数学的方法也会出错The Methods of Mathematics Are Fallible
Brett：如果把数学和物理作个比较：我们有个领域叫粒子物理学，而粒子物理学中最深刻的理论叫作标准模型。它描述了所有存在的基本粒子、粒子之间的相互作用、粒子之间的力，以及规范玻色子——正是它们传递着电子、质子和中子等粒子之间的力。
那么，物质是由什么构成的？我们会说，物质是由物理标准模型所描述的这些粒子构成的。但这是否就排除了这样一种可能——这些基本粒子本身又由更小的粒子组成？我们还有一种可能更深刻的理论，叫作弦理论。因此，我们对"什么是最基本粒子"的认知，与现实中最基本的粒子究竟是什么，是两回事。
数学也是如此。多伊奇解释说，数学是一个我们试图揭示必然真理的领域。数学的研究对象是必然真理，就像粒子物理学的研究对象是基本粒子一样。
但既然基础粒子物理学的研究对象是基本粒子，这并不意味着你真的找到了基本粒子。它只意味着你找到了你最大的粒子加速器所能分辨出的最小粒子。
而如果你有一台更大的粒子加速器，你或许会在这些粒子内部发现更小的粒子。
这就是粒子物理学的历史。我们曾经以为原子是基本的。后来，我们当然发现原子内部还有原子核和电子。在原子核里，我们发现存在质子和中子。而在质子和中子内部，我们发现它们由夸克构成。我们目前就停留在这里——现在我们认为夸克是基本的，电子也是基本的。
但这并不意味着粒子物理学就此终结。我们需要的是进一步的理论，去探索那些极小粒子内部还可能藏着什么。
把这一点和数学对照来看：如果必然真理是数学的研究对象，那么数学家所从事的，就是创造关于必然真理的知识。因为数学家也有一颗大脑——大脑是一个物理对象——而所有物理对象都会因热力学第二定律而发生退化性的差错，或者干脆说，任何人都会犯寻常的心智错误和失误——所以数学家和别人一样容易出错。因此，他们最终证明出来的东西也可能是错的。
Naval：如果我没理解错的话，即便是数学也可能出错，因为数学是一种创造性的活动。我们永远无法真正完工。你的某个公理里可能一直藏着错误。
Brett: If I compare math to physics: We have this domain called particle physics, and the deepest theory we have in particle physics is called the standard model. This describes all of the fundamental particles that exist and the interactions between them, the forces that exist between them, and the gauge bosons, which mediate the force between particles like electrons, protons and neutrons.
Now, what is matter made of? We would say matter is made of these particles described by the standard model of physics. But does that rule out the fact that these fundamental particles might themselves consist of even smaller particles? We have a possibly deeper theory called string theory. So our knowledge of what the most fundamental particles are and what, in reality, the most fundamental particles are, is different.
So, too in mathematics. Deutsch explains that mathematics is a field where what we’re trying to uncover is necessary truth. The subject matter of mathematics is necessary truth, in the same way that the subject matter of particle physics is the fundamental particles.
But since the subject matter of fundamental particle physics is the fundamental particles, that doesn’t mean you actually find the fundamental particles. All it means is that you have found the smallest particles that your biggest particle accelerators are able to resolve.
But if you had an even bigger particle accelerator, you might find particles within those particles.
This has been the history of particle physics. We used to think that atoms were fundamental. Then, of course, we found they contained nuclei and electrons. In the nuclei, we found out that there were protons and neutrons. Inside the protons and neutrons, we found out they were made up of quarks. And that’s where we’re at right now. We’re at the point where we say that quarks are fundamental and electrons and fundamental.
But that doesn’t mean that we’re going to end particle physics right now. What we need are further theories about what might be inside of those really small particles.
Comparing that to mathematics, if necessary truth is the subject matter of mathematics, mathematicians are engaged in creating knowledge about necessary truth. Because a mathematician has a brain—which is a physical object—and all physical objects are subject to making errors of degradation via the second law of thermodynamics—or simply the usual mental mistakes and errors that any human being makes—a mathematician is just as fallible as anyone else. So what they end up proving could be in error.
Naval: If I understand this point, even mathematics is capable of error because mathematics is a creative act. We’re never quite done. There could have been a mistake in your axiom somewhere.