Will it be a golden age or a dark age for mathematics (and science)?

The ability of AI to generate complex mathematical proofs is outstripping humans' ability to understand them, potentially divorcing knowledge from human comprehension.

Francesco Maggi, in a guest post on Terence Tao's blog, posits that artificial intelligence, specifically large language models (LLMs), presents a dual potential for mathematics and science. On one hand, LLMs demonstrate remarkable ability in generating mathematical proofs, such as solutions to complex problems like the Navier-Stokes equations and the quasi-Riemann Hypothesis. This capability could accelerate discovery and lead to extraordinary scientific advances, creating a "golden age." However, Maggi also warns of a "dark age" scenario where scientific output increases but independent human scientific activity diminishes, with knowledge becoming concentrated and incomprehensible to humans. He notes that current LLMs struggle to translate complex, computer-verified proofs (e.g., in Lean) into human-understandable natural language, raising fundamental questions about the purpose of pursuing knowledge without human comprehension. Maggi emphasizes that mathematics is a human artifact deeply rooted in human cognition, and its continuation depends on human understanding, suggesting a need for more mathematicians.

Maggi raises concerns about scientific skepticism in the face of LLM capabilities, arguing that extraordinary performance should be a subject of study rather than a basis for speculative conclusions about intelligence or creativity. He stresses that the social consequences of AI will depend on human choices regarding its development and integration into existing institutions, rather than being predetermined by the technology's capabilities. The author points out that LLMs excel at recombining existing information and finding connections, but it remains an open scientific question whether this constitutes genuine synthetic reasoning or is purely sophisticated linguistic recombination. The transparency of commercial LLM developers is a barrier to understanding the relationship between model performance and the richness of underlying scientific literature, with potential implications for directing future research.

The concept of "mathematization"—the invention of new mathematical languages and frameworks to understand reality—is presented as a crucial human scientific function that LLMs may not fully replicate. While LLMs might mathematize mathematical structures, their ability to go beyond this and translate it into human understanding is uncertain. Maggi suggests that the accessibility of proofs through AI could free up human mathematicians to focus more on mathematization, a potentially transformative opportunity for natural and social sciences where progress has been hampered by the difficulty of establishing theoretical consequences. This renewed focus on mathematization could lead to vastly increased sophistication and depth of insights into complex systems.