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foundations of mathematics

Illustration of researchers building and writing a digital library, which helps a humanoid robot do advanced math
foundations of mathematics

Building the Mathematical Library of the Future

October 1, 2020

A small community of mathematicians is using a software program called Lean to build a new digital repository. They hope it represents the future of their field.

Q&A

Conducting the Mathematical Orchestra From the Middle

September 2, 2020

Emily Riehl is rewriting the foundations of higher category theory while also working to make mathematics more inclusive.

Gödel’s incompleteness theorems.
Abstractions blog

How Gödel’s Proof Works

July 14, 2020

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.

theoretical physics

Does Time Really Flow? New Clues Come From a Century-Old Approach to Math.

April 7, 2020

The laws of physics imply that the passage of time is an illusion. To avoid this conclusion, we might have to rethink the reality of infinitely precise numbers.

foundations of mathematics

With Category Theory, Mathematics Escapes From Equality

October 10, 2019

Two monumental works have led many mathematicians to avoid the equal sign. The process has not always gone smoothly.

Illustration of Fermat's Last Theorem
Quantized Columns

Why the Proof of Fermat’s Last Theorem Doesn’t Need to Be Enhanced

June 3, 2019

Decades after the landmark proof of Fermat’s Last Theorem, ideas abound for how to make it even more reliable. But such efforts reflect a deep misunderstanding of what makes the proof so important.

Measuring tapes
infinity

Mathematicians Measure Infinities and Find They’re Equal

September 12, 2017

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.

geometry

A Fight to Fix Geometry’s Foundations

February 9, 2017

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.

foundations of mathematics

Mathematicians Bridge Finite-Infinite Divide

May 24, 2016

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