[FOM] Re: Soare's article in the BSLIssue 3

Timothy Y. Chow tchow at alum.mit.edu
Fri Feb 4 13:26:53 EST 2005

Stephen G Simpson <simpson at math.psu.edu> wrote:
> Regarding Gromov's theorem.
> The correct statement of the theorem is: If M is a compact Riemannian
> manifold whose fundamental group has unsolvable word problem, then M
> has infinitely many contractible closed geodesics.

This is how it's stated in Soare's manuscript that Martin Davis mentioned
(Theorem 6.2).

> I don't know whether Gromov ever wrote up a proof of the theorem.  I 
> only know that there is a widespread though perhaps not universal 
> practice of attributing a theorem to the person who first proves it, 
> rather than to the person who first states it.

As I've alluded to in passing a couple of times on FOM, it is not uncommon 
to find papers in which the author makes assertions that are supposed to 
be theorems (sometimes they are even explicitly labeled as such; other 
times they are labeled as exercises or not labeled at all), but where 
there is either no proof or a proof that doesn't satisfy everyone.  This 
practice is more common in certain fields of mathematics but I don't think 
any field is immune.  If the assertion is true and the proof, or the 
filling in of the gaps of the proof, is judged to be sufficiently 
straightforward, then typically the original author is given credit for 
the theorem.  If the assertion is true but the missing details turn out to 
be nontrivial and someone specifically fills them in, then often joint 
credit is given.  (An example that comes to mind is Stark's 1969 paper "On 
the `gap' in a theorem of Heegner," J. Number Theory.)  If the assertion 
is true and original, no proof is given, and the problem turns out to be 
very hard, then sometimes the original author will get credit for stating 
a conjecture (even if he didn't think of it as a conjecture).  Only if the 
assertion is false or if the proof is way off base does the original 
author typically get no credit at all.

I can't say I'm completely happy with this type of situation (who decides 
whether the missing details are trivial or nontrivial?) and I think that 
the Jaffe-Quinn paper that I cited on FOM not long ago articulates the 
problems quite well.  However, that is the reality of mathematical 
practice as it stands today, whether we like it or not.


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