* IV Science and Mathematics 四 科学与数学 89

* IV Science and Mathematics 四 科学与数学  89
** Product and Process 1-58 产品和制造产品的方法

Every day, every hour, every minute,
we feel the accelerating impact of science on our lives.
Science has a hand in nearly everything which affects us—
our food, clothing, shelter, job, amusements, and cultural pursuits.
The very air we breathe has been changed
by the combustion engine and by nuclear fission.
Our whole environment is constantly being altered.

现代人每天，每小时，每分钟都会感受到科学对其生活不断增强的影响。
科学插手于影响现代人的几乎每一件东西和事情–
衣，食，住，工作，娱乐以及文化追求。
即使是现代人呼吸的空气，也已被内燃机和核裂变所改变。
现代人的生活环境正在被不断地改变。

Should we not, then, try to understand science? Why?
Why should we not be content, rather,
simply to read the instructions on the outside of the package,
and forget about the ideas, the techniques,
and the processes that went into the making of the product?
Are not the findings, after all, the only things that need concern us?
We read a poem written two thousand years ago
because it still has something to tell us.
But science is progressive and rapidly outdated;
why read about ideas
that have been superseded by two thousand years, or even two years?
And if science is a mystery story,
why not just read the last page?

我们因此而应该去努力理解科学。
为什么呢？
我们为什么不应只是满足于阅读包装上的说明，
不应忘掉制造产品的那些思想，技术和工艺呢？
与我们有关的不只是结果和结论吗？
我们之所以阅读两千年前写的一首诗，
是因为它仍可以告诉我们一些东西。
但科学却是不断前进和迅速更新的；
为什么我们要去了解已经过时的两千年前甚或两年前的思想呢？
如果科学是一部侦探小说的话，为什么不只是读最后一页呢？

But the last page has not yet been written,
the mystery is still not solved.
Perhaps it will never be;
and if it is, it may turn out that the clues were in the first chapter.
And in all the chapters, from the first to the latest,
there is the story of one of the most important human activities:
the struggle to comprehend the universe.

但最后一页尚未写出来，秘密尚未揭开。
也许秘密永远也揭不开。
假如能揭开的话，线索也很可能就在第一章。
而且在所有篇章里，从第一章至最后一章，
讲的都是人类最重要的一种活动，即人类为理解宇宙而进行的斗争。

Humans are question-asking animals,
and our first questions are about the world we see around us,
the “world of science.”
We wonder, and our wonder will not let us rest.
We feel that we are growing when we are inquiring and investigating
and only when we are inquiring and investigating.
We want to know—and always to push beyond the known.
We are a “natural-born” frontier people.
Let us discover a new world,
and we push on to a new planet, a new universe.
To ask is to be excited.
To be excited is to live.

人是好问的动物，他最先问的，便是有关他所看到的周围世界的问题，
而这个世界就是“科学世界”。
他惊奇，这种惊奇使他不得安宁。
他感到，在探索和研究时他在长大，而且只有在探索和研究时才长大。
他想了解事物–而且总是把目光投向未知的事物。
他是“天生的”拓荒者。
假如他发现了一个新的世界，他会把目光投向一个新的星球，一个新的宇宙。
提出问题可以使人感到激动，而感到激动便是生活。

True, this curiosity is not evenly distributed among all persons.
All of us possess it in insatiable quantity at birth.
Unfortunately, most of us lose it, or stifle it, as we grow older.
A few men and women retain it, and are activated by it, all their lives.
They become the persistently, professionally curious
whose curiosity changes the world.
This is the fraternity to which the great scientists—
like the great poets and philosophers—belong.

固然，这种好奇心并不是平均地分配给所有人的。
我们生来都具有好奇心，渴望了解世界。
不幸的是，大多数人随着年龄的增长都丧失或扼杀了好奇心。
只有少数人保留了好奇心，在整个一生中被好奇心所激励。
他们成了百折不挠的职业好奇者，他们的好奇心改变了世界。
大科学家–以及大诗人和大哲学家–就属于这样的人。

The rest of us need not be content with only the end product of science.
We may all know the why and the how.
True, the terminology of science and technology is “another language,”
but it is a language which we can comprehend,
above all when we tackle the original masterpieces—
the exposition of the fundamental discoveries
upon which the highly technical applications are all based.
And our comprehension is facilitated by the fact
that many such works are expounding theories
which, once strange and spectacular,
have long since become familiar to us.

其余的人不应仅仅满足于得到科学的最终产品。
我们都可以对科学原理和科学方法有所了解。
固然，科学技术术语是“另一种语言”，
但却是一种我们能够理解的语言，
尤其是当我们阅读富于创造性的名作时，更是如此。
这种著作阐述的是最基本的发现，高度技术性的应用依据的都是这种发现。
而且，许多这类著作论述的理论虽然曾经是令人不可思议的，轰动一时的，
但现在已早就为我们所熟悉了，这也有助于我们理解这些著作。

It is in the presentation of a new concept
that scientists are most precise, clear, and simple.
They are not, at this point,
speaking to specialists and technicians—this will come later—
and they are not talking shop with colleagues.
At the great turning points in the advance of science
there is no inside group waiting to receive the word
in language that only it can understand.
Here the scientists come closest to speaking to everyone;
it is here that they are most eloquent and lucid.

正是在表述新的概念时，
科学家所使用的语言最为精确，清晰和简明。
此时，他不是在对专家，技术人员说话（对他们说话是以后的事），
也不是在与同行谈论纯技术问题。
在科学技术进步的各各伟大的转折点，
并没有哪一行业的人等待着聆听只有他们才懂的话语。
此时，科学家最接近于对每一个人说话，
正是在此时，他说的话最有说服力，也最为清楚明白。

A scientific theory is like a shelter against chaos.
It is designed to hold in
some sort of order the known facts of the universe.
New facts require a little juggling of the old,
the addition of a new wing or second story;
some sections may have to be torn down and rebuilt.
Increasingly inadequate, the structure stands, representing a world view,
a comfort to us who cannot rest
until we perceive that nature has a pattern.

科学理论如同躲避混乱的庇护所，
力图按照某种顺序排列宇宙中的已知事实。
新事实必然会使旧理论有所变动，需要盖新的侧厅，加高楼层；
某些部分要拆掉重建。
虽然愈来愈不适应需要，但建筑物依然存在，
这代表了一种世界观，对人是一种安慰，
人在感到自然界具有某种格局以前是不会安静下来的。

Then, one day, there is a crisis.
The foundation begins to crumble.
A hypothesis that has been brought in to “save the appearances”
has become an embarrassment, and is now a skeleton in the closet.
But most pressing of all,
new knowledge has been acquired that does not fit in anywhere.
It is time for a move.

于是，有一天，爆发了危机。
地基开始动摇。
曾经用来“保全面子”的假说，
现在反而让人感到难堪，成了不可外扬的家丑。
但最为紧迫的，是获得新知识放在哪里都不合适。
是采取行动的时候了。

Rescuing materials that will be useful in building anew,
the scientist starts all over again to build a shelter;
never entirely from scratch,
for something is always salvaged, often a great deal.
It is in the salvaged material that the continuity of science exists.
There are no complete breaks in the story.
There are dormant periods, blind alleys, revolutions, and crises,
but no breaks.

科学家一边抢救可以用于重建的材料，
一边着手重建庇护所。
绝不是从零开始，
因为科学家总是设法抢救出一些东西，
常常是设法抢救出许多东西。
正是这种被抢救出来的材料，使科学得以保持连续性。
故事并未完全中断。
虽然有休眠期，有死胡同，有革命，有危机，但却没有中断。

Sometimes the new structure can be built
only because the framework had been created decades,
or even centuries, before by mathematicians,
who, pursuing mathematical knowledge for its own sake,
foresaw no possible applications of their formulas to science.
The idea of conics was for more than 1,500 years
a purely intellectual exercise.
Euclid, Archimedes, and Apollonius
investigated the mathematical properties of these curves,
which were not then known in nature.
Then, early in the seventeenth century,
when Kepler discovered that planets move in elliptical orbits,
the language in which to express his theory was found in conics.

有时之所以能盖起新的建筑物，
只是因为数学家在几十年前甚至数百年前为其竖好了框架，
而他们完全是为了数学知识而追求数学知识，
并未预见到他们的公式有可能应用于科学。
二次曲线这一概念连续1500多年纯粹是一种智力训练。
欧几里德，阿基米德和阿波罗尼奥斯都考察了这种曲线的数学性质，
尽管当时人们并不知道在自然界中存在着这种曲线。
后来，在17世纪初，开普勒发现，行星是沿椭圆轨道运行的，
此时二次曲线就成了表达这一理论的语言。

The story of science could be told in a series of such crises.
“There are no eternal theories in science,”
Albert Einstein and Leopold Infeld write in
“/The Rise and Decline of Classical Physics/” (Vol.8).
“It always happens that some of the facts
predicted by a theory are disproved by experiment.
Every theory has its period of gradual development and triumph,
after which it may experience a rapid decline… .
Nearly every great advance in science
arises from a crisis in the old theory,
through an endeavor to find a way
out of the difficulties created.
We must examine old ideas, old theories,
although they belong to the past,
for this is the only way
to understand the importance of the new ones
and the extent of their validity.”

科学的历史就是由一系列这样的危机构成的。
“并没有永恒的科学理论”，
阿尔伯特.爱因斯坦和列奥波德.英费尔德在
《古典物理学的兴衰》一文中这样写道。（见本套书中译本第7卷）
“一种理论所预言的事实中总有一些被实验所否定。
每种理论都有其逐渐发展并取得胜利的时期，此后，它便会迅速衰落。
……科学的几乎每一次伟大进步都起因于旧理论发生的危机，
依赖于人们为摆脱困境所作的努力。
我们必须考察旧概念，旧理论，尽管它们属于过去，
因为唯有此，我们才能理解新理论的重要性和有效程度。”

The establishment and development
in the seventeenth century of classical physics,
to which Einstein and Infeld refer,
represented just such a crisis—a break with the past.
It began, as so often happens,
with a revolution, the “Copernican revolution.”
The new structure was imposing, and the architects were giants.
Copernicus, Galileo, Kepler, and Newton
outlined the material framework for the world
in which we lived from the second half of the seventeenth century
to the end of the nineteenth.

正如爱因斯坦和英费尔德所说，
古典物理学在17世纪的确立与发展，就代表了这样一种危机，一种与过去的决裂。
同经常发生的情况一样，经典物理学的确立与发展开始于一场革命，
即哥白尼革命。
新的建筑物辉煌壮丽，建筑师们都是顶天立地的巨人。
哥白尼，伽利略，开普勒和牛顿
勾勒出了人类从17世纪后半叶至19世纪末生活于其中的世界的物质框架。

The Greek astronomer Ptolemy was a great synthesizer of knowledge,
and his system of the world also represented a break with the past.
Before Ptolemy, astronomy consisted largely of a mass
of more or less unconnected observations
of the movements of the heavenly bodies.
Ptolemy’s theory imposed a unity upon these masses of data
and was adequate to the phenomena then observable.
The shift from Ptolemy, who believed that the earth was the center of the universe,
to Copernicus, who offered the simpler hypothesis that the sun was at the center,
and therefore that the earth was only a small part of a vast cosmic arrangement,
was a shock to the imagination.
The trust that one placed in one’s senses seemed to be undermined;
things were not as they appeared to be.
Still worse, the belief that the universe had been created especially for mankind
seemed to be shaken by the new theory.

希腊天文学家托勒密是知识的伟大综合者，
他的世界体系也是一种与过去的决裂。
在托勒密之前，天文学主要是由一大堆天体运行观察资料构成的，
这些资料相互之间没有多少联系。
托勒密的理论赋予这些资料以一种统一性，
足以解释当时所能观察到的现象。
托勒密认为，地球是宇宙的中心。
而哥白尼则提出了一种更为简单的假说，即：太阳是宇宙的中心，
因而地球只不过是广漠宇宙的一个很小组成部分。
从托勒密到哥白尼，这一转变强烈地冲击了人的想象力。
人对感觉抱有的信心似乎遭到了破坏；
事物并不是像表面所看到的那种样子。
更为糟糕的是，新理论似乎动摇了宇宙是专门为人创造的这一信念。

Sigmund Freud characterizes this change
as the first “great outrage” upon humanity’s
“naïve self-love … when it realized
that our earth was not the centre of the universe,
but only a tiny speck in a world-system
of a magnitude hardly conceivable;
this is associated in our minds with the name of Copernicus …”
(/A General Introduction to Psycho-Analysis/, in GBWW, Vol.54).
This “outrage” expressed itself in the form of fear and resistance.
Sometimes it grew so strong and so bitter
that it burst forth as outright persecution.
Galileo was a brilliant scientist,
whose monumental contributions in the fields of astronomy and physics
had kept the Copernican revolution surging ever forward.
His investigations were so disturbing
that he was finally accused of defying religious authority.

西格蒙德.弗洛伊德称这种转变为
“对人类的朴素的自恋的第一次巨大伤害……
当时人们终于认识到，我们的地球并不是宇宙的中心，
而只不过是广大的几乎不可想象的宇宙体系中的一颗小微粒；
在我们的头脑中，这种转变是同哥白尼的名字联系在一起的……”
（《精神分析引论》，见《西方世界名著》第54卷，第56c-d页）。
由此而产生的“愤怒”表现为恐惧和反抗。
有时这种愤怒变得非常强烈和刻毒，竟爆发为赤裸裸的迫害。
伽利略是位杰出的科学家，他对天文学和物理学做出的不朽贡献，
使哥白尼革命得以不断向前发展。
他所从事的研究工作使人感到如此不安，
以致他最终被安上了不敬神的罪名。

Tommaso Companella, in his seventeenth-century defense of Galileo,
described the objections to Galileo in the following manner:
“Holy Scripture counsels us to
‘seek nothing higher,
nor attempt to know more than it is necessary to know’;
that we ‘leap not over the bounds which the fathers set’;
and that ‘the diligent searcher of majesty is overcome by vain-glory.’
Galileo disregards this counsel, subjects the heavens to his invention,
and constructs the whole fabric of the world according to his pleasure.
Cato rightly taught us to
‘leave secret things to God, and to permit Heaven to inquire concerning them;
for he who is mortal should concern himself with mortal things’”
(“Arguments for and against Galileo,” Vol.8).
But to Galileo, subjecting the heavens to the “spyglass” he had invented,
it seemed “a matter of no small importance
to have ended the dispute about the milky way
by making its nature manifest
to the very senses as well as to the intellect”
(/The Starry Messenger/, Vol.8).

托马索.康帕内拉在17世纪为伽利略辩护时，
以下述方式叙述了人们反对伽利略的理由：
“《圣经》规劝我们‘不要追求更高尚的东西，也不要力求知道不必要知道的事情’；
规劝我们‘不要逾越教父规定的界限’
‘执意探求尊严的人会被虚荣心所压倒’。
伽利略不理睬这种规劝，用自己发明的东西研究天体，
随心所欲地建立有关整个宇宙的结构。
加图正确地告诫我们，
‘把神秘的事情留给上帝，让上帝去探究它们；
因为会死的人只应关心会死的事物’”
（《赞成与反对伽利略的理由》，见本套书中译本第7卷）。
但对伽利略来说，用他所发明的“小望远镜”来研究天体，
“使感觉和理智都明了银河的性质，从而结束有关银河的争论，
似乎并不是无足轻重的事情”（《星际使者》，见本套书中译本第7卷）。

** “To Venture Forth” 1-61  “奋勇向前”  93

The human spirit is resilient,
and gains courage from knowledge.
Though Pascal says that he is frightened when he sees himself
“engulfed in the infinite immensity of spaces
of which I am ignorant and which know me not,”
he also writes:
“It is not from space that I must seek my dignity,
but from the government of my thought.
I shall have no more if I possess worlds.
By space the universe encompasses and swallows me up like an atom;
by thought I comprehend the world”
(/Pensées/, in GBWW, Vol.30).
Mankind slowly learned to accept the continuing questioning of science,
even when it produced profoundly disquieting and bewildering results.

人类的心灵是富于弹性的，可以从知识中汲取勇气。
帕斯卡说，当他看到自己
“被无限广大的空间所吞没，而自己对他一无所知，他也不知道我时”，
他感到一阵颤栗，但他还写道：
“我不应从空间寻求尊严，而应从对思想的把握寻求尊严。
假如我拥有宇宙，我也就一无所有了。
我像一颗原子那样被宇宙用空间所包围和吞没；
但我却能够用思想了解宇宙”
（《思想录》，见《西方世界名著》，第12卷，第297a页）。
人类慢慢学会了接受科学的不断提问，
尽管这种提问产生了一些令人深深地感到不安和困惑的结果。

Having once and for all been thrown out of the center of the universe,
humankind lost a throne, but achieved mobility.
The modern conception of our status,
located somewhere between the edge of infinity in one direction
and the nucleus of the atom in the other,
is presented pictorially in /Cosmic View/, by Kees Boeke (Vol.8).
The ability to measure ourselves in space is scientific and modern;
the insight that bridges distances and dispels the terror of immensity
is philosophical and ancient.
Centuries ago, Marcus Aurelius wrote:
“Whether the universe is a concourse of atoms, or nature is a system,
let this first be established,
that I am a part of the whole which is governed by nature …”
(Meditations, in GBWW, Vol.11).

人类虽然被永远驱逐出了宇宙中心，丧失了王位，
但却由此获得了能动性。
人的位置从一个方向说是无限的边缘，
从另一个方向说是原子核，
人就处于这两者之间的某个地方，
这种有关人的现代观念
凯斯.博克在《宇宙观》一文中用画图做了说明。（见本套书中译本第7卷）
测量人在空间的位置的能力来自现代科学，
而跨越距离，消除对无限的恐惧的顿悟力则来自于古代哲学。
很久以前，马可.奥勒利乌斯就写道：
“宇宙是原子的汇集也罢，大自然是一个体系也罢，
首先应明确这一点，即我们是受大自然支配的那一整体的一个组成部分……”
（《沉思录》，见《西方世界名著》，第12卷，第297a页）。

Modern scientists claim the
“right to venture forth in the world of ideas [as well as]
to extend … horizons in the physical universe”
(/Beyond the Googol/, Vol.9).
This right has been won.
And the world of ideas includes such once “dangerous” concepts
as the idea of the end of the world.
But nowadays, even such a prediction of ultimate natural catastrophe
arouses no condemnation and produces no martyrs of science.
It is neither delivered, nor received, with terror or anguish.
Arthur Eddington, for instance,
sees the universe as inexorably running down,
with no possibility of its being wound up again to repeat the cycle.
So be it, he says. He would rather
“that the universe should accomplish some great scheme of evolution and,
having achieved whatever may be achieved,
lapse back into chaotic changelessness,
than that its purpose should be banalised by continual repetition.
I am an Evolutionist, not a Multiplicationist”
(/The Running-Down of the Universe/, Vol.8).

现代科学家不仅要求享有扩展……物质世界范围的权利，
而且要求“在思想领域享有奋勇向前的权利”
（《超越10的100次方》，见本套书中译本第8卷）。
他们已经赢得了这一权利。
思想领域中的一些思想，
如有关世界将归于毁灭的思想曾被认为是“危险的”。
但现在，即使是预言最终将发生巨大的自然灾难，也不会遭到谴责，
不会使作这一预言的科学家为此而殉难。
科学家无论在提出还是在接受这一预言时，都不会感到恐惧或痛苦。
譬如，阿瑟.爱丁顿认为，
宇宙正在无可挽回地衰竭，不可能得到恢复，再作一次循环。
他说，情况就是这样。他宁愿
“宇宙完成某一伟大的进化目标，
在达到这一目标之后，重新陷于不变的混沌状态，
而不愿宇宙的目标被无休止地重复，弄得庸俗不堪。
我是‘进化论者’，而不是‘反复论者’……”
（《宇宙的耗散》，见本套书中译本第7卷）。

Modern scientific theory is flexible,
making no claim to being the last word,
or even, except provisionally, the right word.
Modern science says: This is the way we describe things—today.
Copernicus upset a world that had been saying:
This is the way things are— forever.
The world that he upset was the world
that had been ruled by the thought of Aristotle for over 1,500 years.

现代科学理论是灵活的，并不声称是最后的定论，
甚至也不声称是永远正确的。
现代科学只是说：这就是我们现在描述事物的方式。
哥白尼所推翻的世界则一直在说：这就是事物永远存在的方式。
他推翻的是被亚里士多德的思想统治了1500多年的世界。

Since the beginning of the seventeenth century,
almost every scientific advance
has had to begin with a refutation of some Aristotelian doctrine.
Why is Aristotle now regarded as
having long imposed a barrier to scientific progress?
Is it because, as the /Syntopicon/ suggests,
he relied too strongly on his senses?
“Just as Ptolemy’s astronomy conforms to what we see as we look at the heavens,
so Aristotle’s physics
represents a too simple conformity with everyday sense-experience.
We observe fire rising and stones falling.
Mix earth, air, and water in a closed container,
and air bubbles will rise to the top,
while the particles of earth will sink to the bottom.
To cover a multitude of similar observations,
Aristotle develops the theory of the natural motions
and places of the four terrestrial elements—
earth, air, fire, and water.
Since bodies move naturally only to attain their proper places,
the great body which is the earth, already at the bottom of all things,
need not move at all.
Being in its proper place, it is by nature stationary”
(GBWW, Vol.1).

自17世纪初以来，几乎每一次科学进步都不得不以驳斥亚里士多德的某一理论而开始。
人们现在为何认为亚里士多德长久以来阻碍了科学的进步？
是否像《同题文集》指出的那样，他过于依靠自己的感觉了？
“正如托勒密的天文学符合于我们仰望天空所看到的情景那样，
亚里士多德的物理学也过于简单地符合日常的感觉经验。
我们看到火向上升，石头往下落。
把土，气和水混合在一个封闭的容器内，则气泡会升至容器的顶部，而土粒则会沉在容器的底部。
为了概括许许多多次这样的观察，亚里士多德提出了有关地球的四种原素，
即土，气，火和水的自然运动和位置的理论。
既然物体移动只是为了获得其适当位置，
所以地球这一巨大物体已处于所有东西之下，无需再移动了。
由于已处于适当位置，因而地球从性质上说是静止的”
（见《西方世界名著》，第2卷，第92b页）。

The fate that Aristotle suffered at the hands of his successors,
who took his tentative conclusions
as a rounded body of complete knowledge
rather than as incentives to inquiry and further discovery,
indicates that science is the making of knowledge,
and not knowledge as such.
And in the making of knowledge,
modern science uses two tools that Aristotle did not use—
experiment and mathematics.

亚里士多德的后继者们
把亚里士多德得出的暂时性结论看作是一完整的知识体系。
亚里士多德在其后继者手中遭受的这种命运告诉我们，
科学是知识的发展过程，而不是知识本身。
在知识的发展过程中，现代科学使用了两件亚里士多德不曾使用的工具，
即实验与数学。

** Experiment and Observation 1-63  实验与观察  95

While the scientific method employs both experiment and observation,
it is clear that not all branches of science are experimental.
As the /Syntopicon/ points out:
“It is not always possible for the scientist to perform experiments,
as, for example, in astronomy,
where the phenomena can be methodically observed and exactly recorded,
but cannot be manipulated or controlled.
Among the great books of natural science,
the biological writings of Hippocrates, Aristotle, Galen, and Darwin,
the astronomical works of Ptolemy, Copernicus, Kepler, and Newton,
and the clinical studies of Freud are examples of scientific
works which are more or less empirical,
but not experimental”
(GBWW, Vol.1).

虽然科学方法要运用实验和观察这两者，
但很显然，并非所有的科学分支都可以进行实验。
正如《同题文集》所指出的：
“科学家并非总是能进行实验，
例如在天文学中，虽然从方法论上说可以观察到并精确地记录下天文现象，
但却无法操纵或控制天文现象。
在自然科学的伟大著作中，
希波克拉底，亚里士多德，加伦和达尔文的生物学著作，
托勒密，哥白尼，开普勒和牛顿的天文学著作，
以及弗洛伊德的临床研究报告，都是科学研究工作的典范，
但它们虽然是经验性的，却不是实验性的”
（见《西方世界名著》第2卷，第474b页）。

Due, however,
to the great strides made by physics and chemistry
from the middle of the nineteenth century,
experiment has gained so much prestige
that it has been hailed by some as the only method
of obtaining knowledge of any kind, scientific or otherwise.
Furthermore, it has been said
that questions which cannot be answered by scientific methods
are not only unanswerable:
they should not even be asked.

然而，由于物理学和化学自19世纪中期以来取得了飞速发展，
实验赢得了极大声誉，
以致有人把实验称为获取所有知识的唯一方法，
无论是科学知识还是其他方面的知识。
而且有人说，大凡科学方法不能回答的问题，
不仅无法回答，甚至根本就不应该提出来。

In /The Great Conversation/
(a book that accompanies /Great Books of the Western World/),
we find a discussion of the question
whether the scientific method can be extended beyond the natural sciences.
“Consider, for example,
statements about God’s existence or the immortality of the soul.
These are answers to questions that cannot be answered—
one way or the other—by the experimental method.
If that is the only method
by which probable and verifiable knowledge is attainable,
we are debarred from having knowledge
about God’s existence or the immortality of the soul.
If modern man, accepting the view
that he can claim to know only what can be demonstrated by experiment
or verified by empirical research,
still wishes to believe in these things,
he must acknowledge that he does so
by religious faith or by the exercise of his will to believe;
and he must be prepared
to be regarded in certain quarters as hopelessly superstitious.”

在《西方世界名著》第一卷中，《伟大的对话》一文讨论了这样一个问题，
即科学方法的应用范围能否超出自然科学。
“例如，让我们来看关于上帝存在和灵魂不朽的说法。
这类说法回答的是用实验方法无论如何都无法回答的问题。
如果只有通过实验才能获得可检验的知识，
我们也就无法得到有关上帝存在或灵魂不朽的知识。
如果现代人承认
自己只能知道被实验所演示或被经验研究所检验的东西，
而又想相信上帝的存在和灵魂的不朽，
那他就必须承认，
他只能通过宗教信仰或通过运用信仰的意志来做到这一点；
而且他必须准备被某些人看作是迷信透顶的人。”

The author of /The Great Conversation/ goes on
to consider what is meant by the scientific method.
“If all that is meant is that a scientist is honest and careful and precise,
and that he weighs all the evidence with discrimination
before he pronounces judgment,
then we can agree that the scientific method
is the only method of reaching and testing the truth in any field.”
But, he adds, these methods
have been used by historians, philosophers, and theologians
since the beginning of time,
and it is misleading to name after science a method used in so many fields.

《伟大的对话》一文的作者接下来考察了科学方法的含义。
“如果科学方法只不过意味着科学家是诚实的，是仔细而讲究精确的，
他公平地权衡了所有证据以后才下判断，
那我们大家便会同意，
科学方法无论在哪一个领域都是获得和检验真理的唯一方法。”
但是，他接着说，这种方法，
历史学家，哲学家和神学家自有史以来一直在使用，
因而把这么多领域使用的一种方法命名为科学方法，
是会使人产生误解的。

Does the scientific method mean the observing and collecting of facts?
Though facts are indispensable, they are not sufficient.
“To solve a problem it is necessary to think.
It is necessary to think even to decide what facts to collect.
Even the experimental scientist cannot avoid being a liberal artist,
and the best of them, as the great books show,
are men of imagination and of theory
as well as patient observers of particular facts.”

那么，科学方法是否指的是观察和搜集事实呢？
事实是必不可少的，但光有事实还不够。
“要解决一个问题，还必须进行思考。
甚至在决定搜集那些事实时，也得进行思考。
即便是搞实验的科学家也不可避免地是自由的艺术家，
而最杰出的科学家，正如他们的伟大著作所表明的，
不仅是特定事实的耐心观察者，
而且还是富于想象力的人，善于建立理论的人。”

Whatever is meant by scientific method, the writer continues,
“the issue remains whether the method associated with experimental science,
as that has developed in modern times,
is the only method of seeking the truth
about what really exists or about what men and societies should do.”

《伟大的对话》一文的作者继续写道，不管科学方法的含义是什么，
“都依然存在着这样的问题，即：
是否只有依靠当今那种与实验科学相联系的方法，
才能寻求到有关实际存在着什么或人和社会应该做什么的真理”
（见《西方世界名著》，第1卷，第33-37页各处）。

Questions raised by the experimental method
are not confined to its use outside of natural science.
There is also a feeling that even within certain branches of science,
it is no longer sufficient unto itself.
This is the attitude of Max Planck,
one of the giants of contemporary physics.
(He is the “father” of quantum mechanics.)
“Scientists,” says Planck,
“have learned that the starting point of their investigations
does not lie solely in the perceptions of the senses,
and that science cannot exist without some small portion of metaphysics.
Modern physics impresses us particularly
with the truth of the old doctrine
which teaches that there are realities existing apart from our sense perceptions,
and that there are problems and conflicts
where these realities are of greater value for us
than the richest treasures of the world of experience”
(/The Universe in the Light of Modern Physics/,
reprinted in /The Great Ideas Today/ 1962).

实验方法提出的问题，
并非仅仅限于
该方法能否在自然科学以外的领域加以应用这一个问题。
人们还感到，即使是在自然科学的某些分支内，
光有实验方法，也不够了。
当代物理学巨匠之一，量子力学之“父”马克斯.普朗克
就持有这种观点。他说，
“科学家已认识到，他们研究工作的起点并非仅仅是感觉，
科学若没有一点形而上学的成分是无法存在的。
现代物理学使我们特别强烈地感受
到这样一种古老的说法是有道理的，
即在我们的感觉之外，还有一些实在，
对于我们来说，在解决一些问题和矛盾时，
这些实在要比经验世界最为丰富的宝藏具有更大的价值”
（《现代物理学眼光中的宇宙》，重印于《当今的伟大思想》，1962年，第521页）。

All through the great books,
we find examples of the authors’ efforts
to seek the truth about the world and themselves,
and the boundary line
between the scientific and the nonscientific method is not always clear.
Consider, for instance, the collecting of facts.
Since facts are beyond number,
and since a scientist must make a judgment
about which facts to consider,
on what basis is this judgment made?
It is the aesthetic sensibility that chooses, says Henri Poincaré.
“It may be surprising to see
emotional sensibility invoked *à propos* of mathematical demonstrations which,
it would seem, can interest only the intellect.
This would be to forget the feeling of mathematical beauty,
of the harmony of numbers and forms,
of geometric elegance.
This is a true aesthetic feeling that all real mathematicians know,
and surely it belongs to emotional sensibility”
(/Mathematical Creation/, Vol.9).

《西方世界名著》中的每一部著作，
都堪称人类努力探求有关世界和人类自身的真理的典范，
因而科学方法与非科学方法的界限并不总是很清晰。
例如拿搜集事实来说。
既然事实多得数不胜数，
既然科学家在决定考察哪些事实时必须作出判断，
那么他是依据什么作判断的呢？
亨利.庞加莱说，是美感在进行选择。
“人们若认识到数学证明求助的是感性，也许会感到惊讶，
因为数学证明似乎只能引起理性的兴趣。
人们会感到数学的优美，数字与形式的和谐，几何的优雅。
这是所有真正的数学家所知道的一种纯粹的美感，
它毫无疑问属于感性范畴”
（《数学家的创造力》，见本套书中译本第8卷）。

Considering the application of mathematics to science,
Campbell touches on the same point when he says that
“relations which appeal to the sense of the mathematician
by their neatness and simplicity
are found to be important in the external world of experiment… .
The expert mathematician has a sense about symbols, as symbols;
he looks at a page covered with what, to anyone else,
are unintelligible scrawls of ink,
and he immediately realizes whether the argument expressed by them
is such as is likely to satisfy his sense of form;
whether the argument will be ‘neat’ and the results ‘pretty.’”
The author believes that
“the only way to understand what Einstein did
is to look at the symbols in which his theory must ultimately be expressed
and to realize that it was reasons of symbolic form, and such reasons alone,
which led him to arrange the symbols in the way he did and in no other”
(/Numerical Laws and the Use of Mathematics in Science/, Vol.9).

坎贝尔讨论数学在科学中的应用时，也触及了上述问题。他说：
“人们发现，那些由于匀整和简洁而合乎数学家心意的关系，
在外部实验世界中是重要的。
……老练的数学家对符号本身非常敏感：
他只要看一下写满了在旁人看来是乱七八糟而根本无法理解的符号的稿纸，
就会立即知道，用这些符号所做的论证是否有可能满足自己对形式的感觉，
论证是否‘简洁’，结论是否‘漂亮’”
坎贝尔认为，
“理解爱因斯坦的研究工作的唯一方法，
就是看一看最终用来表达他的理论的那些符号，
就是要认识到，他之所以这样而不是那样安排这些符号，
仅仅是由于形式的原因”
（《数值定律和数学在科学中的用途》，见本套书中译本第8卷）。

The same application of mathematical order and neatness to data
resulted in the discovery of another important piece of scientific knowledge—
the periodic law of the chemical elements
(“/The Genesis of a Law of Nature/,” Vol.8).
Are scientists, then,
for all their devotion to cold, hard facts,
essentially artists?
Where do the patterns lie
from which they construct their models of reality?
In nature, or in their minds?

把数学的严整性和简洁性应用于观察所得的事实，
还导致发现了另一套重要的科学知识，
即化学元素的周期规律
（《一条自然定律的产生》，见本套书中译本第7卷）。
因此，科学家虽然致力于研究冷酷严峻的事实，
却可能实际上是艺术家。
他建立现实模型的图样究竟在哪里？
是在自然界中还是在他的心灵中？

** Whose Footprint? 1-66 谁的脚印？   99

The interplay between the mind and the universe—
which governs which?
—may be subject for the philosophers,
but scientists also have their philosophical moods.
Tobias Dantzig (selections from his work are in Vol.9 of this set) asks,
“Has the universe an existence *per se*
or does it exist only in the mind of man?”
Giving a tentative answer to the age-old dilemma, he continues:
“To the man of science,
the acceptance of the one hypothesis or the other
is not at all a question of ‘to be or not to be’;
for from the standpoint of logic either hypothesis is tenable,
and from the standpoint of experience neither is demonstrable.
So the choice will forever remain a matter of expediency and convenience.
The man of science will act
as if this world were an absolute whole controlled
by laws independent of his own thoughts or acts;
but whenever he discovers a law of striking simplicity
or one of sweeping universality
or one which points to a perfect harmony in the cosmos,
he will be wise to wonder
what role his mind has played in the discovery,
and whether the beautiful image he sees in the pool of eternity
reveals the nature of this eternity,
or is but a reflection of his own mind.”
(/Number, The Language of Science/, 4th ed., Macmillan, 1962).

心灵与天地万物的相互作用–究竟谁支配谁–
可能是哲学家关心的问题，
但科学家有时也对这一哲学问题感兴趣。
托比亚斯.丹齐格问道：（选自其著作的片段，见本套书中译本第8卷）
“天地是有其自身的存在，还是只存在于人的心灵中？”
在试着回答这一古老的难题时，他写道：
“对于科学家来说，
接受前一种假说或后一种假说并不是‘生死攸关’的问题，
因为从逻辑观点来看，两种假说都是成立的，
而从经验观点来看，两种假说都无法加以证明。
因此，选择哪一假说，将永远是一个便利与方便的问题。
科学家的所有所为似乎总把这个世界当作一绝对的统一体，
支配这一统一体的规律独立于科学家的思想或行为；
但每当他发现一条简单得令人吃惊的规律，
或发现一条适用性极广的规律，
或发现一条显示出宇宙的完美和谐的规律时，
他都应问一下自己的心灵在这一发现中起了什么作用，
问一下自己在这种永恒之池中看到的美丽景象
究竟是显露了永恒的性质呢，
抑或只不过是自己心灵的倒影”
（《数字，科学的语言》，第四版，百科米伦公司，1954年，第233页）。

Arthur Eddington said, in /Space, Time, and Gravitation/
(Cambridge University Press, 1987 edition),
“We have found a strange footprint on the shores of the unknown.
We have devised profound theories, one after another,
to account for its origin.
At last, we have succeeded
in reconstructing the creature that made the footprint,
And Lo! it is our own.”

A.S.爱丁顿在《空间，时间和引力》一书中说，
（剑桥大学出版社，1959年版，第201页）
“我们在未知世界的海岸上发现了奇怪的脚印。
我们提出了一个又一个深奥的理论来解释脚印的来历。
最终，我们复制出了留下脚印的那个生物。
哦，原来是我们自己！”

The question of the actual existence of the objects of scientific inquiry
is nowhere more puzzling than in the field of mathematics.
Do mathematical truths reside in the external world,
or are they manmade inventions?
The answers reflect the dual role that mathematics plays—
appearing sometimes as servant of science,
sometimes as queen in her own right.

科学研究的对象是否真的存在这一问题，
在数学领域比在任何其他领域都更令人困惑不解。
数学真理是存在于外部世界，还是人的发明创造？
我们既可以回答说数学真理存在于外部世界，
也可以回答说数学真理是人的发明创造，
这反映出数学可以扮演双重角色，
有时似乎是科学的奴仆，有时又似乎是名副其实的女王。

To Bertrand Russell, mathematics is independent of mankind,
and her true role is that of queen.
“To those who inquire as to the purpose of mathematics,
the usual answer will be that it facilitates
the making of machines,
the travelling from place to place, and
the victory over foreign nations, whether in war or commerce.”
Russell regards these ends of doubtful value, and adds:
“As respects those pursuits which contribute only remotely,
by providing the mechanism of life,
it is well to be reminded that
not the mere fact of living is to be desired,
but the art of living in the contemplation of great things… .
Mathematics, rightly viewed,
possesses not only truth, but supreme beauty—
a beauty cold and austere, like that of sculpture,
without appeal to any part of our weaker nature,
without the gorgeous trappings of painting or music,
yet sublimely pure,
and capable of a stern perfection
such as only the greatest art can show… .
The contemplation of what is non-human,
the discovery that our minds are capable
of dealing with material not created by them,
above all, the realization that beauty
belongs to the outer world as to the inner
are the chief means of overcoming the terrible sense
of impotence, of weakness, of exile amid hostile powers . . .”
(/The Study of Mathematics/, Vol.9).

在伯特兰.罗素看来，数学是独立于人类的，实际扮演的是女王的角色。
“对于那些询问数学用途的人，通常的回答是，
数学有助于机器的制造，
有助于从一地到另一地的旅行，
有助于无论是在战争中还是在商业中战胜外国。”
罗素认为这些目的所具有的价值是有疑问的，进而写道：
“至于那些只是通过提供生命机制而间接起作用的追求，
应提醒人们注意，
值得人们向往的，不仅仅是生活的事实，
而且还有生活的艺术，借以思考伟大的事物。
……若以正确的观点看待数学，
则数学不仅是真理，而且还是无比优美的。
这是一种似雕像般冷峻的优美，
丝毫不求助于我们虚弱的本性，
也没有绘画或音乐的那种华丽装饰，
但却异常纯净，
可达到一种只有最伟大的艺术才有可能表现的严峻的完美。
……思考非人类的东西，
发现我们的心灵能够处理并非由心灵创造的材料，
特别是认识到优美既属于内心世界也属于外部世界，
这些是克服软弱无力感的主要方法，
也是克服自己被放逐在敌对力量……间的感觉的主要方法”
（《数学的研究》，见本套书中译本第8卷）。

But it is about mathematics as the servant of science
that Lancelot Hogben writes when he says
that mathematics is a language, a size language,
developed in response to man’s material needs:
“Plato’s exaltation of mathematics
as an august and mysterious ritual
had its roots in
dark superstitions which troubled,
and fanciful puerilities which entranced,
people who were living through the childhood of civilization,
when even the cleverest people could not clearly distinguish
the difference between saying that 13 is a ‘prime’ number
and saying that 13 is an unlucky number.
His influence on education
has spread a veil of mystery over mathematics
and helped to preserve the queer freemasonry
of the Pythagorean brotherhoods,
whose members were put to death for revealing mathematical secrets
now printed in school-books”
(/Mathematics, the Mirror of Civilization/, Vol.9).

但兰斯洛特.霍格本则强调数学是科学的奴仆，
他说，数学是一种语言，是一种度量语言，用以满足人类的物质需要：
“柏拉图把数学奉为一种庄严而神秘的仪式，
根源就在于极端的迷信于幼稚，
文明初期的人们被这种迷信与幼稚围绕着，
当时即便是最富于智慧的人也无法
把作为‘素’数的13
与作为不吉利数字的13区别开来。
柏拉图对教育的影响，给数学披上了一层神秘的面纱，
并帮助保留下了毕达哥拉斯兄弟会的一种奇怪的学说。
该兄弟会的成员若泄露数学秘密便会被处死，
而现在这种所谓数学秘密却赫然印在小学课本上”
（《数学–文明的镜子》，见本套书中译本第8卷）。

It was from the “queer freemasonry” of the Pythagorean teachers
that Plato derived the so-called “Platonic bodies,”
the five regular polyhedra that have equal sides and equal angles.
The teachings of the Pythagoreans,
who held that numbers have a real and separate existence,
may seem fanciful now,
but fancies have often been of value to science.
It was from consideration of the “Platonic bodies”
that Kepler developed the first unitary scheme of the universe
(/Epitome of Copernican Astronomy/, in GBWW, Vol.15).

正是依据毕达哥拉斯派教师的这种“奇怪的学说”，
柏拉图推导出了所谓“柏拉图物体”，既五种正方体。
毕达哥拉斯派认为，数字是独立而实际的存在。
现在看来，毕达哥拉斯派的学说太富于想象了，
但想象力却常常于科学是有价值的。
开普勒正是通过思考所谓“柏拉图物体”，
提出了第一个统一的宇宙模式
（《哥白尼天文学梗概》，见《西方世界名著》，第16卷，第851b-872b页）。

In addition to having a reality apart from man,
mathematics had, for Plato,
“something in them which is necessary and cannot be set aside,”
something of “divine necessity.”
Plato believed that the necessities of knowledge
which are divine and not human are
“those of which he who has no use nor any knowledge
at all cannot be a God, or demi-god, or hero to mankind,
or able to take any serious thought or charge of them.
And very unlike a divine man would he be,
who is unable to count one, two, three,
or to distinguish odd and even numbers …”
(Laws, in GBWW, Vol.6).

在柏拉图看来，
数学除了具有独立于人类的实体外，其本身还具有
“某种不可或缺，不能予以轻视的东西”，即某种
“神圣而不可缺少的东西”。
柏拉图认为，之所以必须掌握神圣而非人类的知识，是
“因为既无用又毫无知识的人，不能成为神，
不能成为像神那样的人或人类的英雄，
不能严肃地思考或照料它们。
不会数1，2，3，不能辨别奇数和偶数的人
不可能成为神圣的人……”
（《法律篇》，见《西方世界名著》，第7卷，第728b-c页）。

An opposing view about the nature of mathematics
is presented by Henri Poincaré.
With the development of non-Euclidean geometries,
it seems reasonable to ask which geometry is true.
But the question is without meaning, says Poincaré.
“Geometry is not true, it is advantageous.”
But does not one geometry come closest to being true?
“One geometry cannot be more true than another;
it can only be more convenient”
(/Space/, Vol.9).
But this seems to say
that the mathematician builds a habitation for himself
rather than (as the Platonists believe) finds one eternally standing.

亨利.庞加莱对数学的性质持相反的观点。
非欧几里得几何学建立起来以后，
人们自然会问究竟哪一种几何学是真实的。
然而庞加莱说，这一问题是毫无意义的。
“几何学不是真实的，而是有益处的。”
但难道一种几何学不比另一种几何学更接近于真实吗？
“一种几何学不会比另一种几何学更为真实，
而只会更为便利”
（《空间》，见本套书中译本第8卷）。
但这似乎是说，数学家是在为自己建造住所，
而不是（像柏拉图主义者所相信的那样）
发现了一永久屹立的住所。

Such questions may be unanswerable,
but even unanswerable questions serve as beacons.
We cannot, for instance, know the why of things, says Claude Bernard.
Absolute knowledge of the very essence of phenomena
will always remain beyond our reach.
“When we know that friction and that chemical action produce electricity,
we are still ignorant of the primary nature of electricity.”
But, he continues, the search is not unfruitful.
If a man of science carries experimental analysis far enough,
he will see that though he is ignorant of the cause of the phenomenon,
he will become its master.
“The instrument at work is unknown, but [he] can use it”
(/Experimental Considerations Common to Living Things and Inorganic Bodies/, Vol.8).

这些问题或许是无法回答的，
但即便是无法回答的问题也可以作为指路的灯塔。
譬如，克劳德.贝尔纳说，我们无法知道事物的“原因”。
我们将永远不会绝对了解现象的本质。
“尽管我们知道摩擦和化学作用可以产生电，可我们却仍然对电的根本性质一无所知。”
不过，他接下来说，探索并不是没有成果的。
如果科学家把实验分析推进的足够远，则他会看到，
虽然他对现象的原因一无所知，但他却会成为现象的主人。
“工具起作用的原因是未知的，但科学家却可以利用工具”
（《生物与非生物的实验共性》，见本套书中译本第7卷）。

Mankind’s reluctance to believe
that the universe may be without plan is centuries old.
“Either it is a well-arranged universe or a chaos huddled together,”
admitted Marcus Aurelius.
“But,” he asked,
“can a certain order subsist in thee,
and disorder in the All?”
(/Meditations/, in GBWW, Vol.11).
The persistence of the question indicates that,
although much of modern science
is directed toward controlling our environment,
scientific inquiry is based, at least partly,
on our desire to know what we ourselves are.

人类自古以来一直不愿相信，宇宙是无计划的。
马可.奥勒利乌斯承认，
“宇宙要么是井井有条的，要么是一团混乱”。
“但是”，他问道，
“难道会这里有某种秩序，而全体是无秩序的吗？”
（《沉思集》，见《西方世界名著》，第12卷，第266a页）
这一问题的持久存在表明，
虽然现代科学主要是力图控制人类的环境，
但科学研究却至少部分是建立在这样的基础之上，
即人类想知道自己究竟是什么。