亚伦道弗尔奖（Carl B. Allendoerfer）是美国数学协会（MAA）每年颁发的一个数学奖项，旨在奖励在《数学杂志》（Mathematics Magazine ）上发表的论述清晰的论文。

《数学杂志》是美国数学协会的双月刊，其受众主要是大学数学老师以及数学系学生，刊登的文章主要为数学概念和数学理论提供例子、应用和介绍理论概念被提出的历史背景。

Ernst Snapper所著的“The Three Crises in Mathematics: Logicism, Intuitionism, and Formalism”获得1980年的亚伦道弗尔奖论文奖。

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题目：数学中的三个危机 ：逻辑主义、直觉主义和形式主义

经典哲学的危机揭示了人们对提供数学基础的数学和哲学标准的困惑。

摘要：题目中提到的三个学派都试图给数学提供一个坚实的基础，三个危机意味着这些学派未能完成其任务。本文“以现代的眼光”审视这些危机，使用今天可用的数学，而不仅仅是创建这些学派的先驱们所用的数学。因此，本文不以严格的历史方式来处理这三个危机，也不讨论目前大量的、技术性的数学，这些数学是由上述三个学派所引入的技术而产生的。原因之一是，这样的讨论需要一本书，而不是一篇短文；原因之二是，所有这些技术性数学与数学哲学关系不大，而在这篇文章中，我想强调逻辑主义、直觉主义和形式主义中那些建立在哲学之上的内容。

逻辑主义

这个学派大约在1884年由德国哲学家、逻辑学家和数学家戈特洛布-弗雷格（1848-1925）开创，在18年之后被伯特兰-罗素重新发现，其他早期的逻辑学家是皮亚诺和罗素的《数学原理》的合作者怀特海（A.N. Whitehead）。逻辑主义的目的是要表明，经典数学是逻辑学的一部分。如果逻辑学家能够成功地执行他们的计划，诸如“为什么经典数学没有矛盾？ ”就会变成“为什么逻辑学没有矛盾？ ”这后一个问题至少是哲学家们可以彻底解决的问题。一般来说，逻辑主义的计划成功完成后，会给经典数学在逻辑方面打下坚实的基础。

显然，为了执行逻辑主义的这个计划，首先必须以某种方式定义什么是 "经典数学"，什么是 "逻辑"，应该指出什么是什么的一部分？恰恰是这两个定义，我们希望通过现代人的眼光来看，想象逻辑主义的先驱们拥有今天所有的数学知识。我们从经典数学开始。

为了实施他们的计划，罗素和怀特海创建了“数学原理”（Principia Mathematica），并于1910年出版，可以看作是一种形式化的集合论。虽然这个形式化理论并未完成，但罗素和怀特海认为它是完整的，并计划用它来证明数学可以被还原为逻辑。他们表明，在他们那个时代已知的所有经典数学，都可以从集合论中推导出来，从而从“数学原理”的公理中推导出来。因此，剩下要做的就是证明“数学原理”的所有公理都属于逻辑。

当然，我们也可以用任何形式化的集合论来代替“数学原理”。由于今天由Zermelo和Frankel（ZF）开发的集合论比“数学原理”要有名，所以我们从现在开始将提到ZF而不是“数学原理”。ZF只有九个公理，虽然其中有几个实际上是公理模式，但我们将把它们全部称为“公理”。逻辑主义者的方案现在变成：证明ZF的所有九个公理都属于逻辑。

这种逻辑主义的表述是基于这样一个论点：经典数学可以被定义为在ZF范围内可以被证明的定理集合。这种对经典数学的定义远非完美，正如在[12]中讨论的那样。然而，上述逻辑主义的表述对于说明这个学派无法实施其计划的目的来说是令人满意的。我们现在来谈谈逻辑的定义。

为了理解逻辑主义，重要的是要看清楚逻辑主义的“逻辑”指什么，无论其所指是什么，肯定超出了经典逻辑的意义。如今，人们可以把经典逻辑定义为所有那些不使用非逻辑公理可以用一阶语言（在下面的形式主义部分讨论）证明的理论。因此，我们把自己限制在一阶逻辑中，并使用该逻辑的演绎规则和逻辑公理。一个定理的例子是排中律：如果p是一个命题，那么p或者它的否定为真；换句话说，命题p ∨ ¬ p总是真的。

如果这个经典逻辑的定义也是逻辑主义者对逻辑的定义，那么，哪怕只有一秒钟就可以看出，认为所有的ZF都可以归结为逻辑，是一种愚蠢的想法。然而，逻辑主义的定义更为广泛，他们对于一个命题何时属于逻辑，也就是说，一个命题何时应该被称为“逻辑命题”，有一个一般的概念。他们说: 一个逻辑命题是一个具有完整的一般意义的命题，它的真实性取决于它的形式而不是内容。这里，“命题”一词被用作“定理”的同义词。

例如；上述排中律 p ∨ ¬ p是一个逻辑命题。也就是说，这个定律并不取决于命题p的任何特殊内容，不管p是数学命题还是物理学命题，都不重要。相反，这个定律以“完全的普遍性”而成立，即对许多命题p都是如此。那为什么它能成立呢？逻辑主义者回答说： “因为它的形式”，这里他们所说的形式是指“句法形式”，p ∨ ¬ p的形式是由两个日常用语的连接词给出的，即“或”和否定的“非”。

一方面，不难论证，所有经典逻辑的理论，如上文所定义的，都是逻辑学意义上的逻辑命题。另一方面，没有先验的理由相信，不可能有经典逻辑之外的逻辑命题。这就是为什么我们说，逻辑主义对逻辑的定义比经典逻辑的定义更广泛。现在，逻辑主义者的任务变得更清楚了 ：包括证明ZF的所有九个公理都是逻辑主义意义上的逻辑命题。

评估逻辑主义在执行这一任务方面成败的唯一方法，就是要考察ZF的九条公理，并确定每条公理在逻辑主义的逻辑命题概念下是否失败。这将需要一篇单独的文章，只有对ZF完全熟悉的读者才会感兴趣。因此，我们只想说，这些公理中至少有两个，即无穷公理和选择公理，不可能被视为逻辑命题。例如，无穷公理说，存在着无限集。为什么我们接受这个公理？原因是每个人都熟悉许多无限集，例如自然数集或欧几里得空间的点集。因此，我们接受这个公理是基于我们对集合的日常经验，这清楚地表明，我们接受它是由于它的内容，而不是由于它的句法形式。一般来说，当一个公理声称我们根据日常经验所熟悉的对象的存在时，可以肯定这个公理不是逻辑主义意义上的逻辑命题。

这里是数学的第一个危机：由于ZF的九个公理中至少有两个不是逻辑主义意义上的逻辑命题，可以说这个学派在给数学一个坚实的基础的努力中失败了大约20%。然而，逻辑主义对于现代数学逻辑的发展是重要的。事实上，正是逻辑主义以严格的方式开始了数理逻辑。弗雷格将两个量词，即“所有”和“存在”量词引入了逻辑学，而《数学原理》对数理逻辑发展的影响已成为历史。

重要的是，要认识到逻辑主义是建立在哲学基础上的。例如，当逻辑主义者告诉我们逻辑命题的意思时，他们使用的是哲学语言而不是数学语言。他们必须使用哲学语言来达到这个目的，因为数学根本无法处理如此广泛的定义。

逻辑主义哲学有时被说成是基于被称为“现实主义”（realism）的哲学流派。在中世纪的哲学中， “现实主义”代表了柏拉图的学说，即抽象实体有一个独立于人类思维的存在。当然，数学中充满了抽象的实体，如数、函数、集合等，根据柏拉图的说法，所有这些实体都存在于我们的思想之外，心智可以发现它们，但不能创造它们。这种学说的好处是，人们可以接受“集合”这样的概念，而不必担心智如何构建集合。根据现实主义，集合是为我们所发现的，而不是被我们所构造的，所有其他抽象实体也是如此。简而言之，现实主义允许我们在数学中接受更多的抽象实体，而不是将我们限制在只接受人类思维所能构造的实体的哲学。罗素是一个现实主义者，他接受了经典数学中出现的抽象实体，而没有质疑我们自己的头脑是否能够构造它们。这是逻辑主义和直觉主义之间的根本区别，因为在直觉主义中，只有当抽象实体是人为制造的，才会被接受。

原文：

The Three Crises in Mathematics : Logicism, Intuitionism and Formalism

Crises in classical philosophy reveal doubts about mathematical and philosophical criteria for a satisfactory foundation for mathematics

The three schools, mentioned in the title, all tried to give a firm foundation to mathematics. The three crises are the failures of these schools to complete their tasks. This article looks at these crises « through modern eyes », using whatever mathematics is available today and not just the mathematics which was available to the pioneers who created these schools. Hence, this article does not approach the three crises in a strictly historical way. This article also does not discuss the large volume of current, technical mathematics which has arisen out of the techniques introduced by the three schools in question. One reason is that such a discussion would take a book and not a short article. Another one is that all this technical mathematics has very little to do with the philosophy of mathematics, and in this article I want to stress those aspects of logicisme, intuitionism, and formalism which show clearly that these schools are founded in philosophy.

Logicism

This school was started in about 1884 by the German philosopher, logician and mathematician, Gottlob Frege (1848-1925). The school was rediscover about eighteen years later by Bertrand Russell. Other early logiciels were Peano and Russell’s coauthor of Principia Mathematica, A.N. Whitehead. The purpose of logicism was to show that classical mathematics is part of logic. If the logiciels had been able to carry out their program successfully, such questions as « Why is classical mathematics free of contradictions? » Would have become « Why is logic free of contradictions ? » This latter question is one on which philosophers have at least a thorough handle and one may say in general that the successful completion of the logicists’ program would have given classical mathematics a firm foundation in term of logic. 

Clearly, in order to carry out this program of the logiciels, one must first, somehow, define what « classical mathematics » is and what « logic » is. Otherwise, what are we supposed to show is part of what ? It is precisely at these two definitions that we want yo look through modern eyes, imagining that the pioneers of logicisme had all of present-day mathematics available to them. We begin with classical mathematics.

In order to carry out their program, Russell and Whitehead created Principia Mathematics which was published in 1910. Principia, as we will refer to Principia Mathematics, may be considered as a formal set theory. Although the formalization was not entirely complete, Russell and Whitehead though that it was and planned to use it to show that mathematics can be reduced to logic. They showed that all classical mathematics, known in their time, can be derived from set theory and hence from the axioms of Principia. Consequently, what remained to be done, was to show that all the axioms of Principia belong to logic.

Of course, instead of Principia, one can use any formal set theory just as well. Since today the formal set theory developed by Zermelo and Frankel (ZF）is so much better known than Principia, we shall from now on refer to ZF instead of Principia. ZF has only nine axioms and, although several of them are actually axiom schemas, we shall refer to all of them as « axioms ». The formulation of the logicists’ program now becomes : Show that all nine axioms of ZF belongs to logic.

This formulation of logicism is based on the thesis that classical mathematics can be defined as the set of theorems which can be proved within ZF. This definition of classical mathematics is far from perfect, as is discussed in [12]. However, the above formulation of logicism is satisfactory for the purpose of showing that this school was not able to carry out its program. We now turn to the definition of logic.

In order to understand logicism, it is very important to see clearly what the logicists meant by « logic ». The reason is that whatever they meant, they certainly meant more than classical logic. Nowadays one can define classical logic as consisting of all those theories which can be proven in first order languages (discussed below in the section on formalism) without the use of nonlogical axioms. We are hence restricting ourselves to first order logic and use the deduction rules and logical axioms of the logic. An example of such a theorem is the law of the excluded middle which says that, if p is a proposition, then either p or its negation is true; in other words, the proposition p ou not p is always true where ou is the usual symbol for the inclusive « or ».

If this definition of classical logic had also been the logicists’s definition of logic, it would be a folly to think for even one second that all of ZF can be reduced to logic. However the logicists’ definition was more extensive. They had a general concept as to when a proposition belongs to logic, that is, when a proposition should be called a « logical proposition ». They said : A logical proposition is a proposition which has complete generally and is true in virtue of its form rather than its content. Here, the word « proposition » is used as synonymous with « theorem ». 

For example; the above law of the excluded middle « p ou not p » is a logical proposition. Namely, this law does not hold because of any special content of the proposition p; it does not matter whether ^p is a proposition of mathematics to physics or what have you. On the contrary, this law holds with « complete generality », that isn for many proposition p whatsoever. Why then doest it hold . The logiciels answer : « Because of its form. » Here they mean by form « syntactical form », the form of p ∨ ¬ p being given by the two connectives of everyday speech, the inclusive « or » and the negation « not ».

On the one hand, It is not difficult to argue that all theories of classical logic, as defined above, are logical propositions in the sense of logicisme. On the other hand, there is no a priori reason to believe that there could not be logical propositions which lie outside of classical logic. This is why we said that the logicists’ definition of logic is more extensive than the definition of classical logic. And now the logicists’ task becomes clearer : It consists in showing that all nine axiomes of ZF are logical propositions in the sense of logicism. 

The only way to assess the success or failure of logicism in carrying out this task is by going through all nine axiom of ZF and determining for each of them whether it fails under the logicists’ concept of a logical proposition. This would take a separate article and would be of interest only to readers who are thoroughly familiar with ZF. Hence, instead, we simply state that at least two of these axioms, namely, the axiom of infinity and the axiom of choice, cannot possibly ne considered as logical propositions. Form example, the axiom of infinity says that there exist infinite sets. Why do we accept this axiom as being true ? The reason is that everyone is familiar with so many infinite sets, say, the set of the natural numbers or the set of points in Euclidean 3-space. Hence, we accept this axiom on grounds of our everyday experience with sets, and this clearly shows that we accept it in virtue of its content ad not in virtue of its syntactical form. In general, when an axiom claims the existence of objects with which we are familiar on grounds of ours common everyday experience, it is pretty certain that this axiom is not a logical proposition in the sense of logicism.

And here then is the first crisis in mathematics : Since at least two out of the nine axioms of ZF are not logical propositions in the sense of logicisme, it is fair too say that this school failed by about 20% in its effort to give mathematics a firm foundation. However, logicism has been of the greatest importance for the development of modern mathematical logic. In fact, it was logicisme which started mathematical logic in serious way. The two quantifiers, the ‘for all » quantifiers sans « there exists » quantifier were introduced into logic by Frege, and the influence of Principia on the development of mathematical logic is history.

It is important to realize that logicism is founded in philosophy. For example, when the logiciels tell us what they mean by a logical proposition (above), they use philosophical and not mathematical language. They have tp use philosophical language for the purpose since mathematics simply cannot handle definitions of so wide a scope.

The philosophy of logicism is sometimes said to be based on the philosophical school called « realism ». In medieval philosophy « realism » stood for the Platonic doctrine that abstract entities have an existence independent of the human mind. Mathematics is, of course, full of abstract entities such as numbers, functions, sets etc and according to Plato all such entities exist outside our mind. The mind can discover them but does not create them. This doctrine has the advantage that one can accept such a concept as « set » without worry about how the mind can construct a set. According to realism, sets are there for us to discover, not to be constructed, and the same holds for all other abstract entities. In short, realism allows us to accept many more abstract entities in mathematics than a philosophy which had limited us to accepting only those entities the human mind can construct. Russell was a realist and accepted the abstract entities which occur in classical mathematics without questioning whether our own minds can construct them. This is the fundamental difference between logician and intuitionism, since in intuitionism abstract entities are admitted only if they are man made.