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foundations of mathematics

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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.

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.

Black and white historical photo of eight men in front of a building, including Henri Cartan, André Weil and Szolem Mandelbrojt.
Abstractions blog

Inside the Secret Math Society Known Simply as Nicolas Bourbaki

By Kevin Hartnett
November 9, 2020
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For almost a century, the anonymous members of Nicolas Bourbaki have written books intended as pure expressions of mathematical thought.

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

By Kevin Hartnett
October 1, 2020
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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

By Rachel Crowell
September 2, 2020
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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

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.

theoretical physics

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

By Natalie Wolchover
April 7, 2020
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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

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.

Illustration of Fermat's Last Theorem
Quantized Columns

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

By Michael Harris
June 3, 2019
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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.


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