数学是否存在致命缺陷？ (5-2)

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。

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