微分された世界 | Differential dharma

物理学者・苑田尚之は「数式は言語である」と言った。もし物理現象を数式なしに日常言語だけで述べようとしたら、長さだけでも100倍は下回らないだろう。複雑かつ膨大な説明を 1/100 以下に圧縮・軽量化してしまう数学という言語を、使わない手はない。



収縮している宇宙があるとしよう。その宇宙には中心 (O) があって、恒星も惑星も微小な粒子も、すべての天体が O に向かって収縮している。それは次の微分方程式で記述されるものとする:

x” = – k x                                (1)

意味はこうだ。中心 O から距離 x 離れた天体が、その距離 x に比例する大きさの加速度 x” をもって O に向かっている。中心から離れているところほど、大きな加速度がかかってこの宇宙は収縮しているというわけだ(x の前にマイナスが付いているのは、距離とは逆向きに加速度がかかるから)。k は比例定数。話を簡単にするために、 k = 1 としておこう。

微分方程式 (1) を「解く」とは、(1) を満たす x を時間 t の関数 x = f(t) として求めることだ。= 0 のときの天体の位置 f (0) と初速 f’ (0) –– これを「初期条件」という –– が指定されれば (1) は解かれる。たとえば初期条件を

(0) = 0,   f’ (0) = 1                       (2)


x = sin t                     (3)




As Prof Naoyuki Sonoda of Tōshin Highschool once said, mathematics is a language; a linguistic system that has a nice ability to express lengthy and complicated things in a handful number of equations. Why shouldn’t buddhist scholars use it?

Long ago there was a king in the east India who admired dharma. One day he invited number of monks to his luncheon, where they read a few selected passages from sutra for the king. But Prajnatara did not do that. The king questioned him,

“Why don’t you read sutra for me?”

Prajnatara said,

“I’m reading the sutra as it is, away from conditions and substances, . It contains countless volumes, not just a few passages.”

While it will never be an easy practice to be away from of conditions and substances of the world, it will be even more difficult to express it in words because words belongs to the conditioned world. So let me ask Isaac Newton.

Imagine a contracting cosmos, in which all the celestial bodies including planets, meteorites and particles, are all moving toward its center ‘O’ following the differential equation:

x” = – k x                                (1)

where x denotes the distance between O and the celestial object and x” refers to its magnitude of acceleration. k is a constant. For simplicity, let k be just 1. The equation (1) states that the cosmos is contracting in such a way that the further the object is from O, the greater its acceleration toward O is.

Given a initial condition, say, the object’s position = (t) is 0 and its velocity x’ = f’ (t) is 1 at time = 0:

(0) = 0,   f’ (0) = 1                       (2)

then the equation (1) will be solved (without proof for now) as:

x = sin t                     (3)

So I need some alteration; the cosmos not only contracts but expands in some phases. It actually oscillates between contraction and expansion.

Notice the analogy; for a given set of initial conditions, there is a unique solution to the differential equation. Likewise, for a given collection of substances and conditions, dharma realizes into a few particular passages of sutra. While the differential equation itself describes the universal aspect of the cosmos’ motion irrespective of the initial conditions, Prajnatara reads the sutra of numberless volumes without referring to any particular conditions.

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