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数学中的三个危机 :逻辑主义、直觉主义和形式主义 - 译文(2):直觉主义 精选

已有 3481 次阅读 2021-6-21 05:20 |个人分类:在法国教逻辑课|系统分类:海外观察

这个学派大约在1908年由荷兰数学家布劳威尔(L.E.J.Brouwer1881-1966)开创。直觉主义以一种与逻辑主义完全不同的方式来看待数学基础:逻辑主义从不认为经典数学有什么问题,他们固执想表明经典数学是逻辑学的一部分;而直觉主义则相反,他们认为经典数学存在很多问题。

1908年,康托尔的集合论中出现了悖论,这里,悖论这个词被当作矛盾的同义词来使用。康托尔在1870年左右开创集合论,他的工作是朴素的,即非公理化。因此,他在形成集合时是如此的随意,以至于他自己、罗素和其他人在他的理论中发现了一些悖论。逻辑主义者认为这些悖论是常见的错误,是由数学家犯的错误造成的,而不是由数学的错误造成的。另一方面,直觉主义者认为这些悖论清楚地表明,经典数学本身远非完美,他们认为,数学必须自下而上地重建。

底部,对直觉主义而言就是数学的开端,是对自然数1,2,3…的解释(请注意,在自然数中不包括这个数)。根据直觉主义哲学,人类对自然数有一种内在的原始直觉,这首先意味着,我们对数字1的意义有一种直接的肯定,其次,形成数字1的心理过程可以重复。当我们重复它时,我们得到了数字2的概念;当我们再次重复它时,得到了数字3的概念;以这种方式,人类可以为任何自然数n构造任何有限的初始段12n 如果我们内在没有时间意识,这种一个接一个的自然数的心智构造是永远不可能的。之后(After指的是时间,布劳威尔同意哲学家伊曼纽尔-康德(1724-1804)的观点,即人类对时间有一种直接的认识,康德用直觉intuition)一词来表示即时意识immediate awareness),这就是直觉主义这一名称的由来。

重要的是要注意,自然数的直觉构造只允许人们构造任意长的有限初始段1,2,…n,它不允许构造所有自然数的整个封闭集合,而这在经典数学中是非常熟悉的。同样重要的是要注意到,这种构造既是归纳的inductive),也是能行的effective)。归纳的意义在于,如果一个人想要构造数字3,就必须经历所有的心智步骤,首先构造1,然后是2,最后是3;一个人不可能从天空中抓出3这个数字。它是能行的,因为一旦完成了一个自然数的构造,这个自然数就完全被构造出来了,它作为一个完全完成的心智构造站在我们面前,为我们研究它做好准备。当有人说: “我已经完成了数字3的心智构造,这就像一个砌砖工人说:我已经完成了砌这堵墙,他只有在把每块石头铺好之后才能说出来。

我们现在来谈谈直觉主义对数学的定义。根据直觉主义哲学,数学应该被定义为一种心智活动,而不是一套定理(如上文关于逻辑主义的部分),它是一种活动,一个接一个地进行,那些归纳性的、能行的心智构造,就像自然数的直觉主义构造是归纳的能行的一样。直觉主义认为,人类能够识别一个特定的心智构造是否具有这两个属性。我们将把具有这两个特性的构造称为心智构造。因此,数学的直觉主义定义是: 数学是一种心智活动,一个接一个地进行构造。

这个定义的一个主要后果是,所有的直觉主义数学都是能行的,或者像人们通常所说的那样是构造的。从现在开始,我们将使用构造的这个形容词作为能行的同义词。也就是说,每一个构造都是构造性的,而直觉主义数学不过是在不断地进行构造。例如,如果一个实数r出现在一个直觉主义的证明或定理中,它绝不会仅仅因为存在证明而出现在那里,它出现在那里是因为它被从上到下地构造起来了。例如,这意味着r的每个小数位在原则上是能计算出来的。简言之,所有直觉主义的证明、定理、定义等等,都是完全构造性的。

一旦数学的直觉主义定义被理解和接受,剩下要做的就是以直觉主义的方式做数学。的确,直觉主义已经发展了直觉主义的算术、代数、分析、集合论等。然而,在这些数学的每一个分支中,都会出现一些经典的定理,这些定理并不是由构造组成的,因此,对直觉主义来说,这是毫无意义的文字组合。因此,我们不能说直觉主义已经重构了所有的经典数学,但这并不妨碍直觉主义者,因为他们不能得到的经典数学的部分无论是什么,对他们来说都是无意义的。直观主义的目的并不是要证明经典数学的合理性,而是给出一个数学的有效定义(valid definition),然后等着看从中产生什么数学。对直觉主义者来说,在直觉上经典数学不能实现的,都不是数学。我们在这里观察到,逻辑主义和直觉主义之间的另一个基本区别:逻辑主义者想要证明所有的经典数学。

现在让我们问,直觉主义学派在给我们提供一个良好的数学基础方面有多大的成功,为大多数数学家所接受。再一次,这个问题的回答方式与逻辑主义的回答方式有很大的不同:即使是坚定的逻辑主义者也不得不承认,他们的学派到目前为止大约有20%还没有给数学一个坚实的基础;然而,一个坚定的直觉主义者完全有权利声称直觉主义已经给了数学一个完全满意的基础,有上面讨论的直觉主义的有意义的数学定义,有直觉主义哲学告诉我们为什么构造永远不会产生矛盾,因此,直觉主义数学是没有矛盾的。事实上,不仅是这个问题(没有矛盾),而且所有其他基础性的问题都在直觉主义中得到了完美的解决。

然而,如果从外部来看直觉主义,即从经典数学家的角度来看,就不得不说直觉主义没能给数学一个合适的基础。事实上,数学界几乎毫无例外地拒绝直觉主义,尽管直觉主义有许多非常有吸引力的特点,其中一些已经提到了的,但数学界为什么会这样做呢?

一个原因是,经典数学家断然拒绝放弃许多美丽的定理,而这些定理对直觉主义来说是毫无意义的文字组合。一个例子是拓扑学的布劳威尔定点定理,直觉主义拒绝该定点,因为该定点不能被构造出来,而只能通过存在证明来证明其存在。顺便说一下,这就是创造了直觉主义的布劳威尔,他的工作在(非直觉主义)拓扑学中也同样著名。

第二个原因来自于那些既能以经典方式又能以直觉方式证明的定理。经常发生的情况是,这样一个定理的经典证明很短,很优雅,也很有技巧,但不是构造性的。直观主义者当然会拒绝这样的证明,而用他们自己的构造性证明来代替同一定理。然而,这种构造性证明往往是经典证明的十倍,而且至少在经典数学家看来,失去了所有的优雅。一个例子是代数的基本定理,它在经典数学中的证明约为半页,但在直觉主义数学中的证明需要约10页。同样,经典数学家拒绝相信他们的聪明证明是没有意义的,无论这种证明是不是构造性的。

最后,有一些定理在直觉主义中成立,但在经典数学中是错误的。一个例子是直觉主义定理,该定理说每个为所有实数定义的实数值函数都是连续的。这个定理并不像它听起来那么奇怪,因为它取决于函数的直观概念:一个实数值函数f在直观主义中是为所有实数定义的,只有当对每个实数r的直观构造已经完成时,实数f(r)才能被构造。一个经典数学家可能提到的任何明显不连续的函数都不满足这个构造标准。即便如此,像这样的定理在经典数学家看来是如此遥远,以至于他们拒绝接受任何承认它们的数学。


经典数学家拒绝直觉主义的这三个理由,既不是理性的也不是科学的,甚至也不是实用主义的理由,基于一种信念,即经典数学在物理学或其他科学中的应用比直觉主义更好。它们都是感性的理由,根植于对数学的深刻意义。(如果有读者知道有真正科学的拒绝直觉主义的做法,笔者将非常感谢)。我们现在面临着数学的第二个危机:直觉主义学派未能让至少大多数数学家接受直觉主义。


重要的是要认识到,像逻辑主义一样,直觉主义也是植根于哲学的。例如,当直觉主义者陈述他们对数学的定义时,如前所述,他们使用的是严格意义上的哲学语言,而不是数学语言。事实上,他们不可能用数学来做这样一个定义。数学的心智活动可以用哲学术语来定义,但这个定义必须使用一些不属于它所要定义的活动的术语,这是必然的。


正如逻辑主义与现实主义的关系一样,直觉主义与称为概念主义conceptualism)的哲学有关。这种哲学认为,抽象实体的存在只是因为它们是由人的头脑构建的,这正是直觉主义的态度,它认为数学中出现的抽象实体,无论是序列还是顺序关系,或者其他什么,都是心智构造。这正是为什么人们在直觉主义中找不到在经典数学中以及在逻辑主义中出现的惊人的抽象实体集合。逻辑主义与直觉主义之间的对比和现实主义与概念主义之间的对比非常相似。



原文


Intuitionism

This school was begun about 1908 by the Dutch mathematician, L.E.J. Brouwer (1881-1966). The intuitionists went about the foundations of mathematics in a radically different way from the logiciels. The logiciels never thought that there was anything wring with classical mathematica; then silly wanted to show that classical mathematics is part of logic. The intuitionists, on the contrary, felt that there was plenty wring with classical mathematics. 

By 1908several paradoxes had arisen in Cantor’s set theory. Here, the word « paradoxe » is used as synonymous with « contradiction ». George Cantor created set theory, starting around 1870, and he did his work « naively », meaning non axiomatically. Consequently he formed sets with such abandon that he himself, Russell and others found several paradoxes within his theory. The logicists considered these paradoxes as common errors, caused by erring mathematicians and not by a faulty mathematics. The intuitionists, one the other hand, considered these paradoxes as clear indications that classical mathematics itself is far from perfect. They felt that mathematics had to be rebuilt from the bottom on up.

The « bottom », that is, the beginning of mathematics for the intuitionists, is their explanation of what the natural numbers 1,2,3,… are. (Observe that we do not include the number zero among the natural numbers). According to intuitionistic philosophy all human beings have a primordial intuition for the natural numbers within them. This means in the first place that we have an immediate certainty as to what is meant by the number 1 and, secondly, that the mental process which goes into the formation of the number 1 can be repeated. When we do repeat it, we obtain the concept of the number 2; when we repeat it again, the concept of the number 3; in this way, human beings can construct any finite initial segment 1,2, .., n for any natural number n. This mental construction of one natural number after the other would never have been possible if we did not have an awareness of time within us. « After » refers to time and Brouwer agrees with the philosopher Immanuel Kant (1724-1804) that human beings have an immediate awareness of time. Kant used the word « intuition » for « immediate awareness » and this is where the name « intuitionism » comes from. 

It is important to observe that the intuitionistic construction of natural numbers allows one to construct only arbitrarily long finite initial segments 1,2, …n. It does not allow us to construct that whole closed set of all the natural numbers which is so familiar from classical mathematics. It is equally important to observe that this construction is both « inductive and « effective ». It is inductive in the sense that, if one wants to construct says the number 3, on has to go through all the mental steps of first constructing the 1, then the 2, and finally the 3; one cannot just grab the number 3 out of the sky. It is effective in the sense that, once the construction of a natural number has been finished, that natural number has been constructed in its entirely. It stands before us as a completely finished mental construct, ready for our study of it. When someone says, « I have finished the mental construction of the number 3 », it is like a bricklayer saying, « I have finished that wall », which he can say only after he has laid every stone in place.  

We now turn to the intuitionistic definition of mathematics. According to intuitionistic philosophy, mathematics should be defined as a mental activity and not as a set of theorems (as was done above on the section on logicism). It is the activity which consists in carrying out, one after the other; those mental constructions which are inductive and effective in the sense in which the intuitionistic construction of the natural numbers is inductive effective. Intuitionism maintains that human beings are able to recognize whether a given mental construction has these two properties. We shall refer to a mental construction which has these two properties as a construct and hence the intuitionistic definition of mathematics says : Mathematics is the mental activity which consist in carrying out constructs one after the other.

A major consequence of this definition is that all of intuitionistic mathematics is effective or « constructive » as one usually says. We shall use the adjective « constructive » as synonymous with « effective » from now on. Namely, every construct is constructive, and intuitionistic mathematics is nothing but carrying out constructs over and over. For instance, if a real number r occurs in an intuitionistic proof or theorem, it never occurs there merely on grounds of an existence proof. It occurs there because it has been constructed from top to bottom. This implies for example that each decimal place in the decimal expansion of r can in principle be computed. In short, all intuitionistic proofs, theorems, definitions, etc?, are entirely constructive.

Once the intuitionistic definition of mathematics has been understood and accepted, all there remains to be done is to do mathematics the intuitionistic way. Indeed, the intuitionists have developed intuitionistic arithmetic, algebra, analysis, set theory, etc. However, in each of these branches of mathematics, there occur classical theorems which are not composed of constructs and, hence, are meaningless combinations of words for the intuitionists. Consequently, one cannot say that the intuitionists have reconstructed all of the classical mathematics. This does not bother the intuitionists since whatever parts of classical mathematics they cannot the cannot obtain are meaningless for them anyway. Intuitionsm does not have as its purpose the justification of classical mathematics. Its purpose is to give a valid definition of mathematics and then to « wait and see » what mathematics comes out of it. Whatever classical mathematics cannot be done intuitionistically simply is not mathematics for the intuitionist. We observe here another fundamental difference between logician and intuitionist : The logicists wanted to justify all of classical mathematics.

Let us now ask how successful the intuitionistic schools has been in giving us a good foundation for mathematics, acceptable to the majority of mathematicians. Again, there is a sharp difference between the way this question has to be answered in the present case and in the case go logicism. Even hard-nosed logicists have to admit that their school so far has failed to give mathematics a firm foundation by about 20%. However, a hard-nosed intuitionist has every right in the world to claim that intuitionism has given mathematics an entirely satisfactory foundation. There is the meaningful definition of intuitionistic mathematics, discussed above; there is the intuitionistic philosophy which tells us why constructs can never give rise to contradictions and, hence, that intuitionistic mathematics is free of contradictions. In fact, not only this problem (of freedom from contradiction) but all other problems of a foundational nature as well receive perfectly satisfactory solutions in intuitionism.

Yet if one looks at intuitionism from the outside, namely, from the viewpoint o the classical mathematician, one has to say that intuitionism has failed to give mathematics an adequate foundation. In fact, the mathematical community has almost universally rejected intuitionism. Why has the mathematical community done this, in spite of the many very attractive features of intuitionism, some of which have just been mentioned ?

One reason is that classical mathematicians flatly refuse to do aways with the many beautiful theorems that are meaningless combinations of words for the intuitionists. An example is the Brouwer fixed point theorem of topology which the intuitionists reject because the fixed point confit be constructed, but can only be shown to exist on grounds of an existence proof. This, by the way, is the same Brouwer who created intuitionism; he is equally famous for this work in (nonintuitionistic) topology.

A second reason comes from theorems which can ne proven both classically and intuitionistically. It often happens that the classical proof of such a theorem is short, elegant, and devilishly clever, but not constructive. The intuitionists will of course reject such a proof and replace it by their own constructive proof go the the same theorem. However, this constructive proof frequently turns out to be about ten times as long as the classical proof and often seems, at least to the classical mathematician, to have lost all of its elegance. An example is the fundamental theorem of algebra which in classical mathematics is proved in about half -a page, but takes about ten pages of proof in intuitionistic mathematics. Again, classical mathematicians refuse to believe that their clever proofs are meaningless whenever such proofs are not constructive.

Finally, there are the theorems which hold in intuitionism but are false in classical mathematics. An example is the intuitionistic theorem which says that every real-valued function which is defined for all real numbers is continuous. This theorem is not as strange as it sounds since it depends on the intuitionistic concept of a function : A real-valued function f is defined in intuitionism for all real numbers only if, for every real number r whose intuitionistic construction has been completed, the real number f(r) canoe constructed. Any obviously discontinuous function a classical mathematician may mention does not satisfy this constructive criterion. Even so, theorems such as this one seem so far out to classical mathematicians that they reject any mathematics which accepts them.

These three reasons for the rejection of intuitionism by classical mathematicians are neither rational nor scientific. Nor are they pragmatic reasons, based on a conviction that classical mathematics is better for applications to physics or other science than is intuitions. They are all emotional reasons, grounded in a deep sense as to what mathematics is all about. (If one of the readers knows of a truly scientific rejection of intuitionism, the author would be grateful to hear about it). We now have the second crisis in mathematics in front of us : It consists in the failure of the intuitionistic school to make intuitionism acceptable to at least the majority of mathematicians.


It is important to realize that, like logicism, intuitionism is rooted in philosophy. When, for instance, the intuitionists state their definition of mathematics, given earlier, they use strictly philosophical and not mathematical language. It would, in fact, be quite impossible for them to use mathematics for such a definition. The mental activity which is mathematics can be defined in philosophical terms but this definition must, by necessity, use some terms which do not belong to the activity it is trying to define.


Just as logicism is related to realism, intuitionism is related to the philosophy called « conceptualism ». This is the philosophy which maintains that abstract entities exist only insofar as they are constructed by the human mind. -This is very much the attitude of intuitionism which holds that the abstract entities which occurs in mathematics, whether sequence or order-relations or what have you, are all mental constructions. This is precisely why one does not find in intuitionism the staggering collection of abstract entities which occur in classical mathematics and hence in logicism. The contrast v-between logicism and  intuitionism is very similar to the contrast between realism and conceptualism.





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