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set theory

Three circles that have dots connected by lines, as in a graph.
graph theory

Elegant Six-Page Proof Reveals the Emergence of Random Structure

By Jordana Cepelewicz
April 25, 2022
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Two young mathematicians have astonished their colleagues with a full proof of the Kahn-Kalai conjecture — a sweeping statement about how structure emerges in random sets and graphs.

An orange ball decomposing into points and rematerializing as two balls, each the same size as the first.
explainers

Banach-Tarski and the Paradox of Infinite Cloning

By Max G. Levy
August 26, 2021
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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.

Illustration of people along a path classifying colorful mathematical trees
set theory

Mathematicians Solve Decades-Old Classification Problem

By Steve Nadis
August 5, 2021
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A pair of researchers has shown that trying to classify groups of numbers called “torsion-free abelian groups” is as hard as it can possibly be.

foundations of mathematics

How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer.

By Natalie Wolchover
July 15, 2021
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For 50 years, mathematicians have believed that the total number of real numbers is unknowable. A new proof suggests otherwise.

Gödel’s incompleteness theorems.
Abstractions blog

How Gödel’s Proof Works

By Natalie Wolchover
July 14, 2020
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His incompleteness theorems destroyed the search for a mathematical theory of everything. Nearly a century later, we’re still coming to grips with the consequences.

Multimedia

The Map of Mathematics

By Kevin Hartnett
February 13, 2020
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Explore our surprisingly simple, absurdly ambitious and necessarily incomplete guide to the boundless mathematical universe.

foundations of mathematics

With Category Theory, Mathematics Escapes From Equality

By Kevin Hartnett
October 10, 2019
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Two monumental works have led many mathematicians to avoid the equal sign. The process has not always gone smoothly.

Measuring tapes
infinity

Mathematicians Measure Infinities and Find They’re Equal

By Kevin Hartnett
September 12, 2017
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Two mathematicians have proved that two different infinities are equal in size, settling a long-standing question. Their proof rests on a surprising link between the sizes of infinities and the complexity of mathematical theories.

foundations of mathematics

Mathematicians Bridge Finite-Infinite Divide

By Natalie Wolchover
May 24, 2016
Read Later

A surprising new proof is helping to connect the mathematics of infinity to the physical world.


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