Mathematician Doron Zeilberger argues lemmas, not theorems, are the true engines of mathematical progress, citing Szemeredi’s Regularity Lemma as the prime example.
Key Takeaways
Szemeredi’s Regularity Lemma enabled at least two Fields medals and powered the Green-Tao theorem on primes in arithmetic progressions via hypergraph extension.
Theorems are described as dead ends; a good lemma is reusable across seemingly unrelated problems, outliving philosophical and technological revolutions.
Aigner and Ziegler’s “Proofs from THE BOOK” defines a true lemma by three criteria: wide applicability, obvious-once-seen clarity, and aesthetic beauty in proof.
Paul Taylor’s quote frames the dynamic sharply: “Lemmas do the work in mathematics: Theorems, like management, just take the credit.”
Zeilberger ranks observations above even lemmas as mathematical primitives, flagging that as a separate argument.
