为什么实数有这么好的性质？

实数中蕴藏的数学结构

The set of real numbers has several standard structures:

• An order: each number is either less than or greater than any other number.
• Algebraic structure: there are operations of addition and multiplication, the first of which makes it into a group and the pair of which together make it into a field.
• A measure: intervals of the real line have a specific length, which can be extended to the Lebesgue measure on many of its subsets.
• A metric: there is a notion of distance between points.
• A geometry: it is equipped with a metric and is flat.
• A topology: there is a notion of open sets.

There are interfaces among these:

• Its order and, independently, its metric structure induce its topology.
• Its order and algebraic structure make it into an ordered field.
• Its algebraic structure and topology make it into a Lie group, a type of topological group.

实数集具有几个标准结构：

1. 顺序：每个数字要么小于任何其他数字，要么大于任何其他数字。
2. 代数结构：有加法和乘法运算，其中加法使其成为一个群，而两者一起使其成为一个域。
3. 测度：实数线段具有特定的长度，这可以扩展到其许多子集上的勒贝格测度。
4. 度量：有点之间的距离概念。
5. 几何：它配备了度量并且是平坦的。
6. 拓扑学：有开集的概念。

它们之间存在接口：

• 它的顺序和独立的度量结构导致了它的拓扑结构。
• 它的顺序和代数结构使其成为一个有序域。
• 它的代数结构和拓扑学使其成为李群，这是一种拓扑群。

之后，随着代数拓扑学、微分拓扑学、代数几何学、李群和代数群理论、多复变量函数论、泛函分析等领域趋于平稳，而难以结构化的数学分支，如分析数学、概率论与统计学、离散数学(组合数学)、应用数学、计算数学等开始蓬勃发展，布尔巴基学派逐渐衰微。(p.s.感觉后面这些数学都大都更适于计算机处理，计算机的普及促进了新数学领域的发展。)

不必全部了解这些结构也能做数学（应用数学），因为学习数学不是学习reality,而是模型，脑子里面的模型去解释真实！
如果实数已经拥有这么些好的性质，能够很好地描述现实，that's enough!

Reference

1. Wikipedia Mathematical Structure