Ground and Explanation in Mathematics | UNC Philosophy

volume19,no.33 august2019

Ground and Explanation

in Mathematics

Marc LangeUniversity of North Carolina at Chapel Hill

© 2019 MarcLangeThis work is licensed under a Creative Commons

Attribution-NonCommercial-NoDerivatives 3.0 License. <www.philosophersimprint.org/019033/>

I ncreasedattentionhasrecentlybeenpaidtothefactthatinmath-ematicalpractice,certainmathematicalproofsbutnotothersarerecognized as explaining why the theorems they prove obtain

(Mancosu2008;Lange2010,2015a,2016;Pincock2015).Such“math-ematicalexplanation” ispresumablynotavarietyof causalexplana-tion.Inaddition,theroleofmetaphysicalgroundingasunderwritingavarietyofexplanationshasalsorecentlyreceivedincreasedattention(CorreiaandSchnieder2012;Fine2001,2012;Rosen2010;Schaffer2016).Accordingly,it isnaturaltowonderwhethermathematicalex-planationisavarietyofgroundingexplanation.Thispaperwillofferseveralargumentsthatitisnot.

One obstacle facing these arguments is that there is currentlyno widely accepted account of either mathematical explanation orgrounding.Inthecaseofmathematicalexplanation,Iwilltrytoavoidthisobstaclebyappealingtoexamplesofproofsthatmathematiciansthemselveshavecharacterizedasexplanatory(orasnon-explanatory).I will offer many examples to avoid making my argument toodependent on any single one of them. I will also try to motivatethese characterizations of various proofs as (non-) explanatory byproposinganaccountofwhatmakesaproofexplanatory.Inthecaseof grounding, Iwill try to stickwith features of grounding that arerelativelyuncontroversialamonggroundingtheorists.ButIwillalsolookbrieflyathowsomeofmyargumentswouldfareunderalternativeviewsofgrounding. Ihopeat least to reveal somethingaboutwhatgroundingwouldhavetolooklikeinorderforatheorem’sgroundstomathematicallyexplainwhythattheoremobtains.Ihopetherebytoidentifysomeofthedifficultiesconfrontingatoo-quickidentificationofgroundingrelationsasunderwritingexplanationinmathematics.

1. Proofs that Identify Grounds versus Proofs that Explain.

Supposeyouwritedownallofthewholenumbersfrom1to99,999.Howmanytimeswouldyouwritedownthedigit7?Theanswerturnsout tobe 50,000 times.That is a striking result since (as youhaveprobablynoticed)50,000ishalfof100,000,whichisjustonemorethan99,999.

marclange Ground and Explanation in Mathematics

philosophers’imprint –2– vol.19,no.33(august2019)

outtobealmostexactlyhalfof99,999istreatedbythisproofasifitjustturnedoutthatwayasamatterofmathematicalcoincidence.Ofcourse,itismathematicallynecessarythatthenumberof7’sisalmostequaltohalfof99,999;thisresultisobviouslynotamatterofchance.Nevertheless,asfarasthisproofshows,thefactthatthenumberof7’s isalmostequal tohalfof99,999arises fromtwounrelated facts:thatthereare50,0007’sandthat50,000ishalfof100,000(onemorethan 99,999). As far as this proof reveals, one of these facts is notresponsiblefortheotherandthereisnoimportantcommonreasonbehindboth.2 Theproof arrives at thenumberof 7’swithoutdoinganything like takinghalfof100,000.Thus, theproofmakes it seemlikesheerhappenstancethatthenumberof7’sisverynearlyhalfof99,999.

Bycontrast,considerthisproofofthesameresult:

Include0amongthenumbersunderconsideration—thiswillnotchangethenumberoftimesthedigit7appears.Supposeallthewholenumbersfrom0to99,999(100,000ofthem)arewrittendownwithfivedigitseach,e.g.,1306iswrittenas01306.Allpossiblefive-digitcombinationsare nowwritten down, once each. Because every digitwill take every position equally often, every digitmustoccurthesamenumberoftimesoverall.Sincethereare100,000numberswithfivedigitseach—thatis,500,000digits—eachofthe10digitsappears50,000times.Thatis,100,000x5/10=50,000.(DreyfusandEisenberg1986,3)3

Thisproofdoesnotspecifywhereinthelistofnumberseach7appears;itdoesnotgivetheresult’sground.Nevertheless,thisproofismoreilluminating than thefirstone in that itexplains why thenumberof

2. Therearemanyexamplesofmathematicalcoincidences,suchasthefactthat(inbase10)the13thdigitofπisequaltothe13thdigitofe.Forthatexampleandmoreon“mathematicalcoincidence”,seeLange(2010,2016).

3. ThankstoManyaSundströmforcallingthisarticletomyattention.

Onewaytoprovethisresultwouldbetolistallofthewholenumbersfrom1to99,999;tonote,foreachentryinthelist,thenumberoftimesthedigit 7 appears in it; and to add thesenumbers of appearances,arrivingat50,000intotal. Thisproof,Imaintain,givesgroundsofthefactthat7appears50,000timesintotalfrom1to99,999.Here,Iaimtobeusing thenotionof “ground” in themanner intendedbyFine(2001,2012)andmanyothergroundingtheorists.Onthisview,afact’sgroundsarewhateveritisinvirtueofwhichthatfactobtains,andatruth-bearer(suchasaproposition)isgroundedinitstruth-makers.

In particular, by regarding the fact that 7 appears 50,000 timesinthelistfrom1to99,999asgroundedinfactsabouttheindividualappearancesof7’sinthatlist,Iammakingtheroughpresuppositionthat a mathematical fact is grounded by the atomic (or negatedatomic) truths to which one is led if one starts with that fact andmoves “downward” to logically simpler truths in an obvious way,suchasfromuniversalfacts(i.e.,factsexpressedbygeneralizations)to their instances and from conjunctions to their conjuncts.1 Thispresupposition seems to lie behind many plausible-lookingclaims about the grounds of mathematical facts. For instance, thispresuppositionmotivatesregardingthefactthateverynumberinthesequence31,331,…,3333331isprimeasgroundedbythefactthat31isprime,thefactthat…,andthefactthat3333331isprime.Likewise,thispresuppositionmotivatesregardingthefactthatthereisexactlyoneprimenumberbetween12and15(inclusive)asgroundedinthefact13 isprimebut12,14,and15arecomposite. Inthesameway, itmotivates regarding the fact that 7 appears 50,000 times in the listfrom1 to99,999asgrounded in factsabout theparticular locationswhere7appears(ordoesnotappear)inthatlist.

Let’sreturntotheponderousproofthattallieseveryappearanceofa7inthelistofnumbersfrom1to99,999.Thatthenumberof7’sturns

1. My thanks to a referee who urged me to be more explicit about thesepresuppositions concerning grounding and offered this formulation asa candidate for doing so. In the next section, I will consider briefly howmy arguments would fare under an alternative picture of the grounds ofmathematicalfacts.



has distinct full grounds in each of the disjuncts (e.g., Correia andSchnieder2012,17;Fine2012,58),thatprecedentdoeslittletosuggestthat the second proof and the first proof cite distinct full grounds,sincethefactbeingexplainedisnotadisjunction.Thesecondproofmight be construed as describing the result’s grounds (the decimalexpressionsofthewholenumbersfrom1to99,999)atahighlevelofabstraction:intermsofthedistributionofeachdigitinthelist,withoutspecifyingwhereinthelisteachappears.Butinthenextsection,asmythirdreplytotheobjection,wewillseeexampleswhereexplanatoryproofsarguablydonotevendescribetheresult’sgroundsatsomehighlevelofabstraction.(Thatisbecausetheexplanatoryproofsinthoseexamplesare“impure”.)

Inthispaper,Iwillarguethatthephenomenonondisplayinthisexample iswidespread:Manymathematicalproofsthatexplainwhysome result holds fail to present the groundsof the theorembeingprovedandmanymathematicalproofsthatpresentthegroundsofthetheorembeingprovedfailtoexplainwhythattheoremobtains.Iwillidentifyseveralreasonswhygroundsandexplanationstendtocomeapartinmathematics.

Tobegintoshowwhythisistypical,Iwillgiveanotherexample.5 ConsiderthesetwoTaylorseries:

1/(1–x2)=1+x2+x4+x6+…

1/(1+x2)=1–x2+x4 – x6+…

Theyhavethesameconvergencebehavior:Forrealx,eachconvergeswhen|x|<1butdivergeswhen|x|>1.Thatisaremarkablesimilarity,and it stands out especially strongly against the obvious differencebetween the two functions regardingx= 1:As theirgraphs (below)show,althoughonefunctiongoesundefinedatx=1,theotherfunctionbehavesquitesoberlythere.Thatthetwofunctions’Taylorserieshavethe same convergence behaviormight therefore seem to be utterly

5. ThisexamplehasbeendiscussedbySteiner (1978, 18−19),Waisman (1982,29−30),andWilson(2006,313−314).

7’s is almost exactlyhalf of 99,999. It revealswhere theone-half of100,000comesfrom:Thereare100,000numbersbeingwrittendown,thereare5digits ineachnumber, thereare 10options (inbase 10)foreachofthesedigits,and5dividedby10isone-half.Byvirtueofthisproof,Isuggest,it isnomathematicalcoincidencethatthetotalnumberof7’sisalmostexactlyhalfof99,999.

In section3, Iwillhavemore to saymoreaboutwhatmakes thesecondproofexplanatory,bycontrastwiththefirstproof.Fornow,myaimissimplytopromptyoutoappreciatethatthefirstproof(unlikethesecond)doesnotexplainwhytheresultholds,despitegivingtheresult’s ground, whereas the second proof explains why the resultholdsdespitefailingtoidentifytheresult’sground.

It might be objected that although the first proof displays theresult’s grounds, this fact does not preclude the second proof fromalsodoingso;agivenfactcanhavemanycompletesetsofgrounds.Forinstance,agivenfact’sgroundscanthemselveshavegrounds,andthelattermaythenqualifyasgroundsofthegivenfact.4Ihavethreereplies to this objection. Firstly, part ofwhat I regard this exampleas showing is that the grounds of a mathematical theorem do notautomatically(thatis,simplybyvirtueofbeingitsgrounds)explainit. Even if the secondproof alsogives the result’s grounds, that thefirstproofgivesitsgroundswithoutexplainingwhytheresultholdssufficestoshowthatatheorem’sgroundsdonotautomaticallyexplainit. Ifmathematicalexplanationworkedby tracing theexplanandumtoitsgrounds,thenitwouldbepuzzlingwhythefirstproofdoesnotqualifyasamathematicalexplanation.Secondly,itisnotobviouswhyoneshouldinsistthatthesecondproofgivestheresult’sgrounds.Forinstance,itisnotthecasethatthesecondproofgivesgroundsofanyofthevariousfacts(namely,thatthedigit7appearsnowherein1,…appears twice in7007,…)that thefirstproofdisplaysasgroundingtheresult.Likewise,althoughplausiblythedisjunctionoftwotruths

4. Virtuallyallaccountsofgroundingregardgroundingastransitive(CorreiaandSchnieder2012,32;Fine2012,56)thoughseenote16foronecircumstanceinwhich(accordingtoSchaffer2016)transitivitymaybeviolated.



A ground for the fact that both series have this convergencebehavioristhatthefirstseriesexhibitsthisbehaviorandthesecondseriesdoes,too.Aconjunctionisgroundedinitsconjuncts.6Aproofoftheconjunctionthatgivesentirelyseparateproofsofitstwoconjunctsdoesnothingtosuggestthatitisnocoincidencethatthetwofunctionssharethisconvergencebehavior.

However,itisinfactnocoincidence,aswediscoverbylookingatmatterson thecomplexplane.The“radiusofconvergence theorem”says that a complex Taylor serieswill converge at all and only thepointswithinacertainradiusoftheoriginonthecomplexplane:

Radius of convergence theorem:Foranypowerseries∑anzn (fromn=0to∞),eitheritconvergesforallcomplexnum-bersz,oritconvergesonlyforz=0,orthereisanumberR>0suchthatitconvergesif|z|<Randdivergesif|z|>R.

ThetwoTaylorseriessharethesameconvergencebehaviorbecausethe two functions have another feature in common: They both goundefinedatsomepointon theunitcirclecenteredat theoriginofthecomplexplane.ThissimilarityexplainswhythetwoTaylorseriesbehavealike.AsmathematicianMichaelSpivaksays,aproofof theconvergence theorem “helps explain the behavior of certain Taylorseriesobtainedforrealfunctions”(Spivak1980,528),suchasintheexampleIhavementioned.Thus,aproofthatdisplaysthegroundofthismathematicalresultfailstoexplainwhytheresultholdsbecausesuch a proof fails to give the two functions’ common convergencebehaviorthesameprooffromanotherfeaturethatthetwofunctionssharewhen,infact,theycanbesoproved.

WiththeTaylorseriesexampleandthetallying7’sexample,Ihaveaimed(i)toarguethatamathematicalfact’sgroundsdonot,simplybyvirtueofgrounding it, therebyexplainwhythat factobtainsand

6. AsCorreiaandSchnieder (2012, 17) report, this fact abouta conjunction’sgroundsiswidelyacknowledgedinthegroundingliterature(e.g.,Fine2012,63;Schaffer2016,53).ButshortlyIwilllookbrieflyatanalternativeview.

coincidental, considering that the two functions appear to be quitedissimilarinotherrespects.

f(x)=1/(1+x2)

f(x)=1/(1−x2)



Iwill argue that amathematical fact’s groundsdonot, by virtueofgroundingit,explainwhyitobtains.Wecouldneverthelessinsistthat grounds automatically explain in mathematics by making abare stipulation: that grounds supply a special kind of explanation(“groundingexplanations”).Butthisstipulationwouldnotrevealthatgrounding is connected to explanation of the kind that figures inmathematicalpractice.Thatkindofexplanationwillbemyexclusivefocusthroughout.

2. Purity, Brute Force, and Unification.

Theproof that explainswhy the twoTaylor seriesexhibit the sameconvergencebehaviorplacesthetwoTaylorseriesinabroadercontextbyputting the two functions on the complexplane. By introducingimaginarynumbers,theproofintroducesconceptsexogenoustothetheorem being proved, since the original two Taylor series involveonlyrealnumbers.Thus,one lessonof theTaylorseriesexample isthatanimpureproofcansometimesexplainwhatitproves.

Bya“pure”proof,Imeanroughlyaproofusingonlytheconceptsthatare(inthesenseIwillclarifybelow)“intrinsic”(asmathematiciansputit)tothetheorembeingprovedsothat“theresourcesofproof[are]restrictedtothosewhichdetermine[thetheorem’s]content”(Detlefsen2008, 193).8 Presumably, a proof giving the grounds of the theorembeingprovedmustbeapureproof.Animpureproofiscommonlysaidtointroduce“extraneous”conceptsthatis,presumably,alientothe

anyargumentIwillgive)thatforthetheoremthatthequinticisunsolvable,anexplanatoryproofdoesnotsupplythetheorem’sground.

8. Formoreonpurity,seeDetlefsen(2008),DetlefsenandArana(2011),andreferencestherein.Howtomake“purity”preciseisatopicforanotherpaper(thoughseethenextparagraph).Differentreadingsof“purity”couldcomeapart in some cases, but Iwill argue that even under a liberal reading of“purity”thatdeemstheappealtoimaginarynumberstobeapureproofofthefactaboutthetwoTaylorseries,thisproof’s“purity”doesnotcontributetomakingitexplanatory.

(ii)tosuggestthatthesetwoexamplesaretypicalinthatoftentimes,amathematical proof that specifies a fact’s grounds fails to explainwhy that fact obtainswhereas any explanationof the fact doesnotspecifyitsground.Noneoftheseclaims(forwhichIwillarguefurther)suggeststhatnoproofidentifyingsomemathematicalfact’sgroundseverexplainswhythatfactobtains.ButIcontendthateveninsuchacase,whatmakesagivenproofexplanatoryisnot thatitidentifiesthegroundsofwhatitproves.

In section 2, I will give some further examples that distinguishproofsthatexplainfromproofsthatspecifygrounds,andIwillgivesome further reasonswhy explaining and grounding tend to comeapart.Inparticular,Iwillfocusonthefactthatexplanatoryproofsneednotexhibitpurity,tendnottobebruteforce,andoftenunifyseparatecasesbyidentifyingcommonreasonsbehindthemevenwhenthosecaseshavedistinctgrounds.Insection3,Iwillsketchanaccountofwhatmakesaproofexplanatory.IwillthenusethataccounttodefendthemoralsIhavedrawnfromtheexamplesIhavealreadygiven.Insection4,Iwillusethataccountofexplanationinmathematicstogivetwo further reasonswhyproofs that specifygrounds tendnot tobeproofsthatexplain.Inparticular,Iwillarguethatformanytheorems,there isnoproof that explainswhy theyobtain, even though thesetheorems have grounds. Iwill also argue that the role that contextplaysinmakingaproofexplanatoryensuresthatexplanatorypowerfrequently diverges from grounding (since, unlike having a certainexplanation, a theorem’s having certain grounds is insensitive tocontext).

Thequestionthispaperaimstoaddressiswhetheraproof’spowertoexplainwhatitprovesstandsinanygeneralrelationtoitsspecifyingthegroundofwhatitproves.Accordingly,thispapertakesforgrantedthatinmathematicalpractice,certainproofsdifferfromothersinthedegreetowhichtheyexplainwhyagiventheoremholds.7

7. For arguments that there is such a distinction (or matter of degree) ofexplanatorinessinmathematicalpractice,seeMancosu(2008),Lange(2016),andreferencestherein.Pincock(2015,11)givesabriefargument(distinctfrom



proofdrawsonlyonwhatis“intrinsic”tothetheorembeingproved,whichautomaticallyincludesthetheorem’sgrounds.

ItmightbeobjectedthatalthoughtheTaylorseriestheoremconcernstwoseriescomposedentirelyofrealnumbers,aproofthatappealstoimaginarynumbersshouldstillberegardedaspurebyvirtueofgivinginformationaboutthetheorem’sgroundsbecausefactsaboutthereals and facts about the imaginaries and all of the other complexnumbershavethesamegrounds:infactsaboutsets.However,eveniffactsabouttherealsandfactsabouttheothercomplexnumbersarealikegroundedinsettheory,thosegroundsarenotgivenbytheproofoftheTaylorseriestheoremthatappealstotheradiusofconvergencetheorem.ThatproofdoesnotproceedbyexpressingthetwoTaylorseriesinset-theoreticterms,noristheradiusofconvergencetheoremtypically(orever,asfarasIknow)provedinthatway.Furthermore,wedonothavetoknowthatfactsabouttherealsandfactsabouttheothercomplexnumbersareallgroundedinsettheory(orevensimplythattheyhavethesamegrounds) inorder toappreciate theexplanatorypoweroftheproofoftheTaylorseriestheoremthatproceedsthroughthe complexnumbers. (In thenext section, Iwill sketchaproposalregarding the source of a proof’s explanatory power that accountsforourcapacitytorecognizethisproof’sexplanatorypowerwithoutourhavingtoregardallfactsaboutrealsandothercomplexnumbersas grounded in the sameplace.)Thatwe can recognize theproof’sexplanatory power without recognizing that all complex numbersarealikegroundedinsettheorysuggeststhattheproof’sexplanatorypowerdoesnotdependontheexistenceofthatcommonset-theoreticgroundmakingtheproofqualifyaspure.10

Thatanimpureproofcanbeexplanatory(evenwherethereisapureproofof the same theorem) shows that at least somemathematicalexplanationsdonotworkbysupplyinginformationaboutthegroundsofthetheorembeingexplained.Insofarasmathematicalexplanations

10. Forapowerfulargumentthatthereductionofarithmetictosettheoryisnotexplanatory,seeD’Alessandro(2018).

theoremanditsgrounds.Nevertheless,mathematiciansrecognizethatanimpureproofisinsomecasesmoreexplanatorythanapureproof.9

Itmightbeobjectedthataproofgivingthegroundsofthetheorembeingprovedneednotbeapureproof.Aconceptcouldfailtofigureinthetheorem,inthedefinitionsoftheconceptsfiguringinthetheorem,in the definitions of the concepts figuring in those definitions, andsoon,andneverthelessfigure in the theorem’sground (justasonecouldhavemasteredtheconceptsofheat,temperature,andsoforthwithout having any grasp of the concept of the kinetic energy ofrandommolecularmotion).However,itseemstomethatas“purity”isunderstoodinmathematics,ifaconceptfiguresinthegroundofsomemathematical fact, then a pureproof of that fact can appeal to thatconcept.A“pure”proofdoesnotmeanaproofusingonlytheconceptsthatfigure inthetheorembeingproved,or in thedefinitionsof theconceptsfiguringinthetheorem,orinthedefinitionsoftheconceptsfiguring in thosedefinitions,andsoonor just theresources thatone needs to master in order to understand the theorem beingproved.Evenifonecanmasterthenotionofarealnumberwithouthavingmasteredthecoreconceptsofsettheory,apureproofofafactaboutrealnumberscanappealtosetsiffactsaboutrealnumbersaregrounded in factsaboutsets. Indeed, thedemandsofpurityaresometimes understood in terms of grounding, as when Bolzano isunderstoodasholdingthatpureproofsoftheoremsinanalysis(e.g.,of the intermediate value theorem) cannot appeal to geometricalconcepts because truths of analysis are not grounded in truths ofgeometry(DetlefsenandArana2011,4−5;cf.Bigelow1988,31).Apure

9. BesidestheTaylorseriescase,therearemanyotherexampleswhereimpureproofsexplainwhatpureproofsfailtoexplain.Forexample,Lange(2015a)hasarguedthatmathematiciansregardDesargues’theoremintwo-dimensionalEuclideangeometryasexplainedbyaproofthatnotonlyintroducesathirddimension that isexogenous to the theorem,butalsoputs the theoreminthecontextofprojectivegeometry,whichsupplementsEuclideangeometrywithpointsatinfinity(whereparallelsmeet).ThisadditionaldimensionandthesepointsatinfinityareextraneoustoanytheoremofEuclideangeometry.Theyarenotinvolvedinmakingittrue.(However,thecaseiscomplicated;seeHallett2008,222−228.)



therearepureproofsofthesametheorem,showsthatevenundertheview that a fact canbegrounded inessences, it oftenhappens thatamathematicalproofspecifyingafact’sgroundsfailstoexplainwhythatfactobtains,whereasanexplanationofthefactdoesnotspecifyitsground.

To suggest how common it is for mathematicians to regard animpureproofasexplanatory,Iwouldliketoconsiderbrieflyanothertheorem in Euclidean geometry: that for any two circles, there is alinesuchthat foranypointonthe line, the tangent fromthatpointtoonecircleisequaltothetangentfromthatpointtotheothercircle.This resultholdswhether the twocircles intersecton theEuclideanplaneintwopoints,inonepoint,ornowhere.Thesethreecasesmustbeprovedseparatelyiftheproofistobepure.Suchaproofbycasesmakes itappear tobeacoincidence thatall threecasesarealike inhavingsuchaline.Butinfact,itisnocoincidence.Whatmakesitnocoincidenceis(roughlyspeaking)theexistenceofacommonproofofallthreecases.Butthatproofisimpure:Itinvolvespointswithimaginarycoordinates,whichfallnowhereontheEuclideanplane.Thecommonproofiswidelyregardedasmakingthetheoremnocoincidenceandasexplainingwhythetheoremholds.12Nevertheless,sincethisproofusesimaginarycoordinates,itfailstoidentifythetheorem’sgrounds.

Aproofthatproceedsbydividingthetheoremintothreecasesisusuallycarriedout incoordinategeometry. Itproceedsbyassigning

12. The“commonproof”proceedsbyfindingthelinethroughthetwopointsatwhichthecirclesintersectandthenshowingthatforanypointonthisline,the tangent toonecircleequals the tangent to theother. If the twocirclesintersectatnoEuclideanpoints, thenwhenwesolve for thetwopointsatwhichtheyintersect,wearriveatpointswithimaginarycoordinates.Forin-stance,ifonecircleiscenteredat(0,20)withradius16andtheothercircleiscenteredat(0,−15)withradius9, thentheirequationsarex2+(y−20)2=162andx2+(y+15)2=92,whichhavecommonsolutions(12i,0)and(−12i,0).Althoughtheseintersectionpointshaveimaginaryx-coordinates,theproofproceeds in exactly the sameway as it doeswhen the intersectionpointsareon theEuclideanplane:The twopoints lieon the liney=0,which isontheEuclideanplane,andforanypointonthisline,thetwotangentsareequal(Sawyer1955,180−181).IagreewithSawyer(1943,232)thatcomplexnumbersreveal“thereasonsforresultswhichhadpreviouslyseemedquiteaccidental”.

areoftenimpure,mathematicalexplanationsoftenfailtospecifythegroundsofthetheorembeingexplained.11

Arguably, a pure proof is roughly a proof that proceeds entirelyfrom factsabout theessencesof themathematical itemsfiguring inthetheorembeingproved.Forinstance,thefactthatn!/k!(n−k)!isaninteger(fornaturalnumbersnandkwheren ≥ k)couldbeprovedbyshowingthattheformulagivesthenumberofwaystochooseasubsetofkfromasetofn.Butbecausethisproofdoesnotproceedentirelyinarithmeticterms,it isregardedbymathematiciansasimpure;theessencesofallof the itemsfiguring in the theoremarearithmetical.Somephilosophershaveheldthatifafactfollowsfromtheessencesoftheitemsfiguringinit,thenthatfactisgroundedinfactsabouttheiressences. For instance,Rosen (2010, 119) suggests that the fact thateverytrianglehasthreeanglesisgroundedinthefactthathavingthreeanglesispartofwhatitistobeatriangleisessentialtotriangularity.Thisviewisanalternative(exclusively for thoseuniversal facts thatarenecessary) to thepictureof thegroundsofuniversal facts that Imentionedintheprevioussection,accordingtowhichuniversalfactsaregroundedintheirinstances.

Nevertheless, theexplanatoryproofof theTaylor series theoremdoesnotproceedentirelyfromtheessencesoftheitemsfiguringinthetheorem,becausethoseessencesdonotinvolveimaginarynumbers.Thatitisnotunusualforexplanatoryproofstobeimpure,evenwhen

11. Although a proof’s purity need not contribute to its explanatory power,purityisneverthelessafeaturethatmathematiciansoftenseekandprizeinproofs.Purityandexplanatorypowerarebothvirtuesinproofs,asarebrevity,generalizability,simplicity,visualizability,theoreticalfruitfulness,pedagogicvalue,andsoforth.Withsomany(perhapsincommensurable)virtues,theremaybenosenseinwhichoneproof is“better”,all thingsconsidered,thananother.

Nothing I have said has ruled out the possibility that in some cases, aproof’spuritydoes contribute to itsexplanatorypower or, at least, thatsearchingforapureproofisausefulheuristicforfindinganexplanatoryproof.(DetlefsenandArana(2011,8,fn.31)giveanexamplewhereapureproofissoughtandthissearchiscitedasapossiblemeansforfindinganexplanatoryproof.)Inthesameway,thesearchforavisualizableproofmaysometimesbeausefulheuristicforfindinganexplanatoryproof.Nevertheless,purity(andvisualizability)wouldremaindistinctvirtuesfromexplanatorypower.



motion).A separatederivation fromother fundamental laws showsthatelectrostatic forcesconserveenergy.Theseseparatederivationsidentifythemorefundamentallawsinvirtueofwhichitistruethateachofthesetwointeractionsconservesenergy.Thesemorefundamentallawsarethegroundsofthefactthatthetwoforcesconserveenergy.But these two separate derivations treat it as a coincidence thatthese two forces share this feature. In fact, many physicists havemaintained that it isno coincidence:The two forcesboth conserveenergybecauseeveryforcehasgottoconserveenergy(Lange2016).Energyconservationarisesfromafundamentalspacetimesymmetrymeta-lawthatconstrainsthefundamentalforcelaws(requiringthemall tobe invariantunderarbitrary time translations) so that (withina Lagrangian framework) every fundamental force must conserveenergy.Becausegravitational andelectrostatic forcesboth conserveenergyforthesamereason,itisnocoincidencethattheybothdo.Theseparategroundsofthetwoderivativelawsfailtoexplainwhytheyarealikeinbothconservingenergy.

Scientificexampleslikethismotivategivingasimilarinterpretationto various mathematical examples. Consider a result14 regardingexponentiation (the binomial theorem) and a result regardingdifferentiation(thegeneralLeibnizrule):

A result regarding powers:Iffandgarenumbersandnisanaturalnumber,thenthebinomialtheoremsaysthat

where =n!/k!(n−k)!

14. Ioriginallydiscussed this result inLange2015b, thoughnot inconnectionwithgrounding’srelationtomathematicalexplanation.

coordinates to various points and then showing by calculation thatthetangentfromapointonthegivenlinetoonecircleisequaltothetangentfromapointonthegivenlinetotheothercircle.Itproceedsby(whatmathematicianscall)“bruteforce”.Suchexamples13suggestthatoftentimes,aproof that identifies the theorem’sgroundwillbea brute-force proof and that oftentimes, brute-force proofs do notexplain.Theysimplygrindouttheresultfromtheessentialpropertiesofthesetup.Theirpurityisnotenoughtomakethemexplanatory.(Inthenextsection,Iwillreturntothetensionbetweenaproof’sbeingexplanatoryanditsproceedingbybruteforce.)

Ihavearguedthatsomemathematicalproofsarenotexplanatory,despite identifying the grounds of the theorem being explained,becausetheyproceedbybruteforcefromthosegroundsandtherebyfailtorevealthewaythatthetheorem’svariouscasesareunified.Hence,the proof’s identifying the grounds of the theorembeing explainedgetsinthewayoftheproof’sbeingexplanatory.Another,indirectwaytoarguethatthisoccursinmathematicalexplanationistoshowthatthesamethinghappenssometimesinscientificexplanation.Onewayforsuchanexampletoariseinscienceisforseveralderivativelawsofnaturetopossessthesamefeature,whereeachofthesederivativelaws can be derived separately from its own grounds (that is, fromthe relevant fundamental laws) and where the various derivativelawsdifferintheirgrounds.Thederivationthustreatsthesimilarityamong thederivative lawsas a coincidence; it traces thederivativelaws tono common source that accounts for their common feature.However,ifthesimilarityamongthevariousderivativelawsisinfactnocoincidence,theneachofthoselawsexhibitsthecommonfeatureforthesamereason.Aderivationfromthatsourcefailstoidentifythegroundsofthederivativelaws,butitexplainswhytheyaresimilar.

For example, gravitational andelectrostatic forces are similar inthatbothconserveenergy.Thefactthatgravitationalforcesconserveenergy follows from the gravitational force law (and the laws of

13. Coordinate geometry can likewise prove Desargues’ theorem in two-dimensionalEuclideangeometry(seenote9)by“bruteforce”.



As Gregory (1841, iv) put it, the similarity is not “founded onaccidental analogy”. Just as there is a common reason why bothgravitation and electrostatic forces conserve energy, so there is acommonreasonwhyexponentiationanddifferentiationarealike intherespectexhibitedbytheexpansiontheorem.Thetwooperationsare alike in this respect because they are alike in obeying (whatGregorycalled)thesamethreelawsofcombination:

Fromthefactthatexponentiationobeysthesethreelaws,thebinomial theoremfollows,andintheverysameway,theproductrulefollowsfrom the fact that differentiation obeys these three laws. Thus, asGregory (1841, 237) put it, both of these results “depend only onthe lawsof combination towhich the symbols are subject, and aretherefore true of all symbols,whatever their naturemay be,whichare subject to the same laws of combination”.15 I interpret Gregory,in referring to “dependence”, to be talking about mathematicalexplanation. The grounds of the two separate theorems do notcombinetomathematicallyexplainwhytheirconjunctionholds.16

15. Todaywewouldput thepoint in termsof theexpansion theoremholdingof any commutative ring. (Presumably,Gregory is best understood as notliterallymeaning“symbols”here,sincethesameresultcouldbeexpressedinmanysymbols,andineachcaseitwouldobtainforthesamereason.ShortlyI quote Boole,who does better in referring to the explanation as lying incommonfeaturesofthetwooperations,exponentiationanddifferentiation,notoftheirsymbols.)

16. Regardingthe“dependence”oftheexpansiontheoremonthethree“lawsof

A result regarding derivatives:Iff(x)andg(x)aren-timesdif-ferentiablefunctionsofrealnumbersx,andiff(n)=(dnf)/(dxn)isthenthderivativeof f(andthe0thderivativeof f is f), thenageneralizationof theproductrule(“generalLeibnizrule”)saysthat

Leibniz noticed the striking analogy between these two results asearlyas1695;heevenarguedina1697lettertoWallisthathisnotationwasbetterthanNewton’sbecauseitmakesthisanalogymoresalient(Koppelman 1971, 157−158). As Koppelman (1971, 171) puts it, thesimilarity“certainlycallsforanexplanation”.JohannBernoulliwrotetoLeibnizin1695:“Nothingismoreelegantthantheagreementyouhaveobserved…doubtlessthereissomeunderlyingsecret”(Leibniz2004,398).

Theconjunctionofthesetwoexpansiontheoremsisgroundedinitsconjuncts.Wecouldderivethebinomialtheoremfromitsground,involving what exponentiation is. We could likewise derive thegeneralLeibniz rule from its ground, includingwhatdifferentiationis. But Bernoulli recognized that those derivations do not supplytheunderlying “secret”hewonderedabout.Theseare twoseparatederivations;whataderivativeamountstoplaysnoroleinthegroundofthebinomialtheorem.Sothesetwoseparatederivationstreattheanalogybetween these two results as a coincidence. But in fact, asBernoulli suspected, the analogy has a common origin, and so theanalogy isno coincidence. Its explanation inmathematicsdoesnotworkbytracingthesimilarity’sgrounds.



Onceagain,Itakethekindof“dependence”thatBooleistalkingabout,where thebinomialandgeneralLeibnizrules“depend”onlyonthelawsofcombination,tobeakindofexplanationthatisnotsuppliedbythegrounds.Exponentiationanddifferentiationobeythesamelawsofexpansionbecausetheyobeythesamethreelawsofcombination.

This example illustrates how grounds often fail to explain inmathematicsforsomethinglikethesamereasonasgroundsoftenfailtoexplaininscience.AsImentionedattheendofsection1,wecouldsimplystipulatethatgrounds,invirtueofbeinggrounds,giveaspecialkindofexplanation:groundingexplanation.Butaswehaveseen,thisisnotexplanationasitfiguresinmathematicalpractice.

Itmightbeobjectedthatexponentiation’sobeyingthethreelawsof combination grounds exponentiation’s expansion theorem andthatdifferentiation’sobeyingthethreelawsofcombinationgroundsdifferentiation’s expansion theorem. On this view, the fact thatexponentiationanddifferentiationarealikewithrespect toobeyingthethreelawsofcombinationbothgroundsandexplainstheirbeingalikewithrespecttoobeyingtheexpansiontheorem.

However, I see no reason (beyond the fact that the laws ofcombinationentailtheexpansiontheorem)tosaythatanoperation’sobeying the three laws of combination grounds its expansiontheorem.Iseenoreasontoresistthethoughtthatjustaseachofthethree combination laws for theoperation is grounded separately intheoperation’sessence,so theexpansion theoremisalsogroundedseparately in the operation’s essence with none of these fourhelping to ground another. This characterization is supported by acomparison to the scientific case. A spacetime symmetry principledoesnothelptogroundthefactthatthegravitationalforceconservesenergy. Rather, the gravitational force’s essence (specified by thefundamental gravitational force law) grounds its conserving energy,andtheforcelawneitherhelpstogroundnorispartlygroundedbyaspacetimesymmetryprinciple.Thatthegravitationalforceexhibitsacertainspacetimesymmetry(namely,thatitsforcelawisinvariant

OthercommentatorshavemadethesameobservationsasGregory.For example, François-Joseph Servois (1814−15, 142) said regardingtheanalogy,“Itisnecessarytofindthecause,andeverythingisveryhappily explained”. That the two operations obey the same laws ofcombination is, he said, “la véritable origine” (151) of the analogybetween the two results. Separate, unrelatedderivationsof the tworesultswouldprove themanddisplay theirgrounds,butwouldnotexplainwhytheirconjunctionholds.AsBoole(1841,119)saidoftheanalogydeployedtosolvealineardifferentialequationbysolvinganalgebraicequationandthenexchangingpowersforderivatives:

The analogy … is very remarkable, and unless weemployed a method of solution common to bothproblems, itwouldnotbeeasytoseethereasonforsoclosearesemblanceinthesolutionoftwodifferentkindsofequations.ButtheprocesswhichIhavehereexhibitedshows,thattheformofthesolutiondependssolelyon…processeswhicharecommontothetwooperationsunderconsiderations,beingfoundedonlyonthecommonlawsofthecombinationofthesymbols.

combination”,onemightinitiallybetemptedtointerpretGregoryandBoole(below)astalkingaboutgrounding.However,noticethatGregoryandBooleemphasizethattheexpansiontheoremdependsonlyonthetwooperations’havingthosethree“lawsofcombination”incommon.Thetacitcontrasttheyaredrawing iswith theexpansion theoremdependingalsoonwhat thoseoperationsare−on(asGregoryputsit)the“nature”ofexponentiationandthe“nature”ofdifferentiation.Nowpresumably,thefactthatexponentiation,for instance, obeys a given lawof combination is itself grounded inwhatexponentiationis.Soifthe“dependence”oftheexpansiontheoremonthelawsofcombinationwerealsoamatterofgrounding,thenbythetransitivityofgrounding,theexpansiontheoremwouldalso“depend”onthenaturesofexponentiationanddifferentiation,losingthetacitcontrastthatGregoryandBoolearedrawing.(Virtuallyallaccountsofgroundingregardgroundingastransitive(CorreiaandSchnieder2012,32),aslongasanyimplicitcontrastsproperlylineup(Schaffer2016,55)thatis,aslongasthefactthatGratherthanG´groundsthefactthatHratherthanH´,andthefactthatHratherthanH´groundsthefactthatJratherthanJ´.)



coincidental rather than unifying that result’s various componentsinthewaythatcertainscientificexplanationsunify.Ofcourse,afullaccountof explanation inmathematicsgoeswellbeyond the scopeof this paper. But a sketch of such an account would support myargumentsintheprevioussection.Inaddition,theaccountIwillnowsketch suggests two further arguments thatmathematical theoremsare not automatically explained by their grounds. I will give thoseargumentsinsection4.

To motivate my proposed account of mathematical explanation,considerthistheoremfirstprovedbyd’Alembert:17

Ifthecomplexnumber(whereaandbarereal)isasolu-tiontozn+an-1 z

n-1 +…+a0=0(wheretheai arereal),thenz’s“complexconjugate” z̅ =a – bi isalsoasolution.

Whydothesolutionsofapolynomialequationwithrealcoefficientscomeinthesecomplex-conjugatepairs?Itiseasytoprovethattheydo.First,wecanusestraightforwardcalculationtoshowthatwegetthesameresultwhetherwemultiplytwocomplexnumbersandthentaketheircomplexconjugateorwhetherwego in theoppositeorderbytakingthenumbers’complexconjugatesandthenmultiplyingthem:

Show that z̅ w̅ = z̅w̅:

Hence, ,andlikewiseforallotherpowers.Therefore,. We can likewise use

17. I have given this example, along with a more extended defense of theproposalthatIamabouttomake,inLange(2016).

under arbitrary time translations) and that it conserves energy areseparatelygroundedinwhatgravitationis.

Furthermore, set the above argument aside and supposeexponentiation’s three laws of combination were groundingexponentiation’sexpansiontheoremandweregrounded,inturn,bywhatexponentiationessentiallyisandsimilarlyfordifferentiation.Then itwouldbepuzzling (ifexplanation inmathematicsworksbysupplyinginformationabouthowamathematicalfactarisesfromitsgrounds)whythefactthatthetwooperationsarealikewithrespecttoobeyingtheexpansiontheoremisnotmathematicallyexplainedbywhatexponentiationessentiallyisandwhatdifferentiationessentiallyis. But I have argued that this is no explanation in mathematicalpractice.

3. What Makes a Proof Explanatory?

Ihavenowgivenseveralreasonswhyitistypicalforproofsidentifyingamathematicalresult’sgroundsnottoexplainwhythatresultholds,whereas anexplanatoryproofof the samemathematical result failsto identify itsgrounds.ThereasonsIhavegivenforthisdivergencebetween grounding and mathematically explaining are that proofsidentifying the result’s grounds can incorrectly depict the result ascoincidental,mustbepure,andoftenproceedbybruteforce.Ihavealso argued that the unification that mathematical explanationsfrequently reveal,preciselyby failing to specify the result’sgrounds,is similar to the unification that scientific explanations frequentlyachievebyfailingtoidentifytheresult’sgrounds.

Inthissection,Iwillapproachallofthesemattersfromadifferentdirection: by askingmore generallywhatmakes one proof but notanother explanatory in mathematics. An account of this differenceshould account for the differences in explanatory power betweenthe various proofs we have just looked at. A general account ofmathematicalexplanationshouldalsoaccount forwhatever tensionthere may be between a proof’s explanatory power and its purity,its proceeding by brute force, or its treating the result it proves as



In thisexample,aproof isexplanatorybecause itexploitsasym-metryintheproblemasymmetryofthesamekindasinitiallystruckusinthefactbeingexplained.Iproposetogeneralizethisdiagnosisofwhythetwoproofsofd’Alembert’stheoremdifferintheirexplana-torypower.Although the symmetrybetween i and–i is the featureofd’Alembert’stheoremthatinitiallyjumpedoutatus,aresultcouldhave some other sort of salient feature. That feature of the resultpromptsa“why”questionthatwouldbeansweredbyaproofderiv-ingthatresultfromafeatureofthesetupthatissimilartotheresult’ssalientfeature.Onthisproposal,toaskwhy theresultholdsistoaskforaproofthatexploitsacertainkindoffeatureinthesetup:exactlythesamekindoffeaturethatstandsoutintheresult.Thedistinctionbetweenproofsthatexplainandproofsthatmerelyproveexistsonlywhensome featureoftheresultissalient.Thatfeature’ssaliencecanprivilegesomeproofasexplanatory.19

This proposal accounts for the difference in explanatory powerbetweenthetwoproofs(insection1)ofthefactthat7appears50,000timesbetween1and99,999.Thestrikingthingaboutthisfact,aswesaw,isthatthenumberof7’sisone-halfof100,000,whichisjustonemorethan99,999.Thefirstproofsimplytalliedalloftheappearancesof7’s,andsoalthoughitgavethetheorem’sgrounds,itdidnotderivethetheorembyexploitingafeatureoftheproblem(“Howmanytimesdoes7appear…?”)thatissimilartothesalientfeatureoftheresult.Bycontrast,thesecondproofconstruesthesetupsothatitissolvedbymultiplying100,000by5digitspernumberdividedby10optionsforeachdigitthatis,byone-half.Thisproofexplains,then,byvirtueoftracingtheone-halfof100,000thatistheresult’sstrikingfeaturetoasimilarfeatureofthesetup.

19. Obviously, Icannothopetodefendthisproposalproperlyhere;seeLange(2016). Even if this proposal is not fully satisfactory as it stands, I hope itshowshowamoregeneralaccountofmathematicalexplanationcouldbeused to support the previous section’s argument that mathematical expla-nationandgroundingtypicallydivergeandcansuggestadditionalargu-mentsforthisconclusion(suchasthosegiveninthenextsection).

directcalculationtoshowthatsummationandcomplexconjugationcanbetakenineitherorder:

Thus, , which equals 0andhence0ifzisasolutiontotheoriginalequation.

Althoughwehavejustprovedthatthecomplexconjugateofonesolution isanothersolution, thisproof takesabrute-forceapproach.Thetheorem’sstrikingfeatureisthattheequation’snonrealsolutionscomeinpairswhereonememberofthepaircanbetransformedintotheotherbythereplacementofiwith–i.Whydoesexchangingifor

–i inasolutionleaveuswithanothersolution?Theproofwejustsawdepictsthissymmetryasjustturningoutthatwaywhenweplugandchug.Aderivationofthetheoremfromasimilarsymmetryintheorig-inalproblemiswhatwouldexplain whythetheoremholds.Thatis,anexplanationofthetheoremwouldbeaproofthatexploitstheinvari-anceofthesetupunderthereplacementofi with–i.

Thesought-afterexplanation18isthat–icouldplayexactlythesamerolesintheaxiomsofcomplexarithmeticasiplays.Eachhasthesamedefinition:Eachisexhaustivelycapturedasbeingsuchthatitssquareequals–1.Thereisnothingmoretoi (andto–i)thanthat.Ofcourse,i and–iarenotequal.But theyarenodifferent in theirrelationstothe real numbers.Whatever the axioms of complex arithmetic sayabout one can alsobe truly said about theother. Since the axiomsremaintrueunderthereplacementofiwith–i,somustthetheoremsforexample,anyfactabouttherootsofapolynomialwithrealcoef-ficients.(Thecoefficientsmustberealsothatthetransformationofi into–i leavesthepolynomialunchanged.)Thesymmetryexpressedbyd’Alembert’stheoremarisesfromthesamesymmetryintheaxiomsofcomplexarithmetic.

18. SeeFeynmanetal.(1963,22−7).



The result’s salient feature is that it identifies a property sharedby the two series (namely, their convergence behavior). The proofappealingtotheradiusofconvergencetheoremisexplanatorybecauseinappealingtoeachseries’possessingasingularityonthecomplexplane’sunitcircle,thisproofexploitsapropertythatissharedbythetwoseriesandisthussimilartotheresult’ssalientfeature.

Likewise, the striking thing about the theorem concerning expo-nentiation and differentiation, as Leibniz and Bernoulli remarked,isthatit identifiesarespectinwhichthetwooperationsaresimilar.Therefore, proofs that treat the two operations separately fail to ex-plainwhyexponentiationanddifferentiationarealikeinthisrespect.Bycontrast,thetheoremisexplainedwhenthesimilarityitdisplaysisderiveduniformlyfromanothersimilaritybetweenexponentiationanddifferentiationnamely,thattheyobeythesamethree“rulesofcombination”.

Inthisway,theproposalIhavesketchedconcerningthesourceofaproof’sexplanatorypoweraccountsforthewaythataproof’shavingexplanatory powermay fail to require its purity,may be precludedby its taking a brute-force approach, and may be undercut by itstreating theresult itprovesascoincidental (insteadofunifying thatresult’svariouscomponentsinthesamemannerascertainscientificexplanationsdo).Thisproposalthusaccountsforthedivergencethatwehaveseenbetweengroundingandmathematicallyexplaining.

On thisaccount,aproofderiving the theorem from itsgroundsmayexplain,butitsspecifyingthetheorem’sgroundswouldthenbeincidental to its explanatory power. Itwould derive its explanatorypowernotbyvirtueoftracingthetheoremtoitsgrounds,butratherbyvirtueoftracingthetheoremtoafeatureofthesetupthatislikethetheorem’ssalientfeature.

correspondingsidesareprojectionsoflinesonthesametwoplaneswhichmustintersectinaline.Althoughtheproofexitingtothethirddimensionisimpure,itexplainsbyshowinghowthetheorem’ssalientfeaturearisesfromasimilarsalientfeatureinthesetup.

The proof identifying this theorem’s grounds proceeds by bruteforce.Onthisproposal,nobrute-forceproofisexplanatorywhenthesalientfeatureofthetheorembeingexplainedisitssymmetry(asind’Alembert’s theorem)orsomeothersuchfeature thatabrute-forceproof fails to exploit. A brute-force approach is not selective in itsfocus;itsimplyplugseverythinginandcalculateseverythingout.Bycontrast, anexplanationmust (on thisproposal)pickoutparticularfeaturesof the setup that are similar to the result’s salient features,tracingtheresult’ssalientfeaturebacktothem.

This proposal can also account for the difference in explanatorypowerbetweenthetwoproofsthatthetwoTaylorseriesexhibitthesameconvergencebehavior.WhatstrikesusasremarkableaboutthetheoremisthatitidentifiesafeaturethatiscommontothetwoTaylorseries,despitetheobviousdifferencesbetweenthetwofunctions.Inthiscontext,thepointofaskingforanexplanationistoaskforaproofthattreatsthetwoseriesalike,derivingtheirconvergencebehaviorinthesamewayfromsomeotherfeaturethatthetwofunctionsshare.Aproofthattreatsthetwoseriesseparately,evenifitispureandrevealsthetheorem’sgrounds,failstoexplainbecauseitfailstoderivethetwoseries’ convergencebehavior in the sameway fromanother featurethatthetwofunctionsshare.Bycontrast,theproofusingtheradiusofconvergencetheoremtreatsthetwofunctionstogether,derivingtheircommonconvergencebehaviorfromanotherfeaturethattheyshare namely,thattheygoundefinedsomewhereontheunitcirclecenteredattheoriginofthecomplexplane.Thisproof’simpurityallowsittoexplainwhythetheoremholds.20

20.The caseofDesargues’ theorem (seenotes 9 and 13) is similar (followingLange2015a).Abrute-forceproofusinghomogeneouscoordinatesjustplugseverything inand turns themeatgrinder.Bycontrast,a striking featureofDesargues’ theorem is that it identifies something common to all threeoftheintersectionsofthecorrespondingsidesofthetwotrianglesinperspec-tivenamely, thateachof those intersections fallsonthesame line.Theproof ofDesargues’s theorem that exits to the third dimension allows thetwocoplanar triangles inperspective tobe theprojectionsof trianglesondifferentplanes.Theproofderives the similarityamong the threepairsofcorrespondingsidesfromanothersuchsimilarity:thatineachcase,thetwo



viewthatatheorem’sgroundsdonotgenerallyexplainwhyitobtains.Thisargumentemphasizestherolethattheproposedaccountassignstothesalienceofsomefeaturepossessedbythegiventheorem.

On this account,whether a given proof qualifies as explanatory(indeed, whether there is any distinction between explanatoryand non-explanatory proofs of some theorem) depends on somefeature(s) of the theorem being salient. For example, when weencountered d’Alembert’s theorem (in section 3), its salient featurewasthesymmetrybetweeni and–i thatitidentifiesinthesolutionstopolynomialequations.Thesalientfeatureofthetheoremconcerningexponentiationanddifferentiation (in section2)was its revealingarespect inwhichthetwooperationsarealike.Thesalient featureofthetheorem(insection1)that7appears50,000timesinalistofthenumbersfrom1to99,999isthat50,000ishalfof100,000,whichisnearly99,999.

Whichfeatureofatheoremissalient(andwhetherithasanysalientfeatureatall)dependsonthecontext.Indeed,astheconversationalcontext shifts, a feature that had been salient can retreat into thebackground as a new featurebecomes salient.My account predictsthatwhenthathappens,theremaybeacorrespondingshiftinwhichproof(ifany)qualifiesasexplanatory.

Thispredictionisborneoutinmathematicalpractice.Let’slookatanexample.22

Askyourfriendstoselectanytwonumbersandtoinsertoneinthefirstrowandtheotherinthesecondrowofthefollowingtable:

22. IdiscussedthisexampleinLange2016tomakeadifferentpoint.

4. Two Further Arguments that Grounds do not Explain.

TheaccountofmathematicalexplanationIhavejustsketchedsuggeststwofurtherarguments for theviewthatmathematical theoremsarenotautomaticallyexplainedbytheirgrounds.

According to theaccount Ihave just sketched, if amathematicalresult exhibits no striking feature at all, then there is nothing thatitsexplanationoverandabove itsproofwouldamount to. Inotherwords,thereisnothingthatitwouldmeantoaskwhytheresultholds,overandaboveaskingforaproofoftheresult.Likewise,theaccountentailsthatamathematicalresultexhibitingsomestrikingfeaturemayhavenoproofexploitingasimilarfeatureofthesetup,inwhichcasetheresulthasnomathematicalexplanation.

These consequences of the proposed account seem correct.Mathematicianswhoaretryingtoexplainwhyagiventheoremholds,promptedbythetheorem’sexhibitingsomestrikingfeature,routinelyacknowledgethatthetheoremmayinfacthavenoproofthatexplainswhy it holds. For example, after askingwhy the twoTaylor series Idiscussed earlier have the same convergence behavior, despite thestarkdifferencesbetweenthetwofunctions,Spivak(1980,482)says,“Askingthissortofquestionisalwaysdangerous,sincewemayhavetosettleforanunsympatheticanswer:ithappensbecauseithappens that’sthewaythingsare!”(Ofcourse,hegoesontopointout,“Inthiscasetheredoeshappentobeanexplanation…”namely,involvingtheimpureproofappealingtotheradiusofconvergencetheorem.)21

However,ifamathematicaltheoremwereautomaticallyexplainedbyitsgrounds,theneverytheorem,nomatterhowpedestrian,wouldhaveanexplanation.Thisseemscontrarytomathematicalpractice.ItwouldmakeSpivakincorrectinacknowledgingthatagiventheoremhavingasalientfeaturemightturnouttohavenoexplanation.

Iwillconcludethispaperbydescribinghowtheprevioussection’saccountofexplanatoryproofssuppliesyetanotherargumentforthe

21. Gale(1991,41)alsoacknowledgesthatatheorem,althoughproved,neednothaveanexplanatoryproof.



Tosee ifyoursuccesswas justafluke,your friendsmayaskyouto perform your feat beginning with different numbers (and evenwithfractionsornegativenumbers).Eventually,youwilldivulgeyoursecret:thatforanyinitialtwonumbers,thegrandtotalequals11timestheentryinrow7.Youraudiencewillpresumablythenwanttoknowwhythistheoremholds.Inthissortofcontext,thetheorem’ssalientfeatureisobviouslythatitallowsyourtricktoworkforanytwoinitialnumbers.Inaskingwhythetrickworks,youraudiencewantstoknowwhetherthissimilarityamongthetheorem’scasesisacoincidenceornot.Inotherwords,youraudienceisinterestedinaproofthatdeducesthisfeaturethatiscommontoallofthecasesfromsomeotherfeaturethatiscommontothemall.Thatisthepointinaskingwhythetheoremholds.

Inaskingwhythetheoremholds,themembersofyouraudiencearedemandingthekindofproofthatbooksofmathematical“magic”standardlypresentasexplainingwhythetrickworks.ThefollowingexplanationcomesfromGardner’sMathematical Circus(1979,101−104,167−168).The trick’s explanation is that in any table, the two initialnumbersxandygeneratea“generalizedFibonaccisequence”onthesubsequentrows:x, y, x +y, x+2y,2x+3y,….Gardnerdisplaysthissequenceinthefollowingtable:

1. x

2. y

3. x+y

4. x+2y

5. 2x+3y

6. 3x+5y

7. 5x+8y

8. 8x+13y

9. 13x+21y

10. 21x+34y

total 55x+88y

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

total

Thenaskyourfriendstocompletethetablebyinsertingthesumofthose tworows in row3, thesumof rows2and3 in row4,andsoforth through row 10 and, finally, to compute the grand total bysummingallofthenumbersinrows1through10.Whileyourfriendsarefuriouslycompletingthetable,youlookovertheirshouldersuntiltheyfill in row7.Thenyousimply take thatentry,multiply itby 11inyourhead,andboldlyannouncethegrandtotal longbeforetheyreachit.Here,forexample,isthecompletedtablewhenthetwoinitialnumbersare4and7.

1. 4

2. 7

3. 11

4. 18

5. 29

6. 47

7. 76

8. 123

9. 199

10. 322

total 836



In this new context, a proof explaining why the theorem holdshastotreattheresult’sx-andy-componentsalike,deducingeachinthesamewayfromafeaturetheyshare.Suchaproofmustshowthattheabove twoequationsconcerning theFibonaccisequencearenocoincidencehaveacommonproof.Gardner’soriginalexplanationcannotdothat,sinceittreatsthex-sumseparatelyfromthey-sum.Soitdoesnotexplaininthenewcontext.

Thisshiftisvisibleinthemathematicsliterature.Forinstance,onetextbook(BenjaminandQuinn2003,30–31)beginsbypresentingthetrickunderthetitle“AMagicTrick”,indicatingthekindofproofthatwouldbeexplanatory(namely,onerevealingittobenocoincidencethatthetricksucceedsforanyxandy).ThebookthengivesGardner’sproof,commentingthat“[t]heexplanationofthistrickinvolvesnoth-ingmorethanhighschoolalgebra”.However,oncethetabledecom-posestheresultintothex-andy-components,apreviouslyunrecog-nizedfeatureoftheresultbecomessalient.Thetextbookrecordsthatthegivenexplanationcannothelpraising“why”questions itcannotanswer,remarking:“…thetotalofRows1through10willsumto55x+88y.Asluckwouldhaveit,(actuallybythenextidentity),thenumberinRow7is5x+8y”(BenjaminandQuinn2003,30–31).Theresult’sx-andy-componentsaretheabovetwoequationsregardingthefirsteightandfirstnineFibonaccinumbers,and theproofdepicts theseequationsasiftheirjointlyholdingwereamatterof“luck,”havingnocommonproof:amathematicalcoincidence.Butitisnocoincidence asthetextbookforeshadowedbysaying“actually,bythenextidentity”.

That“nextidentity”concernsFibonaccinumbers.Toexplainwhythe two equations hold, think of the Fibonacci sequence as doubly infinite.WithF1=1andF2=1,itfollowsthatF0=0,F-1=1,etc.NotethatF-1andF0arethex-coefficientsinlines1and2ofGardner’stable.Sothefollowingequationscapturethex-andy-sides,respectively:

F-1+F0+…+F8=11F5

F0+F1+…+F9=11F6

AsGardner’stableproves,thesumofthefirst10membersinanysuchsequenceis55x+88y,whichis11timesthe7thmemberofthesequence(5x+8y).This “commonproof”coveringallcasesalikemakes itnomathematical coincidence that the trickworks in anypossible case.Theproofunifiesallthecasesfallingunderthetheorem.

Nowfortheshiftincontext.LookingatGardner’stable,wethinkofthex’sasformingonesequenceandthey’sasforminganother,separatesequence.Werecognizethatthecoefficientsofthextermsinlines3through10arethefirsteightmembersoftheFibonaccisequenceandthecoefficientsoftheytermsinlines2through10arethefirstninemembers.SoGardner’sexplanation reveals that the result’sholdingonthex-sideconsistsofthesumofthefirst8Fibonaccinumbersplus1equaling11timesthefifthFibonaccinumber(line7’sx-coefficient).Theresult’sholdingonthey-sideconsistsofthesumofthefirstnineFibonaccinumbersequaling11timesthesixthFibonaccinumber(line7’sy-coefficient).Inotherwords(whereFiisthei

thFibonaccinumber):

F1+F2+…+F8+1=11F5

F1+F2+…+F9=11F6

In the wake of Gardner’s explanation, the context has shifted; adifferent featureof the theoremhasbecomesalient.Havingprovedthetheorembyusingthetablefilledinwithx’sandy’ssoastogivea common proof for every instance of the theorem, we now findourselvesconsideringtheoriginalresultashavingtwocomponents:anx-resultanday-result.Thesalient feature isnow that thex-sumworks out so that the coefficient in the grand total is 11 times thecoefficientontheseventhlineandthatthey-sumworksoutintheverysameway.Theresult’ssalientfeatureisnowthatthex-andy-sumsarealike in this respect.This feature couldnotoriginally havebeen theresult’ssalientfeature,sincewecouldnotoriginallyhavedecomposedthetheoremintox-andy-results;wecoulddothatonlyafterhavingGardner’sexplanation.



makershoweverthecontextmayshift.25Thissuggeststhatgroundingandexplaininginmathematicsarefundamentallydistinct.

References

Benjamin, Arthur T., and Jennifer J. Quinn 2003. Proofs that Really Count.WashingtonDC:MathematicalAssociationofAmerica.

Bigelow,John1988.The Reality of Numbers.Oxford:Clarendon.Boole,George1841.Ontheintegrationoflineardifferentialequations

withconstantcoefficients.Cambridge Mathematical Journal2:114−119.Correia, Fabrice and Benjamin Schnieder 2012. Grounding: An

opinionated introduction. In Correia and Schnieder (eds.),Metaphysical Grounding. Cambridge: Cambridge University Press,pp.1−36.

D’Alessandro, William 2018. Arithmetic, set theory, reduction andexplanation.Synthese195(11):5059−5089.

somethingoverandaboveany(context-dependent)contrasttotheexplanan-dum.Althoughinsomecases,acontrastrendersafeatureoftheexplanan-dumsalient,wecanintroduceacontrasteventosomepedestrianmathemati-caltheoremwithoutrenderinganyofitsfeaturessalient.Therethenremainsnothing that itwouldbe to explainwhy that theoremholds. Furthermore,evenunderafixedcontrasttotheexplanandum,ashiftincontextcanshiftwhichof its features issalient.For instance, inaskingwhyGardner’smath-ematicaltrickworks,wepresumablymeantoaskwhyitworksforallinitialnumbersxandyratherthanforonlysome(orfornone).Thiscontrastisnotenoughtopickoutwhetherweseekaproofthattreatseveryx,ypairalikeoraproofthattreatsthex-andy-sumsalike.Evenaproofthatdoesneitherwouldruleoutthecontrastingalternativetothetrick’salwaysworking,de-spitefailingtoexplain(inanyofthesecontexts)whythetrickworks.

25. Thegroundofthefactthatthex-andy-sidesofGardner’stablebothworkoutmaybethatforany10consecutivemembersofthedoubly-infiniteFibonaccisequence,theirsumequals11timestheseventhmemberinthesum.Inthatcase,perhapsoneoftheexplanationsofthemathemagictricksuppliesthegroundsof the trick’s success. (Anaccountofmathematical facts’ groundswouldhavetosaywhethertheregularityregardingevery10-memberFibo-naccisequenceisgroundedinfactsaboutindividual 10-membersequences.)Nevertheless, that proof does not acquire its explanatory power from sup-plyingthegroundsofthetrick’ssuccesssinceinadifferentcontext,thegroundremainsbuttheprooflacksexplanatorypower.

Thispairisnocoincidence.Thetwoequationscanbegiventhesameproofaproofthatforany10consecutivemembersofthedoubly-infinite Fibonacci sequence, their sum equals 11 times the seventhmember in the sum. (I relegate theboringproof to anendnote.23 Itgives the same treatment to every 10-term segment of the doubly-infiniteFibonaccisequence.)

Invirtueofthiscommonproof,itisnocoincidencethatthex-andy-sidesinGardner’stablebothmakethetrickwork.Thetrick’sx-andy-componentsbothworkbecauseofanotherfeaturetheyshare:Eachinvolvesthesumof tenconsecutivemembersof thedoubly-infiniteFibonaccisequence.So(onsection3’saccount)thisproofexplainswhythetrickworks,whenthis“why”questionisaskedinacontextwherethe trick’s salient feature is that the x- and y-sides have somethingincommon.Theproofexplainsby tracingthatsimilarity toanotherfeaturethatthetwosidesshare.

Thisexampleillustrateshowashiftincontextcanalterwhatittakesforaprooftobeexplanatory.Aproof’sexplanatorypowerdependsonthesalienceofparticularfeaturesofthetheorembeingexplained,andtheirsalience iscontext-dependent.Bycontrast,a theorem’sgroundisnotcontextdependent.24Atheorem’struth-makersremainitstruth-

23. Foranyintegern, Fn+1+Fn+2+Fn+3+Fn+4+Fn+5+Fn+6+Fn+7+Fn+8+Fn+9+Fn+10 =2Fn+3+2Fn+4+2Fn+5+0Fn+6+Fn+7+2Fn+8+2Fn+9 =4Fn+5+3Fn+7+4Fn+8 =4Fn+5+4Fn+6+7Fn+7 =11Fn+724.As Imentioned in note 16, some grounding theorists (e.g., Schaffer 2016)

regardgroundingascontrastive: the fact thatG rather than G´grounds thefactthatHrather than H´.Suchcontrastivegroundingfactsarenotcontext-dependent,buttherelevantcontrasttoHmaydifferindifferentcontextsandremainimplicit.So(asarefereehassuggested)thefactthatHgroundsGmaybecontext-dependent:trueforoneimplicit,context-dependentcontrastandfalseforanother.

However,althoughtheexplananduminamathematicalexplanationmayinvolve a (tacit) contrast, the context-dependence of this contrast is insuf-ficienttoaccountforthecontext-dependenceofmathematicalexplanation:The (context-dependent) salience of some feature of the explanandum is



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