A Good Lemma Is Worth a Thousand Theorems
Doron Zeilberger's opinion piece argues that in mathematics, lemmas, often seen as minor stepping stones, are far more significant and impactful than theorems. He posits that good lemmas, despite their perceived triviality, provide foundational utility across diverse problems, outliving and enabling even the deepest theorems. This perspective challenges conventional academic hierarchy, offering a deep dive into the true workhorses of mathematical discovery that resonates with HN's appreciation for insightful, unconventional takes on technical subjects.
The Lowdown
Doron Zeilberger's 82nd Opinion, "A Good Lemma Is Worth a Thousand Theorems," makes a compelling case for the often-underestimated importance of mathematical lemmas over theorems. While theorems often represent the grand conclusions, Zeilberger argues that lemmas are the true engines of progress, providing versatile tools that drive multiple breakthroughs.
- Theorems are described as "deadends," whereas a good lemma, even if easy to prove, holds far greater value due to its broad applicability.
- He cites examples like Schur's Lemma, Lovasz's Local Lemma, and particularly Szemeredi's Regularity Lemma, which has been instrumental in leading to two Fields medals and breakthroughs like the Green-Tao theorem.
- Zeilberger highlights that Endre Szemeredi, famous for his theorem, is arguably more significant for his lemma.
- Supporting quotes from Paul Taylor emphasize that "Lemmas do the work in mathematics: Theorems, like management, just take the credit," and that good lemmas survive philosophical or technological revolutions.
- "Proofs from THE BOOK" further defines a true Lemma by its wide applicability, its obviousness once stated, and the beauty of its proof.
Ultimately, the piece champions the foundational, versatile, and often overlooked role of lemmas as the enduring building blocks that enable profound mathematical advancements, suggesting they are destined to "inherit mathematics."