Proofs By Contradiction And Other Dangers



Some suggestions on how to present proofs, again



Henry Cohn is an expert in many aspects of theory including cryptography, various aspects of combinatorics, and in particular the matrix multiplication problem. One of his most fascinating ideas is on using group theory to approach fast matrix product—this is joint work with Robert Kleinberg Balázs Szegedy, and Chris Umans.

Today I plan on talking about a recurrent theme—can an amateur make a contribution to modern mathematics and theory?



Cohn has written a beautiful piece on exactly this topic, and that is what led me to think about it once more. I frequently get earnest emails from people who think they may have solved the question. They are often amateurs in a sense, but not amateurs in another sense. A typical profile is that the person is a master programmer, with some mathematical training; sometimes the training is informal and sometimes it was formal, yet happened years ago.

Here is a paraphrase of a typical message I get:

I have emailed a number of experts with a version of my solution. Most ignored the email. The ones who responded said that they took a quick glance at the paper, decided it was probably wrong, and refused to discuss it further.

I think right now there are multiple claims, private and public, from some sharp people that they have solved the problem. Some claim equal, others not equal; all claim they have found a proof. At least one side’s proofs must be wrong, else we are all in trouble, since ZF would then be inconsistent. If we insist that something called a “proof” must be a complete argument, then I suspect that all are “wrong”—this quote is relevant:

A mathematician’s reputation rests on the number of bad proofs he has given—Abram Besicovitch.

However, someone could have the correct significant idea for such a proof, or failing that, at least an important idea in a new context that only an outsider to the tight-knit theory community could see.

This has happened many times in science. Let me give a couple of examples where the mainstream specialists thought that the work of an outsider was wrong.

Wrong, Wrong, Right

There are cases throughout science of the conventional wisdom being wrong. This makes things especially hard for an amateur to make a contribution, because the conventional understanding is usually a needed support, but it affects professionals too. Here are three relatively recent examples. In each case the person was right, but the initial reaction was negative. Also, all three were not amateurs but professionals. Thus we all have problems with ideas that do not fit the conventional thinking, even when the thought is coming from a professional, and not an amateur.

Irrational Numbers: Roger Apéry proved that is irrational. This was a major surprise, since it had been open a very long time whether or not this value was rational. I have discussed some of the issues surrounding his wonderful result before—see this. Note there was definitely some doubt surrounding his initial claim; he was right, but that did not make people believers instantly.

Stomach Ulcers:

When I was growing up several friends and relatives unfortunately had stomach ulcers. The belief was that stress was the main cause of their problem, and so most patients were told to reduce their stress. They also were treated with bland diets, and still suffered most of their adult life with the ailment.

Dr. Barry Marshall showed that essentially all the previous ideas about this illness were wrong. He demonstrated that almost all stomach ulcers were caused by a bacterium, Helicobacter pylori, that it could be completely cured by antibiotics in most cases. For this terrific work he was awarded the 2005 Nobel Prize in Physiology or Medicine, along with his co-researcher Robin Warren. I love that the prize title has an “or” in it: “Physiology or Medicine.”

A great story. Marshall and Warren are heroes who saved countless people suffering and worse. Yet it took decades for them to get the mainstream medical profession to take their ideas seriously. Note they were outsiders in a physical sense, since they were based in Australia. But they were physicians, trained in medicine, and based at the Royal Perth Hospital.

At one point Marshall infected himself with the bacterium, which is one way to make a dramatic statement He is quoted as saying “Everyone was against me, but I knew I was right.” See this for an interview with him.

What I find most disturbing about this great success story is that it could have saved lives much earlier. The claim Marshall and Warren made was easily testable: doctors only had to give their stomach ulcer patients standard antibiotics. While no drug is perfectly safe, Marshall points out that this would have been quite safe, especially compared to surgery that many underwent. And it would have quickly shown that he and Warren were right—the patients’ symptoms would have disappeared. But the mainstream profession refused to change for decades.

Prions:

Infections are caused by either bacterium or by viral agents—right? Dr. Stanley Prusiner discovered that there is a third way to cause an “infection.” The third method is based on what he named prions. For this work he was awarded the Nobel Prize in Physiology or Medicine in 1997.

Prions are proteins that have folded incorrectly. They work in an incredibly novel way: when they enter a host they cause the existing correctly folded proteins to also become incorrectly folded. This clearly can cascade and eventually many of the proteins in the host are changed. Unfortunately the proteins that are not folded correctly cause terrible damage to the host. Prions appear to be the cause of mad cow disease in cattle and Creutzfeldt-Jakob disease in humans.

Unlike the work on ulcers, while a breakthrough, this discovery has not helped yet to find a cure for these diseases. Indeed there is still some doubt and debate on whether prions are the cause of these diseases. Read this about the issues.

Danger: Proof By Contradiction

Any proof that is not equal to is likely to be a proof by contradiction, at least in its initial presentation. Since is formally a for-all/there-exists ( ) statement of arithmetic, there is a principle that such a proof should be convertible to one without contradiction—see for instance this and its references. However, recent advances such as Williams’ separation of from have used contradiction extensively. See the posts on such proofs here and related issues here.

Cohn makes an interesting point about proofs by contradiction. Here is a liberal paraphrase of his point, in the context of a proof:

Let’s start by assuming that and we will eventually show that this leads to a contradiction. So far, no problem. Then, Cohn points out there will probably be a long series of definitions and lemmas and theorems. Still fine. Finally, a contradiction will be reached—something wrong will be concluded. This shows that the assumption was wrong, and thus that . Great. Become famous, become acclaimed, become rich, and so on. Great.

The difficulty is what if the contradiction comes not from the assumption that , but rather from some error in the proof of one of the lemmas or theorems? A problem. A serious problem. Then the proof is wrong. The proof is invalid, since the contradiction did not follow from the assumption. The whole point of a proof by contradiction is to show the assumption leads to a falsehood. This is the key; if the contradiction comes from any other source the whole proof is in trouble.

Cohn points this out, but does not have an answer on how to avoid it. Nor do I. It seems inherent in the nature of reasoning about proofs. Especially those that use proof by contradiction.

Danger: Polynomial Time

One misconception that I believe is at the heart of many of the attempts to prove is the nature of polynomial time, . This complexity class is closed under many operations: union, intersection, complement, reductions of various kinds, and much more. However, it not closed under the taking of limits.

Here is what I mean by “taking limits.” Consider a series of Turing machines that run each in polynomial time: let the sequence be , , and so on. They define each a language , , and so on. By definition, all of are in . The problem is the limit of these languages is not always contained in polynomial time. Loosely speaking, even in cases where there is a well-defined limit,

does not converge to a language in , in general.

I see this fact implicitly used in many attempts to solve the problem. If this taking of limits was okay, then there would be lots of simple “proofs” that . Unfortunately it is false. For an easy counter-example, let be the set of so that the Turing machine with input and program accepts in time at most . Here as usual is the length of . Each , but their limit is not.

Note . Suppose that accepts in time , then it certainly still accepts in time . Thus even monotone nested sequences of sets do not converge:

The limiting language leaves polynomial time.

Note this lack of convergence is similar, I believe, to the confusion that occurred in the early days of calculus and analysis. Many famous mathematicians of the time made errors, or “proved” theorems using assumptions about convergence that were either false or at least not justified.

I would guess that at least some major fraction of mistakes in claims about the question stem from this non-convergence property. I know of at least one well respected, well published complexity theorist, who fell to victim to this very misconception. The limiting argument can be well hidden in a long proof, and quite often is subtle to see.

I hope this helps some of those who are trying to resolve . Check carefully that you are not making this convergence mistake. And if you are, be of good cheer, since you are in good company, both recently and historically.

Advice?

The question is how to make a contribution, how to get some to listen to your voice. It is an interesting question. Here is some advice I gave one person recently:

You should try to write a one page or so description of the key idea(s). This page should not say anything about ‘s importance, its background, or its history. We all know that. It should concentrate on what is the new idea that makes your proof work. A very helpful remark would be: I avoid the convergence problem by Or I avoid the usual barriers because As a skilled programmer approach the proof in the same way you write quality code. Define the key notions carefully, lemmas and theorems should be stated exactly. They should be stated so that anyone reading them will understand them. Define the constants, the sets, the notions precisely. Do not use a formal language, but strive to make all crystal clear.

My experience is that if this is done, you may actually discover your own error. But if you still feel it is correct I think we can find people to look at the page.

Open Problems

Any suggestions on how to help with proofs by contradiction? This is the question raised by Cohn. I would like some suggestions on how to avoid the problem he pointed out.