希尔伯特问题

而且，正如他对希尔伯特问题的研究，

But it was also analogous to his attack on the Hilbert problem .

多连通域上二阶非线性椭圆型方程组广义黎曼-希尔伯特问题

The generalized Riemann-Hilbert Problem far a multi-connected region of Second Order Non-linear Elliptic Equations

虽然他解决希尔伯特问题、研究序数逻辑时，都是集中研究机械过程的极限，

Although his solution of the Entscheidungs problem and his work on ordinal logics had focussed attention upon the limitations of mechanical processes ,

当艾伦想到这里时，他马上就知道了希尔伯特问题的答案——不。

As soon as he had made that observation , Alan could see that the answer to Hilbert 's question was ' no ' .

如果艾伦是个遵守规则的学者，他在解决希尔伯特问题之前，就应该读首先遍所有的相关文献，包括丘奇的工作。

If he had been a more conventional worker , he would not have attacked the Hilbert problem without having read up all of the available literature , including Church 's work .

当艾伦意识到这一点，他就知道了希尔伯特的问题的答案，

And once he had seen this , he knew the answer to Hilbert 's question .

他从零开始，想要制造一台机器，来解决希尔伯特的问题，自动判断任意数学命题是否可以被证明。他似乎忘了数学是多么宏伟而复杂。

He started from nothing , and tried to envisage a machine that could tackle Hilbert 's problem , that of deciding the provability of any mathematical assertion presented to it .

丘奇在一年前发现，不可解的问题是存在的，在这篇证明中，他用精确的形式把它描述出来，回答了希尔伯特的问题。

Church 's essential idea , showing the existence of an ' unsolvable problem ' , had been announced a year earlier , but only at this point did he put it exactly in the form of answering Hilbert 's question .

但他的论文不像《可计算数》那么有野心，他既没有构造一个通用的操作者来解决希尔伯特的问题，也没有讨论关于思维的事情。

Post 's paper was much less ambitious than Computable Numbers ; he did not develop a ' universal worker ' nor himself deal with the Hilbert decision problem . Nor was there any argument about ' states of mind ' .

你最近寄给我的定义可算数、并证明希尔伯特判定问题无解的那篇论文，强烈地吸引了这里的一位年轻人。

An offprint which you kindly sent me recently of your paper in which you define ' calculable numbers ' , and shew that the Entscheidungs problem for Hilbert logic is insoluble , had a rather painful interest for a young man , A.M. Turing ,

一个命题用公理进行逻辑推演，却既不能证明，也不能证伪，所以，对于希尔伯特的问题来说，哥德尔已经证明，算术是不完备的。

Nor could it be proved false , for the same reason . It was an assertion which could not be proved or disproved by logical deduction from the axioms , and so Gdel had proved that arithmetic was incomplete , in Hilbert 's technical sense .

五月中旬，当纽曼开始读艾伦的论文时，他几乎无法相信，图灵机如此简单地解决了希尔伯特的问题。在哥德尔解决了前两步之后，这个问题已经让大家花了五年时间了。

When Newman did read it in mid-May , he could hardly believe that so simple and direct an idea as the Turing machine would answer the Hilbert problem over which many had been labouring for the five years since Gdel had disposed of the other Hilbert questions .

这样就不会出现被别人抢先的问题，但他也就永远无法创造出图灵机了。图灵机不仅解答了希尔伯特的问题，而且提出了新的问题——思维状态。

He then might not have been pre-empted - but then , he might never have created the new idea of the logical machine , with its simulation of ' states of mind ' , which not only closed the Hilbert problem but opened up quite new questions .

关于黎曼&希尔伯特边值问题的奇异情形

The Singular Case of Riemann-Hilbert Boundary Value Problem

希尔伯特提出的问题主要是，在原则上，《数学原理》的限制是什么。

In fact Hilbert posed the question as to what were , in principle , the limitations of a scheme such as that of Principia Mathematica .

这一进步，统一了数字和长度的概念，也把希尔伯特的几何问题，转化成了一个算术领域的问题。

This step both unified the concepts of number and length , and had the effect of pushing Hilbert 's questions about geometry into the domain of the integers , or ' arithmetic , " in its technical mathematical sense .

希尔伯特说过好的问题，其标准有二：一是清晰易懂；二是难而又可解决。

Hilbert had said " good question " has two standards : One is clear and understandable ; second is difficult but can be solved .

1936年4月15日，大概是在这天，艾伦告诉纽曼，他证明了希尔伯特的判定性问题是无解的。同样是在这一天，普林斯顿的美国逻辑学家阿隆佐·丘奇也发表了他的证明。

Almost on the same day that Alan announced his discovery to Newman , someone else completed a demonstration that the Hilbert Entscheidungs-problem was unsolvable . It was at Princeton , where the American logician Alonzo Church had finished his argument for publication1 on 15 April 1936 .