The quest to make mathematics rigorous has a long and spotty history — one mathematicians can learn from as they push to formalize everything in the computer program Lean. <p>The post <a href="https://www.quantamagazine.org/in-math-rigor-is-vital-but-are-digitized-proofs-taking-it-too-far-20260325/" target="_blank">In Math, Rigor Is Vital. But Are Digitized Proofs Taking It Too Far?</a> first appeared on <a href="https://www.quantamagazine.org" target="_blank">Quanta Magazine</a></p>
In ancient Greece, Euclid showed that if you agree on a small list of preliminary principles, or axioms, you can use deductive reasoning to reveal all sorts of new mathematical truths. But although these early proofs, as mathematicians call them, were derived using the laws of logic, they sometimes also contained hidden, unstated assumptions or relied on misleading intuitions. In these cases…
Formal proofs are rigorous mathematical arguments that demonstrate the truth of a statement based on axioms and previously established theorems. They are essential for ensuring the validity of mathematical results and are increasingly being implemented through computer-assisted proofs to enhance accuracy and reliability.
Computer-assisted proofs involve the use of computer software to verify the correctness of mathematical proofs. This approach allows mathematicians to handle complex proofs that would be impractical to verify manually, thus expanding the boundaries of what can be rigorously proven.