Doron Zeilberger's 82nd Opinion
Opinion 82:<br>A Good Lemma is Worth a Thousand Theorems
By Doron Zeilberger
Written: Aug. 14, 2007.
Theorems are nice, but they are usually deadends. A lemma may be "trivial", or easy, to prove,<br>once stated, but if it is good, its value far surpasses even the deepest theorems.<br>Wiki has a fairly long<br>list of lemmas,<br>including the powerful Schur's Lemma, Lovasz's Local Lemma, and many others.<br>But my absolute favorite, that lead to at least two Fields medals (so far),<br>is
Szemeredi's Regularity Lemma.<br>(see also the beautiful
Komlos-Simonovits exposition).<br>Its significance was lauded in two consecutive MAA Hedrick lectures, last year<br>by Tim Gowers, and this year<br>by Jennifer Chayes. For example, the recent Green-Tao breakthrough, about<br>primes in arithmetic progressions, used a hypergraph extension of Szemeredi's<br>great lemma.
Endre Szemeredi<br>is most famous for his<br>theorem, but his lemma is<br>yet more significant.
So blessed be the lemmas, for they shall inherit mathematics. Even more important<br>than lemmas are observations , but that is another story.
Added Aug. 22, 2007:<br>Vince Vatter kindly pointed my attention to this funny
economist's critique of lemmas, but of course for them "lemmas" means "complicated".
Added Aug. 23, 2007:<br>Todd Trimble kindly pointed out the following<br>quote from Paul Taylor ("Practical Foundations of Mathematics" ,p. 192):
"Lemmas do the work in mathematics: Theorems, like<br>management, just take the credit. A good lemma also<br>survives a philosophical or technological revolution."
Added Sept. 11, 2007: Kevin Buchin kindly tole me about the<br>beautiful woderful quote from THE BOOK.
Chapter "Lattice Paths and Determinants" in Aigner, Ziegler:"Proofs from THE BOOK"<br>begins with:
The essence of mathematics is proving theorems – and so, that is what mathematicians<br>do: they prove theorems. But to tell the truth, what they really want<br>to prove once in their lifetime, is a Lemma, like the one by Fatou in analysis,<br>the Lemma of Gauss in number theory, or the Burnside-Frobenius Lemma in<br>combinatorics.
Now what makes a mathematical statement a true Lemma? First, it should<br>be applicable to a wide variety of instances, even seemingly unrelated problems.<br>Secondly, the statement should, once you have seen it, be completely obvious.<br>The reaction of the reader might well be one of faint envy: Why haven’t I noticed<br>this before? And thirdly, on an esthetic level, the Lemma including its proof<br>should be beautiful!
(also in: Martin Aigner, Lattice Paths and Determinants,<br>in H. Alt (Ed.): Computational Discrete Mathematics, LNCS 2122, pp. 1–12, 2001.)
Opinions of Doron Zeilberger