The n-Category Café

A more radical and interesting solution would be to have a site arxiv-comments.org where any mistakes noticed, or even potential mistakes, could be recorded

It seems that the arXiv’s trackback system is geared along these lines.

On the other hand, several times I got the impression that arXiv trackbacks seem to be turned off somehow especially in parts of the math archive. (?)

While the issues Nathanson raises are interesting, there are not nearly as interesting as the issues he doesn’t raise.

For example, he assume that we all agree what “mathematical truth” is. In his implicit interpretation a statement is true if it can be deduced from a consistant axiomatic system using informal two valued logic. And he is shocked, shocked! that the judgment of validity of such a deduction amounts to a social consensus. He calls it “political,” but for me “political” is much to loaded a term. My reply to his concern would be
“How else would you do it?”

It is not clear to me that Nathanson’s description of how mathematics is done is realistic. For one thing, the axioms are fluid. There was plenty of interesting mathematics done before ZFC was enshrined as foundation and plenty more will be done after ZFC is replaced by something else (n-category theory, perhaps?). What I am trying to say that we don’t just deduce mathematical truth from axioms. There is also a feedback the other way: our experience of doing math modifies what we consider the foundations/axioms.

As an example of such an evolution look at the definition of orbifolds. There plenty of papers that use Stake’s original definition, but there is an emerging consensus that that definition needs to be abandoned. In what sense then the papers that use the original definition are true?

And finally a question that exposes me as a crank: why do we like two-valued logic so much?

There’s this guy named Carl Hewitt, who invented the Actor model of computation. The creators of the language Scheme, Sussman & Steele, have said they invented it to try to understand Hewitt’s thesis. Milner, who invented the pi calculus, also began from Hewitt’s work. Also Erlang and E are derived from it. He invented the first logic programming language, Planner.

Unfortunately, his work is very hard to read. He likes introducing new terminology and uses wild typography.

His latest work is on something he calls “direct logic” that aims to “contain” the effects of an inconsistency; it’s related to paraconsistent logic. For example, almost none of the programs we write are “correct” in any way, but they still mostly work. What can you say about the behavior of an incorrect program? How do you reason about policies, norms, or practices?

There’s a PDF here.

David wrote:

If a single mathematician “often finds mistakes”, there must be a huge number out there.

Sure. But most of them don’t matter much, since they are easy to fix.

Good math is a lot more like real life than some people think. It’s not a brittle, perfect crystal that’s bound to utterly shatter if it has a single tiny flaw in it. It’s less perfect, and more robust. Some mistakes are impossible to fix, and doom a proof. Lots of them aren’t.

Now that so many papers are posted on the ArXiv, should it not be expected of you that on finding an error you email the author?

That would be the noble thing, but often when I spot the mistake, I spot the fix two minutes later. After this happens, it’s often a bit tricky to tell whether the mistake was a true mistake, or just sloppy writing on the author’s part, or overly pedantic or insufficiently insightful reading on my part!

(Not surprisingly, mistakes occur most often in the passages that are vague or contain phrases like ‘it is easy to see’.)

Given all this, it often seems a bit too tiresome to send someone an email. “Hi! You may have made a small mistake that’s easy to fix, or maybe you just phrased your argument sloppily, or maybe I just didn’t understand you the first time…” I’m not eager to make someone’s acquaintance with an email like this! I’m usually glad that I was able to fix the proof and get on with business.

To increase the reliability of my results, I avoid proving theorems that rely on someone else’s result unless either 1) I understand the proof of that result or 2) the result is sufficiently well-known and time-tested that I feel I can trust it even if I don’t understand the proof. And frankly, I don’t really like 2). When I’m proving stuff, I’m only happy when I know how every little piece works.

(When I write This Week’s Finds, on the other hand, I feel perfectly happy talking about things I just barely understand! Being too constipated there would just slow down the process of learning stuff.)

Great, we’ve achieved something of a consensus that Nathanson is worrying where he largely needn’t, and missing out what is more important, namely, concept change. All of which sits very happily with me, and is taken up in Michael Harris’s contribution to the Princeton Companion.

Nathanson’s article appeared as an opinion in the August edition of Notices. Perhaps we should compose a response.

(By the way did people note Jacob Lurie’s WHAT IS…an $\infty$-Category in the September issue, although what he talks about there is what we call here an $(\infty, 1)$-Category?)

Mistakes in publications? How can this be?

I had 3 questions of 48 in the 11th grade Anatomy and Physiology assessment I just gave (i.e. not affecting grade, start of school year baseline):

6: The peer-review process assures readers that the article they are
reading was reviewed by:
A: A large panel of the general public
B: Journalists and other media personnel
C: Competent scientists working in the field that the article presents
D: The journal’s advertisers, to ensure that they are not offended by
the article’s content

7: Information presented on television news programs is:
A: Always peer-reviewed
B: Not peer-reviewed, but entirely reliable anyway
C: Not peer-reviewed and therefore should be viewed with scepticism
D: Only reliable if it is on a major network, as their advertisers
assure accuracy

8: Internet web pages:
A: Are always peer-reviewed
B: Go through a special peer-review process that differs from that of journals but ensures reliability
C: Are generally not peer-reviewed but can always be considered accurate and impartial
D: Are generally not peer-reviewed and should be viewed with scepticism

I gave that question 6 also to all my 10th grade Biology and 9th grade Chemistry students.

The answers showed that essentially none of them had ever heard the term “peer-review” or had any notion of how truth is assessed in publication. Question 6 had a distribution of roughly equal numbers of (a), (b), (c), (d) answers.

Question 7’s answers show that my students, in the majority, believe what they see on major network, and #8 shows that about half think that web pages undergo peer-review.

Granted, these are impovershed urban teenagers. But it makes me wonder why skepticism is not taught even earlier in the educational process.

Could it be that American public school education has intentionally been damaged, in order to produce more compliant worker bees, consumers, and cannon fodder?

At the high end, I have found and communicated to authors numerous errors in textbooks, journal articles, and conference proceedings.

My parents, with degrees in English Literature cum laude and magna cum laude from Northwestern and Harvard, pointed out errors in the New York times to me as a child. Late in life, my father was perturbed that even the headlines came to have errors.

I’ve worked for Boeing, Burroughs, European Space Agency, Federal Aviation Administration, Ford, General Motors, Hughes, JPL, Lear Astronics, NASA, Systems Development Corporation, U.S. Army, U.S. Navy, U.S. Air Force, Venture Technologies, and Yamaha. I am convinced that, for all the cultish exhibitions of Quality Assurance and Japanese-style Quality Circles, our corporate and government world is awash in errors, mistakes, and defects of every imaginable form.

So now the rubber meets the road: is Mathematics, with its capability of axiomatic perfection, to set a standard for all endeavors? Or is it socially the same as any other human enterprise?

I’m surprised that nobody has suggested using computers to verify correctness. We have proof checkers now, but so far they require details written in tedious detail. But theorem provers are getting better at filling in the obvious gaps using automated tactics. (Incidentally, designing these tactics requires conceptual thinking.)

I will be severely disappointed if we’re unable to subject the vast majority of accepted and proposed proofs to computer verification within a lifetime.