感覚の秩序：
事物は、それらが生体に対して及ぼす効果に応じて分類され、感覚の秩序へ組み込まれる。ある事物がこの秩序内に占める位置は、生体に対してその事物が及ぼす効果を表現している。
物理的世界の秩序：
事物は、それらが互いに及ぼす効果に応じて分類され、物理的世界の秩序へ組み込まれる。事物がこの秩序内に占める位置は、それが他の諸事物と取り結ぶ相互的な関係を表現している。
科学の過程は、前者の感覚秩序による事物の分類を、徐々に、後者の物理的秩序による事物の分類に置き換えて行く過程と見なすことができる。これはすなわち、事物が生体に対して及ぼす効果を度外視していく過程に他ならない。
例えば、水と塩酸は、見た目は同じであるから、（少なくとも視覚刺激に限定すれば）感覚秩序においては同じ事物として分類される。しかし、水は鉄と反応しないが、塩酸は鉄を溶かす、という風に、他の事物との相互関係に基づいて、我々は水と塩酸を別の事物として分類できる。これはごく素朴な例であるが、感覚秩序を徐々に物理的秩序に置き換えて行く過程のエッセンスを捉えていると思われる。
ここで重要なのは、物理的世界の秩序はあくまで関係の秩序である、という点。物理的な事物は、物理的構造の中のノードとしてしか意味を持たず、その構造と無関係であるような如何なる性質も持たない。「塩酸」という概念の意味は、それが他の概念と取り結ぶ関係によって尽くされており、それを超えた「塩酸そのもの性」のようなものはない。
物理的構造の実在性
『実体概念と関数概念』においてCassirerは、変換に対する頑強性を以て構造の客観性（実在性）を定義した。しかし、これだけでは、少なくとも物理的構造の場合、客観性の基準として十分でないように思われる。というのも、同じデータを説明できる構造が、一意的に決まるとは限らないからである。変換に対する頑強性という客観性の捉え方の不十分さは、経験との結び付きが欠けていることに起因している。経験を考慮するには、ある構造の予測能力が、変換によってどの程度影響を蒙るかを考察する必要がある。
2014年7月14日月曜日
2014年7月3日木曜日
異なる次元を持つ物理量同士の乗除がよくて、加減がいけない理由
異なる次元を持つ物理量同士の掛け算や割り算はよくて、足し算や引き算がいけないのは、物理学を習った人なら誰もが知っているだろう。例えば長さ÷時間＝速度だが、長さ＋時間というのは意味を成さない。しかし、このルールは大概、直観的に正しいものとして提示されるだけで、その論拠が示されることは殆どない（少なくとも初等的な教育では）。長さ＋時間が「意味を成さない」とは一体どういうことだろうか？
まず準備として、「単位」(unit)と「次元」(dimension)という二つの概念を区別しておきたい。例として、m、km、parsecなどはそれぞれ別の単位であり、これらはすべて同じ「長さ」という次元を持つ。次元が同じで単位が異なる二つの物理量は、スケールを変換する関係式を利用して単位を揃えることができる（例えば1km = 1000mという関係を利用すれば、1km+100mの単位を揃えて1100mと書くことができる）。
さて、「林檎と蜜柑の足し算」というシンプルな例から始める。「林檎」と「蜜柑」をそれぞれ別の次元の単位と見なし、それぞれの個数の合計を考える。異なる次元を持つ物理量同士の足し算はできないと思われるかもしれないが、何のことはない。「果物」という、「林檎＋蜜柑」の次元を持つ新しい単位を導入して、数値同士を足した結果に付与すればいい。例えば1林檎＋1蜜柑＝2果物、という風に。この新しい単位は全く問題なく使えるし、現に我々は日常生活の中で使っている。この例を見れば、異なる次元を持つ物理量同士の加減を排除する理由はないように見える。確かに、アプリオリに排除する理由は全くないが、上の議論をあらゆる物理量に一般化できない理由がある。
上の林檎と蜜柑の例と全く同じように、
a [m] + b [s] = c [x] …… ① （a、b、cは何らかの数値、mはメートル、sは秒）
と置いて、新しい単位xを導入する。このxがどのような単位であるかは分からないが、長さ＋時間の次元を持つもの、という風に約束しておく。さて、c[x]で表される上の量を、b[s]のスケールをμsに変えて表すとすればどうなるか。このときb[s] = 106・b[μs]である。同様にa[m]もcmに変換する。すなわちa[m] = 100a[cm]である。スケール変換後の量c[x]をc'[x']と置く（x'はxと同じ次元を持つがxとは別の単位である）。すなわち
100a [cm] + 106・b [μs] = c' [x'] …… ②
これら二つの式を用いて、個々のケースにおいて、c[x]とc'[x']をそれぞれ求めることができる。例えばa = b = 1の場合、c = 2、c' = 102+106と求まる。[1] しかし、a、bが分からない状態で、cを直接c'に変換することはできない。時間と長さのスケールの変換の仕方によっては、写像f：c→c'を具体的に求めることができる場合もあるが（例えばa、bが一律に10倍であればcも10倍）、一般的には不可能である。a[m]、b[s]の値が分からないとc'[x']の値は求まらないわけだから、c'[x']は①式で定めたスケールに依存していることが見て取れる。
では乗除の場合はどうであろうか。
a [m] ・ b [s] = c [y] …… ③
という掛け算を考える（割り算の場合も同様）。 yは長さ・時間の次元を持つ単位である。上の足し算の場合と同じように、a[m]とb[s]のそれぞれのスケールをcm、μsに変換し、変換後の量c[y]をc'[y']と置く。すると
c' [y'] = 100a[cm] ・ 106・b [μs] = 108a・b [cm・μs] = 108c [cm・μs] …… ④
③④式より、c[y] (= c'[y']) = 108c[y'] という単位間の変換式が得られ、a、bの値が分からなくても、直接cをc'に、あるいはc'をcに変換できることが分かる（ここではc'=108c）。つまりc'[y']の値（およびc[y]の値）はスケールの取り方に依存していない。
以上まとめると、異なる次元を持つ物理量同士の加減の場合、新しい単位を導入して加減の結果を表すことは可能だが、この新しい単位における値は、最初のスケールの取り方に依存してしまい、そのスケールにおけるそれぞれの要素の値が分かっていないと、別のスケールにおける結果の値を求めることができない。対して乗除の場合、新しく導入された単位はスケールの取り方に依存せず、ある変換式によって異なるスケール間を簡単に往復できる。スケールの取り方に依存しない、というのは恣意性を排除する上で重要である。物理学の法則が、メートルや秒といった特定の単位の取り方に依存していてはまずい。F = maという関係は、どのような単位で測定しようが成り立たなければならない。物理学では、これを対象や法則のスケール不変性(scale invariance)という。そしてスケール不変性が成り立つためには、異なるスケール間を行き来できる変換式が存在しなければならないのだ。
なお、最初の林檎と蜜柑のケースでは、「果物」という新しい単位が、特定のスケールに依存しているが、これはさほど問題にならない。なぜなら、日常観察される離散的な個物の場合、1、2、3、という風に数える単位系が自然であり、あまり恣意性がないからである。
[1] 我々は異なる次元同士の加減に慣れていないので、この求め方は一見変に見えるかもしれないが、等式の両辺の次元が等しいことを念頭に置いて、a+b [m+s] = c [x]、という風に数値と単位を分けて考えれば良い。ただ単位を、通常の代数法則を満たし、数値と同等の身分を持つ記号として扱うならば、もちろんax+by = (a+b)(x+y)といったような関係は成り立たない。これを以て、異なる次元を持つ物理量同士の加減を禁止する論拠と見なすことも可能だろうが、ここでは約束として、単位を数値と同等の記号としては扱わない。というのも、単位を数値と同等の記号と見なすことを正当化するのは、以下において説明するスケール不変性の議論の結果であるように思われるからである。少なくとも、単位が数値と同等の記号と見なされるべきであることは、決して自明ではない。
まず準備として、「単位」(unit)と「次元」(dimension)という二つの概念を区別しておきたい。例として、m、km、parsecなどはそれぞれ別の単位であり、これらはすべて同じ「長さ」という次元を持つ。次元が同じで単位が異なる二つの物理量は、スケールを変換する関係式を利用して単位を揃えることができる（例えば1km = 1000mという関係を利用すれば、1km+100mの単位を揃えて1100mと書くことができる）。
さて、「林檎と蜜柑の足し算」というシンプルな例から始める。「林檎」と「蜜柑」をそれぞれ別の次元の単位と見なし、それぞれの個数の合計を考える。異なる次元を持つ物理量同士の足し算はできないと思われるかもしれないが、何のことはない。「果物」という、「林檎＋蜜柑」の次元を持つ新しい単位を導入して、数値同士を足した結果に付与すればいい。例えば1林檎＋1蜜柑＝2果物、という風に。この新しい単位は全く問題なく使えるし、現に我々は日常生活の中で使っている。この例を見れば、異なる次元を持つ物理量同士の加減を排除する理由はないように見える。確かに、アプリオリに排除する理由は全くないが、上の議論をあらゆる物理量に一般化できない理由がある。
上の林檎と蜜柑の例と全く同じように、
a [m] + b [s] = c [x] …… ① （a、b、cは何らかの数値、mはメートル、sは秒）
と置いて、新しい単位xを導入する。このxがどのような単位であるかは分からないが、長さ＋時間の次元を持つもの、という風に約束しておく。さて、c[x]で表される上の量を、b[s]のスケールをμsに変えて表すとすればどうなるか。このときb[s] = 106・b[μs]である。同様にa[m]もcmに変換する。すなわちa[m] = 100a[cm]である。スケール変換後の量c[x]をc'[x']と置く（x'はxと同じ次元を持つがxとは別の単位である）。すなわち
100a [cm] + 106・b [μs] = c' [x'] …… ②
これら二つの式を用いて、個々のケースにおいて、c[x]とc'[x']をそれぞれ求めることができる。例えばa = b = 1の場合、c = 2、c' = 102+106と求まる。[1] しかし、a、bが分からない状態で、cを直接c'に変換することはできない。時間と長さのスケールの変換の仕方によっては、写像f：c→c'を具体的に求めることができる場合もあるが（例えばa、bが一律に10倍であればcも10倍）、一般的には不可能である。a[m]、b[s]の値が分からないとc'[x']の値は求まらないわけだから、c'[x']は①式で定めたスケールに依存していることが見て取れる。
では乗除の場合はどうであろうか。
a [m] ・ b [s] = c [y] …… ③
という掛け算を考える（割り算の場合も同様）。 yは長さ・時間の次元を持つ単位である。上の足し算の場合と同じように、a[m]とb[s]のそれぞれのスケールをcm、μsに変換し、変換後の量c[y]をc'[y']と置く。すると
c' [y'] = 100a[cm] ・ 106・b [μs] = 108a・b [cm・μs] = 108c [cm・μs] …… ④
③④式より、c[y] (= c'[y']) = 108c[y'] という単位間の変換式が得られ、a、bの値が分からなくても、直接cをc'に、あるいはc'をcに変換できることが分かる（ここではc'=108c）。つまりc'[y']の値（およびc[y]の値）はスケールの取り方に依存していない。
以上まとめると、異なる次元を持つ物理量同士の加減の場合、新しい単位を導入して加減の結果を表すことは可能だが、この新しい単位における値は、最初のスケールの取り方に依存してしまい、そのスケールにおけるそれぞれの要素の値が分かっていないと、別のスケールにおける結果の値を求めることができない。対して乗除の場合、新しく導入された単位はスケールの取り方に依存せず、ある変換式によって異なるスケール間を簡単に往復できる。スケールの取り方に依存しない、というのは恣意性を排除する上で重要である。物理学の法則が、メートルや秒といった特定の単位の取り方に依存していてはまずい。F = maという関係は、どのような単位で測定しようが成り立たなければならない。物理学では、これを対象や法則のスケール不変性(scale invariance)という。そしてスケール不変性が成り立つためには、異なるスケール間を行き来できる変換式が存在しなければならないのだ。
なお、最初の林檎と蜜柑のケースでは、「果物」という新しい単位が、特定のスケールに依存しているが、これはさほど問題にならない。なぜなら、日常観察される離散的な個物の場合、1、2、3、という風に数える単位系が自然であり、あまり恣意性がないからである。
[1] 我々は異なる次元同士の加減に慣れていないので、この求め方は一見変に見えるかもしれないが、等式の両辺の次元が等しいことを念頭に置いて、a+b [m+s] = c [x]、という風に数値と単位を分けて考えれば良い。ただ単位を、通常の代数法則を満たし、数値と同等の身分を持つ記号として扱うならば、もちろんax+by = (a+b)(x+y)といったような関係は成り立たない。これを以て、異なる次元を持つ物理量同士の加減を禁止する論拠と見なすことも可能だろうが、ここでは約束として、単位を数値と同等の記号としては扱わない。というのも、単位を数値と同等の記号と見なすことを正当化するのは、以下において説明するスケール不変性の議論の結果であるように思われるからである。少なくとも、単位が数値と同等の記号と見なされるべきであることは、決して自明ではない。
2014年5月23日金曜日
一人称視点と三人称視点
一人称視点って何なのか、割と分かりやすいと思うんだけど、三人称視点って何なのか、って聞かれると答えに窮する。一人称視点から徐々に生じるのは間違いない。しかしどうやって生じるのか？そして精神と物体は、これら二つの視点とどう関係している？
まず一人称視点について考えてみる。ある被験者に光を見せ、どこに光が見えるかと問うと、彼らは自分の頭を指すのではなく、光源の位置を指す。つまりクオリアは、被験者の視点から見れば脳内にあるのではなく、対象の位置にある。というより、そもそもクオリアとは、一人称視点から見た物質的対象に他ならない。一人称視点から見れば、あらゆる経験は「私」の経験であるという意味で、「世界は私の中にある」と言える（このような反省的了解は必ずしもないにせよ）。ここではクオリア＝物体である。
さて、発生論的に考えると、一人称視点に対して現われる様々な知覚経験の中から、相対的に永続的な要素と、相対的に流動的な要素とが区別され始める。主観と客観の区別の出現である。例えば、空間や時間の座標変換に対して不変に保たれる構造を見出すと、我々はその構造を一つの個物として同定する。実際の主観／客観区別の出現はもっと複雑な要素が色々絡み合っていると思われるので、ここでは割愛する。
いずれ、自分の身体とよく似た構造を知覚経験の中に見出すようになる。すなわち同胞の生命体である。そしてどうやら、これらの同胞たちも、自分と同じように世界を知覚しているらしい、ということを複雑なプロセスを経て学習する。ここに至って「一つの世界」と、それを眺める「複数の主観」という発想が生まれる。これが三人称視点に他ならない。「精神」とは、三人称視点から他者（同胞生命体）の振る舞いのパターンを観察し、それを手掛かりに、他者も自分と同じように世界を経験をしているに違いない、という類推の結果生じる概念。自分に対して「精神」の概念を適用することも可能だが、ここで言う「自分」とは、あくまで三人称視点から見た自分である。さて、問題は次の通りとなる：物体とは何か？
（暫定的回答：おそらく、錯覚や認知上の誤謬に遭遇し、主観と客観の分離が進行したのに伴って、思考によって改変したり破壊したりすることができず、かつ安定的、永続的に持続する経験の要素が「客観」の側に括られ、それ以外が「主観」の側に括られ、そして前者が「物体」と名付けられたうえで実体化された「精神」概念にjuxtaposeされたのだろう）。
まず一人称視点について考えてみる。ある被験者に光を見せ、どこに光が見えるかと問うと、彼らは自分の頭を指すのではなく、光源の位置を指す。つまりクオリアは、被験者の視点から見れば脳内にあるのではなく、対象の位置にある。というより、そもそもクオリアとは、一人称視点から見た物質的対象に他ならない。一人称視点から見れば、あらゆる経験は「私」の経験であるという意味で、「世界は私の中にある」と言える（このような反省的了解は必ずしもないにせよ）。ここではクオリア＝物体である。
さて、発生論的に考えると、一人称視点に対して現われる様々な知覚経験の中から、相対的に永続的な要素と、相対的に流動的な要素とが区別され始める。主観と客観の区別の出現である。例えば、空間や時間の座標変換に対して不変に保たれる構造を見出すと、我々はその構造を一つの個物として同定する。実際の主観／客観区別の出現はもっと複雑な要素が色々絡み合っていると思われるので、ここでは割愛する。
いずれ、自分の身体とよく似た構造を知覚経験の中に見出すようになる。すなわち同胞の生命体である。そしてどうやら、これらの同胞たちも、自分と同じように世界を知覚しているらしい、ということを複雑なプロセスを経て学習する。ここに至って「一つの世界」と、それを眺める「複数の主観」という発想が生まれる。これが三人称視点に他ならない。「精神」とは、三人称視点から他者（同胞生命体）の振る舞いのパターンを観察し、それを手掛かりに、他者も自分と同じように世界を経験をしているに違いない、という類推の結果生じる概念。自分に対して「精神」の概念を適用することも可能だが、ここで言う「自分」とは、あくまで三人称視点から見た自分である。さて、問題は次の通りとなる：物体とは何か？
（暫定的回答：おそらく、錯覚や認知上の誤謬に遭遇し、主観と客観の分離が進行したのに伴って、思考によって改変したり破壊したりすることができず、かつ安定的、永続的に持続する経験の要素が「客観」の側に括られ、それ以外が「主観」の側に括られ、そして前者が「物体」と名付けられたうえで実体化された「精神」概念にjuxtaposeされたのだろう）。
2014年5月11日日曜日
不毛な話
以下、哲学徒らしく不毛な話をば。
「いかなる科学理論もあくまでモデルであって、実在の完全な模写などではない」。確かにその通りだろう。しかしモデルとはなんだろう？「考察対象の重要な性質だけを抽出した概念的な図式」をモデルだとすれば、モデルとは独立に、その考察対象が予め存在する、という二元論的な発想が含意されがちである。
例えば太陽の内部モデルを考えてみる。誰も太陽の内部を見たことなんてないし、我々が太陽の内部について知っていることは、すべてモデルを通してしか知らない。まず太陽の内部を観察して、そこから重要な特徴だけをピックアップしてモデルを作っているのでは決してなく、まず頭の中で構成したモデルがあって、それが現実の方を作り上げているわけだ。太陽の内部のような観察不可能な例に限らず、目に見える対象の場合でも、原理的には同じだろう。
もちろん、常識的に考えれば、天文学者が太陽の内部について語るとき、彼らが語っているのは実際の太陽の内部についてであって、そのモデルについてではない。もし「対象がモデルとは独立に存在する」と言えるとすれば、この常識的な意味合いにおいてだけだろう。しかし、これを哲学的に実体化して、モデルの外側に「物自体」みたいなものが存在するという発想が出てくると、色んな泥沼が生じることになる。
「いかなる科学理論もあくまでモデルであって、実在の完全な模写などではない」。確かにその通りだろう。しかしモデルとはなんだろう？「考察対象の重要な性質だけを抽出した概念的な図式」をモデルだとすれば、モデルとは独立に、その考察対象が予め存在する、という二元論的な発想が含意されがちである。
例えば太陽の内部モデルを考えてみる。誰も太陽の内部を見たことなんてないし、我々が太陽の内部について知っていることは、すべてモデルを通してしか知らない。まず太陽の内部を観察して、そこから重要な特徴だけをピックアップしてモデルを作っているのでは決してなく、まず頭の中で構成したモデルがあって、それが現実の方を作り上げているわけだ。太陽の内部のような観察不可能な例に限らず、目に見える対象の場合でも、原理的には同じだろう。
もちろん、常識的に考えれば、天文学者が太陽の内部について語るとき、彼らが語っているのは実際の太陽の内部についてであって、そのモデルについてではない。もし「対象がモデルとは独立に存在する」と言えるとすれば、この常識的な意味合いにおいてだけだろう。しかし、これを哲学的に実体化して、モデルの外側に「物自体」みたいなものが存在するという発想が出てくると、色んな泥沼が生じることになる。
2014年4月24日木曜日
On Internalism and Externalism in "Epistemic Justification"
The problem of internalism vs. externalism is usually taken up in the context of "justification of belief," but I don't think the question makes any sense. (I put "justification of belief" in scare quotes because I don't share the assumption that belief should be justified in the first place. In my view, doubt, or change in belief, is what requires justification.)
Let internalism denote the idea "that everything necessary to provide justification for a belief must be immediately available to an agent's consciousness," and externalism the view that "factors other than those internal to the believer can affect the justificatory status of a belief" (taken from Wikipedia because I'm lazy). Now, what is meant by the term "consciousness"? The issue persists even if the phrase "immediately available to consciousness" is replaced by concepts such as "access," "mental states," etc. What are these terms supposed to mean?
I think the question of internalism vs. externalism is based on a confusion of two distinct standpoints concerning mind, viz. the first-person standpoint and third-person standpoint. From the third-person standpoint, mind (or consciousness, mental states, or whatever) is a purely natural phenomenon. It is a model that certain organisms, and perhaps non-organisms, build of their surrounding environment. From this standpoint, all minds are equivalent: there is nothing special about my mind compared to that of others. On the other hand, from the first-person standpoint, the world is in my mind, in the sense that everything is experienced as my experience. From this standpoint, "my mind" (if such an expression be tolerated) partakes of a status higher than that of other minds: it is what Kant called the transcendental subject.
Now, the issue of epistemic justification can make sense only from the first-person, or transcendental, standpoint. This is because there can be no such thing as "justification" in a naturalist view of the world, and the third-person standpoint is none other than the naturalist standpoint. On the other hand, phrases such as "internal to the mind" and "external to the mind" can make sense only from the third-person standpoint. This is precisely the manifestation of a naturalist conception of mind; there can be nothing outside of mind from the transcendental standpoint. The internalism vs. externalism debate can thus be understood as the result of applying the naturalist conception of mind to a question that can make sense only from the transcendental standpoint.
Let internalism denote the idea "that everything necessary to provide justification for a belief must be immediately available to an agent's consciousness," and externalism the view that "factors other than those internal to the believer can affect the justificatory status of a belief" (taken from Wikipedia because I'm lazy). Now, what is meant by the term "consciousness"? The issue persists even if the phrase "immediately available to consciousness" is replaced by concepts such as "access," "mental states," etc. What are these terms supposed to mean?
I think the question of internalism vs. externalism is based on a confusion of two distinct standpoints concerning mind, viz. the first-person standpoint and third-person standpoint. From the third-person standpoint, mind (or consciousness, mental states, or whatever) is a purely natural phenomenon. It is a model that certain organisms, and perhaps non-organisms, build of their surrounding environment. From this standpoint, all minds are equivalent: there is nothing special about my mind compared to that of others. On the other hand, from the first-person standpoint, the world is in my mind, in the sense that everything is experienced as my experience. From this standpoint, "my mind" (if such an expression be tolerated) partakes of a status higher than that of other minds: it is what Kant called the transcendental subject.
Now, the issue of epistemic justification can make sense only from the first-person, or transcendental, standpoint. This is because there can be no such thing as "justification" in a naturalist view of the world, and the third-person standpoint is none other than the naturalist standpoint. On the other hand, phrases such as "internal to the mind" and "external to the mind" can make sense only from the third-person standpoint. This is precisely the manifestation of a naturalist conception of mind; there can be nothing outside of mind from the transcendental standpoint. The internalism vs. externalism debate can thus be understood as the result of applying the naturalist conception of mind to a question that can make sense only from the transcendental standpoint.
2014年4月14日月曜日
On Uchii Soshichi’s “Monadology, Information, and Physics Part 1: Metaphysics and Dynamics”
Dr. Uchii Soshichi's paper "Monadology, Information, and Physics" (released recently in preprint: http://philsci-archive.pitt.edu/10599/) explicates his informational interpretation of Leibniz's Monadology. This interpretation attempts to reconstruct Leibniz's dynamics, and the metaphysics upon which it is founded, using the terminology of contemporary information theory. The gist of Dr. Uchii's paper is that such reconstruction is a fruitful approach towards attaining a general understanding of Leibniz's system. Part 1 deals with Leibniz's dynamics and metaphysics, and Part 2 with space and time, though it is uncertain how many parts there will be in all. Here I will only take up Part 1.
The paper's abstract and preliminary overview may at first seem erratic, in that Dr. Uchii does not so much argue for his interpretation as simply assert it in the form of analogy, but the reader who follows the ensuing discussions will gradually find Dr. Uchii's reading to be convincing. While it need hardly be said that Leibniz did not know concepts such as "transition function" and "recursion," Dr. Uchii's paper gives compelling evidence that Leibniz indeed had in mind, albeit in perhaps a vague and incomplete form, something that corresponds neatly to these concepts. In what follows, I would like to address some points of concern, and then conclude with a few remarks on the fruitfulness of reading contemporary concepts into the thought of great minds of the past.
Deriving Vis Viva from Solicitatio
My first concern deals with a technicality I was unable to understand. In Section 5, where Dr. Uchii discusses Leibniz's distinction between vis viva (living force) and vis mortua (dead force), he takes up the concept of solicitatio, or "infinitely small urge (to motion)." Apparently, solicitatio is an infinitesimal increment of velocity, the accruement of which gives rise to macroscopic acceleration. Now, Dr. Uchii expresses the solicitatio of a falling body by dx, speed by x, and from these, attempts to derive the vis viva (macroscopic acceleration) of the body. For the sake of brevity I shall quote the entire paragraph in question, replacing x and dx with v and dv respectively (pp.16-17):
Does Infinite Recursion Give Rise to a Universal Monad?
An important concept in Dr. Uchii's informational interpretation of Leibniz's system is recursion, by which is understood the iteration of a certain program. Of particular significance is the notion of nested recursion, i.e. the application of a program to itself, in such a way as to give rise to a nested hierarchy of programs, "going down from the single dominant program (corresponding to entelechy) to subprograms, which again controls respective subprograms, and ad infinitum" (p.1). According to Dr. Uchii, although Leibniz never used words such as "recursion" or "recursive function," he was well aware of the recursive structure of both the organization of monads and the phenomenal world (p.33). And the function that is iterated in this recursive structure is that which dictates the elastic collision of bodies (pp.43-48).
Now my concern is this: if recursion—the nested hierarchy of programs—is infinite, then how can there be a single dominant program, above which there is no further program controlling it? Just as the microorganisms in the human body may be considered subprograms under the human entelechy, the human entelechy should be a subprogram within the macrocosm of society, which in turn is embraced by the ecosystem, the solar system, and so on ad infinitum. Do each of these comprehensive programs each correspond to an entelechy? If so, then wouldn’t there be an all-embracing entelechy, a kind of world-soul under which all other monads in the universe are subprograms? This seems to be the logical consequence of the view articulated by Dr. Uchii, but is this the view of Leibniz?
On the Fruitfulness of Reading Contemporary Concepts Into the Thought of Past Thinkers
Reading Dr. Uchi's paper, one may be compelled to ask: what is the point of reading contemporary concepts into the thought of a past thinker? Of course, it helps us to gain a multi-faceted understanding of the thinker in question, but should this be considered an end in itself, or are there further reasons why we should interpret past thinkers in light of new ideas? While Dr. Uchii focuses on understanding Leibniz's system as an end in itself, I can think of one reason why contemporary readers should be interested, apart from a sort of antiquarianism: we can gain insights relevant to today's science. It is simply amazing how far geniuses like Leibniz are able to probe into the depths of reality, and there is no reason to think that we have fully plundered the repertory of fruitful ideas that these giants have spun out of their immeasurable minds. Relationist and information-based interpretations of quantum theory, relationist dynamics, digital physics, background-independent approaches to quantum gravity: these are all areas where the pioneering work of Leibniz will serve as a guide light. Seeing how his system can be translated into the language of today's science should aid us in reconsidering our deep-seated intuitions of fundamental concepts, such as space and time.
Perhaps I am getting ahead of myself, as this is only Part 1 of Dr. Uchii's paper. His follow-up is eagerly awaited.
P.S. [04/14] My first concern was found to be based on a misreading on my part. Dr. Uchii's claim is not that energy=distance (of course), but that kinetic energy is to be derived by integrating force by path length (which, in the case of a falling body, depends on the square of time). Also, contrary to what I wrote, for Leibniz solicitatio actually is a kind of force.
Deriving Vis Viva from Solicitatio
My first concern deals with a technicality I was unable to understand. In Section 5, where Dr. Uchii discusses Leibniz's distinction between vis viva (living force) and vis mortua (dead force), he takes up the concept of solicitatio, or "infinitely small urge (to motion)." Apparently, solicitatio is an infinitesimal increment of velocity, the accruement of which gives rise to macroscopic acceleration. Now, Dr. Uchii expresses the solicitatio of a falling body by dx, speed by x, and from these, attempts to derive the vis viva (macroscopic acceleration) of the body. For the sake of brevity I shall quote the entire paragraph in question, replacing x and dx with v and dv respectively (pp.16-17):
Accumulation of solicitatio increases the amount of speed, as we have already seen in Figure 2. And the body continues to fall with increasing speed. Thus, the solicitatio needs time to increase the speed, and the body needs time also, for falling with the increased speed (thus, square of time!). So let solicitatio be expressed by infinitesimal dv, speed by v (which increases with time); then, in order to obtain the living force of the body, we have to consider some quantity that depends on the square of time, and that quantity is nothing but the distance the body has fallen; in short, the square of speed, the crucial quantity, is obtained, technically by an integration of product of solicitatio and (infinitesimal) distance. Since Leibniz often ignores constant factor, he uses "v2" for expressing living force, in the letter to de Volder. Of course this corresponds to the kinetic energy of classical mechanics (in the standard notation, mv2/2).In classical mechanics, kinetic energy is derived by integrating the product of applied force F and infinitesimal distance dx (or dot product in the case of two or more dimensions), i.e. by the operation ∫ F • dx. Using p to denote momentum, this can be rewritten thus: ∫ dp/dt • vdt = ∫ v • dp = m ∫ v • dv. We can therefore see that vis viva is none other than the accruement of the product of velocity v and solicitatio dv (though Leibniz writes v2 instead of v2/2). Multiply this by mass m (derivative passive force) and we get kinetic energy mv2/2. Now what I don't understand is why Dr. Uchii writes that "the square of speed, the crucial quantity, is obtained, technically by an integration of product of solicitatio and (infinitesimal) distance." This sounds as though solicitatio were force F rather than infinitesimal increment of velocity dv. And why does he say that the quantity to be derived is "the distance the body has fallen?" Clearly, distance is not energy. What is going on here?
Does Infinite Recursion Give Rise to a Universal Monad?
An important concept in Dr. Uchii's informational interpretation of Leibniz's system is recursion, by which is understood the iteration of a certain program. Of particular significance is the notion of nested recursion, i.e. the application of a program to itself, in such a way as to give rise to a nested hierarchy of programs, "going down from the single dominant program (corresponding to entelechy) to subprograms, which again controls respective subprograms, and ad infinitum" (p.1). According to Dr. Uchii, although Leibniz never used words such as "recursion" or "recursive function," he was well aware of the recursive structure of both the organization of monads and the phenomenal world (p.33). And the function that is iterated in this recursive structure is that which dictates the elastic collision of bodies (pp.43-48).
Now my concern is this: if recursion—the nested hierarchy of programs—is infinite, then how can there be a single dominant program, above which there is no further program controlling it? Just as the microorganisms in the human body may be considered subprograms under the human entelechy, the human entelechy should be a subprogram within the macrocosm of society, which in turn is embraced by the ecosystem, the solar system, and so on ad infinitum. Do each of these comprehensive programs each correspond to an entelechy? If so, then wouldn’t there be an all-embracing entelechy, a kind of world-soul under which all other monads in the universe are subprograms? This seems to be the logical consequence of the view articulated by Dr. Uchii, but is this the view of Leibniz?
On the Fruitfulness of Reading Contemporary Concepts Into the Thought of Past Thinkers
Reading Dr. Uchi's paper, one may be compelled to ask: what is the point of reading contemporary concepts into the thought of a past thinker? Of course, it helps us to gain a multi-faceted understanding of the thinker in question, but should this be considered an end in itself, or are there further reasons why we should interpret past thinkers in light of new ideas? While Dr. Uchii focuses on understanding Leibniz's system as an end in itself, I can think of one reason why contemporary readers should be interested, apart from a sort of antiquarianism: we can gain insights relevant to today's science. It is simply amazing how far geniuses like Leibniz are able to probe into the depths of reality, and there is no reason to think that we have fully plundered the repertory of fruitful ideas that these giants have spun out of their immeasurable minds. Relationist and information-based interpretations of quantum theory, relationist dynamics, digital physics, background-independent approaches to quantum gravity: these are all areas where the pioneering work of Leibniz will serve as a guide light. Seeing how his system can be translated into the language of today's science should aid us in reconsidering our deep-seated intuitions of fundamental concepts, such as space and time.
Perhaps I am getting ahead of myself, as this is only Part 1 of Dr. Uchii's paper. His follow-up is eagerly awaited.
P.S. [04/14] My first concern was found to be based on a misreading on my part. Dr. Uchii's claim is not that energy=distance (of course), but that kinetic energy is to be derived by integrating force by path length (which, in the case of a falling body, depends on the square of time). Also, contrary to what I wrote, for Leibniz solicitatio actually is a kind of force.
2014年4月9日水曜日
【メモ】 Christopher Hookway / The Pragmatic Maxim: Essays on Peirce and Pragmatism
Introduction
"there is no need to justify current beliefs, only changes in belief" (Isaac Levi, "Pragmatism and Change of View" p.179)
Chapter 1: Peirce and Scepticism
[In the nominalist conception of reality] we know about external things through the mediation of what seem to be intuitions, singular representations ("sensations") which are determined by ("directly influenced by") the transcendental object and which provide the sole basis for our thoughts. (...) This view assigns a fundamental role to intuitions: they are our point of contact with (and only source of information about) reality. The method of doubt can then be seen as a device for identifying what is really given to us in intuition; it helps us to free ourselves from those aspects of our perceptual beliefs that are determined by background substantive assumptions rather than being determined by the transcendental object. (p.31)
"This final opinion, then, is independent, not indeed of thought in general, but of all that is arbitrary and individual in thought; is quite independent of how you, or I, or any number of men think." (CP 8.12)
Chapter 3: Truth, Reality, and Convergence
When a proposition is true, "anyone who investigates" is fated to arrive at belief in it. Now "investigates" is a transitive verb. So, anyone who investigates what? When this is spelled out, a variety of distinct theses emerge.
1. If a proposition is true, then anyone who inquires "into the nature of reality" (well enough and long enough) is fated to believe it.
2. If a proposition is true, then anyone who investigates some question to which that proposition provides the answer is fated to believe it.
The "traditional" reading of Peirce, the reading that leads to claim that he is committed to an absolute conception of reality, suggests that he accepts (1): any truth is, in principle, accessible to any inquirer. Position (2) need not lead to such an interpretation of his views. It is compatible with the recognition that a particular inquirer might be fated never to confront a question to which some true proposition provides the answer, perhaps even with the admission that some inquirer could never even understand such a question. (p.56)
Chapter 5: Normative Logic and Psychology
Suppose that I am justified in believing that someone is an extremely successful poker player. When I learn that they make use of a particular strategy of reasoning in the course of their play, then, we might suppose, I have good, albeit defeasible, reason for thinking that the kinds of reasoning they employ are indeed good ones. I would not have an explanation of what their goodness consists in. But I do have strong inductive evidence that the inferences they use can be trusted. It is important that this depends upon the normative premiss that the poker player is successful. The target of Peirce's criticisms is a position where no such normative information is already available. A similar phenomenon occurs when philosophers of science attempt to construct an account of the scientific method through detailed historical studies of scientific practice. We identify norms for scientific practice through (psychological?) descriptions of how science has actually proceeded. Paul Thagard observed that this is not strictly a case of using descriptions of scientific practice in order to determine how scientific inquiry ought to be. The whole inquiry depends upon the selection of examples of good science as subjects of historical inquiry. A crucial normative judgement is already made at the very beginning. (pp.95-96)
→ 「しかし、ある科学がgoodかbadかの判断は、それがevidenceに合致しているかどうかという事実によって決まるのではないか」という疑問に対しては、科学は「evidenceに従うべき」という規範にやはり従っている、と答えることができる。
Chapter 6: "The Form of a Relation"
The major difficulty for the ante rem [structuralist] position is that it needs to explain what the reality of the required structures consists in when there are no systems that exemplify them. (p.125)
Chapter 9: The Principle of Pragmatism
"Pragmatism is the principle that every theoretical judgment expressible in a sentence in the indicative mood is a confused form of thought whose only meaning, if it has any, lies in its tendency to enforce a corresponding practical maxim expressible as a conditional sentence having its apodosis in the imperative mood." (EP 2: pp.134-135)
→ 条件付きでない確率はconfusedであるという論点との繋がり。「確率とは、ある行為（推論の条件）に対して、ある事象（推論の帰結）が成立する相対頻度である」という定式化は、確率概念のpragmatic clarificationに相当すると言える（？）
プラグマティズムの格率の二つの定式化：
(i) If the circumstances are C and you have desires D, then (you ought to) perform an action of type A.
(ii) In circumstances C, if you were to do A, then you would experience E.
次の補助命題によって、(ii)から(i)を導出できる：
(iii) If you desire to experience E, then you can do so by doing A.
"The occasion of such action would be some sensible perception, the motive of it to produce some sensible result." (W3: pp.256-266)
→ 行為の目的とは常に、何らかの感覚可能な結果をもたらすことである。
概念の使用は常に、（たとえ潜在的であれ）その概念に対する何らかの働きかけ（行為）を伴うとすれば、概念使用がpractical bearingsを持たねばならない理由が分かる。(p.177)
"[The end of an explanatory hypothesis] is, through subjection to the test of experiment, to lead to the avoidance of all surprise and to the establishment of a habit of positive expectation that shall not be disappointed." (EP 2: p.235)
[W]hat is required of pragmatic clarifications is that they can be interpreted in this way [in the form of conditional resolutions]; it is not required that they explicitly take this form. (p.180)
→ 実際、定式化(i)を実際のclarificationで適用するのは、想定可能な場面や欲求が多すぎて実質的に不可能。
One of Peirce's strategies for proving pragmatism, was to argue that the pragmatic clarification of a proposition (or concept) would constitute its "ultimate" or "final" logical interpretant. Even if such an interpretant is not useful to us, is not easy to formulate or apply in practice, the fact that the final logical interpretant takes this form is the key to the correctness of pragmatism. So perhaps the final logical interpretant can take the form of a huge set of conditional resolutions, even if our everyday pragmatic clarifications cannot do so. (p.180)
[James's definition of pragmatism] differs from mine only in that he does not restrict the "meaning," as I do, to a habit, but allows percepts, that is, complex feelings endowed with compulsiveness to be such. If he is willing to do this, I do not quite see how he need give any room at all to habit. (CP 5.494)
→ habitでなければindividual reactionsになってしまい、一般的なパターンを確立できない。
Chapter 11: How Peirce Argued for His Pragmatic Maxim
Peirce talks of the logic of judgement and assertion, and compares this with a psychology of belief. We understand judgement as a practice of belief formation which is governed by logical norms; in making a judgement, I commit myself to the correctness of doing so. (p.204)
The mathematician's notations will typically instantiate the abstract structure that she studies. We obtain knowledge of abstract structures by experimenting upon their instantiations. Thus our knowledge of mathematics doesn't provide counterexamples to the pragmatic maxim. (p.218)
"I will say that a sign is anything, of whatsoever mode of being, which mediates between an object and an interpretant; since it is both determined by the object relatively to the interpretant, and determining the interpretant with reference to the object, in such wise as to cause the interpretant to be determined by the object through the mediation of this 'sign'." (EP 2: p.410)
The two italicized phrases can be best understood as follows: the rash is a sign of measles because it has the power to be understood or interpreted in that way; and the sign can be understood as an indication of measles because the object (the measles) has produced the sign. (p.222)
[O]nly a habit can serve as the ultimate interpretant of an intellectual concept. (p.228)
"It can be proved that the only mental effect that can be so produced and that is not a sign but is of a general application is a habit change; meaning by habit change a modification of a person's tendencies toward action, resulting from previous experiences or from previous exertions of his will or acts, or from a complexus of both kinds of cause." (CP 5.475)
"[E]very man exercises more or less control over himself by means of modifying his own habits; and the way in which he goes to work to bring this effect about in those cases in which circumstances will not permit him to practice reiterations of the desired kind of conduct in the outer world shows that he is virtually well-acquainted with the important principle that reiterations in the inner world—fancied reiterations—if well-intensified by direct effort, produce habits, just as do reiterations in the outer world; and these habits will have power to influence actual behaviour in the outer world; especially, if each reiteration be accompanied by a peculiar strong effort that is usually likened to issuing a command to one's future self." (EP 2: p.413)
→ これはnormativityの自然的説明たりうるのではないか。ただし、実際の場面でnormativityを自然的な記述に還元できる、という意味ではない。
"there is no need to justify current beliefs, only changes in belief" (Isaac Levi, "Pragmatism and Change of View" p.179)
Chapter 1: Peirce and Scepticism
[In the nominalist conception of reality] we know about external things through the mediation of what seem to be intuitions, singular representations ("sensations") which are determined by ("directly influenced by") the transcendental object and which provide the sole basis for our thoughts. (...) This view assigns a fundamental role to intuitions: they are our point of contact with (and only source of information about) reality. The method of doubt can then be seen as a device for identifying what is really given to us in intuition; it helps us to free ourselves from those aspects of our perceptual beliefs that are determined by background substantive assumptions rather than being determined by the transcendental object. (p.31)
"This final opinion, then, is independent, not indeed of thought in general, but of all that is arbitrary and individual in thought; is quite independent of how you, or I, or any number of men think." (CP 8.12)
Chapter 3: Truth, Reality, and Convergence
When a proposition is true, "anyone who investigates" is fated to arrive at belief in it. Now "investigates" is a transitive verb. So, anyone who investigates what? When this is spelled out, a variety of distinct theses emerge.
1. If a proposition is true, then anyone who inquires "into the nature of reality" (well enough and long enough) is fated to believe it.
2. If a proposition is true, then anyone who investigates some question to which that proposition provides the answer is fated to believe it.
The "traditional" reading of Peirce, the reading that leads to claim that he is committed to an absolute conception of reality, suggests that he accepts (1): any truth is, in principle, accessible to any inquirer. Position (2) need not lead to such an interpretation of his views. It is compatible with the recognition that a particular inquirer might be fated never to confront a question to which some true proposition provides the answer, perhaps even with the admission that some inquirer could never even understand such a question. (p.56)
Chapter 5: Normative Logic and Psychology
Suppose that I am justified in believing that someone is an extremely successful poker player. When I learn that they make use of a particular strategy of reasoning in the course of their play, then, we might suppose, I have good, albeit defeasible, reason for thinking that the kinds of reasoning they employ are indeed good ones. I would not have an explanation of what their goodness consists in. But I do have strong inductive evidence that the inferences they use can be trusted. It is important that this depends upon the normative premiss that the poker player is successful. The target of Peirce's criticisms is a position where no such normative information is already available. A similar phenomenon occurs when philosophers of science attempt to construct an account of the scientific method through detailed historical studies of scientific practice. We identify norms for scientific practice through (psychological?) descriptions of how science has actually proceeded. Paul Thagard observed that this is not strictly a case of using descriptions of scientific practice in order to determine how scientific inquiry ought to be. The whole inquiry depends upon the selection of examples of good science as subjects of historical inquiry. A crucial normative judgement is already made at the very beginning. (pp.95-96)
→ 「しかし、ある科学がgoodかbadかの判断は、それがevidenceに合致しているかどうかという事実によって決まるのではないか」という疑問に対しては、科学は「evidenceに従うべき」という規範にやはり従っている、と答えることができる。
Chapter 6: "The Form of a Relation"
The major difficulty for the ante rem [structuralist] position is that it needs to explain what the reality of the required structures consists in when there are no systems that exemplify them. (p.125)
Chapter 9: The Principle of Pragmatism
"Pragmatism is the principle that every theoretical judgment expressible in a sentence in the indicative mood is a confused form of thought whose only meaning, if it has any, lies in its tendency to enforce a corresponding practical maxim expressible as a conditional sentence having its apodosis in the imperative mood." (EP 2: pp.134-135)
→ 条件付きでない確率はconfusedであるという論点との繋がり。「確率とは、ある行為（推論の条件）に対して、ある事象（推論の帰結）が成立する相対頻度である」という定式化は、確率概念のpragmatic clarificationに相当すると言える（？）
プラグマティズムの格率の二つの定式化：
(i) If the circumstances are C and you have desires D, then (you ought to) perform an action of type A.
(ii) In circumstances C, if you were to do A, then you would experience E.
次の補助命題によって、(ii)から(i)を導出できる：
(iii) If you desire to experience E, then you can do so by doing A.
"The occasion of such action would be some sensible perception, the motive of it to produce some sensible result." (W3: pp.256-266)
→ 行為の目的とは常に、何らかの感覚可能な結果をもたらすことである。
概念の使用は常に、（たとえ潜在的であれ）その概念に対する何らかの働きかけ（行為）を伴うとすれば、概念使用がpractical bearingsを持たねばならない理由が分かる。(p.177)
"[The end of an explanatory hypothesis] is, through subjection to the test of experiment, to lead to the avoidance of all surprise and to the establishment of a habit of positive expectation that shall not be disappointed." (EP 2: p.235)
[W]hat is required of pragmatic clarifications is that they can be interpreted in this way [in the form of conditional resolutions]; it is not required that they explicitly take this form. (p.180)
→ 実際、定式化(i)を実際のclarificationで適用するのは、想定可能な場面や欲求が多すぎて実質的に不可能。
One of Peirce's strategies for proving pragmatism, was to argue that the pragmatic clarification of a proposition (or concept) would constitute its "ultimate" or "final" logical interpretant. Even if such an interpretant is not useful to us, is not easy to formulate or apply in practice, the fact that the final logical interpretant takes this form is the key to the correctness of pragmatism. So perhaps the final logical interpretant can take the form of a huge set of conditional resolutions, even if our everyday pragmatic clarifications cannot do so. (p.180)
[James's definition of pragmatism] differs from mine only in that he does not restrict the "meaning," as I do, to a habit, but allows percepts, that is, complex feelings endowed with compulsiveness to be such. If he is willing to do this, I do not quite see how he need give any room at all to habit. (CP 5.494)
→ habitでなければindividual reactionsになってしまい、一般的なパターンを確立できない。
Chapter 11: How Peirce Argued for His Pragmatic Maxim
Peirce talks of the logic of judgement and assertion, and compares this with a psychology of belief. We understand judgement as a practice of belief formation which is governed by logical norms; in making a judgement, I commit myself to the correctness of doing so. (p.204)
The mathematician's notations will typically instantiate the abstract structure that she studies. We obtain knowledge of abstract structures by experimenting upon their instantiations. Thus our knowledge of mathematics doesn't provide counterexamples to the pragmatic maxim. (p.218)
"I will say that a sign is anything, of whatsoever mode of being, which mediates between an object and an interpretant; since it is both determined by the object relatively to the interpretant, and determining the interpretant with reference to the object, in such wise as to cause the interpretant to be determined by the object through the mediation of this 'sign'." (EP 2: p.410)
The two italicized phrases can be best understood as follows: the rash is a sign of measles because it has the power to be understood or interpreted in that way; and the sign can be understood as an indication of measles because the object (the measles) has produced the sign. (p.222)
[O]nly a habit can serve as the ultimate interpretant of an intellectual concept. (p.228)
"It can be proved that the only mental effect that can be so produced and that is not a sign but is of a general application is a habit change; meaning by habit change a modification of a person's tendencies toward action, resulting from previous experiences or from previous exertions of his will or acts, or from a complexus of both kinds of cause." (CP 5.475)
"[E]very man exercises more or less control over himself by means of modifying his own habits; and the way in which he goes to work to bring this effect about in those cases in which circumstances will not permit him to practice reiterations of the desired kind of conduct in the outer world shows that he is virtually well-acquainted with the important principle that reiterations in the inner world—fancied reiterations—if well-intensified by direct effort, produce habits, just as do reiterations in the outer world; and these habits will have power to influence actual behaviour in the outer world; especially, if each reiteration be accompanied by a peculiar strong effort that is usually likened to issuing a command to one's future self." (EP 2: p.413)
→ これはnormativityの自然的説明たりうるのではないか。ただし、実際の場面でnormativityを自然的な記述に還元できる、という意味ではない。
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