纳森（Melvyn Bernard Nathanson，1944年。美国数学家，专门研究数论，纳森是纽约数论研讨会的共同组织者，美国科学促进会和纽约科学院的成员。

纳森的“绝望地寻求数学真理”（Desperately seeking mathematical truth）发表在2008年8月的美国数学学会通讯（Notices of the American Mathematical Society）Opinion栏目上，在这篇文章中纳森讨论数学哲学中的认识论问题和有关数学文献的可靠性问题。

http://www.ams.org/notices/200807/tx080700773p.pdf

一，译文：“绝望地寻求数学真理”

我们数学家对真理有一种天真的信念，我们证明定理。定理是从公理推导出来的，证明中的每一行都是证明的前几行或以前证明的定理的一个简单结果。我们的结论是真实的，无条件的和永恒的。巴比伦人的二次方程和希腊人对√2的非有理数的证明，即使在Large Magellanic Cloud中也是真的。

我们如何知道一个证明是正确的？通过逐行检查它，一台计算机甚至可以被编程来检查它。要发现一个证明，要有想象力和天才来构思一连串的推理，从微不足道的公理引向异常美丽的结论—这是一种罕见的、美妙的才能。这就是数学。但是要检查一个证明—任何傻瓜都可以做到这一点。

不过，这种对数学确定性的信念还是让人感到担忧。2000年，克莱数学研究所宣布为解决七个“千年难题”提供百万美元奖金，解决了其中一个问题，就可以得到一百万美元。根据克莱数学研究所的规则，在解决方案出现在“具有世界声誉的参考数学出版物”上两年后，并在“数学界普遍接受”后，将颁发该奖项。

但为什么要拖延呢？当然，任何有能力的人都可以检查一个证明，它不是对的就是错的，为什么要等两年？原因是，许多伟大而重要的定理实际上并没有证明，它们有证明的草图、论证的大纲、提示和直觉，这些对作者来说是显而易见的（至少在写作时是这样），而且，希望能被数学界的一部分人理解和相信。

但数学界本身是很小的，在大多数数学领域，专家很少。事实上，世界上活跃的研究型数学家非常少，而重要的问题却很多，所以数学家的数量与问题的数量之比很小。每个领域都有“老板（bosses）”，他们宣称一个新结果的正确性或不正确性，以及其重要性或不重要性，有时他们意见不一，就像帮派头目争夺地盘一样。在任何情况下，整个数学界都有一个半证明的定理网。我们对一个定理的真理性的认识，取决于其证明的正确性，以及在其证明中使用的所有定理的正确性。这是一个不稳定的基础。

即使是欧几里德也会出错，因为《几何原本》中的一些陈述（例如第一册，命题1）在逻辑上并不符合公理。在莱布尼茨和牛顿之后的150年里，微分和积分的基础才被正确地制定出来，我们可以回填许多18和19世纪的数学分析的证明。

类似的问题今天也困扰着我们，考虑一下最近两个最广为人知的数学成功。韦尔斯对“费马最后定理”的证明用了几年时间才确认其正确性，在原始论文中发现了一个错误，而且对证明中使用的其他结果的真实性仍有疑问。还有关于佩雷尔曼对庞加莱猜想的证明的完整性的论证，这是第一个被解决的千年难题。有多少数学家同时检查了韦尔斯和佩雷尔曼的证明？

我当然不是声称韦尔斯或佩雷尔曼的工作有漏洞，我不知道。我们（数学界）相信这些证明是正确的，因为已经形成了支持其正确性的政治共识。

有限简单群的分类提供了另一个例子。关于是否有完整的分类证明，以及如果有证明的话，该定理究竟是什么时候被证明的，仍然存在着不确定性。在分类法据说完成后，丹尼-戈伦斯坦想写一篇证明的阐述，他将重读论文并发现错误，他总是可以把事情修补好，但在他这样做之前，根据我们专业的规则，原始论文中的“定理”不是定理，而是未经证实的断言。我们数学家喜欢谈论我们文献的“可靠性”，但事实上，它是不可靠的。

部分问题是审稿，期刊上的许多（我认为是大多数）论文都没有经过审稿。有一个假定的审稿人看了论文，读了导言和结果陈述，看了一下证明，如果一切正常，就建议发表。有些审稿人会逐行检查证明，但许多审稿人不会这样做。当我阅读一篇期刊文章时，我经常发现错误，我是否能修正它们并不重要。文献是不可靠的。

我们如何识别数学真理？如果一个定理有一个简短的完整证明，我们可以检查它。但如果证明很深奥，很困难，写满了100页，如果没有人有时间和精力来填补细节，如果一个“完整”的证明会有10万页长，那么我们就依靠这个领域的老板们的判断。在数学中，一个定理是真实的，否则它就不是一个定理。但即使在数学中，真理也可以是政治性的。

二，原文

We mathematicians have a naive belief in truth. We prove theorems. Theorems are deductions from axioms. Each line in a proof is a simple consequence of the previous lines of the proof, or of previously proved theorems. Our conclusions are true, unconditionally and eternally. The Babylonians’ quadratic formula and the Greeks’ proof of the irrationality of √ 2 are true even in the Large Magellanic Cloud.

How do we know that a proof is correct? By checking it, line by line. A computer might even be programmed to check it. To discover a proof, to have the imagination and genius to conceive a chain of reasoning that leads from trivial axioms to extraordinarily beautiful conclusions— this is a rare and wonderful talent. This is mathematics. But to check a proof—any fool can do this.

Still, there is a nagging worry about this belief in mathematical certitude. In 2000 the Clay Mathematics Institute announced million dollar prizes for the solution of seven “Millennium Problems”. Solve one of the problems and receive a million dollars. According to CMI’s rules, two years after the appearance of the solution in a “refereed mathematics publication of worldwide repute” and after “general acceptance in the mathematics community”, the prize would be awarded.

But why the delay? Surely, any competent person can check a proof. It’s either right or wrong. Why wait two years? The reason is that many great and important theorems don’t actually have proofs. They have sketches of proofs, outlines of arguments, hints and intuitions that were obvious to the author (at least, at the time of writing) and that, hopefully, are understood and believed by some part of the mathematical community.

But the community itself is tiny. In most fields of mathematics there are few experts. Indeed, there are very few active research mathematicians in the world, and many important problems, so the ratio of the number of mathematicians to the number of problems is small. In every field, there are “bosses” who proclaim the correctness or incorrectness of a new result, and its importance or unimportance. Sometimes they disagree, like gang leaders fighting over turf. In any case, there is a web of semi-proved theorems throughout mathematics. Our knowledge of the truth of a theorem depends on the correctness of its proof and on the correctness of all of the theorems used in its proof. It is a shaky foundation.

Even Euclid got things wrong, in the sense that there are statements in the Elements (e.g., Book I, Proposition 1) that do not follow logically from the axioms. It took 150 years after Leibnitz and Newton until the foundations of differential and integral calculus were formulated correctly, and we could backfill proofs of much eighteenth and nineteenth century mathematical analysis.

Similar problems plague us today. Consider the two most highly publicized recent successes of mathematics. It took several years to confirm the correctness of Wiles’ proof of “Fermat’s Last Theorem”. A mistake was found in the original paper, and there still remained questions about the truth of other results used in the proof. There were also arguments about the completeness of Perelman’s proof of the Poincaré conjecture, the first of the Millennium Problems to be solved. How many mathematicians have checked both Wiles’ and Perelman’s proofs?

I certainly don’t claim that there are gaps in Wiles’ or Perelman’s work. I don’t know. We (the mathematical community) believe that the proofs are correct because a political consensus has developed in support of their correctness.

The classification of the finite simple groups provides another example. There is still uncertainty about whether there is a complete proof of the classification, and, if there is a proof, exactly when the theorem was proved. After the classification was supposedly finished, Danny Gorenstein wanted to write an exposition of the proof. He would reread the papers and find mistakes. He could always patch things up, but until he did, the “theorems” in the original papers were, according to the rules of our profession, not theorems but unproven assertions. We mathematicians like to talk about the “reliability” of our literature, but it is, in fact, unreliable.

Part of the problem is refereeing. Many (I think most) papers in most refereed journals are not refereed. There is a presumptive referee who looks at the paper, reads the introduction and the statements of the results, glances at the proofs, and, if everything seems okay, recommends publication. Some referees do check proofs line-by-line, but many do not. When I read a journal article, I often find mistakes. Whether I can fix them is irrelevant. The literature is unreliable.

How do we recognize mathematical truth? If a theorem has a short complete proof, we can check it. But if the proof is deep, difficult, and already fills 100 journal pages, if no one has the time and energy to fill in the details, if a “complete” proof would be 100,000 pages long, then we rely on the judgments of the bosses in the field. In mathematics, a theorem is true, or it’s not a theorem. But even in mathematics, truth can be political.

参考文献：

http://www.ams.org/notices/200807/tx080700773p.pdf

https://fr.wikipedia.org/wiki/Melvyn_Nathanson