Mathematics in the Age of the Turing Machine

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1、MathematicsintheAgeoftheTuringMachineThomasC.Hales?UniversityofPittsburghhales@pitt.eduFig.1.AlanTuring(imagesource[Ima13])arXiv:1302.2898v1[math.HO]12Feb2013“Andwhenitcomestomathematics,youmustrealizethatthisisthehumanmindattheextremelimitofitscapacity.”(H.Robbins)“...soredu

2、cetheuseofthebrainandcalculate!”(E.W.Dijkstra)“Thefactthatabraincandoitseemstosuggestthatthediculties[oftryingwithamachine]maynotreallybesobadastheynowseem.”(A.Turing)?ResearchsupportedinpartbyNSFgrant0804189andtheBenterFoundation.21ComputerCalculation1.1apanoramaofthestatus

3、quoWherestandsthemathematicalendeavor?In2012,manymathematicalutilitiesarereachingconsolidation.Itisanageoflargeaggregatesandlargerepositoriesofmathematics:thearXiv,MathReviews,andeuDML,whichpromisestoaggregatethemanyEuropeanarchivessuchasZentralblattMathandNumdam.Sageaggregat

4、esdozensofmathematicallyorientedcomputerpro-gramsunderasinglePython-scriptedfront-end.BooksalesintheU.S.havebeendroppingforthepastseveralyears.Instead,onlinesourcessuchasWikipediaandMathOverflowarerapidlybecomingstudents’preferredmathreferences.ThePolymathblogorganizesmassivem

5、athematicalcollaborations.Otherblogsorganizepreviouslyisolatedresearchersintonewfieldsofresearch.Theslow,methodicaldeliberationsofrefereesintheoldschoolaregivingway;nowinasinglestroke,Taoblogs,getsfeedback,andpublishes.MachineLearningisinitsascendancy.LogAnswerandWolframAlphaa

6、nswerourelementaryquestionsaboutthequantitativeworld;WatsonourJeopardyquestions.GooglePageranksoursearchesbycalculatingthelargesteigenvalueofthelargestma-trixtheworldhaseverknown.DeepBlueplaysourchessgames.Themillion-dollar-prize-winningPragmaticChaosalgorithmenhancesourNetfli

7、xsearches.Themajorproofassistantsnowcontaintensofthousandsofformalproofsthatarebeingminedforhintsabouthowtoprovethenextgenerationoftheorems.Mathematicalmodelsandalgorithmsrulethequantitativeworld.Withoutappliedmathematics,wewouldbebereftofShor’sfactorizationalgorithmforquantu

8、mcom-puters,Yang-Millstheoriesofstronginteractionsinphysics,invisibi