In the wake of the 40th anniversary of the Apollo 11 landing, here’s a nice piece from the Vanderbilt University News Network about Richard Arenstorf, professor emeritus of mathematics, who solved a major piece of the theoretical puzzle that made that landing possible. Excerpt:

In order to determine the path that the Apollo spacecraft would take in its journey from the earth to the moon, NASA scientists had to come up with a new solution for a difficult mathematical problem, called the three-body problem, that had been studied for more than 300 years by a number of famous mathematicians, including Euler, Lagrange and Poincare. [...]

Using a computer, [Arenstorf] solved a special case of the three-body problem that provided the mission with the information it needed. His solution consisted of a set of closed figure-eight trajectories that pass arbitrarily close to two celestial objects. These are now known as “Arenstorf Periodic Orbits.” In 1966, he was given the NASA Medal for Exceptional Scientific Achievement for his contribution.

I’ve mentioned Prof. Arenstorf here before, since he is not only a famous and prolific mathematician, he was also my Complex Analysis professor in grad school and had a near-miss proof of the Twin Prime Conjecture a few years ago. One of my favorite memories of grad school was sitting with Prof. Arenstorf at our weekly grad student teas — which he regularly attended, because he loved being around graduate students — talking about the space program and comparing his NASA stories with those of my dad, who was an engineer contracted from General Motors working on the Apollo project at around the same time Prof. Arenstorf was at NASA. It’s nice to see him get the recognition he deserves.

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I sat in on a few of Dr. Arenstorf’s lectures while I was in Nashville and enjoyed them thoroughly. I’d taken a lot of “soft” analysis courses in grad school and it was refreshing to see his old-school commitment to “hard” analysis.

I asked him one time about a theorem named after him. I showed him the book where I’d seen it and he complained that he didn’t recognize the theorem, all dressed up in fancy terminology. My impression was that Arenstorf had done all the hard work in a fairly concrete setting and the author had done a cheap generalization by making the statement more abstract.

I’m curious as to what a “near-miss” proof might be? Proofs are either correct or not. In this case, the proof was shown to be flawed, and five years later, the flaw has not been fixed. Thus, this is a non-proof. Not to denigrate the professor, who merely erred, I think it’s silly to call a failed, flawed proof to be a “near-miss” proof. Unless there’s something I am missing, you’re actually doing a disservice to the professor by so describing his erroneous proof. I imagine that he was embarrassed (as Andrew Wiles were, nearly to the point of intellectual paralysis for a short time) when the error was revealed. No one could possibly be happy about having publicly claimed to have a proof, especially of a really famous problem, only to have it shown to be flawed.

I think it’s legitimate to call a proof a “near miss.” A flawed proof by a mathematician of Dr. Arenstorf’s caliber is likely to be a valuable contribution to mathematics, though of course it is not a proof of the twin prime conjecture. For example, much of abstract algebra came out of near misses at proving Fermat’s last theorem. These failed proofs were interesting because they identified variations on the conjecture that were true. A “near miss” proof for an important conjecture might be a bigger contribution to mathematics in the long run than a correct proof of an unimportant result.

I’m not familiar with Dr. Arenstorf’s paper, but I imagine it proves

somethingthat wasn’t known before. If not, it may contain techniques worth investigating.It was a “near miss” in the sense that people actually thought his proof was correct for a not-insignificant period of time, until a flaw in the proof was found. It wasn’t obviously wrong from the get-go like so many proposed “proofs” of the TPC (or the Riemann Hypothesis, etc.).

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