Written: May 21, 2003
On April 25, 2003, Frank Quinn gave a Colloquium talk entitled History and Status of Homology Manifold.
The first 55 minutes were very good and informative, and he pointed out that work in this field is full of incomplete and even erroneous proofs (and to his credit, he also mentioned that one of his published papers had gaps). But I didn't like the punch line, where he claimed that recent work of my Rutgers colleague Steve Ferry, contains gaps, and that Ferry refuses to address and/or admit them.
I am far from qualified to comment about the substance of these accusations. But it was very inappropriate, indeed outright nasty, to make these accusations, whatever their truth-value, in public, especially in the "victim"'s department, and in his absence to boot. "If you don't have anything nice to say, don't say it!" Why not talk about your own (and other's) stuff that you believe is correct, and leave stuff you doubt unsaid? Why this negative campaigning? (unless you were settling old accounts, but even then don't wash Homology's dirty laundry in public!)
Frank Quinn is perhaps the most valiant defender of traditional mathematical rigor, indeed the watchdog of logical rigor against the "theoretical physics riffraff". But after his talk I am no longer sure how really "rigorous" topology is, at least the kind of Homology that he and Ferry do. Even the great Poincare made (what was considered later to be) serious mistakes, and the field is full of proofs that had to be corrected later, and sometimes retracted, as is all too clear from Quinn's talk.
So Frank Quinn is probably right, that Steve Ferry's work contains some gaps. But, with probability 1-epsilon, so do all the other papers in this difficult, heavily human, field, including Quinn's own papers, that use human, semi-formal arguments that have not yet been combinatorialized and computerized. In my humble opinion, only combinatorial and computer proofs are truly reliable. I trust the computer-assisted proofs of the Four Color Theorem (and would have trusted them even more had they been fully computer-generated) much more than any human proof in Homology. If such great experts as Quinn and Ferry can't agree between them on what's correct, then this makes their whole field of questionable rigor.
What Homologists need to do is to perform Computerized Deconstruction, and complete Combinatorialization (like Lou Kauffman did to the Jones invariants). Another way to be 100 percents sure (or close to it), is to split every statement into substatements, subsubstatements, ..., such that group refereeing and checking can be facilitated, like in my proof of the Alternating Sign Matrix Conjecture.
Even so-called "non-rigorous" proofs by physicists, so feared and disdained by Quinn, are MORE reliable than Homology proofs. After all physicists use powerful methods of Quantum Field Theory and Renormalization Group, that while not-yet-rigorous, have a very good track record, (and possibly will be rigorized one day in an unexpected way, say by making it all "formal" (and hence rigorous), just like the use of "divergent series" in combinatorics and number theory, that makes perfect rigorous sense when viewed as formal power series, or Dirac's delta "function" and its derivatives, used freely by Dirac way before Laurent Schwartz invented the notion of distribution).
On the other hand papers in Homology are so error-prone, as even experts like Quinn admit. In a field with a reputation for so many "proofs" that turned out to be "false", what is the probability that the proofs, currently believed to be correct, even by such a careful, meticulous and harsh critic as Frank Quinn, are indeed correct? And frankly, Frank, I don't give a damn (and neither does anyone else except you and at most 100 other specialists).