ! An oh, what grace !
Such rejoicing at an ineffable event.
Oh, admirable nativity! And oh, revered day !
Oh, sublime star of the sea,
Beseeching the eternal sun
That He come to our aid.

Light of years, the soul's sustenance,
Festive light, festive light, festive light shining upon the ages,
O what light, the son of man
Hath taken virginal flesh
In the valley of Jericho
Born of the seed of David.
O great mystery !
O what comfort !
The virgin mother hath given birth
To the God Who hath created all things.

Her childbirth
Hath brought redemption to the world,
The paschal victim feeds us
With eternal grace.

Therefore, brethren, let us rejoice,
And celebrate these festive days,
That unto us this feast
Be bountiful unto all men.
God, amen, amen God, amen !

2019年2月18日月曜日

イ・チャンドン『バーニング』考察(ネタバレあり)

イ・チャンドン監督の『バーニング』を観た.観終わった直後は?だったけど,しばらく反芻しているうちに頭の中で熟成していく作品だ.以下,思い付いたことを綴っていく(ネタバレ注意)

まず,ヘミは最初から実在していなかったんじゃないか.その理由は次の通り.

(1) ヘミは主人公ジョンスにとって不自然なほど都合の良い女性として描かれている.ジョンスにとって,彼女はたまたま再会した幼馴染であり,なぜか自分に好意を持っており,自分にアプローチしてきてすぐにセックスする.ヘミは男の理想を具現化したような存在であり,これは,彼女がジョンスの妄想であることを示唆している.

(2) ジョンスはなぜかヘミの過去について何も覚えていない.昔,ヘミのことをブスだと言ったことを覚えていないし,井戸に落ちた彼女を彼が見つけたエピソードも覚えていない.井戸のエピソードについては,ヘミの家族が,そんなことは起きていないし,そもそも井戸なんてないよ,と言うシーンもある.これもやはりヘミの非実在性を示唆しているように思える.(ヘミの家族が実在しているじゃないかと言われるかもしれないが,それは重要ではないと思う.このような作品において重要なのは,論理的整合性ではなく暗示である)

(3) 序盤で,ヘミが蜜柑の皮を剥くパントマイムをするシーンがあるが,そこで彼女は,「蜜柑が存在しないことを忘れる」のがコツだと言う.これは,ジョンスも同様にヘミが存在しないことを忘れていることの暗示に思える.

(4) ヘミが飼っている猫が存在しているのかどうか,はっきりしない.この猫の存在の曖昧性は,ヘミの存在の曖昧性と重なる.

ヘミが実在しないのであれば,ベンも実在しないんじゃないかという疑いも出てくる.実際,ベンもかなり不自然なキャラクターである.彼はお金持ちのイケメンであり,女性にモテて,ポルシェを乗り回し,仕事をせず遊んで暮らしている.このような外面的な属性を色々持っているものの,彼の内面は全くの空白である.それは彼が,生身の人間というよりは,ジョンスがなりたいけどなれない男性の象徴だからではないか.ヘミがジョンスにとっての女性の理想像であるように,ベンはジョンスにとっての男性の理想像なのだ.

このように,主人公以外の登場人物が,主人公の内面世界の比喩として機能することは,村上春樹の作品によくある.例えば『スプートニクの恋人』におけるKとすみれが同一人物なのは,分かりやすいと思う.あれは行方不明になった自分の分身を探し求め,最後に再会するという物語だ.それに対して『バーニング』では,ジョンスは自分の分身であるベンを最後に殺してしまう(それがどういう意味を持っているのかはよく分からないが)

ところで,ジョンスが最後にベンを殺しに行く直前,ヘミの部屋で,何かを決心したような面持ちでパソコンに文章を打ち込むシーンがある.小説家志望のジョンスは,小説をどのように書くべきか,それまでずっと迷っていたものの,それをようやく書き上げたわけだ.その直後に彼が取る行動は,実は小説の中の出来事なのではないかと思わせるシーンだが,そう考えると,ますます現実と主人公のフィクションの境界が曖昧になっていく.

中盤で,ヘミが「メタファーって何?」とベンに聞くシーンがある.それに対してベンは,「それはジョンスに聞いた方がいいんじゃない?」と答える.両者ともジョンスの内面世界のメタファーだとすれば,この台詞は頷ける.

2018年11月27日火曜日

Charles S. Peirce - The Use of Solitude (MS 891, 1860)

The following is an entry in Peirce's notebook "Private thoughts principally on the conduct of life" (MS 891), dated April 1, 1860. Scans of the original notebook entry are available here and here.



     O, they who know what it is, feel its use! But who does know it? The poet or sentimentalist who shuts himself up for an hour or a day or three days seems to himself to feel the excellence of solitude. But he is mistaken; his condition is not solitude; this: to live in the desert after the two months home-sickness, thoughts of home, and care for home are over, and before any prospect of return to the world has brought them back, this is solitude.
     And these are its properties:
     Negatively, it is a soothing absence of all care for appearances,—it is the normal absence of all thoughts of the fictitious and factitious.

"Sleep, sleep, today, tormenting cares
Of earth and folly born."

     Positively, everybody knows it is drawing nigh unto the personality in nature, and that it is, in an humble sense, walking with God. It is a calmness preparatory to enthusiasm on those things worthy of enthusiasm & the enthusiasm it makes is of calm and noble, unparitzan nature. Thus, rightly used, Solitude has a reference to the world, and if it is rightly used, the mind grows under its climate. Man, certainly, was not made for solitude; hence taken as an end in itself, it only hurts. It enervates. The mind just emerged from it lacks that hardihood which pertains to those who have been grappling with circumstances & exposed to the storms of life. It is, in fact, a green-house and a nursery.
     The Important portion of this earth consists of variegated land, with inland seas (the Atlantic, the Mediterranean, the Indian, the Arctic) all made for the promotion of intercourse. This is man's workshop. But a full half of the globe is nothing but a polynesian ocean—all isolation—with nothing out of the monotony, either to think of or to care for. Thus we see that the idea that man should sometimes be solitary is expressed in the very contour and face of the planet.
     THE MIND, too, is constructed in its contemplative, concentrative faculties, with the same thing in view; and people who live in cities and who violate this physical and mental law of nature, are doomed to a degeneracy, in consequence; and, in my opinion, it should be made one principal point in a boy's or a girl's education, besides instructing them in literature & philosophy & in the accomplishments of discernment and readiness, to teach them also the depth, the power, and the use of solitude.

2018年8月27日月曜日

【読書ノート】 William Seager, Natural Fabrications: Science, Emergence and Consciousness

"The creation of the magnetic field of a bar of iron by spontaneous symmetry breaking is thus a case of emergence which is unpredictable in principle. Similarly, the ratio of the relative strength of the four forces and the masses of the force carrying particles may not be set by nature but emerge through a random process. No matter how much one knew about the symmetric phase, it would be impossible to predict these values" (p. 23)
→ 強磁性体をCurie温度以上に熱すると強磁性が失われるが,それを徐々に冷却していくと再び強磁性が回復する.磁性体は同時に,特定の(当初とは異なる)磁気モーメントの向きを得るが,この向きを決定する法則はなく,ランダムに決まる.これは自発的対称性の破れの一例.強磁性を回復する前は,磁気モーメントの向きに関してすべての空間方向が対称的だが,強磁性の回復によってこの対称性が自発的に(ランダムに)破れる.初期宇宙において,一つの統一的な力から四つの基本的な力が分れていったのも,同様の過程によると考えられている.いずれの場合においても,対称性が破れる前の状態において,破れた後の状態に関わるパラメータ(強磁性体の場合は磁気モーメントの向き,四つの力の分岐の場合は力の相対的な強さや媒介粒子の質量)は原理的に予測できない.したがってこれは創発(正確には下で触れる通時的創発)の例である.

"Picture then the state of the universe when approximately 1 s old. Is there anything missing from our cosmological picture? Considered from the standpoint of the recognized scientific disciplines, almost everything. There is no chemistry (organic or even inorganic), no biology, no psychology and no sociology. Is this surprising? From a purely physical point of view, not at all. Conditions of the universe at this time are, by Earthly standards, very extreme. The temperature was something like one billion degrees and it was quite impossible for atoms to form. The formation of stable atomic nuclei was just becoming possible as the temperature dropped below 1 billion. There was then a small window of opportunity where density and temperature permitted the synthesis of hydrogen and helium nuclei (as described above). Thus it is a natural fact that the only science which applies to the universe at the age of 1 s is physics. As we move back in time closer to the big bang itself, we encounter more exotic, eventually frankly speculative, physics, but it’s physics all the way back. However, as we pursue time away from the big bang towards the present, we obviously enter the domain of application of all the other sciences. Thus, for example, chemical and biological entities and processes (such as molecules, bonding, living organisms and natural selection) must be in some sense emergent phenomena. A little more precisely, any phenomenon which appears in a system which heretofore did not exhibit it can be labeled diachronically emergent. The cosmological tale as we now conceive it must be a tale of diachronic emergence" (p. 23)
→ ビッグバンから1秒後の宇宙には当然,生物学や社会学の研究対象となるようなものはなく,すべて物理学の研究対象となるものである.つまり初期宇宙に適用可能な科学は物理学だけである.そこから時間が経つにつれて,化学や生物学といった他の科学が研究するような対象が徐々に生まれてくる.これも通時的創発である.通時的創発と対比される創発の形態として,共時的創発(synchronic emergence)も考えられる.上で触れたような宇宙の発展は,通時的創発だけでなく共時的創発も示している.というのも,初期宇宙に存在していた対象は,生物学的な性質(例えば自然選択によって進化するといった性質)を持っていなかったと考えられるが,120億年後には,そのような性質を持った対象が存在することが知られているから.

"The weird behavior of some quantum systems called entanglement, in which two systems that have interacted maintain a mysterious kind of connection across any distance so that interaction with one will instantaneously affect the state of the other, might appear to contradict the adjacency requirement of cellular automata. But, again, it is far from clear that this is a real problem for digital physics and for the same reason. The spatial distance separating the ‘parts’ of an entangled system which makes entanglement seem ‘weird’ need not be reflected in the underlying workings of our hypothetical universal CA. In fact, could it be that the phenomenon of quantum entanglement is trying to tell us that what we call spatial separation is not necessarily a ‘real’ separation at the fundamental level of interaction?" (p. 70)

"When we burn books, it looks as though we are destroying information, but of course the information about the letters remains encoded in the correlations between the particles of smoke that remains; it’s just hard to read a book from its smoke. The smoke otherwise looks universal much like the thermal radiation of a black hole. But we know that if we look at the situation in detail, using the full many-body Schrödinger equation, the state of the electrons evolves unitarily" (Luboš Motl, "Hawking and Unitarity").

"Although both the weather and the Solar System are chaotic dynamic systems, the timescale on which chaos reveals itself in the latter case is so long that we can preserve the illusion that the Solar System is easily predictable. The same would be true of the weather if we wished to use our models to make predictions for the next five minutes (though of course it will take longer than five minutes to get any ‘predictions’ out of our weather models). Predictability is a relative notion: relative to the timescales natural to human observers, relative the natural timescales of chaos of the systems in which we are interested and relative to the time it takes for our models to generate their predictions" (p. 104)

自然の階層構造

自然が階層構造を成していることは自明であるという前提でSeagerは議論を進めている:"If anything about the structure of nature seems obvious and irrefutable it is that nature possesses a hierarchical structure within which ‘higher level’ entities have properties lacked by ‘lower level’ entities" (p. 65). しかしこれは決して自明ではない.実際,自然が階層構造を成していることを否定する論者もいる (James Ladyman & Don Ross, Every Thing Must Go).ここでは「自然が階層構造を成している」という描像がどの程度の妥当性を持つか考えてみる.

私たちが「階層」と呼んでいるのは現象のパターンであり,自然をどのように階層に切り分けるかは,どのパターンに注目するかという観測者の視点から切り離せないように思われる.ただ,四つの基本的な力の到達距離と強さの違いが,観測者の視点に依存しない自然な階層を作っているのは事実である.例えば強い力の到達距離は10 -15 [m] くらいなので,それよりも大きいスケールではその影響を無視できる.また重力に比べて電磁気力の方が圧倒的に強いが,大きい物体では電磁気力は打ち消し合う傾向にあるので,天体のスケールまで行くと重力の影響だけを考えることになる.このように,原子核のスケール,電磁気力支配のスケール,そして重力支配のスケールという三つの自然な階層を区別できる(ただし電磁気力支配から重力支配への移行は滑らかである).

しかし私たちはスケールだけで自然を階層に切り分けているわけではない.例えば生物個体や生物集団に固有のスケールは存在しない.私たちが生物個体の階層を階層として同定するのは,生物に特有の振る舞いのパターン(自己増殖,エネルギー変換,恒常性維持など)に注目することによってである.また生物個体の階層が決まれば,生物集団の階層がそれに相対的に決まる.一般に,自然の中のどのパターンに注目するかという観測者の視点から独立に自然の階層構造が厳然と決まっているわけではないと思われる.この点を念頭に置く限り,「下位」レベルからの「上位」レベルの創発を考えることに特に問題はないだろう.

Conservative and radical emergence

Seagerは(通時的・共時的創発の区別とは別に),下位レベルの法則によって上位レベルの性質をすべて「原理的に」記述できるかどうかによって,創発を"conservative emergence" (CE)と"radical emergence" (RE)の二種類に分けている.記述できるのがCEで,記述できないのがREである.問題は,この世界にREが存在するかどうかだが,SeagerはREの存在を否定し,CEだけが存在するという立場を取っている.そして究極的には,世界のすべての現象は「原理的には」基礎物理学の法則によって記述できるという(「原理的には」って便利の良い言葉だなと思う.原理的に記述できるかどうかって一体どうやって判定するんだ?)

しかし私はSeagerの立場に懐疑的で,「原理的にも」(それがどういう意味であれ)基礎物理学によって記述できないような現象があるのではないかと考えている(そもそもSeager自身が取り上げている上記の対称性の自発的破れはREの例ではないのか?).Seagerを含む分析系の哲学者には,上位レベルの現象のモデルは,下位レベルのモデルの単なる「圧縮」だと考えている人が多い印象がある.つまり彼らは,上位レベルのモデルが,予測にとって実際的には有用だったり不可欠だったりするとしても,下位レベルのモデルから情報の一部を削ぎ落としただけのものだと考えている.因果構造の情報論的な分析から,実はそうではないと論じる面白い研究がある:Erik P. Hoel (2017) "When the Map is Better Than the Territory”(テクニカルでない解説としては,同著者の"Agent Above, Atom Below: How Agents Causally Emerge from Their Underlying Microphysics"を参照).それぞれのレベルにおける現象のモデルの因果構造を情報伝送路に見立てると,誤り訂正符号が情報伝送を効率化するのと同じ原理で,上位レベルのモデルは下位レベルのモデルを効率化するらしい.その結果,上位レベルのモデルが下位レベルのモデルよりも大きな情報量を持ち得る.これはとりもなおさず,下位レベルのモデルでは見えないような因果構造が,上位レベルのモデルに現われているということである.これはもちろん,下位レベルの法則によって記述できないような性質が上位レベルにあるということなので,REに相当する.ここで思い出すのが,「考え深い読者よ,政治的党派心のバイアスのかかったオッカム的な先入観——思考においても,存在においても,発展過程においても,不確定なものは,完全な確定性という原初的状態からの退化に由来する,という先入観を取り払いなさい」というPeirceの言葉である.上位レベルのパターンが下位レベルのパターンの単なる「圧縮」だというのは,Peirceの言うところのオッカム的な先入観なのではないだろうか.

下方因果

自己組織化や創発の議論でよく話題になるのが,いわゆる「下方因果」(downward causation / top-down causation)である.しかし,Seagerの本に触発されて下方因果についての研究をいくつか読んでみたところ,用語の使い方が曖昧で,議論が酷く混乱している印象を受ける.Menno Hulswitが指摘するように("How Causal is Downward Causation?"),原因と結果それぞれの存在様式(一般的な法則なのか,個別的な出来事なのか,何らかの実体なのか)にほとんど注意が払われていないことが,混乱を生み出す要因の一つになっているように思われる.また「下方因果」というときの因果性が,作用因なのか,それとも形相因・目的因的なものなのか,判然としないことが多い.

下方因果の捉え方として有望だと思われるものが二つある.一つは,下方因果の「原因」に相当するのは,下位レベルの法則に付加される「境界条件」だというMichael Polanyiの理論である(例えばKnowing and Being: Essays by Michael Polanyi, pp. 233–34).例えばある言語で発話される文章をいくつかの階層に分けるとすると,最下位には音韻があり,次に語彙があり,その次に文法規則があり,その次に文章作成のスタイルがある,といった具合である.音韻は,与えられた言語で許される音の組み合わせを規定するが,でたらめに音を並べただけでは単語にならないので,さらにその言語の語彙を指定する必要がある.語彙は音韻の規則を破ることなく,許される音の組み合わせを制限する.同様に,でたらめに単語を繋ぎ合わせただけでは文にならないので,単語の組み合わせ方に制限を加える文法規則がさらに必要である.このように上位レベルの規則は,下位レベルの規則を破ることなくそれに対して制限を加える「境界条件」になっているというのがPolanyiの理論である.生命の階層と,物理的・化学的な法則によって支配される非生命的物質の階層の間にも同様の関係があると彼は言う.この理論では,下方因果の「原因」に相当するのは,下位レベルの法則に付加される一般的な法則だということになる.しかし,こうした境界条件がどのように生じるのかはあまり明らかでない.

もう一つの捉え方は,下方因果は形相因・目的因的なもので,「原因」に相当するのは一般的な法則だというもの.Peirceの立場がこれに該当すると思われる.例えばスポーツ観戦で熱狂する群衆を考えると,群衆の一人一人のメンバーの振る舞いは,群衆全体の雰囲気の影響を受けるが,この影響は,「一般的な法則ないし習慣が個々の出来事を,共通のパターンに従うように方向付ける」という意味で目的論的な影響である.このような影響が可能であるためには,(単なる無知の表れではないという意味で)客観的な偶然が必要である.というのも,もし各要素の振る舞いがすべて決定論的に決まっているとすれば,目的論的な法則がそれを「方向付ける」余地はなくなってしまうから.目的論的な法則は,そこからの逸脱の可能性があって初めて成立する.

2017年10月15日日曜日

時間の観念は矛盾律の破れを解消するために生じる

Peirceの1908年の論文「不思議な迷宮―結論部」("Some Amazing Mazes: Conclusion")に面白い一節があるので、日本語に訳してみた。原文のpp. 463-64からの抜粋である。



But on analyzing carefully the idea of Time, I find that to say it is continuous is just like saying that the atomic weight of oxygen is 16, meaning that that shall be the standard for all other atomic weights. The one asserts no more of Time than the other asserts concerning the atomic weight of oxygen;—that is, just nothing at all. If we are to suppose the idea of Time is wholly an affair of immediate consciousness, like the idea of royal purple, it cannot be analyzed and the whole inquiry comes to an end. If it can be analyzed, the way to go about the business is to trace out in imagination a course of observation and reflection that might cause the idea (or so much of it as is not mere feeling) to arise in a mind from which it was at first absent. It might arise in such a mind as a hypothesis to account for the seeming violations of the principle of contradiction in all alternating phenomena, the beats of the pulse, breathing, day and night. For though the idea would be absent from such a mind, that is not to suppose him blind to the facts. His hypothesis would be that we are, somehow, in a situation like that of sailing along a coast in the cabin of a steamboat in a dark night illumined by frequent flashes of lightning, and looking out of the windows. As long as we think the things we see are the same, they seem self-contradictory. But suppose them to be mere aspects, that is, relations to ourselves, and the phenomena are explained by supposing our standpoint to be different in the different flashes. Following out this idea, we soon see that it means nothing at all to say that time is unbroken. For if we all fall into a sleeping-beauty sleep, and time itself stops during the interruption, the instant of going to sleep is absolutely unseparated from the instant of waking; and the interruption is merely in our way of thinking, not in time itself.

しかし時間の観念を丁寧に分析してみたところ、時間が連続的であると言うのは、酸素の原子量が16である、つまり酸素を他のすべての原子量の規準にすると言うのと全く同じである、ということを私は見出した。後者が酸素の原子量について何も主張しないのと同様に、前者も時間について何も主張しない。もし時間の観念が、深紫色の観念と同様に、完全に非媒介的的意識の問題だとすれば、それを分析することはできず、探究自体がそこで終わってしまうだろう。もし分析できるなら、我々が行うべきなのは、その観念(あるいは単なる感じではない限りでのその観念)を、当初はそれが不在であった精神において生じさせ得るような一連の観察と反省の過程を、想像の中で辿ってみることである。それは、脈拍、呼吸、昼と夜といった、あらゆる周期的現象における矛盾律の破れに見えるものを説明するための仮説として精神に生じるかもしれない。というのも、そのような精神において[時間の]観念が不在であったとしても、それは、彼[その精神]が諸々の事実に対して盲目であると仮定するのと同じではないからである。彼の仮説は、次のようなものになるだろう。すなわち、なぜか我々は、暗い夜に蒸気船の船室の中にいて、沿岸沿いを航海しているというような事態にいる。外[の景色]は雷の光で頻繁に照らされ、我々は窓からそれを見ている。我々が見ているものが全部同じものだと考える限り、それは自己矛盾したものに見える。しかし、それらのものが単なる側面、つまり我々自身に対する関係であると仮定すると、雷が外を照らす度毎の我々の視点が異なると仮定することによって、現象を説明することができる。この考えを敷衍してみると、時間が途切れていないと言うのは全く無意味であることがすぐに分かる。というのも、もし我々全員が眠り姫的な眠りに落ちて、その空白のあいだ時間そのものが止まるとすると、眠りに落ちる瞬間と起きる瞬間は全く切り離されていないからである。空白は時間そのものにではなく、あくまで我々の考え方にあるだけである。

2016年11月22日火曜日

【メモ】 Vincenzo De Risi / Geometry and Monadology: Leibniz's Analysis Situs and Philosophy of Space

Preface

That space is actually constituted by points, though here abstractly meant as terms of situational relations, is perhaps the highest result of Leibniz’s geometrical investigation and, at the same time, it also marks the core of Leibniz’s theory of phenomenal expression. It shows in fact that a set of non-spatial relations (such as those occurring between monads) can be isomorphic to (“expressed by”) a set of situational relations that per se suffice to produce phenomenal extension and thus, ultimately, faithfully represent the supersensible through the sensible. (p. xii)

Chapter I. Historical Survey

"Having thoroughly inquired, I have found that two things are perfectly similar when they cannot be discerned other than by compresence, for example, two unequal circles of the same matter could not be discerned other than by seeing them together, for in this way we can well see that the one is bigger than the other. You may object: I shall measure the one today, the other tomorrow, and thus I will discern them even without seeing the two of them together. But I say that this still is a way of discerning them not by memory, but by compresence: because the measure of the first one is not stored in your memory, for magnitudes cannot be retained by memory, but in a material measure marked off on a ruler or some other thing. In fact, if all of the things in the world affecting us were diminished by one and the same proportion, it is evident that nobody could make out the change." (Letter to Jean Gallois, September 1677; GM I.180, A III.ii.227-28, A II.i.380) (pp. 58-59)

Chapter II. Geometry

He [Leibniz] says that two figures are congruent not only provided they are only distinguishable through the simultaneous perception of them (which would be the similarity definition), but also provided such a perception requires the presence of a third object (which was not a condition for similarity). For example, two spheres are similar, for you can distinguish them only by seeing them both simultaneously (otherwise, seen one at a time, they prove totally indiscernible). Furthermore, if they are also congruent, you can distinguish them only through their reciprocal situation (one being on the right, the other on the left), which calls in the use of a reference element external to the two spheres such as, for instance, a third object that fixes right and left, or proximity and distance, or an ideal object such as given set of coordinates (Specimen Geometriae luciferae, GM VII.275) (p. 143)

On the notion of homogony and in general on the importance of motion in the foundation of the continuum, Cassirer has forcefully insisted in his Leibniz' System: «Man begreift jedoch, dass es für Leibniz schwierig sein musste, die reine Auffassung der Kontinuität, die er zunächst am Problem der Veränderung gewonnen hatte, auch dann festzuhalten, wenn er für die Definition vom Sein der Ausdehnung ausging. Denn gerade dies erwies sich als das Originale von Leibniz' Gedanken, dass er die Unmöglichkeit zeigt, die Stetigkeit als Eigenschaft an einem gegebenen Sein erschöpfend zu bestimmen. Erst aus dem Gesetz des Werdens verstehen wir das Kontinuum. Die Unbestimmtheit, die wir auch in Kants Definition noch fanden, lässt sich gleichfalls darauf zurückführen, dass Kant die Kontinuität hier noch als Eigenschaft der Grösse, nicht als Prinzip ihrer Entstehung sucht.» [It is understood, however, that it would be difficult for Leibniz to maintain the pure notion of continuity, which he had first gained from the problem of change, even if he had assumed the definition of being as extension. For precisely this proved to be the originality of Leibniz's thinking: that he showed the impossibility of exhaustively determining continuity as a property of a given being. Only from the law of Becoming can we understand the continuum. The indeterminacy which we have also found in Kant's definition can also be attributed to the fact that Kant seeks continuity here as a property of magnitude rather than a principle of its origin.] (Werke, vol. 1, p. 167). Here we need however to remark that, no matter how much you look into kinematics, never will you extract from it nor from its metaphysical foundation even the least proof that it should be continuous. Nor let us be deceived by the fact that Leibniz has assumed it to be true since his young years for definition purposes («motus est mutatio situs continua»), as he should then demonstrate (and he does not) that a continuous change of situation is also possible, which is untrue if one assumes space to be discontinuous. The answer Cassirer seems to suggest to such a puzzle is that continuity of motion is however dynamically founded, i.e. founded on the continuity of force, which in turn may be founded on the continuity of monadic activity. Yet, it is precisely in this way that one goes out of kinematics and thus geometry, so that all objections about using motion in geometry present themselves again—in fact motion cannot be used in geometry because the continuity of changes of a monad is (perhaps) only a contingent one. Hence, it is not motion in itself, let alone the motion generated by the system of forces, that can guarantee continuity of space, but rather, as already suggested, some transcendental operation of production of space. (pp. 184-85n)

Chapter III. Phenomenology

"That is said to express a thing in which there are relations [habitudines] which correspond to the relations of the thing expressed. But there are various kinds of expression; for example, the model of a machine expresses the machine itself, the projective delineation on a plane expresses a solid, speech expresses thoughts and truths, characters express numbers, and an algebraic equation expresses a circle or some other figure. What is common to all these expressions is that we can pass from a consideration of the relations in the expression to a knowledge of the corresponding properties of the thing expressed. Hence it is clearly not necessary for that which expresses to be similar to the thing expressed, if only a certain analogy is maintained between the relations." (G VII: 263–64) (p. 298)

What most matters here about the relational character of the distinctive properties of monads is that an isomorphism which fully or partly expresses intermonadic relations as relations occurring between representational elements whatsoever is clearly a faithful isomorphism, in the sense that beyond such relations there is nothing else that may further characterize monads. In other words, saying that a perceptual isomorphism expresses the relations between monads is equivalent to saying that it expresses monads themselves, because monads consist in those relations or, at least, are perfectly individuated by them. Leibniz's substance, as it were, ends up as a function. (p. 323)

As however supersensible and phenomenal elements, at least in their distinctive features, are reducible to two sets of relations, an isomorphism for which all relations between monads are preserved would only express the logical identity between the noumenal world and the phenomenal one—hence, it would radically deny any phenomenalism. As we can see, some relations must exist that are not preserved by the situational isomorphism or at least (distributive) by the totality of homomorphisms. (p. 326)

[M]ost divergences between Critical Idealism and Leibnizean Phenomenalism can be reduced to the following. In Leibniz, the relation between phenomena and noumena is regulated by a structural morphism, whereas in Kant by a simply functional relation (not a determination, but only the ground of it, the Bestimmungsgrund). Thus, for example, no real intentionality can be recognized in Berkeley's idealism, because here, owing to the lack of any partial isomorphism, the noumenal world is simply overruled (i.e., identified with the phenomenal one): «But, say you, though the ideas themselves do not exist without the mind, yet there may be things like them, whereof they are copies or resemblances, which things exist without the mind in an unthinking substance. I answer, an idea can be like nothing but an idea; a colour or figure can be like nothing but another colour or figure.» (Treatise, §8). It is evident that here Berkeley deals with similarity in a most ingenious way and without having the notion of a structural identity that can be preserved beyond material distinctions of genus between phenomena and noumena. What he misses is the very concept of expression. Relevantly, even though no morphism is involved in Kantian philosophy, some functional relationship however subsists there linking a phenomenon to the thing-in-itself. (p. 327)

[T]he limited expressivity of the perceptual isomorphismwhich is required in order to establish a distance between phenomena and noumena—finds its further specifications in certain limits of aesthetic comprehension. Furthermore, as it is the only representational content distinguishing monads from one another, only the limitedness of aesthetic comprehension guarantees the possibility for a plurality of substances to be determined, which substances at this point can be distinguished through the different limits of their aesthetic comprehension. In effect, since monads also express such a difference phenomenally through each one's different situs, the most accurate specification of the limits of their finite aesthetic comprehension is to be found in the situation of the phenomenon that expresses the representing monad. Clearly, a monad superordinated to another one will have a broader aesthetic comprehension. We have arrived thus at Leibniz's well-known simile of a town multiplied perspectively. ... [E]ach sight sees the whole town (however variously deformed according to different points of view)—which means that it is not different isomorphisms we are dealing with here, but just one isomorphism variously specified into different homomorphisms. (p. 334)

[I]n order for the various (real) homomorphisms to be determined as genuinely expressive, i.e., referred to a unique intentional object (the noumenal world as a given set of relations), or also, say, as harmoniously perceptual homomorphisms, the ideal isomorphism needs to be determined as a regulative limit of the set of such homomorphisms. It plays actually no other role, in Leibniz's system, than that of an indispensable ideal unity that makes universal harmony possible. (pp. 339-40)

"In fact, even though a circle of a foot [pedalis], one of half a foot [semipedalis], etc., all are different between them, no definition of a foot can nevertheless be given, but there must be some fixed and permanent sample; this is why usually measures are made of a long-lasting material, and it has even been suggested that the Egypt Pyramids, which have lasted for many centuries and presumably will still last for a very long time, be employed for this purpose. Thus, as long as we assume that neither the globe of the Earth nor the motions of stars are significantly to change, our descendants will be able to recognize the same quantity of the Earth's degrees as we have. And if any forms keep their sizes throughout the world and the centuries, as is the case, according to many, with the cells of a honeycomb, these may also provide a constant measure. And as long as we assume that nothing is going significantly to change in the causes of gravity and the motions of the stars, our descendants will be able to learn about our measures through a pendulum. If on the contrary, as I have already said elsewhere, God changed everything and yet preserved the proportions, every measure would fail us, nor could we ever know exactly how much things have changed, because measure cannot by any means be defined, nor can it be kept by memory: and its real preservation is necessary. From this argument, I think, the difference between size and form, or between quantity and quality, appears most clearly." (Specimen Geometriae luciferae, GM VII.276) (pp. 356-57)

"If we conceive of two points existing together, and wonder why we say that they exist together, we will think that this is so either because they are perceived together or because it is just possible to perceive them together. When we perceive an object as existent, for this reason we perceive it in space, that is, we perceive that an indefinite number of other objects absolutely indiscernible from it can exist. Or, still in other words: we perceive that such an object can move, and thus come to be either in one place or in another one; but, since it cannot exist in a plurality of places at the same time, nor can it move in just one instant, this is why we perceive this place as continuous." (Characteristica geomterica §108) (p. 412)
→ もし運動がなければ空間もない(?)

"There is, moreover, a definite order in the transition of our perceptions when we pass from one to the other through intervening ones. This order, too, we can call a path. But since it can vary in infinite ways, we must necessarily conceive of one that is most simple, in which the order of proceeding through determinate intermediate states follows from the nature of the thing itself, that is, the intermediate stages are related in the simplest way to both extremes. If this were not the case, there would be no order and no reason for distinguishing among coexisting things, since one could pass from one given thing to another by any path whatever. It is this minimal path from one thing to another whose magnitude is called distance."(Initia rerum mathematicarum metaphysica, GM VII.25) (p. 423)
→ 二点間の移行のしやすさの度合いを「距離」と呼ぶ。

Chapter IV. Metaphysics

In many places we read that the possible must possess an internal tendency to existence, without which nothing would ever exist (De rerum Originatione radicali G VII.303) (p.440)