時間的意義在西方哲學裡是個亙古的話題，而諾貝爾文學獎得主現代數理邏輯創始人之一的柏淳羅素對這個話題的討論卻是獨具特色與眾不同。儘管羅素自己承認他的討論並不完美（卻又聲稱是最安全的—--見下面的【1】），我們仍然可以從中學習領略大師縝密而有創見的思維。下面我們先來看羅素在《我們關於外部世界的知識[1]》一書中用數學性的哲學思維來構造時間的定義的核心部分內容：

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。。。。。。我們所意識到的事件，不論多麼地短暫，總是要持續一定的時間而不會是只存在於一個數學意義上的瞬間。即便是在關於運動的數學理論所考慮的物理世界裡，外界印象使我們產生的感覺也不是嚴格的瞬間，因此我們直接意識到的的感官對象都不是瞬間存在的。所以說，瞬間並不存在於我們的經驗數據中，而只能是（如果是合理的話）推論或構造出的概念。很難看出如何能推論出瞬間這個概念，所以我們只剩下一個選項，就是構造出瞬間概念實體來。應該怎麼做呢？

我們可以從直接的經驗中了解到不同的事件之間的兩種時間關聯性：它們或是同時的，或是一個早些一個晚些。這兩者都是原始數據中的部分；不可能存在有了兩個事件然後再人為主觀地將時間順序加上去的情況。在一定的極限之內，時間順序同所發生的事件一樣地呈現給觀察者。在任何一個探險故事中，你都可以發現下面類似這樣的段落：“帶着譏誚的微笑他用左輪手槍指着那位大膽的年輕人的胸膛。‘我數三下然後就開槍，’他說。他冷酷而清晰地數着那一和二。當那個三字從他那正要張開的嘴唇中突出的剎那間，一個耀眼的閃電劃破了天空。”這裡我們看到了同時性---不是如康德要我們相信的那樣地因為那個大膽的年輕人的大腦器官的功能， 而是由左輪槍和閃電客觀地構成的。時間的這些關係存在於不是嚴格瞬間的事件之間。因而一個事件可能比另一個開始的更早些，但可以一直延續到第二個事件開始之後，因而與第二個事件之間存在着同時性。如果它一直延續到另一個事件結束之後，那麼它就比那第二事件更晚。早些，同時，以及晚些，對於任何延續一定時間（不論多短）的事件來說，總是相互一致的；只有當我們處理瞬間發生的事件時才可能出現不一致性。

我們可以發現，我們無法給出所謂的絕對的時間點，而只能給出由具體事件決定的時間點。我們無法指出某個時間本身，而只能指出在那個時間所發生的事件。所以在我們無法根據經驗來假定有獨立於事件的時間：只有依據同時性和承繼性來排序的事件是我們從經驗所能夠得到的。因此，除非我們再引入多餘的形而上的實體概念來，我們只有藉助僅含有事件及其時序關係的構造來定義數學物理能夠認為是瞬間。

我們怎樣才能如我們所希望的那樣僅通過事件來給出一個準確的時間點呢？如果我們任取一個事件，我們無法準確地給出時間點來，這是因為一個事件不是瞬間的，那等於是要它與兩個不同時的事件同時。為了要給出一個準確的時間點，從理論上來說，我們必須能夠確定是否任一個給定的事件在此時間點之前或之後，而且我們必須知道任何一個其它的時間點必須在此時間點之前或之後，但不能與它同時。現在我們假定，有兩個事件A和B而不是僅一個事件A，而且假定A與B有部分的重疊，但是B比A先結束。這樣一來，另一個與A和B都同時的事件就必須存在於A和B的重疊區裡面；因此我們比僅考慮A或B時更接近於給出一個精確的時間點了。假設C是一個與A和B都同時的事件，但是比A或B都更早地結束。這樣的話，另一個與A和B還有C同時的事件就必須存在於A,B，C三個事件的共同的重疊區裡面，而且這個重疊區比之前的重疊更短。按照這樣的思路繼續下去，不斷地加入新的事件，每一個新事件都和之前加入的所有的事件同時，我們將得到越來越精確地計時的序列。這就給我們提供了一個可以確定完全準確的時間的方法。

A______________________________________

     C___________

我們來取這樣一組事件，它們之間任意兩個都有重疊，所以就一定存在着這樣一段時間，無論多短，在這段時間裡這組事件中的所有事件都存在着。如果我們還能找到任何其它的與這組事件中的所有事件重疊的事件，那麼我們將新找到的事件也加入到這組事件中來；按照這樣的程序操作（譯者註：顯然這是一種理論性的過程而不是實際的操作），最終在這組事件之外將不會再有任何與這組裡的所有事件都重疊的事件，而這組內的所有事件彼此之間都有同時性存在。我們將這樣一個事件組定義為時間上的瞬間。接下來我們需要來演示這樣一個集合具備我們期待一個瞬間所具有的特點。

對於瞬間我們應該期待着什麼樣的特點呢？首先，他們必須構成這樣一個序列：其中的任意兩個中必須有一個比另一個早；如果某個瞬間在另一個之前，而另一個在第三個之前，那麼第一個也必須在第三個之前。其次，每個事件必須經歷一些量的瞬間；如果兩個事件處於同一個瞬間那麼它們是同時的，而如果一個事件所處的瞬間比另一個事件所處的瞬間早，那麼一個事件就在第二個事件之前（譯者註：這裡原作者似乎還應該強調第二個事件不存在於第一個瞬間）.第三，如果我們假設在任何事件存在的期間某些地方總會出現變化，那麼我們需要的瞬間序列應該是緊緻的，也就是說，給出任意兩個瞬間，一定有另外的瞬間存在於它們之間。那麼我們先前所定義的瞬間序列是否具有這些特性呢？

如果一個事件在我們用來如前定義某個瞬間的事件組之內的話，我們將稱這個事件處於那個瞬間；另外，如果定義某個瞬間的集合裡面有某個事件比定義另一個個瞬間的集合裡面的某些事件早而且不具有同時性，我們還將稱第一個瞬間早於第二個瞬間。當某個事件比另一個事件發生的早且不具有任何同時性時，我們稱第一個事件“完全先於”第二個事件。這樣我們就知道當兩個事件之間不存在同時性時，其中一個一定是完全先於另一個，另一個一定不會比前面一個早；我們也知道，如果一個事件完全先於第二個事件，而第二個事件完全先於第三個事件，那麼第一個事件一定完全先於第三個事件。從這些事實來看，我們很容易得出結論說我們之前定義的瞬間構成一個時間序列。

接下來我們需要演示每個事件至少應該處於一個瞬間，也就是說任給一個事件的話，它一定至少是我們定義瞬間時所用的某一個事件組中的成員。為了這個目的，我們來考察與某個事件同時但不在該事件之後發生的的所有事件，即不是完全後於與所給定的事件同時的任何事件。我們將這些事件稱為那個給定事件的“初始同代”事件。可以發現，這樣一組事件便是所給定的事件所存在的第一個瞬間，只要每一個完全後於那個給定的事件的某個同代的事件都完全後於這個事件的最初的同代。

最後，我們只要能證明當給出其中一個完全先於另一個的任意兩個事件時，一定存在一些事件它們完全後於第一個事件而與某些完全先於第二個事件的同時，就能說明我們所構成的瞬間的序列是緊緻的。至於說現實中的事件在時間上的排序是否真是這樣的，那是一個經驗性的問題；但如果它不成立，就沒有理由認為我們所構造的時間序列是緊緻的。[17]（譯者註：大師這裡的論證在邏輯上並不完整）

這樣我們所定義的瞬間序列就滿足（譯者註：由於大師對於緊緻性的論證明不完整，這裡的這個“滿足”二字的意思是要打折扣的，是在假如前面對於緊緻性的解釋符合實際情況的前提下的。因此從論證邏輯上來說多少有些半吊子）了所有的數學要求而不需要假設存在任何有爭議的形而上的概念實體。

瞬間也可以用包含關係（譯者註：這裡原文是enclosure-relation，不知中文應如何譯，故權且用包含一詞）來定義，和我們用點來定義完全一樣。當一個對象與另一個具有同時性但又不早於或晚於那個對象時，它在時間上被那第二個對象所包含。任何有包含或被包含的對象在這裡被稱為“事件”。為了使這種時間包含關係能夠成為“造點機”，我們要求(1)它應該是傳遞性的，即如果一個事件包含了另一個，而那個又包含了第三個，那麼第一個就包含了第三個；（2）每個事件包含它自己，但是如果一個事件包含了另一不同的事件，那麼被包含的事件不能再包含前面那個事件；(3)如果一組事件中至少有一個被所有其它的事件包含，那麼一定有一個事件包含了這組事件所包含的一切，而且它自己也被所有的其它事件所包含；(4) 至少有一個事件是存在的。為了保證時間的無限可分性，我們也要求每個事件應該包含除了自己以外的其它的事件。在假設了這些特徵之後，時間的包含就成為一個無限可分的造點機了。現在我們可以這樣來構造一個關於事件的“包含-序列”：選擇一組事件它們中的任意兩個都是一個包含另一個；如果存在另一個包含-序列它的某些成員被我們的之前選的第一個序列的所有的成員所包含，而第二個序列的所有的成員也包含了第一個序列的某些成員的話，那麼這第一個包含-序列就是一個“精準的包含-序列”。這樣一來，一個“瞬間”就是包含了一個精準的包含-序列的成員的所有事件組成的集合。

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[17] 關於上述的時間關係的假設如下：-

I。 為了確定瞬間能構成一個時間序列，我們假設：

（a） 沒有任何一個事件完全先於它自己。（一個“事件”定義為與某些事同時的任何東西。）

（b） 如果某個事件完全先於另一個，而另一個完全先於第三個，那麼第一個就完全先於第三個。

（c） 如果某個事件完全先於另一個，它們不會有同時性。

（d） 兩個沒有同時性的事件中的一個必須完全先於另一個。

II。 為了確定一個給定事件的最初同代能夠是一個瞬間，我們假設：

（e） 一個完全後於某給定事件的某些同代的事件將完全後於該給定事件的某個最初同代。

III。 為了確定瞬間的序列是緊緻的，我們假設：

（f） 如果一個事件完全後於另一個事件，那麼存在一個完全後於前面的事件而與完全先於後面那個的某些事件同時的事件。

這個假設帶來這樣的結果：如果某個事件覆蓋了直接先於另一個事件的整段時間，那麼它必須和那個事件至少有一個瞬間是重疊的；也就是，不可能有某個事件剛結束另一個事件就開始。我不知道這是否應該被認為是不可接受的。N. Wilner在“對於相對位置的理論的一點貢獻，”中有對於上述問題的數學-邏輯的處理，劍橋哲學學會學報，17刊，5期，441-449頁

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這裡需要對上面所引的羅素的文章做些簡要的解釋。首先，所引的是羅素的《我們關於外部世界的知識》一書的第四講中的一部分，而這部分是針對所謂的私世界(private world)而言的，這裡的私世界指的是每個人自己所觀察到的世界。羅素在該書的第三講中指出了每個所觀察到的私世界是彼此不同的，且不同於科學所研究的物理世界。該書在本文所引的這段討論的後面簡單地指出了，每個人的私世界之間是相關聯（所以我們才能有科學所研究的所謂的客觀的世界），而對於私世界如何彼此相關於一個共同的客觀世界的討論的難度很大，但從哲學理解上來看並不具備嚴重的原則性的難度。他並沒有再具體地深入討論。

另外，我們要注意到羅素的那本書是在愛因斯坦發表了賦予時間與空間一個完全不同傳統觀念的革命性的新意義的狹義相對論之後，而在愛因斯坦還沒有發表廣義相對論之前。但是，由於如羅素自己在書中反覆強調的，他所做的是對於時間的數學處理（應該說是數學性的哲學思辨），沒有引入額外的形而上學的概念實體，因此，它不會象傳統上一些其他哲學大家們對於時間意義的討論那樣會與狹義或廣義相對論的知識發生衝突。

最後，從本文對於緊緻性的討論可以看出，羅素在開始進行時間點的構造時是信心滿滿的，但是在結束時卻發現自己的討論可能有悖於人們生活中對於事件與時間序列之間的關係的經驗常識，如他在[17]中所解釋的，“這個假設（譯者註：這裡的假設指的是任意兩個瞬間點之間存在着其它的瞬間點之說）帶來這樣的結果：如果某個事件覆蓋了直接先於另一個事件的整段時間，那麼它必須和那個事件至少有一個瞬間是重疊的；也就是，不可能有某個事件剛結束另一個事件就開始。我不知道這是否應該被認為是不可接受的。”，因此他以“至於說現實中的事件在時間上的排序是否真是這樣的，那是一個經驗性的問題；但如果它不成立，就沒有理由認為我們所構造的時間序列是緊緻的。”這樣一句聽上去軟軟的話來結束他對於時間序列的構造的討論。

我不知道為什麼羅素這裡會擔心不存在“有某個事件剛結束另一個事件就開始”的可能性。這從數學的無限可分上似乎沒有什麼問題。或許他這裡所擔心的是他自己的討論與前面他提到的拿左輪槍的人所說的數到三之後就開槍這種人們所熟悉的常識性的狀況相違背。因為，一般人們說數到三要幹什麼顯然應該是先數到三，然後再幹什麼。如果他真的在擔心這樣的問題，那麼他應該加一句，他的時間序列構造至少是在大於普朗克常數量級之上的意義上是緊緻的，因為從一個人數到三或是一個電腦指令運行完畢到另一個指令的運行開始之間在這個世界的某處總會有些量子量級上的脈動存在的。

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附錄 文中所引用的羅素的文章的英語原文：

………….

........ 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.

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[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.