{"id":237,"date":"2017-02-09T11:47:53","date_gmt":"2017-02-09T10:47:53","guid":{"rendered":"https:\/\/rosetta.vn\/short\/?p=237"},"modified":"2017-02-09T11:47:53","modified_gmt":"2017-02-09T10:47:53","slug":"shtetl-optimized-ten-signs-a-claimed-mathematical-breakthrough-is-wrong","status":"publish","type":"post","link":"https:\/\/rosetta.vn\/short\/2017\/02\/09\/shtetl-optimized-ten-signs-a-claimed-mathematical-breakthrough-is-wrong\/","title":{"rendered":"Shtetl-Optimized \u00bb Ten Signs a Claimed Mathematical Breakthrough is Wrong"},"content":{"rendered":"

I can’t help laughing while reading the examples for these ten signs. This is very enjoyable!<\/p>\n

Note the context: this blogger is a mathematician, frequently receives review requests, here he discussed about requests to read proofs for famous\u00a0(unsolved) mathematical problems<\/strong>. Quote from his blog (posted on Saturday, January 5th, 2008):<\/p>\n

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?<\/p><\/blockquote>\n

For a blogger like me \u2014 whose opinions are both expected immediately and googlable indefinitely \u2014 this question actually matters. Err in one direction, and I\u2019ll forever be known as the hidebound reactionary who failed to recognize some 21st-century Ramanujan. Err in the other direction, and I\u2019ll spend my whole life proofreading the work of crackpots.<\/p><\/blockquote>\n

…<\/p><\/blockquote>\n

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 \u2014 so in this post, I\u2019ll only be interested in what can be inferred from the text itself.<\/p>\n

Inspired by Sean Carroll\u2019s closely-related\u00a0Alternative-Science Respectability Checklist<\/a>, without further ado I now offer the Ten Signs a Claimed Mathematical Breakthrough is Wrong<\/em>.<\/p>\n

1. The authors don\u2019t use TeX.<\/strong> 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.<\/p>\n

2. The authors don\u2019t understand the question.<\/strong> Maybe they mistake NP\u2260coNP 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\u2019ve seen both.<\/p>\n

3. The approach seems to yield something much stronger and maybe even false (but the authors never discuss that).<\/strong> They\u2019ve proved 3SAT takes exponential time; their argument would go through just as well for 2SAT.<\/p>\n

4. The approach conflicts with a known impossibility result (which the authors never mention).<\/strong> The four months I spent proving the collision lower bound<\/a> actually saved me some time once or twice, when I was able to reject papers violating the bound without reading them.<\/p>\n

5. The authors themselves switch to weasel words by the end.<\/strong> The abstract says \u201cwe show the problem is in P,\u201d but the conclusion contains phrases like \u201cseems to work\u201d and \u201cin all cases we have tried.\u201d Personally, I happen to be a big fan of heuristic algorithms, honestly advertised and experimentally analyzed. But when a \u201cproof\u201d has turned into a \u201cplausibility argument\u201d by page 47 \u2014release the hounds!<\/em><\/p>\n

6. The paper jumps into technicalities without presenting a new idea.<\/strong> If a famous problem could be solved only by manipulating formulas and applying standard reductions, then it\u2019s overwhelmingly likely someone would\u2019ve solved it already. The exceptions to this rule are interesting precisely because they\u2019re rare (and even with the exceptions, a new idea is usually needed to find<\/em> the right manipulations in the first place).<\/p>\n

7. The paper doesn\u2019t build on (or in some cases even refer to) any previous work.<\/strong> Math is cumulative. Even Wiles and Perelman had to stand on the lemma-encrusted shoulders of giants.<\/p>\n

8. The paper wastes lots of space on standard material.<\/strong> If you\u2019d really proved P\u2260NP, then you wouldn\u2019t start your paper by laboriously defining 3SAT, in a manner suggesting your readers might not have heard of it.<\/p>\n

9. The paper waxes poetic about \u201cpractical consequences,\u201d \u201cdeep philosophical implications,\u201d etc.<\/strong> 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\u2260NP, to \u201cmotivate\u201d your result is to insult your readers\u2019 intelligence.<\/p>\n

10. The techniques just seem too wimpy for the problem at hand.<\/strong> Of all ten tests, this is the slipperiest and hardest to apply \u2014 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\u2019t immediately disprove him \u2014 but based on your knowledge of both cars and geography, isn\u2019t it more likely you\u2019re just in Chinatown? I know it\u2019s trite, but this is exactly how I feel when I see (for example) a paper that uses category theory to prove NL\u2260NP. We start in Boston, we end up in Beijing, and at no point is anything resembling an ocean ever crossed.<\/p>\n

Obviously, these are just some heuristics I\u2019ve 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\u2019t necessarily mean it\u2019s wrong; conversely, if it passes all ten that still doesn\u2019t mean it\u2019s 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.<\/p><\/blockquote>\n

Source: Shtetl-Optimized \u00bb Blog Archive \u00bb Ten Signs a Claimed Mathematical Breakthrough is Wrong<\/a><\/em><\/p>\n","protected":false},"excerpt":{"rendered":"

I can’t help laughing while reading the examples for these ten signs. This is very enjoyable! Note the context: this blogger is a mathematician, frequently receives review requests, here he discussed about requests to read proofs for famous\u00a0(unsolved) mathematical problems.<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_mi_skip_tracking":false},"categories":[1],"tags":[164,165,167,166],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p8jhJx-3P","_links":{"self":[{"href":"https:\/\/rosetta.vn\/short\/wp-json\/wp\/v2\/posts\/237"}],"collection":[{"href":"https:\/\/rosetta.vn\/short\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/rosetta.vn\/short\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/rosetta.vn\/short\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/rosetta.vn\/short\/wp-json\/wp\/v2\/comments?post=237"}],"version-history":[{"count":0,"href":"https:\/\/rosetta.vn\/short\/wp-json\/wp\/v2\/posts\/237\/revisions"}],"wp:attachment":[{"href":"https:\/\/rosetta.vn\/short\/wp-json\/wp\/v2\/media?parent=237"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/rosetta.vn\/short\/wp-json\/wp\/v2\/categories?post=237"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/rosetta.vn\/short\/wp-json\/wp\/v2\/tags?post=237"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}