But in 1914 Hardy's collaborator J.E. Littlewood had shown that this was not so, for there existed some point where the simple assumption would underestimate the cumulative total of primes.
但是在1914年，哈代的合作者J.E.列托伍德将此推翻，他认为会存在某些转折点，使这个公式转而低估素数的数量。
Then in 1933 a Cambridge mathematician, S. Skewes, had shown19 that if the Riemann Hypothesis were true, a crossover point would occur before which,
随后在1933年，剑桥的一位数学家S.斯奇乌斯表明，如果黎曼猜想是成立的，那么就会在之前找到一个转折点。
Hardy commented, was probably the largest number ever to serve any definite purpose in mathematics.
哈代评论说，这个数可能是有用的数学研究中出现过的最大的数注。
It could be asked whether this enormous bound might be reduced, or whether one could be found that did not depend upon the truth of the Riemann Hypothesis, and these were the problems that Alan now undertook.
人们现在要问，这个巨大的范围是否有可能缩小，或者说，是否有可能不依赖于黎曼猜想的正确性而找到一个范围。这就是艾伦现在要研究的问题。
One new departure at Cambridge was his acquaintance with Ludwig Wittgenstein, the philosopher.
艾伦在剑桥的另一个收获，是认识了哲学家路德维希·维特根斯坦。
He would have seen Wittgenstein before at the Moral Science Club, and Wittgenstein (like Bertrand Russell) had received a copy of Computable Numbers.
他以前也许在道德科学俱乐部见过维特根斯坦，维特根斯坦（他喜欢伯特兰·罗素）已经读过了《可计算数》。
But it was in this summer of 1937 that Alister Watson, the King's Fellow, introduced them and they met sometimes in the botanical gardens.
1937年夏天，国王学院的研究员艾力斯特·沃特森为他们引见，他们时常会在植物园见面。
Watson had written a paper on the foundations of mathematics for the Moral Science Club, in which he made use of the Turing machine.
沃特森围绕图灵机的概念，给道德科学俱乐部写了一篇关于数学基础的论文。
Wittgenstein, whose first work had been as an engineer, always liked practical, down-to-earth constructions and would have approved of the way that Alan had made a vague idea so definite.
维特根斯坦是工程师出身，他很喜欢实践，他想要通过一种实体的构建，来证明艾伦的想法。
Curiously, the failure of the Hilbert programme had also meant the end of the point of view advanced by Wittgenstein in his first phase, in the Tractatus Logico-Philosophicus, that every well-posed problem could be solved.
有趣的是，希尔伯特计划的失败，终结了维特根斯坦前期的信念，在他的《逻辑哲学论》中，他认为所有表述清楚的问题都是可解的。