by Willie Wong
Besicovitch once said that “A mathematician’s reputation rests on the number of bad proofs he has given.” Of course, originating from someone educated in the Russian school, the word “bad” in the quote should probably be taken to mean “inelegant”. However, lack of beauty is certainly not the only possible deficiency in the quality of a published result. Cases abound where a proof is bad not in the aesthetics, but in something more fundamental.
What drives mathematicians (professional or armchair alike) to proffer incomplete or incorrect proofs is most often the allure of instant fame (and in some, I think small number of, cases, the tantalizing prospect of a promised bounty). And in most cases (at least insofar as professional mathematicians go), the attempts are genuine. Even in the misguided hopes of amateur to provide a result contradictory to known theorems (see the extant efforts in trisecting an angle via straightedge-and-compass) one typically finds real interest in the problem and not a mind to defraud. Under this assumption of good-faith, a mistake can be assumed to be something the mathematician doesn’t understand or something he overlooked.
In many ways, mathematics shares the “many eyes” model with open-source software. The first line of defense to an incorrect proof is of course the author himself. Should the author fail to observe the mistake, it is hoped that in discussion with his peers that the error will be found. The third line of verification is traditionally at the level of the journal referee when the paper is submitted for publication (in modern days, there is often a line of defense between the second and the third, which is when the paper is offered for dissemination as a pre-print on services such as the arXiv). It is unknown how many mistakes were caught at the first and second levels (since such “news” would never be widely propagated), but quite a few high-profile mistakes were caught at the pre–print and referee levels.
What brought about this meditation is a pre-print that appeared recently on arXiv, which is submitted to the Journal of Number Theory, and which points out a mistake that appeared in a published article. While mistakes slipping past referees for a journal publication is not unheard of, I think in this case that the mistake wasn’t caught earlier is rather shocking.
Typically, a mathematician first comes up with a conjecture. The conjecture usually says that under certain conditions, a certain statement will be true. (Let’s not go into how the conjecture is born; that is a totally different can of worms.) Then the conjecture will be tested on trivial cases to make sure it makes sense intuitively, and the conjecture will be tested on pathological examples to make sure the conditions is strong enough to rule out the imaginable counterexamples. At this point the mathematician will start to believe that the conjecture may be true and set out to prove it.
Now there are two modes of failure with a proof. The first is that when the conjecture is ultimately correct, but the method of proof is flawed. Mistakes like this are most easily forgiven in the mathematical community because the error is often subtle, and appears usually as a missing small logical leap (an assumed trivial, and hence omitted, step that turns out not quite trivial), as a misapplication of a prior result, or just as a simple computational error. To find the logical fault in this case is the most difficult, especially for the author who believes in the veracity of the final statement. The second way of failure is that the conjecture itself is incorrect. And in the usual case this is because the conditions asked for in the hypotheses are not strong enough; and due to limited imagination on the part of the original author, he was not able to construct sufficiently pathological examples to falsify his conjecture. In this case once a counterexample is constructed, oftentimes it will be discovered that the author, in his proof, misapplied a prior result by assuming too few conditions. By back-tracing the error from there, the correct conditions can be easily added to the conjecture, and the proof will be complete.
In the case of the paper that inspired this post, the problem is that the conjecture is false, but not for “pathological cases”. Rather, the problem occurs for the “trivial test cases”. The (false) conjecture is the following:
Let be co-prime non-zero integers with . Let be an integer, then the number defined by the infinite sum is transcendental.
Looking at the the conjecture, we first ask: why must ? For the case of , it is simple: if , then the infinite sum is not absolutely summable, so the number is not defined. For the case of , consider the case that it is equal to 1. Then is an integer. So there exists some integer such that , and the infinite sum becomes ill-defined because we need to take zero to a negative power.
With perfect hindsight, we can say that what has been ruled out so far are truly trivial. In fact, the conditions given only rules out the case when the infinite sum is ill-defined. The next natural question (which I think the original authors forgot to ask themselves, and which the referee didn’t consider) should be “are there trivial cases for which the infinite sum is well-defined yet the sum is not transcendental?” With a little bit of thought one should be arrive at a counterexample rather simply.
Define the function for an natural number bigger than 1. There are several rather immediate properties of these functions:
- blows up when is an integer.
- when is not an integer.
- when is not an integer.
- Since is periodic with period 1, we can consider just its restriction to the open interval , on which it is infinitely differentiable.
Combining properties 2 and 3, we see that if were odd, then . This implies in the case , is anti-symmetric about , and in particular by continuity must have , which is not a transcendental number.
As it turns out, these counterexamples () are the only ones for the original conjecture, as shown in the linked arXiv pre-print. They are due to the fact that the lattice not only has translation symmetry, and a reflection symmetry about integer points, but also a reflection symmetry about half-integer points. And in hindsight, while the original authors are probably well aware of the principle that symmetries generate exceptional behaviours, and hence induces trivial examples to check, they may have just overlooked this one symmetry.
All that just goes to show an adage that I live by: “A mathematician should be most skeptical of himself.”