The Evolving Foundations of Math

Math is always reinventing itself.

数学总是在自我重塑。

In the late 19th century, its foundations were still shaky, its definitions and assumptions vague and uncertain.

到了 19 世纪末，它的基础仍然不稳固，它的定义和假设模糊且不确定。

Mathematicians spent decades fixing the problem, setting the course of the modern era.

数学家花了几十年的时间来解决这个问题，设定了现代时代的进程。

But they never stopped questioning their most basic assumptions.

但他们从未停止质疑自己最基本的假设。

This special issue explores how mathematicians are still renovating and rebuilding the core pillars of their field today.

本期特刊探讨了数学家今天如何仍在翻新和重建其领域的核心支柱。

Intuition breaks down once we’re dealing with the endless.

一旦我们面对无穷无尽的事物，直觉就会崩溃。

To begin with: Some infinities are bigger than others.

首先：一些无穷大比其他无穷大。

For 50 years, mathematicians have believed that the total number of real numbers is unknowable.

50 年来，数学家们一直相信实数的总数是不可知的。

A new proof suggests otherwise.

一项新的证据表明事实并非如此。

David Dunning explores the social effects of notation.

大卫·邓宁 (David Dunning) 探讨了符号的社会影响。

Explore the “deplorable evil” that helped transform math.

探索帮助改变数学的“可悲的邪恶”。

Zermelo-Fraenkel set theory is so widely accepted that modern mathematicians hardly think about it.

策梅洛-弗兰克尔集合论被广泛接受，以至于现代数学家几乎不去思考它。

But believing in its core principles didn’t come easily.

但相信其核心原则并不容易。

His two theorems destroyed the search for a mathematical theory of everything.

他的两个定理摧毁了对万物数学理论的探索。

Nearly a century later, we’re still coming to grips with the consequences.

近一个世纪后，我们仍在应对其后果。

One of the strangest results in mathematics explains how it’s possible to turn one sphere into two identical copies, simply by rearranging its pieces.

数学中最奇怪的结果之一解释了如何通过简单地重新排列球体的各个部分，将一个球体变成两个相同的副本。

Grothendieck is revered in the world of math; outside of it, he’s known for his unusual life, if he’s known at all.

格洛腾迪克在数学界备受尊崇。除此之外，他因其不寻常的生活而闻名（如果他有名的话）。

But what were his actual mathematical contributions?

但他实际的数学贡献是什么？

When two mathematicians raised pointed questions about a classic proof that no one really understood, they ignited a years-long debate about how much could be trusted in a new kind of geometry.

当两位数学家对一个没有人真正理解的经典证明提出尖锐的问题时，他们引发了一场长达数年的争论，争论的焦点是一种新的几何学在多大程度上可以被信任。