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How can you have a proof without proving anything? Mathematicians found a way and, in the process, came to blows over it – but 100 years on, this trick is a common part of modern maths, says columnist Jacob Aron

Chain apologises after staff at Hartlepool pub demand proof customer’s companion is genuine guide dogA blind woman felt compelled to leave a Wetherspoons pub after staff demanded proof that her guide dog, Rosie, was a genuine guide dog.Joanne Hewitson, 31, said she was “livid” an

A classic side that effortlessly accompanies almost any summer meal – customise to your taste with homemade mayo and/or fistfuls of fresh herbsThis summer classic is the happy proof that not all things labelled as salad have to feel like penance for helping yourself to all the fr

Clever, emotional – and well worth the hypeI couldn’t get a table at Skof for ages: it was too full, too booked up and far too busy. It seemed there’d be no lightly set miso custard with hen of the woods mushrooms and dashi for me. Jersey royals cooked in chicken fat with pickled

It will come into its own in summer. Until then, try layering it with spring-ready jackets and chill-proof knitwear Continue reading...

Experts say there will still be opportunities ahead in everything from teaching to hotels and the lawEntering the world of work often brings some uncertainty, but now there is another question: how can I AI-proof my career?We asked people from across various industries what they

Midjourney has shown more of its futuristic medical scanner. It still hasn't shown much proof it works. The AI startup, best known for generating images, released a behind-the-scenes video of its dunk-tank ultrasound scanner, which it plans to deploy in spas and hopes will transf

Lyzr, a startup that builds AI agents for enterprises, used its own AI agent to raise a $100 million round — proof, evidently, that the product actually works.

A spray made from a biodegradable polymer capable of capturing and redirecting water to crop seeds could be the key to drought-proofing Western Australian farms.

CSIRO hydrologist Dr. Jai Vaze has lost count of how many times over the past four years he has been asked if he can flood-proof the Northern Rivers region of New South Wales.

Over 300 million years ago, a minnow-sized fish died and fell to the bottom of a prehistoric swamp near the village of Trawden, Lancashire, in northwest England. The remains of this tiny fish—known as Trawdenia planti—became fossilized, embedding proof of its existence in a layer

A mathematical problem that had remained unsolved for more than 10 years in the physics of complex systems has finally been resolved through an unusual collaboration: one involving two theoretical physicists and an artificial intelligence system. In a study published in the Journ