---
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
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---
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:

> 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.

Three commenters wrote in to say the paper looked good. Then the author
found a bug and retracted it.

**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
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