Gödel's seminal work directly contradicted one of the great minds of mathematics and limited the field forever Kurt Gödel, the man who ruined mathematics, was one of the most important thinkers of the 20th century. He was born in 1906, smack-bang in the middle of the greatest crisis that maths has ever known. Just a few decades later, he would help resolve this turmoil, but in doing so doom mathematicians to a smaller world than the one that came before. Mathematics, as an intellectual framework, is incredibly powerful. The entire point is taking one set of logical ideas and using them to build another, making maths the closest thing we have to a cognitive perpetual-motion machine - there is always a new mathematical idea lurking across the horizon, and we just need to assemble the steps to get there.

Fifty years after her death in New York, Hannah Arendt has become the most popular philosopher of our time. For good reason: Her views are just as timely as ever. It must be so nice to play Hannah Arendt. No fewer than five actresses are on stage this evening at the Deutsches Theater Berlin to portray the philosopher. The piece is an adaptation of the graphic novel by American illustrator Ken Krimstein about the philosopher's life, called The Three Escapes of Hannah Arendt," combined with scenes from the famous interview that journalist Günter Gaus conducted with Arendt in 1964 for German public broadcaster ZDF. The article you are reading originally appeared in German in issue 49/2025 (November 28th, 2025) of DER SPIEGEL. They play Arendt and a few of her contemporaries, the philosopher Martin Heidegger, the writer Walter Benjamin, her husband Heinrich Blücher. There is a great deal of speech in the play, especially from Arendt herself. The places of her life are ticked off, her ...

Kant's life was famously dull, but he was less of a hermit than is often supposed.

Can Ukraine restore its pre-war borders? Why are Tomahawk missiles for Ukraine a'red line' for Russia? Is Russia testing NATO with aerial incursions in Europe? Lithuania, a small Baltic state bordering Belarus and Russia's Kaliningrad, is adapting to new tensions between NATO and Moscow. A member of the Lithuanian Riflemen's Union takes part in a military exercise in central Lithuania [Nils Adler/Al Jazeera] Two members of the Lithuanian Riflemen's Union take part in a military exercise in central Lithuania [Nils Adler/Al Jazeera] On a nearby building is an illuminated decorative Z, a symbol used to show support for the Russian military's full-scale invasion of Ukraine, which began in February 2022.

'Kissing Trump's a**': President mocks Canada's obsequious PM as he begs for tariff relief World's most invasive predator terrorizing East Coast is delicious and should be eaten to stop its spread, experts say Clash of the White House titans: Two of Trump's most powerful lieutenants go to WAR with each other - after vicious leak sent shockwaves AMANDA PLATELL: I never thought I'd feel sorry for Harry. There's one thing he'd do anything to defend... and now Meghan's trampled all over it White House insider who says WAR with Venezuela is inevitable... as Trump's lethal options are laid out Jimmy Kimmel's audience boom comes crashing down as he loses 71% of viewers in one week Lynn put her strange symptoms down to being a busy mum. AOC hit by shockingly crude sex insult by White House after she mocked'TINY' Stephen Miller Friends fear for new CBS News boss Bari Weiss, claiming her wife thinks she sold out... and her new job will'consume her life' Biden ordered CIA cover-up of his'corrupt' business ties to Ukraine, astonishing secret files show We've lost FOURTEEN stone on weight-loss jabs... and it's changed our lives in ways you'd NEVER expect. Jerry Jones slapped with fine by NFL for making rude gesture to fans... but Cowboys owner gives baffling excuse And a humiliating lifeline: Backroom secrets of Taylor Swift and Blake Lively... after hit new song Inside the rise of'kidfluencers' and the hidden toll of turning childhood into million-dollar content A US Air Force jet designed to collect intelligence on enemy radar systems was spotted making circles over Russia, following rising tensions with Moscow . Flight tracking data showed the RC-135U'Combat Sent' taking off from England early Tuesday, flying over the Baltic states and looping around Kaliningrad, the Russian exclave between Poland and Lithuania, before returning to the UK.

How is Russia replenishing its military? What is a'coalition of the willing'? How China forgot promises and'debts' to Ukraine How are Europe, the US pulling apart on Ukraine? The Kremlin has announced it has "no alternative" but to continue waging war, as it pushed back on United States President Donald Trump's sudden sea change towards Ukraine that saw him brand Russia a "paper tiger". Kremlin spokesperson Dmitry Peskov hit back at Trump's claim that Ukraine could in effect win the war, declaring on Wednesday that Russia would be continuing its offensive on Ukraine "to ensure our interests and achieve the goals".

How is Russia replenishing its military? What is a'coalition of the willing'? How China forgot promises and'debts' to Ukraine How are Europe, the US pulling apart on Ukraine? Russia and Belarus have begun large-scale military exercises, raising alarm across NATO's eastern flank just days after Warsaw accused Moscow of sending attack drones across Polish airspace, a major escalation that sent shivers through Europe. The Zapad 2025 manoeuvres, which run from Friday until Tuesday, are taking place as Russian forces continue their slow advance in Ukraine and intensify air attacks on Ukrainian cities.

Language models are known to encode a great amount of factual knowledge through pretraining. However, such knowledge might be insufficient to cater to user requests, requiring the model to integrate external knowledge sources and adhere to user-provided specifications. When answering questions about ongoing events, the model should use recent news articles to update its response; when asked to provide recommendations, the model should prioritize user specifications over retrieved product reviews; when some facts are edited in the model, the updated facts should override all prior knowledge learned by the model even if they are conflicting. In all of the cases above, the model faces a decision between its own parametric knowledge, (retrieved) contextual knowledge, and user instruction knowledge. In this paper, we (1) unify such settings into the problem of knowledge preference and define a three-level preference hierarchy over these knowledge sources; (2) compile a collection of existing datasets IfQA, MQuAKE, and MRQA covering a combination of settings (with/without user specifications, with/without context documents) to systematically evaluate how well models obey the intended knowledge preference; and (3) propose a dataset synthesis method that composes diverse question-answer pairs with user assumptions and related context to directly fine-tune LMs for instilling the hierarchy of knowledge. We demonstrate that a 7B model, fine-tuned on only a few thousand examples automatically generated by our proposed method, effectively achieves superior performance (more than 18% improvement across all evaluation benchmarks) in adhering to the desired knowledge preference hierarchy.

Recently, directly using large language models (LLMs) has been shown to be the most reliable method to evaluate QA models. However, it suffers from limited interpretability, high cost, and environmental harm. To address these, we propose to use soft exact match (EM) with entitydriven answer set expansion. Our approach expands the gold answer set to include diverse surface forms, based on the observation that the surface forms often follow particular patterns depending on the entity type. The experimental results show that our method outperforms traditional evaluation methods by a large margin. Moreover, the reliability of our evaluation method is comparable to that of LLM-based ones, while offering the benefits of high interpretability and reduced environmental harm.

Global Navigation Satellite Systems (GNSS)-based positioning plays a crucial role in various applications, including navigation, transportation, logistics, mapping, and emergency services. Traditional GNSS positioning methods are model-based and they utilize satellite geometry and the known properties of satellite signals. However, model-based methods have limitations in challenging environments and often lack adaptability to uncertain noise models. This paper highlights recent advances in Machine Learning (ML) and its potential to address these limitations. It covers a broad range of ML methods, including supervised learning, unsupervised learning, deep learning, and hybrid approaches. The survey provides insights into positioning applications related to GNSS such as signal analysis, anomaly detection, multi-sensor integration, prediction, and accuracy enhancement using ML. It discusses the strengths, limitations, and challenges of current ML-based approaches for GNSS positioning, providing a comprehensive overview of the field.