Mathematics – Quanta Magazine

Zermelo-Fraenkel set theory is so widely accepted that modern mathematicians hardly think about it. But believing in its core principles didn’t come easily. The post Why Math’s Final Axiom Proved So Controversial first appeared on Quanta Magazine

Ultrafinitism, a philosophy that rejects the infinite, has long been dismissed as mathematical heresy. But it is also producing new insights in math and beyond. The post What Can We Gain by Losing Infinity? first appeared on Quanta Magazine

With a newly discovered mathematical tool, researchers are hoping to gain unprecedented insight into the structure of complex knots. The post A Powerful New ‘QR Code’ Untangles Math’s Knottiest Knots first appeared on Quanta Magazine

Archive Latest Articles An Arctic Road Trip Brings Vital Underground Networks into View A vast meshwork of soil-bound fungi governs life aboveground. In Alaska, and at field sites around the world, researchers are racing to understand exactly how, with essential stores of carbon at stake. New Advances Bring the Era of Quantum Computers Closer Than Ever Two research groups say they have significan…

Biggest Breakthroughs in Biology 2025 Biggest Breakthroughs in Biology 2025 2025’s most surprising breakthroughs in biology included a finding that a father’s environmental exposures can impact the development of their offspring, research confirming that intelligence evolved independently in birds and mammals, and a new mathematical model reveals that evolution happens in explosive bursts. Explor…

Videos Biggest Breakthroughs in Biology 2025 2025’s most surprising breakthroughs in biology included a finding that a father’s environmental exposures can impact the development of their offspring, research confirming that intelligence evolved independently in birds and mammals, and a new mathematical model reveals that evolution happens in explosive bursts. Explore All Videos Biggest Breakthrou…

Introduction Some 30 years ago, the mathematician Peter Shor took a niche physics project — the dream of building a computer based on the counterintuitive rules of quantum mechanics — and shook the world. Shor worked out a way for quantum computers to swiftly solve a couple of math problems that classical computers could complete only after many billions of years. Those two math problems happened…

Kristina Armitage/Quanta Magazine In ancient Greece, Euclid showed that if you agree on a small list of preliminary principles, or axioms, you can use deductive reasoning to reveal all sorts of new mathematical truths. But although these early proofs, as mathematicians call them, were derived using the laws of logic, they sometimes also contained hidden, unstated assumptions or relied on misleadi…

Introduction It’s natural to think of math as being fundamentally abstract. Whether it’s invented or discovered, its truths are so literally universal that even aliens would agree (so the thinking goes) that 2 and 2 make 4. The actual work of mathematics, though, typically involves something utterly earthbound: “making marks on paper or blackboards,” said David E. Dunning, a historian of mathemat…

Podcasts Featured Episode The quest to make mathematics rigorous has a long and spotty history — one mathematicians can learn from as they push to formalize everything in the computer program Lean. The quest to make mathematics rigorous has a long and spotty history — one mathematicians can learn from as they push to formalize everything in the computer program Lean.

Introduction No matter where you look, a bell curve is close by. Place a measuring cup in your backyard every time it rains and note the height of the water when it stops: Your data will conform to a bell curve. Record 100 people’s guesses at the number of jelly beans in a jar, and they’ll follow a bell curve. Measure enough women’s heights, men’s weights, SAT scores, marathon times — you’ll alwa…

Introduction The last time I covered the science of humanoid robots, the state of the art looked downright Orwellian — by which I mean, “four legs good, two legs bad.” It was 2015. Boston Dynamics’ first “Spot” quadruped had taken YouTube by storm, confidently trotting up stairs and recovering from vicious kicks. Also popular at the time: humanoids falling down. Constantly. I felt sorrier for tho…

Introduction Picture a bizarre training exercise: A group of runners starts jogging around a circular track, with each runner maintaining a unique, constant pace. Will every runner end up “lonely,” or relatively far from everyone else, at least once, no matter their speeds? Mathematicians conjecture that the answer is yes. The “lonely runner” problem might seem simple and inconsequential, but it …

Introduction “I’ve spent a long time exploring the crystalline beauty of traditional mathematics, but now I’m feeling an urge to study something slightly more earthy,” John Baez wrote on his blog in 2011. An influential mathematical physicist who splits his time between the University of California, Riverside and the University of Edinburgh, Baez had grown increasingly concerned about the state o…

AI Editorial Policy Updated: March 3, 2026 Artificial intelligence is a term with many meanings. These days, it often specifically refers to generative artificial intelligence (GenAI) systems, which generate novel text, images or other media in response to user input. What distinguishes GenAI systems from other computer programs is that they aren’t explicitly programmed to accomplish these tasks.…

Introduction The standard story of the origin of our solar system has gone like this: 4.6 billion years ago, a giant cloud of dust hung frozen in space. Then the explosion of a nearby star caused part of that dust cloud to collapse. Pulled by gravity toward a central point, the dust coalesced into a radiating ball of hydrogen and helium about 1.4 million kilometers in diameter — what would become…

In an 1874 paper, Georg Cantor proved that there are different sizes of infinity and changed math forever. A trove of newly unearthed letters shows that it was also an act of plagiarism. The post The Man Who Stole Infinity first appeared on Quanta Magazine

An exploration of how mathematicians are still renovating and rebuilding the core pillars of their field today. The post The Evolving Foundations of Math first appeared on Quanta Magazine

New Series from Quanta Magazine Explores the Infinite Evolution of Math Math is always reinventing itself. In the late 19th century, its foundations were still shaky, its definitions and assumptions vague and uncertain. Mathematicians spent decades fixing the problem, setting the course of the modern era. But they never stopped questioning their most basic assumptions. In “The Evolving Foundation…

Intuition breaks down once we’re dealing with the endless. To begin with: Some infinities are bigger than others. The post How Can Infinity Come in Many Sizes? first appeared on Quanta Magazine