It would be nice to know where new ideas come from, the kind of thoughts that no one had had before and which, suddenly, solve a problem that has been dragging you down the road of bitterness for weeks or months. I say it would be good because, if we knew where these brilliant ideas came from, maybe we would find a way to stimulate them, and the truth is, we don’t have enough of them. The question is undoubtedly interesting, but also very difficult to explore in depth. How do you do it? You can’t put a thousand people in a bathroom until one of them comes up with Archimedes’ Principle or drops an apple on their head. Brilliant ideas are very rare and very strange things are a decidedly difficult subject to study.
Mathematicians and scientists – at least some of them – have been attentive to the topic and have left us some subjective narratives. The French mathematician Henri Poincaré recounted one of his eureka moments: “The moment I stepped on the step of the bus, the idea came to me suddenly, without anything in my previous thoughts having paved the way for it.” The idea Poincaré was referring to solved the main puzzle about things called “Fuchsian functions” that he and four others would have understood, but let’s forget about it now. The point is that Poincaré had been studying these functions for months, and that the solution came to him while he was thinking about something else.
Cognitive scientist Tyler Marghetis of the University of California, Merced, looked at the question from a unique perspective. He himself must have had an aha inspiration when planning the experiment. He recorded six mathematicians working at the blackboard as they tried to prove a theorem for 40 minutes. He asked them to think aloud and observed their movements, such as when they looked at other parts of the board, pointed to an equation, or erased a drawing. Their results show that, during the two minutes preceding the eureka moment, the mathematician’s movements become unpredictable, as if a certain chaos precedes the final enlightenment. Marghetis is fascinated by the fact that an abstract activity like mathematics can be observed in body movements. Then they tell me that I move my hands too much when I talk.
Data scientist Justo Hidalgo, director of artificial intelligence (AI) of the Spanish Association of Digital Economy, spoke the other day on these pages about what we could call eureka moments of machines. Despite what we humans tend to believe, AI systems have shown signs of emerging abilities, a class of behaviors that are not intended in their programming or made explicit in data. They are emerging concepts, like the one that came to Poincaré’s mind when he put his foot on the step of the bus.
Hidalgo says that when the AI system goes beyond 100 billion parameters, it starts to show signs of emergent properties. He gives the example of languages: if the model learns English and Spanish, it finds common patterns that help it translate other languages it doesn’t know. This happens even to people who know multiple languages, and is an example of an emergent property, because, although it derives from universal features that are in the language, no one has explicitly taught them either to the person or to the machine. In both cases it can be considered an unconscious deduction. I’m going to read your book Contingency models: How complexity drives artificial intelligence (“Emergence Patterns: How Complexity Makes Artificial Intelligence Work”, Amazon, 2025, for now only in English). Or maybe I’ll ask ChatGPT to explain it to me.
