2018-02-23 18:58:03 +00:00
|
|
|
|
---
|
|
|
|
|
created_at: '2012-06-05T13:48:21.000Z'
|
|
|
|
|
title: Signs a Claimed Mathematical Breakthrough is Wrong (2008)
|
|
|
|
|
url: http://www.scottaaronson.com/blog/?p=304
|
|
|
|
|
author: ColinWright
|
|
|
|
|
points: 79
|
|
|
|
|
story_text: ''
|
|
|
|
|
comment_text:
|
|
|
|
|
num_comments: 13
|
|
|
|
|
story_id:
|
|
|
|
|
story_title:
|
|
|
|
|
story_url:
|
|
|
|
|
parent_id:
|
|
|
|
|
created_at_i: 1338904101
|
|
|
|
|
_tags:
|
|
|
|
|
- story
|
|
|
|
|
- author_ColinWright
|
|
|
|
|
- story_4068595
|
|
|
|
|
objectID: '4068595'
|
2018-06-08 12:05:27 +00:00
|
|
|
|
year: 2008
|
2018-02-23 18:58:03 +00:00
|
|
|
|
|
|
|
|
|
---
|
2018-03-03 09:35:28 +00:00
|
|
|
|
Yesterday several people asked my opinion of a
|
|
|
|
|
[preprint](http://arxiv.org/abs/0801.0398) claiming to solve the Graph
|
|
|
|
|
Isomorphism problem in deterministic polynomial time. I responded:
|
2018-02-23 18:19:40 +00:00
|
|
|
|
|
2018-03-03 09:35:28 +00:00
|
|
|
|
> If I read all such papers, then I wouldn’t have time for anything
|
|
|
|
|
> else. It’s an interesting question how you decide whether a given
|
|
|
|
|
> paper crosses the plausibility threshold or not. For me personally,
|
|
|
|
|
> the AKS [“PRIMES in
|
|
|
|
|
> P”](http://www.math.princeton.edu/~annals/issues/2004/Sept2004/Agrawal.pdf)
|
|
|
|
|
> paper somehow crossed it whereas this one somehow doesn’t.
|
|
|
|
|
>
|
|
|
|
|
> Of course, I’d welcome an opinion from anyone who’s actually read the
|
|
|
|
|
> paper.
|
2018-02-23 18:19:40 +00:00
|
|
|
|
|
2018-03-03 09:35:28 +00:00
|
|
|
|
Three commenters wrote in to say the paper looked good. Then the author
|
|
|
|
|
found a bug and retracted it.
|
2018-02-23 18:19:40 +00:00
|
|
|
|
|
2018-03-03 09:35:28 +00:00
|
|
|
|
**Update (1/5):** Laci Babai writes in to tell me that’s not quite what
|
|
|
|
|
happened. See
|
|
|
|
|
[here](http://people.cs.uchicago.edu/~laci/polytope-correspondence.pdf)
|
|
|
|
|
for what did happen, and
|
|
|
|
|
[here](http://people.cs.uchicago.edu/~laci/polytope.pdf) for an argument
|
|
|
|
|
that Friedland’s approach would if sound have implied P=NP.
|
|
|
|
|
|
|
|
|
|
My purpose here is not to heap embarrassment on the author: he’s a
|
|
|
|
|
serious mathematician who had a well-defined and interesting approach,
|
|
|
|
|
and who (most importantly) retracted his claim as soon as a bug was
|
|
|
|
|
discovered. (Would that everyone did the same\!) Though the stakes are
|
|
|
|
|
usually smaller, similar things have happened to most of us, including
|
|
|
|
|
me.
|
|
|
|
|
|
|
|
|
|
Instead I want to explore the following metaquestion: suppose someone
|
|
|
|
|
sends you a complicated solution to a famous decades-old math problem,
|
|
|
|
|
like P vs. NP. How can you decide, in ten minutes or less, whether the
|
|
|
|
|
solution is worth reading?
|
|
|
|
|
|
|
|
|
|
For a blogger like me — whose opinions are both expected immediately and
|
|
|
|
|
googlable indefinitely — this question actually matters. Err in one
|
|
|
|
|
direction, and I’ll forever be known as the hidebound reactionary who
|
|
|
|
|
failed to recognize some 21st-century Ramanujan. Err in the other
|
|
|
|
|
direction, and I’ll spend my whole life proofreading the work of
|
|
|
|
|
crackpots.
|
|
|
|
|
|
|
|
|
|
A few will chime in: “but if everyone wrote out their proofs in
|
|
|
|
|
computer-checkable form, there’d be no need for this absurd dilemma\!”
|
|
|
|
|
Sure, and if everyone buckled up there’d be fewer serious accidents. Yet
|
|
|
|
|
here’s the bloodied patient, and here we are in the emergency room.
|
|
|
|
|
|
|
|
|
|
In deciding whether to spend time on a paper, obviously the identity of
|
|
|
|
|
the authors plays some role. If Razborov says he proved a superlinear
|
|
|
|
|
circuit lower bound for SAT, the claim on our attention is different
|
|
|
|
|
than if Roofus McLoofus says the same thing. But the danger of elitism
|
|
|
|
|
is obvious here — so in this post, I’ll only be interested in what can
|
|
|
|
|
be inferred from the text itself.
|
|
|
|
|
|
|
|
|
|
Inspired by Sean Carroll’s closely-related [Alternative-Science
|
|
|
|
|
Respectability
|
|
|
|
|
Checklist](http://cosmicvariance.com/2007/06/19/the-alternative-science-respectability-checklist/),
|
|
|
|
|
without further ado I now offer the Ten Signs a Claimed Mathematical
|
|
|
|
|
Breakthrough is Wrong.
|
|
|
|
|
|
|
|
|
|
**1. The authors don’t use TeX.** This simple test (suggested by Dave
|
|
|
|
|
Bacon) already catches at least 60% of wrong mathematical breakthroughs.
|
|
|
|
|
David Deutsch and Lov Grover are among the only known false positives.
|
|
|
|
|
|
|
|
|
|
**2. The authors don’t understand the question.** Maybe they mistake
|
|
|
|
|
NP≠coNP for some claim about psychology or metaphysics. Or maybe they
|
|
|
|
|
solve the Grover problem in O(1) queries, under some notion of quantum
|
|
|
|
|
computing lifted from a magazine article. I’ve seen both.
|
|
|
|
|
|
|
|
|
|
**3. The approach seems to yield something much stronger and maybe even
|
|
|
|
|
false (but the authors never discuss that).** They’ve proved 3SAT takes
|
|
|
|
|
exponential time; their argument would go through just as well for 2SAT.
|
|
|
|
|
|
|
|
|
|
**4. The approach conflicts with a known impossibility result (which the
|
|
|
|
|
authors never mention).** The four months I spent proving the [collision
|
|
|
|
|
lower bound](http://www.scottaaronson.com/papers/collision.pdf) actually
|
|
|
|
|
saved me some time once or twice, when I was able to reject papers
|
|
|
|
|
violating the bound without reading them.
|
|
|
|
|
|
|
|
|
|
**5. The authors themselves switch to weasel words by the end.** The
|
|
|
|
|
abstract says “we show the problem is in P,” but the conclusion contains
|
|
|
|
|
phrases like “seems to work” and “in all cases we have tried.”
|
|
|
|
|
Personally, I happen to be a big fan of heuristic algorithms, honestly
|
|
|
|
|
advertised and experimentally analyzed. But when a “proof” has turned
|
|
|
|
|
into a “plausibility argument” by page 47 — release the hounds\!
|
|
|
|
|
|
|
|
|
|
**6. The paper jumps into technicalities without presenting a new
|
|
|
|
|
idea.** If a famous problem could be solved only by manipulating
|
|
|
|
|
formulas and applying standard reductions, then it’s overwhelmingly
|
|
|
|
|
likely someone would’ve solved it already. The exceptions to this rule
|
|
|
|
|
are interesting precisely because they’re rare (and even with the
|
|
|
|
|
exceptions, a new idea is usually needed to find the right manipulations
|
|
|
|
|
in the first place).
|
|
|
|
|
|
|
|
|
|
**7. The paper doesn’t build on (or in some cases even refer to) any
|
|
|
|
|
previous work.** Math is cumulative. Even Wiles and Perelman had to
|
|
|
|
|
stand on the lemma-encrusted shoulders of giants.
|
|
|
|
|
|
|
|
|
|
**8. The paper wastes lots of space on standard material.** If you’d
|
|
|
|
|
really proved P≠NP, then you wouldn’t start your paper by laboriously
|
|
|
|
|
defining 3SAT, in a manner suggesting your readers might not have heard
|
|
|
|
|
of it.
|
|
|
|
|
|
|
|
|
|
**9. The paper waxes poetic about “practical consequences,” “deep
|
|
|
|
|
philosophical implications,” etc.** Note that most papers make exactly
|
|
|
|
|
the opposite mistake: they never get around to explaining why anyone
|
|
|
|
|
should read them. But when it comes to something like P≠NP, to
|
|
|
|
|
“motivate” your result is to insult your readers’ intelligence.
|
|
|
|
|
|
|
|
|
|
**10. The techniques just seem too wimpy for the problem at hand.** Of
|
|
|
|
|
all ten tests, this is the slipperiest and hardest to apply — but also
|
|
|
|
|
the decisive one in many cases. As an analogy, suppose your friend in
|
|
|
|
|
Boston blindfolded you, drove you around for twenty minutes, then took
|
|
|
|
|
the blindfold off and claimed you were now in Beijing. Yes, you do see
|
|
|
|
|
Chinese signs and pagoda roofs, and no, you can’t immediately disprove
|
|
|
|
|
him — but based on your knowledge of both cars and geography, isn’t it
|
|
|
|
|
more likely you’re just in Chinatown? I know it’s trite, but this is
|
|
|
|
|
exactly how I feel when I see (for example) a paper that uses category
|
|
|
|
|
theory to prove NL≠NP. We start in Boston, we end up in Beijing, and at
|
|
|
|
|
no point is anything resembling an ocean ever crossed.
|
|
|
|
|
|
|
|
|
|
Obviously, these are just some heuristics I’ve found successful in the
|
|
|
|
|
past. (The nice thing about math is that sooner or later the truth comes
|
|
|
|
|
out, and then you know for sure whether your heuristics succeeded.) If a
|
|
|
|
|
paper fails one or more tests (particularly tests 6-10), that doesn’t
|
|
|
|
|
necessarily mean it’s wrong; conversely, if it passes all ten that still
|
|
|
|
|
doesn’t mean it’s right. At some point, there might be nothing left to
|
|
|
|
|
do except to roll up your sleeves, brew some coffee, and tell your
|
|
|
|
|
graduate student to read the paper and report back to you.
|
|
|
|
|
|
|
|
|
|
This entry was posted on Saturday, January 5th, 2008 at 12:17 am and is
|
|
|
|
|
filed under [Mirrored on CSAIL
|
|
|
|
|
Blog](https://www.scottaaronson.com/blog/?cat=30), [Mistake of the
|
|
|
|
|
Week](https://www.scottaaronson.com/blog/?cat=9), [Nerd
|
|
|
|
|
Interest](https://www.scottaaronson.com/blog/?cat=11). You can follow
|
|
|
|
|
any responses to this entry through the
|
|
|
|
|
[RSS 2.0](https://www.scottaaronson.com/blog/?feed=rss2&p=304) feed.
|
|
|
|
|
Both comments and pings are currently closed.
|