数学是否存在致命缺陷？

Consider this mathematician, with her standard-issue infinitely sharp knife and a perfect ball. She frantically slices and distributes the ball into an infinite number of boxes. She then recombines the parts into five precise sections. Gently moving and rotating these sections around, seemingly impossibly, she recombines them to form two identical, flawless, and complete copies of the original ball.

这是一位数学家。 她有一把绝对锋利的标准刀 和一个完美球体。 她手起刀落将球切分， 并把切片分别装入无数个盒子中， 然后将这些切片 分成完全平均的五份。 通过仔细地移动和旋转这些切片， 她竟然能将这些切片 重组为两个完整无缝的球体， 且与原本的球体完全一致。

This is a result known in mathematics as the Banach-Tarski paradox. The paradox here is not in the logic or the proof— which are, like the balls, flawless— but instead in the tension between mathematics and our own experience of reality. And in this tension lives some beautiful and fundamental truths about what mathematics actually is. We’ll come back to that in a moment, but first, we need to examine the foundation of every mathematical system: axioms.

这个结果在数学中 被称为巴纳赫-塔尔斯基悖论。 这里的悖论不是 因逻辑或证据产生的—— 毕竟它们就像 那些球一样完美无缺—— 而在于数学 与我们现实体验之间的矛盾。 于此矛盾之中，我们能够窥见 一些迷人的基本真理， 从而理解数学的本质。 这个问题先放在一边。 首先， 我们需要检视一下 所有数学系统的基础：公理。

Every mathematical system is built and advanced by using logic to reach new conclusions. But logic can’t be applied to nothing; we have to start with some basic statements, called axioms, that we declare to be true, and make deductions from there. Often these match our intuition for how the world works— for instance, that adding zero to a number has no effect is an axiom. If the goal of mathematics is to build a house, axioms form its foundation— the first thing that’s laid down, that supports everything else. Where things get interesting is that by laying a slightly different foundation, you can get a vastly different but equally sound structure.

每个数学系统都是通过 使用逻辑得出新结论来构建和推进的。 但是逻辑也需要应用的对象； 我们必须从一些 被称为“公理”的基本陈述开始。 我们声明这些陈述是正确的， 然后基于其进行推理。 公理一般与我们的直觉 对世界的认知是相符的—— 如 “ 0 与数字相加不改变结果” 是一个公理。 如果数学的目标是盖房子， 公理就是地基—— 地基是最早打下的， 支撑其他所有结构。 有趣的是， 通过打一个略有不同的地基， 能够得到一个截然不同 但同样坚实的结构。

For example, when Euclid laid his foundations for geometry, one of his axioms implied that given a line and a point off the line, only one parallel line exists going through that point. But later mathematicians, wanting to see if geometry was still possible without this axiom, produced spherical and hyperbolic geometry. Each valid, logically sound, and useful in different contexts.

例如，欧几里得 为几何学奠定基础时， 由他其中一条公理可推理出： 给定一条直线和一个离线点， 只有一条平行线穿过该点。 但是后来的数学家想知道 几何学没有这条公理是否还能成立， 于是有了球面几何和双曲几何。 它们都有效且符合逻辑， 在不同的情况下非常有用。

One axiom common in modern mathematics is the Axiom of Choice. It typically comes into play in proofs that require choosing elements from sets— which we’ll grossly simplify to marbles in boxes. For our choices to be valid, they need to be consistent, meaning if we approach a box, choose a marble, and then go back in time and choose again, we'd know how to find the same marble. If we have a finite number of boxes, that’s easy. It’s even straightforward when there are infinite boxes if each contains a marble that’s readily distinguishable from the others. It’s when there are infinite boxes with indistinguishable marbles that we have trouble. But in these scenarios, the Axiom of Choice lets us summon a mysterious omniscient chooser that will always select the same marbles— without us having to know anything about how those choices are made. Our stab-happy mathematician, following Banach and Tarski’s proof, reaches a step in constructing the five sections where she has infinitely many boxes filled with indistinguishable parts. So she needs the Axiom of Choice to make their construction possible.

现代数学中常用的一个公理 是选择公理。 需要从集合中选择元素的证明中 通常会用到它—— 这种证明简单来说， 就好比盒装弹珠。 为了使我们的选择有效， 它们必须保持一致， 意味着若从其中一个盒子中 选择一颗弹珠， 然后回到过去、再次选择， 我们就会知道如何找到同一颗弹珠。 如果我们的盒子数量有限， 这便容易做到。 就算有无限个盒子也是小菜一碟， 只要盒中含有一颗 与其他弹珠不同的弹珠。 然而，当有无限个盒子， 且装的全是无法区分的弹珠时， 就会非常难办。 但是，在这种情况下， 选择公理就好比召唤了 一个无所不知的神秘选择者， 总是能选出相同的弹珠， 而无需我们知道 这些选择是如何做出的。 我们的疯狂数学家 效仿巴纳赫和塔尔斯基的证明， 走到了将五个部分重组的那一步， 她已在无限多的盒子中 装入了无法区分的小切片。 因此，她需要选择公理 才有可能将它们复原。

If the Axiom of Choice can lead to such a counterintuitive result, should we just reject it? Mathematicians today say no, because it’s load-bearing for a lot of important results in mathematics. Fields like measure theory and functional analysis, which are crucial for statistics and physics, are built upon the Axiom of Choice. While it leads to some impractical results, it also leads to extremely practical ones.

如果选择公理带来的结果 如此地反直觉， 我们不应该否认它吗？ 今天的数学家不这么认为， 因为它是数学中 许多重要成果的根基。 测度论和泛函分析等领域 对于统计学和物理学至关重要； 而它们都是建立在选择公理上的。 虽然选择公理能推理出 不切实际的结果， 但也能得到极其实用的结果。

Fortunately, just as Euclidean geometry exists alongside hyperbolic geometry, mathematics with the Axiom of Choice coexists with mathematics without it. The question for many mathematicians isn’t whether the Axiom of Choice, or for that matter any given axiom, is right or not, but whether it’s right for what you’re trying to do. The fate of the Banach-Tarski paradox lies in this choice.

幸运的是，就像欧几里得几何 与双曲几何共存一样， 使用选择公理和不使用它的数学 也是共存的。 对于许多数学家来说， 他们关注的并非这些公理的对错， 不论是选择公理还是其他公理， 重要的是它们是否能 为你的目标服务。 巴纳赫-塔斯基悖论的命运 正由这种选择决定。

This is the freedom mathematics gives us. Not only is it a way to model our physical universe using the axioms we intuit from our daily experiences, but a way to venture into abstract mathematical universes and explore arcane geometries and laws unlike anything we can ever experience. If we ever meet aliens, axioms which seem absurd and incomprehensible to us might be everyday common sense to them. To investigate, we might start by handing them an infinitely sharp knife and a perfect ball, and see what they do.

这便是数学赋予我们的自由。 这不仅是使用我们在 日常经历中由直觉得到的公理 模拟我们的物理宇宙的一种方式， 我们还能得以漫游抽象数学宇宙， 探索神秘的几何和定律， 这样的体验是绝无仅有的。 若能遇见外星人， 这些貌似荒谬和难以理解的公理 对他们而言可能是日常常识。 若想知道，我们可以交 给他们一把无限锋利的刀 和一个完美的球， 然后拭目以待。

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