2021-6-19 14:37

《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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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 (ZFis 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.