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foundations of mathematics

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.

foundations of mathematics

Will Computers Redefine the Roots of Math?

May 19, 2015

The Fields medalist Vladimir Voevodsky has died at 51. This 2015 article describes his computer-aided quest to eliminate human error and rewrite the century-old rules underlying all of mathematics.

Image of spiral staircase.
Mathematics

To Settle Infinity Dispute, a New Law of Logic

November 26, 2013

To determine the nature of infinity, mathematicians face a choice between two new logical axioms. What they decide could help shape the future of mathematical truth.

algorithms

In Computers We Trust?

February 22, 2013

As the role of computers in pure mathematics grows, researchers debate their reliability.