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希尔伯特的《几何学基础》与形式主义(Formalism an

E嗨生活 2020-07-08

一般人若对于數学哲学流派中的形式主义(formalism)感兴趣,那幺,最值得參考的经典文献,莫过于希尔伯特(David Hilbert, 1862-1943)的《几何学基础》(The Foundations of Geometry)。这本书源自哥廷根大学 1898-1899年冬季班有关欧氏几何课程的教材。

在英译本的前言中,译者E. J. Townsend 指出本书有下列五个特点:

如果不深入研讀本书,上述这五点恐怕很难体会。不过,正如前述,要想认識或体会形式主义,则本书第1章第1节实在非常「经典」(classical),值得引述如下:

§ I. THE ELEMENTS OF GEOMETRY AND THE FIVE GROUPS OF AXIOMS.

Let us consider three distinct systems of things. The things composing the first system, we will call points and designate them by the letters A, B, C, .…; those of the second, we will call straight lines and designate them by the letters a, b, c, .…; designate them by the Greek letters α, β, γ, .…The points are called the elements of linear geometry; the points and straight lines, the elements of plane geometry; and the points, lines, and planes, the elements of the geometry of space or the elements of space.We think of these points, straight lines, and planes as having certain mutual relations, which we indicate by means of such words as “are situated,” “between,” “parallel,” “congruent,” “continuous,” etc. The complete and exact description of these relations follows as consequence of the axioms of geometry. These axioms may be arranged in five groups. Each of these groups expresses, by itself, certain related fundamental facts of our intuition. We will name these groups as follows:

    1-7. Axioms of connection.1-5. Axioms of order.Axiom of parallels (Euclid’s axiom).1-6. Axiom of congruence.Axiom of continuity (Archimedes axiom).

在上述引文中,希尔伯特所考虑的「事物」(thing)的三个相異系统,后來被他分别称为点、直线和平面。这些事物如点 (point)、直线 (straight line)、平面 (plane),随即被希尔伯特称为几何学的元素 (element)。

然后,他又接着指出說:「我们认为这些点、直线和平面,具有某些相互关係,而我们是藉由有如下列『位于』、『在(兩者)之间』、『平行的』、『全等的』、『連续的』等文字來指出。至于这些关係的一个完备的、确当的描述,则是几何公设推演的结果。」

正如上引,希尔伯特将这些公理分为五群,各自表示了「我们直观的某些基本事实。」所谓「事物」的提法,令人想出欧几里得在他的《几何原本》(The Elements)中「共有概念」(或公理)(common notions)中的「事物」(thing):

不过,这些并非希尔伯特的关注兴趣所在。

因为他所指的欧几里得平行公理是:(Euclid’s axiom):“In a plane α there can be drawn through any point A, lying outside of a straight line a, one and only one straight line which does not intersect the line a. This straight line is called the parallel to a through the given point A.”

这一公理当然与欧几里得《几何原本》中的第五设準 (postulate) 等价:「同平面内任一条直线和另外兩条直线相交,若在某一侧的兩侧内角的和小于兩个直角,这兩条线经任意 (indefinitely) 延长后在这一侧相交。」(That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.)

同时,我们也不应该忘记:在这五条设準之前,有共同的一句话:「令下列被假设成立」(Let the following be postulated) - 其实,这也解释了英文版的第五设準前何以出现英文字 “That”。相反地,在「共有概念」之前,就无此规範了。可見,对于欧几里得而言,共有概念与设準是不一样的公设或公理。但是,对于现代數学而言,那五个共有概念已经没有意义了。

至于所谓的「事物」是否存在,当然不须「操烦」,因为形式主义原來就主张數学不过是一种没有意义的符号游戏罢了 (Mathematics is a symbolic game without meaning.)。既然如此,学习數学本身又如何赋予意义呢?这是形式主义的永恆困境,当然无法解决。希尔伯特原来满心期待他可以解决完备性(completeness)问题,以便至少在结构上对数学知识本质赋予意义,不幸,由于哥德尔 (Godel) 的不完备定理 (incompleteness theorem),而让他功败垂成。

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