Descriptive set theorists study the niche mathematics of infinity. Now, they've shown that their problems can be rewritten in the concrete language of algorithms. All of modern mathematics is built on the foundation of set theory, the study of how to organize abstract collections of objects. But in general, research mathematicians don't need to think about it when they're solving their problems. They can take it for granted that sets behave the way they'd expect, and carry on with their work. Descriptive set theorists are an exception. This small community of mathematicians never stopped studying the fundamental nature of sets--particularly the strange infinite ones that other mathematicians ignore. Their field just got a lot less lonely. In 2023, a mathematician named Anton Bernshteyn published a deep and surprising connection between the remote mathematical frontier of descriptive set theory and modern computer science.

If language is what makes us human, what does it mean now that large language models have gained "metalinguistic" abilities? Among the myriad abilities that humans possess, which ones are uniquely human? Language has been a top candidate at least since Aristotle, who wrote that humanity was "the animal that has language." Even as large language models such as ChatGPT superficially replicate ordinary speech, researchers want to know if there are specific aspects of human language that simply have no parallels in the communication systems of other animals or artificially intelligent devices. In particular, researchers have been exploring the extent to which language models can reason about language itself.

A canonical problem in computer science is to find the shortest route to every point in a network. A new approach beats the classic algorithm taught in textbooks. If you want to solve a tricky problem, it often helps to get organized. You might, for example, break the problem into pieces and tackle the easiest pieces first. But this kind of sorting has a cost.

A fundamental technique lets researchers use a big, expensive model to train another model for less. The Chinese AI company DeepSeek released a chatbot earlier this year called R1, which drew a huge amount of attention. Most of it focused on the fact that a relatively small and unknown company said it had built a chatbot that rivaled the performance of those from the world's most famous AI companies, but using a fraction of the computer power and cost. As a result, the stocks of many Western tech companies plummeted; Nvidia, which sells the chips that run leading AI models, lost more stock value in a single day than any company in history. Some of that attention involved an element of accusation.

The Busy Beaver Challenge, a notoriously difficult question in theoretical computer science, is now producing answers so large they're impossible to write out using standard mathematical notation. Imagine that someone gives you a list of five numbers: 1, 6, 21, 107 and--wait for it--47,176,870. Can you guess what comes next? These are the first five busy beaver numbers. They form a sequence that's intimately tied to one of the most notoriously difficult questions in theoretical computer science. Determining the values of busy beaver numbers is a daunting challenge that has attracted a cult following among both professional and amateur mathematicians for over 60 years. Researchers identified the first four busy beaver numbers in the 1960s and 1970s.

The original version of this story appeared in Quanta Magazine. Large language models work well because they're so large. The latest models from OpenAI, Meta, and DeepSeek use hundreds of billions of "parameters"--the adjustable knobs that determine connections among data and get tweaked during the training process. With more parameters, the models are better able to identify patterns and connections, which in turn makes them more powerful and accurate. But this power comes at a cost.

The Pulitzer Prize-winning book Gödel, Escher, Bach inspired legions of computer scientists in 1979, but few were as inspired as Melanie Mitchell. After reading the 777-page tome, Mitchell, a high school math teacher in New York, decided she "needed to be" in artificial intelligence. She soon tracked down the book's author, AI researcher Douglas Hofstadter, and talked him into giving her an internship. She had only taken a handful of computer science courses at the time, but he seemed impressed with her chutzpah and unconcerned about her academic credentials. Mitchell prepared a "last-minute" graduate school application and joined Hofstadter's new lab at the University of Michigan in Ann Arbor.

The first episode of Sesame Street in 1969 included a segment called "One of These Things Is Not Like the Other." Viewers were asked to consider a poster that displayed three 2s and one W, and to decide--while singing along to the game's eponymous jingle--which symbol didn't belong. Dozens of episodes of Sesame Street repeated the game, comparing everything from abstract patterns to plates of vegetables. Kids never had to relearn the rules. Understanding the distinction between "same" and "different" was enough.

At one time, the AI that beat humans at chess calculated strategies by studying the outcomes of human moves. Joshua Sokol, "Why Artificial Intelligence Like AlphaZero Has Trouble With the Real World" at Quanta Magazine (February 21, 2018) "I feel like that's been neglected by the majority of the AI community." Joshua Sokol, "Why Artificial Intelligence Like AlphaZero Has Trouble With the Real World" at Quanta Magazine (February 21, 2018) Or, as George Gilder says in Gaming AI, in games like chess and Go, the map is the territory. And, as Jeffrey Funk and Gary Smith remind us, failed prophecies of an AI takeover come at a cost: We don't improve what we could improve in human-based services like health care if we are waiting for the phantom AI takeover. At one time, the AI that beat humans at chess calculated strategies by studying the outcomes of human moves.

Reprinted with permission from Quanta Magazine's Abstractions blog. Our sense of time may be the scaffolding for all of our experience and behavior, but it is an unsteady and subjective one, expanding and contracting like an accordion. Emotions, music, events in our surroundings and shifts in our attention all have the power to speed time up for us or slow it down. When presented with images on a screen, we perceive angry faces as lasting longer than neutral ones, spiders as lasting longer than butterflies, and the color red as lasting longer than blue. The watched pot never boils, and time flies when we're having fun.