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Elegant Six-Page Proof Reveals the Emergence of Random Structure
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.
Banach-Tarski and the Paradox of Infinite Cloning
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.
Mathematicians Solve Decades-Old Classification Problem
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.
How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer.
For 50 years, mathematicians have believed that the total number of real numbers is unknowable. A new proof suggests otherwise.
How Gödel’s Proof Works
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.
The Map of Mathematics
Explore our surprisingly simple, absurdly ambitious and necessarily incomplete guide to the boundless mathematical universe.
With Category Theory, Mathematics Escapes From Equality
Two monumental works have led many mathematicians to avoid the equal sign. The process has not always gone smoothly.
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.
Mathematicians Bridge Finite-Infinite Divide
A surprising new proof is helping to connect the mathematics of infinity to the physical world.