Daniel Lemire wrote a blog post this morning that ties together a couple themes previously discussed here.
Most published math papers contain errors, and yet there have been surprisingly few “major screw-ups” as defined by Mark Dominus. Daniel Lemire’s post quotes Doron Zeilberger on why these frequent errors are often benign.
Most mathematical papers are leaves in the web of knowledge, that no one reads, or will ever use to prove something else. The results that are used again and again are mostly lemmas, that while a priori non-trivial, once known, their proof is transparent. (Zeilberger’s Opinion 91)
Those papers that are “branches” rather than “leaves” receive more scrutiny and are more likely to be correct.
Zeilberger says lemmas get reused more than theorems. This dovetails with Mandelbrot’s observation mentioned a few weeks ago.
Many creative minds overrate their most baroque works, and underrate the simple ones. When history reverses such judgments, prolific writers come to be best remembered as authors of “lemmas,” of propositions they had felt “too simple” in themselves and had to be published solely as preludes to forgotten theorems.
There are obvious analogies to software. Software that many people use has fewer bugs than software that few people use, just as theorems that people build on have fewer bugs than “leaves in the web of knowledge.” Useful subroutines and libraries are more likely to be reused than complete programs. And as Donald Knuth pointed out, re-editable code is better than black-box reusable code.
Everybody knows that software has bugs, but not everyone realizes how buggy theorems are. Bugs in software are more obvious because paper doesn’t abort. Proofs and programs are complementary forms of validation. Attempting to prove the correctness of an algorithm certainly reduces the chances of a bug, but proofs are fallible as well. Again quoting Knuth, he once said “Beware of bugs in the above code; I have only proved it correct, not tried it.” Not only can programs benefit from being more proof-like, proofs can benefit from being more program-like.