罗素如何构造时间的定义

时间的意义在西方哲学里是个亘古的话题，而诺贝尔文学奖得主现代数理逻辑创始人之一的柏淳罗素对这个话题的讨论却是独具特色与众不同。尽管罗素自己承认他的讨论并不完美（却又声称是最安全的—--见下面的【1】），我们仍然可以从中学习领略大师缜密而有创见的思维。下面我们先来看罗素在《我们关于外部世界的知识[1]》一书中用数学性的哲学思维来构造时间的定义的核心部分内容：
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。。。。。。我们所意识到的事件，不论多么地短暂，总是要持续一定的时间而不会是只存在于一个数学意义上的瞬间。即便是在关于运动的数学理论所考虑的物理世界里，外界印象使我们产生的感觉也不是严格的瞬间，因此我们直接意识到的的感官对象都不是瞬间存在的。所以说，瞬间并不存在于我们的经验数据中，而只能是（如果是合理的话）推论或构造出的概念。很难看出如何能推论出瞬间这个概念，所以我们只剩下一个选项，就是构造出瞬间概念实体来。应该怎么做呢？

 

我们可以从直接的经验中了解到不同的事件之间的两种时间关联性：它们或是同时的，或是一个早些一个晚些。这两者都是原始数据中的部分；不可能存在有了两个事件然后再人为主观地将时间顺序加上去的情况。在一定的极限之内，时间顺序同所发生的事件一样地呈现给观察者。在任何一个探险故事中，你都可以发现下面类似这样的段落：“带着讥诮的微笑他用左轮手枪指着那位大胆的年轻人的胸膛。‘我数三下然后就开枪，’他说。他冷酷而清晰地数着那一和二。当那个三字从他那正要张开的嘴唇中突出的刹那间，一个耀眼的闪电划破了天空。”这里我们看到了同时性---不是如康德要我们相信的那样地因为那个大胆的年轻人的大脑器官的功能，而是由左轮枪和闪电客观地构成的。时间的这些关系存在于不是严格瞬间的事件之间。因而一个事件可能比另一个开始的更早些，但可以一直延续到第二个事件开始之后，因而与第二个事件之间存在着同时性。如果它一直延续到另一个事件结束之后，那么它就比那第二事件更晚。早些，同时，以及晚些，对于任何延续一定时间（不论多短）的事件来说，总是相互一致的；只有当我们处理瞬间发生的事件时才可能出现不一致性。

 

我们可以发现，我们无法给出所谓的绝对的时间点，而只能给出由具体事件决定的时间点。我们无法指出某个时间本身，而只能指出在那个时间所发生的事件。所以在我们无法根据经验来假定有独立于事件的时间：只有依据同时性和承继性来排序的事件是我们从经验所能够得到的。因此，除非我们再引入多余的形而上的实体概念来，我们只有借助仅含有事件及其时序关系的构造来定义数学物理能够认为是瞬间。

 

我们怎样才能如我们所希望的那样仅通过事件来给出一个准确的时间点呢？如果我们任取一个事件，我们无法准确地给出时间点来，这是因为一个事件不是瞬间的，那等于是要它与两个不同时的事件同时。为了要给出一个准确的时间点，从理论上来说，我们必须能够确定是否任一个给定的事件在此时间点之前或之后，而且我们必须知道任何一个其它的时间点必须在此时间点之前或之后，但不能与它同时。现在我们假定，有两个事件A和B而不是仅一个事件A，而且假定A与B有部分的重叠，但是B比A先结束。这样一来，另一个与A和B都同时的事件就必须存在于A和B的重叠区里面；因此我们比仅考虑A或B时更接近于给出一个精确的时间点了。假设C是一个与A和B都同时的事件，但是比A或B都更早地结束。这样的话，另一个与A和B还有C同时的事件就必须存在于A,B，C三个事件的共同的重叠区里面，而且这个重叠区比之前的重叠更短。按照这样的思路继续下去，不断地加入新的事件，每一个新事件都和之前加入的所有的事件同时，我们将得到越来越精确地计时的序列。这就给我们提供了一个可以确定完全准确的时间的方法。

A______________________________________

B_________________________________

 

C___________

 

我们来取这样一组事件，它们之间任意两个都有重叠，所以就一定存在着这样一段时间，无论多短，在这段时间里这组事件中的所有事件都存在着。如果我们还能找到任何其它的与这组事件中的所有事件重叠的事件，那么我们将新找到的事件也加入到这组事件中来；按照这样的程序操作（译者注：显然这是一种理论性的过程而不是实际的操作），最终在这组事件之外将不会再有任何与这组里的所有事件都重叠的事件，而这组内的所有事件彼此之间都有同时性存在。我们将这样一个事件组定义为时间上的瞬间。接下来我们需要来演示这样一个集合具备我们期待一个瞬间所具有的特点。
 

对于瞬间我们应该期待着什么样的特点呢？首先，他们必须构成这样一个序列：其中的任意两个中必须有一个比另一个早；如果某个瞬间在另一个之前，而另一个在第三个之前，那么第一个也必须在第三个之前。其次，每个事件必须经历一些量的瞬间；如果两个事件处于同一个瞬间那么它们是同时的，而如果一个事件所处的瞬间比另一个事件所处的瞬间早，那么一个事件就在第二个事件之前（译者注：这里原作者似乎还应该强调第二个事件不存在于第一个瞬间）.第三，如果我们假设在任何事件存在的期间某些地方总会出现变化，那么我们需要的瞬间序列应该是紧致的，也就是说，给出任意两个瞬间，一定有另外的瞬间存在于它们之间。那么我们先前所定义的瞬间序列是否具有这些特性呢？

 

如果一个事件在我们用来如前定义某个瞬间的事件组之内的话，我们将称这个事件处于那个瞬间；另外，如果定义某个瞬间的集合里面有某个事件比定义另一个个瞬间的集合里面的某些事件早而且不具有同时性，我们还将称第一个瞬间早于第二个瞬间。当某个事件比另一个事件发生的早且不具有任何同时性时，我们称第一个事件“完全先于”第二个事件。这样我们就知道当两个事件之间不存在同时性时，其中一个一定是完全先于另一个，另一个一定不会比前面一个早；我们也知道，如果一个事件完全先于第二个事件，而第二个事件完全先于第三个事件，那么第一个事件一定完全先于第三个事件。从这些事实来看，我们很容易得出结论说我们之前定义的瞬间构成一个时间序列。

 

接下来我们需要演示每个事件至少应该处于一个瞬间，也就是说任给一个事件的话，它一定至少是我们定义瞬间时所用的某一个事件组中的成员。为了这个目的，我们来考察与某个事件同时但不在该事件之后发生的的所有事件，即不是完全后于与所给定的事件同时的任何事件。我们将这些事件称为那个给定事件的“初始同代”事件。可以发现，这样一组事件便是所给定的事件所存在的第一个瞬间，只要每一个完全后于那个给定的事件的某个同代的事件都完全后于这个事件的最初的同代。

 

最后，我们只要能证明当给出其中一个完全先于另一个的任意两个事件时，一定存在一些事件它们完全后于第一个事件而与某些完全先于第二个事件的同时，就能说明我们所构成的瞬间的序列是紧致的。至于说现实中的事件在时间上的排序是否真是这样的，那是一个经验性的问题；但如果它不成立，就没有理由认为我们所构造的时间序列是紧致的。[17]（译者注：大师这里的论证在逻辑上并不完整）

 

这样我们所定义的瞬间序列就满足（译者注：由于大师对于紧致性的论证明不完整，这里的这个“满足”二字的意思是要打折扣的，是在假如前面对于紧致性的解释符合实际情况的前提下的。因此从论证逻辑上来说多少有些半吊子）了所有的数学要求而不需要假设存在任何有争议的形而上的概念实体。

 

瞬间也可以用包含关系（译者注：这里原文是enclosure-relation，不知中文应如何译，故权且用包含一词）来定义，和我们用点来定义完全一样。当一个对象与另一个具有同时性但又不早于或晚于那个对象时，它在时间上被那第二个对象所包含。任何有包含或被包含的对象在这里被称为“事件”。为了使这种时间包含关系能够成为“造点机”，我们要求(1)它应该是传递性的，即如果一个事件包含了另一个，而那个又包含了第三个，那么第一个就包含了第三个；（2）每个事件包含它自己，但是如果一个事件包含了另一不同的事件，那么被包含的事件不能再包含前面那个事件；(3)如果一组事件中至少有一个被所有其它的事件包含，那么一定有一个事件包含了这组事件所包含的一切，而且它自己也被所有的其它事件所包含；(4) 至少有一个事件是存在的。为了保证时间的无限可分性，我们也要求每个事件应该包含除了自己以外的其它的事件。在假设了这些特征之后，时间的包含就成为一个无限可分的造点机了。现在我们可以这样来构造一个关于事件的“包含-序列”：选择一组事件它们中的任意两个都是一个包含另一个；如果存在另一个包含-序列它的某些成员被我们的之前选的第一个序列的所有的成员所包含，而第二个序列的所有的成员也包含了第一个序列的某些成员的话，那么这第一个包含-序列就是一个“精准的包含-序列”。这样一来，一个“瞬间”就是包含了一个精准的包含-序列的成员的所有事件组成的集合。

 

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[17] 关于上述的时间关系的假设如下：-

 

I。为了确定瞬间能构成一个时间序列，我们假设：

（a）没有任何一个事件完全先于它自己。（一个“事件”定义为与某些事同时的任何东西。）

（b）如果某个事件完全先于另一个，而另一个完全先于第三个，那么第一个就完全先于第三个。

（c）如果某个事件完全先于另一个，它们不会有同时性。

（d）两个没有同时性的事件中的一个必须完全先于另一个。

II。为了确定一个给定事件的最初同代能够是一个瞬间，我们假设：

（e）一个完全后于某给定事件的某些同代的事件将完全后于该给定事件的某个最初同代。

III。为了确定瞬间的序列是紧致的，我们假设：

（f）如果一个事件完全后于另一个事件，那么存在一个完全后于前面的事件而与完全先于后面那个的某些事件同时的事件。

这个假设带来这样的结果：如果某个事件覆盖了直接先于另一个事件的整段时间，那么它必须和那个事件至少有一个瞬间是重叠的；也就是，不可能有某个事件刚结束另一个事件就开始。我不知道这是否应该被认为是不可接受的。N. Wilner在“对于相对位置的理论的一点贡献，”中有对于上述问题的数学-逻辑的处理，剑桥哲学学会学报，17刊，5期，441-449页
 

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这里需要对上面所引的罗素的文章做些简要的解释。首先，所引的是罗素的《我们关于外部世界的知识》一书的第四讲中的一部分，而这部分是针对所谓的私世界(private world)而言的，这里的私世界指的是每个人自己所观察到的世界。罗素在该书的第三讲中指出了每个所观察到的私世界是彼此不同的，且不同于科学所研究的物理世界。该书在本文所引的这段讨论的后面简单地指出了，每个人的私世界之间是相关联（所以我们才能有科学所研究的所谓的客观的世界），而对于私世界如何彼此相关于一个共同的客观世界的讨论的难度很大，但从哲学理解上来看并不具备严重的原则性的难度。他并没有再具体地深入讨论。

另外，我们要注意到罗素的那本书是在爱因斯坦发表了赋予时间与空间一个完全不同传统观念的革命性的新意义的狭义相对论之后，而在爱因斯坦还没有发表广义相对论之前。但是，由于如罗素自己在书中反复强调的，他所做的是对于时间的数学处理（应该说是数学性的哲学思辨），没有引入额外的形而上学的概念实体，因此，它不会象传统上一些其他哲学大家们对于时间意义的讨论那样会与狭义或广义相对论的知识发生冲突。

最后，从本文对于紧致性的讨论可以看出，罗素在开始进行时间点的构造时是信心满满的，但是在结束时却发现自己的讨论可能有悖于人们生活中对于事件与时间序列之间的关系的经验常识，如他在[17]中所解释的，“这个假设（译者注：这里的假设指的是任意两个瞬间点之间存在着其它的瞬间点之说）带来这样的结果：如果某个事件覆盖了直接先于另一个事件的整段时间，那么它必须和那个事件至少有一个瞬间是重叠的；也就是，不可能有某个事件刚结束另一个事件就开始。我不知道这是否应该被认为是不可接受的。”，因此他以“至于说现实中的事件在时间上的排序是否真是这样的，那是一个经验性的问题；但如果它不成立，就没有理由认为我们所构造的时间序列是紧致的。”这样一句听上去软软的话来结束他对于时间序列的构造的讨论。

我不知道为什么罗素这里会担心不存在“有某个事件刚结束另一个事件就开始”的可能性。这从数学的无限可分上似乎没有什么问题。或许他这里所担心的是他自己的讨论与前面他提到的拿左轮枪的人所说的数到三之后就开枪这种人们所熟悉的常识性的状况相违背。因为，一般人们说数到三要干什么显然应该是先数到三，然后再干什么。如果他真的在担心这样的问题，那么他应该加一句，他的时间序列构造至少是在大于普朗克常数量级之上的意义上是紧致的，因为从一个人数到三或是一个电脑指令运行完毕到另一个指令的运行开始之间在这个世界的某处总会有些量子量级上的脉动存在的。
 

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附录 文中所引用的罗素的文章的英语原文：
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......... Events of which we are conscious do not last merely for a
mathematical instant, but always for some finite time, however short. Even if
there be a physical world such as the mathematical theory of motion supposes,
impressions on our sense-organs produce sensations which are not merely and
strictly instantaneous, and therefore the objects of sense of which we are
immediately conscious are not strictly instantaneous. Instants, therefore, are
not among the data of experience, and, if legitimate, must be either inferred
or constructed. It is difficult to see how they can be validly inferred; thus
we are left with the alternative that they must be constructed. How is this to
be done?
 

Immediate experience provides us with two time-relations
among events: they may be simultaneous, or one may be earlier and the other
later. These two are both part of the crude data; it is not the case that only
the events are given, and their time-order is added by our subjective activity.
The time-order, within certain limits, is as much given as the events. In any
story of adventure you will find such passages as the following: “With a
cynical smile he pointed the revolver at the breast of the dauntless youth. ‘At
the word three I shall fire,’ he said. The words one and two had already been
spoken with a cool and deliberate distinctness. The word three was forming on
his lips. At this moment a blinding flash of lightning rent the air.” Here we
have simultaneity—not due, as Kant would have us believe, to the subjective
mental apparatus of the dauntless youth, but given as objectively as the
revolver and the lightning. And it is equally given in immediate experience
that the words one and two come earlier than the flash. These time-relations
hold between events which are not strictly instantaneous. Thus one event may
begin sooner than another, and therefore be before it, but may continue after
the other has begun, and therefore be also simultaneous with it. If it persists
after the other is over, it will also be later than the other. Earlier,
simultaneous, and later, are not inconsistent with each other when we are
concerned with events which last for a finite time, however short; they only
become inconsistent when we are dealing with something instantaneous.
 

It is to be observed that we cannot give what may be called
absolute dates, but only dates determined by events. We cannot point to a time
itself, but only to some event occurring at that time. There is therefore no
reason in experience to suppose that there are times as opposed to events: the
events, ordered by the relations of simultaneity and succession, are all that
experience provides. Hence, unless we are to introduce superfluous metaphysical
entities, we must, in defining what mathematical physics can regard as an
instant, proceed by means of some construction which assumes nothing beyond
events and their temporal relations.

If we wish to assign a date exactly by means of events, how
shall we proceed? If we take any one event, we cannot assign our date exactly,
because the event is not instantaneous, that is to say, it may be simultaneous
with two events which are not simultaneous with each other. In order to assign
a date exactly, we must be able, theoretically, to determine whether any given
event is before, at, or after this date, and we must know that any other date
is either before or after this date, but not simultaneous with it. Suppose,
now, instead of taking one event A, we take two events A and B, and suppose A
and B partly overlap, but B ends before A ends. Then an event which is
simultaneous with both A and B must exist during the time when A and B overlap;
thus we have come rather nearer to a precise date than when we considered A and
B alone. Let C be an event which is simultaneous with both A and B, but which
ends before either A or B has ended. Then an event which is simultaneous with A
and B and C must exist during the time when all three overlap, which is a still
shorter time. Proceeding in this way, by taking more and more events, a new
event which is dated as simultaneous with all of them becomes gradually more
and more accurately dated. This suggests a way by which a completely accurate
date can be defined.

Let us take a group of events of which any two overlap, so
that there is some time, however short, when they all exist. If there is any
other event which is simultaneous with all of these, let us add it to the group;
let us go on until we have constructed a group such that no event outside the
group is simultaneous with all of them, but all the events inside the group are
simultaneous with each other. Let us define this whole group as an instant of
time. It remains to show that it has the properties we expect of an instant.

What are the properties we expect of instants? First, they
must form a series: of any two, one must be before the other, and the other
must be not before the one; if one is before another, and the other before a
third, the first must be before the third. Secondly, every event must be at a
certain number of instants; two events are simultaneous if they are at the same
instant, and one is before the other if there is an instant, at which the one
is, which is earlier than some instant at which the other is. Thirdly, if we
assume that there is always some change going on somewhere during the time when
any given event persists, the series of instants ought to be compact, i.e.
given any two instants, there ought to be other instants between them. Do
instants, as we have defined them, have these properties?

We shall say that an event is “at” an instant when it is a
member of the group by which the instant is constituted; and we shall say that
one instant is before another if the group which is the one instant contains an
event which is earlier than, but not simultaneous with, some event in the group
which is the other instant. When one event is earlier than, but not
simultaneous with another, we shall say that it “wholly precedes” the other.
Now we know that of two events which are not simultaneous, there must be one
which wholly precedes the other, and in that case the other cannot also wholly
precede the one; we also know that, if one event wholly precedes another, and
the other wholly precedes a third, then the first wholly precedes the third.
From these facts it is easy to deduce that the instants as we have defined them
form a series.

We have next to show that every event is “at” at least one
instant, i.e. that, given any event, there is at least one class, such as we
used in defining instants, of which it is a member. For this purpose, consider
all the events which are simultaneous with a given event, and do not begin
later, i.e. are not wholly after anything simultaneous with it. We will call
these the “initial contemporaries” of the given event. It will be found that
this class of events is the first instant at which the given event exists,
provided every event wholly after some contemporary of the given event is
wholly after some initial contemporary of it.

Finally, the series of instants will be compact if, given
any two events of which one wholly precedes the other, there are events wholly
after the one and simultaneous with something wholly before the other. Whether
this is the case or not, is an empirical question; but if it is not, there is
no reason to expect the time-series to be compact. [17]

Thus our definition of instants secures all that mathematics
requires, without having to assume the existence of any disputable metaphysical
entities.

Instants may also be defined by means of the
enclosure-relation, exactly as was done in the case of points. One object will
be temporally enclosed by another when it is simultaneous with the other, but
not before or after it. Whatever encloses temporally or is enclosed temporally
we shall call an “event.” In order that the relation of temporal enclosure may
be a “point-producer,” we require (1) that it should be transitive, i.e. that
if one event encloses another, and the other a third, then the first encloses
the third; (2) that every event encloses itself, but if one event encloses
another different event, then the other does not enclose the one; (3) that
given any set of events such that there is at least one event enclosed by all
of them, then there is an event enclosing all that they all enclose, and itself
enclosed by all of them; (4) that there is at least one event. To ensure infinite
divisibility, we require also that every event should enclose events other than
itself. Assuming these characteristics, temporal enclosure is an infinitely
divisible point-producer. We can now form an “enclosure-series” of events, by
choosing a group of events such that of any two there is one which encloses the
other; this will be a “punctual enclosure-series” if, given any other
enclosure-series such that every member of our first series encloses some
member of our second, then every member of our second series encloses some
member of our first. Then an “instant” is the class of all events which enclose
members of a given punctual enclosure-series.

...........

[17] The assumptions made concerning time-relations in
the above are as follows:—

I. In order to secure that instants form a series, we
assume:

(a) No event wholly precedes itself. (An “event” is
defined as whatever is simultaneous with something or other.)

(b) If one event wholly precedes another, and the other
wholly precedes a third, then the first wholly precedes the third.

(c) If one event wholly precedes another, it is not
simultaneous with it.

(d) Of two events which are not simultaneous, one must
wholly precede the other.

II. In order to secure that the initial contemporaries of
a given event should form an instant, we assume:

(e) An event wholly after some contemporary of a given
event is wholly after some initial contemporary of the given event.

III. In order to secure that the series of instants shall
be compact, we assume:
(f) If one event wholly precedes another, there is an
event wholly after the one and simultaneous with something wholly before the
other.
This assumption entails the consequence that if one event
covers the whole of a stretch of time immediately preceding another event, then
it must have at least one instant in common with the other event; i.e. it is
impossible for one event to cease just before another begins. I do not know
whether this should be regarded as inadmissible. For a mathematico-logical
treatment of the above topics, cf. N. Wilner, “A Contribution to the Theory of
Relative Position,” Proc. Camb. Phil. Soc., xvii. 5, pp. 441–449.

[1]OUR KNOWLEDGE OF THE EXTERNAL WORLD，Bertrand
Russel，URL：http://www.gutenberg.org/files/37090/37090-h/37090-h.htm