THE ARCHITECTURE OF MATHEMATICS 

NICHOLAS BOURBAKI

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1. Mathematic or mathematics? To present a view of the entire field of mathematical science as it exists,-this is an enterprise which presents, at first sight, almost insurmountable difficulties, on account of the extent and the varied character of the subject. As is the case in all other sciences, the number of mathematicians and the number of works devoted to mathematics have greatly increased since the end of the 19th century. The memoirs in pure mathematics published in the world during a normal year cover several thousands of pages. Of course, not all of this material is of equal value; but, after furll allowance has been made for the unavoidable tares, it remains true nevertheless that mathematical science is enriched each year by a mass of new results, that it spreads and branches out steadily into theories, which are subjected to modifications based on new foundations, compared and combined with one another. No mathematician, even were he to devote all his time to the task, would be able to follow all the details of this development. Many mathematicians take up quarters in a corner of the domain of mathematics, which they do not intend to leave; not only do they ignore almost completely what does not concern their special field, but they are unable to understand the language and the terminology used by colleagues who are working in a corner remote from their own. Even among those who have the widest training, there are none who do not feel lost in certain regions of the immense world of mathematics; those who, like Poincar6 or Hilbert, put the seal of their genius on almost every domain, constitute a very great exception even among the men of greatest accomplishment. It must therefore be out of the question to give to the uninitiated an exact picture of that which the mathematicians themselves can not conceive in its totality. Nevertheless it is legitimate to ask whether this exuberant proliferation makes for the development of a strongly constructed organism, acquiring ever greater cohesion and unity with its new growths, or whether it is the external manifestation of a tendency towards a progressive splintering, inherent in the very nature of mathematics, whether the domain of mathematics is not becoming a tower of Babel, in which autonomous disciplines are being more and more widely separated from one another, not only in their aims, but also in their methods and even in their language. In other words, do we have today a mathematic or do we have several mathematics? Although this question is perhaps of greater urgency now than ever before, it is by no means a new one; it has been asked almost from the very beginning of mathematical science. Indeed, quite apart from applied mathematics, there has * Authorized translation by Arnold Dresden of a chapter in "Les grands courants de la pens6e math6matique," edited by F. Le Lionnais (Cahiers du Sud, 1948). t "Professor N. Bourbaki, formerly of the Royal Poldavian Academy, now residing in Nancy, France, is the author of a comprehensive treatise of modern mathematics, in course of publication under the title Elfments de Mathematique (Hermann et Cie, Paris 1939- ), of which ten volumes have appeared so far." 221 222 THE ARCHITECTURE OF MATHEMATICS [April, always existed a dualism between the origins of geometry and of arithmetic (certainly in their elementary aspects), since the latter was at the start a science of discrete magnitude, while the former has always been a science of continuous extent; these two aspects have brought about two points of view which have been in opposition to each other since the discovery of irrationals. Indeed, it is exactly this discovery which defeated the first attempt to unify the science, viz., the arithmetization of the Pythagoreans ("everything is number"). It would carry us too far if we were to attempt to follow the vicissitudes of the unitary conception of mathematics from the period of Pythagoras to the present time. Moreover this task would suit a philosopher better than a mathematician; for it Is a common characteristic of the various attempts to integrate the whole of mathematics into a coherent whole-whether we think of Plato, of Descartes or of Leibnitz, of arithmetization, or of the logistics of the 19th century-that they have all been made in connection with a philosophical system, more or less wide in scope; always starting from a priori views concerning the relations of mathematics with the twofold universe of the external world and the world of thought. We can do no better on this point than to refer the reader to the historical and critical study of L. Brunschvicg [I]. Our task is a more modest and a less extensive one; we shall not undertake to examine the relations of mathematics to reality or to the great categories of thought; we intend to remain within the field of mathematics and we shall look for an answer to the question which we have raised, by analyzing the procedures of mathematics themselves. 

为了呈现整个数学科学领域的现状，这是一项乍一看几乎难以克服的困难的事业，因为学科的范围和性质各不相同。与所有其他科学一样，自19世纪末以来，数学家的数量和致力于数学的著作数量大大增加。在正常年份，世界上出版的纯数学回忆录涵盖数千页。当然，并非所有这些材料都具有同等价值；但是，在考虑了不可避免的皮重之后，尽管如此，数学科学每年都会因大量新的结果而得到丰富，并不断扩展和发展成为理论，这些理论会根据新的基础进行修改，相互比较和结合。没有一位数学家，即使他把所有的时间都投入到这项任务中，也无法了解这一发展的所有细节。许多数学家占据了数学领域的一角，他们不打算离开；他们不仅几乎完全忽略了与他们的专业领域无关的内容，而且无法理解在远离自己的角落工作的同事所使用的语言和术语。即使在那些受过最广泛训练的人中，也没有人不觉得在数学这个广阔世界的某些领域迷失了方向；那些像庞加莱或希尔伯特一样，在几乎每个领域都留下了天才印记的人，即使在最有成就的人中，也是一个非常大的例外。因此，给不熟悉的人一个数学家自己无法想象的整体的准确图景，肯定是不可能的。然而，我们有理由问，这种旺盛的增殖是否有助于一个强大的有机体的发展，与它的新生长获得更大的凝聚力和统一性，或者它是否是一种渐进分裂倾向的外在表现，这是数学本质所固有的，数学领域是否没有成为巴别塔，在巴别塔中，自主学科之间的距离越来越大，不仅在目标上，而且在方法上，甚至在语言上。换句话说，我们今天有一个数学，还是有几个数学？尽管这个问题现在可能比以往任何时候都更紧迫，但它绝不是一个新问题；几乎从数学科学的一开始就有人提出这个问题。事实上，除了应用数学之外，阿诺德·德累斯顿还授权翻译了F.Le Lionnais（Cahiers du Sud，1948年）编辑的《数学数学》一章。N.Bourbaki教授，原皇家波尔达维安学院（Royal Poldavian Academy）教授，现居法国南锡，著有一本综合性的现代数学论文，目前正在出版，书名为Elfments de Mathematique（Hermann et Cie，Paris 1939-），迄今已出版十卷。在几何学和算术的起源之间始终存在着二元论（当然是在它们的基本方面），因为后者一开始是一门离散量级的科学，而前者一直是一门连续范围的科学；这两个方面带来了两种观点，自发现非理性以来，这两种观点一直是对立的。事实上，正是这一发现挫败了统一科学的第一次尝试，即毕达哥拉斯的算术化（“一切都是数字”）。如果我们试图追随从毕达哥拉斯时期到现在的统一数学概念的变迁，那将使我们走得太远。此外，这项任务更适合哲学家而不是数学家；因为这是将整个数学整合成一个连贯的整体的各种尝试的一个共同特点，无论我们想到柏拉图、笛卡尔或莱布尼茨、算术，还是19世纪的逻辑，它们都是与一个哲学体系相联系的，范围或多或少很广；总是从关于数学与外部世界和思想世界的双重宇宙的关系的先验观点出发。在这一点上，我们做得最好的莫过于让读者参考L.Brunschvicg[I]的历史和批判研究。我们的任务是一项更温和、范围更小的任务；我们不应试图研究数学与现实或思想大类的关系；我们打算继续留在数学领域，我们将通过分析数学本身的过程来寻找对我们提出的问题的答案。

2. Logical formalism and axiomatic method. After the more or less evident bankruptcy of the different systems, to which we have referred above, it looked, at the beginning of the present century as if the attempt had just about been abandoned to conceive of mathematics as a science characterized by a definitely specified purpose and method; instead there was a tendency to look upon mathematics as "a collection of disciplines based on particular, exactly specified concepts," interrelated by 'a thousand roads of communication," allowing the methods of any one of these disciplines to fertilize one or more of the others [1, page 447]. Today, we believe however that the internal evolution of mathematical science has, in spite of appearance, brought about a closer unity among its different parts, so as to create something like a central nucleus that is more coherent than it has ever been. The essential aspect of this evolution has been the systematic study of the relations existing between different mathematical theories, and which has led to what is generally known as the "axiomatic method." The words "formalism" and "formalistic method" are also often used; but it is important to be on one's guard from the start against the confusion which may be caused by the use of these ill-defined words, and which is but too frequently made use of by the opponents of the axiomatic method. Everyone knows that superficially mathematics appears as this "long chain of reasons" of 1950]  which Descartes spoke; every mathematical theory is a concatenation of propositions, each one derived from the preceding ones in conformity with the rules of a logical system, which is essentially the one codified, since the time of Aristotle, under the name of "formalogic," conveniently adapted to the particular aims of the mathematician. It is therefore a meaningless truism to say that this "deductive reasoning" is a unifying principle for mathematics. So superficial a remark can certainly not accoutnt for the evident complexity of different mathematical theories, not any more than one could, for example, unite physics and biology into a single science on the ground that both use the experimental method. The method of reasoning by means of chains of syllogisms is nothing but a transforming mechanism, applicable just as well to one set of premises as to another; it could not serve therefore to characterize these premises. In other words, it is the external form which the mathematician gives to his thought, the vehicle which makes it accessible to others,* in short, the language suited to mathematics; this is all, no further significance should be attached to it. To lay down the rules of this language, to set up its vocabulary and to clarify its syntax, all that is indeed extremely useful; indeed this constitutes one aspect of the axiomatic method, the one that can properly be called logical formalism (or "logistics" as it is sometimes called). But we emphasize that it is but one aspect of this method, indeed the least interesting one. What the axiomatic method sets as its essential aim, is exactly that which logical formalism by itself can not supply, namely the profound intelligibility of mathematics. Just as the experimental method starts from the a priori belief in the permanence of natural laws, so the axioma-tic method has its cornerstone in the conviction that, not only is mathematics not a randomly developing concatenation of syllogisms, but neither is it a collection of more or less "astute" tricks, arrived at by lucky combinations, in which purely technical cleverness wins the day. Where the superficial observer sees only two, or several, quite distinct theories, lending one another "unexpected support" [1, page 4461 through the intervention of a mathematician of genius, the axiomatic method teaches us to look for the deep-lying reasons for such a discovery, to find the common ideas of these theories, buried under the accumulation of details properly belonging to each of them, to bring these ideas forward and to put them in their proper light. 

我们的任务是一项更温和、范围更小的任务；我们不应试图研究数学与现实或思想大类的关系；我们打算留在数学领域，我们将通过分析数学本身的过程来寻找对我们提出的问题的答案。2.逻辑形式主义和公理化方法。在我们上面提到的不同系统或多或少地明显破产之后，在本世纪初，人们似乎放弃了将数学视为一门以明确的目的和方法为特征的科学的尝试；相反，有一种倾向将数学视为“基于特定的、确切指定的概念的学科集合”，通过“一千条交流之路”相互关联，“允许其中任何一个学科的方法使其他一个或多个学科受精[1，第447页]然而，今天，我们认为，数学科学的内部进化，尽管表面上如此，却在其不同部分之间带来了更紧密的统一，从而创造出比以往任何时候都更连贯的类似中心核的东西。这种演变的基本方面是对不同数学理论之间存在的关系进行系统研究，这导致了通常所称的“公理化方法”。“形式主义”和“形式主义方法”也经常被使用；但重要的是，从一开始就要警惕这些定义不清的词可能造成的混淆，而公理化方法的反对者经常使用这些词。每个人都知道，从表面上看，数学似乎是笛卡尔所说的“一长串原因”；每一个数学理论都是命题的串联，每一个命题都是根据逻辑系统的规则从前面的命题中派生出来的，而逻辑系统本质上是自亚里士多德时代以来以“形式逻辑”的名称编纂的，方便地适应了数学家的特定目标。因此，说这种“演绎推理”是数学的统一原则是毫无意义的真理。如此肤浅的一句话肯定不能解释不同数学理论的明显复杂性，例如，在物理和生物学都使用实验方法的基础上，人们无法将其统一为一门科学。通过三段论链进行推理的方法只是一种转换机制，既适用于一组前提，也适用于另一组前提；因此，它不能用来描述这些前提。换言之，它是数学家赋予其思想的外在形式，是使他人易于理解的载体，简而言之，是适合数学的语言；这就是一切，不应再赋予它任何意义。要制定这门语言的规则，建立它的词汇表，澄清它的语法，所有这些都是非常有用的；事实上，这构成了公理化方法的一个方面，可以恰当地称之为逻辑形式主义（或有时称之为“物流”）。但我们强调，这只是这种方法的一个方面，实际上是最不有趣的一个。公理化方法所设定的基本目标正是逻辑形式主义本身无法提供的，即数学的深刻可理解性。正如实验方法从对自然规律永久性的先验信念出发一样，公理化方法的基石在于相信，数学不仅不是三段论的随机发展串联，也不是由运气组合得出的或多或少“精明”的技巧的集合，在这种组合中，纯粹的技术聪明赢得了胜利。当肤浅的观察者只看到两个或几个截然不同的理论时，通过一位天才数学家的介入，公理化方法相互给予了“意想不到的支持”[1，第4461页，教导我们寻找这一发现的深层原因，找到这些理论的共同思想，并将其埋藏在属于每一个理论的细节的积累之下，将这些思想提出并以其正确的方式加以阐述。

3. The notion of structure. In what form can this be done? It is here that the axiomatic method comes closest to the experimental method. Like the latter drawing its strength from the source of Cartesianism, it will "divide the difficulties in order to overcome them better." It will try, in the demonstrations of a theory, to separate out the principal mainsprings of its arguments; then, taking each of these separately and formulating it in abstract form, it will develop * Indeed every mathematician knows that a proof has not really been "understood" if one has done nothing more than verifying step by step the correctness ofthe deductions of which it is composed, and has not tried to gain a clear insight into the ideas which have led to the construction of this particular chain of deductions in preference to every other one the consequences which follow from it alone. Returning after that to the theory under consideration, it will recombine the component elements, which had previously been separated out, and it will inquire how these different components influence one another. There is indeed nothing new in this classical going to-andfro between analysis and synthesis; the originality of the method lies entirely in the way in which it is applied. In order to illustrate the procedure which we have just sketched, by an example, we shall take one of the oldest (and also one of the simplest) of axiomatic theories, viz. that of the "abstract groups." Let us consider for example, the three following operations: 1. the addition of real numbers, their sum (positive negative or zero) being defined in the usual manner; 2. the multiplication of integers 'modulo a prime number p," (where the elements under consideration are the whole numbers 1, 2, - - *, p-1) and the "product" of two of these numbers is, by agreement, defined as the remainder of the division of their usual product by p; 3. the "composition" of displacements in three-dimensional Euclidean space, the "resultant" (or "product") of two displacements S, T (taken in this order) being defined as the displacement obtained by carrying out first the displacement T and then the displacement S. In each of these three theories, one makes correspond, by means of a procedure defined for each theory, to two elements x, y (taken in that order) of the set under consideration (in the first case the set of real numbers, in the second the set of numbers 1, 2, p-I, in the third the set of all displacements) a well-determined third element; we shall agree to designate this third element in all three cases by xry (this will be the sum of x and y if x and y are real numbers, their product "modulo p" if they are integers ?p -1, their resultant if they are displacements). If we now examine the various properties of this "operation" in each of the three theories, we discover a remarkable parallelism; but, in each of the separate theories, the properties are interconnected, and an analysis of their logical connections leads us to select a small number of them which are independent (i.e., none of them is a logical consequence of all the others). For example,* one can take the three following, which we shall express by means of our symbolic notation, common to the three theories, but which it would be very easy to translate into tlle particular language of each of them: (a) For all elements x, y, z, one has xr(yTz) (xry)rz ("associativity" of the operation xry); (b) There exists an element e, such that for every element x, one has erx -xTe=x (for the addition of real numbers, it is the number 0; for multiplication "modulo p," it is the number 1; for the composition of displacements, it is the "identical" displacement, which leaves every point of space fixed); (c) Corresponding to every element x, there exists an element x' such that xrx'=X 'rx=e (for the addition of real numbers x' is the number -x; for the * There is nothing absolute in this choice; several systems of axioms are known which are "equivalent" to the one which we are stating explicitly, the axioms of each of these systems being logical consequences of the axioms of any other one. 1950] THE ARCHITECTURE OF MATHEMATICS 225 composition of displacements, x' is the "inverse" displacement of x, i.e. the displacement which replaces each point that had been displaced by x to its original position; for multiplication "modulo p," the existence of x' follows from a very simple arithmetic argument.* It follows then that the properties which can be expressed in the same way in the three theories, by means of the common notation, are consequences of the three preceding ones. Let us try to show, for example that from XTy=XTz follows y = z; one could do this in each of the theories by a reasoning peculiar to it. But, we can proceed as follows by a method that is applicable in all cases: from the relation XTy=XTZ we derive (x' having the meaning which was defined above) x',r(xTy) =X'T(XTZ) ; thence by applying (a), (x'rx)ry= (x',Tx)Tz; by means of (c), this relation takes the form ery=erz, and finally, by applying (b), y=z, which was to be proved. In this reasoning the nature of the elements x, y, z under consideration has been left completely out of account; we have not been concerned to know whether they are real numbers, or integers ; p -1, or displacements; the only premise that was of importance was that the operation XTy on these elements has the properties (a), (b), and (c). Even if it were only to avoid irksome repetitions, it is readily seen that it would be convenient to develop once and for all the logical consequences of the three properties (a), (b), (c) only. For linguistic convenience, it is of course desirable to adopt a common terminology for the three sets. One says that a set in which an operation xry has been defined which has the three properties (a), (b), (c) is provided with a group structure (or, briefly, that it is a group); the properties (a), (b), (c) are called the axioms of ** the group structures, and the devielopment of their consequences constitutes setting up the axiomatic theory of groups. It can now be made clear what is to be understood, in general, by a mathematical structure. The common character of the different concepts designated by this generic name, is that they can be applied to sets of elements whose naturet has not been specified; to define a structure, one takes as given one or * We observe that the remainders left when the numbers x, x2, * * *, xn, * * * are divided by p, can not all be distinct; by expressing the fact that two of these remainders are equal, one shows easily that a power xm of x exists which has a remainder equal to 1; if now x' is the remainder of the division of xm-1 by p, we conclude that the product "modulo p" of x and x' is equal to 1. ** It goes without saying that there is no longer any connection between this interpretation of the word 'axiom" and its traditional meaning of "evident truth." t We take here a naive point of view and do not deal with the thorny questions, half philosophical, half mathematical, raised by the problem of the "nature" of the mathematical "beings" or "objects." Suffice it to say that the axiomatic studies of the nineteenth and twentieth centuries have gra!dually replaced the initial pluralism of the mental representation of these "beings"- thought of at first as ideal "abstractions" of sense experiences and retaining all their heterogeneity-by an unitary concept, gradually reducing all the mathematical notions, first to the concept of the natural number and then, in a second stage, to the notion of set. This latter concept, considered for a long time as "primitive" and "undefinable," has been the object of endless polemics, as a result of its extremely general character and on account of the very vague type of mental representation which it calls forth; the difficulties did not disappear until the notion of set itself disappeared (and with it all the metaphysical pseudo-problems concerning mathematical "beings" in the light of the recent work on logical formalism. From this new point of view, mathematical several relations, into which these elements enter* (in the case of groups, this was the relation z=xry between three arbitrary elements); then one postulatesthat the given relation, or relations, satisfy certain conditions (which are explicitly stated and which are the axioms of the structure under consideration.) t To set up the axiomatic theory of a given structure, amounts to the deduction of the logical consequences of the axioms of the structure, excluding every other hypothesis on the elements under consideration (in particular, every hypotheses as to their own nature). 

3.结构的概念。这可以以什么形式实现？正是在这里，公理化方法最接近实验方法。就像后者从笛卡尔主义的源头汲取力量一样，它将“划分困难，以便更好地克服困难”。在理论论证中，它将试图分离出其论点的主要观点；然后，把每一个都分开，并以抽象的形式表达出来，它就会发展出来。事实上，每个数学家都知道，如果一个人只做了一步一步地验证其组成的推论的正确性，那么一个证明就没有真正被“理解”，并没有试图对导致构建这种特定的推论链而不是其他推论链的想法有一个清晰的见解。它的后果。此后，回到正在考虑的理论，它将重新组合以前分离出来的组成元素，并探究这些不同的组成元素是如何相互影响的。这种在分析和综合之间来回穿梭的经典方法确实没有什么新意；该方法的独创性在于依赖于它的应用方式。为了举例说明我们刚刚概述的过程，我们将采用最古老（也是最简单）的公理化理论之一，即“抽象群”。让我们考虑以下三个操作：1。实数的加法，其和（正负或零）以通常的方式定义；2.整数“模素数p”的乘法（其中考虑的元素是整数1，2，-*，p-1），其中两个数的“积”一致地定义为它们通常的积除以p的余数；3。三维欧氏空间中位移的“组合”，两个位移S、T（按此顺序）的“合成”（或“乘积”）定义为通过首先执行位移T，然后执行位移S而获得的位移，y（按该顺序）是一个确定的第三元素；我们将同意在所有三种情况下用xry来表示第三个元素（如果x和y是实数，这将是x和y的和，如果它们是整数，它们的乘积“模p”？p-1，如果它们为位移，它们的结果）。如果我们现在研究这三种理论中每一种理论的“运算”的各种性质，我们会发现一种显著的平行性；但是，在每一个独立的理论中，属性是相互关联的，对它们的逻辑联系的分析使我们选择了少数独立的属性（即，没有一个属性是所有其他属性的逻辑结果）。例如，*我们可以采用以下三种，我们将通过三种理论共同的符号表示法来表示，但很容易将其翻译成它们各自的特定语言：（a）对于所有元素x，y，z，我们有xr（yTz）（xry）rz（xry操作的“关联性”）；（b） 存在一个元素e，使得对于每个元素x，一个元素具有erx-xTe=x（对于实数的加法，它是数字0；对于乘法“模p”，它是数值1；对于位移的合成，它是“相同”位移，这使得空间的每个点都是固定的）；（c） 对应于每个元素x，存在一个元素x'，使得xrx'=x'rx=e（对于实数的加法，x'是数字-x；对于*，在这个选择中没有绝对的；已知几个公理系统是“等价的”对于我们明确陈述的系统，每个系统的公理都是任何其他系统公理的逻辑结果。位移的组成，x'是x的“逆”位移，即替换已被x位移到其原始位置的每个点的位移；对于乘法“模p”，x'的存在源于一个非常简单的算术论证。*因此，可以在三种理论中以相同的方式，通过共同的表示法来表示的性质是前面三种理论的结果。让我们尝试显示，例如，从XTy=XTz到y=z；在每一个理论中，我们都可以通过其特有的推理来做到这一点。但是，我们可以通过一种适用于所有情况的方法来进行如下操作：从关系式XTy=XTZ中，我们导出（x'具有上述定义的含义）x'，r（XTy）=x T（XTZ）；=erz的形式，最后，通过应用（b），y=z，这将被证明。在这种推理中，所考虑的元素x、y、z的性质被完全排除在外；我们并不关心它们是实数还是整数；p-1或位移；唯一重要的前提是这些元素上的操作XTy具有属性（a）、（b）和（c）。即使只是为了避免令人讨厌的重复，很容易看出，只对（a）、（b）、（c）这三个性质的逻辑结果进行一次彻底的分析是很方便的。为了语言上的方便，当然需要为这三个集合采用一个共同的术语。一种说法是，其中定义了具有三个性质（a）、（b）、（c）的操作xry的集合具有群结构（或者简单地说，它是一个群）；性质（a）、（b）、（c）被称为群结构的公理，其结果的发展构成了群公理理论的建立。现在可以清楚地知道，一般来说，通过数学结构可以理解什么。由该通用名称指定的不同概念的共同特点是，它们可以应用于未指定性质的元素集合；为了定义一个结构，我们将其视为给定的一或*。我们观察到，当数字x_1、x_2、... 、xn、...被p除时剩下的余数不可能都是不同的；通过表达这两个余数相等的事实，可以很容易地证明存在余数等于1的x的幂x^m；如果现在x'是x^{m-1}除以p的余数，我们得出结论，x和x'的乘积“模p”等于1。**不用说，对“公理”一词的这种解释与其“显而易见的真理”的传统含义之间不再有任何联系。我们在这里采取一种天真的观点，不处理数学“存在”或“对象”的“本质”问题所提出的棘手问题，半哲学的，半数学的。可以说，十九世纪和二十世纪的公理化研究已经彻底取代了这些“存在”的心理表征的最初多元主义——最初被认为是理想的“抽象”“感官体验”，并通过一个统一的概念保留其所有异质性，逐渐将所有的数学概念，首先是自然数的概念，然后在第二阶段，是集合的概念，“由于其极为普遍的特点以及它所提出的非常模糊的心理表征，一直是无休止争论的对象；直到集合本身的概念消失（以及与数学“存在”相关的所有形而上学伪问题），困难才消失。”根据最近关于逻辑形式主义的工作，从这个新的观点来看，数学，这些元素进入的几个关系*（在群的情况下，这是三个任意元素之间的关系z=xry）；然后假设给定的关系满足某些条件t建立给定结构的公理化理论，相当于推导结构公理的逻辑结果，排除了关于所考虑元素的所有其他假设（特别是关于其自身性质的所有假设）。

4. The great types of structures. The relations which form the starting point for the definition of a structure can be of very different characters. The one which occurs in the group structure is what one calls a "law of composition," i.e., a relation between three elements which determines the third uniquely as a function of the first two. When the relations which enter the definition of a structure are "laws of composition," the corresponding structure is called an algebraic structure (for example, a field structure is defined by two laws of composition, with suitable axioms: the addition and multiplication of real numbers define a field structure on the set of these numbers). Another important type is furnished by the structures defined by an order relation; this is a relation between two elements x, y which is expressed most frequently in the form "x is at most equal to y," and which we shall represent in general by xRy. It is not at all supposed here that it determines one of the two elements x, y uniquely as a function of the other; the axioms to which it is subjected are the following: (a) for every x we have xRx; (b) from the relations xRy and yRx follows x = y; (c) the relations xRy and yRz have as a consequence xRz. An obvious example of a set with a structure of this kind is the set of integers (or that of real numbers), when the symbol R is replaced by the symbol. We want to say a few words about a third large type of structures, viz. topological structures (or topologies): they furnish an abstract mathematical formulation of the intuitive concepts of neighborhood, limit and continuity, to whichwe are led by our idea of space. The degree of abstraction required for theformulation of the axioms of such a structure is decidedly greater than it was inthe preceding examples; the character of the present article makes it necessaryto refer interested readers to special treatises. See, for example,

4.伟大的结构类型。构成结构定义起点的关系可以具有非常不同的特征。出现在群体结构中的一个是人们所称的“组成法则”，即三个元素之间的关系，它将第三个元素唯一地确定为前两个元素的函数。当进入结构定义的关系是“合成法则”时，对应的结构被称为代数结构（例如，域结构由两个合成法则定义，具有适当的公理：实数的加法和乘法在这些数的集合上定义域结构）。另一种重要类型是由顺序关系定义的结构提供的；这是两个元素x，y之间的关系，最常见的形式是“x至多等于y”，我们通常用xRy表示。这里根本不认为它将两个元素x，y中的一个唯一地确定为另一个的函数；它所遵循的公理如下：（a）对于每个x，我们都有xRx；（b） 根据关系xRy和yRx，x＝y；（c） 关系xRy和yRz的结果是xRz。具有这种结构的集合的一个明显例子是整数集合（或实数集合），我们想谈谈第三大类结构，即拓扑结构（或拓扑）：它们提供了邻域、极限和连续性直观概念的抽象数学公式, 我们被我们的空间观念所引导。所需的抽象程度,这样一个结构的公理的表述显然比在前面的例子深；鉴于本文的特点，有必要让感兴趣的读者阅读专门的论文。

5. The standardization of mathematical technique. We have probably said enough to enable the reader to form a fairly accurate idea of the axiomatic method. It should be clear from what precedes that its most striking feature is to effect a considerable economy of thought. The "structures" are tools for the mathematician; as soon as he has recognized among the elements, which he is studying, relations which satisfy the axioms of a known type, he has at his disposal immediately the entire arsenal of general theorems which belong to the structures of that type. Previously, on the other hand, he was obliged to forge for himself the means of attack on his problems; their power depended on his personal talents and they were often loaded down with restrictive hypotheses, resulting from the peculiarities of the problem that was being studied. One could say that the axiomatic method is nothing but the "Taylor system" for mathematics. This is however, a very poor analogy; the mathematician does not work like a machine, nor as the workingman on a moving belt; we can not over-emphasize the fundamental role played in his research by a special intuition,* which is not the popular sense-intuition, but rather a kind of direct divination (ahead of all reasoning) of the normal behavior, which he seems to have the right to expect of mathematical beings, with whom a long acquaintance has made him as familiar as with the beings of the real world. Now, each structure carries with it its own language, freighted with special intuitive references derived from the theories from which the axiomatic analysis described above has derived the structure. And, for the research worker who suddenly discovers this structure in the phenomena which he is studying, it is like a sudden modulation which orients at one stroke in an unexpected direction the intuitive course of his thought, and which illumines with a new light the mathematical landscape in which he is moving about. Let us think-to take an old example-of the progress made at the beginning of the nineteenth century by the geometric representation of imaginaries. From our point of view, this amounted to discovering in the set of complex numbers a well-known topological structure, that of the Euclidean plane, with all the possibilities for applications which this in- * Like all intuitions, this one also is frequently wrong. 228 THE ARCHITECTURE OF MATHEMATICS [April, volved; in the hands of Gauss, Abel, Cauchy and Riemann, it gave new life to analysis in less than a century. Such examples have occurred repeatedly during the last fifty years; Hilbert space, and more generally, functional spaces, establishing topological structures in sets whose elements are no longer points, but functions; the theory of the Hensel p-adic numbers, where, in a still more astounding way, topology invades a region which had been until then the domain par excellence of the discrete, of the discontinuous, viz. the set of whole numbers; Haar measure, which enlarged enormously the field of application of the concept of integral, and made possible a very profound analysis of the properties of continuous groups;-all of these are decisive instances of mathematical progress, of turning points at which a stroke of genius brought about a new orientation of a theory, by revealing the existence in it of a structure which did not a priori seem to play a part in it. What all this amounts to is that mathematics has less than ever been reduced to a purely mechanical game of isolated formulas; more than ever does intuition dominate in the genesis of discoveries. But henceforth, it possesses the powerful tools furnished by the theory of the great types of structures; in a single view, it sweeps over immense domains, now unified by the axiomatic method, but which were formerly in a completely chaotic state.

5.数学技术的标准化。我们可能已经说了足够多的话，让读者对公理化方法形成一个相当准确的概念。从前面的内容可以清楚地看出，它最显著的特点是实现相当大的思想节约。“结构”是数学家的工具；一旦他在所研究的元素中认识到满足已知类型公理的关系，他就立即掌握了属于该类型结构的所有一般定理。另一方面，此前，他不得不为自己伪造攻击自己问题的手段；他们的权力取决于他的个人才能，他们经常被限制性的假设所拖累，这是由于正在研究的问题的特殊性。可以说，公理化方法只是数学的“泰勒系统”。然而，这是一个非常糟糕的类比；数学家不像机器一样工作，也不像工人在移动的皮带上工作；我们不能过分强调一种特殊的直觉在他的研究中所起的基本作用，这种直觉不是通俗的直觉，而是一种对正常行为的直接占卜（先于一切推理），他似乎有权期待数学存在，与数学存在的长期相识使他与现实世界的存在一样熟悉。现在，每一个结构都有自己的语言，并带有从上述公理分析推导出结构的理论中得出的特殊直觉参考。而且，对于一个突然发现他正在研究的现象中的这种结构的研究工作者来说，这就像一个突然的调制，它一下子将他的直觉思维指向了一个意想不到的方向，并以一种新的光照亮了他所处的数学环境。让我们想一想19世纪初通过想象者的几何表示所取得的进步的一个老例子。从我们的观点来看，这相当于在复数集合中发现了一个众所周知的拓扑结构，欧几里得平面的拓扑结构以及所有应用的可能性，这一点与所有直觉一样，也经常是错误的。在高斯、阿贝尔、柯西和黎曼的手中，它在不到一个世纪的时间里为分析带来了新的生命。在过去的五十年里，这样的例子反复出现；希尔伯特空间，更一般地说，函数空间，在元素不再是点而是函数的集合中建立拓扑结构；亨塞尔p进数理论，其中令人震惊的是，拓扑学侵入了一个区域，在那之前，这个区域一直是离散的、不连续的、即整数集合的卓越领域；哈尔测度极大地扩大了积分概念的应用领域，并使对连续群性质的深入分析成为可能-所有这些都是数学进步的决定性实例，都是一个转折点，在这个转折点上，天才的一笔创造了一个理论的新方向，揭示了理论中存在的一种结构，而这种结构在先验上似乎不起作用。所有这些都意味着，数学比以往任何时候都少被简化为一个孤立公式的纯粹机械游戏；直觉在发现的起源中比以往任何时候都更占主导地位。但从此以后，它拥有由大类型结构理论提供的强大工具；在一个单一的视图中，它席卷了巨大的领域，现在通过公理化方法统一了，但这些领域以前处于完全混乱的状态。

6. A general survey. Let us now try, guided by the axiomatic concept, to look over the whole of the mathematical universe. It is clear that we shall no longer recognize the traditional order of things, which, just like the first nomenclatures of animal species, restricted itself to placing side by side the theories which showed greatest external similarity. In place of the sharply bounded compartments of algebra, of analysis, of the theory of numbers, and of geometry, we shall see, for example, that the theory of prime numbers is a close neighbor of the theory of algebraic curves, or, that Euclidean geometry borders on the theory of integral equations. The organizing principle will be the concept of a hierarchy of structures, going from the simple to the complex, from the general to the particular. At the center of our universe are found the great types of structures, of which the principal ones were mentioned above; they might be called the mother-structures. A considerable diversity exists in each of these types; one has to distinguish between the most general structure of the type under consideration, with the smallest number of axioms, and those which are obtained by enriching the type with supplementary axioms, from each of which comes a harvest of new consequences. Thus, the theory of groups contains, beyond the general conclusions valid for all groups and depending only on the axioms enunciated above, a particular theory of finite groups (obtained by adding the axiom that the number of elements of the group is finite), a particular theory of abelian groups (in which x'ry=yrx for every x and y), as well as a theory of finite abelian groups (where these two axioms are supposed to hold simultaneously). Similarly, in the theory of ordered sets, one notices in particular those sets (as for example, the set of integers, or of real numbers) in which any two elements are comparable, and which are called totally ordered. Among the latter, further attention is given to the sets which are called well-ordered (in which, as in the set of integers greater than 0, every subset has a "least element"). There is an analogous gradation among topological structures. Beyond this first nucleus, appear the structures which might be called multiple structures. They involve two or more of the great mother-structures simultaneously not in simple juxtaposition (which would not produce anything new), but combined organically by one or more axioms which set up a connection between them. Thus, one has topological algebra. This is a study of structures in which occur at the same time, one or more laws of composition and a topology, connected by the condition that the algebraic operations be (for the topology under consideration) continuous functions of the elements on which they operate. Not less important is algebraic topology, in which certain sets of points in space, defined by topological properties (simplexes, cycles, etc.) are themselves taken as elements on which laws of composition operate. The combination of order structures and algebraic structures is also fertile in results, leading, in' one direction to the theory of divisibility and of ideals, and in another to integration and to the "spectral theory" or operators, in which topology also joins in. Farther along we come finally to the theories properly called particular. In these the elements of the sets under consideration, which, in the general structures have remained entirely indeterminate, obtain a more definitely characterized individuality. At this point we merge with the theories of classical mathematics, the analysis of functions of a real or complex variable, differential geometry, algebraic geometry, theory of numbers. But they have no longer their former autonomy; they have become crossroads, where several more general mathematical structures meet and react upon one another. To maintain a correct perspective, we must at once add to this rapid sketch, the remark that it has to be looked upon as only a very rough approximation of the actual state of mathematics, as it exists; the sketch is schematic, and idealized as well as frozen. Schematic-because in the actual procedures, things do not happen in as simple and as systematic a manner as has been described above. There occur, among other things, unexpected reverse movements, in which a specialized theory, such as the theory of real numbers, lends indispensable aid in the construction of a general theory like topology or integration. Idealized-because it is far from true that in all fields of mathematics, the role of each of the great structures is clearly recognized and marked off; in certain theories (for example in the theory of numbers), there remain numerous isolated results, which it has thus far not been possible to classify, nor to connect in a satisfactory way with known structures. Finally frozen,-for nothing is farther from the axiomatic method than a static conception of the science. We do not want to lead the reader to think that we claim to have traced out a definitive state of the science. The structures  are not immutable, neither in number nor in their essential contents. It is quite possible that the future development of mathematics may increase the number of fundamental structures, revealing the fruitfulness of new axioms, or of new combinations of axioms. We can look forward to important progress from the invention of structures, by considering the progress which has resulted from actually known structures. On the other hand, these are by no means finished edifices; it would indeed be very surprising if all the essence had already been extracted from their principles. Thus, with these indispensable qualifications, we can become better aware of the internal life of mathematics, of its unity as well as of its diversity. It is like a big city, whose outlying districts and suburbs encroach incessantly, and in a somewhat chaotic manner, on the surrounding country, while the center is rebuilt from time to time, each time in accordance with a more clearly conceived plan and a more majestic order, tearing down the old sections with their labyrinths of alleys, and projecting towards the periphery new avenues, more direct, broader and more commodious.

6.综述。现在，让我们在公理化概念的指导下，审视整个数学世界。很明显，我们将不再承认传统的事物顺序，就像最初的动物物种命名法一样，它只限于将表现出最大外部相似性的理论放在一起。例如，在代数、分析、数论和几何中，我们将看到素数理论是代数曲线理论的近邻，或者欧几里得几何与积分方程理论接壤。组织原则将是结构层次的概念，从简单到复杂，从一般到特殊。在我们宇宙的中心发现了各种各样的结构，上面提到了其中的主要结构；它们可能被称为母体结构。每一种类型都存在相当大的差异；人们必须区分所考虑的类型的最一般的结构（公理数量最少）和通过用补充公理丰富类型而获得的结构，每一个都会带来新的结果。因此，除了对所有群有效的一般结论之外，群理论还包含一个特定的有限群理论（通过添加群的元素的数量是有限的公理而获得）、一个特定阿贝尔群理论（其中每个x和y的x’ry=yrx）、，以及有限阿贝尔群理论（假设这两个公理同时成立）。类似地，在有序集理论中，我们特别注意到那些集合（例如整数或实数的集合），其中任何两个元素都是可比较的，并且称为完全有序的。在后者中，进一步关注被称为良序的集合（其中，如在大于0的整数集合中，每个子集都有一个“最小元素”）。拓扑结构之间也有类似的层次。在第一个核之外，出现了可以称为多重结构的结构。它们同时涉及两个或多个伟大的母体结构，而不是简单的并置（这不会产生任何新的东西），而是通过一个或多条公理有机地结合在一起，在它们之间建立联系。因此，一个人有拓扑代数。这是对同时出现一个或多个组成法则和拓扑的结构的研究，通过代数运算（对于所考虑的拓扑）是它们所操作的元素的连续函数的条件来连接。同样重要的是代数拓扑，其中由拓扑性质（单纯形、循环等）定义的空间中的某些点集本身被视为构成法则的元素。顺序结构和代数结构的结合在结果上也很丰富，在一个方向上导致了可分性和理想理论，在另一个方向导致了积分和“谱理论”或算子，拓扑也加入其中。在这些方面，所考虑的集合的元素，在一般结构中仍然完全不确定，获得了更明确的个性特征。在这一点上，我们与经典数学理论、实变量或复变量的函数分析、微分几何、代数几何、数论相结合。但他们不再拥有以前的自主权；它们已经成为十字路口，在那里，几个更一般的数学结构相遇并相互反应。为了保持正确的观点，我们必须立即在这幅速写图上加上这样一句话，即它必须被视为数学实际状态的一个非常粗略的近似值，因为它是存在的；草图是示意性的，理想化的，也是冻结的。示意图，因为在实际程序中，事情并不像上面描述的那样简单和系统地发生。除其他外，还出现了意想不到的反向运动，其中一个专门的理论，如实数理论，为构建拓扑或积分等一般理论提供了不可或缺的帮助。理想化是因为在数学的所有领域中，每一个伟大结构的作用都得到了明确的认识和区分，这一点远非事实；在某些理论中（例如，在数论中），仍然有许多孤立的结果，迄今为止还无法将其分类，也无法以令人满意的方式与已知结构联系起来。最后被冻结了，因为没有什么比静态的科学概念更远离公理化方法了，我们不想让读者认为我们声称已经找到了科学的确定状态。这些结构不是一成不变的，无论是在数量上还是在其基本内容上。未来数学的发展很可能会增加基本结构的数量，揭示出新公理或新公理组合的有效性。通过考虑实际已知结构所产生的进展，我们可以期待结构发明的重要进展。另一方面，这些都不是完工的大厦；如果所有的本质都已经从他们的原则中提取出来，那真的会非常令人惊讶。因此，有了这些必不可少的条件，我们可以更好地了解数学的内在生活，了解数学的统一性和多样性。它就像一座大城市，其外围地区和郊区以某种混乱的方式不断侵占周围的国家，而市中心则不时进行重建，每一次都是按照一个更清晰的计划和更宏伟的秩序进行重建，用迷宫般的小巷拆除旧的部分，并向外围地区伸出新的街道，更直接、更广泛、更宽敞

7. Return to the past and conclusion. The concept which we have tried to present in the above paragraphs, was not formed all at once; rather is it a stage in an evolution, which has been in progress for more than a half-century, and which has not escaped serious opposition, among philosophers as well as among mathematicians themselves. Many of the latter have been unwilling for a long time to see in axiomatics anything else than futile logical hairsplitting not capable of fructifying any theory whatever. This critical attitude can probably be accounted for by a purely historical accident. The first axiomatic treatments and those which caused the greatest stir (those of arithmetic by Dedekind and Peano, those of Euclidean geometry by Hilbert) dealt with univalent theories, i.e., theories which are entirely determined by their complete system of axioms; for this reason they could not be applied to any theory except the one from which they had been extracted (quite contrary to what we have seen, for instance, for the theory of groups). If the same had been true for all other structures, the reproach of sterility brought against the axiomatic method, would have been fully justified.* But the further development of the method has revealed its power; and the repugnance which it still meets here and there, can only be explained by the natural difficulty of the mind to admit, in dealing with a concrete problem, that a form of intuition, which is not suggested directly by the given elements (and which often can be arrived at only by a higher and frequently difficult stage of abstraction), can turn out to be equally fruitful. As concerns the objections of the philosophers, they are related to a domain, on which for reasons of inadequate competence we must guard ourselves from * There also occurred, especially at the beginning of axiomatics, awhole crop of monsterstructures, entirely without applications; their sole merit was that of showing the exact bearing of each axiom, by observing what happened if one omitted or changed it. There was of course a temptation to conclude that these were the only results that could be expected from the axiomatic method. 1entering; the great problem of the relations between the empirical world and the mathematical world.* That there is an intimate connection between experimental phenomena and mathematical structures, seems to be fully confirmed in the most unexpected manner by the recent discoveries of contemporary physics. But we are completely ignorant as to the underlying reasons for this fact (supposing that one could indeed attribute a meaning to these words) and we shall perhaps always remain ignorant of them. There certainly is one observation which might lead the philosophers to greater circumspection on this point in the future: before the revolutionary developments of modern physics, a great deal of effort was spent on trying to derive mathematics from experimental truths, especially from immediate space intuitions. But, on the one hand, quantum physics has shown that this macroscopic intuition of reality covered microscopic phenomena of a totally different nature, connected with fields of mathexmatics which had certainly not been thought of for the purpose of applications to experimental science. And, on the other hand, the axiomatic method has shown that the "truths" from which it was hoped to develop mathematics, were but special aspects of general concepts, whose significance was not limited to these domains. Hence it turned out, after all was said and done, that this intimate connection, of which we were asked to admire the harmonious inner necessity, was nothing more than a fortuitous contact of two disciplines whose real connections are much more deeply hidden than could have been supposed a priori. From the axiomatic point of view, mathematics appears thus as a storehouse of abstract forms-the mathematical structures; and it so happens-without our knowing why-that certain aspects of empirical reality fit themselves into these forms, as if through a kind of preadaptation. Of course, it can not be denied that most of these forms had originally a very definite intuitive content; but, it is exactly by deliberately throwing out this content, that it has been possible to give these forms all the power which they were capable of displaying and to prepare them for new interpretations and for the development of their full power. It is only in this sense of the word "form" that one can call the axiomatic method a "formalism." The unity which it gives to mathematics is not the armor of formal logic, the unity of a lifeless skeleton; it is the nutritive fluid of an organism at the height of its development, the supple and fertile research instrument to which all the great mathematical thinkers since Gauss have contributed, all those who, in the words of Lejeune-Dirichlet, have always labored to "substitute ideas for calculations."

7.

我们在上述段落中试图提出的概念并不是一下子形成的；相反，它是进化的一个阶段，进化已经进行了半个多世纪，并且没有逃过哲学家和数学家之间的严重反对。后者中的许多人很长一段时间以来都不愿意在公理论中看到任何其他东西，而只是徒劳的逻辑分裂，无法使任何理论产生结果。这种批判态度很可能是一次纯粹的历史事故造成的。最初的公理化处理和引起最大轰动的处理（Dedekind和Peano的算术处理，Hilbert的欧几里德几何处理）涉及一价理论，即完全由其完整公理体系决定的理论；由于这个原因，它们不能应用于任何理论，除非它们是从中提取出来的（与我们所看到的相反，例如群理论）。如果所有其他结构都是如此，那么对公理化方法的无菌指责将是完全合理的。*但该方法的进一步发展已经揭示了它的威力；而它仍然在这里和那里遇到的反感，只能用头脑在处理具体问题时自然难以承认的一种直觉形式来解释，这种直觉形式不是由给定的元素直接提出的（而且往往只能通过更高且经常困难的抽象阶段来实现），结果可能同样富有成效。至于哲学家的反对意见，它们与一个领域有关，由于能力不足，我们必须防范这个领域。此外，特别是在公理论的初期，也出现了一系列完全没有应用的畸形结构；它们的唯一优点是通过观察如果省略或更改它所发生的事情，来显示每个公理的确切方位。当然，有人倾向于断定这些是公理化方法所能预期的唯一结果。1进入；经验世界和数学世界之间关系的巨大问题。*实验现象和数学结构之间有着密切的联系，这似乎以最意想不到的方式得到了当代物理学的最新发现的充分证实。但我们完全不知道这一事实的根本原因（假设人们确实可以给这些词赋予某种意义），我们可能永远都不知道它们。当然，有一个观察结果可能会让哲学家们在未来对这一点更加谨慎：在现代物理学的革命性发展之前，人们花了大量精力试图从实验真理，特别是从直接的空间直觉中推导出数学。但是，一方面，量子物理学表明，这种宏观的现实直觉涵盖了完全不同性质的微观现象，与数学领域相关，而数学领域肯定没有被认为是用于实验科学的目的。另一方面，公理化方法表明，希望从中发展数学的“真理”只是一般概念的特殊方面，其意义并不局限于这些领域。因此，事实证明，无论说了什么，做了什么，这种亲密的联系，我们被要求欣赏和谐的内在必要性，无非是两个学科之间的偶然接触，而这两个学科的真正联系远比人们想象的要深得多。从公理化的观点来看，数学似乎是一个抽象形式的仓库——数学结构；这是在我们不知道为什么经验现实的某些方面会融入这些形式的情况下发生的，就像是通过一种预适应。当然，不能否认的是，这些形式中的大多数最初都有非常明确的直观内容；但是，正是通过故意抛出这些内容，才有可能赋予这些形式它们所能展现的所有力量，并使它们为新的解释和充分发挥其力量做好准备。只有在“形式”这个词的意义上，人们才能将公理化方法称为“形式主义”。它赋予数学的统一不是形式逻辑的铠甲，而是一个没有生命的骨架的统一；它是一个处于发育高峰的有机体的营养液，是自高斯以来所有伟大的数学思想家所贡献的柔软而肥沃的研究工具，用莱杰恩·狄利克雷的话来说，所有这些人都致力于“用思想代替计算.

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