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.
Jenga is a game where you start with a tower of wooden pegs and take turns removing pegs until someone makes the tower collapse. A style of mathematics analogous to Jenga reached the height of its popularity about 40 years ago and then fell out of fashion. I use the phrase “Jenga mathematics” to refer to generalizing a well-known theorem by weakening its hypotheses, seeing how many pegs you can pull out before it falls.
Many 20th century mathematicians spent their careers going over the work of 19th century mathematicians, removing every hypothesis they could. Sometimes a 20th century mathematician would get his name tacked on to a 19th century theorem due to his Jenga accomplishments.
Taken to extremes, Jenga mathematics turns theorems inside-out and proofs become hypotheses. Natural hypotheses are replaced with a laundry list of properties necessary to make the proof work. Start with some theorem of the form “Let X be a widget. Then X has a foozle.” Go back over the proof and see just what features of a widget are needed for the proof. Then restate the theorem as “Let X have the following apparently arbitrary list of properties necessary for my proof to work. Then X has a foozle.” Never mind whether anybody can think of anything other that a widget that satisfies the hypotheses of the new theorem.
Jenga mathematics is no longer fashionable. Mathematicians still value removing unneeded hypotheses, but they’re not as willing to go to extremes to do so. They are more interested in building new towers than in removing every piece possible from old towers.
The book Out of Their Minds quotes Leslie Lamport on proofs:
The proofs have been carried out to an excruciating level of detail … The reader may feel that we have given long, tedious proofs of obvious assertions. However, what he has not seen are the many equally obvious assertions that we discovered to be wrong only by trying to write similarly long, tedious proofs.
See Lamport’s paper How to Write a Proof. See also Complementary validation.
Mark Dominus brought up an interesting question last month: have there been major screw-ups in mathematics? He defines a “major screw-up” to be a flawed proof of an incorrect statement that was accepted for a significant period of time. He excludes the case of incorrect proofs of statements that were nevertheless true.
It’s remarkable that he can even ask the question. Can you imagine someone asking with a straight face whether there have ever been major screw-ups in, say, software development? And yet it takes some hard thought to come up with examples of really big blunders in math.
No doubt there are plenty of flawed proofs of false statements in areas too obscure for anyone to care about. But in mainstream areas of math, blunders are usually uncovered very quickly. And there are examples of theorems that were essentially correct but neglected some edge case. Proofs of statements that are just plain wrong are hard to think of. But Mark Dominus came up with a few.
Yesterday he gave an example of a statement by Kurt Gödel that was flat-out wrong but accepted for over 30 years. Warning: reader discretion advised. His post is not suitable for those who get queasy at the sight of symbolic logic.