IdeaMakers:PersonalPerspectivesontheLives&IdeasofSomeNotablePeopleCopyright©2016StephenWolfram,LLCWolframMedia,Inc.|wolfram-media.comISBN978-1-57955003-5(hardback)ISBN978-1-57955005-9(ebook)Biography/Science

LibraryofCongressCataloging-in-PublicationDataWolfram,Stephen,author.Ideamakers:personalperspectivesonthelives&ideasofsomenotablepeoplebyStephenWolfram.Firstedition.|Champaign:WolframMedia,Inc.,[2016]LCCN2016025380(print)|LCCN2016026486(ebook)ISBN9781579550035(hardcover:acid-freepaper)ISBN9781579550059(ebook)|ISBN9781579550110(ePub)ISBN9781579550059(kindle)LCSH:Scientists—Biography.|Science—History.LCCQ141.W6785622016(print)|LCCQ141(ebook)DDC509.2/2—dc23LCrecordavailableathttps://lccn.loc.gov/2016025380

Sourcesforphotosandarchivalmaterialsthatarenotfromtheauthor’scollectionorinthepublicdomain:pp.45,66,67:TheCarlH.PforzheimerCollectionofShelleyandHisCircle,TheNewYorkPublicLibrary;pp.46,55,56,63,65,67,79:AdditionalManuscriptsCollection,CharlesBabbagePapers,BritishLibrary;p.49:NationalPortraitGallery;pp.50,60:MuseumoftheHistoryofScience,Oxford;pp.52,53,85,86:ScienceMuseum/Science&SocietyPictureLibrary;p.54:ThePowerHouseMuseum,Sydney;p.68:LordLytton,TheBodleianLibrary;pp.99–120:Leibniz-Archiv/LeibnizResearchCenterHannover,StateLibraryofLowerSaxony;p.138:AlisaBokulich;p.167–170,178:CambridgeUniversityLibrary;pp.192,193:TrinityCollegeLibrary;p.194:TataInstituteofFundamentalResearch

PrintedbyFriesens,Manitoba,Canada. Acid-freepaper.Firstedition.

Preface

I’ve spentmost ofmy life working hard to build the futurewith science andtechnology. But two of my other great interests are history and people. Thisbook is a collection of essays I’ve written that indulge those interests. All ofthemare in the formof personal perspectives onpeople—describing frommypointofviewthestoriesoftheirlivesandoftheideastheycreated.I’vewrittendifferentessaysfordifferentreasons:sometimestocommemorate

ahistoricalanniversary,sometimesbecauseofacurrentevent,andsometimes—unfortunately—because someone justdied.Thepeople I’vewrittenabout spanthreecenturiesintime—andrangefromtheveryfamoustothelittle-known.Allofthemhadintereststhatintersectinsomewayoranotherwithmyown.Butit’sendeduparathereclecticlist—that’sgivenmetheopportunitytoexploreawiderangeofverydifferentlivesandideas.WhenIwasyounger,Ireallydidn’tpaymuchattentiontohistory.Butasthe

decades have gone by, and I’ve seen so many different things develop, I’vebecome progressivelymore interested in history—and inwhat it can teach usaboutthepatternofhowthingswork.AndI’velearnedthatdecodingtheactualfactsandpathofhistory—likesomanyotherareas—isafascinatingintellectualprocess.There’sa stereotype that someone focusedonscienceand technologywon’t

beinterestedinpeople.Butthat’snotme.I’vealwaysbeeninterestedinpeople.I’vebeen fortunateover thecourseofmy life toget toknowavery largeanddiversesetofthem.AndasI’vegrownmycompanyoverthepastthreedecadesI’vehadthepleasureofworkingwithmanywonderfulindividuals.Ialwaysliketogivehelpandadvice.ButI’malsofascinatedjusttowatchthetrajectoriesofpeople’slives—andtoseehowpeopleendupdoingthethingstheydo.It’sbeengreattopersonallywitnesssomanylifetrajectoriesoverthepasthalf

century.AndinthisbookI’vewrittenaboutafewofthem.ButI’vealsobeeninterestedtolearnaboutthelifetrajectoriesofthosefromthemoredistantpast.UsuallyIknowquitea lotabout theendof thestory: the legacyof theirworkandideas.ButIfinditfascinatingtoseehowthesethingscametobe—andhowthepathsofpeople’slivesledtowhattheydid.Partofmyinterestispurelyintellectual.Butpartofitismorepractical—and

more selfish.What can I learn fromhistorical examples abouthow things I’minvolvedinnowwillworkout?HowcanIusepeoplefromthepastasmodelsfor people I know now?What can I learn for my own life from what these

peopledidintheirlives?To be clear: this book is not a systematic analysis of great thinkers and

creators through history. It is an eclectic collection of essays about particularpeoplewho forone reasonor anotherwere topical forme towrite about. I’vetried to give both a sketch of each person’s life in its historical context and adescriptionof their ideas—and then I’ve tried to relate those ideas tomyownideas,andtothelatestscienceandtechnology.In the process of writing these essays I’ve ended up doing a considerable

amountoforiginal research.When theessaysareaboutpeople I’vepersonallyknown, I’ve been able to drawon interactions I hadwith them, aswell as onmaterialI’vepersonallyarchived.Forotherpeople,I’vetriedwhenit’spossibleto seekout individualswhoknew them—and in all cases I’veworkedhard tofindoriginaldocumentsandotherprimarysources.Manypeopleandinstitutionshave been very forthcoming with their help—and also it’s been immenselyhelpful that inmodern times somanyhistoricaldocumentshavebeen scannedandputontheweb.Butwithallofthis,I’mstillconstantlystruckbyhowharditistodohistory.

Sooftenthere’sbeensomestoryoranalysisthatpeoplerepeatallthetime.Butsomehow something about it hasn’t quite rung true with me. So I’ve gonediggingtotrytofindouttherealstory.Occasionallyonejustcan’t tellwhatitwas.ButatleastforthepeopleI’vewrittenaboutinthisbook,thereareusuallyenough records and documents—or actual people to talk to—that one caneventuallyfigureitout.Mystrategy is tokeepondiggingandgetting informationuntil thingsmake

sensetome,basedonmyknowledgeofpeopleandsituationsthataresomehowsimilartowhatI’mstudying.It’scertainlyhelpedthatinmyownlifeI’veseenallsortsofideasandotherthingsdevelopoverthecourseofyears—whichhasgivenmesomeintuitionabouthowsuchthingswork.Andoneoftheimportantlessonsof this is thathoweverbrilliantonemaybe,every idea is the resultofsomeprogressionorpath—oftenhard-won. If there seems tobea jump in thestory—amissinglink—thenthat’sjustbecauseonehasn’tfigureditout.AndIalways try to go on until there aren’tmysteries anymore, and everything thathappenedmakessenseinthecontextofmyownexperiences.Sohavingtracedthelivesofquiteafewnotablepeople,whathaveIlearned?

Perhaps the clearest lesson is that serious ideas that people have are alwaysdeeplyentwinedwiththetrajectoriesoftheirlives.Thatisnottosaythatpeoplealwayslivetheparadigmstheycreate—infact,often,almostparadoxically,theydon’t.But ideas arise out of the context of people’s lives. Indeed,more oftenthan not, it’s a very practical situation that someone finds themselves in that

leadsthemtocreatesomestrong,new,abstractidea.Whenhistoryiswritten,allthat’susuallysaidisthatso-and-socameupwith

such-and-such an idea.But there’s alwaysmore to it: there’s always a humanstorybehindit.Sometimesthatstoryhelpsilluminatetheabstractidea.Butmoreoften, it instead gives us insight about how to turn some human situation orpracticalissueintosomethingintellectual—andperhapssomethingthatwillliveon,abstractly,longafterthepersonwhocreateditisgone.This book is the first time I’ve systematically collected what I’ve written

aboutpeople.I’vewrittenmoregenerallyabouthistoryinafewotherplaces—forexampleinthehundredpagesorsoofdetailedhistoricalnotesatthebackofmy2002bookANewKindofScience.Ihappenedtostartmycareeryoung,somyearlycolleagueswereoftenmucholderthanme—makingitdemographicallylikelythattheremaybemanyobituariesformetowrite.ButsomehowIfinditcathartictoreflectonhowaparticularlifeaddedstones—largeorsmall—tothegreattowerthatrepresentsourcivilizationanditsachievements.IwishIcouldhavepersonallyknownallthepeopleIwriteaboutinthisbook.

Butforthosewhodiedlongagoitfeelslikeagoodsecondbesttoreadsomanydocuments theywrote—andsomehow toget inandunderstand their lives.Mygreatest personal passion remains trying to build the future. But I hope thatthroughunderstandingthepastImaybeabletodoitalittlebetter—andperhapshelp build it on a more informed and solid basis. For now, though, I’m justhappy tohavebeenable tospenda little timeonsomeremarkablepeopleandtheirremarkablelives—andIhopethatwe’llallbeabletolearnsomethingfromthem.

Forexternallinksandreferences,seestephenwolfram.com,ortheebookversionofthisbook.

RichardFeynmanApril20,2005*

IfirstmetRichardFeynmanwhenIwas18,andhewas60.Andoverthecourseoftenyears,IthinkIgottoknowhimfairlywell.FirstwhenIwasinthephysicsgroup at Caltech. And then later when we both consulted for a once-thrivingBostoncompanycalledThinkingMachinesCorporation.Iactuallydon’t think I’veever talkedaboutFeynman inpublicbefore.And

there’sreallysomuchtosay,I’mnotsurewheretostart.But if there’s one moment that summarizes Richard Feynman and my

relationship with him, perhaps it’s this. It was probably 1982. I’d been atFeynman’shouse,andourconversationhad turned tosomekindofunpleasantsituationthatwasgoingon.Iwasabouttoleave.AndFeynmanstoppedmeandsaid,“Youknow,youandIareverylucky.Becausewhateverelseisgoingon,we’vealwaysgotourphysics.”Feynmanloveddoingphysics.Ithinkwhathelovedmostwastheprocessof

it.Ofcalculating.Offiguringthingsout.Itdidn’tseemtomattertohimsomuchifwhatcameoutwasbigandimportant.Oresotericandweird.Whatmatteredtohimwastheprocessoffindingit.Andhewasoftenquitecompetitiveaboutit.Somescientists(myselfprobablyincluded)aredrivenbytheambitiontobuild

grandintellectualedifices.IthinkFeynman—atleastintheyearsIknewhim—wasmuchmore driven by the pure pleasure of actually doing the science.Heseemedtolikebesttospendhistimefiguringthingsout,andcalculating.Andhewasagreatcalculator.Allaroundperhapsthebesthumancalculatorthere’severbeen.Here’sapagefrommyfiles:quintessentialFeynman.CalculatingaFeynman

diagram:

It’skindofinterestingtolookat.Hisstylewasalwaysverymuchthesame.Healways justusedregularcalculusand things.Essentiallynineteenth-centurymathematics.Henevertrustedmuchelse.Butwhereveronecouldgowiththat,Feynmancouldgo.Likenooneelse.I always found it incredible.Hewould startwith someproblem, and fill up

pages with calculations. And at the end of it, he would actually get the rightanswer!Butheusuallywasn’tsatisfiedwiththat.Oncehe’dgottentheanswer,he’dgobackandtrytofigureoutwhyitwasobvious.Andoftenhe’dcomeupwithoneof thoseclassicFeynmanstraightforward-soundingexplanations.Andhe’dnevertellpeopleaboutallthecalculationsbehindit.Sometimesitwaskindof a game for him: having people be flabbergasted by his seemingly instantphysical intuition, not knowing that really it was based on some long, hardcalculationhe’ddone.He always had a fantastic formal intuition about the innards of his

calculations.Knowingwhat kind of result some integral should have,whethersomespecialcaseshouldmatter,andsoon.Andhewasalwaystryingtosharpenhisintuition.You know, I remember a time—it must have been the summer of 1985—

whenI’djustdiscoveredathingcalledrule30.That’sprobablymyownall-timefavorite scientificdiscovery.And that’swhat launcheda lot of thewholenewkindofsciencethatI’vespent20yearsbuilding(andwroteaboutinmybookANewKindofScience).

Well,FeynmanandIwerebothvisitingBoston,andwe’dspentmuchofanafternoon talking about rule 30. About how it manages to go from that littleblacksquareat thetoptomakeall thiscomplicatedstuff.Andaboutwhat thatmeansforphysicsandsoon.

Well,we’d justbeencrawlingaround the floor—withhelp fromsomeotherpeople—tryingtousemeterrulestomeasuresomefeatureofagiantprintoutofit.AndFeynman tookmeaside, ratherconspiratorially,andsaid,“Look, I justwant toaskyouone thing:howdidyouknowrule30woulddoall this crazystuff?” “You know me,” I said. “I didn’t. I just had a computer try all thepossible rules.And I found it.” “Ah,”he said, “now I feelmuchbetter. Iwasworriedyouhadsomewaytofigureitout.”FeynmanandItalkedabunchmoreaboutrule30.Hereallywantedtogetan

intuitionforhowitworked.Hetriedbashingitwithallhisusualtools.Likehetriedtoworkoutwhattheslopeofthelinebetweenorderandchaosis.Andhecalculated.Usingallhisusualcalculusandsoon.HeandhissonCarlevenspentabunchoftimetryingtocrackrule30usingacomputer.

And one day he callsme and says, “OK,Wolfram, I can’t crack it. I thinkyou’reontosomething.”Whichwasveryencouraging.FeynmanandItriedtoworktogetheronabunchofthingsovertheyears.On

quantumcomputersbeforeanyonehadeverheardofthose.Ontryingtomakeachip thatwould generate perfect physical randomness—or eventually showingthat that wasn’t possible. On whether all the computation needed to evaluateFeynmandiagramsreallywasnecessary.Onwhetheritwasacoincidenceornotthatthere’sane–HtinstatisticalmechanicsandaneiHtinquantummechanics.Onwhatthesimplestessentialphenomenonofquantummechanicsreallyis.I remember oftenwhenwewere both consulting for ThinkingMachines in

Boston,Feynmanwouldsay,“Let’shideawayanddosomephysics.”Thiswasatypicalscenario.Yes,Ithinkwethoughtnobodywasnoticingthatwewereoffatthebackofapressconferenceaboutanewcomputersystemtalkingaboutthenonlinear sigmamodel. Typically, Feynmanwould do some calculation.Withmecontinuallyprotestingthatweshouldjustgoanduseacomputer.EventuallyI’ddothat.ThenI’dgetsomeresults.Andhe’dgetsomeresults.Andthenwe’dhaveanargumentaboutwhoseintuitionabouttheresultswasbetter.

Ishouldsay,by theway, that itwasn’t thatFeynmandidn’t likecomputers.He even had gone to some trouble to get an early Commodore PET personalcomputer,andenjoyeddoingthingswithit.Andin1979,whenIstartedworkingontheforerunnerofwhatwouldbecomeMathematica,hewasvery interested.We talked a lot about how it should work. He was keen to explain hismethodologies for solving problems: for doing integrals, for notation, fororganizinghiswork.Ievenmanagedtogethimalittleinterestedintheproblemof language design. Though I don’t think there’s anything directly fromFeynman that has survived inMathematica. But his favorite integrals we cancertainlydo.

You know, it was sometimes a bit of a liability having Feynman involved.LikewhenIwasworkingonSMP—theforerunnerofMathematica—Iorganizedsomeseminarsbypeoplewho’dworkedonothersystems.AndFeynmanusedtocome. And one day a chap from a well-known computer science departmentcame to speak. I thinkhewas a little tired, andhe endedupgivingwhatwasadmittedly not a good talk. And it degenerated at some point into essentiallytellingpunsaboutthenameofthesystemthey’dbuilt.Well,Feynmangotmoreandmoreannoyed.Andeventuallystoodupandgaveawholespeechabouthow“Ifthisiswhatcomputerscienceisabout,it’sallnonsense....”IthinkthechapwhogavethetalkthoughtI’dputFeynmanuptothis.Andhashatedmefor25years...You know, in many ways, Feynman was a loner. Other than for social

reasons, he really didn’t like to work with other people. And he was mostlyinterested in his ownwork.He didn’t read or listen toomuch; hewanted thepleasure of doing things himself. He did used to come to physics seminars,though. Although he had rather a habit of using them as problem-solvingexercises. And he wasn’t always incredibly sensitive to the speakers. In fact,therewasaperiodoftimewhenIorganizedthetheoreticalphysicsseminarsatCaltech.Andheofteneggedmeon tocompete to find fatal flaws inwhat thespeakerswere saying.Which led to somevery unfortunate incidents.But alsoledtosomeinterestingscience.OnethingaboutFeynmanisthathewenttosometroubletoarrangehislifeso

thathewasn’tparticularlybusy—andsohecouldjustworkonwhathefeltlike.Usually he had a good supply of problems. Though sometimes his long-timeassistant would say: “You should go and talk to him. Or he’s going to startworkingontryingtodecodeMayanhieroglyphsagain.”Healwayscultivatedanair of irresponsibility. Though I would say more towards institutions thanpeople.And I was certainly very grateful that he spent considerable time trying to

givemeadvice—evenifIwasnotalwaysgreatattakingit.Oneofthethingsheoftensaidwasthat“peaceofmindisthemostimportantprerequisiteforcreativework.”Andhethoughtoneshoulddoeverythingonecouldtoachievethat.Andhe thought thatmeant, among other things, that one should always stay awayfromanythingworldly,likemanagement.Feynman himself, of course, spent his life in academia—though I think he

foundmostacademicsratherdull.AndIdon’tthinkhelikedtheirstandardviewof theoutsideworldverymuch.Andhehimselfoftenpreferredmoreunusualfolk.

Quite often he’d introduce me to the odd characters who’d visit him. Irememberonceweendeduphavingdinnerwiththerathercharismaticfounderofasemi-cultcalledEST.Itwasacuriousdinner.Andafterwards,FeynmanandItalkedforhoursaboutleadership.AboutleaderslikeRobertOppenheimer.AndBrighamYoung.Hewasfascinated—andmystified—bywhatitisthatletsgreatleadersleadpeopletodoincrediblethings.Hewantedtogetanintuitionforthat.Youknow, it’s funny.ForallFeynman’s independence,hewas surprisingly

diligent. I rememberoncehewaspreparingsomefairlyminorconference talk.Hewasquiteconcernedaboutit. Isaid,“You’reagreatspeaker;whatareyouworrying about?”He said, “Yes, everyone thinks I’m a great speaker. So thatmeans they expect more from me.” And in fact, sometimes it was thosethrowawayconferencetalksthathaveendedupbeingsomeofFeynman’smostpopularpieces.Onnanotechnology.Orfoundationsofquantumtheory.Orotherthings.You know, Feynman spent most of his life working on prominent current

problems inphysics.Buthewasaconfidentproblemsolver.Andoccasionallyhe would venture outside, bringing his “one can solve any problem just bythinkingaboutit”attitudewithhim.Itdidhavesomelimits,though.Ithinkhenever really believed it applied to human affairs, for example. Likewhenwewere both consulting for Thinking Machines in Boston, I would always bejumpingupanddownabouthowif themanagementof thecompanydidn’tdothisorthat,theywouldfail.Hewouldjustsay,“Whydon’tyouletthesepeoplerun theircompany;wecan’t figureout thiskindofstuff.”Sadly, thecompanydidintheendfail.Butthat’sanotherstory.

*AtalkgivenontheoccasionofthepublicationofFeynman’scollectedletters.

KurtGödelMay1,2006

When Kurt Gödel was born—one hundred years ago today—the field ofmathematics seemed almost complete.Twomillennia of development had justbeen codified into a few axioms, from which it seemed one should be ablealmost mechanically to prove or disprove anything in mathematics—and,perhapswithsomeextension,inphysicstoo.Twenty-fiveyears later thingswereproceeding apace,when at the endof a

small academicconference, aquietbut ambitious freshPhD involvedwith theViennaCircle ventured that he hadproved a theorem that thiswhole programmustultimatelyfail.Intheseventy-fiveyearssincethen,whatbecameknownasGödel’stheorem

has been ascribed almost mystical significance, sowed the seeds for thecomputer revolution, and meanwhile been practically ignored by workingmathematicians—andviewedasirrelevantforbroaderscience.TheideasbehindGödel’stheoremhave,however,yettoruntheircourse.And

in fact I believe that today we are poised for a dramatic shift in science andtechnologyforwhichitsprincipleswillberemarkablycentral.Gödel’s originalworkwas quite abstruse.He took the axioms of logic and

arithmetic, and asked a seemingly paradoxical question: can one prove thestatement“thisstatementisunprovable”?Onemightnotthinkthatmathematicalaxiomsalonewouldhaveanythingto

say about this. But Gödel demonstrated that in fact his statement could beencodedpurelyasastatementaboutnumbers.Yet the statement says that it is unprovable. So here, then, is a statement

within mathematics that is unprovable by mathematics: an “undecidablestatement”. And its existence immediately shows that there is a certainincompleteness to mathematics: there are mathematical statements thatmathematicalmethodscannotreach.It could have been that these ideas would rest here. But from within the

technicalitiesofGödel’sproof thereemergedsomethingof incrediblepracticalimportance.ForGödel’sseeminglybizarretechniqueofencodingstatementsintermsofnumberswasacriticalsteptowardstheideaofuniversalcomputation—which implied the possibility of software, and launched the whole computerrevolution.Thinking in terms of computers gives us amodernway to understandwhat

Gödel did: although he himself in effect only wanted to talk about one

computation,heprovedthatlogicandarithmeticareactuallysufficienttobuildauniversal computer, which can be programmed to carry out any possiblecomputation.Not all areas of mathematics work this way. Elementary geometry and

elementaryalgebra,forexample,havenouniversalcomputation,andnoanalogofGödel’s theorem—andweevennowhavepractical software that canproveanystatementaboutthem.But universal computation—when it is present—has many deep

consequences.TheexactscienceshavealwaysbeendominatedbywhatIcallcomputational

reducibility: the idea of finding quickways to computewhat systemswill do.Newtonshowedhowtofindoutwherean(idealized)Earthwillbeinamillionyears—we just have to evaluate a formula—wedonothave to trace amillionorbits.Butifwestudyasystemthatiscapableofuniversalcomputationwecanno

longer expect to “outcompute” it like this; instead, to find outwhat itwill domaytakeusirreduciblecomputationalwork.Andthisiswhyitcanbesodifficulttopredictwhatcomputerswilldo—orto

provethatsoftwarehasnobugs.Itisalsoattheheartofwhymathematicscanbedifficult:itcantakeanirreducibleamountofcomputationalworktoestablishagivenmathematicalresult.Anditiswhatleadstoundecidability—forifwewanttoknow,say,whether

anynumberofanysizehasacertainproperty,computationalirreducibilitymaytellusthatwithoutcheckinginfinitelymanycaseswemaynotbeabletodecideforsure.Working mathematicians, though, have never worried much about

undecidability. For certainly Gödel’s original statement is remote, beingastronomically long when translated into mathematical form. And the fewalternatives constructed over the years have seemed almost as irrelevant inpractice.But my own work with computer experiments suggests that in fact

undecidabilityismuchcloserathand.AndindeedIsuspectthatquiteafewofthe famous unsolved problems in mathematics today will turn out to beundecidablewithintheusualaxioms.The reason undecidability has not been more obvious is just that

mathematicians—despite their reputation for abstract generality—like mostscientists,tendtoconcentrateonquestionsthattheirmethodssucceedwith.Back in 1931, Gödel and his contemporaries were not even sure whether

Gödel’s theoremwas somethinggeneral,or just aquirkof their formalism for

logic and arithmetic. But a few years later, when Turing machines and othermodels for computers showed the same phenomenon, it began to seemmoregeneral.Still,Gödelwonderedwhether therewouldbe an analogof his theorem for

humanminds,orforphysics.Westilldonotknowthecompleteanswer,thoughIcertainlyexpectthatbothmindsandphysicsareinprinciplejustlikeuniversalcomputers—withGödel-liketheorems.Oneofthegreatsurprisesofmyownworkhasbeenjusthoweasyitistofind

universal computation. If one systematically explores the abstract universe ofpossiblecomputationalsystemsonedoesnothavetogofar.Oneneedsnothinglikethebilliontransistorsofamodernelectroniccomputer,oreventheelaborateaxioms of logic and arithmetic. Simple rules that can be stated in a shortsentence—orsummarizedinathree-digitnumber—areenough.And it is almost inevitable that such rules are common in nature—bringing

withthemundecidability.Isasolarsystemultimatelystable?Canabiochemicalprocess ever go out of control? Can a set of laws have a devastatingconsequence? We can now expect general versions of such questions to beundecidable.ThismighthavepleasedGödel—whooncesaidhehadfoundabugintheUS

Constitution,whogavehis friendEinsteinaparadoxicalmodelof theuniverseforhisbirthday—andwhotoldaphysicistIknewthatfortheoreticalreasonshe“didnotbelieveinnaturalscience”.Eveninthefieldofmathematics,Gödel—likehisresults—wasalwaystreated

assomewhatalientothemainstream.Hecontinuedfordecadestosupplycentralideas tomathematical logic, even as “the greatest logician sinceAristotle” (asJohn von Neumann called him) became increasingly isolated, worked toformalize theology using logic, became convinced that discoveries of Leibnizfromthe1600shadbeensuppressed,andin1978,withhiswife’shealthfailing,diedofstarvation,suspiciousofdoctorsandafraidofbeingpoisoned.Heleftusthelegacyofundecidability,whichwenowrealizeaffectsnotjust

esoteric issues about mathematics, but also all sorts of questions in science,engineering,medicineandmore.Onemight think of undecidability as a limitation to progress, but in many

ways it is instead a sign of richness. For with it comes computationalirreducibility,andthepossibilityforsystemstobuildupbehaviorbeyondwhatcan be summarized by simple formulas. Indeed, my own work suggests thatmuchofthecomplexityweseeinnaturehaspreciselythisorigin.Andperhapsitisalso theessenceofhowfromdeterministicunderlyinglawswecanbuildupapparentfreewill.

Inscienceandtechnologywehavenormallycraftedourtheoriesanddevicesbycarefuldesign.ButthinkingintheabstractcomputationaltermspioneeredbyGödel’smethodswecanimagineanalternative.Forifwerepresenteverythinguniformly in terms of rules or programs, we can in principle just explicitlyenumerateallpossibilities.Inthepast,though,nothinglikethiseverseemedevenfaintlysensible.Forit

wasimplicitlyassumedthattocreateaprogramwithinterestingbehaviorwouldrequireexplicithumandesign—orat least theeffortsofsomething likenaturalselection. But when I started actually doing experiments and systematicallyrunning the simplest programs,what I found instead is that the computationaluniverseisteemingwithdiverseandcomplexbehavior.Already there is evidence that many of the remarkable forms we see in

biology just come from sampling this universe.And perhaps by searching thecomputationaluniversewemayfind—evensoon—theultimateunderlyinglawsforourownphysicaluniverse.(Todiscoveralltheirconsequences,though,willstillrequireirreduciblecomputationalwork.)Exploring the computational universe puts mathematics too into a new

context. For we can also now see a vast collection of alternatives to themathematicsthatwehaveultimatelyinheritedfromthearithmeticandgeometryofancientBabylon.Andforexample,theaxiomsofbasiclogic,farfrombeingsomething special, now just appear as roughly the 50,000th possibility. Andmathematics, long a purely theoretical science, must adopt experimentalmethods.The exploration of the computational universe seems destined to become a

core intellectual framework in the future of science. And in technology thecomputationaluniverseprovides avast new resource that canbe searchedandmined for systems that serve our increasingly complex purposes. It isundecidability that guarantees an endless frontier of surprising and usefulmaterialtofind.And so it is that from Gödel’s abstruse theorem about mathematics has

emergedwhatIbelievewillbethedefiningthemeofscienceandtechnologyinthetwenty-firstcentury.

AlanTuringJune23,2012

InevermetAlanTuring;hediedfiveyearsbeforeIwasborn.ButsomehowIfeelIknowhimwell—notleastbecausemanyofmyownintellectual interestshavehadanalmosteerieparallelwithhis.And by a strange coincidence,Mathematica’s “birthday” (June 23, 1988) is

alignedwithTuring’s—sothattodayisnotonlythecentenaryofTuring’sbirth,butisalsoMathematica’s24thbirthday.I think I first heard aboutAlan Turingwhen Iwas about eleven years old,

rightaroundthetimeIsawmyfirstcomputer.Throughafriendofmyparents,Ihadgotten toknowarathereccentricoldclassicsprofessor,who,knowingmyinterest in science, mentioned to me this “bright young chap named Turing”whomhehadknownduringtheSecondWorldWar.One of the classics professor’s eccentricities was that whenever the word

“ultra”cameupinaLatintext,hewouldrepeatitoverandoveragain,andmakecommentsaboutrememberingit.Atthetime,Ididn’tthinkmuchofit—thoughIdidrememberit.OnlyyearslaterdidIrealizethat“Ultra”wasthecodenamefortheBritishcryptanalysiseffortatBletchleyParkduringthewar.InaveryBritishway, the classics professor wanted to tell me something about it, withoutbreakinganysecrets.AndpresumablyitwasatBletchleyParkthathehadmetAlanTuring.Afewyearslater,IheardscatteredmentionsofAlanTuringinvariousBritish

academic circles. I heard that he had done mysterious but important work inbreakingGermancodesduringthewar.AndIhearditclaimedthatafterthewar,he had been killed by British Intelligence. At the time, at least some of theBritishwartimecryptographyeffortwasstillsecret,includingTuring’sroleinit.Iwonderedwhy.SoIaskedaround,andstartedhearingthatperhapsTuringhadinvented codes that were still being used. (In reality, the continued secrecyseems tohavebeen intended toprevent it beingknown that certain codeshadbeenbroken—soothercountrieswouldcontinuetousethem.)I’mnotsurewhere InextencounteredAlanTuring.Probably itwaswhenI

decided to learn all I could about computer science—and saw all sorts ofmentionsof“Turingmachines”.ButIhaveadistinctmemoryfromaround1979ofgoingtothelibraryandfindingalittlebookaboutAlanTuringwrittenbyhismother,SaraTuring.And gradually I built up quite a picture ofAlanTuring and hiswork.And

overthe30+yearsthathavefollowed,IhavekeptonrunningintoAlanTuring,

ofteninunexpectedplaces.In theearly1980s, forexample, Ihadbecomevery interested in theoriesof

biologicalgrowth—onlytofind(fromSaraTuring’sbook)thatAlanTuringhaddoneallsortsoflargelyunpublishedworkonthat.And for example in 1989, when we were promoting an early version of

Mathematica,IdecidedtomakeaposteroftheRiemannzetafunction—onlytodiscoverthatAlanTuringhadatonetimeheldtherecordforcomputingzerosofthezetafunction.(Earlierhehadalsodesignedagear-basedmachinefordoingthis.)Recently I even found out that Turing had written about the “reform of

mathematical notation and phraseology”—a topic of great interest to me inconnectionwithbothMathematicaandWolfram|Alpha.AndatsomepointIlearnedthatahighschoolmathteacherofmine(Norman

Routledge) hadbeen a friendofTuring’s late in his life.But even thoughmyteacherknewmyinterestincomputers,henevermentionedTuringorhisworktome.Andindeed,35yearsago,AlanTuringandhisworkwerelittleknown,anditisonlyfairlyrecentlythatTuringhasbecomeasfamousasheistoday.Turing’sgreatestachievementwasundoubtedlyhisconstructionin1936ofa

universal Turing machine—a theoretical device intended to represent themechanization ofmathematical processes.And in some sense,Mathematica ispreciselyaconcreteembodimentof thekindofmechanization thatTuringwastryingtorepresent.In1936,however,Turing’simmediatepurposewaspurelytheoretical.Indeed

itwas to shownotwhat couldbemechanized inmathematics, butwhat couldnot.In1931,Gödel’stheoremhadshownthattherewerelimitstowhatcouldbeprovedinmathematics,andTuringwantedtounderstandtheboundariesofwhatcouldeverbedonebyanysystematicprocedureinmathematics.TuringwasayoungmathematicianinCambridge,England,andhisworkwas

couchedintermsofmathematicalproblemsofhistime.Butoneofhisstepswasthe theoretical construction of a universal Turing machine capable of being“programmed” to emulate any other Turing machine. In effect, Turing hadinvented the idea of universal computation—which was later to become thefoundationonwhichallofmoderncomputertechnologyisbuilt.Atthetime,though,Turing’sworkdidnotmakemuchofasplash,probably

largely because the emphasis of Cambridge mathematics was elsewhere. JustbeforeTuringpublishedhispaper,he learnedaboutasimilar resultbyAlonzoChurchfromPrinceton,formulatednot intermsof theoreticalmachines,but intermsof themathematics-like lambdacalculus.Andasa resultTuringwent toPrinceton for ayear to studywithChurch—andwhilehewas there,wrote the

mostabstrusepaperofhislife.Thenext fewyears forTuringweredominatedbyhiswartimecryptanalysis

work. I learned a few years ago that during the war Turing visited ClaudeShannonatBellLabsinconnectionwithspeechencipherment.Turinghadbeenworkingonakindofstatisticalapproachtocryptanalysis—andIamextremelycurious to know whether Turing told Shannon about this, and potentiallylaunched the idea of information theory, which itself was first formulated forsecretcryptanalysispurposes.After the war, Turing got involvedwith the construction of the first actual

computers in England. To a large extent, these computers had emerged fromengineering, not from a fundamental understanding of Turing’s work onuniversalcomputation.There was, however, a definite, if circuitous, connection. In 1943 Warren

McCullochandWalterPitts inChicagowrote a theoretical paper aboutneuralnetworks that used the idea of universal Turing machines to discuss generalcomputationinthebrain.JohnvonNeumannreadthispaper,anduseditinhisrecommendations about how practical computers should be built andprogrammed.(JohnvonNeumannhadknownaboutTuring’spaperin1936,butat the time did not recognize its significance, instead describing Turing in arecommendation letter as having done interesting work on the central limittheorem.)It isremarkablethatinjustoveradecadeAlanTuringwastransportedfrom

writing theoretically about universal computation, to being able to writeprograms for an actual computer. I have to say, though, that from today’svantage point, his programs look incredibly “hacky”—with lots of specialfeaturespackedin,andencodedasstrangestringsofletters.Butperhapstoreachtheedgeofanewtechnologyit’sinevitablethattherehastobehackiness.And perhaps too it required a certain hackiness to construct the very first

universal Turing machine. The concept was correct, but Turing quicklypublishedanerratumtofixsomebugs,andinlateryears,it’sbecomeclearthatthereweremorebugs.ButatthetimeTuringhadnointuitionabouthoweasilybugscanoccur.Turingalsodidnotknowjusthowgeneralornothis resultsaboutuniversal

computation might be. Perhaps the Turing machine was just one model of acomputationalprocess,andothermodels—orbrains—mighthavequitedifferentcapabilities.But gradually over the course of several decades, it became clearthat a wide range of possible models were actually exactly equivalent to themachinesTuringhadinvented.It’s strange to realize that Alan Turing never appears to have actually

simulated a Turing machine on a computer. He viewed Turing machines astheoretical devices relevant for proving general principles. But he does notappeartohavethoughtaboutthemasconcreteobjectstobeexplicitlystudied.And indeed, when Turing came to make models of biological growth

processes, he immediately started using differential equations—and appearsnever to have considered the possibility that something like aTuringmachinemightberelevanttonaturalprocesses.WhenIbecameinterested insimplecomputationalprocessesaround1980, I

also didn’t considerTuringmachines—and instead started off studyingwhat Ilaterlearnedwerecalledcellularautomata.AndwhatIdiscoveredwasthatevencellularautomatawithincrediblysimplerulescouldproduceincrediblycomplexbehavior—which I soon realized could be considered as corresponding to acomplexcomputation.Iprobably simulatedmy first explicitTuringmachineonly in1991.Tome,

Turingmachineswerebuiltalittlebittoomuchlikeengineeringsystems—andnotlikesomethingthatwouldlikelycorrespondtoasysteminnature.ButIsoonfound that even simple Turing machines, just like simple cellular automata,couldproduceimmenselycomplexbehavior.Inasense,AlanTuringcouldeasilyhavediscoveredthis.Buthisintuition—

likemyoriginalintuition—wouldhavetoldhimthatnosuchphenomenonwaspossible. So it would likely only have been luck—and access to easycomputation—thatwouldhaveledhimtofindthephenomenon.Had he done so, I am quite sure hewould have become curious about just

whatthethresholdforhisconceptofuniversalitywouldbe,andjusthowsimplea Turing machine would suffice. In the mid-1990s, I searched the space ofsimpleTuringmachines,andfoundthesmallestpossiblecandidate.AndafterIput up a $25,000 prize, in 2007 Alex Smith showed that indeed this Turingmachineisuniversal.NodoubtAlanTuringwouldquitequicklyhavegrasped the significanceof

such results for thinking about both natural processes and mathematics. Butwithouttheempiricaldiscoveries,histhinkingdidnotprogressinthisdirection.Instead,hebegantoconsiderfromamoreengineeringpointofviewtowhat

extentcomputersshouldbeabletoemulatebrains,andheinventedideasliketheTuring test.Reading throughhiswritings today, it is remarkablehowmanyofhis conceptual arguments about artificial intelligence still need to be made—though some, like his discussion of extrasensory perception, have becomequaintlydated.And looking at his famous 1950 article on “Computing Machinery and

Intelligence”oneseesadiscussionofprogrammingintoamachinethecontents

of Encyclopædia Britannica—which he estimates should take 60 workers 50years.IwonderwhatAlanTuringwouldthinkofWolfram|Alpha,which,thankstoprogressoverthepast60years,andperhapssomecleverness,hassofartakenatleastslightlylesshumaneffort.In addition to his intellectual work, Turing has in recent times become

somethingof a folkhero,mostnotably through the storyofhisdeath.Almostcertainly it will never be known for sure whether his death was in factintentional.ButfromwhatIknowandhaveheardImustsaythatIratherdoubtthatitwas.Whenone first hears thatAlanTuringdiedby eating an apple impregnated

withcyanideoneassumes itmusthavebeen intentionalsuicide.Butwhenonelater discovers that hewasquite a tinkerer, had recentlymade cyanide for thepurpose of electroplating spoons, kept chemicals alongside his food, and wasratheramessyindividual,thepicturebecomesalotlessclear.IoftenwonderwhatAlanTuringwouldhavebeenliketomeet.Idonotknow

ofany recordingofhisvoice (thoughhedidoncedoaBBCradiobroadcast).ButIgatherthatevenneartheendofhislifehegiggledalot,andtalkedwithakindofstutterthatseemedtocomefromthinkingfasterthanhewastalking.Heseemed to have found it easiest to talk tomathematicians.He thought a littleaboutphysics, thoughdoesn’t seem tohaveevergottendeeply into it.Andheseemed to have maintained a child-like enthusiasm and wonder for manyintellectualquestionsthroughouthislife.Hewassomethingofaloner,workingsuccessivelyonhisownonhisvarious

projects.Hewasgay,andlivedalone.Hewasnoorganizationalpolitician,andtowardstheendofhislifeseemstohavefoundhimselflargelyignoredbothbypeople working on computers and by people working on his new interest ofbiologicalgrowthandmorphogenesis.HewasinsomerespectsaquintessentialBritishamateur,dippinghisintellect

into different areas. He achieved a high level of competence in puremathematics, and used that as his professional base. His contributions intraditional mathematics were certainly perfectly respectable, though notspectacular.But in every area he touched, therewas a certain crispness to theideas he developed—even if their technical implementation was sometimesshroudedinarcanenotationandmassesofdetail.In somewayshewas fortunate to livewhenhedid.Forhewasat the right

timetobeabletaketheformalismofmathematicsasithadbeendeveloped,andtocombineitwiththeemergingengineeringofhisday,toseeforthefirsttimethegeneralconceptofcomputation.It is perhaps a shame that he died 25 years before computer experiments

became widely feasible. I certainly wonder what he would have discoveredtinkeringwithMathematica. I don’t doubt that hewould have pushed it to itslimits,writingcodethatwouldhorrifyme.ButIfullyexpectthatlongbeforeIdid,hewouldhavediscoveredthemainelementsofANewKindofScience,andbeguntounderstandtheirsignificance.Hewouldprobablybedisappointedthat60yearsafterheinventedtheTuring

test,thereisstillnofullhuman-likeartificialintelligence.Andperhapslongagohe would have begun to campaign for the creation of something likeWolfram|Alpha,toturnhumanknowledgeintosomethingcomputerscanhandle.Ifhehadlivedafewdecadeslonger,hewouldnodoubthaveappliedhimself

toahalfdozenmoreareas.ButthereisstillmuchtobegratefulforinwhatAlanTuringdidachieve inhis41years, andhismodern reputationas the foundingfatheroftheconceptofcomputation—andtheconceptualbasisformuchofwhatI,forexample,havedone—iswelldeserved.

JohnvonNeumannDecember28,2003

Itwouldhavebeen JohnvonNeumann’s 100thbirthday today—if hehadnotdiedatage54in1957.I’vebeeninterestedinvonNeumannformanyyears—not leastbecausehiswork touchedon someofmymost favorite topics.He ismentioned in12separateplaces inANewKindofScience—second innumberonlytoAlanTuring,whoappears19times.Ialwaysfeelthatonecanappreciatepeople’sworkbetterifoneunderstands

thepeoplethemselvesbetter.Andfromtalkingtomanypeoplewhoknewhim,IthinkI’vegraduallybuiltupadecentpictureofJohnvonNeumannasaman.He would have been fun to meet. He knew a lot, was very quick, always

impressedpeople,andwaslively,socialandfunny.One video clip of him has survived. In 1955 he was on a television show

calledYouthWantstoKnow,whichtodayseemspainfullyhokey.Surroundedbyteenage kids, he is introduced as a commissioner of the Atomic EnergyCommission—whichinthosedayswasabigdeal.Heisaskedaboutanexhibitof equipment. He says very seriously that it’s mostly radiation detectors. Butthen a twinkle comes into his eye, and he points to another item, and saysdeadpan,“Exceptthis,whichisacarryingcase.”Andthat’stheendoftheonlyvideorecordofJohnvonNeumannthatexists.Somescientists(suchasmyself)spendmostoftheirlivespursuingtheirown

grand programs, ultimately in a fairly isolated way. John von Neumann wasinsteadsomeonewhoalwayslikedtointeractwiththelatestpopularissues—andthepeoplearound them—andthencontribute to theminhisowncharacteristicway.Heworkedhard,oftenonmanyprojectsatonce,andalwaysseemedtohave

fun.Inretrospect,hechosemostofhistopicsremarkablywell.Hestudiedeachofthemwithadefinitepracticalmathematicalstyle.Andpartlybybeingthefirstperson to try applying seriousmathematicalmethods in various areas, hewasabletomakeimportantanduniquecontributions.ButI’vebeentoldthathewasnevercompletelyhappywithhisachievements

becausehethoughthemissedsomegreatdiscoveries.Andindeedhewascloseto a remarkable number of important mathematics-related discoveries of thetwentieth century: Gödel’s theorem, Bell’s inequalities, information theory,Turingmachines,computerlanguages—aswellasmyownmorerecentfavoritecoreANewKindofSciencediscoveryofcomplexityfromsimplerules.Butsomehowheneverquitemadetheconceptualshiftsthatwereneededfor

anyofthesediscoveries.Therewere,Ithink,twobasicreasonsforthis.First,hewassogoodatgetting

newresultsbythemathematicalmethodsheknewthathewasalwaysgoingoffto get more results, and never had a reason to pause and see whether somedifferentconceptual frameworkshouldbeconsidered.Andsecond,hewasnotparticularly one to buck the system: he liked the socialmilieu of science andalwaysseemedtotakebothintellectualandotherauthorityseriously.Byallreports,vonNeumannwassomethingofaprodigy,publishinghisfirst

paper(onzerosofpolynomials)attheageof19.Byhisearlytwenties,hewasestablishedasapromisingyoungprofessionalmathematician—workingmainlyin the then-popular fields of set theory and foundations of math. (One of hisachievementswasalternateaxiomsforsettheory.)LikemanygoodmathematiciansinGermanyatthetime,heworkedonDavid

Hilbert’s program for formalizingmathematics, and for examplewrote papersaimedatfindingaproofofconsistencyfortheaxiomsofarithmetic.ButhedidnotguessthedeeperpointthatKurtGödeldiscoveredin1931:thatactuallysucha proof is fundamentally impossible. I’ve been told that von Neumann wasalwaysdisappointedthathehadmissedGödel’stheorem.Hecertainlyknewallthemethodsneededtoestablishit(andunderstooditremarkablyquicklyoncehehearditfromGödel).ButsomehowhedidnothavethebrashnesstodisbelieveHilbert,andgolookingforacounterexampletoHilbert’sideas.Inthemid-1920sformalizationwasalltherageinmathematics,andquantum

mechanics was all the rage in physics. And in 1927 vonNeumann set out tobring these together—by axiomatizing quantum mechanics. A fair bit of theformalismthatvonNeumannbuilthasbecomethestandardframeworkforanymathematicallyorientedexpositionofquantummechanics.ButImustsaythatIhavealwaysthoughtthatitgavetoomuchofanairofmathematicaldefinitenesstoideas(particularlyaboutquantummeasurement)thatinrealitydependonallsorts of physical details.And indeed someof vonNeumann’s specific axiomsturnedouttobetoorestrictiveforordinaryquantummechanics—obscuringforanumberofyears thephenomenonofentanglement,andlaterofcriteriasuchasBell’sinequalities.But von Neumann’s work on quantum mechanics had a variety of fertile

mathematical spinoffs, and particularly what are now called von Neumannalgebras have recently become popular in mathematics and mathematicalphysics.Interestingly, von Neumann’s approach to quantum mechanics was at first

very much aligned with traditional calculus-based mathematics—investigatingpropertiesofHilbertspaces,continuousoperators,etc.Butgraduallyitbecame

more focused on discrete concepts, particularly early versions of “quantumlogic”.InasensevonNeumann’squantumlogicideaswereanearlyattemptatdefiningacomputationalmodelofphysics.Buthedidnotpursuethis,anddidnotgo in thedirections thathaveforexample led tomodern ideasofquantumcomputing.Bythe1930svonNeumannwaspublishingseveralpapersayear,onavariety

of popular topics in mainstream mathematics, often in collaboration withcontemporaries of significant later reputation (Wigner, Koopman, Jordan,Veblen, Birkhoff, Kuratowski, Halmos, Chandrasekhar, etc.). VonNeumann’sworkwasunquestionablygoodandinnovative,thoughverymuchintheflowofdevelopmentofthemathematicsofitstime.DespitevonNeumann’searlyinterestinlogicandthefoundationsofmath,he

(likemostofthemathcommunity)movedawayfromthisbythemid-1930s.InCambridgeandtheninPrincetonheencounteredtheyoungAlanTuring—evenofferinghimajobasanassistantin1938.ButheapparentlypaidlittleattentiontoTuring’sclassic1936paperonTuringmachinesandtheconceptofuniversalcomputation,writinginarecommendationletteronJune1,1937that“[Turing]has done goodwork on… theory of almost periodic functions and theory ofcontinuousgroups”.Asitdidformanyscientists,vonNeumann’sworkontheManhattanProject

appearstohavebroadenedhishorizons,andseemstohavespurredhiseffortstoapplyhismathematicalprowesstoproblemsofallsorts—notjustintraditionalmathematics. His pure mathematical colleagues seem to have viewed suchactivitiesasapeculiarandsomewhatsuspecthobby,butonethatcouldgenerallybetoleratedinviewofhisrespectablemathematicalcredentials.At the Institute for Advanced Study in Princeton, where von Neumann

worked,therewasstrain,however,whenhestartedaproject tobuildanactualcomputerthere.Indeed,evenwhenIworkedattheInstituteintheearly1980s,therewerestillpainedmemoriesoftheproject.ThepuremathematiciansattheInstitutehadneverbeenkeenonit,andthestorywasthatwhenvonNeumanndied, theyhadbeenhappy toacceptThomasWatsonof IBM’soffer tosendatrucktotakeawayallofvonNeumann’sequipment.(Amusingly,the6-inchon-offswitchfor thecomputerwas leftbehind,bolted to thewallof thebuilding,and has recently become a prized possession of a computer industryacquaintanceofmine.)Ihadsomesmall interactionwithvonNeumann’sheritageat theInstitute in

1982whenthethen-director(HarryWoolf)wasrecruitingme.(Harry’soriginalconceptwas togetme to start aSchool ofComputation at the Institute, to goalongwiththeexistingSchoolofNaturalSciencesandSchoolofMathematics.

But for various reasons, thiswas notwhat happened.) I was concerned aboutintellectualpropertyissues,havingjusthadsomedifficultywiththematCaltech.Harry’sresponse—thatheattributedtothechairmanoftheirboardoftrustees—was, “Look, von Neumann developed the computer here, but we insisted ongivingitaway;afterthat,whyshouldweworryaboutanyintellectualpropertyrights?” (The practical result was a letter disclaiming any rights to anyintellectualpropertythatIproducedattheInstitute.)Among several of von Neumann’s interests outside of mainstream pure

mathematicswashis attempt todevelopamathematical theoryofbiologyandlife.Inthemid-1940stherehadbeguntobe—particularlyfromwartimeworkonelectronic control systems—quite a bit of discussion about analogies between“naturalandartificialautomata”,and“cybernetics”.AndvonNeumanndecidedto apply hismathematicalmethods to this. I’ve been told he was particularlyimpressedbytheworkofMcCulloughandPittsonformalmodelsoftheanalogybetweenbrains and electronics. (Therewere undoubtedlyother influences too:JohnMcCarthytoldmethataround1948hevisitedvonNeumann,andtoldhimabout applying information theory ideas to thinking about the brain as anautomaton;vonNeumann’smainresponseatthetimewasjust,“Writeitup!”)Von Neumann was in many ways a traditional mathematician, who (like

Turing)believedheneededtoturntopartialdifferentialequationsindescribingnaturalsystems.I’vebeentoldthatatLosAlamosvonNeumannwasverytakenwithelectricallystimulatedjellyfish,whichheappearstohaveviewedasdoingsomekindofcontinuousanalogof the informationprocessingofanelectroniccircuit. Inanycase,byabout1947,hehadconceived the ideaofusingpartialdifferentialequationstomodelakindoffactorythatcouldreproduceitself,likealivingorganism.VonNeumannalwaysseemstohavebeenverytakenwithchildren,andIam

told that itwas in playingwith an erector set owned by the son of his game-theorycollaboratorOskarMorgensternthatvonNeumannrealizedthathisself-reproducing factory could actually be built out of discrete robotic-like parts.(There was already something of a tradition of building computers out ofMeccano—and indeed for example some ofHartree’s early articles on analogcomputersappearedinMeccanoMagazine.)AnelectricalengineernamedJulianBigelow,whoworkedonvonNeumann’s

IAScomputerproject,pointedoutthat3Dpartswerenotnecessary,andthat2Dwouldworkjustaswell.(WhenIwasattheInstituteintheearly1980sBigelowwas still there, thoughunfortunately viewed as a slightly peculiar relic of vonNeumann’sproject.)Stan Ulam told me that he had independently thought about making

mathematical models of biology, but in any case, around 1951 he appears tohave suggested to von Neumann that one should be able to use a simplified,essentiallycombinatorialmodel—basedonsomething like the infinitematricesthat Ulam had encountered in the so-called Scottish Book of math problems(namedafteracafeinPoland)towhichhehadcontributed.Theresultofallthiswasamodelthatwasformallyatwo-dimensionalcellular

automaton. Systems equivalent to two-dimensional cellular automata werearising in several other contexts around the same time (see A New Kind ofScience). Von Neumann seems to have viewed his version as a convenientframework in which to construct a mathematical system that could emulateengineeredcomputersystems—especiallytheEDVAConwhichvonNeumannworked.Intheperiod1952–53vonNeumannsketchedanoutlineofaproofthatitwas

possibleforaformalsystemtosupportselfreproduction.Wheneverheneededadifferentkindofcomponent(wire,oscillator,logicelement,etc.)hejustaddeditasanewstateofhiscellularautomaton,withnewrules.Heendedupwitha29-statesystem,anda200,000-cellconfigurationthatcouldreproduceitself.(VonNeumannhimselfdidnotcompletetheconstruction.Thiswasdoneintheearly1960sby a former assistant of vonNeumann’s namedArthurBurks,whohadleft the IAS computer project to concentrate on his interests in philosophy,thoughwhomaintainseventodayaninterestincellularautomata.[ArthurBurksdiedin2008,attheageof92.])From the point of view ofANewKind of Science, vonNeumann’s system

now seems almost grotesquely complicated.But vonNeumann’s intuition toldhim that one could not expect a simpler system to show something assophisticated and biological as self reproduction. What he said was that hethought that below a certain level of complexity, systems would always be“degenerative”, and always generate what amounts to behavior simpler thantheir rules.But then, from seeing the exampleof biology, andof systems likeTuring machines, he believed that above some level, there should be an“explosive”increaseincomplexity,withsystemsabletogenerateothersystemsmorecomplexthanthemselves.Buthesaidthathethoughtthethresholdforthiswouldbesystemswithmillionsofparts.Twenty-fiveyearsagoImightnothavedisagreedtoostronglywiththat.And

certainlyformeittookseveralyearsofcomputerexperimentationtounderstandthat in fact it takes only very simple rules to produce even themost complexbehavior.SoIdonot thinkitsurprising—orunimpressive—thatvonNeumannfailedtorealizethatsimpleruleswereenough.Ofcourse,asoneoften realizes in retrospect,hedidhavesomeotherclues.

Heknewabouttheideaofgeneratingpseudorandomnumbersfromsimplerules,suggestingtheso-called“middlesquaremethod”.Hehadthebeginningsoftheidea of doing computer experiments in areas like number theory.He analyzedthe first 2000 digits of π and e, computed on the ENIAC, finding that theyseemed random—though making no comment on it. (He also looked atContinuedFraction[21/3].)IhaveaskedmanypeoplewhoknewhimwhyvonNeumannneverconsidered

simplerrules.MarvinMinskytoldmethatheactuallyaskedvonNeumannaboutthis directly, but that von Neumann had been somewhat confused by thequestion. ItwouldhavebeenmuchmoreUlam’sstyle thanvonNeumann’s tohave come up with simpler rules, and Ulam indeed did try making a one-dimensionalanalogof2Dcellularautomata,butcameupnotwith1Dcellularautomata,butwithacuriousnumber-theoreticalsystem.In the last tenyearsofhis life,vonNeumanngot involved inan impressive

arrayofissues.Someofhiscolleaguesseemtohavefeltthathespenttoolittletimeoneachone,butstillhiscontributionswereusuallysubstantial—sometimesdirectly in terms of content, and usually at least in terms of lending hiscredibilitytoemergingareas.Hemademistakes,ofcourse.Hethoughtthateachlogicalstepincomputation

wouldnecessarilydissipateacertainamountofheat,whereasinfactreversiblecomputation is in principle possible. He thought that the unreliability ofcomponents would be a major issue in building large computer systems; heapparentlydidnothaveanidealikeerror-correctingcodes.Heisreputedtohavesaid that no computer programwould everbemore than a few thousand lineslong. He was probably thinking about proofs of theorems—but did not thinkaboutsubroutines,theanalogoflemmas.VonNeumannwasagreatbeliever in theefficacyofmathematicalmethods

andmodels,perhapsimplementedbycomputers.In1950hewasoptimisticthataccuratenumericalweatherforecastingwouldsoonbepossible.Inaddition,hebelievedthatwithmethodslikegametheoryitshouldbepossibletounderstandmuchofeconomicsandotherformsofhumanbehavior.VonNeumannwas always quite a believer in using the latestmethods and

tools(I’msurehewouldhavebeenabigMathematicausertoday).Hetypicallyworked directly with one or two collaborators, sometimes peers, sometimesassistants, thoughhemaintained contactwith a largenetworkof scientists. (AtypicalcommunicationwasaletterhewrotetoAlanTuringin1949,inwhichheasks,“Whataretheproblemsonwhichyouareworkingnow,andwhatisyourprogram for the immediate future?”) In his later years he often operated as adistinguished consultant, brought in by the government, or other large

organizations.Hisworkwasthenoftenpresentedasareport,thatwasaccordedparticular weight because of his distinguished consultant status. (It was alsooftenagoodandclearpieceofwork.)Hewasoftenviewedalittleambivalentlyas an outsider in the fields he entered—positively because he brought hisdistinctiontothefield,negativelybecausehewasnotinthecliqueofexpertsinthefield.Particularly in the early 1950s, von Neumann became deeply involved in

militaryconsulting,and indeed Iwonderhowmuchof the intellectual styleofColdWarUSmilitarystrategicthinkingactuallyoriginatedwithhim.Heseemstohavebeenquiteflatteredthathewascalledupontodothisconsulting,andhecertainlytreatedthegovernmentwithconsiderablymorerespectthanmanyotherscientists of his day. Except sometimes in his exuberance to demonstrate hismathematical and calculational prowess, he seems to have always been quitemature and diplomatic. The transcript of his testimony at the Oppenheimersecurityhearingcertainlyforexamplebearsthisout.Nevertheless, von Neumann’s military consulting involvements left some

factions quite negative about him. It’s sometimes said, for example, that vonNeumannmighthavebeenthemodelfor thesinisterDr.Strangelovecharacterin StanleyKubrick’smovie of that name (and indeed vonNeumannwas in awheelchairforthelastyearofhislife).AndvaguenegativefeelingsaboutvonNeumann surface for example in a typical statement I heard recently from ascience historian of the period—that “somehow I don’t like von Neumann,thoughIcan’trememberexactlywhy”.IrecentlymetvonNeumann’sonlychild—hisdaughterMarina,whoherself

has had a distinguished career, mostly at GeneralMotors. She reinforced myimpression that until his unpleasant final illness, John von Neumann was ahappy and energetic man, working long hours on mathematical topics, andalwayshavingfun.Shetoldmethatwhenhedied,heleftaboxthathedirectedshouldbeopenedfiftyyearsafterhisdeath.Whatdoesitcontain?Hislastsoberpredictionsofafuturewehavenowseen?Orajoke—likeafunnypartyhatofthetypehelikedtowear?It’llbemostinterestingin2007tofindout.

JohnvonNeumann’sBoxFebruary8,2007

At the end of my piece about the 100th anniversary of John von Neumann’sbirth,Imentionedthathisdaughterhadtoldmeaboutaboxtobeopenedonthe50thanniversaryofhisdeath.Thatanniversaryistoday.

Andbeingremindedofthislastweek,Isentmailtohisdaughtertoaskwhathadbecomeofthebox.Disappointingly,sheresponded,“TheBigBoxopeningturnedouttobeaBig

Bust.” Apparently she and her children and their grandchildren had allassembled…onlytodiscoverthatitwasallabigmistake:theboxwasactuallynotJohnvonNeumann’satall!Perhaps it was for the best. Von Neumann’s daughter sent me a piece she

wrote about him, pointing out the unpredictability (irreducibility?) oftechnologicalandotherchange—andexpressingconcernthatourspeciesmightwipeitselfoutbytheyear1980.Fortunatelythatofcoursehasn’thappened.Anditwasrecentlysuggestedto

methatperhapsTheBoxmightcontainsomeCold-War-inspiredplanforaDr.-Strangelove-like Doomsday Machine. So—“von Neumann machine” self-replicatorsnotwithstanding—it’sperhaps justaswell that in theend therewasnobox.

GeorgeBooleNovember2,2015

Today is the 200th anniversary of the birth of George Boole. In our moderndigitalworld,we’realwayshearingabout“Booleanvariables”—1or0,trueorfalse. And one might think, “What a trivial idea! Why did someone evenexplicitlyneedtoinventit?”Butasissooftenthecase,there’sadeeperstory—forBooleanvariableswere really justa sideeffectofan important intellectualadvancethatGeorgeBoolemade.When George Boole came onto the scene, the disciplines of logic and

mathematics had developed quite separately for more than 2000 years. AndGeorge Boole’s great achievement was to show how to bring them together,throughtheconceptofwhat’snowcalledBooleanalgebra.Andindoingsoheeffectivelycreatedthefieldofmathematicallogic,andsetthestageforthelongseriesofdevelopmentsthatledforexampletouniversalcomputation.WhenGeorgeBooleinventedBooleanalgebra,hisbasicgoalwastofindaset

ofmathematical axioms that could reproduce theclassical resultsof logic.Hisstartingpointwasordinaryalgebra,withvariables likexandy,andoperationslikeadditionandmultiplication.Atfirst,ordinaryalgebraseemsalotlikelogic.Afterall,pandqisthesame

as q and p, just as p×q = q×p. But if one looks in more detail, there aredifferences.Likep×p=p²,butpandpisjustp.Somewhatconfusingly,Booleusedthenotationofstandardalgebra,butaddedspecialrulestocreateanaxiomsystemthathethenshowedcouldreproducealltheusualresultsoflogic.Boole was rather informal in the way he described his axiom system. But

withinafewdecades,ithadbeenmorepreciselyformalized,andoverthecourseofthecenturythatfollowed,afewprogressivelysimplerformsofitwerefound.Andthen,asithappens,16yearsagoIendedupfinishingthis150-yearprocess,byfinding—largelyasasideeffectofotherscienceIwasdoing—theprovablyverysimplestpossibleaxiomsystemforlogic,thatactuallyhappenstoconsistofjustasingleaxiom.

Ithoughtthisaxiomwasprettyneat,andlookingatwhereitliesinthespaceof possible axioms has interesting implications for the foundations ofmathematicsandlogic.ButinthecontextofGeorgeBoole,onecansaythatit’sa minimal version of his big idea: that one can have a mathematical axiomsystem that reproduces all the results of logic just by what amount to simplealgebra-liketransformations.

WhoWasGeorgeBoole?But let’s talk about George Boole, the person.Whowas he, and how did hecometodowhathedid?George Boole was born in 1815, in England, in the fairly small town of

Lincoln, about120milesnorthofLondon.His fatherhada serious interest inscience and mathematics, and had a small business as a shoemaker. GeorgeBoolewassomethingofaself-taughtprodigy,whofirstbecamelocallyfamousat age 14 with a translation of a Greek poem that he published in the localnewspaper.At age16hewashired as a teacher at a local school, andby thattimehewas readingcalculusbooks, andapparently starting to formulatewhatwouldlaterbehisideaaboutrelationsbetweenmathematicsandlogic.Atage19,GeorgeBooledidastartup:hestartedhisownelementaryschool.

Itseemstohavebeendecentlysuccessful,andinfactBoolecontinuedmakinghislivingrunning(or“conducting”asitwasthencalled)schoolsuntilhewasinhis thirties. He was involved with a few people educated in places likeCambridge,notablythroughthelocalMechanics’Institute(alittlelikeamoderncommunitycollege).Butmostlyheseemsjusttohavelearnedbyreadingbooksonhisown.Hetookhisprofessionasaschoolteacherseriously,anddevelopedallsortsof

surprisingly modern theories about the importance of understanding anddiscovery(asopposedtorotememorization),andthevalueoftangibleexamplesin areas likemathematics (he surelywould have been thrilled bywhat’s nowpossiblewithcomputers).Whenhewas23,Boolestartedpublishingpapersonmathematics.Hisearly

paperswereabouthottopicsofthetime,suchascalculusofvariations.Perhapsit was his interest in education and exposition that led him to try creatingdifferent formalisms, but soon he became a pioneer in the “calculus ofoperations”: doing calculus by manipulating operators rather than explicitalgebraicexpressions.Itwasn’t longbeforehewasinteractingwithleadingBritishmathematicians

oftheday,andgettingpositivefeedback.HeconsideredgoingtoCambridgetobecomea“universityperson”,butwasputoffwhentoldthathewouldhavetostartwiththestandardundergraduatecourse,andstopdoinghisownresearch.

MathematicalAnalysisofLogicLogicasafieldofstudyhadoriginatedinantiquity,particularlywiththeworkofAristotle. Ithadbeenastapleofeducation throughout theMiddleAgesandbeyond,fittingintothepracticeofrotelearningbyidentifyingspecificpatternsof logical arguments (“syllogisms”) with mnemonics like “bArbArA” and“cElArEnt”.Inmanyways,logichadn’tchangedmuchinoverathousandyears,though by the 1800s there were efforts to make it more streamlined and“formal”. But the question was how. And in particular, should this happenthroughthemethodsofphilosophy,ormathematics?Inearly1847,Boole’sfriendAugustusDeMorganhadbecomeembroiledin

a piece of academic unpleasantness over the question. And this led Boolequickly to go off and work out his earlier ideas about how logic could beformulatedusingmathematics.Theresultwashisfirstbook,TheMathematicalAnalysisofLogic,publishedthesameyear:

The book was not long—only 86 pages. But it explained Boole’s idea ofrepresenting logic using a formof algebra.Thenotion that one couldhave analgebrawithvariablesthatweren’tjustordinarynumbershappenedtohavejustarisen in Hamilton’s 1843 invention of quaternion algebra—and Boole wasinfluencedbythis.(Galoishadalsodonesomethingsimilarin1832workingongroups and finite fields.) 150 years before Boole, Gottfried Leibniz had alsothoughtaboutusingalgebra to represent logic.Buthe’dnevermanaged to seequitehow.AndtheideaseemstohavebeenallbutforgottenuntilBoolefinallysucceededindoingitin1847.LookingatBoole’sbooktoday,muchofitisquiteeasytounderstand.Here,

for example, is him showing how his algebraic formulation reproduces a fewstandardresultsinlogic:

Atasurfacelevel,thisallseemsfairlystraightforward.“And”isrepresentedbymultiplicationofvariablesxy,“not”by1–x,and“(exclusive)or”byx+y–2xy.There are also extra constraints like x²=x. Butwhen one tries digging deeper,things become considerably murkier. Just what are x and y supposed to be?Todaywe’dcalltheseBooleanvariables,andimaginetheycouldhavediscretevalues1or0,representingtrueorfalse.ButBooleseemstohaveneverwantedtotalkaboutanythingthatexplicit,oranythingdiscreteorcombinatorial.Allheever seemed to discusswas algebraic expressions and equations—even to thepointofusingseriesexpansionstoeffectivelyenumeratepossiblecombinationsofvaluesforlogicalvariables.

TheLawsofThoughtWhenBoolewrotehisfirstbookhewasstillworkingasateacherandrunningaschool.Buthehadalsobecomewellknownas amathematician, and in1849,whenQueen’sCollege,Cork(nowUniversityCollegeCork)openedinIreland,Boolewashiredasitsfirstmathprofessor.AndonceinCork,Boolestartedtowork on what would become his most famous book, An Investigation of theLawsofThought:

His preface began, “The design of the following treatise is to investigate thefundamental laws of those operations of the mind by which reasoning isperformed;togiveexpressiontotheminthesymbolicallanguageofaCalculus,and upon this foundation to establish the science of Logic and construct itsmethod;…”Boole appears to have seen himself as trying to create a calculus for the

“science of intellectual powers” analogous to Newton’s calculus for physicalscience.ButwhileNewtonhadbeenabletorelyonconceptslikespaceandtimetoinformthestructureofhiscalculus,Boolehadtobuildonthebasisofamodelofhowthemindworks,whichforhimwasunquestionablylogic.The first part of Laws of Thought is basically a recapitulation of Boole’s

earlierbookonlogic,butwithadditionalexamples—suchasachaptercoveringlogicalproofsabouttheexistenceandcharacteristicsofGod.Thesecondpartofthe book is in a sense more mathematically traditional. For instead ofinterpreting his algebraic variables as related to logic, he interprets them astraditionalnumberscorrespondingtoprobabilities—andindoingsoshowsthatthe laws for combining probabilities of events have the same structure as thelawsforcombininglogicalstatements.For the most part Laws of Thought reads like a mathematical work, with

abstractdefinitionsandformalconclusions.ButinthefinalchapterBooletriesto connectwhat hehasdone to empirical questions about theoperationof themind. He discusses how free will can be compatible with definite laws ofthought.He talks about how imprecise human experiences can lead to preciseconcepts. He discusses whether there is truth that humans can recognize thatgoesbeyondwhatmathematicallawscaneverexplain.Andhetalksabouthowanunderstandingofhumanthinkingshouldinformeducation.

TheRestofBoole’sLifeAfter thepublicationofLawsofThought,GeorgeBoolestayedinCork, livinganotherdecadeanddyingin1864ofpneumoniaattheageof49.Hecontinuedtopublishwidelyonmathematics,butneverpublishedonlogicagain,thoughheprobablyintendedtodoso.Inhislifetime,Boolewasmuchmorerecognizedforhisworkontraditional

mathematicsthanonlogic.Hewrotetwotextbooks,onein1859ondifferentialequations,andonein1860ondifferenceequations.Botharecleanandelegantexpositions. And interestingly, while there are endless modern alternatives toBoole’sDifferential Equations, sufficiently little has been done on difference

equations that when we were implementing them in Mathematica in the late1990s,Boole’s1860bookwasstillanimportantreference,notableespeciallyforitsniceexamplesofthefactorizationoflineardifferenceoperators.

WhatWasBooleLike?WhatwasBoolelikeasaperson?There’squiteabitofinformationonthis,notleast fromhiswife’swritings and from correspondence and reminiscences hissistercollectedwhenhedied.Fromwhatonecantell,Boolewasorganizedanddiligent, with careful attention to detail. He worked hard, often late into thenight, and could be so engrossed in his work that he became quite absentminded.Despitehowhelooksinpictures,heappearstohavebeenrathergenialinperson.Hewaswelllikedasateacher,andwasatalentedlecturer,thoughhisblackboard writing was often illegible. He was a gracious and extensivecorrespondent, andmademany visits to different people and places.He spentmanyyearsmanagingpeople,firstatschools,andthenattheuniversityinCork.Hehadastrongsenseofjustice,andwhilehedidnotlikecontroversy,hewasoccasionallyinvolvedinit,andwasnotshytomaintainhisposition.Despite his successes, Boole seems to have always thought of himself as a

self-taught schoolteacher, rather than a member of the academic elite. Andperhaps this helped in his ability to take intellectual risks. Whether it wasplayingfastandloosewithdifferentialoperatorsincalculus,orfindingwaystobend the laws of algebra so they could apply to logic, Boole seems to havealwaystakentheattitudeofjustmovingforwardandseeingwherehecouldgo,trustinghisownsenseofwhatwascorrectandtrue.Boolewassinglemostofhislife,thoughfinallymarriedattheageof40.His

wife,MaryEverestBoole,was17yearshisjunior,andsheoutlivedhimby52years,dyingin1916.Shehadaninterestingstoryinherownright,laterinherlifewritingbookswithtitleslikePhilosophyandFunofAlgebra,LogicTaughtbyLove,ThePreparationoftheChildforScienceandTheMessageofPsychicSciencetotheWorld.GeorgeandMaryBoolehadfivedaughters—who,alongwiththeirownchildren,hadawiderangeofcareersandaccomplishments,somequitemathematical.

LegacyIt is something of an irony that George Boole, committed as he was to themethodsofalgebra,calculusandcontinuousmathematics,shouldhavecometo

symbolize discrete variables. But to be fair, this took awhile. In the decadesafterhedied,theprimaryinfluenceofBoole’sworkonlogicwasonthewaveofabstractionandformalizationthatsweptthroughmathematics—involvingpeoplelikeFrege,Peano,Hilbert,Whitehead,RussellandeventuallyGödelandTuring.And it was only in 1937, with the work of Claude Shannon on switchingnetworks,thatBooleanalgebrabegantobeusedforpracticalpurposes.TodaythereisalotonBooleancomputationinMathematicaandtheWolfram

Language,andinfactGeorgeBooleisthepersonwiththelargestnumber(15)ofdistinctfunctionsinthesystemnamedafterthem.ButwhathasmadeBoole’snamesowidelyknownisnotBooleanalgebra,it’s

themuchsimplernotionofBooleanvariables,whichappearinessentiallyeverycomputer language—leading toaprogressive increase inmentionsof theword“Boolean”inpublicationssincethe1950s:

Was this inevitable? In somesense, I suspect itwas.Forwhenone looksathistory, sufficiently simple formal ideas have a remarkable tendency toeventuallybewidelyused,eveniftheyemergeonlyslowlyfromquitecomplexorigins. Most often what happens is that at some moment the ideas becomerelevanttotechnology,andquicklythengofromcuriositiestomainstream.MyworkonANewKindofSciencehasmademethinkaboutenumerationsof

whatamount toallpossible“simpleformal ideas”.Somehavealreadybecomeincorporated in technology, but many have not yet. But the story of GeorgeBoole and Boolean variables provides an interesting example of what canhappenover thecourseofcenturies—andhowwhatat firstseemsobscureandabstrusecaneventuallybecomeubiquitous.

AdaLovelaceDecember10,2015

AdaLovelacewasborn200yearsagotoday.Tosomesheisagreatherointhehistoryofcomputing;toothersanoverestimatedminorfigure.I’vebeencuriousforalongtimewhattherealstoryis.Andinpreparationforherbicentennial,Idecidedtotrytosolvewhatformehasalwaysbeenthe“mysteryofAda”.ItwasmuchharderthanIexpected.Historiansdisagree.Thepersonalitiesin

thestoryarehardtoread.Thetechnologyisdifficulttounderstand.Thewholestory is entwined with the customs of 19th-century British high society. Andthere’sasurprisingamountofmisinformationandmisinterpretationoutthere.But after quite a bit of research—including going to see many original

documents—Ifeel likeI’vefinallygotten toknowAdaLovelace,andgottenagrasponherstory.Insomewaysit’sanennoblingandinspiringstory;insomewaysit’sfrustratingandtragic.It’s a complex story, and tounderstand it,we’llhave to startbygoingover

quitealotoffactsandnarrative.

TheEarlyLifeofAdaLet’s begin at the beginning.AdaByron, as shewas then called,was born inLondon on December 10, 1815 to recently married high-society parents. Herfather, Lord Byron (George Gordon Byron) was 27 years old, and had justachieved rock-star status in England for his poetry. Her mother, AnnabellaMilbanke, was a 23-year-old heiress committed to progressive causes, whoinherited the title BaronessWentworth.Her father said he gave her the name“Ada”because“Itisshort,ancient,vocalic”.Ada’sparentsweresomethingofastudyinopposites.Byronhadawildlife—

andbecameperhapsthetop“badboy”ofthe19thcentury—withdarkepisodesinchildhood,andlotsoflaterromanticandotherexcesses.Inadditiontowritingpoetryandfloutingthesocialnormsofhistime,hewasoftendoingtheunusual:keepingatamebearinhiscollegeroomsinCambridge,livingitupwithpoetsinItaly and “five peacocks on the grand staircase”, writing a grammar book ofArmenian,and—hadhenotdiedtoosoon—leadingtroopsintheGreekwarofindependence (as celebrated by a big statue in Athens), despite having nomilitarytrainingwhatsoever.Annabella Milbanke was an educated, religious and rather proper woman,

interested in reform and good works, and nicknamed by Byron “Princess ofParallelograms”.HerverybriefmarriagetoByronfellapartwhenAdawasjust5weeksold,andAdaneversawByronagain(thoughhekeptapictureofheronhisdeskandfamouslymentionedherinhispoetry).Hediedattheageof36,atthe height of his celebrityhood, when Ada was 8. There was enough scandalaroundhimtofuelhundredsofbooks,andthePRbattlebetweenthesupportersofLadyByron(asAda’smotherstyledherself)andofhimlastedacenturyormore.Ada led an isolated childhood on hermother’s rented country estates, with

governessesandtutorsandherpetcat,Mrs.Puff.Hermother,oftenabsentforvarious(quitewacky)healthcures,enforcedasystemofeducationforAdathatinvolvedlonghoursofstudyandexercisesinselfcontrol.Adalearnedhistory,literature, languages, geography, music, chemistry, sewing, shorthand andmathematics (taught in part through experiential methods) to the level ofelementarygeometryandalgebra.WhenAdawas11,shewentwithhermotherand an entourage on a year-long tour of Europe.When she returned she wasenthusiastically doing things like studying what she called “flyology”—andimagininghowtomimicbirdflightwithsteam-poweredmachines.

Butthenshegotsickwithmeasles(andperhapsencephalitis)—andendedupbedriddenandinpoorhealthfor3years.Shefinallyrecoveredintimetofollowthecustomforsocietygirlsoftheperiod:onturning17shewenttoLondonforaseason of socializing. On June 5, 1833, 26 days after she was “presented atCourt”(i.e.mettheking),shewenttoapartyatthehouseof41-year-oldCharlesBabbage(whoseoldestsonwasthesameageasAda).Apparentlyshecharmedthehost,andheinvitedherandhermothertocomebackforademonstrationofhis newly constructed Difference Engine: a 2-foot-high hand-crankedcontraption with 2000 brass parts, now to be seen at the ScienceMuseum inLondon:

Ada’s mother called it a “thinking machine”, and reported that it “raisedseveral Nos. to the 2nd & 3rd powers, and extracted the root of a QuadraticEquation”.ItwouldchangethecourseofAda’slife.

CharlesBabbageWhat was the story of Charles Babbage? His father was an enterprising andsuccessful (if personally distant) goldsmith and banker. After various schoolsand tutors, Babbage went to Cambridge to study mathematics, but soon wasintent on modernizing the way mathematics was done there, and with hislifelong friends John Herschel (son of the discoverer of Uranus) and GeorgePeacock (later a pioneer in abstract algebra), founded the Analytical Society(which laterbecame theCambridgePhilosophicalSociety) topushfor reformslike replacing Newton’s (“British”) dot-based notation for calculus withLeibniz’s(“Continental”)function-basedone.BabbagegraduatedfromCambridgein1814(ayearbeforeAdaLovelacewas

born),wenttoliveinLondonwithhisnewwife,andstartedestablishinghimselfontheLondonscientificandsocialscene.Hedidn’thaveajobassuch,butgavepubliclecturesonastronomyandwroterespectableifunspectacularpapersaboutvariousmathematical topics (functional equations, continuedproducts, numbertheory, etc.)—and was supported, if modestly, by his father and his wife’sfamily.

In1819BabbagevisitedFrance,andlearnedaboutthelarge-scalegovernmentproject there to make logarithm and trigonometry tables.Mathematical tableswereofconsiderablemilitaryandcommercialsignificanceinthosedays,beingusedacrossscience,engineeringandfinance,aswellasinareaslikenavigation.It was often claimed that errors in tables could make ships run aground orbridgescollapse.BackinEngland,BabbageandHerschelstartedaprojecttoproducetablesfor

theirnewAstronomicalSociety,anditwasintheefforttocheckthesetablesthatBabbageissaidtohaveexclaimed,“IwishtoGodthesetableshadbeenmadebysteam!”—andbeganhislifelongefforttomechanizetheproductionoftables.

StateoftheArtThere were mechanical calculators long before Babbage. Pascal made one in1642,andwenowknowtherewasevenoneinantiquity.ButinBabbage’sdaysuch machines were still just curiosities, not reliable enough for everydaypractical use. Tables weremade by human computers, with the work dividedacross a team, and the lowest-level computations being based on evaluatingpolynomials(sayfromseriesexpansions)usingthemethodofdifferences.What Babbage imagined is that there could be a machine—a Difference

Engine—thatcouldbesetuptocomputeanypolynomialuptoacertaindegreeusingthemethodofdifferences,andthenautomaticallystepthroughvaluesandprinttheresults,takinghumansandtheirpropensityforerrorsentirelyoutoftheloop.

Byearly1822,the30-year-oldBabbagewasbusystudyingdifferenttypesofmachinery, andproducingplansandprototypesofwhat theDifferenceEnginecouldbe.TheAstronomicalSocietyhe’dco-foundedawardedhimamedalfortheidea,andin1823theBritishgovernmentagreedtoprovidefundingfor theconstructionofsuchanengine.Babbage was slightly distracted in 1824 by the prospect of joining a life

insurancestartup,forwhichhedidacollectionoflife-tablecalculations.Buthesetupaworkshop inhisstable (his“garage”),andkeptonhaving ideasabouttheDifferenceEngineandhowitscomponentscouldbemadewiththetoolsofhistime.In 1827, Babbage’s table of logarithms—computed by hand—was finally

finished,andwouldbereprintedfornearly100years.Babbagehadthemprintedonyellowpaperonthetheorythatthiswouldminimizeusererror.(WhenIwasin elementary school, logarithm tables were still the fast way to domultiplication.)

Alsoin1827,Babbage’sfatherdied,leavinghimabout£100k,orperhaps$14million today, settingupBabbage financially for the restofhis life.Thesameyear,though,hiswifedied.Shehadhadeightchildrenwithhim,butonlythreesurvivedtoadulthood.Dispiritedbyhiswife’sdeath,BabbagetookatriptocontinentalEurope,and

beingimpressedbywhathesawofthesciencebeingdonethere,wroteabookentitledReflectionson theDeclineofScience inEngland, that endedupbeingmainlyadiatribeagainsttheRoyalSociety(ofwhichhewasamember).

Though often distracted, Babbage continued to work on the DifferenceEngine,generatingthousandsofpagesofnotesanddesigns.Hewasquitehandson when it came to personally drafting plans or doing machine-shopexperiments.Buthewasquitehandsoff inmanaging theengineershehired—andhedidnotdowellatmanagingcosts.Still,by1832aworkingprototypeofasmall Difference Engine (without a printer) had successfully been completed.AndthisiswhatAdaLovelacesawinJune1833.

BacktoAdaAda’s encounter with the Difference Engine seems to be what ignited herinterestinmathematics.ShehadgottentoknowMarySomerville,translatorofLaplace and a well-known expositor of science—and partly with herencouragement,wassoon,forexample,enthusiasticallystudyingEuclid.Andin1834,Adawentalongonaphilanthropic tourofmills in thenorthofEnglandthat her mother was doing, and was quite taken with the then-high-techequipmenttheyhad.

Onthewayback,Adataughtsomemathematicstothedaughtersofoneofhermother’s friends. She continued by mail, noting that this could be “thecommencementof‘ASentimentalMathematicalCorrespondencecarriedonforyears between two ladies of rank’ to be hereafter published no doubt for theedificationofmankind,orwomankind.”Itwasn’tsophisticatedmath,butwhatAda said was clear, and complete with admonitions like, “You should neverselect an indirect proof, when a direct one can be given.” (There’s a lot ofunderlining,hereshownasitalics,inallAda’shandwrittencorrespondence.)Babbage seemsat first tohaveunderestimatedAda, trying to interesther in

theSilverLadyautomatontoythatheusedasaconversationpieceforhisparties(andnotinghisadditionofaturbantoit).ButAdacontinuedtointeractwith(assheputit)Mr.BabbageandMrs.Somerville,bothseparatelyandtogether.AndsoonBabbagewasopeninguptoheraboutmanyintellectualtopics,aswellasabout the trouble he was having with the government over funding of theDifferenceEngine.In thespringof1835,whenAdawas19,shemet30-year-oldWilliamKing

(or, more accurately, William, Lord King). He was a friend of MarySomerville’sson,hadbeeneducatedatEton(thesameschoolwhereIwent150yearslater)andCambridge,andthenhadbeenacivilservant,mostrecentlyatan outpost of theBritishEmpire in theGreek islands.William seems to havebeenaprecise,conscientiousanddecentman,ifsomewhatstiff.Butinanycase,Adaandhehititoff,andtheyweremarriedonJuly8,1835,withAdakeepingthenewsquietuntilthelastminutetoavoidpaparazzi-likecoverage.ThenextseveralyearsofAda’slifeseemtohavebeendominatedbyhaving

threechildrenandmanagingalargehousehold—thoughshehadsometimeforhorseriding,learningtheharp,andmathematics(includingtopicslikesphericaltrigonometry). In1837,QueenVictoria (then18)came to the throne, andasamemberofhighsociety,Adamether.In1838,Williamwasmadeanearlforhisgovernmentwork,andAdabecometheCountessofLovelace.

Withinafewmonthsofthebirthofherthirdchildin1839,Adadecidedtogetmore seriousaboutmathematicsagain.She toldBabbageshewanted to finda“mathematicalInstructor”inLondon,thoughaskedthatinmakingenquirieshenotmentionhername,presumablyforfearofsocietygossip.The person identified was Augustus De Morgan, first professor of

mathematics at University College London, noted logician, author of severaltextbooks, and not only a friend of Babbage’s, but also the husband of thedaughterofAda’smother’smainchildhoodteacher.(Yes,itwasasmallworld.De Morgan was also a friend of George Boole’s—and was the person whoindirectlycausedBooleanalgebratobeinvented.)In Ada’s correspondence with Babbage, she showed interest in discrete

mathematics, and wondered, for example, if the game of solitaire “admits ofbeing put into amathematical Formula, and solved.” But in keepingwith themath education traditionsof the time (and still today),DeMorgan setAdaonstudyingcalculus.

HerletterstoDeMorganaboutcalculusarenotunlikelettersfromacalculusstudent today—except for theVictorianEnglish.Evenmanyof theconfusionsarethesame—thoughAdawasmoresensitivethansometothebadnotationsofcalculus(“whycan’tonemultiplybydx?”,etc.).Adawasa tenaciousstudent,andseemedtohavehadagreattimelearningmoreandmoreaboutmathematics.Shewaspleasedbythemathematicalabilitiesshediscoveredinherself,andbyDeMorgan’spositivefeedbackaboutthem.ShealsocontinuedtointeractwithBabbage,andononevisittoherestate(inJanuary1841,whenshewas25),shecharminglytoldthethen-49-year-oldBabbage,“IfyouareaSkater,praybringSkatestoOckham;thatbeingthefashionableoccupationherenow,&oneIhavemuchtakento.”

Ada’s relationship with her mother was a complex one. Outwardly, Adatreatedhermotherwithgreatrespect.Butinmanywayssheseemstohavefoundhercontrollingandmanipulative.Ada’smotherwasconstantlyannouncingthatshehadmedicalproblemsandmightdie imminently (sheactually lived toage64). And she increasingly criticized Ada for her child rearing, householdmanagement and deportment in society. But by February 6, 1841, Ada wasfeeling good enough about herself and her mathematics to write a very openlettertohermotheraboutherthoughtsandaspirations.She wrote, “I believe myself to possess a most singular combination of

qualities exactly fitted to make me pre-eminently a discoverer of the hiddenrealitiesofnature.”Shetalkedofherambitiontodogreatthings.Shetalkedofher “insatiable& restless energy”which she believed she finally had found apurpose for. And she talked about how after 25 years she had become less“secretive&suspicious”withrespecttohermother.But then, threeweeks later, hermother dropped a bombshell, claiming that

beforeAdawasborn,Byronandhishalf-sisterhadhadachildtogether.Incestlike thatwasn’t actually illegal in England at the time, but itwas scandalous.Adatookthewholethingveryhard,anditderailedherfrommathematics.Ada had had intermittent health problems for years, but in 1841 they

apparentlyworsened,andshestartedsystematicallytakingopiates.Shewasverykeentoexcelinsomething,andbegantogettheideathatperhapsitshouldbemusicand literature rather thanmath.ButherhusbandWilliamseems tohavetalkedheroutofthis,andbylate1842shewasbacktodoingmathematics.

BacktoBabbageWhathadBabbagebeenuptowhileallthishadbeengoingon?He’dbeendoingallsortsofthings,withvaryingdegreesofsuccess.After several attempts, he’d rather honorifically been appointed Lucasian

Professor ofMathematics at Cambridge—but never really even spent time inCambridge.Still,hewrotewhatturnedouttobeafairlyinfluentialbook,OntheEconomyofMachineryandManufactures, dealingwith such things ashow tobreakuptasksinfactories(anissuethathadactuallycomeupinconnectionwiththehumancomputationofmathematicaltables).

In 1837, he weighed in on the then-popular subject of natural theology,appending hisNinth Bridgewater Treatise to the series of treatiseswritten byotherpeople.Thecentralquestionwaswhetherthereisevidenceofadeityfromtheapparentdesignseeninnature.Babbage’sbookisquitehardtoread,openingforexamplewith,“Thenotionsweacquireofcontrivanceanddesignarisefromcomparingourobservationsontheworksofotherbeingswiththeintentionsofwhichweareconsciousinourownundertakings.”In apparent resonancewith some ofmy ownwork 150 years later, he talks

abouttherelationshipbetweenmechanicalprocesses,naturallawsandfreewill.Hemakesstatementslike“computationsofgreatcomplexitycanbeeffectedbymechanicalmeans”,butthengoesontoclaim(withratherweakexamples)thatamechanical engine can produce sequences of numbers that show unexpectedchangesthatarelikemiracles.

Babbage tried his hand at politics, running for parliament twice on amanufacturing-oriented platform, but failed to get elected, partly because ofclaimsofmisuseofgovernmentfundsontheDifferenceEngine.Babbagealsocontinued tohaveupscalepartiesathis largeand increasingly

disorganized house in London, attracting such luminaries as Charles Dickens,Charles Darwin, Florence Nightingale, Michael Faraday and the Duke ofWellington—withhisagedmotherregularlyinattendance.Buteventhoughthedegrees and honors that he listed after his name ran to 6 lines, he wasincreasinglybitterabouthisperceivedlackofrecognition.

Central tothiswaswhathadhappenedwiththeDifferenceEngine.Babbagehadhiredoneof the leadingengineersofhisday to actuallybuild the engine.But somehow, after a decade ofwork—and despite lots of precisionmachinetooldevelopment—theactualenginewasn’tdone.Backin1833,shortlyafterhemetAda,Babbagehad tried to rein in the project—but the resultwas that hisengineer quit, and insisted that he got to keep all the plans for theDifferenceEngine,eventheonesthatBabbagehimselfhaddrawn.But rightaround this time,Babbagedecidedhe’dhadabetter ideaanyway.

Insteadofmakingamachinethatwouldjustcomputedifferences,heimaginedan “Analytical Engine” that supported a whole list of possible kinds ofoperations,thatcouldineffectbedoneinanarbitrarilyprogrammedsequence.Atfirst,hejustthoughtabouthavingthemachineevaluatefixedformulas,butashestudieddifferentusecases,headdedothercapabilities,likeconditionals—andfiguredoutoftenverycleverwaystoimplement themmechanically.But,mostimportant, he figured out how to control the steps in a computation usingpunched cards of the kind that had been invented in 1801 by Jacquard forspecifyingpatternsofweavingonlooms.

Babbage created some immensely complicated designs, and today it seemsremarkable that they could work. But back in 1826 Babbage had inventedsomething he called Mechanical Notation—that was intended to provide asymbolicrepresentationfortheoperationofmachineryinthesamekindofwaythatmathematicalnotationprovidesasymbolicrepresentationforoperationsinmathematics.Babbagewas disappointed already in 1826 that people didn’t appreciate his

invention.Undoubtedlypeopledidn’tunderstandit,sinceevennowit’snotclearhow it worked. But it may have been Babbage’s greatest invention—becauseapparentlyit’swhatlethimfigureoutallhiselaboratedesigns.Babbage’soriginalDifferenceEngineprojecthadcosttheBritishgovernment

£17,500 or the equivalent of perhaps $2 million today. It was a modest sumrelativetoothergovernmentexpenditures,buttheprojectwasunusualenoughtolead to a fair amount of discussion.Babbagewas fond of emphasizing that—unlikemanyofhiscontemporaries—hehadn’ttakengovernmentmoneyhimself(despitechargebacksforrenovatinghisstableasafireproofworkshop,etc.).Healso claimed that he eventually spent £20,000 of his own money—or themajorityofhisfortune(no,Idon’tseehowthenumbersaddup)—onhisvariousprojects.Andhekepton trying toget furthergovernmentsupport,andcreatedplansforaDifferenceEngineNo.2,requiringonly8000partsinsteadof25,000.By1842,thegovernmenthadchanged,andBabbageinsistedonmeetingwith

the new prime minister (Robert Peel), but ended up just berating him. Inparliament the idea of funding the Difference Engine was finally killed withquips like that themachineshouldbeset tocomputewhen itwouldbeofuse.(ThetranscriptsofdebatesabouttheDifferenceEnginehaveacertaincharm—especiallywhen they discuss its possible uses for state statistics that strangelyparallelcomputable-countryopportunitieswithWolfram|Alphatoday.)

Ada’sPaperDespite the lack of support in England, Babbage’s ideas developed somepopularity elsewhere, and in 1840 Babbage was invited to lecture on theAnalyticalEngineinTurin,andgivenhonorsbytheItaliangovernment.BabbagehadneverpublishedaseriousaccountoftheDifferenceEngine,and

hadneverpublishedanythingatallabout theAnalyticalEngine.Buthe talkedabout theAnalyticalEngine inTurin, andnoteswere takenby a certainLuigiMenabrea,whowasthena30-year-oldarmyengineer—butwho,27yearslater,becameprimeministerofItaly(andalsomadecontributionstothemathematics

ofstructuralanalysis).InOctober1842,Menabreapublishedapaper inFrenchbasedonhisnotes.

WhenAdasawthepaper,shedecidedtotranslateitintoEnglishandsubmitittoaBritish publication.Many years later Babbage claimed he suggested toAdathat she write her own account of the Analytical Engine, and that she hadrespondedthatthethoughthadn’toccurredtoher.Butinanycase,byFebruary1843,Adahadresolvedtodothetranslationbutaddextensivenotesofherown.

Over the months that followed she worked very hard—often exchanginglettersalmostdailywithBabbage(despitesometimeshavingother“pressingandunavoidableengagements”).Andthoughinthosedayslettersweresentbypost(whichdidcome6 timesaday inLondonat the time)or carriedbya servant(AdalivedaboutamilefromBabbagewhenshewasinLondon),theyreadalotlikeemailsaboutaprojectmight today,apart frombeinginVictorianEnglish.AdaasksBabbagequestions;heresponds;shefiguresthingsout;hecommentson them. She was clearly in charge, but felt she was first and foremostexplainingBabbage’swork, sowanted to check thingswith him—though shegotannoyedwhenBabbage, forexample, tried tomakehisowncorrections tohermanuscript.It’scharmingtoreadAda’sletterassheworksondebugginghercomputation

ofBernoullinumbers:“MyDearBabbage.Iaminmuchdismayathavinggotinto so amazing a quagmire& botherationwith theseNumbers, that I cannotpossibly get the thing done today.….I am nowgoing out on horseback. Tantmieux.” Later she told Babbage, “I have worked incessantly, & mostsuccessfully, all day. Youwill admire the Table&Diagram extremely. Theyhave been made out with extreme care, & all the indices most minutely &scrupulously attended to.” Then she added thatWilliam (or “Lord L.” as shereferredtohim)“isatthismomentkindlyinkingitalloverforme.Ihadtodoitinpencil…”

William was also apparently the one who suggested that she sign thetranslationandnotes.AsshewrotetoBabbage,“It isnotmywishtoproclaimwhohaswrittenit;at thesametimeIratherwishtoappendanythingthatmaytend hereafter to individualize,& identify it,with the other productions of thesaidA.A.L.”(for“AdaAugustaLovelace”).BytheendofJuly1843,Adahadprettymuchfinishedwritinghernotes.She

wasproudofthem,andBabbagewascomplimentaryaboutthem.ButBabbagewanted one more thing: he wanted to add an anonymous preface (written byhim)thatexplainedhowtheBritishgovernmenthadfailedtosupporttheproject.Adathoughtitabadidea.Babbagetriedtoinsist,evensuggestingthatwithoutthe preface thewhole publication should bewithdrawn.Adawas furious, andtoldBabbageso.Intheend,Ada’stranslationappeared,signed“AAL”,withoutthepreface,followedbyhernotesheaded“Translator’sNote”.Ada was clearly excited about it, sending reprints to her mother, and

explaining that “No one can estimate the trouble & interminable labour ofhaving to revise the printing of mathematical formulae. This is a pleasantprospect for the future, as I suppose many hundreds & thousands of suchformulaewillcomeforthfrommypen,inonewayoranother.”ShesaidthatherhusbandWilliamhadbeenexcitedlygivingawaycopiestohisfriendstoo,andAdawrote,“Williamespeciallyconceivesthat itplacesmeinamuch juster&truerposition&light,thananythingelsecan.Andhetellsmethatithasalreadyplacedhiminafarmoreagreeablepositioninthiscountry.”Withindays,therewasalsoapparentlysocietygossipaboutAda’spublication.

SheexplainedtohermotherthatsheandWilliam“arebynomeansdesirousofmaking it a secret, altho’ I do not wish the importance of the thing to beexaggeratedandoverrated”.ShesawherselfasbeingasuccessfulexpositorandinterpreterofBabbage’swork,settingitinabroaderconceptualframeworkthatshehopedcouldbebuilton.There’slotstosayabouttheactualcontentofAda’snotes.Butbeforeweget

tothat,let’sfinishthestoryofAdaherself.WhileBabbage’sprefacewasn’t itselfagreatidea,onegoodthingitdidfor

posteritywastocauseAdaonAugust14,1843towriteBabbageafascinating,andveryforthright,16-pageletter.(Unlikeherusualletters,whichwereonlittlefoldedpages,thiswasonlargesheets.)Init,sheexplainsthatwhileheisoften“implicit”inwhathesays,sheisherself“alwaysavery‘explicitfunctionofx’”.Shesaysthat“Youraffairshavebeen,&are,deeplyoccupyingbothmyselfandLord Lovelace…. And the result is that I have plans for you…” Then sheproceedstoask,“IfIamtolaybeforeyouinthecourseofayearortwo,explicit

& honorable propositions for executing your engine … would there be anychance of allowing myself … to conduct the business for you; your ownundividedenergiesbeingdevotedtotheexecutionofthework…”

In other words, she basically proposed to take on the role of CEO, withBabbage becoming CTO. It wasn’t an easy pitch to make, especially givenBabbage’spersonality.Butshewasskillfulinmakinghercase,andaspartofit,she discussed their different motivation structures. She wrote, “My ownuncompromisingprinciple is toendeavour to love truth&Godbefore fame&glory …”, while “Yours is to love truth & God… but to love fame, glory,honours, yet more.” Still, she explained, “Far be it from me, to disclaim theinfluenceofambition&fame.Nolivingsouleverwasmoreimbuedwithitthanmyself…butIcertainlywouldnotdeceivemyselforothersbypretendingitisotherthanaveryimportantmotive&ingredientinmycharacter&nature.”Sheended the letter,“Iwonder ifyouwillchoose toretain the lady-fairy in

yourserviceornot.”AtnoonthenextdayshewrotetoBabbageagain,askingifhewouldhelpin

“the final revision”. Then she added, “You will have hadmy long letter thismorning.Perhapsyouwillnotchoosetohaveanythingmoretodowithme.ButIhopethebest…”

At 5 pm that day, Ada was in London, and wrote to her mother, “I amuncertainasyethowtheBabbagebusinesswillend….Ihavewrittentohim…very explicitly; statingmyownconditions…Hehas so strong an idea of theadvantageofhavingmypenashisservant,thathewillprobablyyield;thoughIdemandvery strong concessions. If hedoes consent towhat I propose, I shallprobablybeenabledtokeephimoutofmuchhotwater;&tobringhisenginetoconsummation, (which all I have seen of him& his habits the last 3months,makesmescarcelyanticipate iteverwillbe,unlesssomeonereallyexercisesastrongcoerciveinfluenceoverhim).Heisbeyondmeasurecareless&desultoryat times.—IshallbewillingtobehisWhipper-induringthenext3yearsifIseefairprospectofsuccess.”But on Babbage’s copy of Ada’s letter, he scribbled, “Saw A.A.L. this

morningandrefusedalltheconditions.”

YetonAugust18,BabbagewrotetoAdaaboutbringingdrawingsandpaperswhenhewouldnext come tovisit her.Thenextweek,Adawrote toBabbagethat “We are quite delighted at your (somewhat unhoped for) proposal” [of alongvisitwithAdaandherhusband].AndAdawrotetohermother,“Babbage&Iare I thinkmore friends thanever. Ihaveneverseenhimsoagreeable, soreasonable,orinsuchgoodspirits!”

Then,onSeptember9,Babbagewrote toAda,expressinghisadmirationforher and (famously) describing her as “Enchantress ofNumber” and “my dearand much admired Interpreter”. (Yes, despite what’s often quoted, he wrote“Number”not“Numbers”.)

The next day, Ada responded to Babbage, “You are a brave man to giveyourself wholly up to Fairy-Guidance!” and Babbage signed off on his nextletter as “Your faithful Slave.” And Ada described herself to her mother asservingasthe“High-PriestessofBabbage’sEngine.”

AfterthePaperButunfortunatelythat’snothowthingsworkedout.ForawhileitwasjustthatAdahadtotakecareofhouseholdandfamilythingsthatshe’dneglectedwhileconcentratingonherNotes.But thenherhealthcollapsed,andshespentmanymonths going between doctors and various “cures” (her mother suggested“mesmerism”,i.e.hypnosis),allthewhilewatchingtheireffectson,assheputit,“thatportionofthematerialforcesoftheworldentitledthebodyofA.A.L.”She was still excited about science, though. She’d interacted withMichael

Faraday, who apparently referred to her as “the rising star of Science”. Shetalked about her first publication as her “first-born”, “with a colouring &undercurrent (ratherhintedat& suggested thandefinitely expressed)of large,general,&metaphysicalviews”,andsaidthat“He[thepublication]willmakeanexcellenthead(Ihope)ofalargefamilyofbrothers&sisters”.Whenhernoteswerepublished,Babbagehadsaid,“Youshouldhavewritten

anoriginal paper.The postponement of thatwill however only render itmoreperfect.”ButbyOctober1844, it seemed thatDavidBrewster (inventorof thekaleidoscope,amongotherthings)wouldwriteabouttheAnalyticalEngine,andAda asked if perhapsBrewster could suggest another topic for her, saying, “Iratherthinksomephysiologicaltopicswouldsuitmeaswellasany.”Andindeedlaterthatyear,shewrotetoafriend(whowasalsoherlawyer,as

well as beingMarySomerville’s son), “It does not appear tome that cerebralmatterneedbemoreunmanageabletomathematiciansthansidereal&planetarymatter&movements;iftheywouldbutinspectitfromtherightpointofview.Ihope to bequeath to the generations a Calculus of the Nervous System.” Animpressive vision—10 years before, for example, George Boole would talkaboutsimilarthings.Both Babbage and Mary Somerville had started their scientific publishing

careers with translations, and she saw herself as doing the same, saying thatperhaps her nextworkswould be reviews ofWhewell andOhm, and that shemighteventuallybecomeageneral“prophetofscience”.Therewere roadblocks, to be sure.Like that, at that time, as awoman, she

couldn’t get access to theRoyalSociety’s library inLondon, even thoughher

husband,partlythroughherefforts,wasamemberofthesociety.ButthemostseriousissuewasstillAda’shealth.Shehadawholeseriesofproblems,thoughin1846shewasstillsayingoptimistically,“Nothingisneededbutayearortwomoreofpatience&cure.”Therewerealsoproblemswithmoney.Williamhadanever-endingseriesof

elaborate—andoftenquiteinnovative—constructionprojects(heseemstohavebeenparticularlykeenontowersandtunnels).Andtofinancethem,theyhadtoturntoAda’smother,whooftenmadethingsdifficult.Ada’schildrenwerealsoapproaching teenage-hood, and Ada was exercised by many issues that werecomingupwiththem.Meanwhile, she continued to have a good social relationshipwithBabbage,

seeinghimwithconsiderablefrequency,thoughinherletterstalkingmoreaboutdogsandpetparrots thantheAnalyticalEngine.In1848Babbagedevelopedahare-brainedschemetoconstructanenginethatplayedtic-tac-toe,andtotouritaroundthecountryasawaytoraisemoneyforhisprojects.Adatalkedhimoutof it. The idea was raised for Babbage to meet Prince Albert to discuss hisengines,butitneverhappened.William also dipped his toe into publishing. He had already written short

reportswith titles like “Method of growingBeans andCabbages on the sameGround”and“On theCultureofMangold-Wurzel”.But in1848hewroteonemoresubstantialpiece,comparingtheproductivityofagricultureinFranceandEngland,basedondetailedstatistics,withobservationslike,“Itisdemonstrable,notonlythattheFrenchmanismuchworseoffthantheEnglishman,butthatheislesswellfedthanduringthedevastatingexhaustionoftheempire.”

1850wasanotableyearforAda.SheandWilliammovedintoanewhouseinLondon, intensifying their exposure to theLondon scientific social scene. Shehadahighlyemotionalexperiencevisitingforthefirsttimeherfather’sfamily’sformerestateinthenorthofEngland—andgotintoanargumentwithhermotherabout it.Andshegotmoredeeply involved inbettingonhorseracing,and lostsomemoneydoingit.(Itwouldn’thavebeenoutofcharacterforBabbageorherto invent somemathematical scheme for betting, but there’s no evidence thattheydid.)InMay 1851 theGreat Exhibition opened at theCrystal Palace in London.

(When Ada visited the site back in January, Babbage advised, “Pray put onworstedstockings,corksolesandeveryotherthingwhichcankeepyouwarm.”)TheexhibitionwasahighpointofVictorianscienceandtechnology,andAda,Babbageandtheirscientificsocialcirclewereallinvolved(thoughBabbagelesssothanhe thoughtheshouldbe).BabbagegaveoutmanycopiesofaflyeronhisMechanicalNotation.Williamwonanawardforbrick-making.Butwithin a year,Ada’s health situationwas dire. For awhile her doctors

were just telling her to spend more time at the seaside. But eventually theyadmitted shehad cancer (fromwhatweknownow, probably cervical cancer).Opium no longer controlled her pain; she experimented with cannabis. ByAugust1852,shewrote,“IbegintounderstandDeath;whichisgoingonquietly&graduallyeveryminute,&willneverbea thingofoneparticularmoment.”AndonAugust19,sheaskedBabbage’sfriendCharlesDickenstovisitandreadheranaccountofdeathfromoneofhisbooks.Hermothermovedintoherhouse,keepingotherpeopleawayfromher,and

on September 1, Ada made an unknown confession that apparently upsetWilliam.Sheseemedclosetodeath,butshehungon,ingreatpain,fornearly3moremonths, finallydyingonNovember27,1852, at theageof36.FlorenceNightingale,nursingpioneer and friendofAda’s,wrote, “They said shecouldnot possibly have lived so long,were it not for the tremendous vitality of thebrain,thatwouldnotdie.”AdahadmadeBabbagetheexecutorofherwill.And—muchtohermother’s

chagrin—she had herself buried in the Byron family vault next to her father,who, likeher, died at age36 (Ada lived266days longer).Hermother built amemorialthatincludedasonnetentitled“TheRainbow”thatAdawrote.

TheAftermathAda’s funeral was small; neither her mother nor Babbage attended. But theobituarieswerekind,ifVictorianintheirsentiments:

William outlived her by 41 years, eventually remarrying. Her oldest son—withwhomAdahadmanydifficulties—joinedthenavyseveralyearsbeforeshedied, but deserted. Ada thought he might have gone to America (he wasapparently in San Francisco in 1851), but in fact he died at 26 working in ashipyardinEngland.Ada’sdaughtermarriedasomewhatwildpoet,spentmanyyears in theMiddleEast,andbecametheworld’sforemostbreederofArabianhorses.Ada’syoungestsoninheritedthefamilytitle,andspentmostofhislifeonthefamilyestate.Ada’s mother died in 1860, but even then the gossip about her and Byron

continued,withbooksandarticlesappearing,includingHarrietBeecherStowe’s1870LadyByronVindicated.In1905,ayearbeforehedied,Ada’syoungestson—whohadbeenlargelybroughtupbyAda’smother—publishedabookaboutthewholething,withsuchchoicelinesas“LordByron’slifecontainednothingofanyinterestexceptwhatoughtnottohavebeentold”.WhenAdadied,therewasacertainairofscandalthatseemedtohangaround

her.Had shehad affairs?Had she runuphugegamblingdebts?There’s scantevidenceofeither.Perhapsitwasareflectionofherfather’s“badboy”image.Butbeforelongtherewereclaimsthatshe’dpawnedthefamilyjewels(twice!),or lost, some said, £20,000, or maybe even £40,000 (equivalent to about $7milliontoday)bettingonhorses.It didn’t help thatAda’smother and her youngest son both seemed against

her.OnSeptember1,1852—thesamedayasherconfession toWilliam—Adahadwritten, “It ismy earnest and dying request that allmy friendswho haveletters from me will deliver them to my mother Lady Noel Byron after mydeath.” Babbage refused. But others complied, and, later on, when her sonorganizedthem,hedestroyedsome.But many thousands of pages of Ada’s documents still remain, scattered

around the world. Back-and-forth letters that read like a modern text stream,setting upmeetings, ormentioning colds and other ailments.CharlesBabbagecomplainingabout thepostal service.ThreeGreeksisters seekingmoney fromAda because their dead brother had been a page for Lord Byron. CharlesDickens talking about chamomile tea. Pleasantries from a person Ada met atPaddington Station. And household accounts, with entries for note paper,musicians, and ginger biscuits. And then, mixed in with all the rest, seriousintellectualdiscussionabouttheAnalyticalEngineandmanyotherthings.

WhatHappenedtoBabbage?

SowhathappenedtoBabbage?Helived18moreyearsafterAda,dyingin1871.He triedworking on the Analytical Engine again in 1856, butmade no greatprogress.Hewrote paperswith titles like “On theStatistics ofLight-Houses”,“Table of the Relative Frequency of Occurrences of the Causes of BreakingPlate-GlassWindows”,and“OnRemainsofHumanArt,mixedwiththeBonesofExtinctRacesofAnimals”.Then in 1864 he published his autobiography,Passages from the Life of a

Philosopher—a strange and rather bitter document. The chapter on theAnalyticalEngine openswith a quote froma poembyByron—“Manwrongs,andTimeavenges”—andgoesonfromthere.Therearechapterson“Theatricalexperience”,“Hintsfortravellers”(includingonadviceabouthowtogetanRV-like carriage in Europe), and, perhaps most peculiar, “Street nuisances”. Forsome reason Babbage waged a campaign against street musicians who heclaimedwokehimupat6am,andcausedhimtoloseaquarterofhisproductivetime. One wonders why he didn’t invent a sound-masking solution, but hiscampaignwassonotable,andsoodd,thatwhenhedieditwasaleadingelementofhisobituary.Babbageneverremarriedafterhiswifedied,andhislastyearsseemtohave

beenlonelyones.Agossipcolumnofthetimerecordsimpressionsofhim:

Apparentlyhewasfondofsayingthathewouldgladlygiveuptheremainderofhislifeifhecouldspendjust3days500yearsinthefuture.Whenhedied,hisbrainwaspreserved,andisstillondisplay…Even though Babbage never finished his Difference Engine, a Swedish

company did, and even already displayed part of it at the Great Exhibition.When Babbage died, many documents and spare parts from his DifferenceEngineprojectpassedtohissonMajor-GeneralHenryBabbage,whopublishedsomeofthedocuments,andprivatelyassembledafewmoredevices,includingpart of the Mill for the Analytical Engine. Meanwhile, the fragment of theDifferenceEngine that had been built inBabbage’s timewas deposited at theScienceMuseuminLondon.

RediscoveryAfterBabbagedied,hislifeworkonhisengineswasallbutforgotten(thoughhedid, for example, get a mention in the 1911 Encyclopædia Britannica).Mechanicalcomputersneverthelesscontinuedtobedeveloped,graduallygivingway to electromechanical ones, and eventually to electronic ones. And whenprogrammingbegantobeunderstoodinthe1940s,Babbage’swork—andAda’sNotes—wererediscovered.People knew that “AAL” was Ada Augusta Lovelace, and that she was

Byron’s daughter. Alan Turing read her Notes, and coined the term “LadyLovelace’s Objection” (“an AI can’t originate anything”) in his 1950 TuringTestpaper.ButAdaherselfwasstilllargelyafootnoteatthatpoint.ItwasacertainBertramBowden—aBritishnuclearphysicistwhowentinto

thecomputerindustryandeventuallybecameMinisterofScienceandEducation—who“rediscovered”Ada.Inresearchinghis1953bookFasterThanThought(yes, about computers), he locatedAda’s granddaughter LadyWentworth (thedaughter of Ada’s daughter), who told him the family lore about Ada, bothaccurateandinaccurate,andlethimlookatsomeofAda’spapers.Charmingly,BowdennotesthatinAda’sgranddaughter’sbookThoroughbredRacingStock,thereisuseofbinaryincomputingpedigrees.Ada,andtheAnalyticalEngine,ofcourse,useddecimal,withnobinaryinsight.But even in the 1960s, Babbage—and Ada—weren’t exactly well known.

Babbage’sDifferenceEngineprototypehadbeengiventotheScienceMuseuminLondon,buteventhoughIspentlotsoftimeattheScienceMuseumasachildinthe1960s,I’mprettysureIneversawitthere.Still,bythe1980s,particularlyafter theUSDepartment ofDefense named its illfated programming language

afterAda,awarenessofAdaLovelaceandCharlesBabbagebeganto increase,andbiographiesbegantoappear,thoughsometimeswithhair-raisingerrors(myfavorite is that themention of “the problem of three bodies” in a letter fromBabbage indicated a romantic triangle between Babbage, Ada andWilliam—whileitactuallyreferstothethree-bodyproblemincelestialmechanics!).AsinterestinBabbageandAdaincreased,sodidcuriosityaboutwhetherthe

Difference Engine would actually have worked if it had been built fromBabbage’s plans.A projectwasmounted, and in 2002, after a heroic effort, acomplete Difference Engine was built, with only one correction in the plansbeingmade.Amazingly, themachineworked.Building itcostabout thesame,inflationadjusted,asBabbagehadrequestedfromtheBritishgovernmentbackin1823.WhatabouttheAnalyticalEngine?Sofar,norealversionofithaseverbeen

built—orevenfullysimulated.

WhatAdaActuallyWroteOK, so now that I’ve talked (at length) about the life ofAda Lovelace,whatabouttheactualcontentofherNotesontheAnalyticalEngine?They start crisply: “The particular function whose integral the Difference

Enginewasconstructedtotabulate,is….”ShethenexplainsthattheDifferenceEngine can compute values of any 6th degree polynomial—but theAnalyticalEngine is different, because it canperformany sequenceof operations.Or, asshesays,“TheAnalyticalEngineisanembodyingofthescienceofoperations,constructedwith peculiar reference to abstract number as the subject of thoseoperations.TheDifferenceEngineis theembodyingofoneparticularandverylimitedsetofoperations…”Charmingly, at least forme, considering the years I have spentworking on

Mathematica,shecontinuesatalaterpoint,“Wemayconsidertheengineasthematerialandmechanicalrepresentativeofanalysis,andthatouractualworkingpowersinthisdepartmentofhumanstudywillbeenabledmoreeffectuallythanheretofore to keep pace with our theoretical knowledge of its principles andlaws,throughthecompletecontrolwhichtheenginegivesusovertheexecutivemanipulationofalgebraicalandnumericalsymbols.”Alittlelater,sheexplainsthatpunchedcardsarehowtheAnalyticalEngineis

controlled, and then makes the classic statement that “the Analytical Engineweaves algebraical patterns just as the Jacquard-loom weaves flowers andleaves”.

Ada then goes through how a sequence of specific kinds of computationswould work on the Analytical Engine, with “Operation Cards” defining theoperations to be done, and “Variable Cards” defining the locations of values.Ada talksabout“cycles”and“cyclesofcycles,etc”,nowknownas loopsandnestedloops,givingamathematicalnotationforthem:

There’sa lotofmodern-seemingcontent inAda’snotes.Shecomments that“ThereisinexistenceabeautifulwovenportraitofJacquard,inthefabricationof which 24,000 cards were required.” Then she discusses the idea of usingloops to reduce the number of cards needed, and the value of rearrangingoperations to optimize their execution on the Analytical Engine, ultimatelyshowingthatjust3cardscoulddowhatmightseemlikeitshouldrequire330.AdatalksaboutjusthowfartheAnalyticalEnginecangoincomputingwhat

waspreviouslynotcomputable,at leastwithanyaccuracy.Andasanexampleshediscussesthethree-bodyproblem,andthefactthatinhertime,of“about295coefficientsoflunarperturbations”thereweremanyonwhichdifferentpeople’scomputationsdidn’tagree.FinallycomesAda’sNoteG.Earlyon,shestates,“TheAnalyticalEnginehas

nopretensionswhatevertooriginateanything.Itcandowhateverweknowhowtoorderittoperform.…Itsprovinceistoassistusinmakingavailablewhatwearealreadyacquaintedwith.”Ada seems to have understood with some clarity the traditional view of

programming:thatweengineerprogramstodothingsweknowhowtodo.Butshealsonotes that inactuallyputting“the truthsandtheformulaeofanalysis”intoaformamenabletotheengine,“thenatureofmanysubjectsinthatscienceare necessarily thrown into new lights, andmore profoundly investigated.” Inotherwords—asIoftenpointout—actuallyprogrammingsomethinginevitablyletsonedomoreexplorationofit.She goes on to say that “in devising formathematical truths a new form in

whichtorecordandthrowthemselvesoutforactualuse,viewsarelikelytobeinduced,whichshouldagainreactonthemoretheoreticalphaseofthesubject”,or inotherwords—asIhavealsooftensaid—representingmathematical truthsin a computable form is likely to help one understand those truths themselvesbetter.Ada seems to have understood, though, that the “science of operations”

implemented by the engine would not only apply to traditional mathematicaloperations.Forexample,shenotesthatif“thefundamentalrelationsofpitchedsounds in the science of harmony”were amenable to abstract operations, thentheenginecouldusethemto“composeelaborateandscientificpiecesofmusicof any degree of complexity or extent.”Not a bad level of understanding for1843.

TheBernoulliNumberComputation

What’sbecomethemostfamouspartofwhatAdawroteis thecomputationofBernoullinumbers,inNoteG.ThisseemstohavecomeoutofalettershewrotetoBabbage,inJuly1843.Shebeginstheletterwith,“Iamworkingveryhardforyou; like the Devil in fact; (which perhaps I am)”. Then she asks for somespecific references, and finally ends with, “I want to put in something aboutBernoulli’s Numbers, in one of myNotes, as an example of how an implicitfunctionmaybeworkedoutbytheengine,withouthavingbeenworkedoutbyhumanhead&handsfirst….Givemethenecessarydata&formulae.”

Ada’schoiceofBernoullinumberstoshowofftheAnalyticalEnginewasaninteresting one. Back in the 1600s, people spent their lives making tables ofsums of powers of integers—in other words, tabulating values of

for differentm and n. But Jacob Bernoulli pointedout that all such sums can be expressed as polynomials in m, with thecoefficients being related to what are now called Bernoulli numbers. And in1713 Bernoulli was proud to say that he’d computed the first 10 Bernoullinumbers“inaquarterofanhour”—reproducingyearsofotherpeople’swork.Today, of course, it’s instantaneous to do the computation in theWolfram

Language:

And,asithappens,afewyearsago,justtoshowoffnewalgorithms,weevencomputed10millionofthem.But, OK, so how did Ada plan to do it? She started from the fact that

Bernoullinumbersappearintheseriesexpansion:

Thenbyrearrangingthisandmatchinguppowersofx,shegotasequenceofequationsfortheBernoullinumbersBn—whichshethen“unravelled”togivearecurrencerelationoftheform:

Now Ada had to specify how to actually compute this on the AnalyticalEngine.First, sheused the fact thatoddBernoullinumbers (other thanB1) arezero,thencomputedBn,whichisourmodernB2n(orBernoulliB[2n]inWolframLanguage).ThenshestartedfromB0,andsuccessivelycomputedBnforlargern,storingeachvalueshegot.Thealgorithmsheusedforthecomputationwas(inmodernterms):

On the Analytical Engine, the idea was to have a sequence of operations(specifiedby“OperationCards”)performedbythe“Mill”withoperandscomingfromthe“Store”(withaddressesspecifiedby“VariableCards”). (In theStore,each number was represented by a sequence of wheels, each turned to theappropriatevalue for eachdigit.)TocomputeBernoulli numbers thewayAdawantedtakestwonestedloopsofoperations.WiththeAnalyticalEnginedesignthatexistedatthetime,Adahadtobasicallyunrolltheseloops.ButintheendshesuccessfullyproducedadescriptionofhowB8(whichshecalledB7)couldbecomputed:

Thisiseffectivelytheexecutiontraceofaprogramthatrunsfor25steps(plusaloop)ontheAnalyticalEngine.Ateachstep,thetraceshowswhatoperationisperformed on which Variable Cards, and which Variable Cards receive theresults.Lacking a symbolicnotation for loops,Ada just indicated loops in theexecutiontraceusingbraces,notinginEnglishthatpartsarerepeated.Andintheend,thefinalresultofthecomputationappearsinlocation24:

Asit’sprinted,there’sabuginAda’sexecutiontraceonline4:thefractionisupsidedown.Butifyoufixthat,it’seasytogetamodernversionofwhatAdadid:

Andhere’swhatthesameschemegivesforthenexttwo(nonzero)Bernoullinumbers. As Ada figured out it doesn’t ultimately take any more workingvariables (specified byVariable Cards) to compute higher Bernoulli numbers,justmoreoperations.

The Analytical Engine, as it was designed in 1843, was supposed to store100040-digitnumbers,whichwouldinprinciplehaveallowedittocomputeupto perhaps B50 (=495057205241079648212477525/66). It would have beenreasonablyfasttoo;theAnalyticalEnginewasintendedtodoabout7operationspersecond.SoAda’sB8wouldhavetakenabout5secondsandB50wouldhavetakenperhapsaminute.Curiously, even in our record-breaking computation ofBernoulli numbers a

fewyearsago,wewerebasicallyusingthesamealgorithmasAda—thoughnowthere are slightly faster algorithms that effectively compute Bernoulli numbernumerators modulo a sequence of primes, then reconstruct the full numbersusingtheChineseRemainderTheorem.

Babbagevs.Ada?TheAnalyticalEngine and its constructionwere allBabbage’swork.SowhatdidAdaadd?Adasawherselffirstandforemostasanexpositor.Babbagehadshownher lotsofplansandexamplesof theAnalyticalEngine.Shewanted toexplainwhattheoverallpointwas—aswellasrelateit,assheputit,to“large,general,&metaphysicalviews”.In the surviving archive of Babbage’s papers (discovered years later in his

lawyer’s family’s cowhide trunk), there are a remarkable number of drafts ofexpositionsof theAnalyticalEngine, starting in the1830s, and continuing fordecades,withtitleslike“OftheAnalyticalEngine”and“TheScienceofNumberReducedtoMechanism”.WhyBabbageneverpublishedanyoftheseisn’tclear.Theyseemlikeperfectlydecentdescriptionsofthebasicoperationoftheengine—thoughtheyaredefinitelymorepedestrianthanwhatAdaproduced.When Babbage died, he was writing a “History of the Analytical Engine,”

which his son completed. In it, there is a dated list of “446 Notations of theAnalyticalEngine,”eachessentiallya representationofhowsomeoperation—like division—could be done on theAnalytical Engine. The dates start in the1830s,andrunthroughthemid-1840s,withnotmuchhappeninginthesummerof1843.

Meanwhile, in the collection of Babbage’s papers at the ScienceMuseum,therearesomesketchesofhigher-leveloperationsontheAnalyticalEngine.Forexample, from 1837 there’s “Elimination between two equations of the firstdegree”—essentiallytheevaluationofarationalfunction:

Thereareafewverysimplerecurrencerelations:

Then from1838, there’s a computationof the coefficients in theproduct oftwopolynomials:

But there’s nothing as sophisticated—or as clean—asAda’s computation ofthe Bernoulli numbers. Babbage certainly helped and commented on Ada’swork,butshewasdefinitelythedriverofit.Sowhat didBabbage say about that? In his autobiographywritten 26years

later,hehadahardtimesayinganythingniceaboutanyoneoranything.AboutAda’s Notes, he writes, “We discussed together the various illustrations thatmightbeintroduced:Isuggestedseveral,buttheselectionwasentirelyherown.Soalsowasthealgebraicworkingoutofthedifferentproblems,except,indeed,thatrelatingtothenumbersofBernoulli,whichIhadofferedtodotosaveLadyLovelace the trouble. This she sent back to me for an amendment, havingdetectedagravemistakewhichIhadmadeintheprocess.”When I first read this, I thought Babbage was saying that he basically

ghostwrote all of Ada’s Notes. But reading what he wrote again, I realize itactuallysaysalmostnothing,otherthanthathesuggestedthingsthatAdamayormaynothaveused.Tome,there’slittledoubtaboutwhathappened:Adahadanideaofwhatthe

Analytical Engine should be capable of, and was asking Babbage questionsabouthowitcouldbeachieved.Ifmyownexperienceswithhardwaredesignersinmodern timesare anything togoby, the answerswill oftenhavebeenverydetailed.Ada’sachievementwastodistillfromthesedetailsaclearexpositionoftheabstractoperationofthemachine—somethingwhichBabbageneverdid.(Inhisautobiography,hebasicallyjustreferstoAda’sNotes.)

Babbage’sSecretSauceForallhisvariousshortcomings,theveryfactthatBabbagefiguredouthowtobuildevenafunctioningDifferenceEngine—letaloneanAnalyticalEngine—isextremelyimpressive.Sohowdidhedoit?I thinkthekeywaswhathecalledhisMechanicalNotation.He firstwrote about it in1826under the title “OnaMethodofExpressingbySignstheActionofMachinery”.Hisideawastotakeadetailedstructureofamachineandabstractakindofsymbolicdiagramofhowitspartsactoneachother.Hisfirstexamplewasahydraulicdevice:

Thenhegavetheexampleofaclock,showingontheleftakindof“executiontrace”of how the components of the clock change, andon the right a kindof“blockdiagram”oftheirrelationships:

It’saprettynicewaytorepresenthowasystemworks,similarinsomewaysto amodern timing diagram—but not quite the same.Andover the years thatBabbageworkedon theAnalyticalEngine,hisnotes showevermorecomplexdiagrams.It’snotquiteclearwhatsomethinglikethismeans:

But it looks surprisingly like a modern Modelica representation—say inWolframSystemModeler. (One difference inmodern times is that subsystemsare represented much more hierarchically; another is that everything is nowcomputable, so that actual behavior of the system can be simulated from therepresentation.)

But even though Babbage used his various kinds of diagrams extensivelyhimself, he didn’twrite papers about them. Indeed, his only other publicationabout “Mechanical Notation” is the flyer he had printed up for the GreatExhibition in 1851—apparently a pitch for standardization in drawings ofmechanical components (and indeed these notations appear on Babbage’sdiagramsliketheoneabove).

I’mnotsurewhyBabbagedidn’tdomoretoexplainhisMechanicalNotationandhisdiagrams.Perhapshewasjustbitteraboutpeople’sfailuretoappreciateitin1826.Orperhapshesawitasthesecretthatlethimcreatehisdesigns.Andeven though systems engineering has progressed a longway since Babbage’stime,theremayyetbeinspirationtobehadfromwhatBabbagedid.

TheBiggerPictureOK,sowhat’sthebiggerpictureofwhathappenedwithAda,BabbageandtheAnalyticalEngine?CharlesBabbagewas an energeticmanwhohadmany ideas, someof them

good.At theageof30he thoughtofmakingmathematical tablesbymachine,and continued to pursue this idea until he died 49 years later, inventing theAnalytical Engine as a way to achieve his objective. He was good—eveninspired—attheengineeringdetails.Hewasbadatkeepingaprojectontrack.AdaLovelacewas an intelligentwomanwho became friendswithBabbage

(there’szeroevidencetheywereeverromanticallyinvolved).AssomethingofafavortoBabbage,shewroteanexpositionoftheAnalyticalEngine,andindoingsoshedevelopedamoreabstractunderstandingofitthanBabbagehad—andgotaglimpseoftheincrediblypowerfulideaofuniversalcomputation.TheDifferenceEngineandthingslikeitarespecial-purposecomputers,with

hardwarethat’sbuilttodoonlyonekindofthing.Onemighthavethoughtthattodolotsofdifferentkindsofthingswouldnecessarilyrequirelotsofdifferentkindsofcomputers.Butthisisn’ttrue.Andinsteadit’safundamentalfactthatit’spossible tomakegeneral-purposecomputers,whereasingle fixedpieceofhardware can be programmed to do any computation. And it’s this idea ofuniversal computation that for example makes software possible—and thatlaunchedthewholecomputerrevolutioninthe20thcentury.GottfriedLeibnizhadalreadyhadaphilosophicalconceptofsomething like

universal computation back in the 1600s. But it wasn’t followed up. AndBabbage’sAnalyticalEngineisthefirstexplicitexampleweknowofamachinethatwouldhavebeencapableofuniversalcomputation.Babbagedidn’t thinkof it in these terms, though.He justwantedamachine

thatwas as effective as possible at producingmathematical tables. But in theefforttodesignthis,heendedupwithauniversalcomputer.WhenAdawroteaboutBabbage’smachine,shewantedtoexplainwhatitdid

in theclearestway—and todo thisshe lookedat themachinemoreabstractly,with the result that she ended up exploring and articulating something quite

recognizableasthemodernnotionofuniversalcomputation.WhatAdadidwaslostformanyyears.Butasthefieldofmathematicallogic

developed, the idea of universal computation arose again, most clearly in theworkofAlanTuringin1936.Thenwhenelectroniccomputerswerebuiltinthe1940s, it was realized they too exhibited universal computation, and theconnectionwasmadewithTuring’swork.Therewas still, though,a suspicion thatperhaps someotherwayofmaking

computersmightleadtoadifferentformofcomputation.Anditactuallywasn’tuntil the1980s thatuniversalcomputationbecamewidelyacceptedasa robustnotion.Andbythattime,somethingnewwasemerging—notablythroughworkIwasdoing: that universal computationwasnotonly something that’spossible,butthatit’sactuallycommon.And what we now know (embodied for example in my Principle of

ComputationalEquivalence)isthatbeyondalowthresholdaverywiderangeofsystems—even of very simple construction—are actually capable of universalcomputation.ADifferenceEnginedoesn’t get there.But as soon as one adds just a little

more,onewillhaveuniversalcomputation.So inretrospect, it’snotsurprisingthattheAnalyticalEnginewascapableofuniversalcomputation.Today, with computers and software all around us, the notion of universal

computationseemsalmostobvious:ofcoursewecanuse software tocomputeanythingwewant.Butintheabstract,thingsmightnotbethatway.AndIthinkonecanfairlysay thatAdaLovelacewas thefirstpersonever toglimpsewithanyclaritywhathasbecomeadefiningphenomenonofourtechnologyandevenourcivilization:thenotionofuniversalcomputation.

WhatIf…?What if Ada’s health hadn’t failed—and she had successfully taken over theAnalyticalEngineproject?Whatmighthavehappenedthen?I don’t doubt that the Analytical Engine would have been built. Maybe

Babbagewouldhavehad to revisehisplansabit,but I’msurehewouldhavemade itwork.The thingwouldhavebeen the sizeof a train locomotive,withmaybe50,000movingparts.Andnodoubtitwouldhavebeenabletocomputemathematicaltablesto30-or50-digitprecisionattherateofperhapsoneresultevery4seconds.Wouldtheyhavefiguredthat themachinecouldbeelectromechanicalrather

thanpurelymechanical? I suspect so.After all,CharlesWheatstone,whowas

intimately involved in the development of the electric telegraph in the 1830s,wasagoodfriendoftheirs.Andbytransmittinginformationelectricallythroughwires, rather than mechanically through rods, the hardware for the machinewould have been dramatically reduced, and its reliability (which would havebeenabigissue)wouldhavebeendramaticallyincreased.Another major way that modern computers reduce hardware is by dealing

withnumbers inbinaryrather thandecimal.Wouldtheyhavefiguredthat ideaout?Leibnizknewaboutbinary.And ifGeorgeBoolehad followeduponhismeetingwith Babbage at theGreat Exhibition,maybe thatwould have led tosomething. Binary wasn’t well known in the mid-1800s, but it did appear inpuzzles,andBabbage,atleast,wasquiteintopuzzles:anotableexamplebeinghisquestionofhowtomakeasquareofwordswith“bishop”alongthetopandside(whichnowtakesjustafewlinesofWolframLanguagecodetosolve).Babbage’sprimaryconceptionoftheAnalyticalEnginewasasamachinefor

automatically producing mathematical tables—either printing them out bytypesetting,orgiving themasplotsbydrawingontoaplate.He imagined thathumanswouldbe themainusersof these tables—althoughhedid thinkof theidea of having libraries of pre-computed cards that would provide machine-readableversions.Today—intheWolframLanguageforexample—weneverstoremuchinthe

wayofmathematical tables;we just computewhatweneedwhenweneed it.ButinBabbage’sday—withtheideaofamassiveAnalyticalEngine—thiswayofdoingthingswouldhavebeenunthinkable.So, OK: would the Analytical Engine have gotten beyond computing

mathematical tables? I suspect so. If Ada had lived as long as Babbage, shewould still have been around in the 1890swhenHermanHollerithwas doingcard-based electromechanical tabulation for the census (and founding whatwouldeventuallybecomeIBM).TheAnalyticalEnginecouldhavedonemuchmore.Perhaps Ada would have used the Analytical Engine—as she began to

imagine—to produce algorithmic music. Perhaps they would have used it tosolve things like the three-bodyproblem,maybeevenby simulation. If they’dfiguredoutbinary,maybe theywouldevenhave simulated things like cellularautomata.NeitherBabbagenorAdaevermademoneycommercially (and,asBabbage

took pains to point out, his government contracts just paid his engineers, nothim). If they had developed the Analytical Engine, would they have found abusiness model for it? No doubt they would have sold some engines togovernments.Maybetheywouldevenhaveoperatedakindofcloudcomputing

serviceforVictorianscience,technology,financeandmore.But none of this actually happened, and instead Ada died young, the

AnalyticalEnginewasneverfinished,andittookuntilthe20thcenturyforthepowerofcomputationtobediscovered.

WhatWereTheyLike?IfonehadmetCharlesBabbage,whatwouldhehavebeenlike?Hewas,Ithink,agoodconversationalist.Earlyinlifehewasidealistic(“domybesttoleavetheworldwiser thanI found it”); laterhewasalmostaDickensiancaricatureofabitteroldman.Hegavegoodparties,andputgreatvalueinconnectingwiththehigheststrataofintellectualsociety.Butparticularlyinhislateryears,hespentmost of his time alone in his large house, filled with books and papers andunfinishedprojects.Babbagewasnevera terriblygood judgeofpeople,orofhowwhathesaid

wouldbeperceivedbythem.Andeveninhiseighties,hewasstillquitechild-likeinhispolemics.Hewasalsonotoriouslypooratstayingfocused;healwayshadanewideatopursue.Theonebigexceptiontothiswashisalmost-50-yearpersistenceintryingtoautomatetheprocessofcomputation.Imyselfhavesharedamodernversionofthisverygoalinmyownlife(…,

Mathematica,Wolfram|Alpha,WolframLanguage,…)—thoughsofaronlyfor40years.Iamfortunatetohavelivedinatimewhenambienttechnologymadethismuch easier to achieve, but in every large project I have done it has stilltakenacertainsinglemindednessandgrittytenacity—aswellasleadership—toactuallygetitfinished.So what about Ada? From everything I can tell, she was a clear speaking,

clear thinking individual. She came from the upper classes, but didn’t wearespeciallyfashionableclothes,andcarriedherselfmuchlesslikeastereotypicalcountessthanlikeanintellectual.Asanadult,shewasemotionallyquitemature—probably more so than Babbage—and seems to have had a good practicalgraspofpeopleandtheworld.LikeBabbage,shewasindependentlywealthy,andhadnoneedtoworkfora

living. But she was ambitious, and wanted to make something of herself. Inperson, beyond the polished Victorian upper-class exterior, I suspect she wassomething of a nerd, completewithmath jokes and everything. Shewas alsocapable of great and sustained focus, for example over the months she spentwritingherNotes.Inmathematics,shesuccessfullylearneduptothestateoftheartinhertime

—probablyaboutthesamelevelasBabbage.UnlikeBabbage,wedon’tknowofanyspecificresearchshedidinmathematics,soit’shardtojudgehowgoodshewouldhavebeen;Babbagewasrespectablethoughunremarkable.Whenone readsAda’s letters,what comes through is a smart, sophisticated

person,withaclear, logicalmind.What shesays isoftendressed inVictorianpleasantries—butunderneath,theideasareclearandoftenquiteforceful.Ada was very conscious of her family background, and of being “Lord

Byron’s daughter”. At some level, his story and success no doubt fueled herambition, and her willingness to try new things. (I can’t help thinking of herleadingtheengineersoftheAnalyticalEngineasabitlikeLordByronleadingthe Greek army.) But I also suspect his troubles loomed over her. For manyyears, partly at hermother’s behest, she eschewed things like poetry.But shewasdrawntoabstractwaysofthinking,notonlyinmathematicsandscience,butalsoinmoremetaphysicalareas.And she seems to have concluded that her greatest strength would be in

bridging the scientific with the metaphysical—perhaps in what she called“poeticalscience”.Itwaslikelyacorrectselfperception.Forthat is inasenseexactly what she did in the Notes she wrote: she took Babbage’s detailedengineering,andmadeitmoreabstractand“metaphysical”—andintheprocessgaveusafirstglimpseoftheideaofuniversalcomputation.

TheFinalStoryThe story of Ada and Babbage has many interesting themes. It is a story oftechnical prowess meeting abstract “big picture” thinking. It is a story offriendshipbetweenoldandyoung.Itisastoryofpeoplewhohadtheconfidencetobeoriginalandcreative.It isalsoa tragedy.A tragedy forBabbage,who lost somanypeople inhis

life, and whose personality pushed others away and prevented him fromrealizing his ambitions. A tragedy for Ada, who was just getting started insomethingshelovedwhenherhealthfailed.WewillneverknowwhatAdacouldhavebecome.AnotherMarySomerville,

famousVictorianexpositorofscience?ASteve-Jobs-likefigurewhowouldleadthe vision of the Analytical Engine? Or an Alan Turing, understanding theabstractideaofuniversalcomputation?ThatAdatouchedwhatwouldbecomeadefiningintellectualideaofourtime

was good fortune. Babbage did not know what he had; Ada started to seeglimpsesandsuccessfullydescribedthem.

ForsomeonelikemethestoryofAdaandBabbagehasparticularresonance.LikeBabbage,Ihavespentmuchofmylifepursuingparticulargoals—thoughunlikeBabbage,Ihavebeenabletoseeafairfractionofthemachieved.AndIsuspectthat,likeAda,IhavebeenputinapositionwhereIcanpotentiallyseeglimpsesofsomeofthegreatideasofthefuture.ButthechallengeistobeenoughofanAdatograspwhat’sthere—oratleast

to find anAdawhodoes.But at least now I think I have an ideaofwhat theoriginalAdaborn200yearsagotodaywaslike:afittingpersonalityontheroadto universal computation and the present and future achievements ofcomputationalthinking.It’sbeenapleasuregettingtoknowyou,Ada.

Quiteafeworganizationsandpeoplehelpedingettinginformationandmaterialforthispiece.I’dliketothanktheBritishLibrary;theMuseumoftheHistoryofScience,Oxford,ScienceMuseum,London;theBodleianLibrary,Oxford(withpermissionfromtheEarlofLytton,Ada’sgreat-greatgrandson,andoneofher10livingdescendants);theNewYorkPublicLibrary;St.MaryMagdaleneChurch,Hucknall,Nottinghamshire(Ada’sburialplace);andBettyToole(authorofacollectionofAda’sletters);aswellastwooldfriends:TimRobinson(re-creatorofBabbageengines)andNathanMyhrvold(funderofDifferenceEngine#2re-creation).

GottfriedLeibnizMay14,2013

I’vebeencuriousaboutGottfriedLeibnizforyears,notleastbecauseheseemsto havewanted to build something likeMathematica andWolfram|Alpha, andperhapsANewKind of Science aswell—though three centuries too early. Sowhen I took a trip recently to Germany, I was excited to be able to visit hisarchiveinHanover.Leafingthroughhisyellowed(butstillrobustenoughformetotouch)pages

ofnotes,Ifeltacertainconnection—asItriedtoimaginewhathewasthinkingwhen hewrote them, and tried to relatewhat I saw in them towhatwe nowknowafterthreemorecenturies:

Some things, especially in mathematics, are quite timeless. Like here’sLeibnizwritingdownaninfiniteseriesfor√2(thetextisinLatin):

Orhere’sLeibniztryingtocalculateacontinuedfraction—thoughhegotthearithmeticwrong,eventhoughhewroteitallout(theΠwashisearlierversionofanequalsign):

Or here’s a little summary of calculus, that could almost be in a moderntextbook:

Butwhatwaseverythingelseabout?Whatwas the larger storyofhisworkandthinking?I have always found Leibniz a somewhat confusing figure. He did many

seemingly disparate and unrelated things—in philosophy, mathematics,theology,law,physics,history,andmore.Andhedescribedwhathewasdoinginwhatseemtousnowasstrange17thcenturyterms.ButasI’velearnedmore,andgottenabetterfeelingforLeibnizasaperson,

I’ve realized that underneath much of what he did was a core intellectualdirection that is curiously close to the modern computational one that I, forexample,havefollowed.GottfriedLeibnizwasborninLeipziginwhat’snowGermanyin1646(four

yearsafterGalileodied,andfouryearsafterNewtonwasborn).Hisfatherwasaprofessor of philosophy; hismother’s familywas in the book trade. Leibniz’sfatherdiedwhenLeibnizwas6—andaftera2-yeardeliberationonitssuitabilityfor one so young, Leibnizwas allowed into his father’s library, and began toread his way through its diverse collection of books. He went to the localuniversity at age 15, studying philosophy and law—and graduated in both ofthematage20.Evenasateenager,Leibnizseemstohavebeeninterestedinsystematization

andformalizationofknowledge.Therehadbeenvagueideasforalongtime—forexampleinthesemi-mysticalArsMagnaofRamonLlullfromthe1300s—that one might be able to set up some kind of universal system in which allknowledgecouldbederivedfromcombinationsofsignsdrawnfromasuitable(as Descartes called it) “alphabet of human thought”. And for his philosophygraduation thesis, Leibniz tried to pursue this idea. He used some basiccombinatorialmathematicstocountpossibilities.Hetalkedaboutdecomposingideas into simple components on which a “logic of invention” could operate.And, for good measure, he put in an argument that purported to prove theexistenceofGod.AsLeibnizhimselfsaidinlateryears,thisthesis—writtenatage20—wasin

many ways naive. But I think it began to define Leibniz’s lifelong way ofthinkingaboutallsortsofthings.Andso,forexample,Leibniz’slawgraduationthesis about “perplexing legal cases” was all about how such cases couldpotentiallyberesolvedbyreducingthemtologicandcombinatorics.Leibniz was on a track to become a professor, but instead he decided to

embarkon a lifeworking as an advisor for various courts andpolitical rulers.Some of what he did for them was scholarship, tracking down abstruse—butpolitically important—genealogyandhistory.Someof itwasorganization and

systematization—of legal codes, libraries and so on. Some of it was practicalengineering—like trying to work out better ways to keep water out of silvermines. And some of it—particularly in earlier years—was “on the ground”intellectualsupportforpoliticalmaneuvering.Onesuchactivityin1672tookLeibniztoParisforfouryears—duringwhich

timehe interactedwithmany leading intellectual lights.Before then,Leibniz’sknowledge of mathematics had been fairly basic. But in Paris he had theopportunitytolearnallthelatestideasandmethods.Andforexamplehesoughtout Christiaan Huygens, who agreed to teach Leibniz mathematics—after hesucceeded in passing the test of finding the sum of the reciprocals of thetriangularnumbers.Over the years, Leibniz refined his ideas about the systematization and

formalizationofknowledge,imaginingawholearchitectureforhowknowledgewould—in modern terms—be made computational. He saw the first step asbeing thedevelopmentofanarscharacteristica—amethodologyforassigningsigns or symbolic representations to things, and in effect creating a uniform“alphabet of thought”. And he then imagined—in remarkable resonance withwhatwenowknowaboutcomputation—thatfromthisuniformrepresentationitwouldbepossibletofind“truthsofreasoninanyfield…throughacalculus,asinarithmeticoralgebra”.He talked about his ideas under a variety of rather ambitious names like

scientia generalis (“general method of knowledge”), lingua philosophica(“philosophical language”), mathematique universelle (“universalmathematics”), characteristica universalis (“universal system”) and calculusratiocinator (“calculusof thought”).He imaginedapplicationsultimately inallareas—science,law,medicine,engineering,theologyandmore.Buttheoneareainwhichhehadclearsuccessquitequicklywasmathematics.Tomeit’sremarkablehowrarelyinthehistoryofmathematicsthatnotation

has been viewed as a central issue. It happened at the beginning of modernmathematicallogicinthelate1800swiththeworkofpeoplelikeGottlobFregeandGiuseppePeano.Andinrecenttimesit’shappenedwithmeinmyeffortstocreate Mathematica and the Wolfram Language. But it also happened threecenturies ago with Leibniz. And I suspect that Leibniz’s successes inmathematicswereinnosmallpartduetotheeffortheputintonotation,andtheclarityofreasoningaboutmathematicalstructuresandprocessesthatitbrought.WhenonelooksatLeibniz’spapers,it’sinterestingtoseehisnotationandits

development. Many things look quite modern. Though there are charmingdashes of the 17th century, like the occasional use of alchemical or planetarysymbolsforalgebraicvariables:

There’s Π as an equals sign instead of =, with the slightly hacky idea ofhavingitbelikeabalance,withalongerlegononesideortheotherindicatinglessthan(“<”)orgreaterthan(“>”):

Thereareoverbarstoindicategroupingofterms—arguablyabetterideathanparentheses,thoughhardertotype,andtypeset:

We do use overbars for roots today. But Leibniz wanted to use them inintegrals too—alongwith the rather nice “tailed d,”which remindsme of thedouble-struck “differential d” that we invented for representing integrals inMathematica.

Particularlyinsolvingequations,it’squitecommontowanttouse±,andit’salways confusing how the grouping is supposed towork, say ina±b±c.Well,Leibniz seems to have found it confusing too, but he invented a notation tohandleit—whichweactuallyshouldconsiderusingtodaytoo:

I’mnotsurewhatsomeofLeibniz’snotationmeans—thoughthoseovertildesarerathernice-looking:

Asarethesethingswithdots:

Orthisinteresting-lookingdiagrammaticform:

Ofcourse,Leibniz’smostfamousnotationsarehisintegralsign(long“s”for“summa”)andd,heresummarizedinthemarginforthefirsttime,onNovember11, 1675 (the “5” in “1675” was changed to a “3” after the fact, perhaps byLeibniz):

Ifinditinterestingthatdespiteallhisnotationfor“calculational”operations,Leibniz apparently did not invent similar notation for logical operations. “Or”wasjusttheLatinwordvel,“and”waset,andsoon.Andwhenhecameupwiththeideaofquantifiers(modern∀and∃),hejustrepresentedthembytheLatinabbreviationsU.A.andP.A.:

It’salwaysstruckmeasaremarkableanomalyinthehistoryofthoughtthatittookuntil the1930sfor theideaofuniversalcomputationtoemerge.AndI’veoften wondered if lurking in the writings of Leibniz there might be an earlyversion of universal computation—maybe even a diagram that we could nowinterpretasasystemlikeaTuringmachine.ButwithmoreexposuretoLeibniz,it’sbecomeclearertomewhythat’sprobablynotthecase.Onebigpiece,Isuspect,isthathedidn’ttakediscretesystemsquiteseriously

enough. He referred to results in combinatorics as “self-evident”, presumablybecauseheconsideredthemdirectlyverifiablebymethodslikearithmetic.Andit was only “geometrical”, or continuous, mathematics that he felt needed tohaveacalculusdevelopedfor it. Indescribing things likepropertiesofcurves,Leibnizcameupwithsomethinglikecontinuousfunctions.Butheneverseemstohaveappliedtheideaoffunctionstodiscretemathematics—whichmightforexample have led him to think about universal elements for building upfunctions.Leibnizrecognizedthesuccessofhisinfinitesimalcalculus,andwaskeento

comeupwithsimilar“calculi”forotherthings.Andinanother“nearmiss”withuniversalcomputation,Leibnizhadtheideaofencodinglogicalpropertiesusingnumbers.Hethoughtaboutassociatingeverypossibleattributeofathingwithadifferent prime number, then characterizing the thing by the product of theprimes for its attributes—and then representing logical inferenceby arithmeticoperations. But he only considered static attributes—and never got to an idealikeGödelnumberingwhereoperationsarealsoencodedinnumbers.ButeventhoughLeibnizdidnotgettotheideaofuniversalcomputation,he

didunderstandthenotionthatcomputationisinasensemechanical.Andindeedquite early in life he seems to have resolved to build an actual mechanicalcalculatorfordoingarithmetic.Perhapsinpartitwasbecausehewantedtouseithimself(alwaysagoodreasontobuildapieceoftechnology!).Fordespitehisprowess at algebra and the like, his papers are charmingly full of basic (andsometimes incorrect) school-level arithmetic calculations written out in themargin—andnowpreservedforposterity:

There were scattered examples of mechanical calculators being built inLeibniz’s time, andwhen hewas in Paris, Leibniz no doubt saw the additioncalculatorthathadbeenbuiltbyBlaisePascalin1642.ButLeibnizresolvedtomake a “universal” calculator, that could for the first time do all four basicfunctionsofarithmeticwithasinglemachine.Andhewantedtogiveitasimple“user interface”, where one would for example turn a handle one way formultiplication,andtheoppositewayfordivision.In Leibniz’s papers there are all sorts of diagrams about how the machine

shouldwork:

Leibniz imagined thathiscalculatorwouldbeofgreatpracticalutility—andindeedheseemstohavehopedthathewouldbeabletoturnitintoasuccessfulbusiness.But inpractice,Leibniz struggled toget thecalculator toworkat allreliably. For like other mechanical calculators of its time, it was basically aglorified odometer. And just like in Charles Babbage’s machines nearly 200years later, it wasmechanically difficult tomakemanywheelsmove at oncewhenacascadeofcarriesoccurred.Leibniz at first had a wooden prototype of his machine built, intended to

handlejust3or4digits.ButwhenhedemoedthistopeoplelikeRobertHookeduringavisittoLondonin1673itdidn’tgoverywell.Buthekeptonthinkinghe’d figured everything out—for example in 1679 writing (in French) of the“lastcorrectiontothearithmeticmachine”:

Notesfrom1682suggestthatthereweremoreproblems,however:

ButLeibnizhadplansdraftedupfromhisnotes—andcontractedanengineertobuildabrassversionwithmoredigits:

It’sfuntoseeLeibniz’s“marketingmaterial”forthemachine:

Aswellaspartsofthe“manual”(with365×24asa“workedexample”):

Completewithdetailedusagediagrams:

Butdespiteallthiseffort,problemswiththecalculatorcontinued.Andinfact,for more than 40 years, Leibniz kept on tweaking his calculator—probablyaltogetherspending(intoday’scurrency)morethanamilliondollarsonit.So what actually happened to the physical calculator? When I visited

Leibniz’sarchive,Ihadtoask.“Well,”myhostssaid,“wecanshowyou.”Andthereinavault,alongwithshelvesofboxes,wasLeibniz’scalculator,lookingasgoodasnewinaglasscase—herecapturedbymeinastrangejuxtapositionofancientandmodern:

All the pieces are there. Including a convenient wooden carrying box.Completewithacrankinghandle.And,if itworkedright, theabilitytodoanybasicarithmeticoperationwithafewminutesofcranking:

Leibnizclearlyviewedhiscalculatorasapracticalproject.Buthestillwantedtogeneralize fromit, forexample trying tomakeageneral“logic” todescribegeometries of mechanical linkages. And he also thought about the nature ofnumbersandarithmetic.Andwasparticularlystruckbybinarynumbers.Bases other than 10 had been used in recreational mathematics for several

centuries.ButLeibniz latchedon tobase2ashavingparticular significance—andperhapsbeingakeybridgebetweenphilosophy,theologyandmathematics.Andhewasencouragedinthisbyhisrealizationthatbinarynumberswereatthecore of the I Ching, which he’d heard about frommissionaries to China, andviewedasrelatedinspirittohischaracteristicauniversalis.Leibnizworkedout that itwould be possible to build a calculator basedon

binary. But he appears to have thought that only base 10 could actually beuseful.It’s strange to readwhatLeibnizwroteaboutbinarynumbers.Someof it is

clearandpractical—andstillseemsperfectlymodern.Butsomeofitisvery17thcentury—talking for example about how binary proves that everything can bemadefromnothing,with1beingidentifiedwithGod,and0withnothing.AlmostnothingwasdonewithbinaryforacoupleofcenturiesafterLeibniz:

infact,untiltheriseofdigitalcomputinginthelastfewdecades.Sowhenonelooks at Leibniz’s papers, his calculations in binary are probably what seemmost“outofhistime”:

Withbinary,Leibnizwasinasenseseekingthesimplestpossibleunderlyingstructure.Andnodoubthewasdoingsomethingsimilarwhenhe talkedaboutwhat he called “monads”. I have to say that I’ve never really understoodmonads.AndusuallywhenIthinkIalmosthave,there’ssomementionofsoulsthatjustthrowsmecompletelyoff.Still,I’vealwaysfoundittantalizingthatLeibnizseemedtoconcludethatthe

“best of all possible worlds” is the one “having the greatest variety ofphenomena from the smallest number of principles”. And indeed, in theprehistory of my work on A New Kind of Science, when I first startedformulating and studying one-dimensional cellular automata in 1981, Iconsiderednamingthem“polymones”—butatthelastminutegotcoldfeetwhenIgotconfusedagainaboutmonads.There’salwaysbeenacertainmystiquearoundLeibnizandhispapers.Kurt

Gödel—perhaps displaying his paranoia—seemed convinced that Leibniz haddiscovered great truths that had been suppressed for centuries. Butwhile it istrue thatLeibniz’spapersweresealedwhenhedied, itwashisworkon topicslikehistoryandgenealogy—andthestatesecretstheymightentail—thatwastheconcern.Leibniz’spaperswereunsealedlongago,andafterthreecenturiesonemight

assumethateveryaspectofthemwouldhavebeenwellstudied.Butthefactisthatevenafterallthistime,nobodyhasactuallygonethroughallofthepapersinfull detail. It’s not that there are so many of them. Altogether there are onlyabout 200,000 pages—filling perhaps a dozen shelving units (and only a littlelargerthanmyownpersonalarchivefromjustthe1980s).Buttheproblemisthediversity of material. Not only lots of subjects. But also lots of overlappingdrafts,notesandletters,withunclearrelationshipsbetweenthem.Leibniz’sarchivecontainsabewilderingarrayofdocuments.From thevery

large:

To the very small (Leibniz’s writing got smaller as he got older andmorenear-sighted):

Most of the documents in the archive seem very serious and studious. Butdespite the high cost of paper in Leibniz’s time, one still finds preserved forposteritytheoccasionaldoodle(isthatSpinoza,byanychance?):

Leibnizexchangedmailwithhundredsofpeople—famousandnot-so-famous—alloverEurope.Sonow,300yearslater,onecanfindinhisarchive“randomletters”fromthelikesofJacobBernoulli:

What did Leibniz look like? Here he is, both in an official portrait, andwithout his rather oversized wig (that was mocked even in his time), that hepresumablyworetocoverupalargecystonhishead:

Asaperson,Leibnizseemstohavebeenpolite,courtierlyandeven-tempered.Insomeways,hemayhavecomeacrossassomethingofanerd,expoundingatgreatdepthonallmanneroftopics.Heseemstohavetakengreatpains—ashedidinhisletters—toadapttowhoeverhewastalkingto,emphasizingtheologywhenhewastalkingtoatheologian,andsoon.Likequiteafewintellectualsofhis time,Leibniznevermarried, thoughheseems tohavebeensomethingofafavoritewithwomenatcourt.Inhiscareerasacourtier,Leibnizwaskeentoclimbtheladder.Butnotbeing

intohuntingordrinking,heneverquitefitinwiththeinnercirclesoftherulershe worked for. Late in his life, when George I of Hanover became king ofEngland, itwouldhavebeennatural forLeibniz to joinhis court.ButLeibnizwastoldthatbeforehecouldgo,hehadtostartwritingupahistoryprojecthe’dsupposedly beenworking on for 30 years.Had he done so before he died, hemightwell have gone toEngland and had a very different kind of interactionwithNewton.AtLeibniz’sarchive, thereare lotsofpapers,hismechanicalcalculator,and

onemorething:afoldingchairthathetookwithhimwhenhetraveled,andthathe had suspended in carriages so he could continue to write as the carriagemoved:

Leibnizwasquiteconcernedaboutstatus(heoftenstyledhimself“GottfriedvonLeibniz”,thoughnobodyquiteknewwherethe“von”camefrom).Andasaform of recognition for his discoveries, hewanted to have amedal created tocommemorate binary numbers. He came up with a detailed design, completewiththetaglineomnibusexnihiloducendis;sufficitunum(“everythingcanbederivedfromnothing;allthatisneededis1”).Butnobodyevermadethemedalforhim.In2007,though,Iwantedtocomeupwitha60thbirthdaygiftformyfriend

GregChaitin,whohasbeen a long-timeLeibniz enthusiast.And so I thought:whynotactuallymakeLeibniz’smedal?Sowedid.Thoughontheback,insteadofthepictureofadukethatLeibnizproposed,weputaLatininscriptionaboutGreg’swork.Andwhen I visited theLeibniz archive, Imade sure to bring a copyof the

medal,soIcouldfinallyputarealmedalnexttoLeibniz’sdesign:

Itwould have been interesting to knowwhat pithy statementLeibnizmighthavehadonhis grave.But as itwas,whenLeibniz died at the ageof 70, hispolitical fateswere at a low ebb, and no elaboratememorialwas constructed.Still,when Iwas inHanover, Iwaskeen toseehisgrave—which turnsout tocarryjustthesimpleLatininscription“bonesofLeibniz”:

Across town, however, there’s another commemoration of a sort—an outletstoreforcookiesthatcarrythename“Leibniz”inhishonor:

So what should we make of Leibniz in the end? Had history developeddifferently, there would probably be a direct line from Leibniz to moderncomputation.Butasitis,muchofwhatLeibniztriedtodostandsisolated—tobeunderstoodmostlybyprojectingbackwardfrommoderncomputationalthinkingtothe17thcentury.Andwithwhatweknownow,itisfairlyclearwhatLeibnizunderstood,and

what he did not. He grasped the concept of having formal, symbolicrepresentationsforawiderangeofdifferentkindsof things.Andhesuspectedthat theremight be universal elements (maybe even just 0 and 1) fromwhichthese representations could be built. And he understood that from a formalsymbolic representation of knowledge, it should be possible to compute itsconsequences inmechanical ways—and perhaps create new knowledge by anenumerationofpossibilities.Some of what Leibniz wrote was abstract and philosophical—sometimes

maddeninglyso.ButatsomelevelLeibnizwasalsoquitepractical.Andhehadsufficient technicalprowess tooftenbeable tomakerealprogress.His typicalapproach seems to have been to start by trying to create a formal structure toclarify things—with formalnotation ifpossible.Andafter thathisgoalwas tocreatesomekindof“calculus”fromwhichconclusionscouldsystematicallybedrawn.Realistically he only had true success with this in one specific area:

continuous“geometrical”mathematics.It’sapityhenevertriedmoreseriouslyin discrete mathematics, because I think he might have been able to makeprogress, and might conceivably even have reached the idea of universalcomputation.Hemightwellalsohaveendedupstarting toenumeratepossiblesystemsinthekindofwayIhavedoneinthecomputationaluniverse.One area where he did try his approach was with law. But in this he was

surelyfartooearly,anditisonlynow—300yearslater—thatcomputationallawisbeginningtoseemrealistic.Leibniz also tried thinking about physics.Butwhile hemade progresswith

somespecificconcepts(likekineticenergy),henevermanagedtocomeupwithanysortoflarge-scale“systemoftheworld”,ofthekindthatNewtonineffectdidinhisPrincipia.Insomeways, I thinkLeibnizfailed tomakemoreprogressbecausehewas

trying toohard tobepractical, and—likeNewton—todecode theoperationofactual physics, rather than just looking at related formal structures. For hadLeibniz tried todoat least thebasickindsofexplorations that Idid inANewKindofScience,Idon’tthinkhewouldhavehadanytechnicaldifficulty—butI

thinkthehistoryofsciencecouldhavebeenverydifferent.AndIhavecometorealizethatwhenNewtonwonthePRwaragainstLeibniz

over the inventionofcalculus, itwasnot justcredit thatwasat stake, itwasawayofthinkingaboutscience.Newtonwasinasensequintessentiallypractical:he invented tools, then showed how these could be used to compute practicalresults about the physical world. But Leibniz had a broader and morephilosophicalview,andsawcalculusnotjustasaspecifictoolinitself,butasanexample that should inspire efforts at other kinds of formalization and otherkindsofuniversaltools.I have often thought that the modern computational way of thinking that I

follow is somehow obvious—and somehow an inevitable feature of thinkingaboutthingsinformal,structuredways.Butithasneverbeenverycleartomewhetherthisapparentobviousnessisjusttheresultofmoderntimes,andofourexperiencewithmodernpracticalcomputertechnology.ButlookingatLeibniz,wegetsomeperspective.Andindeedwhatweseeisthatsomecoreofmoderncomputational thinking was possible even long before modern times. But theambient technology and understanding of past centuries put definite limits onhowfarthethinkingcouldgo.Andofcoursethisleadstoasoberingquestionforustoday:howmucharewe

failingtorealizefromthecorecomputationalwayofthinkingbecausewedonothave the ambient technologyof thedistant future?Forme, looking atLeibnizhasputthisquestioninsharperfocus.Andatleastonethingseemsfairlyclear.InLeibniz’swholelife,hebasicallysawlessthanahandfulofcomputers,and

all theydidwasbasic arithmetic.Today there arebillionsof computers in theworld,andtheydoallsortsofthings.Butinthefuturetherewillsurelybefarfarmore computers (made easier to create by the Principle of ComputationalEquivalence).Andnodoubtwe’llgettothepointwherebasicallyeverythingwemakewillexplicitlybemadeofcomputersateverylevel.Andtheresultisthatabsolutelyeverythingwillbeprogrammable,downtoatoms.Ofcourse,biologyhasinasensealreadyachievedarestrictedversionofthis.Butwewillbeabletodoitcompletelyandeverywhere.At some level we can already see that this implies some merger of

computationalandphysicalprocesses.Butjusthowmaybeasdifficultforustoimagine as things likeMathematica andWolfram|Alpha would have been forLeibniz.LeibnizdiedonNovember14,1716. In2016 that’ll be300years ago.And

it’llbeagoodopportunity tomake sureeverythingwehave fromLeibnizhasfinally been gone through—and to celebrate after three centuries how manyaspectsofLeibniz’scorevisionarefinallycomingtofruition,albeitinwayshe

couldneverhaveimagined.

BenoitMandelbrotNovember22,2012*

Innature,technologyandartthemostcommonformofregularityisrepetition:asingle element repeated many times, as on a tile floor. But another form ispossible,inwhichsmallerandsmallercopiesofapatternaresuccessivelynestedinsideeachother,sothatthesameintricateshapesappearnomatterhowmuchyou “zoom in” to the whole. Fern leaves and Romanesco broccoli are twoexamplesfromnature.Onemighthavethoughtthatsuchasimpleandfundamentalformofregularity

wouldhavebeenstudiedforhundreds,ifnotthousands,ofyears.Butitwasnot.Infact,itrosetoprominenceonlyoverthepast30orsoyears—almostentirelythroughtheeffortsofoneman,themathematicianBenoitMandelbrot,whodiedin2010justbeforecompletingthisautobiography.BorntoaJewishfamilyinPolandin1924,hewasthesonofadentistmother

andabusinessmanfather.Oneofhisuncles,SzolemMandelbrojt,wasafairlysuccessfulpuremathematicianinFrance,anditwastherethattheMandelbrotsfledin1936asthesituationforJewsinPolandworsened.Afterhidingoutduringthewar in thecountryside,Mandelbrotwoundupas

an outstanding math student at a top technical college in Paris. At the time,FrenchmathematicswasdominatedbythehighlyabstractBourbakimovement,whichwasnamedforthecollectivepseudonymwithwhichitsmemberssignedtheir work.WhenMandelbrot graduated in 1947, instead of joining them, hedecided to study aeronautical engineering at Caltech. He was determined, hewrites, to stay away from established mathematics so that he could “feel theexcitementofbeingthefirsttofindadegreeoforderinsomereal,concrete,andcomplexareawhereeveryoneelsesawalawlessmess.”ReturningtoFrancetwoyearslater,hegotinvolvedwithinformationtheory

and, from that, statistical physics and the structure of language. He ended upwritinghisPhDon“GamesofCommunication,”notablyinvestigatinghowthefrequencies with which different words were used in a text followed adistributionknownasa“powerlaw”.Hedidmilitaryserviceasakindofscienceresearchscout,spenttimeataPhilipscolortelevisionlab,andthenbeganlifeasa“wanderingscientist”,partlysupportedbytheFrenchgovernment.His early destinations includedPrinceton (where heworkedwith the game-

theory pioneer John von Neumann) and Geneva (with the psychologist JeanPiaget). And then, in 1958, he visited IBM, ostensibly to work on a newautomated-translationproject—andendedupstayingfor35years.

IBMsawitsresearchdivision,inpart,asawaytospreaditsreputation,anditallowed Mandelbrot to continue wandering and spend time as a visitor atuniversities likeHarvardandYale (whereheeventuallyspent the lastyearsofhiscareer).Hisworkmovedfromthestatisticsof languagesto thestatisticsofeconomicsystemsand,bythemid-1960s,tohydrodynamicsandhydrology.Histypical mode of operation was to apply fairly straightforward mathematics(typicallyfromthetheoryofrandomprocesses)toareasthathadbarelyseenthelight of serious mathematics before. He calls himself a “would-be Kepler ofcomplexity”,evoking(ashedoescontinually)JohannesKepler,the17th-centuryscientistwhodeterminedthelawsthatdescribethemovementoftheplanets.But in the early 1970s a mathematician friend (Mark Kac) made a crucial

suggestion:Stopwritinglotsofpapersaboutstrangelydiversetopicsandinsteadtiethemtogetherinabook.Theunifyingthemecouldhavebeenatechnicalone—inwhich case few people todaywould have probably ever heard of BenoitMandelbrot.Butinstead,perhapsjustthroughtheveryactofexposition(hisautobiography

doesnotmakeitclear),Mandelbrotendedupdoingagreatpieceofscienceandidentifyingamuchstrongerandmorefundamentalidea—putsimply,thattherearesomegeometricshapes,whichhecalled“fractals”,thatareequally“rough”atallscales.Nomatterhowcloseyoulook,theynevergetsimpler,muchasthesectionofarockycoastlineyoucanseeatyourfeetlooksjustasjaggedasthestretch you can see from space. This insight formed the core of his breakout1975book,Fractals.Before this book, Mandelbrot’s work had been decidedly numerical, with

simplegraphsbeingaboutasvisualasmostofhispapersgot.Butbetweenhisaccess to computer graphics at IBM and publishers with distinctly visualorientations,hisbookendedupfilledwithstrikingillustrations,withhisthemepresentedinahighlygeometricway.Therewasnobooklikeit.Itwasanewparadigm,bothinpresentationandin

the informal style of explanation it employed.Papers slowly started appearing(quiteoftenwithMandelbrotasco-author) thatconnected it todifferent fields,suchasbiologyandsocialscience.Resultsweremixedandoftencontroversial.Did the paths traced by animals or graphs of stock prices really have nestedstructuresorfollowexactpowerlaws?EvenMandelbrothimselfbegantowaterdownhismessage,introducingconceptslike“multifractals”.But independent of science, fractals began to thrive in computer graphics,

particularlyinmakingwhatMandelbrotcalled“forgeries”ofnaturalphenomena—surprisingly lifelike images of trees or mountains. Some mathematiciansbegan to investigate fractals in abstract terms, connecting them to a branch of

mathematicsconcernedwithso-callediteratedmaps.Thesehadbeenstudiedinthe early 1900s—notably by French mathematicians whom Mandelbrot hadknownasa student.But after a few results, their investigationhad largely runoutofsteam.Armed with computer graphics, however, Mandelbrot was able to move

forward,discovering in1979 the intricate shapeknownas theMandelbrot set.Theset,withitsdistinctivebulb-likelobes,lentitselftocolorfulrenderingsthathelpedfractalstakeholdinboththepopularandscientificmind.AndalthoughIconsidertheMandelbrotsetinsomewaysaratherarbitrarymathematicalobject,ithasbeenafertilesourceofpuremathematicalquestions—aswellasastrikingexampleofhowvisualcomplexitycanarisefromsimplerules.Inmanyways,Mandelbrot’slifeisaheroicstoryofdiscovery.Hewasagreat

scientist,whosevirtuesIamkeentoextol.Forallhisachievements,however,hecouldbeadifficultman,asIdiscoveredininteractionsoverthecourseofnearly30years.Hewasconstantlyseekingvalidationandconstantlyfightingtogethisdue,athemethatisalreadyclearonthefirstpageofhisautobiography:“Letmeintroduce myself. A scientific warrior of sorts, and an old man now, I havewrittenagreatdealbutneveracquiredapredictableaudience.”HecampaignedfortheNobelPrizeinphysics;lateritwaseconomics.Iused

toaskhimwhyhecaredsomuch.Ipointedout thatreallygreatscience—likefractals—tends to be too original for there to be prizes defined for it. But hewouldsloughoffmycommentsandrecitesomeotherevidenceforthegreatnessofhisachievements.Inhisway,Mandelbrotpaidmesomegreatcompliments.WhenIwasinmy

20s, and he in his 60s, hewould ask aboutmy scientificwork: “How can somanypeople take someone soyoung so seriously?” In2002,mybookANewKindofScience—inwhichIarguedthatmanyphenomenaacrosssciencearethecomplexresultsofrelativelysimple,program-likerules—appeared.Mandelbrotseemed to see it asadirect threat,oncedeclaring that“Wolfram’s ‘science’ isnot new except when it is clearly wrong; it deserves to be completelydisregarded.”Inprivate,though,severalmutualfriendstoldme,hefrettedthatinthelongviewofhistoryitwouldoverwhelmhiswork.EverytimeIsawhim,IwouldexplainwhyIthoughthewaswrongtoworry

andwhy,even though in the“computationaluniverse” thatmybookdescribedfractals appear in just a small corner, they will nevertheless always befundamentally important—not least as perhaps the unique intermediate stepbetween simple repetitive regularity and the apparent randomness of morecomplexcomputationalprocesses.I sawMandelbrot quite frequently in the years before his death, andmany

questionsheansweredwiththephrase,“Readmyautobiography.”Andindeeditdoes answer somequestions—especially about his early life—though it leavesquiteafewunanswered,suchashowexactlyhisbreakthrough1975bookcametogether. But knowing more about Benoit Mandelbrot as a person helps toilluminate hiswork and to illustratewhat it takes for great new science to becreated.TheFractalist is awell-written tale of a scientific life, completewithfirst-personaccountsofasurprisingrangeofscientificgreats.

*OriginallywrittenasareviewofMandelbrot’sposthumousautobiography,TheFractalist.

SteveJobsOctober6,2011

I’m so sad this evening—as millions are—to hear of Steve Jobs’s death.Scatteredoverthelastquartercentury,IlearnedmuchfromSteveJobs,andwasproudtoconsiderhimafriend.Andindeed,hecontributedinvariouswaystoallthreeofmymajorlifeprojectssofar:Mathematica,ANewKindofScienceandWolfram|Alpha.I firstmetSteveJobs in1987,whenhewasquietlybuildinghis firstNeXT

computer,andIwasquietlybuildingthefirstversionofMathematica.Amutualfriendhadmadetheintroduction,andSteveJobswastednotimeinsayingthathewasplanning tomake thedefinitive computer for higher education, andhewantedMathematica to be part of it. I don’t now remember the details of ourfirstmeeting,butattheendofit,Stevegavemehisbusinesscard,whichtonightIfounddulystillsittinginmyfiles:

Over themonths after our firstmeeting, I had all sorts of interactionswithSteveaboutMathematica.Actually, itwasn’tyetcalledMathematica then,andoneof thebigtopicsofdiscussionwaswhat itshouldbecalled.Atfirst ithadbeen Omega (yes, like Alpha) and later PolyMath. Steve thought those werelousynames.IgavehimlistsofnamesI’dconsidered,andpressedhimforhissuggestions.Forawhilehewouldn’tsuggestanything.Butthenonedayhesaidtome:“YoushouldcallitMathematica.”I’d actually considered that name, but rejected it. I asked Steve why he

thoughtitwasgood,andhetoldmehistheoryforanamewastostartfromthegenerictermforsomething,thenromanticizeit.HisfavoriteexampleatthetimewasSony’sTrinitron.Well,itwentbackandforthforawhile.ButintheendIagreed that, yes,Mathematicawas a good name.And so it has been now fornearly24years.AsMathematicawasbeingdeveloped,weshowedittoSteveJobsquiteoften.

He always claimedhe didn’t understand themath of it (though I later learnedfromagoodfriendofminewhohadknownSteveinhighschoolthatStevehaddefinitelytakenatleastonecalculuscourse).Buthemadeallsortsof“makeitsimpler”suggestionsabouttheinterfaceandthedocumentation.Withoneslightexception,perhapsofat least curiosity interest toMathematicaaficionados:hesuggested that cells in Mathematica notebook documents (and now CDFs)should be indicated not by simple vertical lines—but instead by bracketswithlittleserifsattheirends.Andasithappens,thatideaopenedthewaytothinkingofhierarchiesofcells,andultimatelytomanyfeaturesofsymbolicdocuments.InJune1988wewere ready to releaseMathematica.ButNeXThadnotyet

released its computer, Steve Jobs was rarely seen in public, and speculationabout what NeXT was up to had become quite intense. So when Steve Jobsagreedthathewouldappearatourproductannouncement, itwasahuge thingforus.Hegave a lovely talk, discussinghowhe expectedmore andmore fields to

become computational, and to need the services of algorithms and ofMathematica.Itwasaverycleanstatementofavisionwhichhasindeedworkedoutashepredicted.(Andnowit’snicewhenIhear throughthegrapevinethatthereareallsortsofalgorithmscentral to the iPhonethatweredevelopedwiththehelpofMathematica.)Awhile later, theNeXTwasdulyreleased,andacopyofMathematicawas

bundledwith every computer.Although theNeXTwas not in its own right acommercialsuccess,Steve’sdecisiontobundleMathematicaturnedouttobeareally good idea, and was often quoted as the #1 reason people had bought

NeXTs.Andasacuriousfootnotetohistory(whichIlearnedyearslater),onebatchof

NeXTs bought for the purpose of running Mathematica went to CERN inGeneva, Switzerland—where they ended up having no less distinction thanbeingthecomputersonwhichthewebwasfirstdeveloped.IusedtoseeSteveJobswithsomeregularityinthosedays.OnetimeIwentto

seehiminNeXT’sswanknewofficesinRedwoodCity.IparticularlywantedtotalktohimaboutMathematicaasacomputerlanguage.Healwayspreferreduserinterfaces to languages,buthewas trying tobehelpful.Theconversationwasgoing on, but he said he couldn’t go to dinner, and actually he was quitedistracted,becausehewasgoingoutonadatethatevening—andhehadn’tbeenon a date for a long time.He explained that he’d justmet thewomanhewasseeingafewdaysearlier,andwasverynervousabouthisdate.TheSteveJobs—soconfidentasabusinessmanandtechnologist—hadmeltedaway,andhewasaskingme—hardlyanotedauthorityonsuchthings—abouthisdate.As it turnedout, thedateapparentlyworkedout—andwithin18months the

womanhemetbecamehiswife,andremainedsountiltheend.MydirectinteractionswithSteveJobsdecreasedduringthedecadethatIwas

forallpracticalpurposesahermitworkingonANewKindofScience.Formostofthattime,though,IusedaNeXTcomputerinalmosteverywakinghour—andinfactmymaindiscoveriesweremadeonit.Andwhenthebookwasfinished,Steveaskedforapre-releasecopy,whichIdulysent.Atthetime,allsortsofpeopleweretellingmethatIneededtoputquoteson

thebackcoverofthebook.SoIaskedSteveJobsifhe’dgivemeone.Variousquestions came back. But eventually Steve said, “Isaac Newton didn’t haveback-coverquotes;whydoyouwantthem?”Andthat’show,atthelastminute,thebackcoverofANewKindofScienceendedupwithjustasimpleandelegantarrayofpictures.AnothercontributionfromSteveJobs,thatInoticeeverytimeIlookatmybigbook.Inmy life, I havehad thegood fortune to interactwith all sorts of talented

people.Tome,SteveJobsstandsoutmostforhisclarityof thought.Overandoveragainhe tookcomplexsituations,understood theiressence,andused thatunderstandingtomakeabolddefinitivemove,ofteninacompletelyunexpecteddirection.Imyselfhavespentmuchofmylife—inscienceandintechnology—tryingto

workinsomewhatsimilarways.AndtryingtobuildtheverybestpossiblethingsIcan.Yet looking at the practical world of technology and business there are

certainlytimeswhenithasnotbeenobviousthatanyofthisisagoodstrategy.

Indeed,sometimesit looksasifallthatclarity,understanding,qualityandnewideasaren’treallythepoint—andthatthewinnersarethosewithquitedifferentinterests.So for me—and our company—it has been immensely inspiring to watch

Steve Jobs’s—and Apple’s—amazing success in recent years. It validates somany of the principles that I have long believed in. And encourages me topursuethemwithevengreatervigor.I think thatover theyearsSteve Jobs appreciated the approach I’ve tried to

takewithourcompany.Hewascertainlyalwaysagreatsupporter.(Justtonight,for example, I was reminded of a terrific video that he sent us for the 10th-anniversaryMathematica user conference.)And hewas always keen for us toworkfirstwithNeXT,andlaterwithApple.I thinkMathematicamayhold thedistinctionofhavingbeen theonlymajor

software system available at launch on every single computer that Steve Jobscreated since 1988. Of course, that’s often led to highly secretive emergencyMathematica porting projects—culminating a couple of times in Theo GraydemoingtheresultsinSteveJobs’skeynotespeeches.WhenApple started producing the iPod and iPhone Iwasn’t sure how they

wouldrelatetoanythingwedid.ButafterWolfram|Alphacameout,westartedrealizingjusthowpowerfulitwastohavecomputationalknowledgeonthisnewplatformthatSteveJobshadcreated.AndwhentheiPadwascomingout,TheoGray—atSteveJobs’surging—insistedthatwehadtodosomethingsignificantforit.The result was the formation last year of Touchpress, the publication of

Theo’s The Elements iPad ebook, and now a string of other iPad ebooks. AwholenewdirectionmadepossiblebySteveJobs’screationoftheiPad.It’s hard to remember tonight all the ways Steve Jobs has supported and

encouraged us over the years. Big things and small things. Looking at myarchiveIrealizeI’dforgottenjusthowmanydetailedproblemshejumpedintosolve.From theglitches inversionsofNeXTSTEP, to thepersonalphonecallnot longago toassureus that ifweportedMathematicaandCDF to iOS theywouldn’tbebanned.ThereismuchthatIamgratefultoSteveJobsfor.Buttragically,hisgreatest

contribution to my latest life project—Wolfram|Alpha—happened justyesterday: the announcement that Wolfram|Alpha will be used in Siri on theiPhone4S.It is somehow a quintessential Steve Jobsmove.To realize that people just

want direct access to knowledge and actions on their phones.Without all theextrastepsthatpeoplewouldusuallyassumehavetobethere.

I’mproudthatweareinapositiontoprovideanimportantcomponentforthatvisionwithWolfram|Alpha.What’scomingoutnow is justabeginning,and IlookforwardtowhatwewilldowithAppleinthisdirectioninthefuture.I’mjustsadthatSteveJobswillnownotbepartofit.When I first met Steve Jobs nearly 25 years ago I was struck by him

explainingtomethatNeXTwaswhathe“wantedtodowithhisthirties”.Atthetime,Ithoughtitwasaboldthingtoplanone’slifeindecadeslikethat.And—particularly for those of us who spend their lives doing large projects—it’sincredibly inspiring to see what Steve Jobs was able to achieve in his smallnumberofdecades,sotragicallycutshorttoday.Thankyou,Steve,foreverything.

MarvinMinskyJanuary26,2016

Ithinkitwas1979whenIfirstmetMarvinMinsky,whileIwasstillateenagerworking on physics at Caltech. It was a weekend, and I’d arranged to seeRichardFeynmantodiscusssomephysics.ButFeynmanhadanothervisitorthatday as well, who didn’t just want to talk about physics, but insteadenthusiasticallybroughtuponeunexpectedtopicafteranother.That afternoonwewere driving throughPasadena,California—andwith no

apparent concern to the actual process of driving, Feynman’s visitor wasenergeticallypointingoutallsortsofthingsanAIwouldhavetofigureifitwasto be able to do the driving. I was a bit relieved when we arrived at ourdestination, but soon the visitor was on to another topic, talking about howbrainswork,andthensayingthatassoonashe’dfinishedhisnextbookhe’dbehappytoletsomeoneopenuphisbrainandputelectrodesinside,iftheyhadagoodplantofigureouthowitworked.Feynmanoftenhadeccentricvisitors,butIwasreallywonderingwhothisone

was. It took a couple more encounters, but then I got to know that eccentricvisitorasMarvinMinsky,pioneerofcomputationandAI—andwaspleasedtocounthimasafriendformorethanthreedecades.Just a fewdays ago Iwas talking about visitingMarvin—and Iwas so sad

whenIheardhedied.Istartedreminiscingaboutallthewaysweinteractedoverthe years, and all the interests we shared. Every major project of my life IdiscussedwithMarvin, fromSMP,myfirstbigsoftwaresystemback in1981,throughMathematica,ANewKindofScience,Wolfram|AlphaandmostrecentlytheWolframLanguage.ThispictureisfromoneofthelasttimesIsawMarvin.Hishealthwasfailing,

but he was keen to talk. Having watched more than 35 years of my life, hewanted to tellmehisassessment:“Youreallydid it,Steve.”Well, sodidyou,Marvin!(I’malways“Stephen”,butsomehowAmericansofacertainagehaveahabitofcallingme“Steve”.)

TheMarvinthatIknewwasawonderfulmixtureofseriousandquirky.Aboutalmost any subject he’d have something to say, most often quite unusual.Sometimes it’d be really interesting; sometimes it’d just be unusual. I’mremindedofatimeintheearly1980swhenIwasvisitingBostonandsublettinganapartmentfromMarvin’sdaughterMargaret(whowasinJapanatthetime).Margaret had a large and elaborate collectionofplants, andoneday I noticedthatsomeofthemhaddevelopednasty-lookingspotsontheirleaves.Beingnoexpertonsuchthings(andwithoutthewebtolookanythingup!),I

calledMarvintoaskwhat todo.Whatensuedwasalongdiscussionabout thepossibility of developing microrobots that could chase mealybugs away.Fascinatingthoughitwas,attheendofitIstillhadtoask,“ButwhatshouldIactuallydoaboutMargaret’splants?”Marvinreplied,“Oh,Iguessyou’dbettertalktomywife.”Formanydecades,Marvinwasperhapstheworld’sgreatestenergysourcefor

artificialintelligenceresearch.Hewasafountofideas,whichhefedtohislongsequence of students atMIT.And though the details changed, he always kepttruetohisgoaloffiguringouthowthinkingworks,andhowtomakemachinesdoit.

MarvintheComputationTheoristBythetimeIknewMarvin,hetendedtotalkmostlyabouttheorieswherethingscould be figured out by what amounts to common sense, perhaps based onpsychological or philosophical reasoning. But earlier in his life, Marvin hadtaken a different approach. His 1954 PhD thesis from Princeton was aboutartificial neural networks (“Theory of Neural-Analog Reinforcement Systemsand Its Application to the Brain Model Problem”) and it was a mathematicsthesis, full of technical math. And in 1956, for example,Marvin published apaper entitled “Some Universal Elements for Finite Automata”, in which hetalked about how “complicated machinery can be constructed from a smallnumberofbasicelements”.This particular paper considered only essentially finite machines, based

directlyon specificmodelsof artificialneuralnetworks.But soonMarvinwaslooking at more general computational systems, and trying to see what theycoulddo. Inasense,Marvinwasbeginning just thekindofexplorationof thecomputationaluniversethatyearslaterIwouldalsodo,andeventuallywriteANew Kind of Science about. And in fact, as early as 1960, Marvin cameextremelyclosetodiscoveringthesamecorephenomenonIeventuallydid.

In 1960, as now, Turingmachineswere used as a standard basicmodel ofcomputation.Andinhisquesttounderstandwhatcomputation—andpotentiallybrains—couldbebuiltfrom,MarvinstartedlookingattheverysimplestTuringmachines(withjust2statesand2colors)andusingacomputertofindoutwhatall4096ofthemactuallydo.Mosthediscoveredjusthaverepetitivebehavior,and a few have what we’d now call nested or fractal behavior. But none doanythingmore complicated, and indeedMarvin based the final exercise in hisclassic1967bookComputation:FiniteandInfiniteMachinesonthis,notingthat“D.G.Bobrowandtheauthordidthisforall(2,2)machines[1961,unpublished]byatediousreductiontothirty-oddcases(unpublishable).”Years later,Marvin toldme that after all the effort he’d spent on the (2,2)

Turingmachineshewasn’tinclinedtogofurther.ButasIfinallydiscoveredin1991,ifonejustlooksat(2,3)Turingmachines,thenamongthe3millionorsoof them, there are a few that don’t just show simple behavior anymore—andinsteadgenerateimmensecomplexityevenfromtheirverysimplerules.Back in the early 1960s, even though he didn’t find complexity just by

searchingsimple“naturallyoccurring”Turingmachines,Marvinstillwantedtoconstruct the simplest one he could that would exhibit it. And throughpainstaking work, he came up in 1962 with a (7,4) Turing machine that heproved was universal (and so, in a sense, capable of arbitrarily complexbehavior).Atthetime,Marvin’s(7,4)Turingmachinewasthesimplestknownuniversal

Turingmachine.Anditkeptthatrecordessentiallyunbrokenfor40years—untilIfinallypublisheda(2,5)universalTuringmachineinANewKindofScience.Ifelta littleguilty takingtherecordawayfromMarvin’smachineaftersolong.ButMarvin was very nice about it. And a few years later he enthusiasticallyagreed tobeon thecommittee foraprize Iputup toestablishwhethera (2,3)Turing machine that I had identified as the simplest possible candidate foruniversalitywasinfactuniversal.Itdidn’ttakelongforaproofofuniversalitytobesubmitted,andMarvingot

quite involved in some of the technical details of validating it, noting thatperhapsweshouldallhaveknownsomething like thiswaspossible,given thecomplexitythatEmilPosthadobservedwiththesimplerulesofwhathecalledatagsystem—backin1921,beforeMarvinwasevenborn.

MarvinandNeuralNetworksWhenitcametoscience,itsometimesseemedasifthereweretwoMarvins.One

was the Marvin trained in mathematics who could give precise proofs oftheorems.TheotherwastheMarvinwhotalkedaboutbigandoftenquirkyideasfarawayfromanythinglikemathematicalformalization.I thinkMarvinwasultimatelydisappointedwithwhat couldbeachievedby

mathematics and formalization. In his early years he had thought that withsimple artificial neural networks—andmaybe things like Turingmachines—itwouldbeeasytobuildsystemsthatworkedlikebrains.Butitneverseemedtohappen.And in 1969,with his long-timemathematician collaborator SeymourPapert,Marvinwrote a book that proved that a certain simple class of neuralnetworks known as perceptrons couldn’t (in Marvin’s words) “do anythinginteresting”.To Marvin’s later chagrin, people took the book to show that no neural

networkofanykindcouldeverdoanythinginteresting,andresearchonneuralnetworks all but stopped.But a bit likewith the (2,2)Turingmachines,muchricherbehaviorwasactuallylurkingjustoutofsight.Itstartedbeingnoticedinthe1980s,butit’sonlybeeninthelastcoupleofyears—withcomputersabletohandlealmost-brain-scalenetworks—that the richnessofwhatneuralnetworkscandohasbeguntobecomeclear.AndalthoughIdon’t thinkanyonecouldhaveknownit then,wenowknow

thattheneuralnetworksMarvinwasinvestigatingasearlyas1951wereactuallyonapaththatwouldultimatelyleadtojustthekindofimpressiveAIcapabilitieshewashoping for. It’s apity it took so long, andMarvinbarelygot to see it.(Whenwe released our neural-network-based image identifier last year, I sentMarvin a pointer saying “I never thought neural networks would actuallywork…but…”Sadly, Ineverendedup talking toMarvinabout it.)MarvinandSymbolicAIMarvin’searliestapproachestoAIwerethroughthingslikeneuralnetworks.Butperhaps through the influence of John McCarthy, the inventor of LISP, withwhom Marvin started the MIT AI Lab, Marvin began to consider more“symbolic”approachestoAIaswell.Andin1961Marvingotastudentofhistowrite a program in LISP to do symbolic integration. Marvin told me that hewanted the program to be as “human like” as possible—so every so often itwouldstopandsay“Givemeacookie”,andtheuserwouldhavetorespond“Acookie”.By the standards of Mathematica or Wolfram|Alpha, the 1961 integration

programwasveryprimitive.ButI’mcertainlygladMarvinhaditbuilt.BecauseitstartedasequenceofprojectsatMITthatledtotheMACSYMAsystemthatIendedupusing in the1970s—that inmanyways launchedmyeffortsonSMP

andeventuallyMathematica.Marvin himself, though, didn’t go on thinking about using computers to do

mathematics,butinsteadstartedworkingonhowtheymightdothekindoftasksthat all humans—including children—routinely do. Marvin’s collaboratorSeymourPapert,whohadworkedwithdevelopmentalpsychologistJeanPiaget,was interested in how children learn, and Marvin got quite involved inSeymour’s project of developing a computer language for children.The resultwas Logo—a direct precursor of Scratch—and for a brief while in the 1970sMarvinandSeymourhadacompanythattriedtomarketLogoandahardware“turtle”toschools.Forme therewasalwaysacertainmystiquearoundMarvin’s theoriesabout

AI.Insomewaystheyseemedlikepsychology,andinsomewaysphilosophy.But occasionally there’d actually be pieces of software—or hardware—thatclaimedtoimplementthem,ofteninwaysthatIdidn’tunderstandverywell.Probably the most spectacular example was the Connection Machine,

developed by Marvin’s student Danny Hillis and his company ThinkingMachines (for which Richard Feynman and I were both consultants). It wasalways in the air that theConnectionMachinewas built to implement one ofMarvin’s theories about the brain, and might be seen one day as like the“transistor of artificial intelligence”. But I, for example, ended up using itsmassivelyparallelarchitecturetoimplementcellularautomatonmodelsoffluids,andnotanythingAI-ishatall.Marvin was always having new ideas and theories. And even as the

ConnectionMachinewasbeingbuilt,hewasgivingmedraftsofhisbookTheSocietyofMind,which talkedaboutnewanddifferentapproaches toAI.Everone to do the unusual,Marvin told me he thought about writing the book inverse.ButinsteadthebookisstructuredabitlikesomanyconversationsIhadwithMarvin:withone ideaoneachpage,oftengood,but sometimesnot—yetalwayslively.IthinkMarvinviewedTheSocietyofMindashismagnumopus,andIthink

he was disappointed that more people didn’t understand and appreciate it. Itprobablydidn’t help that the book cameout in the 1980s,whenAIwas at itslowest ebb. But somehow I think to really appreciatewhat’s in the book onewould needMarvin there, presenting his ideaswith his characteristic personalenergyandrespondingtoanyobjectionsonemighthaveaboutthem.

MarvinandCellularAutomata

Marvinwasusedtohavingtheoriesaboutthinkingthatcouldbefiguredoutjustby thinking—a bit like the ancient philosophers had done. But Marvin wasinterested in everything, including physics. He wasn’t an expert on theformalism of physics, though he did make contributions to physics topics(notablypatentingaconfocalmicroscope).Andthroughhislong-timefriendEdFredkin,hehadalreadybeenintroducedtocellularautomataintheearly1960s.Hereallylikedthephilosophyofhavingphysicsbasedonthem—andendedupfor examplewriting a paper entitled “NatureAbhors an EmptyVacuum” thattalkedabouthowonemight ineffectengineercertainfeaturesofphysics fromcellularautomata.Marvindidn’tdoterriblymuchwithcellularautomata,thoughin1970heand

FredkinusedsomethinglikethemintheTriadexMusedigitalmusicsynthesizerthattheypatentedandmarketed—anearlyprecursorofcellular-automaton-basedmusiccomposition.Marvinwasverysupportiveofmyworkoncellularautomataandothersimple

programs,thoughIthinkhefoundmyorientationtowardsnaturalscienceabitalien.During thedecade that IworkedonANewKindof Science I interactedwithMarvinwith some regularity. Hewas startingwork on a book then too,about emotions, that he toldme in 1992 he hoped “might reform how peoplethink about themselves”. I talked to himoccasionally about his book, trying Isupposetounderstandtheepistemologicalcharacterofit(Ionceaskedifitwasabit likeFreud in this respect, andhe said yes). It took15years forMarvin tofinishwhatbecameTheEmotionMachine.Iknowhehadotherbooksplannedtoo;in2006,forexample,hetoldmehewasworkingonabookontheologythatwas“acoupleofyearsaway”—butwhichsadlyneversawthelightofday.

MarvininPersonItwas always a pleasure to seeMarvin.Often itwouldbe at his bighouse inBrookline,Massachusetts.As soon as one entered,Marvinwould start sayingsomethingunusual.Itcouldbe,“Whatwouldweconcludeif thesundidn’tsettoday?”Or,“You’vegottocomeseetheactualbinarytreeinmygreenhouse.”OncesomeonetoldmethatMarvincouldgiveatalkaboutalmostanything,butif one wanted it to be good, one should ask him an interesting question justbeforehestarted,andthenthat’dbewhathewouldtalkabout.IrealizedthiswashowtohandleconversationswithMarvintoo:bringupatopicandthenhecouldbecountedontosaysomethingunusualandofteninterestingaboutit.I rememberafewyearsagobringingupthetopicof teachingprogramming,

andhowIwashoping theWolframLanguagewouldbe relevant to it.Marvinimmediately launched into talking about how programming languages are theonlyonesthatpeopleareexpectedtolearntowritebeforetheycanread.Hesaidhe’d been trying to convince Seymour Papert that the best way to teachprogrammingwastostartbyshowingpeoplegoodcode.Hegavetheexampleofteachingmusic by giving peopleEine kleine Nachtmusik, and asking them totransposeittoadifferentrhythmandseewhatbugsoccur.(Marvinwasalong-time enthusiast of classical music.) In just this vein, one way the WolframProgramming Lab that we launched just last week lets people learnprogrammingisbystartingwithgoodcode,andthenhavingthemmodifyit.TherewasalwaysacertainwarmthtoMarvin.Helikedandsupportedpeople;

heconnectedwithallsortsofinterestingpeople;heenjoyedtellingnicestoriesaboutpeople.Hishousealwaysseemedtobuzzwithactivity,evenas,overtheyears,itpiledupwithstufftothepointwheretheonlyfreespacewasatinypartofakitchentable.Marvinalsohadagreatloveofideas.Onesthatseemedimportant.Onesthat

werestrangeandunusual.ButIthinkintheendMarvin’sgreatestpleasurewasinconnectingideaswithpeople.Hewasahackerofideas,butIthinktheideasbecamemeaningfultohimwhenheusedthemasawaytoconnectwithpeople.I shallmiss all those conversations about ideas—both ones I thoughtmade

senseandonesIthoughtdidn’t.Ofcourse,Marvinwasalwaysagreatenthusiastof cryonics, so perhaps this isn’t the end of the story. But at least for now,farewell,Marvin,andthankyou.

RussellTowleOctober10,2008

A few times a year theywould arrive. Email dispatches from an adventurousexplorer in the world of geometry. Sometimes with subject lines like“Phenomenaldiscoveries!!!”Usuallywithimagesattached.AndstoriesofhowRussell Towle had just usedMathematica to discover yet another strange andwonderfulgeometricalobject.

Then,thisAugust,anotheremailarrived,thistimefromRussellTowle’sson:“…lastnight,myfatherdiedinacaraccident.”I first heard from Russell Towle thirteen years ago, when he wrote to me

suggestingthatMathematica’sgraphicslanguagebeextendedtohaveprimitivesnot just for polygons and cubes, but also for “polar zonohedra”. I donot nowrecall,butIstronglysuspectthatatthattimeIhadneverheardofzonohedra.ButRussell Towle’s letter included some intriguing pictures, and we wrote backencouragingly.There soon emerged more information. That Russell Towle lived in a

hexagonal house of his own design, in a remote part of the Sierra Nevadamountains of California. That he was a fan of Archimedes, and had learnedGreek to be able to understand hiswork better. And that hewas not only anindependent mathematician, but also a musician and an accomplished localhistorian.In the years that followed, Russell Towle wrote countless Mathematica

programs, published his work in The Mathematica Journal, created videos(Regular Polytopes: The Movie, Joyfully Bitten Zonotope, …) and recentlybeganpublishingonTheWolframDemonstrationsProject.

In 1996, he sent me probably the first image I had ever seen of digitalelevationdatarenderedinMathematica.His last letter tomewas thisMay,where he explained, “Along theway it

occurredtomethatitwouldbeinterestingtotreatthezonogonsofaplanetilingaspixels;anexampleisattached,inwhichImappedaphotoofabutterflyontoapatchofPenrosetiling.”

Throughout the years, though, zonohedra were Russell Towle’s greatestpassion.Everyone knows about the fivePlatonic solids,where every face and every

vertex configuration has the same regular form. Then there are the 13Archimedean solids (PolyhedronData[“Archimedean”]; the cuboctahedron,icosidodecahedron, truncated cube, etc.), constructed by requiring the sameconfigurationateachvertex,butallowingmorethanonekindofregularface.The zonohedra are basedon a different approach to constructingpolyhedra.

They start from a collection of vectors vi at the origin, then simply consist ofthoseregionsofspacecorrespondingto∑aiviwherethe0<ai<1.Withtwovectors,thisconstructionalwaysgivesaparallelogram.Andin3D,

withthreevectors,itgivesaparallelepiped.Andasoneincreasesthenumberofvectors,oneseesalotoffamiliar(andnot-so-familiar)polyhedra.I’m not sure how the “known” polyhedra (included for example in

PolyhedronData) aredistributed in zonohedron space.Thatwouldhavebeenagood question to ask Russell Towle. My impression is that many of the“famous”polyhedrahavesimpleminimalrepresentationsaszonohedra.Butthecomplete space of zonohedra contains all sorts of remarkable forms that, forwhatever reason,havenever arisen in the traditionalhistoricaldevelopmentofpolyhedra.Russell Towle identified some particular families of zonohedra, with

interestingmathematical and aesthetic properties.Zonohedra not only have allsortsofmathematicalconnections(theyare,forexample,thefiguresformedbyprojectionsofhigher-dimensionalcubes),butmay,asRussellTowlesuggestedinhisfirstlettertome,havepracticalimportanceasconvenientparametrizationsofsymmetricalgeometricforms.In recent years, zonohedra have for example begun to find their way into

architecture.Indeed,a600-footapproximationtoazonohedronnowgracestheLondonskylineastheSwissRebuilding(“Gherkin”).There is a certain wonderful timelessness to polyhedra. We see ancient

Egyptiandice that aredodecahedra.We seepolyhedra inLeonardodaVinci’sillustrations. But somehow all these polyhedra, whenever they are from, lookmodern.Thatthereismoretoexploreintheworldofpolyhedraaftertwothousandor

moreyearsmight seem remarkable.Partof it is thatwe live in a timeofnewtools—when we can use Mathematica to explore the universe of geometricalforms.Andpartofitisthattherehavebeenratherfewindividualswhohavethekindofpassion,intuitionandtechnicalskillaboutpolyhedraofaRussellTowle.

It’sgoodthatRussellTowlehadtheopportunitytoshowusalittlemoreabouttheworldofzonohedra;it’ssadthatheleftussosoon,nodoubtwithsomanyfascinatingkindsofzonohedrastilllefttodiscover.

BertrandRussell&AlfredWhiteheadNovember25,2010

A hundred years ago thismonth the first volume ofWhitehead and Russell’snearly-2000-pagemonumental workPrincipiaMathematica was published. Adecadeinthemaking,itcontainedpageafterpageliketheonebelow,devotedtoshowinghowthetruthsofmathematicscouldbederivedfromlogic.

PrincipiaMathematica is inspiringfor theobviouseffortput into it—andassomeone who has spent much of his life engaged in very large intellectualprojects,Ifeelacertainsympathytowardsit.Inmyownwork,MathematicashareswithPrincipiaMathematicathegoalof

formalizing mathematics—but by building on the concept of computation, ittakesaratherdifferentapproach,withaquitedifferentoutcome.AndinANewKindofScience,oneofmyobjectives,alsolikePrincipiaMathematica,wastounderstandwhatliesbeneathmathematics—thoughagainmyconclusionisquitedifferentfromPrincipiaMathematica.EversinceEuclidtherehadbeenthenotionofmathematicalproofasaformal

activity.But therehadalwaysbeena tacit assumption thatmathematics—withitsnumbersandgeometricalfigures—wasstillatsomeleveltalkingaboutthingsinthenaturalworld.In the mid-1800s, however, that began to change, notably with the

introduction of non-Euclidean geometries and algebras other than those ofordinarynumbers.Andbytheendofthe1800s,therewasageneralmovementtowards thinking of mathematics as abstract formalism, independent of thenaturalworld.Meanwhile, ever sinceAristotle, there had in a sense been another kind of

formalism—logic—whichwasoriginallyintendedtorepresentspecifickindsofidealizedhumanarguments,buthadgraduallybecomeassumedtorepresentanyvalid form of reasoning. Formost of its history, logic had been studied, andtaught,quite separately frommathematics.But in the1800s, therebegan tobeconnections.GeorgeBooleshowedhowbasiclogiccouldbeformulatedinalgebraicterms

(“Boolean algebra”). And then Gottlob Frege, working in some isolation inGermany,developedpredicate logic (“forall”, “thereexists”, etc.), andusedaversion of set theory to try to describe numbers and mathematics in purelylogicalterms.And it was into this context thatPrincipiaMathematica was born. Its two

authorsbroughtdifferent thingsto theproject.AlfredNorthWhiteheadwasanestablishedCambridgeacademic,who,in1898,attheageof38,hadpublishedA Treatise on Universal Algebra “to present a thorough investigation of thevarious systemsofSymbolicReasoningallied toordinaryAlgebra”.ThebookdiscussedBooleanalgebra,quaternionsandthetheoryofmatrices—usingthemas thebasis for a clean, if fairly traditional, treatmentof topics in algebra andgeometry.BertrandRussell was a decade younger. He had studiedmathematics as an

undergraduate in Cambridge, and by 1900, at the age of 28, he had alreadypublishedbooksrangingintopicfromGermansocialdemocracy,tofoundationsofgeometry,andthephilosophyofLeibniz.Thenatureofmathematicsandmathematicaltruthwasacommonsubjectof

debate among philosophers—as it had been at some level since Plato—andRussell seems to have believed that bymaking use of the latest advances, hecouldonceandforallresolvethedebates.In1903,hepublishedThePrinciplesof Mathematics, volume 1 (no volume 2 was ever published)—in essence asurvey, without formalism, of how standard areas of mathematics could beviewedinlogicalterms.His basic concept was that by tightening up all relevant definitions using

logic, it should be possible to derive every part ofmathematics in a rigorousway,andtherebyimmediatelyanswerquestionsaboutitsnatureandphilosophy.But in1901, ashe tried tounderstand the conceptof infinity in logical terms,and thought about ancient logical problems like the liar’s paradox (“thisstatement is false”), he came across what seemed to be a fundamentalinconsistency:aparadoxof self reference (“Russell’sparadox”)aboutwhetherthesetofallsetsthatdonotcontainthemselvesinfactcontainsitself.Toresolvethis,Russellintroducedwhatisoftenviewedashismostoriginal

contribution tomathematical logic:his theoryof types—which inessence triesto distinguish between sets, sets of sets, etc. by considering them to be ofdifferent“types”,andthenrestrictshowtheycanbecombined.ImustsaythatIconsidertypestobesomethingofahack.AndindeedIhavealwaysfeltthattherelated idea of “data types” has very much served to hold up the long-termdevelopmentofprogramminglanguages.(Mathematica,forexample,getsgreatflexibility precisely from avoiding the use of types—even if internally it doesuse something like them to achieve various practical efficiencies.) But backaround 1900, as both Russell and Whitehead were trying to extend theirformalizationsofmathematics,theydecidedtolaunchintotheprojectthatwouldconsumeadecadeoftheirlivesandbecomePrincipiaMathematica.Particularly since theworkofGottfriedLeibniz in the late1600s, therehad

beendiscussionof developing a notation formathematics that transcended theimprecision of human language. In 1879 Gottlob Frege introduced hisBegriffsschrift (“conceptscript”)—whichwasamajoradvance inconceptsandfunctionality, but had a strange two-dimensional layout that was almostimpossibletoread,ortoprinteconomically.Andinthe1880s,GiuseppePeanodevelopedacleanerandmorelinearnotation,muchofwhichisstillinusetoday.Itdidnothelp thedisseminationofPeano’swork thathechose towritehis

narrative text in a languageof his own construction (basedon classicalLatin)

called Interlingua. But still, in 1900, Russell went to a back-to-back pair ofconferencesinParisaboutphilosophyandmathematics(notableforbeingwhereHilbert announced his problems), met Peano and became convinced that hisattempt to formalize mathematics should be based on Peano’s notation andapproach. (Russell gave a distinctly pre-relativistic philosophical talk at theconferenceaboutabsoluteorderingofspatiotemporalevents,whilehiswifeAlysgaveatalkabouttheeducationofwomen.)TheideathatmathematicscouldbebuiltupfromasmallinitialsetofaxiomshadexistedsincethetimeofEuclid.ButRussell andWhiteheadwanted tohave the smallestpossible set, andhavethembebasednotonideasderivedfromobservingthenaturalworld,butinsteadonwhattheyfeltwasthemoresolidanduniversalgroundoflogic.Withpresent-dayexperienceofcomputersandprogrammingitdoesnotseem

surprising that with enough “code” one should be able to start from basicconcepts of logic and sets, and successfully build up numbers and the otherstandard constructs of mathematics. And indeed Frege, Peano and others hadalreadystartedthisprocessbefore1900.Butbyitsveryweightiness,PrincipiaMathematicamadethepointseembothsurprisingandimpactful.Ofcourse,itdidnothurtthewholeimpressionthatittookuntilmorethan80

pagesintovolume2tobeabletoprove(as“Proposition*110.643”)that1+1=2(withthecomment“[This]propositionisoccasionallyuseful”).

IdonotknowifRussellandWhiteheadintendedPrincipiaMathematicatobereadable by humans—but in the end Russell estimated years later that onlyperhaps sixpeoplehadever read thewhole thing.Tomoderneyes, theuseofPeano’s dot notation instead of parentheses is particularly difficult. And thenthereisthematterofdefinitions.Attheendofvolume1,PrincipiaMathematicalistsabout500“definitions”,

eachwithaspecialnotation.Inmanyways,thesearetheanalogsofthebuilt-infunctionsofMathematica.ButinPrincipiaMathematica,insteadofbeinggivenEnglish-basednames, all theseobjects areassigned specialnotations.The firsthandfularenottoodifficulttounderstand.Butbythesecondpageone’sseeingall sorts of strange glyphs, and I, at least, lose hope of ever being able todecipherwhatiswritteninthem.

Beyondthesenotationalissues,thereisamuchmorefundamentaldifferencebetween the formalization of mathematics in Principia Mathematica and inMathematica. For in Principia Mathematica the objective is to exhibit truetheorems of mathematics, and to represent the processes involved in provingthem. But in Mathematica, the objective is instead to compute: to takemathematicalexpressions,andevaluatethem.(These differences in objective lead to many differences in character. For

example, Principia Mathematica is constantly trying to give constraints thatindirectlyspecifywhateverstructureitwantstotalkabout.InMathematica,thewhole idea is to have explicit symbolic structures that can then be computedwith.) In the hundred years sincePrincipiaMathematica, there has been slowprogressinpresentingtheoremsofmathematicsinformalways.Buttheideaofmathematicalcomputationhastakenoffspectacularly—andhastransformedtheuseofmathematics,andmanyareasofitsdevelopment.Butwhat about the conceptual purposes ofPrincipiaMathematica?Russell

explained in the introduction to his The Principles of Mathematics that heintended to “reduce thewhole of [the propositions ofmathematics] to certainfundamentalnotionsoflogic.”Indeed,heevenmadewhatheconsideredtobeaverygeneraldefinitionof“puremathematics”asalltruelogicalstatementsthatcontainonlyvariableslikepandq,andnotliteralslike“thecityofNewYork”.(Appliedmathematics, he suggested,would come from replacing thevariablesbyliterals.)Butwhystartfromlogic?IthinkRusselljustassumedthatlogicwasthe most fundamental possible thing—the ultimate incontrovertiblerepresentation for all formal processes. Traditional mathematical constructs—likenumbersandspace—heimaginedwereassociatedwiththeparticularsofourworld. But logic, he imagined, was a kind of “pure thought”, and somethingmoregeneral,andwhollyindependentoftheparticularsofourworld.InmyownworkleadinguptoANewKindofScience,Istartedbystudying

thenaturalworld,yetfoundmyselfincreasinglybeingledtogeneralizebeyondtraditionalmathematicalconstructs.ButIdidnotwindupwithlogic.Instead,Ibegantoconsiderallpossiblekindsofrules—orasIhavetendedtodescribeit(makinguseofmodernexperience), thecomputationaluniverseofallpossibleprograms.Some of these programs describe parts of the natural world. Some give us

interesting fodder for technology. And some correspond to traditional formalsystemslikelogicandmathematics.Onethingtodoistolookatthespaceofallpossibleaxiomsystems.Thereare

some technical issues about modern equational systems compared to

implicationalsystemsofthekindconsideredinPrincipiaMathematica.Buttheessentialresultisthatdottedaroundthespaceofallpossibleaxiomsystemsaretheparticularaxiomsystemsthathavehistoricallyariseninthedevelopmentofmathematicsandrelatedfields.Islogicsomehowspecial?Ithinknot.In Principia Mathematica, Russell and Whitehead originally defined logic

usingafairlycomplicatedtraditionalsetofaxioms.Inthesecondeditionofthebook,theymadeapointofnotingthatbywritingeverythingintermsofNAND(Sheffer stroke) rather thanAND,OR andNOT, it is possible to use amuchsimpleraxiomsystem.In2000,bydoingasearchofthespaceofpossibleaxiomsystems,Iwasable

to find the very simplest (equational) axiom system for standard propositionallogic: just the single axiom ((a.b).c).(a.((a.c).a))=c.And from this result,wecan tellwhere logic lies in the space of possible formal systems: in a naturalenumerationofaxiomsystems inorderof size, it isabout the50,000th formalsystemthatonewouldencounter.Afewothertraditionalareasofmathematics—likegrouptheory—occurina

comparableplace.Butmostrequiremuchlargeraxiomsystems.AndintheendthepictureseemsverydifferentfromtheoneRussellandWhiteheadimagined.It is not that logic—as conceived by human thought—is at the root ofeverything. Instead, there are a multitude of possible formal systems, somepicked by the natural world, some picked by historical developments inmathematics,butmostoutthereuninvestigated.InwritingPrincipiaMathematica,oneofRussell’sprincipalobjectiveswasto

give evidence that all ofmathematics really couldbederived from logic.Andindeedtheveryheftofthebookgaveimmediatesupporttothisidea,andgavesuchcredibilitytologic(andtoRussell)thatRussellwasabletospendmuchofthe rest of his long life confidently presenting logic as a successful way toaddressmoral,socialandpoliticalissues.Of course, in 1931 Kurt Gödel showed that no finite system—logic or

anything else—canbeused toderive all ofmathematics.And indeed theverytitle of his paper refers to the incompleteness of none other than the formalsystem of Principia Mathematica. By this time, however, both Russell andWhiteheadhadmovedon tootherpursuits,andneither returned toaddress theimplicationsofGödel’stheoremfortheirproject.So can one say that the idea of logic somehow underlying mathematics is

wrong?Ataconceptuallevel,Ithinkso.Butinastrangetwistofhistory,logiciscurrentlypreciselywhatisactuallyusedtoimplementmathematics.For inside all current computers are circuits consisting of millions of logic

gates—each typically performing a NAND operation. And so, for example,when Mathematica runs on a computer and implements the operations ofmathematics, it does so precisely by marshalling the logic operations in thehardwareofthecomputer.(Tobeclear,thelogicimplementedbycomputersisbasic, propositional logic—not the more elaborate predicate logic, combinedwith set theory, that Principia Mathematica ultimately uses.)We know fromcomputational universality—and more precisely from the Principle ofComputationalEquivalence—thatthingsdonothavetoworkthisway,andthatthere aremany very different bases for computation that could be used. Andindeed,ascomputersmovetoamolecularscale,standardlogicwillmostlikelynolongerbethemostconvenientbasistouse.Butsowhyislogicusedintoday’scomputers?Isuspectitactuallyhasquitea

bittodowithnoneotherthanPrincipiaMathematica.ForhistoricallyPrincipiaMathematicadidmuchtopromotetheimportanceandprimacyoflogic,andtheglow that it left is in many ways still with us today. It is just that we nowunderstandthatlogicisjustonepossiblebasisforwhatwecando—nottheonlyconceivableone.(Peoplefamiliarwithtechnicalaspectsoflogicmayprotestthatthenotionof

“truth”issomehowintimatelytiedtotraditionallogic.Ithinkthisisamatterofdefinition,but inanycase,whathasbecomeclear formathematics is that it isvastly more important to compute answers than merely to state truths.) Ahundred years afterPrincipiaMathematica there is still much that we do notunderstandevenaboutbasicquestionsinthefoundationsofmathematics.Anditis humbling to wonder what progress could be made over the next hundredyears.WhenwelookatPrincipiaMathematica, itemphasizesexhibitingparticular

truthsofmathematics that itsauthorsderived.But todayMathematica ineffectevery day automatically delivers millions of truths of mathematics, made toorderforamultitudeofparticularpurposes.Yet it is still the case that it operates with just a few formal systems that

happentohavebeenstudied inmathematicsorelsewhere.AndeveninANewKindofScience, I concentratedonparticularprogramsor systems that foronereasonoranotherIthoughtwereinteresting.Inthefuture,however,Isuspectthattherewillbeanotherlevelofautomation.

Probablyitwilltakemuchlessthanahundredyears,butintimeitwillbecomecommonplacenotjust tomakecomputationstoorder,buttomaketoordertheverysystemsonwhich thosecomputationsarebased—ineffect in theblinkofaneyeinventinganddevelopingsomethinglikeawholePrincipiaMathematicatorespondtosomeparticularpurpose.

RichardCrandallDecember30,2012

RichardCrandall liked tocallhimself a “computationalist”, for thoughhewastrained in physics (and served formany years as a physics professor at ReedCollege), computation was at the center of his life. He used it in physics, inengineering,inmathematics,inbiology…andintechnology.Hewasapioneerinexperimentalmathematics,andwasassociatedformanyyearswithAppleandwithSteveJobs,andwasproudofhavinginvented“atleast5algorithmsusedintheiPhone”.HewasalsoanextremelyearlyadopterofMathematica,andawell-known figure in the Mathematica community. And when he died just beforeChristmas at the age of 64 hewas hard atwork on his latest, rather differentproject: an “intellectual biography” of Steve Jobs that I had suggested he call“ScientisttoMr.Jobs”.IfirstmetRichardCrandallin1987,whenIwasdevelopingMathematica,and

hewasChiefScientistatSteveJobs’scompanyNeXT.Richardhadpioneeredusing Pascal onMacintoshes to teach scientific computing.But as soon as hesawMathematica,heimmediatelyadoptedit,andforaquarterofacenturyusedittoproduceawonderfulrangeofdiscoveriesandinventions.HealsocontributedgreatlytoMathematicaanditsusage.Indeed,evenbefore

Mathematica1.0in1988,heinsistedonvisitingourcompanytocontributehisexpertise in numerical evaluation of special functions (his favorites werepolylogarithmsandzeta-likefunctions).Andthen,aftertheNeXTcomputerwasreleased,hewrotewhatmayhavebeenthefirst-everMathematica-basedapp:a“supercalculator” named Gourmet that he said “eats other calculators forbreakfast”.AcoupleofyearslaterhewroteabookentitledMathematicafortheSciences, that pioneered the use of Mathematica programs as a form ofexposition.Overtheyears,IinteractedwithRichardaboutagreatmanythings.Usuallyit

would start with a “call me” message. And I would get on the phone, neverknowingwhattoexpect.AndRichardwouldbetalkingabouthislatestresultinnumbertheory.OrthelatestAppleGPU.Orhismodelsoffluepidemiology.OrtheimportanceofrunningMathematicaoniOS.Oranewwaytomultiplyverylong integers. Or his latest achievements in image processing. Or a way toreconstructfractalbraingeometries.Richardmadecontributions—fromhighlytheoreticaltohighlypractical—toa

widerangeoffields.Hewasalwaysalittletoooriginaltobeinthemainstream,with the result that there are few fields where he is widely known. In recent

years,however,hewasbeginning tobe recognized forhispioneeringwork inexperimental mathematics, particularly as applied to primes and functionsrelatedtothem.Buthealwaysknewthathisworkwiththegreatest immediatesignificance for the world at large was what he did for Apple behind closeddoors.RichardwasborninAnnArbor,Michigan,in1947.Hisfatherwasanactuary

whobecameasought-afterexpertwitnessoncomplexcorporateinsurance-fraudcases,andwho,Richard toldme, taughthim“anabsolute lackof fearof largenumbers”.RichardgrewupinLosAngeles,studyingfirstatCaltech(whereheencounteredRichardFeynman),thenatReedCollegeinOregon.FromtherehewenttoMIT,wherehestudiedthemathematicalphysicsofhigh-energyparticlescattering(Regge theory),andgothisPhDin1973.On thesidehebecameanelectronicsentrepreneur,workingparticularlyonsecuritysystems,andinventing(and patenting) a new type of operational amplifier and a new form of alarmsystem. After his PhD these efforts led him to New York City, where hedesigned a computerized fire safety and energy control system used inskyscrapers.Asahobbyheworkedonquantumphysics andnumber theory—andaftermovingbacktoOregontoworkforanelectronicscompanythere,hewashiredin1978atReedCollegeasaphysicsprofessor.SteveJobshadendedhisshortstayatReedsomeyearsearlier,butthroughhis

effort to get Reed computerized, Richard got connected to him, and began arelationshipthatwouldlasttherestofSteve’slife.Idon’tknowevenafractionofwhatRichardworkedon forNeXTandApple.ForawhilehewasApple’sChief Cryptographer—notably inventing a fast form of elliptic curveencryption. And later on, he was also involved in compression, imageprocessing,touchdetection,andmanyotherthings.Through most of this, Richard continued as a practicing physics

professor. Early on, hewon awards for creatingminimal physics experiments(“measurethespeedoflightonatabletopwith$10ofequipment”).Bythemid-1980s, he began to concentrate on using computers for teaching—andincreasinglyforresearch.OneparticulardirectionthatRichardhadpursuedformany years was to use computers to study properties of numbers, and forexample search for primes of particular types. And particularly once he hadMathematica, he got involved in more and more sophisticated number-theoretical mathematics, particularly around primes, among other things co-authoring the (Mathematica-assisted) definitive textbook Prime Numbers: AComputationalPerspective.Heinventedfastermethodsfordoingarithmeticwithverylongintegers,that

were instrumental, for example, in early crowdsourced prime discoveries, and

that are in fact used today in modified form in Mathematica. And by doingexperimental mathematics with Mathematica he discovered a wonderfulcollection of zeta-function-related results and identitiesworthy of Ramanujan.Hewas particularly proud of his algorithms for the fast evaluation of variouszeta-like functions (notably polylogarithms and Madelung sums), and indeedearlier this year he sent me the culmination of his 20 years of work on thesubject, in the form of a paper dedicated to Jerry Keiper, the founder of thenumericsgroupatWolframResearch,whodiedinanaccidentin1995butwithwhomRichardhadworkedatlength.Richardwasalwayskeenonpresentation,albeitinhisownsomewhatunique

way. Through his “industrial algorithm” company Perfectly Scientific, hepublished a poster of the digits of every new Mersenne prime that wasdiscovered.Thepriceoftheposterincreasedwiththenumberofdigits,andforconvenience his company also sold a watchmaker’s loupe to allow people toreadthedigitsontheposters.Richard always had a certain charming personal ponderousness to him, his

conversation peppered with phrases like “let me commend to yourattention”.And indeedas Iwrite this, I find a classic exampleofover-the-topRichardness in the opening to hisMathematica for the Sciences: “It has beensaid that the evolution of humankind took a substantial, discontinuous swerveabout the timewhen our forepaws left the ground.Once in the air, our handswerefreefor‘otherthings’.Toolmaking.…”,andeventually,asheexplainsafterhis“admittedlyconjecturalrambling”,computersandMathematica…Richard regularlyvisitedSteveJobsandhis family,withhis lastvisitbeing

just a fewdaysbeforeStevedied.Hewas alwaysdeeply impressedbySteve,and frustrated that he felt people didn’t understand the strength of Steve’sintellect.HewasdisappointedbyWalterIsaacson’shighlysuccessfulbiographyof Steve, and had embarked on writing his own “intellectual biography” ofSteve. He had years of interesting personal anecdotes about Steve and hisinteractions with him, but he was adamant that his book should tell “the realstory”,aboutideasandtechnology,andshouldatallcostsavoidwhatheatleastconsidered“gossip”.Atfirst,hewasgoingtotrytotakehimselfcompletelyoutofthestory,butIthinkIsuccessfullyconvincedhimthatwithhisuniqueroleas“scientisttoSteveJobs”,hehadnochoicebuttobeinthestory,andindeedtotellhisownstoryalongtheway.Richardwas inmanyways a rather solitary individual.But he always liked

talkingabouthisnow-15-year-olddaughter,whomhewouldinvariablyrefertorather formally as “Ellen Crandall”. He had theories about many things,includingchildrearing,andconsideredoneofhissignaturequotes tobe,“The

mostefficientway to raiseanatheistkid is tohaveapriest fora father”.Andindeedaspartof the lastexchange Ihadwithhim justa fewweeksbeforehedied, he marveled that his daughter from a “pure blank, white start”… “hassuddenly taken up filling giant white poster boards with minutely detaileddrawing”.Whilehisoverallhealthwasnotperfect,Richardwasinmanywaysstillinthe

prime of his life. He had ambitious plans for the future, in mathematics, inscienceandintechnology,nottomentioninwritinghisbiographyofSteveJobs.Butafewweeksago,hesuddenlyfellill,andwithintendayshedied.Alifecutofffartoosoon.Butauniquelifeinwhichmuchwasinventedthatwouldlikelyneverhaveexistedotherwise.I shall miss Richard’s flow of wonderfully eccentric ideas, as well as the

mysterious “call me” messages, and of late the practically monthlyencouragement speech about the importance of having Mathematica on theiPhone.(I’msosorry,Richard,thatwedidn’tgetitdoneintime.)Richardwasalwaysimaginingwhatmightbepossible,theninhisuniqueway

doggedlytryingtobuildtowardsit.Aroundtheworldatanytimeofdayornightmillionsofpeople areusing their iPhones.Andunknown to them, somewhereinside,algorithmsarerunningthatonecanimaginerepresentalittlepieceofthesoulofthatinterestingandcreativehumanbeingnamedRichardCrandall,nowcastintheformofcode.

SrinivasaRamanujanApril27,2016

Theyused tocomebyphysicalmail.Nowit’susuallyemail.Fromaround theworld, I have formany years received a steady trickle ofmessages thatmakeboldclaims—aboutprimenumbers,relativitytheory,AI,consciousnessorahostofother things—butgive littleornobackup forwhat they say. I’malways sobusywithmyown ideasandprojects that I invariablyputoff lookingat thesemessages. But in the end I try to at least skim them—in large part because IrememberthestoryofRamanujan.On about January 31, 1913 a mathematician named G. H. Hardy in

Cambridge,Englandreceivedapackageofpaperswithacoverletterthatbegan:“Dear Sir, I beg to introduce myself to you as a clerk in the AccountsDepartment of the Port Trust Office at Madras on a salary of only £20 perannum.Iamnowabout23yearsofage....”andwentontosaythatitsauthorhadmade “startling” progress on a theory of divergent series inmathematics, andhad all but solved the longstanding problem of the distribution of primenumbers.Thecoverletterended:“Beingpoor,ifyouareconvincedthatthereisanything of value I would like to have my theorems published.... BeinginexperiencedIwouldveryhighlyvalueanyadviceyougiveme.Requestingtobe excused for the trouble I give you. I remain, Dear Sir, Yours truly, S.Ramanujan”.What followed were at least 11 pages of technical results from a range of

areasofmathematics (at least2of thepageshavenowbeen lost).Thereareafewthingsthatonfirstsightmightseemabsurd,likethatthesumofallpositiveintegerscanbethoughtofasbeingequalto–1/12:

Then there are statements that suggest a kind of experimental approach tomathematics:

Butsomethingsgetmoreexotic,withpagesofformulaslikethis:

Whatarethese?Wheredotheycomefrom?Aretheyevencorrect?Theconceptsare familiar fromcollege-level calculus.But thesearenot just

complicated college-level calculus exercises. Instead, when one looks closely,each one has something more exotic and surprising going on—and seems toinvolveaquitedifferentlevelofmathematics.Todaywe can useMathematica orWolfram|Alpha to check the results—at

least numerically. And sometimes we can even just type in the question and

immediatelygetouttheanswer:

Andthefirstsurprise—justasG.H.Hardydiscoveredbackin1913—isthat,yes,theformulasareessentiallyallcorrect.Butwhatkindofpersonwouldhavemade them?And how?And are they all part of some bigger picture—or in asensejustscatteredrandomfactsofmathematics?

TheBeginningoftheStoryNeedless to say, there’s a human story behind this: the remarkable story ofSrinivasaRamanujan.HewasborninasmallishtowninIndiaonDecember22,1887(whichmade

him not “about 23”, but actually 25, when he wrote his letter to Hardy). HisfamilywasoftheBrahmin(priests,teachers,...)castebutofmodestmeans.TheBritish colonial rulers of India had put in place a very structured system ofschools,andbyage10Ramanujanstoodoutbyscoringtopinhisdistrictinthestandard exams. He also was known as having an exceptional memory, andbeing able to recite digits of numbers like pi as well as things like roots ofSanskrit words. When he graduated from high school at age 17 he wasrecognizedforhismathematicalprowess,andgivenascholarshipforcollege.WhileinhighschoolRamanujanhadstartedstudyingmathematicsonhisown

—anddoinghis own research (notablyon the numerical evaluationofEuler’sconstant,andonpropertiesof theBernoullinumbers).Hewasfortunateatage16(inthosedayslongbeforetheweb!)togetacopyofaremarkablygoodandcomprehensive (at least as of 1886) 1055-page summary of high-endundergraduate mathematics, organized in the form of results numbered up to6165.The bookwaswritten by a tutor for the ultra-competitiveMathematicalTripos exams in Cambridge—and its terse “just the facts” format was verysimilartotheoneRamanujanusedinhislettertoHardy.

BythetimeRamanujangottocollege,allhewantedtodowasmathematics—andhefailedhisotherclasses,andatonepointranaway,causinghismothertosendamissing-personlettertothenewspaper:

Ramanujan moved to Madras (now Chennai), tried different colleges, hadmedicalproblems,andcontinuedhisindependentmathresearch.In1909,whenhewas21,hismotherarranged(inkeepingwithcustomsofthetime)forhimtomarry a then-10-year-old girl named Janaki, who started living with him acoupleofyearslater.Ramanujan seems to have supported himself by doing math tutoring—but

soonbecameknownaroundMadrasasamathwhiz,andbeganpublishingintherecentlylaunchedJournaloftheIndianMathematicalSociety.Hisfirstpaper—publishedin1911—wasoncomputationalpropertiesofBernoullinumbers(thesameBernoullinumbers thatAdaLovelacehadused inher1843paperon theAnalytical Engine). Though his results weren’t spectacular, Ramanujan’sapproachwasaninterestingandoriginalonethatcombinedcontinuous(“what’sthe numerical value?”) and discrete (”what’s the prime factorization?”)mathematics.

When Ramanujan’s mathematical friends didn’t succeed in getting him ascholarship,Ramanujanstarted lookingfor jobs,andwoundupinMarch1912as an accounting clerk—or effectively, a human calculator—for the Port ofMadras (which was then, as now, a big shipping hub). His boss, the ChiefAccountant,happenedtobeinterestedinacademicmathematics,andbecamealifelong supporter of his. The head of the Port of Madras was a ratherdistinguishedBritishcivilengineer,andpartly throughhim,Ramanujanstartedinteracting with a network of technically oriented British expatriates. Theystruggled to assess him, wondering whether “he has the stuff of greatmathematicians”orwhether“hisbrainsareakintothoseofthecalculatingboy”.They wrote to a certain Professor M. J. M. Hill in London, who looked atRamanujan’s rather outlandish statements about divergent series and declaredthat“Mr.RamanujanisevidentlyamanwithatasteforMathematics,andwithsomeability,buthehasgotontowronglines.”HillsuggestedsomebooksforRamanujantostudy.Meanwhile, Ramanujan’s expat friendswere continuing to look for support

for him—and he decided to start writing to British mathematicians himself,thoughwith some significant help at composing theEnglish in his letters.Wedon’t know exactly who all he wrote to first—although Hardy’s long-timecollaborator John Littlewoodmentioned two names shortly before he died 64years later: H. F. Baker and E. W. Hobson. Neither were particularly goodchoices: Baker worked on algebraic geometry and Hobson on mathematicalanalysis, both subjects fairly far fromwhatRamanujanwas doing.But in anyevent,neitherofthemresponded.AndsoitwasthatonThursday,January16,1913,Ramanujansenthisletter

toG.H.Hardy.

WhoWasHardy?

G.H.Hardywas born in 1877 to schoolteacher parents based about 30milessouth of London. He was from the beginning a top student, particularly inmathematics.EvenwhenIwasgrowingupinEnglandintheearly1970s,itwastypicalforsuchstudentstogotoWinchesterforhighschoolandCambridgeforcollege.And that’s exactlywhatHardydid. (Theother, slightlymore famous,track—less austere and less mathematically oriented—was Eton and Oxford,whichhappenstobewhereIwent.)Cambridgeundergraduatemathematicswasat the time very focused on solving ornately constructed calculus-relatedproblems as a kind of competitive sport—with the final event being theMathematicalTriposexams,whichrankedeveryonefromthe“SeniorWrangler”(top score) to the “Wooden Spoon” (lowest passing score). Hardy thought heshouldhavebeentop,butactuallycamein4th,anddecidedthatwhathereallylikedwasthesomewhatmorerigorousandformalapproachtomathematicsthatwasthenbecomingpopularinContinentalEurope.ThewaytheBritishacademicsystemworkedatthattime—andbasicallyuntil

the1960s—wasthatassoonastheygraduated,topstudentscouldbeelectedto“collegefellowships”thatcouldlasttherestoftheirlives.HardywasatTrinityCollege—thelargestandmostscientificallydistinguishedcollegeatCambridgeUniversity—andwhenhe graduated in 1900, hewas duly elected to a collegefellowship.Hardy’s first research paper was about doing integrals like these:

ForadecadeHardybasicallyworkedonthefinerpointsofcalculus,figuringouthowtododifferentkindsof integralsandsums,andinjectinggreaterrigorintoissueslikeconvergenceandtheinterchangeoflimits.Hispapersweren’tgrandorvisionary,buttheyweregoodexamplesofstate-

of-the-artmathematicalcraftsmanship.(AsacolleagueofBertrandRussell’s,hedippedintothenewareaoftransfinitenumbers,butdidn’tdomuchwiththem.)Then in 1908, hewrote a textbook entitledACourse of PureMathematics—whichwasagoodbook,andwasverysuccessfulinitstime,evenifitsprefacebeganbyexplainingthatitwasforstudents“whoseabilitiesreachorapproachsomethinglikewhatisusuallydescribedas‘scholarshipstandard’”.By 1910 or so, Hardy had pretty much settled into a routine of life as a

Cambridgeprofessor,pursuingasteadyprogramofacademicwork.Butthenhemet JohnLittlewood.Littlewoodhadgrownup inSouthAfrica andwas eightyearsyounger thanHardy, a recentSeniorWrangler, and inmanywaysmuchmoreadventurous.Andin1911Hardy—whohadpreviouslyalwaysworkedonhisown—beganacollaborationwithLittlewoodthatultimatelylastedtherestofhislife.Asaperson,Hardygivesmetheimpressionofagoodschoolboywhonever

fully grew up. He seemed to like living in a structured environment,concentrating on his math exercises, and displaying cleverness whenever hecould.Hecouldbeverynerdy—whetheraboutcricketscores,provingthenon-existenceofGod,orwritingdown rules forhis collaborationwithLittlewood.And in a quintessentially British way, he could express himself with wit andcharm, but was personally stiff and distant—for example always theminghimselfas“G.H.Hardy”,with“Harold”basicallyusedonlybyhismotherandsister.

Soinearly1913therewasHardy:arespectableandsuccessful,ifpersonallyreserved,Britishmathematician,whohadrecentlybeenenergizedbystartingtocollaboratewithLittlewood—andwasbeingpulled in thedirectionof numbertheory by Littlewood’s interests there. But then he received the letter fromRamanujan.

TheLetterandItsEffectsRamanujan’s letter began in a somewhat unpromising way, giving theimpressionthathethoughthewasdescribingforthefirsttimethealreadyfairlywell-known technique of analytic continuation for generalizing things like thefactorial function to non-integers. He made the statement that “My wholeinvestigations are based upon this and I have been developing this to aremarkable extent so much so that the local mathematicians are not able tounderstandme inmyhigher flights.”But after the cover letter, there followedmorethanninepagesthatlistedover120differentmathematicalresults.Again,theybeganunpromisingly,withrathervaguestatementsabouthavinga

methodtocountthenumberofprimesuptoagivensize.Butbypage3,thereweredefiniteformulasforsumsandintegralsandthings.Someofthemlookedat least from a distance like the kinds of things that were, for example, inHardy’s papers. But some were definitely more exotic. Their general texture,though, was typical of these types of math formulas. But many of the actualformulaswerequitesurprising—oftenclaimingthatthingsonewouldn’texpecttoberelatedatallwereactuallymathematicallyequal.Atleasttwopagesoftheoriginalletterhavegonemissing.Butthelastpage

we have again seems to end inauspiciously—with Ramanujan describingachievementsofhis theoryofdivergent series, including the seeminglyabsurdresultaboutaddingupallthepositiveintegers,1+2+3+4+...,andgetting–1/12.SowhatwasHardy’sreaction?FirstheconsultedLittlewood.Wasitperhaps

apracticaljoke?Weretheseformulasallalreadyknown,orperhapscompletelywrong?Sometheyrecognized,andknewwerecorrect.Butmanytheydidnot.ButasHardylatersaidwithcharacteristicclevergloss,theyconcludedthatthesetoo “must be true because, if they were not true, no one would have theimaginationtoinventthem.”BertrandRussellwrotethatbythenextdayhe“foundHardyandLittlewood

in a state of wild excitement because they believe they have found a secondNewton, a Hindu clerk inMadras making 20 pounds a year.” Hardy showedRamanujan’s letter to lots of people, and started making enquiries with the

governmentdepartmentthathandledIndia.Ittookhimaweektoactuallyreplyto Ramanujan, opening with a certain measured and precisely expressedexcitement: “I was exceedingly interested by your letter and by the theoremswhichyoustate.”Then he went on: “You will however understand that, before I can judge

properly of the value of what you have done, it is essential that I should seeproofsofsomeofyourassertions.”Itwasaninterestingthingtosay.ToHardy,itwasn’tenoughtoknowwhatwastrue;hewantedtoknowtheproof—thestory—ofwhyitwastrue.Ofcourse,Hardycouldhavetakenituponhimselftofindhisownproofs.ButIthinkpartofitwasthathewantedtogetanideaofhowRamanujanthought—andwhatlevelofmathematicianhereallywas.His letter went on—with characteristic precision—to group Ramanujan’s

results into threeclasses:alreadyknown,newand interestingbutprobablynotimportant, and new and potentially important. But the only things heimmediately put in the third category were Ramanujan’s statements aboutcountingprimes,addingthat“almosteverythingdependsonthepreciserigourofthemethodsofproofwhichyouhaveused.”HardyhadobviouslydonesomebackgroundresearchonRamanujanby this

point,sinceinhisletterhemakesreferencetoRamanujan’spaperonBernoullinumbers.Butinhisletterhejustsays,“Ihopeverymuchthatyouwillsendmeasquicklyaspossible...afewofyourproofs,”thencloseswith,“Hopingtohearfromyouagainassoonaspossible.”RamanujandidindeedrespondquicklytoHardy’s letter,andhisresponseis

fascinating.First,hesayshewasexpectingthesamekindofreplyfromHardyashehadfromthe“MathematicsProfessoratLondon”,whojusttoldhim“not[to]fall into the pitfalls of divergent series.” Then he reacts toHardy’s desire forrigorousproofsbysaying,“If Ihadgivenyoumymethodsofproof Iamsureyouwill follow theLondonProfessor.”Hementionshis result 1+2+3+4+...=–1/12andsaysthat“IfItellyouthisyouwillatoncepointouttomethelunaticasylumasmygoal.”Hegoesontosay,“Idilateonthissimplytoconvinceyouthatyouwillnotbeable to followmymethodsofproof... [basedon] a singleletter.”HesaysthathisfirstgoalisjusttogetsomeonelikeHardytoverifyhisresults—sohe’llbeabletogetascholarship,since“Iamalreadyahalfstarvingman.TopreservemybrainsIwantfood...”

Ramanujan makes a point of saying that it was Hardy’s first category ofresults—onesthatwerealreadyknown—thathe’smostpleasedabout,“Formyresults are verified to be true even though Imay takemy stand upon slenderbasis.”Inotherwords,Ramanujanhimselfwasn’tsureiftheresultswerecorrect—andhe’sexcitedthattheyactuallyare.Sohowwashegettinghisresults?I’llsaymoreaboutthislater.Buthewas

certainly doing all sorts of calculationswith numbers and formulas—in effectdoing experiments. And presumably he was looking at the results of thesecalculations togetan ideaofwhatmightbe true. It’snotclearhowhefiguredoutwhatwasactuallytrue—andindeedsomeoftheresultshequotedweren’tintheendtrue.Butpresumablyheusedsomemixtureoftraditionalmathematicalproof,calculationalevidence,andlotsofintuition.Buthedidn’texplainanyofthistoHardy.Instead,he just startedconductinga correspondenceabout thedetailsof the

results,andthefragmentsofproofshewasabletogive.HardyandLittlewoodseemedintentongradinghisefforts—withLittlewoodwritingaboutsomeresult,forexample,“(d)isstillwrong,ofcourse,ratherahowler.”Still,theywonderedifRamanujanwas“anEuler”,ormerely“aJacobi”.ButLittlewoodhadtosay,“The stuff about primes is wrong”—explaining that Ramanujan incorrectlyassumed the Riemann zeta function didn’t have zeros off the real axis, eventhough itactuallyhasan infinitenumberof them,whichare thesubjectof thewholeRiemannhypothesis.(TheRiemannhypothesisisstillafamousunsolvedmathproblem,eventhoughanoptimisticteachersuggestedittoLittlewoodasaproject when he was an undergraduate...) What about Ramanujan’s strange1+2+3+4+... = –1/12?Well, that has to dowith theRiemann zeta function aswell. For positive integers, ζ(s) is defined as the sum 1/1s+1/2s+1/3s+....Andgiven those values, there’s a nice function—called Zeta[s] in the WolframLanguage—thatcanbeobtainedbycontinuingtoallcomplexs.Nowbasedontheformulaforpositivearguments,onecanidentifyZeta[–1]with1+2+3+4+...But one can also just evaluate Zeta[–1]:

It’saweirdresult,tobesure.Butnotascrazyasitmightatfirstseem.Andinfact it’s a result that’s nowadays considered perfectly sensible for purposes ofcertain calculations in quantum field theory (in which, to be fair, all actualinfinitiesareintendedtocanceloutattheend).Butbacktothestory.HardyandLittlewooddidn’treallyhaveagoodmental

model for Ramanujan. Littlewood speculated that Ramanujan might not begiving the proofs they assumed he had because hewas afraid they’d steal hiswork. (Stealingwas amajor issue in academia then as it is now.)Ramanujansaidhewas“pained”bythisspeculation,andassuredthemthathewasnot“inthe least apprehensive of my method being utilised by others.” He said thatactuallyhe’d invented themethodeightyearsearlier,buthadn’t foundanyonewhocouldappreciateit,andnowhewas“willingtoplaceunreservedlyinyourpossessionwhatlittleIhave.”Meanwhile,evenbeforeHardyhadrespondedtoRamanujan’sfirstletter,he’d

been investigating with the government department responsible for IndianstudentshowhecouldbringRamanujantoCambridge.It’snotquiteclearquitewhatgotcommunicated,butRamanujanrespondedthathecouldn’tgo—perhapsbecauseofhisBrahminbeliefs,orhismother,orperhapsbecausehejustdidn’tthinkhe’dfitin.Butinanycase,Ramanujan’ssupportersstartedpushinginsteadforhimtogetagraduatescholarshipattheUniversityofMadras.Moreexpertswereconsulted,whoopinedthat“Hisresultsappear tobewonderful;butheisnot,now,able topresentany intelligibleproofof someof them,”but“HehassufficientknowledgeofEnglishandisnottoooldtolearnmodernmethodsfrombooks.”The university administration said their regulations didn’t allow a gradate

scholarship to be given to someone like Ramanujan who hadn’t finished anundergraduatedegree.Buttheyhelpfullysuggestedthat“SectionXVoftheActofIncorporationandSection3oftheIndianUniversitiesAct,1904,allowofthe

grant of such a scholarship [by the Government Educational Department],subject to the express consentof theGovernorofFortStGeorge inCouncil.”And despite the seemingly arcane bureaucracy, things moved quickly, andwithina fewweeksRamanujanwasdulyawardedascholarship for twoyears,withthesolerequirementthatheprovidequarterlyreports.

AWayofDoingMathematicsBythetimehegothisscholarship,Ramanujanhadstartedwritingmorepapers,and publishing them in the Journal of the Indian Mathematical Society.Comparedtohisbigclaimsaboutprimesanddivergentseries,thetopicsofthesepaperswerequitetame.Butthepaperswereremarkablenevertheless.What’simmediatelystrikingaboutthemishowcalculationaltheyare—fullof

actual,complicatedformulas.Mostmathpapersaren’tthatway.Theymayhavecomplicated notation, but they don’t have big expressions containingcomplicatedcombinationsofroots,orseeminglyrandomlongintegers.

In modern times, we’re used to seeing incredibly complicated formulasroutinelygeneratedbyMathematica.Butusuallythey’rejustintermediatesteps,and aren’t what papers explicitly talk much about. For Ramanujan, though,complicated formulaswereoftenwhat really told the story.Andof course it’sincrediblyimpressivethathecouldderivethemwithoutcomputersandmoderntools.(As an aside, back in the late 1970s I started writing papers that involved

formulas generated by computer. And in one particular paper, the formulashappenedtohavelotsofoccurrencesofthenumber9.Buttheexperiencedtypistwho typed thepaper—yes, fromamanuscript—replacedevery“9”witha“g”.WhenIaskedherwhy,shesaid,“Well,thereareneverexplicit9’sinpapers!”)LookingatRamanujan’spapers, another striking feature is the frequentuseofnumericalapproximations inarguments leadingtoexactresults.People tendtothinkofworkingwith algebraic formulas as an exact process—generating, forexample, coefficients that are exactly 16, not just roughly 15.99999. But forRamanujan,approximationswereroutinelypartofthestory,evenwhenthefinalresultswereexact.In somesense it’snot surprising that approximations tonumbersareuseful.

Let’s say we want to know which is larger: or

.Wecanstartdoingallsortsoftransformationsamongsquareroots,and trying to derive theorems from them. Or we can just evaluate eachexpression numerically, and find that the first one (2.9755...) is obviouslysmallerthanthesecond(3.322...).InthemathematicaltraditionofsomeonelikeHardy—or, for thatmatter, in a typicalmoderncalculuscourse—suchadirectcalculationalwayofanswering thequestionseemssomehowinappropriateandimproper.And of course if the numbers are very close one has to be careful about

numerical round-off and so on. But for example in Mathematica and theWolframLanguagetoday—particularlywiththeirbuilt-inprecisiontrackingfornumbers—weoftenusenumericalapproximationsinternallyaspartofderivingexactresults,muchlikeRamanujandid.WhenHardyaskedRamanujanforproofs,partofwhathewantedwastogeta

kind of story for each result that explained why it was true. But in a senseRamanujan’smethodsdidn’tlendthemselvestothat.Becausepartofthe“story”wouldhavetobethatthere’sthiscomplicatedexpression,andithappenstobenumerically greater than this other expression. It’s easy to see it’s true—butthere’snorealstoryofwhyit’strue.And the same happens whenever a key part of a result comes from pure

computation of complicated formulas, or in modern times, from automatedtheoremproving.Yes,onecan trace thestepsandsee that they’recorrect.Butthere’snobiggerstorythatgivesoneanyparticularunderstandingoftheresults.Formostpeopleit’dbebadnewstoendupwithsomecomplicatedexpression

orlongseeminglyrandomnumber—becauseitwouldn’ttellthemanything.ButRamanujan was different. Littlewood once said of Ramanujan that “everypositiveintegerwasoneofhispersonalfriends.”Andbetweenagoodmemoryand good ability to notice patterns, I suspectRamanujan could conclude a lotfromacomplicatedexpressionoralongnumber.Forhim,justtheobjectitselfwouldtellastory.Ramanujanwasofcoursegeneratingallthesethingsbyhisowncalculational

efforts. But back in the late 1970s and early 1980s I had the experience ofstartingtogeneratelotsofcomplicatedresultsautomaticallybycomputer.Andafter I’d beendoing it awhile, something interesting happened: I started beingable to quickly recognize the “texture” of results—and often immediately seewhatmight be likely to be true. If Iwas dealing, say,with some complicatedintegral,itwasn’tthatIknewanytheoremsaboutit.Ijusthadanintuitionabout,forexample,whatfunctionsmightappearintheresult.Andgiventhis,Icouldthenget thecomputertogoinandfill inthedetails—andcheckthat theresultwascorrect.ButIcouldn'tderivewhytheresultwastrue,ortellastoryaboutit;itwasjustsomethingthatintuitionandcalculationgaveme.Now of course there’s a fair amount of pure mathematics where one can’t

(yet)justroutinelygoinanddoanexplicitcomputationtocheckwhetherornotsome result is correct. And this often happens for example when there areinfinite or infinitesimal quantities or limits involved. And one of the thingsHardyhad specialized inwasgivingproofs thatwerecareful inhandling suchthings.In1910he’devenwrittenabookcalledOrdersofInfinitythatwasaboutsubtle issues that comeup in taking infinite limits. (In particular, in a kindofalgebraicanalogofthetheoryoftransfinitenumbers,hetalkedaboutcomparinggrowth rates of things like nested exponential functions—and we even makesomeuseofwhatarenowcalledHardyfieldsindealingwithgeneralizationsofpowerseriesintheWolframLanguage.)SowhenHardysawRamanujan’s“fastand loose” handling of infinite limits and the like, itwasn’t surprising that he

reacted negatively—and thought he would need to “tame” Ramanujan, andeducatehiminthefinerEuropeanwaysofdoingsuchthings,ifRamanujanwasactuallygoingtoreliablygetcorrectanswers.

SeeingWhat’sImportantRamanujan was surely a great human calculator, and impressive at knowingwhether a particular mathematical fact or relation was actually true. But hisgreatest skillwas, I think, something in a sensemoremysterious: an uncannyabilitytotellwhatwassignificant,andwhatmightbededucedfromit.Take for example his paper “ModularEquations andApproximations to π”,

publishedin1914,inwhichhecalculates(withoutacomputerofcourse):

Mostmathematicianswouldsay,“It’sanamusingcoincidence that that’s soclosetoaninteger—butsowhat?”ButRamanujanrealizedtherewasmoretoit.Hefoundotherrelations(those“=”shouldreallybe“≅”):

Then he began to build a theory—that involves elliptic functions, thoughRamanujandidn’tknowthatnameyet—andstartedcomingupwithnewseriesapproximationsforπ:

Previousapproximations toπhad ina sensebeenmuchmoresober, thoughthe best one beforeRamanujan’s (Machin’s series from1706) did involve theseeminglyrandomnumber239:

ButRamanujan’sseries—bizarreandarbitraryastheymightappear—hadanimportantfeature:theytookfarfewertermstocomputeπtoagivenaccuracy.In1977,BillGosper—himselfaratherRamanujan-likefigure,whomI’vehadthepleasureofknowingformorethan35years—tookthelastofRamanujan’sseriesfromthelistabove,andusedittocomputearecordnumberofdigitsofπ.Theresoonfollowedothercomputations,allbaseddirectlyonRamanujan’sidea—asisthemethodweuseforcomputingπinMathematicaandtheWolframLanguage.It’s interesting tosee inRamanujan’spaper thatevenheoccasionallydidn’t

knowwhatwasandwasn’tsignificant.Forexample,henoted:

Andthen—inprettymuchhisonlypublishedexampleofgeometry—hegaveapeculiargeometricconstructionforapproximately“squaringthecircle”basedonthisformula:

TruthversusNarrativeTo Hardy, Ramanujan’s way of working must have seemed quite alien. ForRamanujan was in some fundamental sense an experimental mathematician:goingoutintotheuniverseofmathematicalpossibilitiesanddoingcalculationsto find interestingand significant facts—andonly thenbuilding theoriesbasedonthem.Hardy on the other hand worked like a traditional mathematician,

progressively extending the narrative of existing mathematics. Most of hispapers begin—explicitly or implicitly—by quoting some result from themathematicalliterature,andthenproceedbytellingthestoryofhowthisresultcan be extended by a series of rigorous steps. There are no sudden empiricaldiscoveries—andnoseeminglyinexplicablejumpsbasedonintuitionfromthem.It’smathematicscarefullyargued,andbuilt,inasense,brickbybrick.Acenturylaterthisisstillthewayalmostallpuremathematicsisdone.And

evenif it’sdiscussingthesamesubjectmatter,perhapsanythingelseshouldn’tbe called “mathematics”, because its methods are too different. In my owneffortstoexplorethecomputationaluniverseofsimpleprograms,I’vecertainlydoneafairamount thatcouldbecalled“mathematical” in thesensethat it, forexample,exploressystemsbasedonnumbers.Over the years, I’ve found all sorts of results that seem interesting. Strange

structuresthatarisewhenonesuccessivelyaddsnumberstotheirdigitreversals.Bizarrenestedrecurrencerelationsthatgenerateprimes.Peculiarrepresentationsof integers using trees of bitwise XORs. But they’re empirical facts—demonstrably true, yet not part of the tradition and narrative of existingmathematics.Formanymathematicians—likeHardy—the process of proof is the core of

mathematical activity. It’s not particularly significant to come up with aconjectureaboutwhat’strue;what’ssignificantistocreateaproofthatexplainswhy something is true, constructing a narrative that othermathematicians canunderstand.Particularly today,aswestart tobeable toautomatemoreandmoreproofs,

they can seem a bit like mundane manual labor, where the outcome may beinteresting but the process of getting there is not. But proofs can also beilluminating.Theycanineffectbestories that introducenewabstractconceptsthat transcend the particulars of a given proof, and provide raw material tounderstandmanyothermathematicalresults.

ForRamanujan,though,Isuspectitwasfactsandresultsthatwerethecenterofhismathematical thinking, andproofs felt abit like somestrangeEuropeancustomnecessary to takehisresultsoutofhisparticularcontext,andconvinceEuropeanmathematiciansthattheywerecorrect.

GoingtoCambridgeButlet’sreturntothestoryofRamanujanandHardy.Intheearlypartof1913,HardyandRamanujancontinuedtoexchangeletters.

Ramanujandescribedresults;HardycritiquedwhatRamanujansaid,andpushedforproofsandtraditionalmathematicalpresentation.Thentherewasalonggap,butfinallyinDecember1913,Hardywroteagain,explainingthatRamanujan’smost ambitious results—about the distribution of primes—were definitelyincorrect, commenting that “…the theory of primes is full of pitfalls, tosurmountwhich requires the fullest of trainings inmodern rigorousmethods.”HealsosaidthatifRamanujanhadbeenabletoprovehisresultsitwouldhavebeen “about the most remarkable mathematical feat in the whole history ofmathematics.”In January 1914 a young Cambridge mathematician named E. H. Neville

came to give lectures inMadras, and relayed themessage thatHardywas (inRamanujan’swords) “anxious to get [Ramanujan] to Cambridge”. Ramanujanresponded that back in February 1913 he’d had a meeting, along with his“superior officer”, with the Secretary to the Students Advisory Committee ofMadras,whohadaskedwhetherhewaspreparedtogotoEngland.Ramanujanwrote that he assumed he’d have to take exams like the other Indian studentshe’d seen go to England,which he didn’t think he’d dowell enough in—andalsothathissuperiorofficer,a“veryorthodoxBrahmanhavingscruplestogotoforeignlandrepliedatoncethatIcouldnotgo”.ButthenhesaidthatNevillehad“cleared[his]doubts”,explainingthatthere

wouldn’t be an issue with his expenses, that his English would do, that hewouldn’thavetotakeexams,andthathecouldremainavegetarianinEngland.HeendedbysayingthathehopedHardyandLittlewoodwould“begoodenoughtotakethetroubleofgettingme[toEngland]withinaveryfewmonths.”Hardyhadassumed itwouldbebureaucratically trivial togetRamanujan to

England,butactuallyitwasn’t.Hardy’sownTrinityCollegewasn’tpreparedtocontributeanyrealfunding.HardyandLittlewoodofferedtoputupsomeofthemoney themselves. But Neville wrote to the registrar of the University ofMadras saying that “the discovery of the genius of S. Ramanujan ofMadras

promises to be the most interesting event of our time in the mathematicalworld”—and suggested the university come up with themoney. Ramanujan’sexpat supporters swung into action, with the matter eventually reaching theGovernor of Madras—and a solution was found that involved taking moneyfrom a grant that had been given by the government five years earlier for“establishing University vacation lectures”, but that was actually, in thebureaucratic languageof “DocumentNo.182of theEducationalDepartment”,“notbeingutilisedforanyimmediatepurpose”.Therearestrangelittlenotesinthebureaucraticrecord,likeonFebruary12:

“Whatcaste ishe?Treatasurgent.”Buteventuallyeverythingwassortedout,andonMarch17,1914,afterasend-offfeaturinglocaldignitaries,RamanujanboardedashipforEngland,sailingup through theSuezCanal,andarriving inLondon on April 14. Before leaving India, Ramanujan had prepared forEuropean life by gettingWestern clothes, and learning things like how to eatwithaknifeand fork,andhowto tiea tie.ManyIndianstudentshadcome toEnglandbefore,andtherewasawholeprocedureforthem.Butafterafewdaysin London, Ramanujan arrived in Cambridge—with the Indian newspapersproudlyreportingthat“Mr.S.Ramanujan,ofMadras,whoseworkinthehighermathematics has excited the wonder of Cambridge, is now in residence atTrinity.”(In addition to Hardy and Littlewood, two other names that appear in

connectionwithRamanujan’searlydaysinCambridgeareNevilleandBarnes.They’re not especially famous in the overall history ofmathematics, but it sohappensthat in theWolframLanguagethey’rebothcommemoratedbybuilt-infunctions:NevilleThetaSandBarnesG.)RamanujaninCambridge

WhatwastheRamanujanwhoarrivedinCambridgelike?Hewasdescribedasenthusiasticandeager, thoughdiffident.Hemade jokes, sometimesathisownexpense.Hecouldtalkaboutpoliticsandphilosophyaswellasmathematics.Hewas never particularly introspective. In official settings he was polite anddeferentialandtriedtofollowlocalcustoms.HisnativelanguagewasTamil,andearlier in his life he had failed English exams, but by the time he arrived inEngland, his English was excellent. He liked to hang out with other Indianstudents,sometimesgoingtomusicalevents,orboatingontheriver.Physically,hewas described as short and stout—with hismain notable feature being thebrightnessofhiseyes.Heworkedhard,chasingonemathematicalproblemafteranother.Hekepthislivingspacesparse,withonlyafewbooksandpapers.Hewas sensible about practical things, for example in figuring out issues withcookingandvegetarianingredients.Andfromwhatonecantell,hewashappytobeinCambridge.ButthenonJune28,1914—twoandahalfmonthsafterRamanujanarrivedin

England—ArchdukeFerdinandwasassassinated,andonJuly28,WorldWarIbegan.TherewasanimmediateeffectonCambridge.Manystudentswerecalledupformilitaryduty.Littlewoodjoinedthewareffortandendedupdevelopingways to compute range tables for anti-aircraft guns. Hardy wasn’t a bigsupporterof thewar—not leastbecausehe likedGermanmathematics—buthevolunteeredfordutytoo,thoughwasrejectedonmedicalgrounds.Ramanujan described thewar in a letter to hismother, saying for example,

“Theyflyinaeroplanesatgreatheights,bombthecitiesandruinthem.Assoonasenemyplanesaresightedinthesky,theplanesrestingonthegroundtakeoffandflyatgreatspeedsanddashagainstthemresultingindestructionanddeath.”Ramanujan nevertheless continued to pursuemathematics, explaining to his

motherthat“wariswagedinacountrythat isasfarasRangoonisawayfrom[Madras]”. There were practical difficulties, like a lack of vegetables, whichcausedRamanujan toaska friend in India to sendhim“some tamarind (seedsbeingremoved)andgoodcocoanutoilbyparcelpost”.Butofmoreimportance,asRamanujanreportedit,wasthatthe“professorshere…havelosttheirinterestinmathematicsowingtothepresentwar”.Ramanujan told a friend that hehad “changed [his] planofpublishing [his]

results”. He said that he would wait to publish any of the old results in hisnotebooksuntilthewarwasover.ButhesaidthatsincecomingtoEnglandhehadlearned“theirmethods”,andwas“tryingtogetnewresultsbytheirmethodssothatIcaneasilypublishtheseresultswithoutdelay”.In 1915 Ramanujan published a long paper entitled “Highly Composite

Numbers” about maxima of the function (DivisorSigma in the WolframLanguage)thatcountsthenumberofdivisorsofagivennumber.Hardyseemstohavebeenquite involved in thepreparationof thispaper—anditservedas thecenterpieceofRamanujan’sanalogofaPhDthesis.For the next couple of years, Ramanujan prolifically wrote papers—and

despite the war, they were published. A notable paper he wrote with Hardyconcerns the partition function (PartitionsP in the Wolfram Language) thatcounts the number of ways an integer can be written as a sum of positiveintegers. The paper is a classic example of mixing the approximate with theexact.Thepaperbeginswiththeresultforlargen:

But then, using ideas Ramanujan developed back in India, it progressivelyimproves the estimate, to the point where the exact integer result can beobtained. In Ramanujan’s day, computing the exact value of PartitionsP[200]wasabigdeal—and theclimaxofhispaper.Butnow, thanks toRamanujan’smethod, it’s instantaneous:

Cambridgewasdispiritedbythewar—withanappallingnumberofitsfineststudentsdying,oftenwithinweeks,atthefrontlines.TrinityCollege’sbigquadhadbecomeawarhospital.Butthroughallofthis,Ramanujancontinuedtodohismathematics—andwithHardy’shelpcontinuedtobuildhisreputation.ButtheninMay1917,therewasanotherproblem:Ramanujangotsick.From

whatweknownow, it’s likely thatwhathehadwasaparasitic liver infectionpicked up in India.But back then nobody could diagnose it.Ramanujanwentfrom doctor to doctor, and nursing home to nursing home. He didn’t believemuch of what he was told, and nothing that was done seemed to helpmuch.Some months he would be well enough to do a significant amount ofmathematics; others not. He became depressed, and at one point apparentlysuicidal.Itdidn’thelpthathismotherhadpreventedhiswifebackinIndiafromcommunicatingwithhim,presumablyfearingitwoulddistracthim.Hardy tried to help—sometimes by interacting with doctors, sometimes by

providing mathematical input. One doctor told Hardy he suspected “someobscure Oriental germ trouble imperfectly studied at present”. Hardy wrote,“LikeallIndians,[Ramanujan]isfatalistic,andit is terriblyhardtogethimtotakecareofhimself.”Hardylatertoldthenow-famousstorythatheoncevisitedRamanujanatanursinghome,tellinghimthathecameinataxicabwithnumber1729, and saying that it seemed to him a rather dull number—to whichRamanujan replied: “No, it is a very interesting number; it is the smallestnumber expressible as the sum of two cubes in two different ways”:1729=1³+12³=9³+10³. (Wolfram|Alpha now reports some other properties too.)Butthroughallofthis,Ramanujan’smathematicalreputationcontinuedtogrow.He was elected a Fellow of the Royal Society (with his supporters including

HobsonandBaker,bothofwhomhadfailedtorespondtohisoriginalletter)—and inOctober 1918hewas elected a fellowofTrinityCollege, assuringhimfinancial support.Amonth laterWorldWar Iwas over—and the threat ofU-boatattacks,whichhadmadetraveltoIndiadangerous,wasgone.AndsoonMarch13,1919,RamanujanreturnedtoIndia—nowveryfamous

andrespected,butalsoveryill.Throughitall,hecontinuedtodomathematics,writing a notable letter toHardy about “mock” theta functions on January 12,1920.Hechosetolivehumbly,andlargelyignoredwhatlittlemedicinecoulddoforhim.AndonApril26,1920,at theageof32,and threedaysafter the lastentryinhisnotebook,hedied.

TheAftermathFromwhenhefirststarteddoingmathematicsresearch,Ramanujanhadrecordedhis results in a series of hardcover notebooks—publishing only a very smallfractionof them.WhenRamanujandied,Hardybegan toorganizeaneffort tostudy and publish all 3000 or so results in Ramanujan’s notebooks. Severalpeoplewereinvolvedinthe1920sand1930s,andquiteafewpublicationsweregenerated.Butthroughvariousmisadventurestheprojectwasnotcompleted—tobetakenupagainonlyinthe1970s.

In1940,HardygaveallthelettershehadfromRamanujantotheCambridgeUniversityLibrary,buttheoriginalcoverletterforwhatRamanujansentin1913was not among them—so now the only record we have of that is thetranscriptionHardy later published.Ramanujan’s threemainnotebooks sat formany years on top of a cabinet in the librarian’s office at the University ofMadras,wheretheysuffereddamagefrominsects,butwereneverlost.Hisothermathematical documents passed through several hands, and some of themwound up in the incrediblymessy office of a Cambridgemathematician—butwhen he died in 1965 they were noticed and sent to a library, where theylanguisheduntiltheywere“rediscovered”withgreatexcitementasRamanujan’slostnotebookin1976.When Ramanujan died, it took only days for his various relatives to start

askingfor financialsupport.Therewere largemedicalbills fromEngland,andtherewastalkofsellingRamanujan’spaperstoraisemoney.Ramanujan’swifewas 21when he died, but aswas the custom, she never

remarried.Shelivedverymodestly,makingherlivingmostlyfromtailoring.In1950 she adopted the son of a friend of hers who had died. By the 1960s,Ramanujanwasbecoming somethingof ageneral Indianhero, and she startedreceiving various honors and pensions. Over the years, quite a fewmathematicianshadcometovisither—andshehadsuppliedthemforexamplewiththepassportphotothathasbecomethemostfamouspictureofRamanujan.She lived a long life, dying in 1994 at the age of 95, having outlived

Ramanujanby73years.

WhatBecameofHardy?Hardywas 35whenRamanujan’s letter arrived, andwas 43whenRamanujandied.Hardyviewedhis“discovery”ofRamanujanashisgreatestachievement,anddescribedhisassociationwithRamanujanasthe“oneromantic incidentof[his]life”.AfterRamanujandied,HardyputsomeofhiseffortsintocontinuingtodecodeanddevelopRamanujan’sresults,butforthemostparthereturnedtohis previous mathematical trajectory. His collected works fill seven largevolumes(whileRamanujan’spublicationsmakeupjustonefairlyslimvolume).ThewordcloudsofthetitlesofhispapersshowonlyafewchangesfrombeforehemetRamanujantoafter:

Shortly beforeRamanujan entered his life,Hardy had started to collaboratewith John Littlewood, who he would later say was an even more importantinfluenceonhis life thanRamanujan.AfterRamanujandied,Hardymoved towhatseemedlikeabetterjobinOxford,andendedupstayingtherefor11yearsbeforereturningtoCambridge.Hisabsencedidn’taffecthiscollaborationwithLittlewood,though—sincetheyworkedmostlybyexchangingwrittenmessages,evenwhen their roomswere less than a hundred feet apart.After 1911Hardyrarely did mathematics without a collaborator; he worked especially withLittlewood,publishing95paperswithhimoverthecourseof38years.Hardy’smathematicswasalwaysof the finestquality.Hedreamedofdoing

something like solving the Riemann hypothesis—but in reality never didanythingtrulyspectacular.Hewrotetwobooks,though,thatcontinuetobereadtoday: An Introduction to the Theory of Numbers, with E. M. Wright; andInequalities,withLittlewoodandG.Pólya.Hardy lived his life in the stratum of the intellectual elite. In the 1920s he

displayedapictureofLenin inhis apartment, andwasbrieflypresidentof the“scientific workers” trade union. He always wrote elegantly, mostly aboutmathematics, and sometimes about Ramanujan. He eschewed gadgets andalways livedalongwith students andotherprofessors inhis college.Henevermarried, though near the end of his life his younger sister joined him inCambridge(shealsohadnevermarried,andhadspentmostofherlifeteachingatthegirls’schoolwhereshewentasachild).In 1940 Hardy wrote a small book called A Mathematician’s Apology. I

rememberwhenIwasabout12beinggivenacopyof thisbook. I thinkmanypeoplevieweditasakindofmanifestooradvertisementforpuremathematics.But I must say it didn’t resonate with me at all. It felt to me at oncesanctimoniousandaustere,andIwasn’timpressedbyitsattempttodescribetheaesthetics andpleasures ofmathematics, or by thepridewithwhich its authorsaidthat“nothingIhaveeverdoneisoftheslightestpracticaluse”(actually,heco-invented the Hardy-Weinberg law used in genetics). I doubt I would havechosenthepathofapuremathematiciananyway,butHardy’sbookhelpedmakecertainofit.

To be fair, however, Hardywrote the book at a low point in his own life,when he was concerned about his health and the loss of his mathematicalfaculties. And perhaps that explains why he made a point of explaining that“mathematics…isayoungman’sgame”.(AndinanarticleaboutRamanujan,hewrote that“amathematician isoftencomparativelyoldat30,andhisdeathmaybelessofacatastrophethanitseems.”)Idon’tknowifthesentimenthadbeen expressed before—but by the 1970s it was taken as an established fact,extending to science as well as mathematics. Kids I knew would tell me I’dbettergetonwiththings,becauseit’dbealloverbyage30.

Isthatactuallytrue?Idon’tthinkso.It’shardtogetclearevidence,butasoneexample I took the data we have on notable mathematical theorems inWolfram|AlphaandtheWolframLanguage,andmadeahistogramoftheagesofpeoplewhoprovedthem.It’snotacompletelyuniformdistribution(thoughthepeak just before 40 is probably just a theorem-selection effect associatedwithFieldsMedals),butparticularlyifonecorrectsforlifeexpectanciesnowandinthepast it’s a far cry from showing thatmathematical productivityhas all butdriedupbyage30.

Myownfeeling—assomeonewho’sgettingoldermyself—isthatatleastuptomyage,manyaspectsofscientificandtechnicalproductivityactuallysteadilyincrease.Forastart,itreallyhelpstoknowmore—andcertainlyalotofmybestideashavecomefrommakingconnectionsbetweenthingsI’velearneddecadesapart.Italsohelpstohavemoreexperienceandintuitionabouthowthingswillworkout.Andifonehasearliersuccesses,thosecanhelpprovidetheconfidenceto move forward more definitively, without second guessing. Of course, onemustmaintain the constitution to focuswith enough intensity—andbe able toconcentrateforlongenough—tothinkthroughcomplexthings.Ithinkinsomeways I’ve gotten slower over the years, and in someways faster. I’m slowerbecause I know more about mistakes I make, and try to do things carefullyenough to avoid them. But I’m faster because I knowmore and can shortcutmanymore things.Of course, forme in particular, it also helps that over theyearsI’vebuiltallsortsofautomationthatI’vebeenabletomakeuseof.A quite different point is that while making specific contributions to an

existing area (as Hardy did) is something that can potentially be done by theyoung, creating awholenew structure tends to require thebroader knowledgeandexperiencethatcomeswithage.ButbacktoHardy.Isuspectitwasalackofmotivationratherthanability,but

inhislastyears,hebecamequitedispiritedandallbutdroppedmathematics.Hediedin1947attheageof70.Littlewood, who was a decade younger than Hardy, lived on until 1977.

LittlewoodwasalwaysalittlemoreadventurousthanHardy,alittlelessaustere,and a little less august. LikeHardy, he nevermarried—though he did have adaughter(withthewifeofthecouplewhosharedhisvacationhome)whomhedescribed as his “niece” until she was in her forties. And—giving a lie toHardy’s claimaboutmathbeingayoungman’sgame—Littlewood (helpedbygettingearlyantidepressantdrugsat theageof72)had remarkablyproductiveyearsofmathematicsinhis80s.

Ramanujan’sMathematicsWhat became of Ramanujan’s mathematics? For many years, not too much.Hardypursueditsome,butthewholefieldofnumbertheory—whichwaswherethemajorityofRamanujan’sworkwasconcentrated—wasoutoffashion.Here’saplotofthefractionofallmathpaperstaggedas“numbertheory”asafunctionoftimeintheZentralblattdatabase:

Ramanujan’sinterestmayhavebeentosomeextentdrivenbythepeakintheearly1900s(whichwouldprobablygoevenhigherwithearlierdata).Butbythe1930s, the emphasis ofmathematics had shifted away fromwhat seemed likeparticular results inareas likenumber theoryandcalculus, towards thegreatergeneralityandformalitythatseemedtoexistinmorealgebraicareas.In the 1970s, though, number theory suddenly becamemore popular again,

driven by advances in algebraic number theory. (Other subcategories showingsubstantialincreasesatthattimeincludeautomorphicforms,elementarynumbertheory and sequences.) Back in the late 1970s, I had certainly heard ofRamanujan—thoughmoreinthecontextofhisstorythanhismathematics.AndIwaspleased in1982,whenIwaswritingabout thevacuuminquantumfieldtheory, that I could use results of Ramanujan’s to give closed forms forparticularcases(ofinfinitesumsinvariousdimensionsofmodesofaquantumfield—correspondingtoEpsteinzetafunctions):

Starting in the1970s, therewasabigeffort—stillnotentirelycomplete—toproveresultsRamanujanhadgiveninhisnotebooks.Andtherewereincreasingconnections being found between the particular results he’d got, and generalthemesemerginginnumbertheory.A significant part of what Ramanujan did was to study so-called special

functions—and to invent some new ones. Special functions—like the zetafunction, elliptic functions, theta functions, and so on—can be thought of asdefiningconvenient“packets”ofmathematics.Therearean infinitenumberofpossible functions one can define, but what get called “special functions” areoneswhosedefinitionssurvivebecausetheyturnouttoberepeatedlyuseful.Andtoday,forexample,inMathematicaandtheWolframLanguagewehave

RamanujanTau,RamanujanTauL,RamanujanTauThetaandRamanujanTauZasspecial functions. Idon’tdoubt that in the futurewe’llhavemoreRamanujan-inspired functions. In the last year of his life, Ramanujan defined someparticularlyambitiousspecialfunctionsthathecalled“mockthetafunctions”—and that are still in the process of being made concrete enough to routinelycompute.If one looks at the definition of Ramanujan’s tau function it seems quite

bizarre(noticethe“24”):

Andtomymind,themostremarkablethingaboutRamanujanisthathecoulddefinesomethingasseeminglyarbitraryasthis,andhaveitturnouttobeusefulacenturylater.

AreTheyRandomFacts?Inantiquity, thePythagoreansmademuchof thefact that1+2+3+4=10.But toustoday,thisjustseemslikearandomfactofmathematics,notofanyparticularsignificance.WhenIlookatRamanujan’sresults,manyofthemalsoseemlikerandomfactsofmathematics.Buttheamazingthingthat’semergedoverthepastcentury,andparticularlyoverthepastfewdecades,isthatthey’renot.Instead,more and more of them are being found to be connected to deep, elegantmathematicalprinciples.To enunciate these principles in a direct and formalway requires layers of

abstract mathematical concepts and language which have taken decades todevelop. But somehow, through his experiments and intuition, Ramanujanmanagedtofindconcreteexamplesoftheseprinciples.Oftenhisexampleslookquitearbitrary—fullofseeminglyrandomdefinitionsandnumbers.Butperhapsit’snotsurprisingthatthat’swhatittakestoexpressmodernabstractprinciplesintermsoftheconcretemathematicalconstructsoftheearlytwentiethcentury.It’sabitlikeapoettryingtoexpressdeepgeneralideas—butbeingforcedtouseonlytheimperfectmediumofhumannaturallanguage.It’sturnedouttobeverychallengingtoprovemanyofRamanujan’sresults.

And part of the reason seems to be that to do so—and to create the kind ofnarrativeneeded foragoodproof—oneactuallyhasnochoicebut tobuildupmuchmoreabstractandconceptuallycomplexstructures,ofteninmanysteps.So how is it that Ramanujan managed in effect to predict all these deep

principlesoflatermathematics?Ithinktherearetwobasiclogicalpossibilities.Thefirstisthatifonedrillsdownfromanysufficientlysurprisingresult,sayinnumbertheory,onewilleventuallyreachadeepprincipleintheefforttoexplainit. And the second possibility is that while Ramanujan did not have thewherewithaltoexpressitdirectly,hehadwhatamountstoanaestheticsenseofwhich seemingly random factswould turn out to fit together and have deepersignificance.I’m not sure which of these possibilities is correct, and perhaps it’s a

combination. But to understand this a little more, we should talk about theoverall structure of mathematics. In a sense mathematics as it’s practiced isstrangelyperchedbetweenthetrivialandtheimpossible.Atanunderlyinglevel,

mathematics isbasedonsimpleaxioms.And itcouldbe—as it is, say, for thespecific case of Boolean algebra—that given the axioms there’s astraightforwardproceduretofigureoutwhetheranyparticularresultistrue.ButeversinceGödel’stheoremin1931(whichHardymusthavebeenawareof,butapparentlynevercommentedon) it’sbeenknown that foranarea likenumbertheory the situation isquitedifferent: thereare statementsonecangivewithinthecontextofthetheorywhosetruthorfalsityisundecidablefromtheaxioms.Itwasprovedintheearly1960sthattherearepolynomialequationsinvolving

integerswhereit’sundecidablefromtheaxiomsofarithmetic—orineffectfromthe formal methods of number theory—whether or not the equations havesolutions.Theparticularexamplesofclassesofequationswhereit’sknownthatthis happens are extremely complex. But from my investigations in thecomputational universe, I’ve long suspected that there are vastly simplerequations where it happens too. Over the past several decades, I’ve had theopportunitytopollsomeoftheworld’sleadingnumbertheoristsonwheretheythink the boundary of undecidability lies. Opinions differ, but it’s certainlywithin the realm of possibility that for example cubic equations with threevariablescouldexhibitundecidability.Sothequestionthenis,whyshouldthetruthofwhatseemlikerandomfacts

ofnumbertheoryevenbedecidable?Inotherwords,it’sperfectlypossiblethatRamanujancouldhavestateda result that simplycan’tbeproved trueor falsefrom the axioms of arithmetic.Conceivably theGoldbach conjecturewill turnouttobeanexample.AndsocouldmanyofRamanujan’sresults.SomeofRamanujan’sresultshavetakendecadestoprove—butthefact that

they’reprovableatallisalreadyimportantinformation.Foritsuggeststhatinasensethey’renotjustrandomfacts; they’reactuallyfactsthatcansomehowbeconnectedbyproofsbacktotheunderlyingaxioms.AndImustsaythattomethistendstosupporttheideathatRamanujanhad

intuition and aesthetic criteria that in some sense captured someof the deeperprincipleswenowknow,evenifhecouldn’texpressthemdirectly.

AutomatingRamanujanIt’sprettyeasytostartpickingmathematicalstatements,sayatrandom,andthengetting empirical evidence for whether they’re true or not. Gödel’s theoremeffectivelyimpliesthatyou’llneverknowhowfaryou’llhavetogotobecertainof anyparticular result. Sometimes itwon’t be far, but sometimes itmay in asensebearbitrarilyfar.

Ramanujan no doubt convinced himself of many of his results by whatamount to empirical methods—and often it worked well. In the case of thecountingofprimes,however,asHardypointedout, things turnout tobemoresubtle,andresultsthatmightworkuptoverylargenumberscaneventuallyfail.So let’ssayone looksat thespaceofpossiblemathematicalstatements,and

picksstatements thatappearempiricallyat least tosome level tobe true.Nowthenextquestion:arethesestatementsconnectedinanyway?Imagine one could find proofs of the statements that are true. These proofs

effectively correspond to paths through a directed graph that starts with theaxioms,andleadstothetrueresults.Onepossibilityisthenthatthegraphislikea star—with every result being independently proved from the axioms. Butanotherpossibilityis that therearemanycommon“waypoints”ingettingfromthe axioms to the results. And it’s these waypoints that in effect representgeneralprinciples.Ifthere’sacertainsparsitytotrueresults,thenitmaybeinevitablethatmany

of them are connected through a small number of general principles. Itmightalsobethatthereareresultsthataren’tconnectedinthisway,buttheseresults,perhaps just because of their lack of connections, aren’t considered“interesting”—andsoareeffectivelydroppedwhenonethinksaboutaparticularsubject.Ihavetosaythattheseconsiderationsleadtoanimportantquestionforme.I

have spent many years studying what amounts to a generalization ofmathematics: the behavior of arbitrary simple programs in the computationaluniverse.AndI’vefoundthatthere’sahugerichnessofcomplexbehaviortobeseen in suchprograms.But I have also foundevidence—not least throughmyPrincipleofComputationalEquivalence—thatundecidabilityisrifethere.Butnowthequestionis,whenonelooksatallthatrichandcomplexbehavior,

arethereineffectRamanujan-likefactstobefoundthere?Ultimatelytherewillbemuchthatcan’treadilybereasonedaboutinaxiomsystemsliketheonesinmathematics.Butperhapstherearenetworksoffactsthatcanbereasonedabout—andthatallconnecttodeeperprinciplesofsomekind.Weknow from the idea around thePrinciple ofComputationalEquivalence

that therewillalwaysbepocketsof“computationalreducibility”:placeswhereone will be able to identify abstract patterns and make abstract conclusionswithoutrunningintoundecidability.Repetitivebehaviorandnestedbehavioraretwo almost trivial examples. But now the question is whether among all thespecific details of particular programs there are other general forms oforganizationtobefound.Ofcourse,whereasrepetitionandnestingareseeninagreatmanysystems,it

could be that another form of organization would be seen only much morenarrowly.Butwedon’tknow.Andasofnow,wedon’treallyhavemuchofahandleon findingout—at leastuntilorunless there’s aRamanujan-like figurenotfortraditionalmathematicsbutforthecomputationaluniverse.

ModernRamanujans?Will there ever be another Ramanujan? I don’t know if it’s the legend ofRamanujanorjustanaturalfeatureofthewaytheworldissetup,butforatleast30yearsI’vereceivedasteadystreamoflettersthatreadabitliketheoneHardygot from Ramanujan back in 1913. Just a few months ago, for example, Ireceivedanemail(fromIndia,asithappens)withanimageofanotebooklistingvarious mathematical expressions that are numerically almost integers—very

muchlikeRamanujan’s .

Are these numerical facts significant? I don’t know. Wolfram|Alpha cancertainlygenerate lotsofsimilar facts,butwithoutRamanujan-like insight, it’shardtotellwhich,ifany,aresignificant.

Over the years I’ve received countless communications a bit like this one.Numbertheoryisacommontopic.Soarerelativityandgravitationtheory.Andparticularly in recent years,AI and consciousness have been popular too.Thenice thing about letters related to math is that there’s typically somethingimmediately concrete in them: some specific formula, or fact, or theorem. InHardy’sdayitwashardtochecksuchthings;todayit’saloteasier.But—asinthe case of the almost integer above—there’s then the question of whetherwhat’s being said is somehow “interesting”, or whether it’s just a “randomuninterestingfact”.Needlesstosay,thedefinitionof“interesting”isn’taneasyorobjectiveone.

AndinfacttheissuesareverymuchthesameasHardyfacedwithRamanujan’sletter.Ifonecanseehowwhat’sbeingpresentedfitsintosomebiggerpicture—somenarrative—thatoneunderstands,thenonecantellwhether,atleastwithinthat framework, something is “interesting”.But if onedoesn’t have thebiggerpicture—orifwhat’sbeingpresented is just“toofarout”—thenonereallyhasnowaytotellifitshouldbeconsideredinterestingornot.When I first started studying the behavior of simple programs, there really

wasn’t a context forunderstandingwhatwasgoingon in them.Thepictures Igot certainly seemed visually interesting. But it wasn’t clear what the biggerintellectual story was. And it took quite a few years before I’d accumulatedenough empirical data to formulate hypotheses and develop principles that letone go back and see what was and wasn’t interesting about the behavior I’dobserved.I’ve put a few decades into developing a science of the computational

universe. But it’s still young, and there is much left to discover—and it’s ahighlyaccessiblearea,withnothresholdofelaboratetechnicalknowledge.Andone consequence of this is that I frequently get letters that show remarkablebehaviorinsomeparticularcellularautomatonorothersimpleprogram.OftenIrecognizethegeneralformofthebehavior,becauseitrelatestothingsI’veseenbefore,butsometimesIdon’t—andsoIcan’tbesurewhatwillorwon’tendupbeinginteresting.Back in Ramanujan’s day, mathematics was a younger field—not quite as

easy toenteras the studyof thecomputationaluniverse,butmuchcloser thanmodernacademicmathematics.Andtherewereplentyof“randomfacts”beingpublished:aparticulartypeofintegraldoneforthefirsttime,oranewclassofequations that couldbe solved.Manyyears laterwewouldcollect asmanyofthese as we could to build them into the algorithms and knowledgebase ofMathematica and the Wolfram Language. But at the time probably the most

significantaspectoftheirpublicationwastheproofsthatweregiven:thestoriesthatexplainedwhy the resultswere true.Because in theseproofs, therewasatleastthepotentialthatconceptswereintroducedthatcouldbereusedelsewhere,andbuilduppartofthefabricofmathematics.Itwouldtakeustoofarafieldtodiscussthisatlengthhere,butthereisakind

of analog in the study of the computational universe: the methodology forcomputerexperiments.Justasaproofcancontainelementsthatdefineageneralmethodology for getting a mathematical result, so the particular methods ofsearch,visualizationoranalysiscandefinesomethingincomputerexperimentsthat is general and reusable, and can potential give an indication of someunderlyingideaorprinciple.Andso,abitlikemanyofthemathematicsjournalsofRamanujan’sday,I’ve

tried to provide a journal and a forum where specific results about thecomputationaluniversecanbereported—thoughthereismuchmorethatcouldbedonealongtheselines.When a letter one receives contains definite mathematics, in mathematical

notation,thereisatleastsomethingconcreteonecanunderstandinit.Butplentyofthingscan’tusefullybeformulatedinmathematicalnotation.Andtoooften,unfortunately,lettersareinplainEnglish(orworse,forme,otherlanguages)andit’salmostimpossibleformetotellwhatthey’retryingtosay.Butnowthere’ssomething much better that people increasingly do: formulate things in theWolframLanguage.Andinthatform,I’malwaysabletotellwhatsomeoneistryingtosay—althoughIstillmaynotknowifit’ssignificantornot.Over the years, I’ve been introduced to many interesting people through

letters they’ve sent. Often they’ll come to our Summer School, or publishsomething inoneofourvariouschannels. Ihavenostory (yet)asdramaticasHardy and Ramanujan. But it’s wonderful that it’s possible to connect withpeopleinthisway,particularlyintheirformativeyears.AndIcan’tforgetthatalong time ago, I was a 14-year-oldwhomailed papers about the research I’ddonetophysicistsaroundtheworld…

WhatIfRamanujanHadMathematica?Ramanujandidhiscalculationsbyhand—withchalkonslate,orlaterpencilonpaper.TodaywithMathematicaandtheWolframLanguagewehaveimmenselymore powerful tools with which to do experiments and make discoveries inmathematics(nottomentionthecomputationaluniverseingeneral).It’s fun to imagine what Ramanujan would have done with these modern

tools.Iratherthinkhewouldhavebeenquiteanadventurer—goingoutintothemathematicaluniverseandfindingallsortsofstrangeandwonderfulthings,thenusinghisintuitionandaestheticsensetoseewhatfitstogetherandwhattostudyfurther.Ramanujanunquestionablyhadremarkableskills.ButIthinkthefirststepto

followinginhisfootstepsisjusttobeadventurous:nottostayinthecomfortofwell-established mathematical theories, but instead to go out into the widermathematicaluniverseandstartfinding—experimentally—what’strue.It’stakenthebetterpartofacenturyformanyofRamanujan’sdiscoveriesto

be fitted into a broader and more abstract context. But one of the greatinspirationsthatRamanujangivesusisthatit’spossiblewiththerightsensetomakegreatprogressevenbeforethebroadercontexthasbeenunderstood.AndIforonehope thatmanymorepeoplewill take advantageof the toolswehavetoday to followRamanujan’s lead andmake great discoveries in experimentalmathematics—whethertheyannouncetheminunexpectedlettersornot.

SolomonGolombMay25,2016

TheMost-UsedMathematicalAlgorithmIdeainHistoryAnoctillion.Abillionbillionbillion.That’safairlyconservativeestimateofthenumber of times a cellphone or other device somewhere in the world hasgeneratedabitusingamaximum-lengthlinear-feedbackshiftregistersequence.It’sprobably thesinglemost-usedmathematicalalgorithmidea inhistory.AndthemainoriginatorofthisideawasSolomonGolomb,whodiedonMay1—andwhomIknewfor35years.SolomonGolomb’sclassicbookShiftRegisterSequences,publishedin1967

—basedonhiswork in the1950s—wentoutofprint longago.But itscontentlives on in pretty much every modern communications system. Read thespecificationsfor3G,LTE,Wi-Fi,Bluetooth,orforthatmatterGPS,andyou’llfindmentions of polynomials that determine the shift register sequences thesesystemsuse toencode thedata theysend.SolomonGolombis thepersonwhofiguredouthowtoconstructallthesepolynomials.HealsowasinchargewhenradarwasfirstusedtofindthedistancetoVenus,

andofworkingouthowtoencodeimagestobesentfromMars.Heintroducedtheworldtowhathecalledpolyominoes,whichlaterinspiredTetris(“tetrominotennis”).Hecreatedandsolvedcountlessmathandwordplaypuzzles.And—asIlearned about 20 years ago—he came very close to discovering my all-time-favoriterule30cellularautomatonallthewaybackin1959,theyearIwasborn.

HowIMetSolGolombMostofthescientistsandmathematiciansIknowImetfirstthroughprofessionalconnections.ButnotSolGolomb.Itwas1981,andIwasatCaltech,a21-year-oldphysicistwho’djustreceivedsomemediaattentionfrombeingtheyoungestinthefirstbatchofMacArthurawardrecipients.Igetaknockatmyofficedoor—andayoungwomanisthere.Alreadythiswasunusual,becauseinthosedaystherewerehopelesslyfewwomentobefoundaroundatheoreticalhigh-energyphysicsgroup.IwasashelteredEnglishmanwho’dbeeninCaliforniaacoupleofyears,buthadn’treallyventuredoutsidetheuniversity—andwasillpreparedfortheburstofSouthernCalifornianenergythatdroppedintoseemethatday.She introducedherself asAstrid, and said that she’dbeenvisitingOxford andknew someone I’d been at kindergarten with. She explained that she had a

personalmandatetocollectinterestingacquaintancesaroundthePasadenaarea.I think sheconsideredmeadifficult case,butpersistednevertheless.AndonedaywhenItriedtoexplainsomethingabouttheworkIwasdoingshesaid,“Youshouldmeetmyfather.He’sabitold,buthe’sstillassharpasatack.”Andsoitwas thatAstridGolomb,oldestdaughterofSolGolomb, introducedme toSolGolomb.TheGolombs lived in a houseperched in thehills nearPasadena. I learned

that they had two daughters—Astrid, a little older than me, an aspiringHollywood person, andBeatrice, aboutmy age, a high-powered science type.TheGolombsistersoftenhadparties,usuallyattheirfamily’shouse.Therewerethemes,liketheflamingoes&hedgehogscroquetgardenparty(“recognitionwillbe given to the person who appears most appropriately attired”), or theStonehenge party with instructions written using runes. The parties had aninteresting cross-section of young and not-so-young people, including variouslocal luminaries. And always there, hanging back a little, was SolGolomb, asmall man with a large beard and a certain elf-like quality to him, typicallywearingadarksuitcoat.I gradually learned a little about Sol Golomb. That he was involved in

“information theory”. That he worked at USC (the University of SouthernCalifornia). That he had various unspecified but apparently high-levelgovernmentandotherconnections.I’dheardofshiftregisters,butdidn’treallyknowanythingmuchaboutthem.Then in the fall of 1982, I visitedBellLabs inNew Jersey andgave a talk

aboutmy latest resultsoncellularautomata.One topic Idiscussedwaswhat Icalled“additive”or“linear”cellularautomata—andtheirbehaviorwithlimitednumbersofcells.Wheneveracellularautomatonhasalimitednumberofcells,it’sinevitablethatitsbehaviorwilleventuallyrepeat.Butasthesizeincreases,themaximumrepetitionperiod—sayfortherule90additivecellularautomaton—bouncesaroundseeminglyquiterandomly:1,1,3,2,7,1,7,6,31,4,63,….A few days before my talk, however, I’d noticed that these periods actuallyseemedtofollowaformulathatdependedonthingsliketheprimefactorizationofthenumberofcells.ButwhenImentionedthisduringthetalk,someoneatthebackputuptheirhandandasked,“Doyouknowifitworksforthecasen=37?”Myexperimentshadn’tgottenas faras thesize-37caseyet, so Ididn’tknow.Butwhywouldsomeoneaskthat?ThepersonwhoaskedturnedouttobeacertainAndrewOdlyzko,anumber

theoristatBellLabs.Iaskedhim,“Whatonearthmakesyouthinktheremightbesomethingspecialaboutn=37?”“Well,”hesaid,“Ithinkwhatyou’redoingisrelatedtothetheoryoflinear-feedbackshiftregisters,”andhesuggestedthatI

look at Sol Golomb’s book (“Oh yes,” I said, “I know his daughters…”).Andrewwas indeed correct: there is a very elegant theory of additive cellularautomata basedonpolynomials that is similar to the theorySol developed forlinear-feedback shift registers. Andrew and I ended up writing a now-rather-well-cited paper about it (it’s interesting because it’s a rare case wheretraditional mathematical methods let one say things about nontrivial cellularautomaton behavior). And for me, a side effect was that I learned somethingaboutwhatthesomewhatmysteriousSolGolombactuallydid.(Remember,thiswas before theweb, so one couldn’t just instantly look everything up.)TheStoryofSolGolombSolomonGolombwas born inBaltimore,Maryland in 1932.His family camefromLithuania.His grandfather hadbeen a rabbi; his fathermoved to theUSwhen he was young, and got a master’s degree in math before switching tomedievalJewishphilosophyandalsobecomingarabbi.Sol’smothercamefromaprominentRussianfamilythathadmadebootsfortheTsar’sarmyandthenranabank.Soldidwellinschool,notablybeingaforceinthelocaldebatingscene.Encouragedbyhisfather,hedevelopedaninterestinmathematics,publishingaproblem he invented about primes when he was 17. After high school, SolenrolledatJohnsHopkinsUniversitytostudymath,narrowlyavoidingaquotaon Jewish students by promising he wouldn’t switch to medicine—and tooktwicetheusualcourseload,graduatingin1951afterhalftheusualtime.FromtherehewouldgotoHarvardforgraduateschoolinmath.Butfirsthe

tookasummerjobattheGlennL.MartinCompany,anaerospacefirmfoundedin1912thathadmovedtoBaltimorefromLosAngelesinthe1920sandmostlybecomeadefensecontractor—and thatwouldeventuallymerge intoLockheedMartin. At Harvard, Sol specialized in number theory, and in particular inquestionsabout characterizationsof setsofprimenumbers.But every summerhewouldreturntotheMartinCompany.Ashelaterdescribedit,hefoundthatatHarvard “the question of whether anything that was taught or studied in themathematicsdepartmenthadanypracticalapplicationscouldnotevenbeasked,let alone discussed”. But at theMartinCompany, he discovered that the puremathematicsheknew—evenaboutprimesandthings—didindeedhavepracticalapplications,andveryinterestingones,especiallytoshiftregisters.The first summer he was at the Martin Company, Sol was assigned to a

control theory group. But by his second summer, he’d been put in a groupstudyingcommunications.AndinJune1954itsohappenedthathissupervisorhadjustgonetoaconferencewherehe’dheardaboutstrangebehaviorobservedin linear-feedback shift registers (he called them “tapped delay lines with

feedback”)—andheaskedSolifhecouldinvestigate.Itdidn’ttakeSollongtorealize thatwhatwas goingon could be very elegantly studiedusing the puremathematics he knew about polynomials over finite fields.Over the year thatfollowed, he split his timebetweengraduate school atHarvard and consultingfortheMartinCompany,andinJune1955hewrotehisfinalreport,“SequenceswithRandomnessProperties”—whichwouldbasicallybecomethefoundationaldocumentofthetheoryofshiftregistersequences.Sol liked math puzzles, and in the process of thinking about a puzzle

involvingarrangingdominoesonacheckerboard,heendedupinventingwhathecalled “polyominoes”. He gave a talk about them in November 1953 at theHarvard Mathematics Club, published a paper about them (his first researchpublication),wonaHarvardmathprize forhisworkon them,and, ashe latersaid,then“found[himself]irrevocablycommittedtotheircareandfeeding”fortherestofhislife.

In June 1955, Sol went to spend a year at the University of Oslo on aFulbrightFellowship—partlysohecouldworkwithsomedistinguishednumbertheoriststhere,andpartlysohecouldaddNorwegian,SwedishandDanish(andsomerunicscripts) tohiscollectionof languageskills.Whilehewas there,hefinished a long paper on prime numbers, but also spent time traveling aroundScandinavia,andinDenmarkmetayoungwomannamedBo(BodilRygaard)—whocamefromalargefamilyinaruralareamostlyknownforitspeatmoss,buthadmanaged to get into university andwas studying philosophy. Sol andBoapparentlyhititoff,andwithinmonths,theyweremarried.WhentheyreturnedtotheUSinJuly1956,Solinterviewedinafewplaces,

then accepted a job at JPL—the Jet Propulsion Lab that had spun off fromCaltech,initiallytodomilitarywork.SolwasassignedtotheCommunicationsResearchGroup,asaSeniorResearchEngineer.ItwasatimewhenthepeopleatJPLwereeagertotrylaunchingasatellite.Atfirst,thegovernmentwouldn’tletthemdoit,fearingitwouldbeviewedasamilitaryact.ButthatallchangedinOctober1957whentheSovietUnionlaunchedSputnik,ostensiblyaspartoftheInternationalGeophysicalYear.Amazingly,ittookonly3monthsfortheUSto launch Explorer 1. JPL built much of it, and Sol’s lab (where he hadtechniciansbuildingelectronic implementationsof shift registers)wasdivertedintodoing things likemaking radiationdetectors (including, as it happens, theonesthatdiscoveredtheVanAllenradiationbelts)—whileSolhimselfworkedonusingradartodeterminetheorbitofthesatellitewhenitwaslaunched,takingalittletimeouttogobacktoHarvardforhisfinalPhDexam.ItwasatimeofgreatenergyaroundJPLandthespaceprogram.InMay1958

anewInformationProcessingGroupwasformed,andSolwasputincharge—and in the samemonth,Sol’s first child, the aforementionedAstrid,wasborn.Solcontinuedhisresearchonshiftregistersequences—particularlyasappliedtojamming-resistant radio control of missiles. In May 1959, Sol’s second childarrived—andwasnamedBeatrice,forminganiceA,Bsequence.Inthefallof1959,SoltookasabbaticalatMIT,wherehegottoknowClaudeShannonandanumberofotherMITluminaries,andgotinvolvedininformationtheoryandthetheoryofalgebraiccodes.Asithappens,he’dalreadydonesomeworkoncodingtheory—intheareaof

biology. The digital nature of DNA had been discovered by JimWatson andFrancis Crick in 1953, but it wasn’t yet clear just how sequences of the fourpossiblebasepairs encoded the20aminoacids. In1956,MaxDelbrück—JimWatson’s former postdoc advisor at Caltech—asked around at JPL if anyonecouldfigureitout.SolandtwocolleaguesanalyzedanideaofFrancisCrick’s

andcameupwith“comma-freecodes”inwhichoverlappingtriplesofbasepairscould encode amino acids. The analysis showed that exactly 20 amino acidscouldbeencodedthisway.Itseemedlikeanamazingexplanationofwhatwasseen—but unfortunately it isn’t how biology actually works (biology uses amorestraightforwardencoding,wheresomeofthe64possibletriplesjustdon’trepresentanything).In addition to biology, Sol was also pulled into physics. His shift register

sequenceswereusefulfordoingrangefindingwithradar(muchasthey’reusednowinGPS),andatSol’ssuggestion,hewasputinchargeoftryingtousethemtofindthedistancetoVenus.Andsoitwasthatinearly1961—whentheSun,Venus, and Earth were in alignment—Sol’s team used the 85-foot Goldstoneradio dish in the Mojave Desert to bounce a radar signal off Venus, anddramatically improve our knowledge of the Earth-Venus and Earth-Sundistances.With his interest in languages, coding and space, it was inevitable that Sol

wouldget involvedinthequestionofcommunicationswithextraterrestrials.In1961 he wrote a paper for the Air Force entitled “A Short Primer forExtraterrestrialLinguistics”,andoverthenextseveralyearswroteseveralpaperson the subject for broader audiences. He said that “There are two questionsinvolvedincommunicationwithExtraterrestrials.Oneisthemechanicalissueofdiscoveringamutuallyacceptablechannel.Theotheristhemorephilosophicalproblem (semantic, ethic, and metaphysical) of the proper subject matter fordiscourse. In simpler terms,we first require a common language, and thenwemust think of something clever to say.” He continued, with a touch of hischaracteristic humor: “Naturally, we must not risk telling too much until weknow whether the Extraterrestrials’ intentions toward us are honorable. TheGovernment will undoubtedly set up a Cosmic Intelligence Agency (CIA) tomonitor Extraterrestrial Intelligence. Extreme security precautions will bestrictlyobserved.AsH.G.Wellsoncepointedout[orwasitanepisodeofTheTwilight Zone?], even if the Aliens tell us in all truthfulness that their onlyintentionis‘toservemankind,’wemustendeavortoascertainwhethertheywishtoserveusbakedorfried.”While at JPL, Sol had also been teaching some classes at the nearby

universities: Caltech, USC and UCLA. In the fall of 1962, following somechanges at JPL—andperhaps because hewanted to spendmore timewith hisyoungchildren—hedecidedtobecomeafull-timeprofessor.Hegotoffersfromall three schools. He wanted to go somewhere where he could “make adifference”.HewastoldthatatCaltech“noonehasanyinfluenceiftheydon’tatleasthaveaNobelPrize”,whileatUCLA“theUCbureaucracyissuchthatno

one ever has any ability to affect anything”. The result was that—despite itsmuch-inferior reputation at the time—Sol chose USC. He went there in thespringof1963asaProfessorofElectricalEngineering—andendedupstayingfor53years.

ShiftRegistersBefore going on with the story of Sol’s life, I should explain what a linear-feedback shift register (LFSR)actually is.Thebasic idea is simple. Imaginearowofsquares,eachcontainingeither1or0(say,blackorwhite).Inapureshiftregisterall thathappens is thatat eachstepallvalues shiftoneposition to theleft.The leftmostvalue is lost, andanewvalue is “shifted in” from the right.The idea of a feedback shift register is that the value that’s shifted in isdetermined(or“fedback”)fromvaluesatotherpositionsintheshiftregister.Inalinear-feedbackshiftregister,thevaluesfrom“taps”atparticularpositionsintheregisterarecombinedbybeingaddedmod2(sothat1⊕1=0insteadof2),orequivalentlyXOR’ed(“exclusiveor”,trueifeitheristrue,butnotboth).

Ifonerunsthisforawhile,here’swhathappens:

Obviouslytheshiftregisterisalwaysshiftingbitstotheleft.Andithasaverysimple rule for how bits should be added at the right.But if one looks at thesequenceofthesebits,itseemsratherrandom—though,asthepictureshows,itdoes eventually repeat. What Sol Golomb did was to find an elegantmathematicalwaytoanalyzesuchsequences,andhowtheyrepeat.If a shift register has size n, then it has 2ⁿ possible states altogether

(corresponding to all possible sequences of 0s and 1s of length n). Since therules for the shift register aredeterministic, anygiven statemust alwaysgo tothesamenextstate.Andthatmeansthemaximumpossiblenumberofstepstheshift registercouldconceivablygothroughbefore it repeats is2ⁿ (actually, it’s2ⁿ–1,becausethestatewithall0scan’tevolveintoanythingelse).Intheexampleabove,theshiftregisterisofsize7,anditturnsouttorepeat

afterexactly2⁷–1=127steps.Butwhichshiftregisters—withwhichparticulararrangementsoftaps—willproducesequenceswithmaximallengths?Thisisthefirst question Sol Golomb set out to investigate in the summer of 1954. Hisanswerwassimpleandelegant.Theshiftregisterabovehastapsatpositions7,6and1.Solrepresentedthis

algebraically,usingthepolynomialx⁷+x⁶+1.Thenwhatheshowedwasthatthesequencethatwouldbegeneratedwouldbeofmaximallengthifthispolynomialis “irreducible modulo 2”, so that it can’t be factored, making it sort of theanalogofaprimeamongpolynomials—aswellashavingsomeotherpropertiesthatmake it a so-called “primitive polynomial”.Nowadays,withMathematicaandtheWolframLanguage,it’seasytotestthingslikethis:

Backin1954,Solhadtodoall thisbyhand,butcameupwithafairlylongtable of irreducible polynomials corresponding to shift registers that givemaximallengthsequences:

ThePrehistoryofShiftRegistersThe idea of maintaining short-term memory by having “delay lines” thatcirculate digital pulses (say in an actual column ofmercury) goes back to theearliest days of electronic computers.By the late 1940s suchdelay lineswereroutinelybeingimplementedpurelydigitally,usingsequencesofvacuumtubes,andwerebeingcalled“shiftregisters”.It’snotclearwhenthefirstfeedbackshiftregisterswerebuilt.Perhapsitwasattheendofthe1940s.Butit’sstillshroudedinmystery—becausethefirstplacetheyseemtohavebeenusedwasinmilitarycryptography.The basic idea of cryptography is to take meaningful messages, and then

randomize them so they can’t be recognized, but in such a way that therandomization can always be reversed if you know the key that was used tocreate it. So-called stream ciphers work by generating long sequences ofseemingly random bits, then combining thesewith some representation of themessage—then decoding by having the receiver independently generate thesamesequenceofseeminglyrandombits,and“backingthisout”oftheencodedmessagereceived.Linear-feedback shift registers seem at first to have been prized for

cryptography because of their long repetition periods. As it turns out, themathematicalanalysisSolusedtofindthingsliketheseperiodsalsomakesclearthat such shift registers aren’t good for secure cryptography. But in the earlydays, theyseemedprettygood—particularlycomparedto,say,successiverotorpositions inanEnigmamachine—and there’sbeenapersistent rumor that, forexample,Sovietmilitarycryptosystemswerelongbasedonthem.Backin2001,whenIwasworkingonhistorynotesformybookANewKind

ofScience, I hada longphoneconversationwithSol about shift registers.Soltoldmethatwhenhestartedout,hedidn’tknowanythingaboutcryptographicworkonshiftregisters.HesaidthatpeopleatBellLabs,LincolnLabsandJPLhadalsostartedworkingonshiftregistersaroundthesametimehedid—thoughperhaps through knowing more pure mathematics, he managed to get furtherthantheydid,andintheendhis1955reportbasicallydefinedthefield.Overtheyearsthatfollowed,Solgraduallyheardaboutvariousprecursorsof

his work in the pure mathematical literature. Way back in the year 1202FibonacciwasalreadytalkingaboutwhatarenowcalledFibonaccinumbers—andwhich are generated by a recurrence relation that can be thought of as ananalog of a linear-feedback shift register, but working with arbitrary integers

ratherthan0sand1s.Therewasalittleworkonrecurrenceswith0sand1sdoneintheearly1900s,butthefirstlarge-scalestudyseemstohavebeenbyØysteinOre,who,curiously,camefromtheUniversityofOslo, thoughwasby thenatYale.Orehada studentnamedMarshallHall—whoSol toldmeheknewhadconsultedforthepredecessoroftheNationalSecurityAgencyinthelate1940s—possiblyaboutshiftregisters.Butwhateverhemayhavedonewaskeptsecret,and so it fell to Sol to discover and publish the story of linear-feedback shiftregisters—even though Sol did dedicate his 1967 book on shift registers toMarshallHall.

WhatAreShiftRegisterSequencesGoodFor?Over the years I’ve noticed the principle that systems defined by sufficientlysimplerulesalwayseventuallyenduphavinglotsofapplications.Shiftregistersfollow this principle in spades. And for example modern hardware (andsoftware)systemsarebristlingwithshiftregisters:atypicalcellphoneprobablyhasadozenortwo,implementedusuallyinhardwarebutsometimesinsoftware.(When I say “shift register” here, I mean linear-feedback shift register, orLFSR.) Most of the time, the shift registers that are used are ones that givemaximum-length sequences (otherwise known as “m-sequences”). And thereasonsthey’reusedaretypicallyrelatedtosomeveryspecialpropertiesthatSoldiscoveredaboutthem.Onebasicpropertytheyalwayshaveisthattheycontainthesametotalnumberof0sand1s(actually,there’salwaysexactlyoneextra1).Solthenshowedthattheyalsohavethesamenumberof00s,01s,10sand11s—andthesameholdsforlargerblockstoo.This“balance”propertyisonitsownalreadyveryuseful,forexampleifone’stryingtoefficientlytestallpossiblebitpatternsasinputtoacircuit.ButSoldiscoveredanother,evenmoreimportantproperty.Replaceeach0in

asequenceby–1,thenimaginemultiplyingeachelementinashiftedversionofthesequencebythecorrespondingelementintheoriginal.WhatSolshowedisthat if one adds up these products, they’ll always sum to zero, except whenthere’snoshiftatall.Saidmoretechnically,heshowedthatthesequencehasnocorrelationwithshiftedversionsofitself.Both this and the balance property will be approximately true for any

sufficiently longrandomsequenceof0sand1s.But thesurprising thingaboutmaximum-length shift register sequences is that these properties are alwaysexactly true. The sequences in a sense have some of the signatures ofrandomness—but inaveryperfectway,madepossibleby the fact that they’renotrandomatall,butinsteadhaveaverydefinite,organizedstructure.It’s this structure that makes linear-feedback shift registers ultimately not

suitable for strong cryptography.But they’re great for basic “scrambling” and“cheapcryptography”—andthey’reusedallovertheplaceforthesepurposes.Avery common objective is just to “whiten” (as in “white noise”) a signal. It’sprettycommontowanttotransmitdatathat’sgotlongsequencesof0sinit.Buttheelectronicsthatpicktheseupcangetconfusedif theyseewhatamountsto“silence” for too long.One can avoid the problem by scrambling the originaldatabycombiningitwithashiftregistersequence,sothere’salwayssomekindof “chattering” going on.And that’s indeedwhat’s done inWi-Fi, Bluetooth,USB,digitalTV,Ethernetandlotsofotherplaces.It’softenanicesideeffectthattheshiftregisterscramblingmakesthesignal

hardertodecode—andthis issometimesusedtoprovideat leastsomelevelofsecurity. (DVDsuse a combination of a size-16 and a size-24 shift register toattempttoencodetheirdata;manyGSMphonesuseacombinationofthreeshiftregisterstoencodealltheirsignals,inawaythatwasatfirstsecret.)GPSmakescrucial use of shift register sequences too. Each GPS satellite continuouslytransmitsashiftregistersequence(fromasize-10shiftregister,asithappens).Areceivercantellatexactlywhattimeasignalit’sjustreceivedwastransmittedfrom a particular satellite by seeingwhat part of the sequence it got. And bycomparingdelay times fromdifferent satellites, the receiver can triangulate itsposition. (There’s also a precision mode of GPS, that uses a size-1024 shiftregister.)

A quite different use of shift registers is for error detection. Say one’stransmitting a block of bits, but each one has a small probability of error. Asimplewaytoletonecheckforasingleerroristoincludea“paritybit”thatsayswhetherthereshouldbeanoddorevennumberof1sintheblockofbits.Thereare generalizations of this called CRCs (cyclic redundancy checks) that cancheck for a larger number of errors—and that are computed essentially byfeedingone’sdata intononeother thana linear-feedback shift register. (Thereare also error-correcting codes that let one not only detect but also correct acertain number of errors, and some of these, too, can be computedwith shiftregistersequences—andinfactSolGolombusedaversionofthesecalledReed–Solomon codes to design the video encoding forMars spacecraft.) The list ofusesforshiftregistersequencesgoesonandon.Afairlyexoticexample—morepopular in the past than now—was to use shift register sequences to jitter theclock in a computer to spread out the frequency at which the CPU wouldpotentially generate radio interference (“select Enable Spread Spectrum in theBIOS”).One of the single most prominent uses of shift register sequences is in

cellphones, for what’s called CDMA (Code Division Multiple Access).Cellphonesgottheirnamebecausetheyoperatein“cells”,withallphonesinagivencellbeingconnectedtoaparticulartower.Buthowdodifferentcellphonesin a cell not interfere with each other? In the first systems, each phone justnegotiatedwiththetowertouseaslightlydifferentfrequency.Later,theyuseddifferent time slices (TDMA, orTimeDivisionMultipleAccess).ButCDMAusesmaximum-lengthshiftregistersequencestoprovideacleveralternative.Theideaistohaveallphonesessentiallyoperateonthesamefrequency,but

tohave eachphone encode its signal using (in the simplest case) a differentlyshiftedversionofashiftregistersequence.AndbecauseofSol’smathematicalresults, thesedifferentlyshiftedversionshavenocorrelation—sothecellphonesignals don’t interfere. And this is how, for example, most 3G cellphonenetworksoperate.Sol created the mathematics for this, but he also brought some of the key

people together. Back in 1959, he’d gotten to know a certain Irwin Jacobs,who’d recentlygottenaPhDatMIT.Meanwhile,heknewAndyViterbi,whoworkedatJPL.Sol introducedthe twoof them—andby1968they’dformedacompany called Linkabit which did work on coding systems, mostly for themilitary.Linkabithadmanyspinoffsanddescendents,andin1985JacobsandViterbi

started a new company calledQualcomm. It didn’t immediately do especially

well,butbytheearly1990sitbeganameteoricrisewhenitstartedmakingthecomponents to deploy CDMA in cellphones—and in 1999 Sol became the“ViterbiProfessorofCommunications”atUSC.

WhereAreThereShiftRegisters?It’ssortofamazingthat—althoughmostpeoplehaveneverheardofthem—shiftregistersequencesareactuallyusedinonewayoranotheralmostwheneverbitsaremovedaroundinmoderncommunicationsystems,computersandelsewhere.It’s quite confusing sometimes, because there are lots of thingswith differentnames and acronyms that all turn out to be linear-feedback shift registersequences (PN, pseudonoise, M-, FSR, LFSR sequences, spread spectrumcommunications,MLS,SRS,PRBS,…).Ifonelooksatcellphones,shiftregistersequenceusagehasgoneupanddown

over the years. 2G networks are based on TDMA, so don’t use shift registersequences to encode theirdata—but still oftenuseCRCs tovalidateblocksofdata.3GnetworksarebigusersofCDMA—sothereareshiftregistersequencesinvolvedinprettymucheverybitthat’stransmitted.4Gnetworkstypicallyuseacombination of time and frequency slots which don’t directly involve shiftregistersequences—thoughtherearestillCRCsused,forexampletodealwithdata integrity when frequency windows overlap. 5G is designed to be moreelaborate—with large arrays of antennas dynamically adapting to use optimaltimeandfrequencyslots.Buthalftheirchannelsaretypicallyallocatedto“pilotsignals” that are used to infer the local radio environment—and work bytransmittingnoneotherthanshiftregistersequences.Throughoutmostkindsofelectronicsit’scommontowanttousethehighest

dataratesandthelowestpowersthatstillgetbitstransmittedcorrectlyabovethe“noisefloor”.Andtypicallythewayonepushestotheedgeis todoautomaticerrordetection—usingCRCsandthereforeshiftregistersequences.Andinfactprettymucheverykindofbus(PCIe,SATA,etc.) insideacomputerdoesthis:whetherit’sconnectingpartsofCPUs,gettingdataoffdevices,orconnectingtoa displaywithHDMI.And on disks and inmemory, for example, CRCs andother shift-register-sequence-based codes are pretty much universally used tooperateatthehighestpossibleratesanddensities.Shiftregistersaresoubiquitous,it’salittledifficulttoestimatejusthowmany

ofthemare inuse,andhowmanybitsarebeinggeneratedbythem.Thereareperhaps 10 billion computers, slightly fewer cellphones, and an increasingnumberofbillionsofembeddedandIoT(“InternetofThings”)devices. (Even

many of the billion cars in the world, for example, have at least 10microprocessors in them.) At what rate are the shift registers running? Here,again,thingsarecomplicated.Incommunicationssystems,forexample,there’sa basic carrier frequency—usually in theGHz range—and then there’swhat’scalled a “chipping rate” (or, confusingly, “chip rate”) that says how fastsomething likeCDMA is done, and this is usually in theMHz range.On theotherhand,inbusesinsidecomputers,orinconnectionstoadisplay,allthedataisgoingthroughshiftregisters,atthefulldatarate,whichiswellintotheGHzrange.Soitseemssafetoestimatethatthereareatleast10billioncommunications

links,runningforatleast1/10billionseconds(whichis3years),thatuseatleast1 billion bits from a shift register every second—meaning that to date Sol’salgorithmhasbeenusedatleastanoctilliontimes.Isitreallythemost-usedmathematicalalgorithmideainhistory?Ithinkso.I

suspect the main potential competition would be from arithmetic operations.These days processors are doing perhaps a trillion arithmetic operations persecond—and such operations are needed for pretty much every bit that’sgeneratedbyacomputer.Buthowisarithmeticdone?Atsomelevelit’sjustadigital electronics implementation of the way people have done arithmeticforever.Buttherearesomewrinkles—some“algorithmicideas”—thoughthey’requite

obscure,excepttomicroprocessordesigners.JustaswhenBabbagewasmakinghisDifferenceEngine,carriesareabignuisance indoingarithmetic. (Onecanactually think of a linear-feedback shift register as being a system that doessomethinglikearithmetic,butdoesn’tdocarries.)Thereare“carrypropagationtrees” that optimize carrying. There are also little tricks (“Booth encoding”,“Wallacetrees”,etc.)thatreducethenumberofbitoperationsneededtodotheinnards of arithmetic. But unlike with LFSRs, there doesn’t seem to be onealgorithmicideathat’suniversallyused—andsoIthinkit’sstilllikelythatSol’smaximum-lengthLFSRsequenceideaisthewinnerformost-used.

CellularAutomataandNonlinearShiftRegistersEventhoughit’snotobviousatfirst,itturnsoutthere’saverycloserelationshipbetweenfeedbackshiftregistersandsomethingI’vespentmanyyearsstudying:cellular automata. The basic setup for a feedback shift register involvescomputingonebitatatime.Inacellularautomaton,onehasalineofcells,andat each step all the cells areupdated inparallel, basedon a rule that depends,

say,onthevaluesoftheirnearestneighbors.Toseehowthesearerelated,thinkaboutrunningafeedbackshiftregisterof

sizen,butdisplayingitsstateonlyeverynsteps—inotherwords,lettingallthebitsberewrittenbeforeonedisplaysagain.Ifonedisplayseverystepofalinear-feedbackshiftregister(herewithtwotapsnexttoeachother),asinthefirsttwopanelsbelow,nothingmuchhappensateachstep,exceptthatthingsshifttotheleft. But if one makes a compressed picture, showing only every n steps,suddenlyapatternemerges.

It’sanestedpattern,and it’sveryclose tobeing theexactsamepattern thatonegetswitha cellular automaton that takesacell and itsneighbor, andaddsthemmod2(orXORsthem).Here’swhathappenswiththatcellularautomaton,ifonearrangesitscellssothey’reinacircleofthesamesizeastheshiftregisterabove:

Atthebeginning,thecellularautomatonandshiftregisterpatternsareexactlythe same—though when they “hit the edge” they become slightly differentbecause the edges are handled differently. But looking at these pictures itbecomes less surprising that the math of shift registers should be relevant tocellular automata. And seeing the regularity of the nested patterns makes itclearerwhy theremightbeanelegantmathematical theoryofshift registers inthefirstplace.Typical shift registers used in practice don’t tend to make such obviously

regular patterns, though.Here are a few examples of shift registers that yieldmaximum-length sequences. When one’s doing math, like Sol did, it’s verymuchthesamestoryasforthecaseofobviousnesting.Butherethefactthatthetapsarefarapartmakesthingsgetmixedup,leavingnoobviousvisualtraceofnesting.

So how broad is the correspondence between shift registers and cellularautomata?Incellularautomatatherulesforgeneratingnewvaluesofcellscanbeanythingonewants.Inlinear-feedbackshiftregisters,however, theyalwayshave tobebasedonaddingmod2 (orXOR’ing).But that’swhat the “linear”partof“linear-feedbackshiftregister”means.Andit’salsoinprinciplepossibleto have nonlinear-feedback shift registers (NFSRs) that usewhatever rule onewantsforcombiningvalues.And in fact, once Sol had worked out his theory for linear-feedback shift

registers,hestartedinonthenonlinearcase.WhenhearrivedatJPLin1956hegotanactual lab,completewithracksof littleelectronicmodules.Sol toldmeeachmodulewasaboutthesizeofacigarettepack—andwasbuiltfromaBellLabsdesigntoperformaparticular logicoperation(AND,OR,NOT,…).Themodules could be strung together to implement whatever nonlinear-feedbackshift register onewanted, and they ran pretty fast—producing about amillionbits per second. (Sol toldme that someone tried doing the same thingwith ageneral-purposecomputer—andwhat took1 secondwith thecustomhardwaremodulestook6weeksonthegeneral-purposecomputer.)WhenSolhadlookedatlinear-feedbackshiftregisters,thefirstbigthinghe’dmanagedtounderstandwas their repetitionperiods.Andwithnonlinearonesheputmostofhiseffortintotryingtounderstandthesamething.Hecollectedallsortsofexperimentaldata.Hetoldmeheeventestedsequencesoflength2⁴⁵—whichmusthavetakena year. He made summaries, like the one below (notice the visualizations ofsequences,shownasoscilloscope-liketraces).Buthenevermanagedtocomeupwithanykindofgeneraltheoryashehadwithlinear-feedbackshiftregisters.

It’s not surprising he couldn’t do it. Becausewhen one looks at nonlinear-feedback shift registers, one’s effectively sampling the whole richness of thecomputational universe of possible simple programs. Back in the 1950s therewere already theoretical results—mostly based on Turing’s ideas of universalcomputation—aboutwhatprogramscouldinprincipledo.ButIdon’tthinkSoloranyoneelseeverthoughttheywouldapplytotheverysimple—ifnonlinear—functionsinNFSRs.Andintheenditbasicallytookuntilmyworkaround1981forittobecome

clearjusthowcomplicatedthebehaviorofevenverysimpleprogramscouldbe.My all-time favorite example is rule 30—a cellular automaton in which thevalues of neighboring cells are combined using a function that can berepresented as p+q+r+qr mod 2 (or p XOR (q OR r)). And, amazingly, Sollookedatnonlinear-feedbackshiftregistersthatwerebasedonincrediblysimilarfunctions—like, in 1959, p+r+s+qr+qs+rsmod 2. Here’s what Sol’s function(whichcanbethoughtofas“rule29070”),rule30,andacoupleofothersimilarruleslooklikeinashiftregister:

Andhere’swhattheylooklikeascellularautomata,withoutbeingconstrainedtoafixed-sizeregister:

Ofcourse,Solnevermadepictureslikethis(anditwould,realistically,havebeenalmostimpossibletodosointhe1950s).Instead,heconcentratedonakindofaggregatefeature:theoverallrepetitionperiod.Sol wondered whether nonlinear-feedback shift registers might make good

sourcesof randomness.Fromwhatwenowknowabout cellular automata, it’scleartheycan.Andforexampletherule30cellularautomatoniswhatweusedtogeneraterandomnessforMathematicafor25years(thoughwerecentlyretiredit in favor of a more efficient rule that we found by searching trillions ofpossibilities).Soldidn’ttalkaboutcryptographymuch—thoughIsuspecthedidquiteabit

of government work on it. He did tell me though that in 1959 he’d found a“multi-dimensional correlation attack on nonlinear sequences”, though he saidthat at the time he “carefully avoided stating that the application was tocryptanalysis”.The fact is that cellular automata like rule 30 (andpresumablyalso nonlinear-feedback shift registers) do seem to be good cryptosystems—thoughpartlybecauseofconfusionsaboutwhetherthey’resomehowequivalenttolinear-feedbackshiftregisters(they’renot),they’veneverbeenusedasmuchastheyshould.Beingahistoryenthusiast,I’vetriedoverthepastfewdecadestoidentifyall

precursorstomyworkon1Dcellularautomata.2Dcellularautomatahadbeenstudiedabit,buttherewasonlyquitetheoreticalworkonthe1Dcase,togetherwith a few specific investigations in the cryptography community (that I’veneverfullyfoundoutabout).Andintheend,ofallthethingsI’veseen,IthinkSolGolomb’snonlinear-feedbackshiftregisterswereinasenseclosesttowhatIactuallyendedupdoingaquartercenturylater.

PolyominoesMentionthename“Golomb”andsomepeoplewill thinkofshiftregisters.Butmanymorewill thinkofpolyominoes.Soldidn’t inventpolyominoes—thoughhe did invent the name. But what he did was to make systematic what hadappearedonlyinisolatedpuzzlesbefore.Themain question Sol was interested in was how andwhen collections of

polyominoes can be arranged to tile particular (finite or infinite) regions.Sometimes it’s fairly obvious, but often it’s very tricky to figure out. Solpublished his first paper on polyominoes in 1954, but what really launchedpolyominoes into the public consciousness was Martin Gardner’s 1957MathematicalGamescolumnontheminScientificAmerican.AsSolexplained

in the introduction tohis1964book, theeffectwas thatheacquired“a steadystream of correspondents from around the world and from every stratum ofsociety—board chairmen of leading universities, residents of obscuremonasteries,inmatesofprominentpenitentiaries…”Gamecompaniestooknoticetoo,andwithinmonths,forexample,the“New

Sensational Jinx Jigsaw Puzzle” had appeared—followed over the course ofdecadesbya long sequenceofotherpolyomino-basedpuzzles andgames (no,the sinister bald guy doesn’t look anything like Sol):

Sol was still publishing papers about polyominoes 50 years after he firstdiscussedthem.In1961heintroducedgeneralsubdividable“rep-tiles”,whichitlater became clear canmake nested, fractal (“infin-tile”), patterns. But almosteverythingSoldidwithpolyominoes involved solving specific tilingproblemswiththem.Forme, polyominoes aremost interesting not for their specifics but for the

examples they provide of more-general phenomena. One might have thoughtthatgivenafewsimpleshapesitwouldbeeasytodecidewhethertheycantilethe whole plane. But the example of polyominoes—with all the games andpuzzlestheysupport—makesitclearthatit’snotnecessarilysoeasy.Andinfactit was proved in the 1960s that in general it’s a theoretically undecidableproblem.If one’s only interested in a finite region, then in principle one can just

enumerateallconceivablearrangementsoftheoriginalshapes,andseewhetheranyofthemcorrespondtosuccessfultilings.Butifone’sinterestedinthewhole,infinite plane then one can’t do this.Maybe onewill find a tiling of size onemillion,butthere’snoguaranteehowfarthetilingcanbeextended.ItturnsoutitcanbelikerunningaTuringmachine—oracellularautomaton.

You start froma lineof tiles.Then thequestionofwhether there’s an infinitetiling is equivalent to thequestionofwhether there’s a setup for someTuringmachine that makes it never halt. And the point then is that if the Turingmachineisuniversal(sothatitcanineffectbeprogrammedtodoanypossiblecomputation) then thehaltingproblem for it canbeundecidable,whichmeansthatthetilingproblemisalsoundecidable.Ofcourse,whetheratilingproblemisundecidabledependsontheoriginalset

ofshapes.Andformeanimportantquestionishowcomplicatedtheshapeshavetobe so that theycanencodeuniversalcomputation,andyieldanundecidabletiling problem. Sol Golomb knew the literature on this kind of question, butwasn’t especially interested in it. But I start thinking about materials formedfrompolyominoeswhosepatternof“crystallization”canineffectdoanarbitrarycomputation, or occur at a “melting point” that seems “random” because itsvalueisundecidable.Complicated, carefully crafted sets of polyominoes are known that in effect

support universal computation. But what’s the simplest set—and is it simpleenoughthatonemightrunacrossbyaccident?Myguessisthat—justlikewithotherkindsofsystemsI’vestudiedinthecomputationaluniverse—thesimplestsetisinfactsimple.Butfindingitisverydifficult.Aconsiderablyeasierproblemistofindpolyominoesthatsuccessfullytilethe

plane,butcan’tdosoperiodically.RogerPenrose(ofPenrosetilesfame)found

an example in 1994.MybookANewKindof Science gave a slightly simplerexamplewith3polyominoes:

TheRestoftheStoryBythetimeSolwasinhisearlythirties,he’destablishedhistwomostnotablepursuits—shift registers and polyominoes—and he’d settled into life as auniversityprofessor.Hewasconstantlyactive,though.Hewrotewhatendedupbeing a couple of hundred papers, some extending his earlier work, somestimulatedbyquestionspeoplewouldaskhim,andsomewritten, it seems, forthe sheerpleasureof figuringout interesting thingsaboutnumbers, sequences,cryptosystems,orwhatever.Shift registers and polyominoes are both big subjects (they even each have

their own category in the AMS classification of mathematical publicationtopics).Bothhavehadacertaininjectionofenergyinthepastdecadeortwoasmoderncomputerexperimentsstartedtobedoneonthem—andSolcollaboratedwithpeopledoingthese.Butbothfieldsstillhavemanyunansweredquestions.Evenforlinear-feedbackshiftregisterstherearebiggerHadamardmatricestobefound. And very little is known even now about nonlinear-feedback shiftregisters.Not tomentionall the issuesaboutnonperiodicandotherwiseexoticpolyominotilings.Solwasalways interested inpuzzles,bothwithmathandwithwords.Fora

whilehewroteapuzzlecolumnfortheLosAngelesTimes—andfor32yearshewrote “Golomb’s Gambits” for the Johns Hopkins alumni magazine. Heparticipated inMegaIQ tests—earninghimselfa trip to theWhiteHousewhenheanditschiefofstaffhappenedtobothscoreinthetopfiveinthecountry.Hepoured immenseeffort intohisworkat theuniversity,notonly teaching

undergraduate courses andmentoring graduate students but also ascending theranksofuniversityadministration(presidentof thefacultysenate,viceprovostfor research, etc.)—and occasionally opining more generally about universitygovernance(forexamplewritingapaperentitled“FacultyConsulting:ShouldItBe Curtailed?”; answer: no, it’s good for the university!) At USC, he wasparticularlyinvolvedinrecruiting—andoverhistimeatUSChehelpeditascendfromaschoolessentiallyunknowninelectricalengineeringtoonethatmakesitontolistsoftopprograms.Andthentherewasconsulting.Hewasmeticulousatnotdisclosingwhathe

didforgovernmentagencies,thoughatonepointhedidlamentthatsomenewlypublished work had been anticipated by a classified paper he had written 40yearsearlier. In the late1960s—frustrated thateveryonebuthimseemedtobeselling polyomino games—Sol started a company called Recreational

Technology, Inc. It didn’tgoparticularlywell, butone sideeffectwas thathegot involved in business with Elwyn Berlekamp—a Berkeley professor andfellowenthusiastofcoding theoryandpuzzles—whomhepersuaded to start acompany calledCyclotomics (in honorof cyclotomicpolynomials of the formxⁿ–1)whichwas eventually sold toKodak for a respectable sum. (Berlekampalsocreatedanalgorithmic tradingsystemthathesold toJimSimonsand thatbecameastartingpoint forRenaissanceTechnologies,nowoneof theworld’slargest hedge funds.) More than 10,000 patents refer to Sol’s work, but Solhimself got only one patent: on a cryptosystem based on quasigroups—and Idon’tthinkheeverdidmuchtodirectlycommercializehiswork.Sol was for many years involved with the Technion (Israel Institute of

Technology)andquitedevoted to Israel.Hecharacterizedhimself asan“non-observant orthodox Jew”—but occasionally did things like teach a freshmanseminar on theBookofGenesis, aswell asworking on decoding parts of theDeadSeaScrolls.Sol and his wife traveled extensively, but the center of Sol’s world was

definitelyLosAngeles—hisofficeatUSC,and thehouse inwhichheandhiswifelivedfornearly60years.Hehadacircleoffriendsandstudentswhoreliedonhimformanythings.Andhehadhisfamily.HisdaughterAstridremainedalocalpersonality,evenbeingportrayedinfictionafewtimes—asastudentinaplayaboutRichardFeynman(shesatasadrawingmodelforhimmanytimes),andasacharacterinanovelbyafriendofmine.BeatricebecameanMD/PhDwho’s spent her career applying an almostmathematical level of precision tovarious kinds of medical reasoning and diagnosis (Gulf War illness, statineffects, hiccups, etc.)—even as she often quotes “Beatrice’s Law”, that“everything in biology is more complicated than you think, even taking intoaccountBeatrice’sLaw”.(I’mhappytohavemadeat leastonecontribution toBeatrice’s life: introducing her to her husband, now of 26 years, TerrySejnowski,oneof the foundersofmoderncomputationalneuroscience.) In theyears I knew Sol, there was always a quiet energy to him. He seemed to beinvolvedinlotsofthings,evenifheoftenwasn’tparticularlyforthcomingaboutthe details. Occasionally I would talk to him about actual science andmathematics; usually he was more interested in telling stories (often veryengagingones)aboutpersonalitiesandorganizations(“Canyoubelievethat[in1985]afternotgoingtoconferencesforyears,ClaudeShannonjustshowedupunannouncedatthebarattheannualinformationtheoryconference?”,“DoyouknowhowmuchtheyhadtopaythepresidentofCaltechtogethimtomovetoSaudiArabia?”,etc.)Inretrospect,IwishI’ddonemoretogetSolinterestedinsomeofthemathquestionsbroughtupbymyownwork.Idon’tthinkIproperly

internalized theextent towhichhe likedcrackingproblemssuggestedbyotherpeople.Andthentherewasthematterofcomputers.Despiteallhiscontributionsto the infrastructure of the computational world, Sol himself basically neverseriouslyusedcomputers.Hetookparticularprideinhisownmentalcalculationcapabilities.And he didn’t really use email until hewas in his seventies, andnever used a computer at home—though, yes, he did have a cellphone. (Atypical email from him was short. I had mentioned last year that I wasresearching Ada Lovelace; he responded: “The story of Ada Lovelace asBabbage’s programmer is so widespread that everyone seems to accept it asfactual, but I’ve never seen original sourcematerial on this.”) Sol’s daughtersorganized a party for his 80thbirthday a fewyears ago, creating an invitationwithcharacteristicmathematicalfeatures:

Solhadafewmedicalproblems,thoughtheydidn’tseemtobeslowinghimdownmuch.Hiswife’s health, though,was failing, and a fewweeks ago herconditionsuddenlyworsened.SolstillwenttohisofficeasusualonFriday,butonSaturdaynight,inhissleep,hedied.HiswifeBodiedjusttwoweekslater,twodaysbeforewhatwouldhavebeentheir60thweddinganniversary.Though Sol himself is gone, the work he did lives on—responsible for an

octillion bits (and counting) across the digital world. Farewell, Sol. And onbehalfofallofus,thanksforallthosecleverlycreatedbits.