Computers Are Rewriting Mathematics

For centuries, mathematicians have grappled with a fundamental tension: the need for creative leaps to discover new ideas versus the demand for logical rigor to ensure those ideas are correct. This push and pull has shaped the history of mathematics, from Euclid's early axioms to the 19th-century formalization of calculus. Today, this ancient debate is entering a new phase as mathematicians attempt to rewrite all of mathematics in a computer language called Lean, which can automatically verify proofs with machine precision.

Lean represents the most ambitious formalization project in history, aiming to put mathematics on what proponents call 'the most solid foundation one can imagine.' The program has already verified more than 260,000 theorems using a library of over 120,000 definitions, funded in part by billionaire financier Alex Gerko. This effort breaks proofs into reusable components that can be independently verified, much like engineers using pre-made parts rather than mining their own ore for every new spaceship.

Ology behind Lean involves mathematicians writing every line of a proof in a language the computer can understand, with the program checking each logical step. If even a single deduction doesn't follow from preceding ones, the proof fails. This process requires enormous time and effort—sometimes months or even years to formalize a single proof—but offers complete certainty. The approach builds on proof assistant technology developed since the 1960s, with Lean emerging as the current leading system.

From Lean's implementation show both practical successes and philosophical . In 2019, mathematician Peter Scholze wrote a complex proof by hand that was later formalized in Lean by a team led by Johan Commelin and Adam Topaz. The verification not only confirmed the proof's correctness but led to a cleaner version and refined some of Scholze's original ideas. Kevin Buzzard of Imperial College London, who is using Lean to formalize the famously complex proof of Fermat's Last Theorem, describes how the process 'forces you to think about mathematics in the right way' and 'forces you to become an artist.'

Despite these successes, Lean faces significant limitations and concerns. The program currently works better for some areas of mathematics than others—it's a good fit for number theory and algebraic geometry but less suited to graph theory and category theory. This technical bias could subtly influence which mathematical questions researchers pursue, potentially making the field more homogeneous. As Stephanie Dick of Simon Fraser University notes, technologies like Lean might 'shift the focus and the intuition and the understanding away from the mathematical problem domain toward the behavior of this system.'

The historical context reveals why these concerns matter. In the mid-20th century, the secret mathematical society Bourbaki prioritized abstraction and formal rigor above all else, influencing mathematics worldwide but sidelining more concrete fields like combinatorics for decades. Graph theory was once considered 'the slum of topology,' according to mathematician Béla Bollobás, thriving only where Bourbaki's influence was limited. Some mathematicians worry Lean could create similar distortions, with Aravind Asok of the University of Southern California warning against any mathematics 'where one mode dominates.'

Looking forward, the integration of artificial intelligence adds another layer of complexity. As mathematicians begin using AI to generate informal proofs, tools like Lean become crucial for verification. AI is already helping write Lean proofs more efficiently, suggesting a future where human creativity and machine verification work in tandem. However, this future requires careful navigation to avoid the pitfalls of past formalization efforts that, as historian Jesper Lützen observed, sometimes caused mathematics to gain 'rigor and generality' but lose 'elegance and simplicity and was estranged from intuition.'

The ultimate question isn't whether mathematics should be rigorous—that's been settled since the days of Cauchy and Weierstrass—but how much formalization serves versus constraint. As David Hilbert wrote in 1905, mathematics develops like a dwelling where comfortable rooms are explored first, with foundations fortified only when needed. Whether Lean represents necessary fortification or excessive constraint remains mathematics' current great debate, one that will shape how we discover truth for generations to come.