set theory – Quanta Magazine

‘A-Team’ of Math Proves a Critical Link Between Addition and Sets

Leila Sloman — Wed, 06 Dec 2023 15:40:14 +0000

In a randomly chosen set of numbers, addition can go wild. Add together every pair from such a set, and you’ll end up with a new list that’s likely to have a lot more numbers than what you started with. Start with 10 random numbers, and this new list (called the sumset) will have about 50 elements. Start with 100 and the sumset will probably have around 5,000; 1,000 random initial numbers will...

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Ninth Dedekind Number Found by Two Independent Groups

Rachel Crowell — Tue, 01 Aug 2023 14:36:19 +0000

Richard Dedekind was a 19th-century mathematical giant, responsible for reshaping the field right down to its foundations. He was the first to give a rigorous definition of infinity; he also came up with the definition of the real numbers that form the basis of much of modern mathematics. In 1897, he published an investigation into a certain numerical pattern. That work led him to define a...

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Is There Math Beyond the Equal Sign?

Steven Strogatz — Wed, 22 Mar 2023 06:53:03 +0000

Is there more to mathematics than the equal sign? A pair of shoes is different from a pair of gloves, yet we recognize a sameness in them because both meet our definition of a pair. Category theory is a branch of mathematics that examines how things can be essentially the same without being exactly equal. It’s what the mathematician Eugenia Cheng uses to make connections between abstract concepts...

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Google Researcher, Long Out of Math, Cracks Devilish Problem About Sets

Kevin Hartnett — Tue, 03 Jan 2023 14:50:47 +0000

In mid-October, Justin Gilmer flew from California to New York to attend a friend’s wedding. While on the East Coast he visited his former adviser, Michael Saks, a mathematician at Rutgers University, where Gilmer had received his doctorate seven years earlier. Saks and Gilmer caught up over lunch, but they didn’t talk about math. In fact, Gilmer had not thought seriously about math since...

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Elegant Six-Page Proof Reveals the Emergence of Random Structure

Jordana Cepelewicz — Mon, 25 Apr 2022 15:42:42 +0000

When the mathematicians Jeff Kahn and Gil Kalai first posed their “expectation threshold” conjecture in 2006, they didn’t believe it themselves. Their claim — a broad assertion about mathematical objects called random graphs — seemed too strong, too all-encompassing, too bold to possibly be true. It felt more like wishful thinking than a reflection of mathematical truth. Even so...

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