一切知识都是猜想All Knowledge Is Conjectural
Brett： 所有的知识都是猜测性的。它始终在被猜测。它是我们在任何特定时刻的最佳理解。
你说得对，公理有可能是错的。那我们怎么知道一个公理是错的呢？传统上的回答是：“因为它显而易见、明摆着就是这样。”你如何证明 x 加零必须等于 x？你只能接受它是真的。
但想想欧几里得的《几何原本》这样的东西吧。任何人都可以自己做做这个实验：拿一张纸、一支笔，在纸上点两个点。现在，你能画出多少条穿过这两个点的不同的直线？答案对你来说应该相当明显：只能画出一条。然而，我们知道这是错的。
回想一下：当你盯着那张纸——看着那条唯一的直线从中穿过——你会生出确定无疑的感觉。你绝对相信自己没有错。对于这种感受，我们应该始终心存怀疑。即使在一个像数学这样看似充满确定性的领域，那些曾绝对笃定的人，也已被证明是错的。
那么我们怎么证明它是错的呢？你可能会觉得我在耍花招，但话说回来，你得回想一下，当我最初让你“过两点画一条直线”时，你是否理解了我的意思。把纸弯起来。用三维的方式思考。如果你有篮球，就把纸包在篮球上。现在想想，你能用哪些方式在两点之间画出一条直线。
你可以用笔在其中一点上戳一个洞，然后从纸的另一面、穿过另一个洞把笔推出来——于是你就得到了一条不同的直线。你既有用笔画出的那条直线，也有这样一条直线：它实实在在就是你的笔穿过这两个点。
你最初那种“过这两点只能画出唯一一条直线”的绝对确定感是错的。你可能会想：“这不公平，这是作弊。”你是在二维空间里思考。我不是。我想的维度比那更多。
卡尔·波普尔有句绝妙的话：“不可能用某种方式说话而不被误解。”情况总是如此。
即使在数学中——这个我们力求尽可能精确的领域——人们也可能犯错，可能对自己想要论证的论点抱持错误的前提。
欧几里得几何的这个具体例子——因为几何传统上是在纸上的二维平面里进行的——被不同的人先后解决，并催生了弯曲空间中的几何学，进而让爱因斯坦提出了广义相对论。
所以，正是对这些最深层的假设——那些我们认为自己绝无可能出错的地方——发起质疑，才带来了真正的进步，才带来了科学以及所有其他领域里真正根本性的变革。
Brett: All knowledge is conjectural. It’s always being guessed. It’s our best understanding at any given time.
You’re right to say that the axioms might be incorrect. How do we know that an axiom is incorrect? Traditionally the answer has been, “Because it’s clearly and obviously the case.” How can you prove that x plus zero must equal x? You just have to accept that it’s true.
But consider something like Euclid’s Elements. Anyone might want to try this experiment for themselves: Take a piece of paper, take a pen, draw two dots on the piece of paper. Now, how many unique straight lines can you draw through those two dots? It should be fairly obvious to you that only one line can be drawn. However, we know that’s false.
Reflect on the fact that as you’re staring at the piece of paper, through which only one straight line is being drawn, you have the feeling of certainty. You are absolutely sure that you’re not wrong. This feeling is something we should always be skeptical of. When people have been absolutely certain, even in a domain as apparently full of certainty as mathematics, they’ve been shown to be wrong.
So how can we show it’s wrong? You might think that I’m cheating, but, then again, you have to reflect on whether you understood what I was saying when I first told you to draw a unique straight line through two points. Bend the piece of paper. Think in three dimensions. Wrap the piece of paper around a basketball if you have one. Now consider the ways in which you could draw a straight line through those two points.
You could punch a hole through one of those dots with your pen and push it out through the other side through the other hole—and now you have a different straight line. You have the straight line that is drawn with your pen, and you have a straight line that is literally your pen pushed through these two dots.
Your initial feeling of absolute certainty that only a unique line could be drawn through these two dots is false. You might be thinking, “That’s unfair, that’s cheating.” You were thinking in two dimensions. I wasn’t. I was thinking in more dimensions than that.
Karl Popper has this wonderful saying, “It is impossible to speak in such a way that you cannot be misunderstood.” This is always the case.
Even in mathematics, where we try to be as precise as possible, it’s possible for people to make errors, to think false premises about what argument they’re trying to make.
This particular example of Euclidean geometry—because geometry was traditionally done in two dimensions on a piece of paper—was resolved by various people and led to geometry in curved space, which led to Einstein coming up with the general theory of relativity.
So it is questioning these deepest assumptions we have—where we think there’s no possible way we could be mistaken—that leads to true progress and to a genuine, fundamental change in the sciences and everywhere else.