数学问题 - 1900年在巴黎国际数学家大会上的演讲

大卫-希尔伯特

我们谁不愿意揭开隐藏在未来的面纱，瞥一眼我们科学的下一步进展和未来几个世纪发展的秘密？未来几代人的主要数学精神将朝着什么特别的目标而努力？在广泛而丰富的数学思想领域中，新的世纪将披露哪些新方法和新事实？

历史告诉我们，科学的发展是连续性的。我们知道，每个时代都有自己的问题，而下一个时代要么解决这些问题，要么认为这些问题无益而弃之不顾，代之以新的问题。如果我们想了解数学知识在不久的将来的可能发展，我们就必须让那些尚未解决的问题在我们的脑海中掠过，并审视今天的科学所提出的问题，以及我们所期待的未来的解决方案。对于这样的问题回顾，在我看来，今天是几个世纪的交汇点，很适合。因为一个伟大时代的结束不仅邀请我们回顾过去，而且还将我们的思想引向未知的未来。

某些问题对整个数学科学发展的深远意义以及它们在个别研究者的工作中所发挥的重要作用是不容否定的。只要一个科学分支能提供大量的问题，它就有生命力；缺乏问题就预示着灭亡或独立发展的停止。正如人类的每一项事业都在追求某些目标一样，数学研究也需要问题。正是通过解决问题，研究者检验了他的韧性；他发现了新的方法和新的前景，并获得了更广泛和更自由的视野。

要事先正确判断一个问题的价值是困难的，而且往往是不可能的；因为最终的奖励取决于科学从问题中获得的收益。然而，我们可以问，是否有一般的标准来标志一个好的数学问题。一位法国老数学家说“一个数学理论只有在你把它说得如此清楚，以至于你能向你在街上遇到的第一个人解释它时，才能被认为是完整的。”这里所坚持的数学理论的这种清晰性和易懂性，如果要做到完美，我更应该对数学问题提出要求；因为清晰和易懂的东西会吸引我们，复杂的东西会排斥我们。

此外，一个数学问题应该是困难的，可以吸引我们，但又不是完全不可触及的，以免嘲笑我们的努力。对我们来说，它应该是指引通往隐秘真理的一条路，并最终提醒我们在成功解决中的快乐。过去几个世纪的数学家们习惯于以极大的热情投入到解决特殊的困难问题中，他们知道困难问题的价值。

我只提醒你约翰-伯努利（John Bernoulli）提出的“最陡下降问题”。伯努利在公开宣布这个问题时解释说，经验告诉我们，引导崇高的思想为科学的进步而努力，不外乎是把困难的同时又有用的问题摆在他们面前，因此他希望通过效仿梅森（Mersenne）、帕斯卡（Pascal）、费马（Fermat）、维维亚尼（Viviani ）等人的榜样，把一个问题摆在他那个时代的杰出分析家面前，作为试金石，他们可以检验自己方法的价值并衡量其力量，从而赢得数学界的感谢。变分学的起源要归功于伯努利的这个问题和类似的问题。

众所周知，费马曾断言，丢番图方程

xⁿ + yⁿ = zⁿ

(x、y和z的整数)是不可解的—除了某些不言而喻的情况外。证明这种不可能性的尝试提供了一个突出的例子，说明这样一个非常特殊和明显不重要的问题可能对科学产生的启发作用。因为在费马问题的激励下，库默（Kummer）被引向了理想数的引入，并发现了圆形领域的数字独特地分解为理想素数的规律—这一规律在今天被戴德金（Dedekind）和克罗内克（Kronecker）推广到任何代数领域，成为现代数论的中心，其意义远远超出了数论的界限，进入了代数和函数理论的领域。

说到一个非常不同的研究领域，我提醒你们注意三体（three bodies）的问题。庞加莱（Poincaré）为天体力学带来的富有成效的方法和意义深远的原则，以及今天在实际天文学中得到承认和应用的情况，都是由于他承诺重新处理这个困难的问题并接近解决。

最后提到的两个问题--费马的问题和三体的问题--在我们看来几乎是相反的两极--前者是纯粹理性的自由发明，属于抽象数论的范畴，后者是天文学强加给我们的，对于理解自然界最简单的基本现象是必要的。 

但经常发生的情况是，同样的特殊问题在数学知识的最不相似的分支中也能找到应用。因此，例如，最短线的问题在几何学的基础上，在曲线和曲面的理论中，在力学和变化微积分中起着主要和历史性的重要作用。克莱因（F. Klein）在他关于二十面体的工作中，描绘了正规多面体问题在初等几何学、群论、方程理论和线性微分方程中的重要性，是多么令人信服。

为了说明某些问题的重要性，我还可以提到魏尔斯特拉斯（Weierstrass），他说他很幸运，在他的科学生涯的一开始就发现了一个像雅可比（Jacobi）的反转问题这样重要的问题，可以在上面工作。

在回顾了数学中问题的普遍重要性之后，让我们转向这个问题，即这门科学的问题来自哪些方面。当然，数学的每一个分支的最初和最古老的问题都来自于经验，并由外部现象的世界提出。即使是整数的计算规则也一定是在人类文明的低级阶段以这种方式发现的，就像今天的孩子通过经验方法学习这些规律的应用。几何学的第一批问题也是如此，这些问题是古代留给我们的，如立方体的重复，圆的平方；还有数值方程解理论中最古老的问题，曲线理论和微分与积分微积分，变分微积分，傅里叶级数理论和势能理论，更不用说更多的属于力学、天文学和物理学的问题。

但是，在一个数学分支的进一步发展中，人类的思维在其解决方案的成功鼓励下，开始意识到其独立性。它通过逻辑组合、概括化、专业化，通过以幸运的方式分离和收集思想，从自身出发，往往在没有明显影响的情况下，独自演化出新的和富有成效的问题，并在那时作为真正的提问者出现。这样就出现了素数问题和数论的其他问题，伽罗瓦的方程理论，代数不变量理论，阿贝尔函数和自动函数的理论；事实上，现代算术和函数理论中几乎所有较好的问题都是这样产生的。

同时，当纯粹理性的创造能力在工作时，外部世界又开始发挥作用，从实际经验中迫使我们提出新的问题，开辟新的数学分支，而当我们为纯粹思想的领域寻求征服这些新的知识领域时，我们常常发现旧的未解决的问题的答案，从而同时最成功地推进旧理论。在我看来，数学家在他的科学的各个分支的问题、方法和思想中经常看到的无数令人惊讶的类比和那种明显预先安排好的和谐，都是源于思想和经验之间这种不断发生的相互作用。

我们还需要简单讨论一下，对于一个数学问题的解决，有哪些一般要求是可以合理规定的。首先，我要说的是：通过基于有限数量的假设的有限步骤来确定解决方案的正确性是可能的，这些假设隐含在问题的陈述中，而且必须始终准确地制定。这种通过有限数量的过程进行逻辑推导的要求只是推理中的严谨性要求。事实上，严格的要求在数学中已成为一句谚语，它对应于我们理解的普遍的哲学需要；另一方面，只有满足这一要求，问题的思想内容和暗示性才能达到其全部效果。一个新的问题，特别是当它来自外部经验世界时，就像一根年轻的树枝，只有当它按照严格的园艺规则小心翼翼地嫁接在我们数学科学的既定成就这一老茎上时，它才会茁壮成长，结出果实。

此外，认为证明中的严格性是简单性的敌人也是一个错误。恰恰相反，我们发现有许多例子证实，严格的方法同时也是更简单和更容易被理解的。对严格性的努力迫使我们找出更简单的证明方法。它也经常把我们引向比不那么严格的旧方法更有能力发展的方法。因此，代数曲线的理论经历了相当大的简化，并通过更严格的函数理论方法和对超越装置的持续引入而获得了更大的统一性。此外，幂级数允许应用四种基本算术运算以及逐项微分和积分的证明，以及对幂级数的效用的认识取决于这一证明，这对所有分析的简化，特别是消除理论和微分方程理论的简化，以及这些理论所要求的存在性证明，都有很大的帮助。但对我的陈述来说，最突出的例子是变分的计算。对定积分的第一和第二变化的处理，部分需要极其复杂的计算，而旧数学家所应用的过程没有必要的严格性。魏尔斯特拉斯为我们指明了通向变迁微积分的新的和可靠的基础的道路。通过简单积分和双重积分的例子，我将在讲座结束时简要说明，这种方法是如何一下子导致变化微积分的惊人简化的。因为在证明最大和最小发生的必要和充分标准时，第二个变化的计算以及与第一个变化有关的部分令人厌烦的推理可以完全免除--更不用说取消对变化的限制所涉及的进步了，对于这些变化，函数的微分系数只是略有变化。

在坚持把证明的严格性作为一个问题的完美解决的要求的同时，我想在另一方面反对这样的观点，即只有分析的概念，甚至只有算术的概念，才有可能得到完全严格的处理。我认为，这种偶尔由知名人士倡导的观点是完全错误的。对严格要求的这种片面解释，很快就会导致对几何学、力学和物理学产生的所有概念的忽视，导致来自外部世界的新材料的流动停止，最后，实际上，作为最后的结果，导致对连续统和无理数的概念的拒绝。但是，如果几何学和数学物理学被消灭，那么对数学科学至关重要的一条重要的神经就会被切断！相反，我认为，无论从知识理论或几何学方面，或从自然科学或物理科学的理论方面，出现了数学思想，数学科学的问题就在于研究这些思想的基本原理，并把它们建立在一个简单而完整的公理系统上，使新思想的精确性和它们对推理的适用性在任何方面都不逊于旧的算术概念。

新的概念必然对应着新的符号。我们选择这些符号的方式，是为了使它们提醒我们注意那些形成新概念的现象。因此，几何图形是空间直觉的标志或记忆符号，所有数学家都是这样使用的。谁不总是使用双不等式a>b>c的图片，作为 “之间”这一概念的几何图片，在一条直线上相互跟随的三个点？当需要非常严格地证明一个关于函数连续性或凝结点的存在的困难定理时，谁不使用线段和矩形相互包围的图画？谁能舍弃三角形的图形、有中心的圆或三个垂直轴的交叉？或者谁会放弃矢量场的表示，或者曲线族或曲面及其包络的图片，这些在微分几何、微分方程理论、变化微积分的基础和其他纯数学科学中发挥了如此重要的作用？

使用几何符号作为严格的证明手段的前提是准确了解和完全掌握作为这些形象（figure）基础的公理；为了使这些几何形象能够被纳入数学符号的一般宝库，有必要对其概念内容进行严格的公理研究。正如在两个数字相加时，人们必须把数字按正确的顺序放在对方下面，因此只有计算规则，即算术公理，才能决定形象的正确使用，所以几何符号的使用是由几何概念的公理和它们的组合决定。

几何思想和算术思想之间的一致性还表现在，我们在算术中并不习惯于沿着推理链回到公理，比在几何讨论中更甚。相反，特别是在第一次处理一个问题时，我们应用快速的、无意识的、不是绝对确定的组合，相信对算术符号行为的某种算术感觉，在算术中我们可以免除这种感觉，就像在几何中免除几何想象力一样。作为严格运用几何思想和符号的算术理论的一个例子，我可以提到闵可夫斯基（Minkowski）的作品《Zahlen的几何》。

关于数学问题可能带来的困难，以及克服这些困难的方法，在此可以做一些评论。

如果我们不能成功地解决一个数学问题，原因往往在于我们没有认识到一个更普遍的观点，从这个观点来看，我们面前的问题只是一连串相关问题中的一个环节。在找到这个观点之后，不仅这个问题经常更容易被我们研究，而且同时我们也掌握了一种也适用于相关问题的方法。Cauchy提出的复杂积分路径和Kummer提出的数论中的IDEALS概念可以作为例子。这种寻找一般方法的方法当然是最实用和最确定的；因为在没有明确问题的情况下寻找方法的人，大部分都是徒劳的。

我认为，在处理数学问题时，专门化比一般化更重要。也许在大多数情况下，当我们徒劳地寻求一个问题的答案时，失败的原因在于比眼前这个问题更简单和更容易的问题要么根本没有解决，要么不完全解决。因此，一切都取决于找到这些更简单的问题，并通过尽可能完善的设备和能够概括的概念来解决这些问题。这条规则是克服数学困难的最重要的杠杆之一，在我看来，它几乎总是被使用，尽管可能是无意识的。

偶尔会发生这样的情况：我们在不充分的假设下或在不正确的意义上寻求解决方案，由于这个原因而没有成功。这时问题就来了：证明在给定的假设下，或在所考虑的意义上，解是不可能的。古人曾做过这种不可能的证明，例如，他们证明了等腰直角三角形的斜边与边的比率是无理的。在后来的数学中，关于某些解决方案的不可能性的问题起着突出的作用，我们通过这种方式看到，古老而困难的问题，如平行公理的证明，圆的平方，或用根式解决五度方程的问题，最终找到了完全令人满意和严格的解决方案，尽管是在另一种意义上比原来的意图。可能正是这个重要的事实和其他哲学原因，导致了这样一种信念（每个数学家都有这种信念，但至今还没有人用证明来支持这种信念），即每一个明确的数学问题都必须能够得到确切的解决，要么是以对所问问题的实际答案的形式，要么是通过证明其解决方案的不可能性，从而导致所有尝试的必然失败。以任何明确的未解决的问题为例，如欧拉-马舍罗尼常数C的非理性问题，或2n+1形式的无限多的素数的存在。无论这些问题在我们看来多么遥不可及，无论我们在它们面前多么束手无策，我们还是坚信，它们的解决必须通过有限的纯逻辑过程。

这个关于每个问题的可解性的公理是否仅仅是数学思维的特点，还是可能是思维本质中固有的一般规律，即它提出的所有问题都必须有答案？因为在其他科学中，人们也会遇到一些古老的问题，这些问题通过证明它们的不可能性，以一种最令人满意和对科学最有益的方式得到解决。我举例说，永动机的问题，在徒劳地寻求制造永动机之后，人们研究了如果这种机器不可能的话，自然界的各种力量之间必须存在的关系；这个颠倒的问题导致了能量守恒定律的发现，这又解释了永动机在最初意义上的不可能性。

这种对每一个数学问题的可解性的信念是对人的强大激励。我们在内心深处听到了永恒的呼唤，问题就在那里，寻找它的解决方案。你可以通过纯粹的理性找到它，因为在数学中没有无知的人。

数学中的问题是取之不尽用之不竭的，一旦一个问题被解决了，就会有无数的问题出现。请允许我在下文中暂且提到一些特定的问题，这些问题来自于数学的各个分支，通过对这些问题的讨论，可以期待科学的进步。

让我们来看看分析和几何的原则。在我看来，上个世纪在这一领域取得的最有启发性和最引人注目的成就是考奇（Cauchy）、博尔扎诺（Bolzano）和康托尔（Cantor）作品中连续体概念的算术表述，以及高斯（Gauss）、博莱（Bolyai）和洛巴切夫斯基（Lobachevsky）对非欧几里得几何的发现。因此，我首先引导大家注意属于这些领域的一些问题。

1. 康托尔的连续统假设

2. 算术公理之相容性

3. 两四面体有相同体积之证明法

4. 直线是两点之间最短距离的问题

5. 所有连续群是否皆为可微群

6. 对物理学公理的数学处理

7. 超越数

8. 素数的问题

9. 任意代数数域的一般互反律

10. 丢番图方程的可解性

11. 代数系数之二次形式

12.一般代数数域的阿贝尔扩张

13. 以二元函数解任意七次方程

14. 某些完整函数系统的有限性的证明

15. 舒伯特演算的严格基础

16. 代数曲线和曲面的拓扑学问题

17. 把有理函数写成平方和分式

18. 非正多面体能否密铺空间、球体最紧密的排列

19. 拉格朗日系统（Lagrangian）之解是否皆可解析

20. 边界值的一般问题

21. 证明有线性微分方程有给定的单值群

22. 通过自动函数实现分析关系的统一化

23. 进一步发展变化微积分的方法

所提到的问题只是一些问题的样本，但它们足以说明今天的数学科学是多么丰富、多样和广泛，有人向我们提出这样的问题：数学是否注定要像那些分裂成独立分支的其他科学一样，其代表几乎不了解彼此，其联系变得越来越松散。我不相信也不希望如此。在我看来，数学科学是一个不可分割的整体，是一个有机体，其生命力取决于各部分的联系。因为尽管数学知识种类繁多，我们仍然清楚地意识到逻辑装置的相似性，数学整体思想的关系，以及不同部门中的大量类比。我们还注意到，一个数学理论发展得越远，它的构造就越和谐统一，而且在这门科学的不同分支之间还透露出未曾预见的关系。因此，随着数学的扩展，它的有机特性并没有丧失，而只是更清楚地表现出来。

但是，我们要问，随着数学知识的扩展，单一的研究者是否最终不可能囊括这一知识的所有部门？在回答这个问题时，让我指出，每一个真正的进步都是与更锐利的工具和更简单的方法的发明同时进行的，这些工具和方法有助于理解早期的理论，并把更复杂的旧发展抛在一边，这在数学科学中是多么的根深蒂固。因此，当个人研究者将这些更锐利的工具和更简单的方法作为自己的工具时，他就有可能在数学的各个分支中比在任何其他科学中更容易找到自己的方向。

数学的有机统一性是这门科学的本质所固有的，因为数学是所有关于自然现象的精确知识的基础。为了使它能完全完成这一崇高的使命，愿新世纪产生有天赋的大师和许多热心和热情的弟子!

参考资料：

David Hilbert: Mathematical Problems, http://aleph0.clarku.edu/~djoyce/hilbert/problems.html

原文：

Mathematical Problems

Lecture delivered before the International Congress of Mathematicians at Paris in 1900

By Professor David Hilbert

Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be toward which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?

History teaches the continuity of the development of science. We know that every age has its own problems, which the following age either solves or casts aside as profitless and replaces by new ones. If we would obtain an idea of the probable development of mathematical knowledge in the immediate future, we must let the unsettled questions pass before our minds and look over the problems which the science of today sets and whose solution we expect from the future. To such a review of problems the present day, lying at the meeting of the centuries, seems to me well adapted. For the close of a great epoch not only invites us to look back into the past but also directs our thoughts to the unknown future.

The deep significance of certain problems for the advance of mathematical science in general and the important role which they play in the work of the individual investigator are not to be denied. As long as a branch of science offers an abundance of problems, so long is it alive; a lack of problems foreshadows extinction or the cessation of independent development. Just as every human undertaking pursues certain objects, so also mathematical research requires its problems. It is by the solution of problems that the investigator tests the temper of his steel; he finds new methods and new outlooks, and gains a wider and freer horizon.

It is difficult and often impossible to judge the value of a problem correctly in advance; for the final award depends upon the gain which science obtains from the problem. Nevertheless we can ask whether there are general criteria which mark a good mathematical problem. An old French mathematician said: "A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street." This clearness and ease of comprehension, here insisted on for a mathematical theory, I should still more demand for a mathematical problem if it is to be perfect; for what is clear and easily comprehended attracts, the complicated repels us.

Moreover a mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts. It should be to us a guide post on the mazy paths to hidden truths, and ultimately a reminder of our pleasure in the successful solution.

The mathematicians of past centuries were accustomed to devote themselves to the solution of difficult particular problems with passionate zeal. They knew the value of difficult problems. 

I remind you only of the "problem of the line of quickest descent," proposed by John Bernoulli. Experience teaches, explains Bernoulli in the public announcement of this problem, that lofty minds are led to strive for the advance of science by nothing more than by laying before them difficult and at the same time useful problems, and he therefore hopes to earn the thanks of the mathematical world by following the example of men like Mersenne, Pascal, Fermat, Viviani and others and laying before the distinguished analysts of his time a problem by which, as a touchstone, they may test the value of their methods and measure their strength. The calculus of variations owes its origin to this problem of Bernoulli and to similar problems.

Fermat had asserted, as is well known, that the diophantine equation

xⁿ + yⁿ = zⁿ

(x, y and z integers) is unsolvable—except in certain self evident cases. The attempt to prove this impossibility offers a striking example of the inspiring effect which such a very special and apparently unimportant problem may have upon science. For Kummer, incited by Fermat's problem, was led to the introduction of ideal numbers and to the discovery of the law of the unique decomposition of the numbers of a circular field into ideal prime factors—a law which today, in its generalization to any algebraic field by Dedekind and Kronecker, stands at the center of the modern theory of numbers and whose significance extends far beyond the boundaries of number theory into the realm of algebra and the theory of functions.

To speak of a very different region of research, I remind you of the problem of three bodies. The fruitful methods and the far-reaching principles which Poincaré has brought into celestial mechanics and which are today recognized and applied in practical astronomy are due to the circumstance that he undertook to treat anew that difficult problem and to approach nearer a solution.

The two last mentioned problems—that of Fermat and the problem of the three bodies—seem to us almost like opposite poles—the former a free invention of pure reason, belonging to the region of abstract number theory, the latter forced upon us by astronomy and necessary to an understanding of the simplest fundamental phenomena of nature.

But it often happens also that the same special problem finds application in the most unlike branches of mathematical knowledge. 

So, for example, the problem of the shortest line plays a chief and historically important part in the foundations of geometry, in the theory of curved lines and surfaces, in mechanics and in the calculus of variations. And how convincingly has F. Klein, in his work on the icosahedron, pictured the significance which attaches to the problem of the regular polyhedra in elementary geometry, in group theory, in the theory of equations and in that of linear differential equations.

In order to throw light on the importance of certain problems, I may also refer to Weierstrass, who spoke of it as his happy fortune that he found at the outset of his scientific career a problem so important as Jacobi's problem of inversion on which to work.

Having now recalled to mind the general importance of problems in mathematics, let us turn to the question from what sources this science derives its problems. Surely the first and oldest problems in every branch of mathematics spring from experience and are suggested by the world of external phenomena. Even the rules of calculation with integers must have been discovered in this fashion in a lower stage of human civilization, just as the child of today learns the application of these laws by empirical methods. The same is true of the first problems of geometry, the problems bequeathed us by antiquity, such as the duplication of the cube, the squaring of the circle; also the oldest problems in the theory of the solution of numerical equations, in the theory of curves and the differential and integral calculus, in the calculus of variations, the theory of Fourier series and the theory of potential—to say nothing of the further abundance of problems properly belonging to mechanics, astronomy and physics.

But, in the further development of a branch of mathematics, the human mind, encouraged by the success of its solutions, becomes conscious of its independence. It evolves from itself alone, often without appreciable influence from without, by means of logical combination, generalization, specialization, by separating and collecting ideas in fortunate ways, new and fruitful problems, and appears then itself as the real questioner. Thus arose the problem of prime numbers and the other problems of number theory, Galois's theory of equations, the theory of algebraic invariants, the theory of abelian and automorphic functions; indeed almost all the nicer questions of modern arithmetic and function theory arise in this way.

In the meantime, while the creative power of pure reason is at work, the outer world again comes into play, forces upon us new questions from actual experience, opens up new branches of mathematics, and while we seek to conquer these new fields of knowledge for the realm of pure thought, we often find the answers to old unsolved problems and thus at the same time advance most successfully the old theories. And it seems to me that the numerous and surprising analogies and that apparently prearranged harmony which the mathematician so often perceives in the questions, methods and ideas of the various branches of his science, have their origin in this ever-recurring interplay between thought and experience.

It remains to discuss briefly what general requirements may be justly laid down for the solution of a mathematical problem. I should say first of all, this: that it shall be possible to establish the correctness of the solution by means of a finite number of steps based upon a finite number of hypotheses which are implied in the statement of the problem and which must always be exactly formulated. This requirement of logical deduction by means of a finite number of processes is simply the requirement of rigor in reasoning. Indeed the requirement of rigor, which has become proverbial in mathematics, corresponds to a universal philosophical necessity of our understanding; and, on the other hand, only by satisfying this requirement do the thought content and the suggestiveness of the problem attain their full effect. A new problem, especially when it comes from the world of outer experience, is like a young twig, which thrives and bears fruit only when it is grafted carefully and in accordance with strict horticultural rules upon the old stem, the established achievements of our mathematical science.

Besides it is an error to believe that rigor in the proof is the enemy of simplicity. On the contrary we find it confirmed by numerous examples that the rigorous method is at the same time the simpler and the more easily comprehended. The very effort for rigor forces us to find out simpler methods of proof. It also frequently leads the way to methods which are more capable of development than the old methods of less rigor. Thus the theory of algebraic curves experienced a considerable simplification and attained greater unity by means of the more rigorous function-theoretical methods and the consistent introduction of transcendental devices. Further, the proof that the power series permits the application of the four elementary arithmetical operations as well as the term by term differentiation and integration, and the recognition of the utility of the power series depending upon this proof contributed materially to the simplification of all analysis, particularly of the theory of elimination and the theory of differential equations, and also of the existence proofs demanded in those theories. But the most striking example for my statement is the calculus of variations. The treatment of the first and second variations of definite integrals required in part extremely complicated calculations, and the processes applied by the old mathematicians had not the needful rigor. Weierstrass showed us the way to a new and sure foundation of the calculus of variations. By the examples of the simple and double integral I will show briefly, at the close of my lecture, how this way leads at once to a surprising simplification of the calculus of variations. For in the demonstration of the necessary and sufficient criteria for the occurrence of a maximum and minimum, the calculation of the second variation and in part, indeed, the wearisome reasoning connected with the first variation may be completely dispensed with—to say nothing of the advance which is involved in the removal of the restriction to variations for which the differential coefficients of the function vary but slightly.

While insisting on rigor in the proof as a requirement for a perfect solution of a problem, I should like, on the other hand, to oppose the opinion that only the concepts of analysis, or even those of arithmetic alone, are susceptible of a fully rigorous treatment. This opinion, occasionally advocated by eminent men, I consider entirely erroneous. Such a one-sided interpretation of the requirement of rigor would soon lead to the ignoring of all concepts arising from geometry, mechanics and physics, to a stoppage of the flow of new material from the outside world, and finally, indeed, as a last consequence, to the rejection of the ideas of the continuum and of the irrational number. But what an important nerve, vital to mathematical science, would be cut by the extirpation of geometry and mathematical physics! On the contrary I think that wherever, from the side of the theory of knowledge or in geometry, or from the theories of natural or physical science, mathematical ideas come up, the problem arises for mathematical science to investigate the principles underlying these ideas and so to establish them upon a simple and complete system of axioms, that the exactness of the new ideas and their applicability to deduction shall be in no respect inferior to those of the old arithmetical concepts.

To new concepts correspond, necessarily, new signs. These we choose in such a way that they remind us of the phenomena which were the occasion for the formation of the new concepts. So the geometrical figures are signs or mnemonic symbols of space intuition and are used as such by all mathematicians. Who does not always use along with the double inequality a > b > c the picture of three points following one another on a straight line as the geometrical picture of the idea "between"? Who does not make use of drawings of segments and rectangles enclosed in one another, when it is required to prove with perfect rigor a difficult theorem on the continuity of functions or the existence of points of condensation? Who could dispense with the figure of the triangle, the circle with its center, or with the cross of three perpendicular axes? Or who would give up the representation of the vector field, or the picture of a family of curves or surfaces with its envelope which plays so important a part in differential geometry, in the theory of differential equations, in the foundation of the calculus of variations and in other purely mathematical sciences?

The use of geometrical signs as a means of strict proof presupposes the exact knowledge and complete mastery of the axioms which underlie those figures; and in order that these geometrical figures may be incorporated in the general treasure of mathematical signs, there is necessary a rigorous axiomatic investigation of their conceptual content. Just as in adding two numbers, one must place the digits under each other in the right order, so that only the rules of calculation, i. e., the axioms of arithmetic, determine the correct use of the digits, so the use of geometrical signs is determined by the axioms of geometrical concepts and their combinations.

The agreement between geometrical and arithmetical thought is shown also in that we do not habitually follow the chain of reasoning back to the axioms in arithmetical, any more than in geometrical discussions. On the contrary we apply, especially in first attacking a problem, a rapid, unconscious, not absolutely sure combination, trusting to a certain arithmetical feeling for the behavior of the arithmetical symbols, which we could dispense with as little in arithmetic as with the geometrical imagination in geometry. As an example of an arithmetical theory operating rigorously with geometrical ideas and signs, I may mention Minkowski's work, Die Geometrie der Zahlen.²

Some remarks upon the difficulties which mathematical problems may offer, and the means of surmounting them, may be in place here.

If we do not succeed in solving a mathematical problem, the reason frequently consists in our failure to recognize the more general standpoint from which the problem before us appears only as a single link in a chain of related problems. After finding this standpoint, not only is this problem frequently more accessible to our investigation, but at the same time we come into possession of a method which is applicable also to related problems. The introduction of complex paths of integration by Cauchy and of the notion of the IDEALS in number theory by Kummer may serve as examples. This way for finding general methods is certainly the most practicable and the most certain; for he who seeks for methods without having a definite problem in mind seeks for the most part in vain.

In dealing with mathematical problems, specialization plays, as I believe, a still more important part than generalization. Perhaps in most cases where we seek in vain the answer to a question, the cause of the failure lies in the fact that problems simpler and easier than the one in hand have been either not at all or incompletely solved. All depends, then, on finding out these easier problems, and on solving them by means of devices as perfect as possible and of concepts capable of generalization. This rule is one of the most important levers for overcoming mathematical difficulties and it seems to me that it is used almost always, though perhaps unconsciously.

Occasionally it happens that we seek the solution under insufficient hypotheses or in an incorrect sense, and for this reason do not succeed. The problem then arises: to show the impossibility of the solution under the given hypotheses, or in the sense contemplated. Such proofs of impossibility were effected by the ancients, for instance when they showed that the ratio of the hypotenuse to the side of an isosceles right triangle is irrational. In later mathematics, the question as to the impossibility of certain solutions plays a preeminent part, and we perceive in this way that old and difficult problems, such as the proof of the axiom of parallels, the squaring of the circle, or the solution of equations of the fifth degree by radicals have finally found fully satisfactory and rigorous solutions, although in another sense than that originally intended. It is probably this important fact along with other philosophical reasons that gives rise to the conviction (which every mathematician shares, but which no one has as yet supported by a proof) that every definite mathematical problem must necessarily be susceptible of an exact settlement, either in the form of an actual answer to the question asked, or by the proof of the impossibility of its solution and therewith the necessary failure of all attempts. Take any definite unsolved problem, such as the question as to the irrationality of the Euler-Mascheroni constant C, or the existence of an infinite number of prime numbers of the form 2ⁿ + 1. However unapproachable these problems may seem to us and however helpless we stand before them, we have, nevertheless, the firm conviction that their solution must follow by a finite number of purely logical processes.

Is this axiom of the solvability of every problem a peculiarity characteristic of mathematical thought alone, or is it possibly a general law inherent in the nature of the mind, that all questions which it asks must be answerable? For in other sciences also one meets old problems which have been settled in a manner most satisfactory and most useful to science by the proof of their impossibility. I instance the problem of perpetual motion. After seeking in vain for the construction of a perpetual motion machine, the relations were investigated which must subsist between the forces of nature if such a machine is to be impossible;³ and this inverted question led to the discovery of the law of the conservation of energy, which, again, explained the impossibility of perpetual motion in the sense originally intended.

This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus.

The supply of problems in mathematics is inexhaustible, and as soon as one problem is solved numerous others come forth in its place. Permit me in the following, tentatively as it were, to mention particular definite problems, drawn from various branches of mathematics, from the discussion of which an advancement of science may be expected.

Let us look at the principles of analysis and geometry. The most suggestive and notable achievements of the last century in this field are, as it seems to me, the arithmetical formulation of the concept of the continuum in the works of Cauchy, Bolzano and Cantor, and the discovery of non-euclidean geometry by Gauss, Bolyai, and Lobachevsky. I therefore first direct your attention to some problems belonging to these fields.

1. Cantor's problem of the cardinal number of the continuum

2. The compatibility of the arithmetical axioms

3. The equality of two volumes of two tetrahedra of equal bases and equal altitudes

4. Problem of the straight line as the shortest distance between two points

5. Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining the group

6. Mathematical treatment of the axioms of physics

7. Irrationality and transcendence of certain numbers

8. Problems of prime numbers

9. Proof of the most general law of reciprocity in any number field

10. Determination of the solvability of a diophantine equation

Given a diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: to devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers.

12. Extension of Kroneker's theorem on abelian fields to any algebraic realm of rationality

13. Impossibility of the solution of the general equation of the 7-th degree by means of functions of only two arguments

14. Proof of the finiteness of certain complete systems of functions

15. Rigorous foundation of Schubert's enumerative calculus

16. Problem of the topology of algebraic curves and surfaces

17. Expression of definite forms by squares

18. Building up of space from congruent polyhedra

19. Are the solutions of regular problems in the calculus of variations always necessarily analytic?

20. The general problem of boundary values

21. Proof of the existence of linear differential equations having a prescribed monodromic group

22. Uniformization of analytic relations by means of automorphic functions

23. Further development of the methods of the calculus of variations

The problems mentioned are merely samples of problems, yet they will suffice to show how rich, how manifold and how extensive the mathematical science of today is, and the question is urged upon us whether mathematics is doomed to the fate of those other sciences that have split up into separate branches, whose representatives scarcely understand one another and whose connection becomes ever more loose. I do not believe this nor wish it. Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts. For with all the variety of mathematical knowledge, we are still clearly conscious of the similarity of the logical devices, the relationship of the ideas in mathematics as a whole and the numerous analogies in its different departments. We also notice that, the farther a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separate branches of the science. So it happens that, with the extension of mathematics, its organic character is not lost but only manifests itself the more clearly.

But, we ask, with the extension of mathematical knowledge will it not finally become impossible for the single investigator to embrace all departments of this knowledge? In answer let me point out how thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which at the same time assist in understanding earlier theories and cast aside older more complicated developments. It is therefore possible for the individual investigator, when he makes these sharper tools and simpler methods his own, to find his way more easily in the various branches of mathematics than is possible in any other science.

The organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of all exact knowledge of natural phenomena. That it may completely fulfil this high mission, may the new century bring it gifted masters and many zealous and enthusiastic disciples!