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.