There\u2019s an absolute\u00adly fas\u00adci\u00adnat\u00ading series of respons\u00ades by lead\u00ading sci\u00aden\u00adtists and intel\u00adlec\u00adtu\u00adals to the ques\u00adtion, \u201cWHAT DO YOU BELIEVE IS TRUE EVEN THOUGH YOU CANNOT PROVE IT?\u201d<\/p>\n
Like this fun\u00adny dia\u00adlog<\/a> by Stan\u00adford prof Leonard Susskind.<\/p>\n Con\u00adver\u00adsa\u00adtion With a Slow Stu\u00addent<\/p>\n Stu\u00addent: Hi Prof. I\u2019ve got a prob\u00adlem. I decid\u00aded to do a lit\u00adtle prob\u00ada\u00adbil\u00adi\u00adty experiment\u2014you know, coin flipping\u2014and check some of the stuff you taught us. But it did\u00adn\u2019t work.<\/p>\n Pro\u00adfes\u00adsor: Well I\u2019m glad to hear that you\u2019re inter\u00adest\u00aded. What did you do?<\/p>\n Stu\u00addent: I flipped this coin 1,000 times. You remem\u00adber, you taught us that the prob\u00ada\u00adbil\u00adi\u00adty to flip heads is one half. I fig\u00adured that meant that if I flip 1,000 times I ought to get 500 heads. But it did\u00adn\u2019t work. I got 513. What\u2019s wrong?<\/p>\n Pro\u00adfes\u00adsor: Yeah, but you for\u00adgot about the mar\u00adgin of error. If you flip a cer\u00adtain num\u00adber of times then the mar\u00adgin of error is about the square root of the num\u00adber of flips. For 1,000 flips the mar\u00adgin of error is about 30. So you were with\u00adin the mar\u00adgin of error.<\/p>\n Stu\u00addent: Ah, now I get if. Every time I flip 1,000 times I will always get some\u00adthing between 970 and 1,030 heads. Every sin\u00adgle time! Wow, now that\u2019s a fact I can count on.<\/p>\n Pro\u00adfes\u00adsor: No, no! What it means is that you will prob\u00ada\u00adbly get between 970 and 1,030.<\/p>\n Stu\u00addent: You mean I could get 200 heads? Or 850 heads? Or even all heads?<\/p>\n Pro\u00adfes\u00adsor: Prob\u00ada\u00adbly not.<\/p>\n Stu\u00addent: Maybe the prob\u00adlem is that I did\u00adn\u2019t make enough flips. Should I go home and try it 1,000,000 times? Will it work bet\u00adter?<\/p>\n Pro\u00adfes\u00adsor: Prob\u00ada\u00adbly.<\/p>\n Stu\u00addent: Aw come on Prof. Tell me some\u00adthing I can trust. You keep telling me what prob\u00ada\u00adbly means by giv\u00ading me more prob\u00ada\u00adblies. Tell me what prob\u00ada\u00adbil\u00adi\u00adty means with\u00adout using the word prob\u00ada\u00adbly.<\/p>\n Pro\u00adfes\u00adsor: Hmmm. Well how about this: It means I would be sur\u00adprised if the answer were out\u00adside the mar\u00adgin of error.<\/p>\n Stu\u00addent: My god! You mean all that stuff you taught us about sta\u00adtis\u00adti\u00adcal mechan\u00adics and quan\u00adtum mechan\u00adics and math\u00ade\u00admat\u00adi\u00adcal prob\u00ada\u00adbil\u00adi\u00adty: all it means is that you\u2019d per\u00adson\u00adal\u00adly be sur\u00adprised if it did\u00adn\u2019t work?<\/p>\n Pro\u00adfes\u00adsor: Well, uh\u2026\n<\/p><\/blockquote>\n Or the some\u00adwhat more heady essay on the lim\u00adit\u00aded util\u00adi\u00adty of for\u00admal proofs by Stan\u00adford math\u00ademeti\u00adcian Kei\u00adth Devlin<\/a><\/p>\n \nBefore we can answer this ques\u00adtion we need to agree what we mean by proof. (This is one of the rea\u00adsons why its good to have math\u00ade\u00admati\u00adcians around. We like to begin by giv\u00ading pre\u00adcise def\u00adi\u00adn\u00adi\u00adtions of what we are going to talk about, a pedan\u00adtic ten\u00adden\u00adcy that some\u00adtimes dri\u00adves our physi\u00adcist and engi\u00adneer\u00ading col\u00adleagues crazy.) For instance, fol\u00adlow\u00ading Descartes, I can prove to myself that I exist, but I can\u2019t prove it to any\u00adone else. Even to those who know me well there is always the pos\u00adsi\u00adbil\u00adi\u00adty, how\u00adev\u00ader remote, that I am mere\u00adly a fig\u00adment of their imag\u00adi\u00adna\u00adtion. If it\u2019s rock sol\u00adid cer\u00adtain\u00adty you want from a proof, there\u2019s almost noth\u00ading beyond our own exis\u00adtence (what\u00adev\u00ader that means and what\u00adev\u00ader we exist as) that we can prove to our\u00adselves, and noth\u00ading at all we can prove to any\u00adone else.<\/p>\n Math\u00ade\u00admat\u00adi\u00adcal proof is gen\u00ader\u00adal\u00adly regard\u00aded as the most cer\u00adtain form of proof there is, and in the days when Euclid was writ\u00ading his great geom\u00ade\u00adtry text Ele\u00adments that was sure\u00adly true in an ide\u00adal sense. But many of the proofs of geo\u00admet\u00adric the\u00ado\u00adrems Euclid gave were sub\u00adse\u00adquent\u00adly found out to be incorrect\u2014David Hilbert cor\u00adrect\u00aded many of them in the late nine\u00adteenth cen\u00adtu\u00adry, after cen\u00adturies of math\u00ade\u00admati\u00adcians had believed them and passed them on to their students\u2014so even in the case of a ten line proof in geom\u00ade\u00adtry it can be hard to tell right from wrong.<\/p>\n When you look at some of the proofs that have been devel\u00adoped in the last fifty years or so, using incred\u00adi\u00adbly com\u00adpli\u00adcat\u00aded rea\u00adson\u00ading that can stretch into hun\u00addreds of pages or more, cer\u00adtain\u00adty is even hard\u00ader to main\u00adtain. Most math\u00ade\u00admati\u00adcians (includ\u00ading me) believe that Andrew Wiles proved Fer\u00admat\u2019s Last The\u00ado\u00adrem in 1994, but did he real\u00adly? (I believe it because the experts in that branch of math\u00ade\u00admat\u00adics tell me they do.)<\/p>\n In late 2002, the Russ\u00adian math\u00ade\u00admati\u00adcian Grig\u00adori Perel\u00adman post\u00aded on the Inter\u00adnet what he claimed was an out\u00adline for a proof of the Poin\u00adcare Con\u00adjec\u00adture, a famous, cen\u00adtu\u00adry old prob\u00adlem of the branch of math\u00ade\u00admat\u00adics known as topol\u00ado\u00adgy. After exam\u00adin\u00ading the argu\u00adment for two years now, math\u00ade\u00admati\u00adcians are still unsure whether it is right or not. (They think it \u201cprob\u00ada\u00adbly is.\u201d)<\/p>\n Or con\u00adsid\u00ader Thomas Hales, who has been wait\u00ading for six years to hear if the math\u00ade\u00admat\u00adi\u00adcal com\u00admu\u00adni\u00adty accepts his 1998 proof of astronomer Johannes Keplers 360-year-old con\u00adjec\u00adture that the most effi\u00adcient way to pack equal sized spheres (such as can\u00adnon\u00adballs on a ship, which is how the ques\u00adtion arose) is to stack them in the famil\u00adiar pyra\u00admid-like fash\u00adion that green\u00adgro\u00adcers use to stack oranges on a counter. After exam\u00adin\u00ading Hales\u2019 argu\u00adment (part of which was car\u00adried out by com\u00adput\u00ader) for five years, in spring of 2003 a pan\u00adel of world experts declared that, where\u00adas they had not found any irrepara\u00adble error in the proof, they were still not sure it was cor\u00adrect.<\/p>\n With the idea of proof so shaky\u2014in practice\u2014even in math\u00ade\u00admat\u00adics, answer\u00ading this year\u2019s Edge ques\u00adtion becomes a tricky busi\u00adness. The best we can do is come up with some\u00adthing that we believe but can\u00adnot prove to our own sat\u00adis\u00adfac\u00adtion. Oth\u00aders will accept or reject what we say depend\u00ading on how much cre\u00addence they give us as a sci\u00aden\u00adtist, philoso\u00adpher, or what\u00adev\u00ader, gen\u00ader\u00adal\u00adly bas\u00ading that deci\u00adsion on our sci\u00aden\u00adtif\u00adic rep\u00adu\u00adta\u00adtion and record of pre\u00advi\u00adous work. At times it can be hard to avoid the whole thing degen\u00ader\u00adat\u00ading into a slang\u00ading match. For instance, I hap\u00adpen to believe, firm\u00adly, that sta\u00adples of pop\u00adu\u00adlar-sci\u00adence-books and breath\u00adless TV-spe\u00adcials such as ESP and mor\u00adphic res\u00ado\u00adnance are com\u00adplete non\u00adsense, but I can\u2019t prove they are false. (Nor, despite their repeat\u00aded claims to the con\u00adtrary, have the pro\u00adpo\u00adnents of those crack\u00adpot the\u00ado\u00adries proved they are true, or even worth seri\u00adous study, and if they want the sci\u00aden\u00adtif\u00adic com\u00admu\u00adni\u00adty to take them seri\u00adous\u00adly then the onus if very much on them to make a strong case, which they have so far failed to do.)<\/p>\n Once you rec\u00adog\u00adnize that proof is, in prac\u00adti\u00adcal terms, an unachiev\u00adable ide\u00adal, even the old math\u00ade\u00admati\u00adcians stand\u00adby of G\u00cfdel\u2019s Incom\u00adplete\u00adness The\u00ado\u00adrem (which on first blush would allow me to answer the Edge ques\u00adtion with a state\u00adment of my belief that arith\u00admetic is free of inter\u00adnal con\u00adtra\u00addic\u00adtions) is no longer avail\u00adable. G\u00cfdel\u2019s the\u00ado\u00adrem showed that you can\u00adnot prove an axiomat\u00adi\u00adcal\u00adly based the\u00ado\u00adry like arith\u00admetic is free of con\u00adtra\u00addic\u00adtion with\u00adin that the\u00ado\u00adry itself. But that does\u00adn\u2019t mean you can\u2019t prove it in some larg\u00ader, rich\u00ader the\u00ado\u00adry. In fact, in the stan\u00addard axiomat\u00adic set the\u00ado\u00adry, you can prove arith\u00admetic is free of con\u00adtra\u00addic\u00adtions. And per\u00adson\u00adal\u00adly, I buy that proof. For me, as a liv\u00ading, human math\u00ade\u00admati\u00adcian, the con\u00adsis\u00adten\u00adcy of arith\u00admetic has been proved\u2014to my com\u00adplete sat\u00adis\u00adfac\u00adtion.<\/p>\n So to answer the Edge ques\u00adtion, you have to take a com\u00admon sense approach to proof\u2014in this case proof being, I sup\u00adpose, an argu\u00adment that would con\u00advince the intel\u00adli\u00adgent, pro\u00adfes\u00adsion\u00adal\u00adly skep\u00adti\u00adcal, trained expert in the appro\u00adpri\u00adate field. In that spir\u00adit, I could give any num\u00adber of spe\u00adcif\u00adic math\u00ade\u00admat\u00adi\u00adcal prob\u00adlems that I believe are true but can\u00adnot prove, start\u00ading with the famous Rie\u00admann Hypoth\u00ade\u00adsis. But I think I can be of more use by using my math\u00ade\u00admati\u00adcian\u2019s per\u00adspec\u00adtive to point out the uncer\u00adtain\u00adties in the idea of proof. Which I believe (but can\u00adnot prove) I have.\n<\/p><\/blockquote>\n Or Seth Lloyd from MIT<\/a>:<\/p>\n \nI can\u00adnot prove that elec\u00adtrons exist, but I believe fer\u00advent\u00adly in their exis\u00adtence. And if you don\u2019t believe in them, I have a high volt\u00adage cat\u00adtle prod I\u2019m will\u00ading to apply as an argu\u00adment on their behalf. Elec\u00adtrons speak for them\u00adselves.\n<\/p><\/blockquote>\n And the appar\u00adent\u00adly lone the\u00adist, David Myers<\/a>:<\/p>\n \nAs a Chris\u00adt\u00adian monothe\u00adist, I start with two unproven axioms:<\/p>\n 1. There is a God.<\/p>\n 2. It\u2019s not me (and it\u2019s also not you).<\/p>\n Togeth\u00ader, these axioms imply my surest con\u00advic\u00adtion: that some of my beliefs (and yours) con\u00adtain error. We are, from dust to dust, finite and fal\u00adli\u00adble. We have dig\u00adni\u00adty but not deity.<\/p>\n And that is why I fur\u00adther believe that we should<\/p>\n a) hold all our unproven beliefs with a cer\u00adtain ten\u00adta\u00adtive\u00adness (except for this one!),<\/p>\n b) assess oth\u00aders\u2019 ideas with open-mind\u00aded skep\u00adti\u00adcism, and<\/p>\n c) freely pur\u00adsue truth aid\u00aded by obser\u00adva\u00adtion and exper\u00adi\u00adment.<\/p>\n This mix of faith-based humil\u00adi\u00adty and skep\u00adti\u00adcism helped fuel the begin\u00adnings of mod\u00adern sci\u00adence, and it has informed my own research and sci\u00adence writ\u00ading. The whole truth can\u00adnot be found mere\u00adly by search\u00ading our own minds, for there is not enough there. So we also put our ideas to the test. If they sur\u00advive, so much the bet\u00adter for them; if not, so much the worse.<\/p>\n With\u00adin psy\u00adchol\u00ado\u00adgy, this \u201cever-reform\u00ading\u201d process has many times changed my mind, lead\u00ading me now to believe, for exam\u00adple, that new\u00adborns are not so dumb, that elec\u00adtro con\u00advul\u00adsive ther\u00ada\u00adpy often alle\u00advi\u00adates intractable depres\u00adsion, that Amer\u00adi\u00adca\u2019s eco\u00adnom\u00adic growth has not improved our morale, that the auto\u00admat\u00adic uncon\u00adscious mind dwarfs the con\u00adscious mind, that trau\u00admat\u00adic expe\u00adri\u00adences rarely get repressed, that most folks don\u2019t suf\u00adfer low self-esteem, and that sex\u00adu\u00adal ori\u00aden\u00adta\u00adtion is not a choice.\n<\/p><\/blockquote>\n Any\u00adway, inter\u00adest\u00ading stuff.<\/p>\n","protected":false},"excerpt":{"rendered":" There\u2019s an absolute\u00adly fas\u00adci\u00adnat\u00ading series of respons\u00ades by lead\u00ading sci\u00aden\u00adtists and intel\u00adlec\u00adtu\u00adals to the ques\u00adtion, \u201cWHAT DO YOU BELIEVE IS TRUE EVEN THOUGH YOU CANNOT PROVE IT?\u201d Like this fun\u00adny dia\u00adlog by Stan\u00adford prof Leonard Susskind. Con\u00adver\u00adsa\u00adtion With a Slow Stu\u00addent Stu\u00addent: Hi Prof. I\u2019ve got a prob\u00adlem. I decid\u00aded to do a lit\u00adtle prob\u00ada\u00adbil\u00adi\u00adty \u2026 Con\u00adtin\u00adue read\u00ading \u201cInsights From Bril\u00adliant Peo\u00adple\u201d<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"wp_typography_post_enhancements_disabled":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"jetpack_post_was_ever_published":false},"categories":[1],"tags":[],"class_list":["post-319","post","type-post","status-publish","format-standard","hentry","category-of-random-interest"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p6Ded-59","jetpack_sharing_enabled":true,"jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/glenandpaula.com\/wordpress\/wp-json\/wp\/v2\/posts\/319","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/glenandpaula.com\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/glenandpaula.com\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/glenandpaula.com\/wordpress\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/glenandpaula.com\/wordpress\/wp-json\/wp\/v2\/comments?post=319"}],"version-history":[{"count":0,"href":"https:\/\/glenandpaula.com\/wordpress\/wp-json\/wp\/v2\/posts\/319\/revisions"}],"wp:attachment":[{"href":"https:\/\/glenandpaula.com\/wordpress\/wp-json\/wp\/v2\/media?parent=319"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/glenandpaula.com\/wordpress\/wp-json\/wp\/v2\/categories?post=319"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/glenandpaula.com\/wordpress\/wp-json\/wp\/v2\/tags?post=319"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}\n