Chapter 9 Chapter 8 "Principles of Mathematics": Mathematical Aspects

Both Whitehead and I were disappointed that Principia Mathematica was viewed only from a philosophical point of view.There is great interest in the discussion of contradictions and the question of whether ordinary mathematics is correctly deduced from purely logical premises, but not in the mathematical tricks found in this book.I used to know only half a dozen people who read the later parts of this book.Three of them were Poles who were later (I believe) liquidated by Hitler.The other three were Texans who were later assimilated to satisfaction.There are even people, who work on exactly the same problems as ours, who don't think it's worth looking up what Principia Mathematica has to say about them.Let me give two examples: About ten years after the publication of "Principia Mathematics", "The Chronicle of Mathematics" published a long article, some of the results of which we have already produced by coincidence in the fourth part of our book.There are some mistakes in this article, which we have avoided, but there is not a single right place which we have not published.The author of this article is obviously completely unaware that his kind of work has already been done before him.A second example arose when I was a colleague of Reichenbach at the University of California.He told me that he had an invention in which he extended the method of mathematical induction.He called it "transfinite induction".I told him that this problem is fully discussed in the third volume of Principia Mathematica.After a week he told me that he had confirmed it.I want to try not to be too specialized in this chapter, but from a mathematical point of view, not from a philosophical point of view, to explain several aspects of "Principles of Mathematics" that I think are important.

Let me start with a question, which is a philosophical question as well as a mathematical question, namely, the importance of relationships.In my book on Leibniz, I have emphasized the importance of relational facts and propositions, as opposed to facts by noumenon-and-properties and by subject- And -- a proposition made of predicates.I've found that the bias toward relationships has had a bad effect in philosophy and mathematics.Like Leibniz's unsuccessful efforts, Boole's mathematical logic is discussion-like and is only a development of the syllogism.Pierce once worked out a relational logic, but he regarded relation as a class made of pairs.This is technically possible, but doesn't automatically draw attention to what's important.What's important in relational logic is something different than class logic.With regard to relations, my philosophical opinions help to focus me on something that turns out to be extremely useful.

At that time, I almost only thought of relationships as insourcing.Sentences like this come to mind: "x is before y", "x is greater than y", "x is north of y".I felt then (and I still feel) that although from a formal algorithmic point of view we can treat relations as a set of ordered pairs, it is only inclusion that makes the set a unity.Of course, the same goes for classes.What makes a class a unity is only that inclusion which is common to and specific to the terms of the class.Whenever we deal with a class whose items we cannot enumerate, the above reasoning is obvious.The inability to enumerate is obvious for infinite classes, but it is also true for most finite classes.For example, who can enumerate the items in the class of earwigs?Nevertheless, we can say some propositions (true or false) about all earwigs, and we can do so because of the inclusiveness that makes this class possible.The above points can also be applied to relations.We have a lot to say about the order in time, because we understand the meaning of the word "before", although we cannot enumerate all the occasions of x and y in which x comes before y.But there is an objection to the view that relations are classes of pairs: the pairs must be ordered pairs, that is, we must be able to distinguish between the pair x,y and the pair y,x.This cannot be done without a certain relationship in the internal package.As long as we are confined to classes and predicates, it is impossible to explain order, or to distinguish an ordered pair from an unordered two-term class.

All of this is the philosophical background for the relational algorithms we develop in Principia Mathematica.We have to symbolize various concepts which mathematical logicians did not make obvious before.The most important of these concepts are: (1) a class of terms that have a relation R to a given term y; (2) a class of terms for a given term x (3) the "scope" of the relationship, which is made up of a class, and all the items in this class have the R relationship with something; (4) the "opposite scope" of R, which is is made of a class, and something has an R relation to all the terms of this class; (5) the "field" of R, which is made up of the "range" and "reverse range" mentioned above. (6) the "opposite" of an R relation, which is a relation between y and x when there is an R relation between x and y; (7) the "relationship" of the two relations R and S product", which is a relation between x and z when there is a term in y, x has an R relation to y, and y has an S relation to z; (8) plurals are defined as follows: class a, we form a class of terms all of which have a relation R to some term of a.We can look at the relationship between people as an example of the above concepts.For example, suppose R is a parent-child relationship.Then, (1) is the parent of y; (2) is the child of x; (3) is the class of all those who have children; (4) is the class of all those who have parents, that is, except Adam and Eve, everyone is included; (5) the domain of the "parent" relationship includes everyone, who is either someone's parent or someone's child; (6) the "parent" relationship The opposite is a relationship of "children"; (7) "grandparents" is the product of the relationship between parents and parents, "brother or ae? sister" is the product of the relationship between "children" and "parents", "cousins ​​or brothers or "ae? sister" is the product of the relationship between grandson and grandparents, and I can be analogized; (8) "Parents of Yitong College students" is a plural in this sense.

Different kinds of relationships have different kinds of usefulness.We can first talk about a relationship, which produces a thing, which I call "narrative function".This is a relation at most that only one item can have to a given item.This relation produces phrases using the word "the" in the singular, such as "the father of x" (the father of x), "thedou-bleof x" (twice the x), "the sine of x" (the sine of x), and the mathematical All ordinary functions in .Such functions can only arise from what I call "one-to-many" relations, that is, the relations that at most one item can have to any other item.For example, if you are talking about a Christian country, you can say "the wife of x", but if applied to a polygamous country, the meaning of this phrase is not clear.In mathematics you can say "the square of x", but not "the square root of x", because x has two square roots."Range", "Reverse Range" and "Region" in the tables listed above all generate narrative functions.

The second extremely important relationship is the one that establishes a mutual relationship between two classes.I call this relationship a "one-to-one" relationship.This is a relation in which not only at most one x has a relation R to a given y, but also at most one y to which a given x has a relation R.To give an example: Prohibition of polygamous marriages. Whenever such a mutual relation exists between two classes, the number of items of the two classes is the same.For example: without counting, we know that the number of wives is the same as the number of husbands, and that the number of noses of people is the same as the number of people.There is a special form of interrelationship, which is also extremely important.

The cause of this mutual relation is that there are two classes which are the domain of the two relations P and Q, and which have a mutual relation between them, and whenever two terms have the relation P, their correlators have Q this relationship, and vice versa.An example is the ranking of married officials and the ranking of their wives.If the wives were not connected with nobles, or if the officials were not bishops, the rank of the wives was the same as that of the husbands.This correlation-generating thing is called a "correlation-producer of order," because whatever order the items in the field of P have, this order is always preserved in their correlates in the field of Q.

The third important type of relationship is the one that produces a series. "Series" is an old, familiar term, but I think I was the first to give it a definite meaning.A series is a group consisting of terms which have an order derived from a relation which has three properties: (a) the relation must be asymmetric, that is, if x has this relation to y, y has no such relation to x; (b) it must be transitive, that is, if x has this relation to y, and y has it to z, x has it to z has this relation; (c) it must be connected, that is, if x and y are any distinct terms in the field of this relation, then either x has this relation to y or y There is such a relation for x.If a relation possesses these three properties, it arranges the items in its domain in a series.

All these properties are easily illustrated by the example of human relations. ? Husband? Husband This relationship is asymmetrical, because if A is B's husband, B is not A's husband.Conversely, mates are symmetrical.Ancestors are transitive because an ancestor of an ancestor of A is an ancestor of A; but father? father is intransitive.Of the three properties necessary for a serial relationship, ancestors have two, and the third, "connection," that property, for not one of any two persons is necessarily the ancestor of the other.On the other hand, if we look at the succession of a royal family, for example, the son always inherits from the father, the ancestry relationship limited to this royal line is connected, so these kings form a series.

The above three relations are extremely important relations in the transition between logic and general mathematics. Now I would like to go on to talk about several developments in general. The above-mentioned logical set is very useful for these developments.But before I speak, let me say a few general remarks. When I was young, I was told that mathematics is the science of numbers and quantities, another way of saying that mathematics is the science of numbers and measurements.This definition is too narrow.First: the many different kinds of numbers taught in traditional mathematics represent only a small fraction of the range to which the mathematical method is applied, and the reasoning we cannot do without in order to establish the foundations of arithmetic is that numbers do not very closely related.Second: In speaking of arithmetic and its introduction, we must not forget that there are theorems which are equally true for finite and infinite classes or numbers.Whenever possible, we should not prove these theorems for the former only.To put it more generally, if we can prove some theorems in a more general range, we think that it is a time-consuming thing to prove these theorems in a special class of instances.Third: Some traditional formal laws in arithmetic, namely, the law of associativity, the law of (a+b)+c=a+(b+c) reciprocation, a+b=b+a and similar laws of multiplication and the law of distribution a x (b+c)=( a × b) + (a × c) We consider it part of our purpose to verify these laws.Beginners in mathematics only learn these laws without proofs, or else, if they have proofs, they use mathematical induction, which is therefore valid only for finite numbers.Common definitions on addition and multiplication assume that the number of factors is finite.We have tried to get rid of some restrictions including the one mentioned above.

Using the so-called "selection" method, we can extend multiplication to an infinite number of factors.It is easiest to see what the concept of choice is by using the example of electing members of Parliament.Assuming that every elected member of the country must be a member of the voters, the entire parliament is a so-called "choice" from the voters.The general idea is this: if there is a class made up of several classes, none of the classes is zero, and selection is a relationship, picking an item from each class to be the "representative" of that class.The number of doing so (assuming that there is no item common to both classes) is the product of the numbers of these classes.Suppose, for example, that we have three classes, the first consisting of x1, x2, x3, the second consisting of y1, y2, y3, the third consisting of z1, z2, z3, each containing a A class of x, a y, and a z is a choice from a class of three.There are twenty-seven ways to make this choice without difficulty to any reader. After we have adopted this definition of multiplication, we encounter an unexpected difficulty.If the number of classes is infinite, it seems we cannot be sure that selection is possible.If the number of classes were finite, we could choose a representative from each class arbitrarily, as is the case in general elections; but if the number of classes were infinite, we could not have an infinite number of arbitrary choices. , and we cannot know with certainty that a choice can be made unless there is an in-pack to get the desired result.Let me give you an example: Once upon a time there was a millionaire who bought countless pairs of shoes, and as long as he bought a pair of shoes, he also bought a pair of socks.We can make a choice to choose one of each pair of shoes, since we can always choose the right shoe or the left shoe.So, when it comes to shoes, the choice exists.But with socks, since there is no right and left, we cannot use this rule of choice.If we want to be able to choose among socks, we have to take a much more sophisticated approach.For example, we can find a characteristic that one pair of socks is closer to this characteristic than the other. In this way, we choose the sock that is closer to this characteristic from each pair, and we have selected a set.I once posed this riddle to a German mathematician who occasionally sat next to me at the staff table at Trinity College, but his only comment was: "Why millionaire?" Some people think that it goes without saying that if none of these classes is zero, it must be possible to choose one from each class.Others think otherwise.On this point Peano said it best: "Is this a principle true? Our opinion is of no value." Our definition of what we call the "axiom of multiplication" is this: it is assumed that it is always possible A representative is selected from each of a set of several classes, none of which is zero.We can find no arguments for or against this axiom, so we include this axiom explicitly in the assumptions of any theorems to which it applies.At the same time as we were running into this problem, Zermeler came up with what he called the "principle of selection," a slightly different but logically equivalent assumption.He and some others took it as a self-evident truth.Since we do not take this opinion, we try to find some way of dealing with multiplication without assuming that this axiom is true. The logic theory of choice does not depend on the concept of "number" at any point. In "Principles of Mathematics", we put forward the theory of choice before defining "number".This sense applies also to another conception of great importance, viz., that of C which is denoted in ordinary language by the words "etc." Suppose you want to illustrate the concept "ancestor" with the concept "parent". You can say that A is the ancestor of Z, if A is the parent of B, B is the parent of C, etc., and thus after a finite number of steps, you arrive at the person Y , he is the father (or mother) of Z.There is nothing wrong with this, except for one thing, which includes the words "limited", which must be defined. It is only possible to define "limited" by a special application of a quite general concept, namely, the concept of ancestral relation from any given relation.This concept of ancestry relations was first developed by Frege as far back as 1879, but until the time Whitehead and I developed it, Frege's work had remained largely unnoticed.The concept we want to define can be preliminarily explained as follows: If x has a relationship R with respect to y, let us call the step from x to y "R step".You can take another R steps from y to z.Whatever you can reach by taking those R steps from x, we say to be a "descendant" of x with respect to R.We cannot say that everything you can achieve by a "finite number of R steps", because we do not yet have the knowledge of "finite" This term is defined.We can only define it by borrowing the concept of "offspring".The descendants of x of R can be defined as follows: Let us first define a "hereditary" class of R. This is a class having the property that whatever is reached from an item of this class by one R step is also an item of this class.For example, the quality of the name "Smith" is transmitted in the relationship between father and son, and the quality of human nature is transmitted in the relationship of parent to child. "If y belongs to every hereditary class about R to which x belongs, y belongs to the descendants of x about R", I now explain what this means.Let us now apply this to ordinary integers, replacing R by the relation of a number to the number immediately below it.If we now look at the descendant of 0 with respect to this number, it is clear that 1 belongs to this descendant, since 1=0+1; and, since 1 belongs to 0, so does 2; in this way.In this way, we get a whole set of descendant numbers that belong to 0.We can apply the proof by the so-called "mathematical induction" to all these numbers.Mathematical induction is such a principle: If a property belongs to 0, and belongs to the number immediately below any number that has this property, then this property belongs to all finite numbers.The specification of "finite" numbers as descendants of 0 is a direct consequence of this definition.In the past, everyone thought that mathematical induction was a principle, because they used to think that all numbers must be finite.This is a mistake.Mathematical induction is not a principle, but a definition.It is correct for some numbers and incorrect for others.Any number to which it applies is a finite number.For example, adding 1 to a finite number increases the finite number; this is not the case with an infinite number. The whole doctrine of ancestry is not only of great importance for numbers.For this reason, we created the doctrine before we came up with a definition of number. Now let me talk about something, which I call "Relational Arithmetic", which occupies the second half of the second volume of "Principles of Mathematics".From a mathematical point of view, this is my most important contribution to this book.What I call "Relationship Number" Is a completely new number, ordinary number is an extremely specialized example of this kind of number.I found that all the formal laws that apply to ordinary ordinal numbers apply to this much more general number.I also found that the number of relations is very important to understand the structure. Some words ("structure" being one of them), like "etc." or "series," are familiar but have no definite meaning.By means of relational arithmetic, the concept of "structure" can be precisely defined. The basic definition in this question is the definition of "similarity of order" or "similarity" mentioned above. Where relations are concerned, this plays the same role as analogy does between classes. The similarity between classes is the existence of a one-to-one relationship, linking each item of one class with a related person in another class. The similarity of the order between the two relations of P and Q means that there is a mutual relationship between the field of P and the field of Q. If there is a relationship between two items, their relatives have a relationship of Q, and vice versa.Let us give an example: Suppose P is the rank relation of married government officials, and Q is the rank relation of their wives. The relation of wife and husband makes the fields of P and Q have such a mutual relation: as long as these Wives have a Q relationship, their husbands have a P relationship, and vice versa.When the two relations P and Q are similar in order, if S is the relation that produces the mutual relation, Q is the product of the relation between S and P, and is the inverse of S.For example, in the example given above, if x and y are two wives, and x is related Q to y, and if S is the relationship of wife to husband, then x is related P to y's husband. The wife of such a man, that is to say, the product of the relation of Q and S to P is the same relation, and is the inverse of S; the inverse of S is the relation of husband to wife.Where P and Q are serially related, their resemblance lies in the fact that their terms can be related to each other without changing their order.But the notion of similarity applies to all domained relations, that is, to all relations in which scope and inverse scope are a type. We now say that the number of relations of a P-relation is the class of those relations which are similar in order to P.This is exactly like substituting ordinal analogy for class analogy, and relational substitution for class cardinality arithmetic.The definitions of addition, multiplication, and exponents are somewhat similar to those in cardinal arithmetic.Both addition and multiplication obey associative laws.The laws of distribution apply in one form but, generally speaking, do not apply in another.Except that the domain of relations is limited, interaction laws do not apply.For example, there is a series like the series of natural numbers, and two terms are added to this series.If you add these two items at the beginning, the new series is like the old series; but if you add these two items at the end, the new series is different.Whenever x has a relation P to y, or x has a relation Q to y, or if x belongs to the domain of P and y to the domain of Q, then the sum of the two relations P and Q is said to apply to A relationship between x and y.According to this definition, generally speaking, the sum of P and Q is not the same as the sum of Q and P.This is true not only of relational numbers in general, but also of ordinal numbers, if one or both of them are infinite. Ordinal numbers are a subclass of relational numbers, that is, they can be applied to "orderly" series, "orderly" The nature of a series is that any subclass in which there are several items has a first item.Kanter had studied transfinite ordinal numbers, but, as far as I know, general relational numbers were first defined and studied in Principia Mathematica. One or two illustrations may help us.Suppose you have a series of pairs, and you want to form a series of choices from these pairs in the sense explained above for the axiom of choice.This procedure is very similar to the procedure in radix arithmetic, except that we now want to arrange the choices into an order, whereas before we just counted them as a class.Suppose furthermore, as we did when we discussed the choice of classes, that we have three groups, (x1, x2, x3), (y1, y2, y3) and (z1, z2, z3), from which we wish to make A selection of series to come.There are various ways of doing this.Perhaps the simplest approach is this: any choice that includes x1 appears before any choice that does not.Among those choices that both contain x1 or neither contain x1, those that contain y1 appear before those that do not.Among those choices that either contain x1 or neither contain y1, those that contain z1 appear before those that do not.We make similar rules for mantissa 2 and mantissa 3.In this way we get all possible choices arranged in a series, the series starts with (x1, y1, z1) and ends with (x3, y3, z3).Obviously this series has twenty-seven items, but the number twenty-seven here is no longer a cardinal number like in our previous example, but an ordinal number, that is to say, it is a special kind of relational number.Since an order is established among those choices, it is distinguished from a cardinality, which does not establish an order.As long as we are restricted to finite numbers, there is no important formal distinction between ordinal and cardinal numbers; but with infinite numbers, the distinction becomes important because the law of interaction does not work. In proving formal laws of relational arithmetic, we often have occasion to discuss series of series of series.You can get a concrete picture in your mind by using the following example: Suppose you are going to pile up some bricks, and, to make the matter more interesting, suppose they are gold bricks, Kesburg works.I'm assuming now that you make a row of bricks, placing each brick due east of the previous one; Many lines, as far as is appropriate.Then you make a second layer on top of the first layer, a third layer on top of the second layer, and so on until all the bricks are stacked.Then each row is a series, each layer is a series of series, and this whole pile is a series of series of series.We can represent this process notationally as follows: Suppose P is the relation of the upper layer to the lower layer; the domain of P is made up of layers; each layer is a series of rows.Assume that Q1 is the relationship between the rows of the highest layer and the rows from south to north, Q2 is the relationship of the rows of the second layer, and so on. The field of Q is a sequence of rows.In the southernmost row of the highest layer, we call the relationship between east and west R11; in the second row of the highest layer, we call the relationship between east and west R12; and so on, the last is Rmm , assuming m is the number of layers and n is the number of rows in each layer.In this example, I'm assuming that the number of layers and rows is limited, but this is a completely unnecessary limitation, which is only there to make this example easier.In ordinary language, all this is quite complicated and lengthy, but it becomes easy to use its symbols.Suppose E is the relation of x to P (this relation is a term of the field where x is P).Then, F3 is the product of the relationship between F and F and F.For example, individual bricks are items that have an F3 relationship to P, that is, each brick is an item of an item of the domain of P's domain.In proving the associative laws of addition and multiplication we need such series of series of series. If two relations are similar in order, we can say that they produce the same "structure", but structure is a somewhat broader concept than this, since it is not limited to relations of two, that is, between two terms Relationship.In geometry, the relations between three or four terms are important, and Whitehead was to discuss these relations in the fourth volume of Principia Mathematica.But after he had done much preparatory work, his interest slackened, he abandoned the project, and turned to philosophy. But it is not difficult to see how the concept of structure can be generalized.Assume that P and Q are no longer the relationship of two, but the relationship of three. There are many popular examples of such a relationship, such as "between" and "jealousy".Concerning P and Q, we may say that they have the same structure, and if we can make them relate to each other, whenever xyz in that order has a relation of P, their correlators have a relation of Q in the same order, and vice versa. Of course.Structure is important for empirical reasons, but it is also important for purely logical reasons.If two relations have the same structure, their logical properties are identical except for those properties that depend on the terms of their fields.By "properties of logic" I mean those properties that can be expressed in logical terms, not just those properties that can be proved by logic.An example is the three features that define serial relations, that is, they are asymmetrical, transitive, and conjunctive.These characteristics can be expressed in logical terms; if a relation has any of these characteristics, every relation similar in order to it also has this characteristic.Every relation number, whether finite or infinite, is a logical property of any relation with that number.In general, everything that you can say about a relation, without mentioning the terms that have it, or any quality that cannot be expressed in logical terms, is perfectly applicable to any relation similar to the one you set out to .The distinction between logical and other properties is important.For example, if P is a relation between colors (such as the order of the colors in a rainbow), is a relation between colors. Such a property does not belong to all relations similar to P in order; Such a property is so.To take another more complicated example: the gramophone and the original music at the time of filming are indistinguishable in their logical nature, even though the actual materials of which the two things are made are very different. Another example may help us clarify the concept of structure. Suppose you know the rules of sentence construction in a language, but you don't know a single word except those used for logic, and suppose you are given a sentence written in that language: the sentence What are the different meanings it can have?What do these meanings have in common?You can assign any meaning to each individual word as long as it makes sense (that is, makes sense logically) for the entire sentence.The sentence, then, has many possible meanings, perhaps infinitely many, but they all have the same logical structure.If your language has certain logical requisites, the facts that make some of your sentences true have the same structure. I think relational arithmetic is important not just because it is an interesting generalization, but also because it gives one of the symbolic techniques necessary to deal with structures. I have always thought that people who are not familiar with mathematical logic have a hard time understanding the meaning of "structure," and that, because of this difficulty, they can easily go astray when trying to understand the world of experience.It is only because of this reason that the theory of relational arithmetic has not been paid much attention to by the world, and I feel very sorry for this. I know that this doctrine has not been completely ignored because in 1956 I unexpectedly received a letter from Professor Jürgen Smith of the Hamburg University in Berlin.He told me that parts of this doctrine have been used in what is called the "lexicographic problem," which consists in specifying the alphabetical arrangement of words in a language whose alphabet is infinite. THE DEVELOPMENT OF MY PHILOSOPHY CHAPTER 9 THE EXTERNAL WORLD Not long after Principia Mathematica was finished and still in print, I was asked by Gilbert Marais to write a short book for the Home College Language gives an outline of my philosophy.The invitation came at the right time.I long to escape the rigor of symbolic deductive reasoning.And at that time my proposition was clearer than ever before and since then, and it was easy to explain it in a simple and plain way.这本书很成功，现在销路仍然很广。我觉得多数哲学家仍然认为这书是充分说明了我的主张。 把那本书重读一遍，我发现里边有很多东西是我现在仍然相信的。我仍然承认“知识”不是一个精确的概念，而是混入到“或然的意见”中。我仍然承认自明是有不同的程度的，了解一个普遍命题而不知道其真理的任何个别的例子是可能的，例如：“所有从未乘到一起的成对的数其积是大于１，０００”。但是另有一些问题我的意见已经起了很大的变化。我不再以为逻辑定理是事物的规律；适得其反，我现在把逻辑定理看做纯是属于语言性质的。我不再以为点、瞬和质点是世界原料部分。我在那本小书里所讲的关于归纳法的话，我现在看来是很粗疏的。我讲到普遍和我们关于普遍的知识讲得很有把握，我现在没有那种把握了，虽然我关于这个问题没有什么新的意见象从前那样自信地提出来。 关于点、瞬和质点，我是被怀特海从我的“独断的睡梦”中唤醒的。怀特海发明了一个方法，把点、瞬和质点构成一组一组的事件，每一个的范围都是有限定的。这就有了可能象我们以前在算术中用奥卡姆剃刀那样，把它用在物理学里。我很喜欢数理逻辑方法上的这种新应用。这似乎是暗示，用于理论物理学里的那些概念，其光滑顺溜与其说是由于世界的性质，倒不如说是由于数学家的巧妙手段造成的。而且在知觉问题上这也好象是开辟了一个全新的前景。我受聘于一九一四年春季要在波士顿作劳威尔讲演，我选择了“我们关于外界的知识”做我的题目，并且就这个问题我开始利用怀特海的新工具做研究。 知觉是我们外界知识的源泉这个问题，在我看来是很麻烦的。如果两个人看一样东西，由于透视和光线射下来的方向，他们之所见就有所不同。没有理由单挑出一个知觉者来，说他才是看见了那件东西的真相。所以我们不能认为外界的物就是人之所见。物理学家认为这是老生常谈：我们看不见原子和分子。物理学家向我们保证原子和分子是物的构成成分。生理学家也一样使人气馁。他讲明从眼到脑有一个复杂的因果连环，而且你之所见是有赖于脑子里的变化。如果这个脑的状态能够被非平时的原因所引起，你就会有一种视觉，这个视觉不像平时那样和一个外界的物体相牵连。这类的事不专是牵涉到视觉。这可以由一个大家都知道的例子来说明：一个人觉得他的大脚趾疼，虽然他的腿已经被切断了。这种论证说明，我们直接所经验到的不可能是物理学所讨论的外界的物，可是只有我们直接所经验到的才给我们理由相信有个物理学的世界。 要想解决这个问题，有各种方法。最简便的是唯我论的方法。我是把唯我论当做一种假设，而不是当做一种定论。那就是说，我是考量一个学说，就是，除了我自己的经验以外，没有正当的理由对于任何东西加以肯定或否定。我不认为这个学说可以驳得倒，但是我也不认为任何人能认真相信它。 有些人主张，承认经验是合理的，不管是自己的或是别人的，但是相信没人经验得到的事情则是不合理的。这个学说是承认来自别人的证明，但是拒绝相信有无生命的物质。 最后就是朴素实在论者和物理学家所都同意的那个羽翼已成的学说。据这一个学说的说法，有些东西是活的，是一簇一簇的经验，另一些东西是无生命的。 这些学说中的第二个和第三个是需要从我所经验到的推论到我所不能经验到的东西。 这些推论不能按照逻辑加以证明。只有承认演绎逻辑范围以外的一些原则，这些推论才能算确实。在和所有我以前的思想里，我是承认物理学中所讲的那样的物质的。可是这就留下了一条介乎物理学和知觉（也可以说心与物）之间的令人不快的鸿沟。在最初我热心要放弃物理学家的那个“物质”的时候，我希望能揭示出那些假设的实体来，这些实体一个知觉者不能知觉为一些完全由他所知觉到的成分所组成的结构。 我头一回把罗威尔讲演里所提出的学说加以解说的时候，我提议这是一件可能的事。这头一回的解说是在一篇题为《感觉材料对物理学的关系》的文章里，发表在一九一四年的《科学》里。在这篇文章里我说：“如果科学要是可以证实的，我们就要遇到以下的这个问题：物理学把感觉材料证明为物体的作用，但是只有在物体能证明为感觉材料的作用的时候，科学的证实才是可能的。因此我们就不能不解决那些用物体来表示感觉材料的方程式，为的是使这些方程式倒是用感觉材料来表示物体”。但是没有多久，我就相信这是一个行不通的计划，物体不能解释为由实际上经验到的成分所组成的结构。也是在这一篇文章里，在后边的一段里，我说明我容许我有两种推断：（甲）别人的感觉材料和（乙），我所谓“感相”，我假定这是指物在没人知觉它们的地方所呈的现象。 我接着说，我倒高兴能把这两种推断废除，“这样就把物理学建立在一个唯我论的基础上；可是毫无疑问，那些人性比要求逻辑经济更强的人（我恐怕是大多数）就不会和我一样要把唯我论弄得能满足科学上的条件。”因此我就断念不再想只用经验的材料来构成“物质”，并且安于一个把物理学和知觉和谐地配合为一个整体的世界的图形。 一九一四年元旦日我忽然想到的那个关于我们的外界的学说有几件新奇的东西。其中最重要的是空间有六度而不是有三度的那个学说。我得到的结论是，在物理学的空间里，认为是一个点的，说得更正确一些，认为是一个“极微地域”的，实际上是一个由三度而成的复合体。一个人的知觉对象的全体就是这个复合体的一个实例。我之所以有这个主张是有种种理由的。也许最有力的理由是可以造出一些仪起来，这些仪器在没有活着的知觉者的地方能把一些东西记录下来，那些东西如果一个人在那儿是可以知觉到的。一个照相感光板可以把多星的天空任何选出来的一部分制出一个相起来。一个口授留声机可以把近旁的人所说的话记下来。象这样制做机械的记录（这些记录有类乎如果一个人也在那里他所得到的知觉）在学理上是没有限制的。给繁星闪烁的天空照相也许是说明所牵涉到的东西的最好的例子。无论哪个星都可以在任何地方（若是有一个人的眼在那里也看得见那个星）照下相来。因此，在照相板那个地方，有些事情发生，这些事情是和在那里能照下相来的所有那些不同的星有关系。因此在物理空间的一个微小的地域里随时都有无数的事情发生，与一个人在那里所能看见的或一件仪器所能记录的一切事情相应。不但如此，这些事情彼此有空间关系，这些空间关系多多少少正与物理空间中的那些对立的物体相应。在一张星体照相中所出现的那个复杂世界是在拍照的那个地方。同样，知觉之心的内容那个复杂世界是在我所在的那个地方。这两种情形不拘哪一个都是从物理学的观点来讲的。照这一个学说来讲，在我看见一颗星的时候，里边牵涉到三个地方：两个在物理空间里，一个在我私人的空间里。有星所处于物理空间中的那个地方；有我所处于物理空间中的那个地方；又有关于这颗星的我的知觉内容所处于我的别的知觉内容中的那个地方。 在这个学说里有两种方法把事件一束一束地收集起来。 一方面，你可以把所有那些可以认为是一件“东西”的现象的事件弄成一束。例如，假定这项东西是太阳，首先你就有正在看见太阳的那些人的所有视觉内容。其次你有正在被天文学家拍照下来的所有那些关于太阳的照片。最后，你有所有那些在各处发生的事情，正因为有这些事情，才有在那些地方看见太阳或给太阳照相的可能。这一整束的事件是和物理学的太阳有因果关系的。这些事件以光的速度从物理空间中太阳所在的地方向外进行。在它们从太阳向外进行的时候，它们的性质发生变化有两种情形。第一可以称之为“正规”的情形，这就是大小和强度依反平方律减少。在相当切近的程度上来说，这种变化只是发生在空虚的空间里。但是太阳在有物质的地方所呈现的光景是依物质的性质而有不同的变化。雾就要使太阳显得红，薄的云彩就要使太阳显得暗，完全不透明的物质就要使太阳完全不现任何现象。（我说现象的时候，我不只是指人们之所见，也是指没有知觉者的地方与太阳有关的那些所发生的事。）如果插进来的那个媒介物包含一只眼睛和一个视神经，则太阳因此所呈的现象就是某人实际上所看见的了。 某件东西从不同的地方所呈的现象（只要这些现象是“规则的”）如果是属于视觉的，就为透视定律所连结，如果是由别种感觉透露出来的，这些现象也为不是全然不同的定律所连结。 前面我曾说过，还有另外一个方法把事件集为一些束。按照这一个方法，我们不是把一件东西所呈的现象的那些事件集合起来，而是把在一个物理上的处所所呈的现象的所有那些事件都集合起来。在一个物理上的处所的事件其全体我称之为一个“配景”。 在某一个时间我的知觉内容的总体构成一个“配景”。仪器在某一个处所能够记录下来的所有事件之总体也是如此。在我们以前制束的方法中，我们曾有一束是由太阳的许多现象所组成。但是在这第二个方法中，一束只包含太阳的一种现象，那种现象和从那个地方所能知觉到的每个“物”的一种现象相联。在心理学中特别合适的乃是这第二种制束的方法。一个配景，如果碰巧是在一个脑子里，就是由该脑所属的那个人临时所有那些知觉之心的内容所组成。所有这些，从物理学的观点来看，都是在一个地方，但是，在这个配景里有若干空间关系，由于这些空间关系，原来物理学上说是一个地方的，现在却变成一个三度的复合体了。 不同的人对于一件东西有不同的知觉这个谜，关于一件物理上的物和它在不同的地方所呈的现象二者之间的因果关系这个谜，最后，（也许是最重要的）心与物之间的因果关系这个谜，都被这一个学说一扫而光了。这些谜之所以发生，都是由于不能把与某一个知觉的心之内容相连的三个处所加以区分。这三个处所就是（我再说一遍）：（１）“东西”所在的物理空间中的处所；（２）我所在的物理空间中的处所；（３）在我的配置中，我的知觉之心的内容对于别的知觉之心的内容所占据的处所。 我之提出上面的学说并不是认为那是唯一能解释事实的学说，或者认为一定是正确的。我之把它提出来是认为那是一个与所有既知的事实相符合的学说，并且认为，讫今为止，这是唯一能这样说的学说。在这一方面，这个学说是和（举例来说）爱因斯坦的广义相对论并列的。所有这些学说都超出事实所能证明的以外，并且，如果解决了一些谜，并且不论在哪一点上都和既知的事实不相矛盾，则这些学说都是可以接受的，至少暂时是可以的。我认为这就是以上那个学说所具备的条件，也就是任何有普遍性的科学上的学说所应有的条件。 怀特海把点解释为一类一类的事件，这个方法对于我求得以上那个学说是一个很大的帮助。可是我认为，是否事件实际上真适合于解释具有几何学上的点所应有的特性的任何东西，是可怀疑的。怀特海假定每个事件都是具有有限度的范围的，但是一个事件的范围并没有最小的限度。我找到了一种方法，从一类一类的事件来构成一个点，这些事件没有一个是小于一个指定的最小限度；但是他的和我的方法只能靠一些假定才有效。 没有这些假定，虽然我们能够达到很小的地域，我们也许不能达到点。在以上的叙述中，我之所以说“最小的地域”而不说点，正是因为这个理由。我不认为这有什么重大的关系。