All posts tagged: Mathematicians

Mathematicians figured out the perfect espresso

Mathematicians figured out the perfect espresso

Published by Jenni Sidey

Get the Popular Science daily newsletter💡 Breakthroughs, discoveries, and DIY tips sent six days a week. People love a good cup of coffee, but how do you get a perfect brew? Barring philosophical deep dives into the nature of perfection, an international team of mathematicians and environmental scientists believe that it’s entirely possible to calculate the ideal espresso. Not only that, but they now have the formulas to back it up. The math detailed in their study published in the journal Royal Society Open Science is dense. But the short answer is that’s all about puck size. Picture the typical espresso machine at your favorite cafe. The small dish into which your friendly barista tamps coffee grounds is called the puck. After inserting it into the machine, hot water flows through the receptacle and molecularly absorbs the beans’ flavor, hue, and (most importantly) caffeine.  The quality of the final espresso depends on many aspects, including how the grounds are packed, how long water passes through the coffee, and the size of the grounds themselves. It’s …

This startup wants to change how mathematicians do math

This startup wants to change how mathematicians do math

Published by skeptic

Geordie Williamson, a mathematician at the University of Sydney, who worked on PatternBoost with Charton, has not yet tried Axplorer. But he is curious to see what mathematicians do with it. (Williamson still occasionally collaborates with Charton on academic projects but says he is not otherwise connected to Axiom Math.) Williamson says Axiom Math has made several improvements to PatternBoost that (in theory) make Axplorer applicable to a wider range of mathematical problems. “It remains to be seen how significant these improvements are,” he says. “We are in a strange time at the moment, where lots of companies have tools that they’d like us to use,” Williamson adds. “I would say mathematicians are somewhat overwhelmed by the possibilities. It is unclear to me what impact having another such tool will be.” Hong admits that there are a lot of AI tools being pitched at mathematicians right now. Some also require mathematicians to train their own neural networks. That’s a turnoff, says Hong, who is a mathematician herself. Instead, Axplorer will walk you through what you …

The success of machine mathematicians shows us how to be OK with AI

The success of machine mathematicians shows us how to be OK with AI

Published by skeptic

Have you ever received an email and had a sneaking suspicion it was written by AI, rather than lovingly handcrafted? Mathematicians have been wrestling with similar feelings for half a century, and have some lessons for the rest of us. It all began in 1976, when Kenneth Appel and Wolfgang Haken announced a proof of the four colour theorem, which states it takes a maximum of four shades to colour any map so that no two adjacent regions match. The theorem’s simplicity meant mathematicians were expecting an elegant proof revealing a greater mathematical truth. Instead, they got 60,000 lines of impenetrable computer code. Appel and Haken had solved the problem by programming a machine to systematically go through nearly 2000 kinds of map, representing every possible configuration. At the time, it felt unsatisfactory. But over the decades, mathematicians came to terms with using code in this way and resolved many of the philosophical objections. This meant when the current AI wave arrived, mathematics was ready. As we report here, AI is improving at such a …

Amateur mathematicians solve long-standing maths problems with AI

Amateur mathematicians solve long-standing maths problems with AI

Published by skeptic

AI tools are helping to decipher long-standing maths problems andresr/Getty Images Amateur mathematicians are using artificial intelligence chatbots to solve long-standing problems, in a move that has taken professionals by surprise. While the problems in question aren’t the most advanced in the mathematical canon, the success of AI models in tackling them shows that their mathematical performance has passed a significant threshold, say researchers, and could fundamentally change the way we do mathematics. The questions being solved by AI originate from Hungarian mathematician Paul Erdős, who was famous for his ability to pose useful but difficult questions during a career that spanned over six decades. “The questions tended to be very simple, but very hard,” says Thomas Bloom at the University of Manchester, UK. By his death in 1996, there were more than 1000 of these unsolved Erdős problems, spanning a wide range of mathematical disciplines, from combinatorics (the study of combinations) to number theory. Today, they are seen as signposts for progress in these fields, says Bloom, who runs a website that catalogues the …

Mathematicians unified key laws of physics in 2025

Mathematicians unified key laws of physics in 2025

Published by skeptic

The equations that govern fluids can be tricky to handle Vladimir Veljanovski / Alamy In 1900, mathematician David Hilbert presented his colleagues with a list of problems he believed both captured the present state of mathematics and the shape of its future. This year, 125 years later, Zaher Hani at the University of Michigan and his colleagues solved one of Hilbert’s problems – and unified several laws of physics in the process. Hilbert was a proponent of deriving all laws of physics from mathematical axioms – statements that mathematicians take to be basic truths. The sixth problem on his list was to derive laws of physics that dictate the behaviour of fluids from such axioms. Until 2025, physicists actually had three distinct ways of describing fluids, depending on their scale. Different rules governed the microscopic scale of single particles, the mesoscopic world populated by collections of particles and the macroscopic realm filled with fully fledged fluids like water flowing in a sink. Researchers had made strides in finding links between them, but the three were …

Behold the Manifold, the Concept that Changed How Mathematicians View Space

Behold the Manifold, the Concept that Changed How Mathematicians View Space

Published by skeptic

The original version of this story appeared in Quanta Magazine. Standing in the middle of a field, we can easily forget that we live on a round planet. We’re so small in comparison to the Earth that from our point of view, it looks flat. The world is full of such shapes—ones that look flat to an ant living on them, even though they might have a more complicated global structure. Mathematicians call these shapes manifolds. Introduced by Bernhard Riemann in the mid-19th century, manifolds transformed how mathematicians think about space. It was no longer just a physical setting for other mathematical objects, but rather an abstract, well-defined object worth studying in its own right. This new perspective allowed mathematicians to rigorously explore higher-dimensional spaces—leading to the birth of modern topology, a field dedicated to the study of mathematical spaces like manifolds. Manifolds have also come to occupy a central role in fields such as geometry, dynamical systems, data analysis, and physics. Today, they give mathematicians a common vocabulary for solving all sorts of problems. …

Mathematicians spent 2025 exploring the edge of mathematics

Mathematicians spent 2025 exploring the edge of mathematics

Published by skeptic

When numbers get large, things get weird Jezper / Alamy In 2025, the edges of mathematics came a little more sharply into view when members of the online Busy Beaver Challenge community closed in on a huge number that threatens to defy the logical underpinnings of the subject. This number is the next in the “Busy Beaver” sequence, a series of ever-larger numbers that emerges from a seemingly simple question – how do we know if a computer program will run forever? To find out, researchers turn to the work of mathematician Alan Turing, who showed that any computer algorithm can be mimicked by imagining a simplified device called a Turing machine. More complex algorithms correspond to Turing machines with larger sets of instructions or, in mathematical parlance, more states. Each Busy Beaver number BB(n) captures the longest possible run-time for a Turing machine with n states. For example BB(1) is 1 and BB(2) is 6, so making the algorithm twice as complex increases its runtime sixfold. But the rate of this increase turns out …