数学没有定论There Is No Settled Mathematics
Naval： 还有两位我喜欢的科学思想家，得出了与多伊奇相似的结论。
一位是纳西姆·塔勒布，他普及了黑天鹅的概念——无论有多少只白天鹅，都不能证明黑天鹅不存在。你永远无法确凿地说所有的天鹅都是白的。你永远无法确立最终的真理。你所能做的，就是用你今天拥有的最好解释去工作，而这仍然远胜于无知。任何时候都可能冒出一只黑天鹅来推翻你的理论，然后你就得去寻找一个更好的理论。
另一位我觉得很迷人的人是格雷戈里·柴廷。他是一位与库尔特·哥德尔一脉相承的数学家，因为他探索的是数学中可能性的极限与边界。他提出的一个观点是：哥德尔不完备定理并不是说数学是垃圾；这个定理不值得绝望。哥德尔不完备定理说的是：任何形式系统——包括数学在内——都不可能既完备又一致。要么存在一些为真却无法在系统内被证明为真的命题，要么系统内部某处必然出现矛盾。
对于那些把数学看作一种抽象、完美、完全自洽之物的人来说，这可能成为绝望的理由。但柴廷论证说，实际上，这为数学打开了创造性的空间。它意味着，即使在数学中，你也总是离"推翻某个东西、然后为它找到更好的解释"只有一步之遥。它把人类、人类的创造力以及人类寻找好解释的努力，重新放回了核心位置。
在某个深层意义上，数学仍然是一门艺术。当然，数学会产出非常有用的东西。你仍然在建造一座知识的大厦，但根本不存在什么最终的、尘埃落定的真理。没有定论的科学，也没有定论的数学。有的只是好的解释，它们会随着时间被更好的、能解释更多世界的解释所取代。
布雷特： 这是我们主要从学校教育中继承来的东西。它是我们学术文化的一部分，并渗透进了更广泛的文化。人们有这样的观念：数学是一块纯净的知识领域，在那里被证明为真的东西就一定是真的。
然后是科学，它不给你确定无疑的真理，但你对所发现的东西可以高度确信。你可以用实验来确认你的说法看起来是正确的，但你仍然可能出错。然后，当然还有哲学，它仅仅是观点之争。
这就是一些人从学校继承来的等级体系：数学是确定的，科学几乎是确定的，其余的一切或多或少都是观点问题。这就是多伊奇所说的"数学家的误解"。数学家有一种直觉式的认知方式，认为他们的证明——通过这种证明方法得出的定理——是绝对、确定无疑的。
事实上，这是把研究对象和对研究对象的知识混为一谈了。
Naval: There are two other scientific thinkers who I like who come to similar conclusions as Deutsch.
One is Nassim Taleb, who popularized the idea of the black swan, which is that no number of white swans disproves the existence of a black swan. You can never conclusively say all swans are white. You can never establish a final truth. All you can do is work with the best explanation you have today, which is still far better than ignorance. At any time a black swan can show up and disprove your theory, and then you have to go find a better one.
The other one I find fascinating is Gregory Chaitin. He is a mathematician very much in the vein of Kurt Gödel because he explores the limits and boundaries of what is possible in mathematics. One of the points that he makes is that Gödel’s incompleteness theorem doesn’t say that mathematics is junk; the theorem isn’t a cause for despair. Gödel’s incompleteness theorem says that no formal system—including mathematics—can be both complete and correct. Either there are statements that are true that cannot be proven true in the system, or there will be a contradiction somewhere inside the system.
This could be a cause of despair for mathematicians who view mathematics as this abstract, perfect, fully self-contained thing. But Chaitin makes the argument that, actually, it opens up for creativity in mathematics. It means that even in mathematics you are always one step away from falsifying something and then finding a better explanation for it. It puts humans and their creativity and their bid to find good explanations back at the core of it.
At some deep level, mathematics is still an art. Of course, very useful things come out of mathematics. You’re still building an edifice of knowledge, but there is no such thing as a conclusive, settled truth. There is no settled science, there is no settled mathematics. There are good explanations that will be replaced over time with more good explanations that explain more of the world.
Brett: This is something that we inherit from our schooling more than anything else. It’s part of our academic culture, and it bleeds into the wider culture as well. People have this idea that mathematics is this pristine area of knowledge where what is proved to be true is certainly true.
Then you have science, which doesn’t give you certain truth but you can be highly confident in what you discover. You can use experiments to confirm that what you’re saying appears to be correct, but you might be wrong. And then, of course, there’s philosophy, which is a mere matter of opinion.
This is the hierarchy that some people inherit from school: Mathematics is certain, science is almost certain, and the rest of it is more or less a matter of opinion. This is what Deutsch calls the mathematician’s misconception. Mathematicians have this intuitive way of realizing that their proof—the theorem they have reached by this method of proof—is absolutely, certainly true.
In fact, it’s a confusion between the subject matter and their knowledge of the subject matter.