Peer Review and Proof

Suppose that I today, below, in this post, prove to you a certain mathematics theorem, say, “The Symmetry of Individual Constants.”

Is that theorem true?

Before you answer, consider that you probably aren’t a professional mathematicians or logician. Since you are not, you might not understand each step in the proof. Or if you are a mathematician, because you are inexperienced in the particular field, you might not understand the shorthand or the “givens” routinely used.

Also, and most importantly, the proof has not been “peer reviewed.” Which is to say, it has not been vetted by others who do understand the history and shorthand of this particular field.

Given all that, I repeat the question: is that theorem true?

Of course it is. This was not a trick question. I said—my main premise was—that I proved the theorem. A proof means that the truth of a theorem has been demonstrated. Quot erat demonstradum, and so forth.

Specifically, it does not matter one whit whether you believe the result, nor does it matter whether scores of people believe it. Even stronger, it does not matter whether experts in the field know of, approve of, or have reviewed the result.

We operate under a simple tautology: either my proof is valid or it is not. Whether that proof has been vetted is irrelevant. The theorem is true or not. Moreover, it has always been true or not. It was even true before I discovered a proof of its validity. Human agency does not—and cannot—change the truth status of the theorem.

Obviously, then, where a valid proof appeared is irrelevant to the proof’s validity. Another way to say this is that it is a logical fallacy to claim, “Because this result was not peer reviewed and published in an academic journal, it is false.”

This fallacy is routinely used by academics who cannot refute a distasteful result. “If So-and-So is so confident in his work, let him publish it!”

Our friend Steve McIntyre hears this fallacy a lot. McIntyre has published on his blog many true statements about the climate that are disliked or are unwanted by mainstream climate scientists. Instead of trying to answer, or to refute, McIntyre’s results, these global warming advocates dismiss them because they have not been peer reviewed and published.

Unfortunately, this fallacy—as shockingly obvious as it is—is usually accepted as proof of refutation. That is, it is thought that a refutation has been demonstrated merely because a credentialed scientist cast an aspersion.

Journalists—God love ’em—are particularly eager to swallow this faulty line of reasoning. I have yet to see a case where a reporter rebuts, “Yes, McIntyre’s work hasn’t been published. But how is it wrong?” That is, of course, the only question worth asking.

This fallacy does not just find a home in climate theory: it is warmly accepted everywhere. It’s familiarity is proof that scientists have egos just as large as Hollywood stars.

Have you heard of Grigory Perelman, the bushy eyebrow-bearing Russian mathematician who has been turning down prizes?

In 2002, Perelman wrote up a valid proof of the Poincaré conjecture, probably the most lusted after unsolved problem in mathematics. Briefly, imagine a two-dimensional surface on which is placed a loop of string. If you can shrink that loop continuously until it reaches a point such that the loop nowhere becomes kinked or stuck on a hole on the surface, then that 2D surface is really the same as a sphere; albeit a sphere that might be pulled, crushed, and twisted out of shape.

The Poincaré conjecture says that this same loop-shrinking is true for three-dimensional spaces. And what’s the most familiar 3D space? Well, Space. The universe. Not for the first time, Poincaré had said something deep about life, the universe, and everything.

Point is, the non-conformist Perelman submitted his results to the non-peer reviewed, non-journal Arxiv.org. This is a place where scientists (you have to be recommended to win publishing privileges) place their thoughts in advance of, or in lieu of, peer review. It’s a place to publish not-so-polished, non-popular thoughts in the hope of winning First to Discover status or to avoid the tediousness of peer review.

It turns out that a group of Chinese mathematicians wanted in on Perelman’s success. They re-wrote Perelman’s results and published them in a peer-reviewed, academically approved journal. They also strongly hinted (in a 2003 Science article) that theirs was the real proof—so that credit properly belonged to them—because their proof had been formally published.

The incident is non-technically retold in the New Yorker, where it now seems that Chinese group have recognized their use of the not-published fallacy. They are now claiming “they couldn’t follow” Perelman’s shorthand, even though other mathematicians could. Well, academics was always a blood-stained enterprise.

Anyway, let’s get to work. How many instances of the not-published fallacy can we find?