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foundations of mathematics

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Mathematicians Measure Infinities and Find They’re Equal

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.


A Fight to Fix Geometry’s Foundations

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

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?

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.

To Settle Infinity Dispute, a New Law of Logic

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.


In Computers We Trust?

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