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I just wrote a math journal article, and it only took me thirty seconds. Here it is: Scalars and Monge’s Conjecture. Pretty good, considering I can’t read any of it. I did it with the just-for-fun random mathematics journalizer, MathGen. You can write one too!

Okay, it’s idle fun that means nothing—except that the journal *Advances in Pure Mathematics* reportedly got Sokal-hoaxed with MathGen. The journal is reported to have accepted a random MathGen paper for publication, with just a few recommendations for improvement from the anonymous peer reviewer(s). The author replied to one of them,

4. The author believes the proofs given for the referenced propositions are entirely sufficient [they read, respectively, “This is obvious” and “This is clear”]. However, she respects the referee’s opinion and would consider adding a few additional details.

You’ll get a good laugh out of the rest of the article.

Meanwhile you might also take a moment to pause and wonder whether this kind of thing happens more often than we realize.The whole enterprise of academics rests on the competence and trustworthiness of its practitioners. Its pronouncements come with a heavy weight of authority, which the rest of us have to accept on the basis of trust.

Now, I’m confident that if the journal had actually published the paper, its error would have been exposed within hours. That’s possible in mathematics, but not in most research-based fields. Sokal-style hoaxes aside, I’m hardly the first to ask, do we really know what we think we know?

One of the great virtues of science is that its results can be checked and verified. That process really is effective for confirming the authority of scientific findings. There’s a caution to go along with that, though. Most people think that process goes on virtually all the time. Most of us are inclined—you might even say *trained*—to grant every journal article, or every media report of a new finding, the full authority that comes with a thoroughly tested result. Is that trust deserved?

This MathGen incident is silly and inconsequential, something to get a good laugh from and move on—except that it exposes once again the weak underbelly of scientific authority. *Caveat lector, caveat emptor.*

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@Tom Gilson:

Mathematics does have that advantage over the empirical sciences: one only needs the requisite knowledge and a sufficient amount of pen and paper. Not only that, but with the advent of proof-checkers (Coq, Mizar, HOL, etc.), a sizable chunk of foundational mathematics has been computer-checked for validity so these portions have the highest degree of certainty that human knowledge can attain. It is reasonable to expect that in the not-so distant future the process of submitting a mathematical paper will be accompanied by the corresponding proof-checks produced by these programs.

Having said, and no matter how eager I am to defend the honor of my lady, a few caveats need to be said.

1. The errors would be caught in hours *if* the paper was deemed important enough to be so scrutinized. The simple matter of fact is with a culture of publish-or-perish, there are literally thousands of papers that fall to the side with no significant scrutiny, because no one bothers to read them.

2. In a paper of 2000, the great Russian mathematician Novikov wrote

So it seems that about 25 years, mathematicians were confiding in a result that had no complete and proper proof in the literature. The above quote was lifted from a paper by Y. Rudiak of 2001. After the quote he says,

and then embarks on the project of putting all the pieces together to give a complete proof.

3. Y. Rudyak’s paper is about 70 pages. This is already a long paper by the normal standards. Now consider the case of the classification of finite simple groups.

The proof of the theorem is spread through more than 300 papers, by over 100 authors, running over 10000 journal pages. Gorenstein announced in 1983 that the classification was finished; he was wrong. The quasi-thin case was only settled in 2004 by Aschbacher and Smith, adding about 1220 journal pages to the total tally.

The second generation proof worked on by Gorenstein, Lyons & Solomon runs over 5000 pages spread out over more than 6 volumes. A third generation proof is already being worked upon to simplify and condense even more, but it is still expected to run over a couple of thousand pages.

2000 pages? This is still a frightening amount. There are not many people that can hold in the head not only the structure of the proof, but the sheer amount of details. It is a full time job.

The classification of finite simple groups is an extreme case, but it points to an inherent limitation in our cognitive abilities. It also explains why mathematicians will go for more and more abstract notions: they are just *necessary* if we are to keep the cognitive load at a reasonable, humane amount.

note: A similar problem is faced by computer programmers and engineers by the way.

Tom,

Advances in Pure Mathematics is an “open access” journal which charges authors $500 to publish their papers. As you can imagine, this creates a financial incentive for the journal to accept as many papers as possible, even if the papers contain mistakes. I think these Sokal-type hoaxes are less likely to happen with traditional journals that make their money by selling subscriptions.

I’m curious to know why you’re writing about this here…

Thanks for the info on APM. That does make a difference.

My main concern here is what I closed with: “the weak underbelly of scientific authority,” which is too often raised as a club against other forms of knowledge, especially regarding religious belief. “

Caveat lector. Caveat emptor.“