Mathematics as a Science of Patterns Ontology and Reference (2)

7/29/2019 Mathematics as a Science of Patterns Ontology and Reference (2)

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Mathematics as a Science of Patterns: Ontology and ReferenceAuthor(s): Michael D. ResnikReviewed work(s):Source: Nos, Vol. 15, No. 4, Special Issue on Philosophy of Mathematics (Nov., 1981), pp. 529-550Published by: Blackwell PublishingStable URL: http://www.jstor.org/stable/2214851 .

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Mathematicss A Science fPatterns:OntologyndReferenceMICHAEL D. RESNIK

UNiVERSITYOF NORTH CAROLINA

1. INTRODUCTIONI seekan accountof mathematics nwhichthe ogical forms f mathe-matical tatementsre takenat face value and their emantics sstand-ardlyreferential, ay, n themannerof Tarski. This togetherwith airlyuncontested assumptions entails that mathematics s a science ofabstract ntities, hat s, mmaterial nd nonmental hingswhichdo notexist n space and time. So I am a platonist.1Many philosophers f mathematicswho would like to be platonistsare bothered by two rather deep problems. The first s that sinceplatonic mathematical bjects do not exist in space or time the verypossibility f our acquiring knowledgeand beliefs about themcomesinto question. The second arises from the factthat no mathematicaltheorycan do more than determine ts objects up to isomorphism.Thus theplatonist eems to be in theparadoxical positionof claimingthat a given mathematical heory s about certainthingsand yetbeunable to make any definitive tatement f whatthese things re.2I think hat heseproblems rise n part because of a fundamentalmisconceptionof what mathematics s about. If we conceive of thenumbers, ay, s objects ach one of which anbe givento us in solationfrom the others as we thinkof, say,chairs or automobiles, then it isdifficult o. void conceivingofknowledgeof a number as dependentupon some sortof interaction etweenus and thatnumber. The sameline ofthought eads us to think hatthe identity f a numbervis visany other object should be completely determined. For some timemathematicians ave emphasized thatmathematicss concerned withstructuresnvolvingmathematical bjectsand notwiththe "internal"nature of the objectsthemselves. heyhave recognizedthatweare notgivenmathematical bjects n solationbut rather nstructures. hat 13is a primenumber s not determinedby some internalproperty f 13but ratherby its place in the structure f thenatural numbers. Some

529


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530 NOO Sphilosophershave alreadytakennote ofthismovement nmathemat-ical thought.3 ut I propose to take this dea more seriously hantheyhave. In mathematics, claim,wedo nothaveobjectswith n "internal"compositon arranged in structures,we have only structures.Theobjects of mathematics, hat is, the entitieswhich our mathematicalconstants nd quantifiers enote, are structureless oints or positionsin structures.As positions n structures,heyhave no identity r fea-turesoutside of a structure. urthermore, hevarious results fmath-ematics which seem to show that mathematicalobjects such as thenumbersdo haveinternal tructures,.g., their dentification ith ets,are in fact nterstructuralelationships.I find t moresuggestive orepistemologicalpurposes to speak ofmathematicalpatterns nd theirpositionsratherthanofstructures.viewpatterns nd theirpositions s abstract ntities.Most ofthispaperwillbe devotedtodevelopingan extensional ccount ofpatterns nd toexplaininghowviewingmathematical bjectsas positions n patternsleads to a reconception of mathematicalobjects which defuses theobjectiontoplatonismbased upon our inabilityocompletely ix heiridentity.I also thinkthatviewingmathematics s a science of patternspromisesto solve theplatonist's pistemologicalproblems as well-orat least to make them less urgent-by showing that mathematicalknowledge has a fairly entralplace in our general epistemologicalpicture. expectto develop a moredetailed epistemology lsewhere,but beforeproceedingto the maintasksof thepaper I want to sketchmycurrentview.4When we are exposed to several nstancesof a patternofcertainkinds and are intherightpsychological et and have theapprorpriatepurposes)wearebynature truck ythesimilarityf the nstances-wesee thattheyfit pattern.5Thus thedetective ees severalstrangula-tions na month's ime nd thinks there sa pattern othis".Ofcourse,this snot to know whatthepattern s nor tobe able tomakeanyuse ofit. However, after further xposure to instances of the pattern orfurther houghton theprevious nstances)we are likely o attempt odescribe thepattern tself. his is oftendone via role talkor positionaltalk. For example, the detectivewillsay: thevictims strangledwithscarf, he murdererapproaches the victimfrombehind,thecrime scommitted n a Wednesday,etc. (Reflection n thismay ead us to anaccount of how reference nmathematicsspossible.)Suchdescriptionsofa pattern an then be checkedagainstthe data whichgeneratedthedescriptionand which are supposed to fitthe pattern.Describingapattern s thusclosely kintodevelopinga theory f it and ofthedatafittingt.Once we have a partialdescriptionof a patternwe can alsoinfer therfeatures f t.Thus the detective educes that hemurderer


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MATHEMATICS AS A SCIENCE OF PATTERNS 531has thesize and strength ostrangle hevictimwith scarf.Moreover,once we have represented pattern oourselves we can easilythink fvariations n it.Thus murderon FridayratherthanWednesday,twovictims imultaneouslytrangledwith ne longscarf, tc.This route totheknowledgeof newpatterns snotonlyquite familiar omathemat-iciansbutabsolutely rucial tomy ccount ofmathematical nowledge.For it is not implausibleto thinkof our comingto knowsmall finitepatterns hroughexposure toconcrete nstancesofthem,but one hasto strainquite hard to believe thatwe arrive at our knowledge ofinfinite r complicatedpatterns n thisway. nfinitepatterns re firstthoughtof, I would suggest,bythinking f finitepatterns s indefi-nitely xtended. There is more thanthisto knowledgeof an infinitepattern, f course; our representation fthe infinite atternmust beconsistent; tmustrelate to the finite atterns textends n theappro-priateway nd so on. But I think hat longthese inesan epistemologyfornumber theoryand the higherbranches of mathematics an bedeveloped. If so, thiswould place the epistemologyof mathematicsalong sideepistemologiesfor inguisticnd musicalknowledge.Fornodoubt weperform nalogous operations nthosefields.We internalizepatterns flinguistic tterances r muscialperformances;we are ableto generate new patternswhich have never before been utteredorperformed, nd we are able toclassifynd recognizegeneral featuresofthesepatterns. f we could not,we could not writenewnovelsandsongsor recognizenonsense and dissonance. Knowingthatadding iscommutativemaybe likeknowing hat theactive-passive ransforma-tiongenerallypreservescognitivemeaning; knowingthatthere s nogreatestnatural number maybe like knowingthatgiven a song inwhicha measure is repeated, say,twice, here s (could be) another nwhich t is repeated threetimes.Muchmoreneeds tobe saidabout an epistemology fmathematicsbased upon our knowledgeof patterns.But I hope thatmysketchmakes tplausiblethat heepistemology fmathematicssno more ornotmuchmore)mysterioushantheepistemologiesoflinguisticsndmusic.Like mathematics heybeginwith xperience,abstractfrom tand arrive ttheunexperienced and, perhaps, ikemathematics,venthe unexperientable).

2. PATTERNSAlthough knowofno developed philosophicalaccountofpatternshave encounteredseveralsuggestions s towhattheymight e.6Theyall take offfrom herelationshipbetween pattern nd its nstances.focus instead upon how patterns are related to each other and getinstantiations a specialcase. The approach seemsquitenaturaltome,


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532 NOUSbut myexperiencehas been that ntuitions ary onsiderably n thesematters.t may help you understandmy pproach ifyoubear inmindthat thasbeen generatedbyreflecting n twomathematical heories fstructure, amely,geometry nd model theory.Myview makes modeltheory ntoa geometry.On myview, patterns a complexentity onsisting f one or moreobjects, which I call positions,tanding n various relationships andhaving various characteristics,distinguished positions and oper-ations.)7A position s like a geometricalpointin that t has no distin-guishingfeatures therthan those thas in virtue f being that ositionin the patternto which it belongs. Thus relative to the equilateraltriangle ABC the three points A, B, C can be differentiated, utconsidered n isolation hey re indistinguishable rom ach otherandtheverticies f any triangle ongruent o ABC. Indeed, considered asan isolated riangle, ABC cannot be differentiated rom any otherequilateral triangle. In geometrystructuralrelationships-such ascongruence and similarity-are paramount while claims of identityand distinctness f points, ines, planes,etc. are reserved forcontextswheretheseentities re relatedto severalother geometrical bjects.transfer his geometrical nalogyto the various structures tudied bymathematics.Within structure r patternpositionsmaybe identifiedor distinguished, incethe structure r pattern ontainingthempro-vides the contextfor o doing. However,the paramount relationshipsamong patternsare those of structural imilarity congruence andequivalence introduced below) and structural ontainment occur-rence and subpattern).Letus apply the analogyto the naturalnumbersequence (N, S). I take this o be a patternwith singlebinary elation(successor)and the natural numbersto be itspositions.Viewed thusthere sno moreperplexityn the fact hat he naturalnumbershave noidentifying eatures beyond those definable in terms of the patternthan there is to the correspondingfact about the points in triangleABC.Patterns re relatedto each otherbybeingcongruentrstructurallyisomorphic.Congruence san equivalence relationwhosefield take toinclude both bstract tructuresnd arrangements f concreteobjects.Thinkingof patterns s models of formal ystems,t stherelationshipwhichholdsbetween somorphicodels f formal ystems. onsider,forexample, a first rder system with singletwo-place predicate"R"and axioms stating hatR is a totalordering.This systemhas manymodels-all the totalorderings-but they re not allcongruent oeachother.Onlythose whosedomainshave thesamecardinalityre. The setof numbers from one to ten taken in their natural order and tenpuppies in a litter aken norder of theirbirth re isomorphicmodelsof the system ; so I count the abstractnumerical structure nd thearrangementof puppies as congruent.


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MATHEMATICS AS A SCIENCE OF PATTERNS 533Whena pattern nd an arrangement f so-calledconcreteobjectssuch as the puppies are congruentthen I say thatthe arrangement

instantiateshe pattern. nstantiation henis a special case of congru-ence in which theobjects"occupyingthepositions"of a patternhaveidentifyingfeatures over and above those conferred by the ar-rangementsto whichtheybelong. The puppies thus instantiate heone-to-tenpattern.The occurrenceof patternswithin ach otherhas playeda morefundamental olein my hinkinghanhas instantiation.fyouthink fthe sentence"A catbit the catwho ate thecat" as a linguistic attern,thenthe word "cat" is a patternwhichoccurs threetimes n it. Thenaturalnumbersequence occurswithin he patternof thereal num-bers taken in theirnatural order; it also occurs withinthe iterativehierarchy fsets nd withintself. ive pointedsolidstarpatterns ccurin boththe patterns ftheUnited Statesflagand theCaliforniaflag.The fieldoftheoccurrencerelation sthesame as thatofcongruence;hence concretearrangements an occurwithinpatterns r othercon-crete arrangements.My interest,however,is mainlywithpatternsoccurringwithin atterns.More formally,ccurrence sa reflexive ndtransitive elationwhichholds betweenstructures and Q whenP isisomorphic oa structure efinable nQ. Thus (N, S) occurswithinN,


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534 NOUSa b cO 0 0

o~~~0-(a%d ,' ,e %% fg,00 h "i? ---------lo---------1?o

Nowgivenmyunderstanding f thediagram and I admit hatyoumaynot see things s I do) the rectangleof dots containsa triangle f dotsgbi. It containsotherdot triangles swell,ofcourse.) Thinkingnow ofmydrawing as also a representation f a spatialpattern one which srathermorecomplicate'd han the originalpattern), concludethat hespatialtriangle sa subpattern f the whole spatial rray. then ake thisas evidence that correspondingto the dot triangle re subpatterns ftheoriginalpattern nvolving hepositionsg, b, and i. These patternscan be characterizedbyrelationsdefinable n termsofA and Ljust asthespatial dottriangle an be described nterms frelationsdefinableinterms fabove nd left.ndeed the former efinitions an be read offthe atter uitestraightforwardly.evertheless,he relations fneitherthespatial nor thenon-spatial ubpattern re simple restrictionsf therelationsof the originalpattern.Thus ifwe count the spatialtriangleand its orrespondent s subpatterns f the originalpattern-as I thinkwe should-then the subpatternrelationmust be defined along thelines which I have suggested.Myvisual ntuitionswith espect osubpatterns nd pattern ccur-rences can be transferred o some ofthe model theoretic ases as well.Let us think f the natural numbersequence as representedbymeansof an unending linearsequence of dots.000000000000000 ....Clearly a sequence of dots which we obtain fromthis sequence bystartingwith the nth dot from the left s a subpatternof theoriginalpattern.These correspondto all thoseprogressionswhich are obtain-able from N, S) by restricing to a subsetof N. But there re infinitelymanymodels of numbertheory btainable from N, S) which are leftout bec'ause theirrelationsof successionare not subrelationsof thesuccessorrelation.The even numbersequence isone, the odd numbersequence isanother, nd theprimenumber equence a third.These allcorrespond to selectingsome progressionof dots from the originalsequence of dots. These sequences occur within the originalsequence-at least as I see it-and to account forthatboth thesubpat-tern nd pattern ccurrencerelationsmustbe characterizednterms fdefinability.


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MATHEMATICS AS A SCIENCE OF PATTERNS 535The natural number sequence (N, S) occurs withinthe naturalnumbersordered under less than, N,


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536 NOUStheory ut thiswillhelp onlywhen we can form categorical heory f apattern s well. For patterns re specificmodels oftheories-specific atleast up to isomrorphism,nd complete theories with nonisomorphicmodels abound. Despite this, heconceptionofdefinitionally quiva-lenttheories as distinct ncapsulations of a common set of theoremssuggests a similarapproach forpatterns.Intuitively,wo patterns re like "essentially he same" iftheybothencapsulate some "bigger"pattern romwhich hey anbe obtained bydeleting ome of tsrelationships.Moreformally,et us call a patternPa truncationf a patternQ ifeveryposition, haracteristic nd relationof P is also one ofQ. Then a patternP willbe said to be equivalent o apatternQjust incase there sa patternRwhich sa subpattern f both Pand Q and ofwhichP and Q are respective runcations.To illustratethis oncept et us return o N, S) and (N,


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MATHEMATICS AS A SCIENCE OF PATTERNS 537Suppose then thatwe ignorethisaspect of thedrawingand considerthem s isolatedfigureswithno informationuppliedas totheir patialrelationships.Then the figurescould be spatiallyseparate, or theycouldbe contiguous nd A could be A'-as wouldbe the case ifthe inesegmentsAB and A'B' are identical.Of course, this s no reason toconcludethat here s nofact fthematterhereconcerning he dentityof A and A'. They are, afterall, pointsin space and have adequateidentityonditions-such as beingon thesame lines.We simply o nothave enough information o determinethe facts n thisexample.This is quite true.But it s part and parcelof our developmentofgeometry s a theoryofspace.Had geometrydeveloped insteadas atheory r collectionoftheoriesoffigures r shapes, henpointsmightonlyplaytheir oleofmarkingocationswithin igureswithout layingtheirrole of marking ocationsin the containingspace. This wouldrequire us to viewthe triangle nd rectanglepresented bove (with hespatialrelationships s indicatedbythedrawing)as a singlecompositefigureand to take a similarviewof othercomplexes of geometricalfigures, ut I see no obstacleto doing thisor to interpretingmuchofelementarygeometry long these lines. Geometrydeveloped in thiswaywould attributeno being to points independent of the figurescontaining hem. dentitymight e relativized oelementsof thesamefigureor each figure figurekind) mightbe treatedby means of aseparatetheorywith n unrelativizeddentity redicaterangingoverall elementsof its universe.The latteralternative s the one whichmathematics akes withrespect to the structures tudied by its theories. Natural numbertheory, or example, deals with certainpattern; t has themeans toraiseand answerquestionsconcerning he dentityf variousnumbers,but t annotvenformulatehe uestions towhether he numberone is T.On the otherhand,withinnumbertheorydentitysabsolute:"(x)(y)(x= yV x / y)" s one of itstheorems.Similarremarkshold forthe realnumbersand real numbertheory, unctions nd analysis, ets and settheory. n each case the theory peaks onlyof elementsof a certainstructure nd has no means to identify r distinguishthese fromelementsof another structure. ike the variantofgeometrywithdif-ferent heories f differenthapes,mathematicssa conglomeration ftheories achdealingwith tsownstructure rpattern nd each forego-ing identities eading outside of itspattern.However, f we exclude identities nvolvingpositionsofdifferentpatterns this xclusiondoes notapplytosubpatterns) henwecannotuse any of the equivalence relationsintroduced earlier as identityconditions orpatterns. o seewhy onsidercongruence.Sinceit sthestrongest f these relations,we can restrict ur attention o it. Thenaturalnumbersequence is congruentto the even numbersequence.


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538 NOUSIf we take theircongruenceas sufficent or their dentity,hen theircorresponding positions should count as identical too. But then 1would be 2! Wemight ry oavoid thisbycapitalizing n one's being thesecond natural number and two'sbeing the second even numberandrelativizedentity redicates ike this:2 7NW1, =E1. The trouble s that4 is the even numbercorresponding o the naturalnumber 2 but quaeven numbers hey re distinct,o we would have: 4 =E2, 4 #E2. I knowof no other means for identifying istinct ongruent,equivalent orco-occurringpatternswhichwill not fall prey to similardifficulties.This suggeststhat we identify atternsust in case theyhave thesame positions,the same relations,etc. We could (and I would) useextensional dentity onditionsfor relationsand characteristicsf wetook this route.But even this s not open to me. For it would requiresomething ike:P1 = P2 (x) (x is a positionof P1 x is a positionof P2) & etc.and thatwouldrequireadmitting ositionsof different atterns nto asingleuniverseof discourse contrary omypreviousmoves.So neitheridentity etweenpatternsnor betweenpositions fdifferent atterns s(everywhere)defined.The exclusionof patterns hemselves romthefield ofidentitysconsistentwith the practice of mathematics tself. Number theoryquantifiesover ust the numbers but not over the number theoreticstructure,ettheory uantifiesover setsbut not over the set theoretichierarchy.Even model theory,which purportsto treatof arbitrarymathematical tructures, oes so by positionalizing hem,thatis, byidentifying tructureswith ets or ordered n-tuples.Nonetheless,thereasoning which began withthe restriction f identity o positionswithin he same pattern s now in danger of underminingmyentiretheory.For itappears thateven the variousequivalences among pat-ternsmustbe abandoned ifwe cannotspeakofpositionsfrom he samepatternwithin he same breath. began bytaking tructural r modeltheoretic somorphismas criterialforpattern congruence,but suchisomorphisms re functions rom hepositionsof one pattern o thoseofanother. This seems to require a universewhich containspositionsfrom different atterns.To speakofrelationships mongvariouspatternswe must volve acomprehensivetheory n whichpositionsfromdifferent atterns anbe dealtwith n the same breath.There are severaloptionsforsuch acomprehensive heory.First, ne could use a many-sortedheorywithseparateuniversesforeach pattern nd unreduced functions etweenthese universesto serveas isomorphisms. his would be closesttothespiritof my nformal xposition.A second (ontologically leaner) approach would constructpat-terntheory long the ines ofgeometry y positing space ofpositions


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MATHEMATICS AS A SCIENCE OF PATTERNS 539fromwhichpatterns ould be constructed. his theorywould stablishfacts of the matter for identities nvolvingpositionsof itsdifferentpatterns ince identitywould apply to all positions n its space. How-ever, ikegeometry,we would notexpectthetheory ohave theoremsdealing with particularpositions.A third pproach would refine thesecond one step further y reducing patterns o setsofpositionsjust smodel theory onstruesmodels as setsbuiltup from he domain of themodel. One could push thiseven one stepfurther nd construe posi-tions hemselves s puresets, hereby educing o-called pattern heoryto set theory.The last threemoves do not do awaywithpatterns. nstead for hemotley fpatternsdealt withbythevarious branches of mathematicsand bymy nitial pproach to patterns hey ubstitute ne big patternand its positionsand ask mathematics o make do with t. For on myview,the set theoretichierarchy, or nstance, s a pattern long withothers.Pure setsare positions n it whoseapparentinternal tructuresare a fabrication f theirrelationships o other positions n the hierar-chy.Moving to one big pattern n which all positionsand "internal"patternsfall within he scope of identity patternsbecome setson thethirdand fourthapproaches) seems to involve a retractionof myearlierclaims about identity. think,however, hat can explain whythis s not so. I did not mean to suggest ndiscussing hesecond,thirdand fourth pproaches thatthe various patternsof mathematics reidenticalwith hepatternsdeveloped within ne of thecomprehensiveaccounts; I onlymeant to suggestthatthey ould be reduced to them.To appreciate the differencewe must turn now to the subject ofreduction n mathematics.

3. REDUCTIONSAccording to the definitions have given,the natural number se-quence has multiple ccurrenceswithin patternwhich willrefer oas the iterativeettheoreticierarchy.t follows that even categoricalversionsof numbertheory an be interpreted s dealing with nyofthesepatterns nd the set of ts rue entenceswillremain ntact. his isas itshouldbe when we speak ofpatterns. f I completely escribetherelationships f a set ofpoints na geometrical igure, hatdescriptioncontinuesto hold if think f thatfigure s embedded ina largerone.The difference s that my descriptionwill not cover therelationshipsthesepointshave tonewpointsof thecontainingfigure nd willfailtocharacterize thatfigure.But whyshould that be demanded of theoriginaldescription?The settheoretic ierarchy, yvirtue f ts ontaining ccurrencesof the naturalnumbersequence, can be viewed as an extention f the


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540 NOUSlatter pattern. This interplaybetween occurrence and extention hasbeen useful n guidingthe development f set theory tself. orrowingthe notation nd axioms of number theory o describe a portionof theset theoretic atternwill ntroduceno falsehoods nto set theory nd itwillcall attention o the occurrenceofthe natural number sequence inquestion. That there is such an occurrence is surely as interestingmathematically s it is philosophically.Bringing number theory nsuggests developments in set theory itself too-transfinite gen-eralizationsof numbertheory.Thus the practicenow standard n settheory fusingthenumerals o designate ertain ets sjustified rompragmatic point of view.Although only one of the many occurrences of the natural se-quence is referred o in thedevelopmentofnumber theorywithin ettheory, here s no embarrassment or mathematicsn this. Choosingone occurrence,rather than another,will certainly ffectwhich settheoretic heorems get reported in our monographs but it does notchange thesetoftheoremsof set theory tself. t is important o showthat numbertheory an be done within et theory.Giventhat hereareseveralways nwhich t can be done, the simplest,mostelegant or mostnovel ways are what deserve attention.Doing mathematicswell re-quires selectivity.The development f number heorywithin ettheorys known s areduction of number theory o set theory.More generallywhen thepattern r patterns fone theory ccurwithin hepattern r patternsof another theory, he former s reducible to that of the latter.Al-thoughthe existenceofreductions smathematicallynteresting, heyhave had little ffect n the practice f general mathematicstself.Mostbooks on settheorydevelop thenatural numbers n termsofsets,butbookson numbertheory, opology, lgebraoranalysisbeginwithbothnumbers and sets. Nor do set theoristshave an atypical nterest nontological economy, although probablysome has rubbed off fromtheirphilosophical colleagues. Instead we findnumbers reduced tosets n settheory ecause, as mentionedearlier,doing so promotes ettheory tself.There is another phenomenon which has greatly hanged math-ematics nd which ould be calleda reduction.This is the settheorizingof mathematics. have in mindthe use of the anguage ofsettheory sthebackground language ofworkingmathematics nd the attendantobjectification or, in my terms,positionalization)of mathematicalstructures. This is reflected in the different mphases of pre-settheoretic nd post-set heoreticmathematics.Courant and Robbins'famous book, What s Mathematics?,llustrates he pre-settheoreticapproach tomathematicsalthough t does containdiscussions fsets).The discussion n thebook is almostexclusively bout theelementsof


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MATHEMATICS AS A SCIENCE OF PATTERNS 541traditionalmathematical tructures-the various numbers,geomet-ricalobjectsand functions.Mostof itstheorems an be formalized nthefirst rdertheory ftheseobjects.Compare thiswithrecentbookson topologyor algebra. Here theemphasis is upon thevarious struc-turesthemselves,whattheirsubstructures re, how they can be ex-tended,and towhat they re isomorphic.To formalize uchtheoremswe mustuse theories n whichstructures hemselves an be takenasindividuals, nd thishas been greatly acilitated ythe ntroduction fset theoreticmethods.Nonetheless,everymathematical heorydealswithone or more structureswhich it describes by showing how theelements f each are nternallyelated.Post-set heoreticmathematics sno exception. t speaks ofstructuresnlybytaking hem s elements fa "larger"structure-the hierarchy f sets.We now have two kinds of reduction.Withthe first patternoccurs in another and the theoryof the former s reiterated n thetheory fthe atter.With he secondpatterns hemselves re identifiedwith lements fanotherpattern, herebynablingus toachieveresultsaboutthesepatternswhichwere notevenstateablepreviously.t is thesecond sortof reductionwhichhas had a genuine effectupon thepractice of mathematics.Philosophers have been fascinatedbythefirst ind of reduction.They wanted toknowwhether ome oftheseestablished hatnumbersare sets. Now we see thattheymight lso wonderwhether henaturalnumbersequence is a set.Both questions are ambiguous. Taken one way theysimply skwhether ertain etsare sets. To be interestinghey houldbe takenasasking whethercertainthingsknownto us before the advent of settheory re sets.Since,on myview, ets re positionsna certainpattern,thequestionsaskwhether hingswhich renot known obe positions fthatpatternare positions n it. The question makeprima acie sensesince it appears thatthe entities n questionmight e positionsof thepattern n question.But howcould we findout?Notby mathematicalmeans. No newtheoremswillsettle hem,nor willnewaxioms.Thesequestions are not likethequestionof the generalizedcontinuumhy-pothesis, Is each aleph thecardinality fthepower set of tspredeces-sor?",despite theirgrammatical imilarity. aken in the non-trivialsense theyare not in the language of a single mathematical heory.They can, to be sure, be "resolved"by extending the notion of settheory nd adding newidentities r non-identities o this anguage asaxioms.13 However,eventheseresolutions annot be decidedbymath-ematicalevidence,since tmakesno difference o thecontent fmath-ematicswhetherwedo settheorywithnumbertheory n topof t, o tospeak, or whetherwe do itwithnumbertheoryreduced to it.


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542 NOUSOf course, t does make a difference o the elegance ofour mathe-matical theories.Elegance is highlyprized in mathematics ut it hasnothing to do withmathematical ruth; fterwe have settleda givenmathematical uestion it is fitting o ask whetherwe can do it moreelegantly. From the mathematicalpoint of view the questions aboutnumbers, the natural number sequence and sets are not questionsabout factsbut ratherquestions about how we should present mathe-maticaltheories.14 This, in my opinion, is the onlynon-trivial ensewhich an be given othesequestions.The view hat here s a fact f thematterhere is equivalent to assuming that given anyobject x and aposition f na patternyeitherx = yor x 7 yThis assumptionfailswithrespect to my conceptionof patterns.Mathematicalreductionscannot tellus what numbers,functions,sets, tc. are in any factual e'nse,but they an lead us from defectiveconception or theory f mathematical bjects of a givenkind to moreadequate ones. For example, to someone who objects to functions nthe grounds that they nvolve intensional notions such as rules, wemight respond that functions re nothing but sets of ordered pairs.There is nothingwrongwith his ortof response,so long as we realizethatwe are not claiming hatfunctions onceivedas rulesare thesameas functions onceivedas sets.Whatwe aredoing spointing ut thatwecan discard worries bout the intensionalityf functions y usinganextensionaltheorywhichcan servethemathematical nds of theorigi-nal one.Something ike this can be said about my proposals forpatterntheory-although I do not think fmyview as a mathematical heory.Those whofindpatterns nd positionsmysteriousmight indreducingthemtopure setsreassuring.Ofcourse, given myviewthatmathemat-ical objects are positions n patterns, hiswould be another face to thereductionof mathematics o set theory. n both cases we would switchtheorieswith herebeingno fact fthe matter s to whether heold andnew theorieshave the same ontology.On the otherhand, I would stillmaintain hat hereductionofpatterns opuresets sbothunnecessaryand undesirable.Unnecessary, ecause my onceptionofpositions ndpatternss clearenoughas is; undesirable,because thereduction oone"big'"patternwould tend topushtherelationships mong patterns ndtheirpositionsoutoffocus.And thediscomfortingnterpatternues-tions annot be avoidedbythe move to one pattern nyway, ecause wewill tillwonderhow t relatesto the old patterns rhowit willrelate topatternsdiscoveredin the future. f I am right bout patterns, heseworries are misguided. If I rightabout mathematicalobject's beingpositions n patterns, hen so too are worries boutwhethernumbersare reallysets.


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MAIHEMATI(CS AS A SCIENCE OF PATTERNS 5434. REFERENCE

We describe patternsn a number ofways.One wayconsists n statinghowmanypositions patternhas and howthey re related.Another stopresent n instance fa pattern nd add that hepatternnstantiatedis thepatternwe have nmind. A third onsistsn ntroducingabelsforpositions nd stating ow thepositions rerelated nterms f these. Nodescription fa pattern-not evenone bymeansofa categorical et ofaxioms--will ifferentiatehepattern romtsoccurrenceswithin therpatterns r itsotheroccurrences n isolation;Similarly description fa square will not differentiatet from other squares to which it iscongruent.This isentirelyn accord withmypointof view.For it snota question of whether one patternor many patternsare being de-scribed,althoughwe cannot tell which.There is ust one factof thematter oncerninghow many patterns re there.15*Mathematicaltheoriesdescribe various mathematical atterns.16Their truth onsists n doing so accurately.But ifpatterns nd theirpositions annotbe fullydentifiedhow is a correspondencesufficientfor truth stablishedbetweena theory nd itspattern?17 o fixourideas let us ask how number theorydescribes the natural numbersequence.

The standard approach to reference nd truth ssumes that tispossibleto assignexactlyone position number)to each numeralanddo so without ualification.Truth is then characterizedrecursivelynthemanner of Tarski. This approach failsforpatterns, or tpresup-poses thatwe can makeunqualified dentifications fpositionswhileclaimthat dentifications an onlybe made within pattern.Nicholas White in [13] developed a view which in effect reatsnumeralsas functional ermswith rgumentplaces for progressions.Instead of "3" we have "3(P)" which denotes the fourth object inwhateverprogressionis chosen as a value of P. The view can beextended tootherkindsofstructures s well with he result, orexam-ple, that thereare progressionsP and set theoretichierarchiesS forwhich1 (P) = { 0 } (S) whileforother P and S thisfails.This approachrequires a common universe for all mathematical tructures-or atleastrequires a universeofall progressions-and functions efinedonit. It thus sees factsof thematterwhere I see none.HartryField's [4] comes muchcloser to myview. He allows num-erals to refermultiply y using the notion of partial predicate. Thenumeral"0",forexample, partially efers o 0, to0,to { 0 }, and to thefirst lementofeveryprogression.The othernumeralspartially eferto thecorresponding lementsofprogressionswith tsbeingrequiredthat hey e linked ntheirpartialreference ssignments o thatdistinctnumeralspartially efer odifferent lements fthe sameprogression.


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544 NOUSThus although"2" partially enotesboth { {0}} of (0, {0}, {{ 0}}, . . )and I{ 0} of { 0}, { 0} , I {0} ,. . .),the bvious ontraditionsreavoided. The relation symbolsof number theoryare assigned thecorresponding elationsna progression s theirpartialdenotations.Astructureforwhich thisobtains is said to partially ccord withthesemantics fnumbertheory.A sentenceofnumbertheory ssaid tobetrue respectively,alse) fand only f t strue false) neverytructurewhichpartially ccordswith hesemantics fnumbertheory.Truth na structuresunderstood ntheTarskisense. It thus turnsoutthatthetruths fthe anguage ofnumbertheory re exactly hosewhichholdin everyprogression.Furthermore, n a mixed numbertheoretic-settheoreticanguage, thesemanticswill ountsentences uchas "1 = {0}"as neither ruenorfalse-so long as boththenumbertheoretic nd thesettheoretic erms re takenas primitive. hus Field sees truth, alsityand lack of truth-value xactlywhere I do.Despite thisField sees factsof thematterwhere I see none. Thepartial referencerelation is one which holds absolutelybetween anumeral and elementsofprogressions.Nowsuppose that 2" partiallydenotes an elementx ofa progressionP1while"3" partially enotesanelementy of a progressionP2. Field's semanticsrequires thatx bedistinct rom6 ifP1and P2are thesame progression. t thusrequiresthattherebe a fact o thematter oncerningwhetherx = y evenwhenP1 nd P2 re not ncluded none and thesamestructure. n this espectour viewsdiffer.'8The only lternativeeft hat knowof ssomeversionofreferen-tialrelativity. think variantofQuine's viewsfitswellwithmine,butbefore explainhow I wanttoaddress thequestionofwhy theory freference s needed to completemyaccount. The answer, think sfairlytraightforward.espouse realism, hus believethatmathemat-ical statementsre true or false nvirtue fthefeatures fan objectivereality. o far I have presentedan account of whatthatreality s. A"theory"of reference s needed to explain the connectionbetweenmathematical tatements nd thatreality.'9Let us suppose thatwe havefixedan occurrencef the naturalnumbersequence. Then, relative o taking hisoccurrenceas fixed,standardTarskian approach to truth nd reference s possible.Thesymbol O" s assignedthefirst osition n thesequence, thesuccessorsymbol s assignedthesuccessorrelation, hequantifiers ssignedthetotalityf thepositions nthesequence as their ange,and soon. Truthis definedrecursivelynthefamiliarway.The setofnumbertheoreticstatementshereby erifiedwillbe thesamenomatterwhatoccurrenceof thenatural numbersequence is fixed.Thus determiningnumbertheoretic ruth ia one occurrencedetermines tforall occurrences-exceptthere s no fact fthematter oncerningwhether here re oneor manynaturalnumbersequences in question.


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MATHEMATICS AS A SCIENCE OF PATTERNS 545Nextconsidera language in whichnumber theoretic anguage isembedded. To be specific et thisbe the anguage ofset theory.Thereare two cases to distinguish.n thefirst henumbertheoretic ymbolsare primitive ymbolsof the larger language. Then we mustfix anoccurrenceof a set theoretichierarchyn whichthe naturalnumbersequence occurs "at the bottom" so that the naturalnumbers can betakenas ur-elements.We thenassign thesepositionsto numeralsandothers o setsymbols nd proceed as usual. This givesrise tocombinedtruths f numbertheory nd settheory uchas "34EI2,3, 4}" but createsno newones denoting henumbers lone such as "2E3". No one hasanycomplaintsabout this.In the secondcase thenumbertheoretic ymbols re defined.Thenumerals thuscome torefer o a particular ccurrenceofthe naturalnumbersequence withinnot"atthe bottom") f the occurrenceof thesettheoretichierarchy ixed.20 ome truths pparently nvolvingustthe numbers-such as "2E3"-are thereby reated.My positionhere isthat this s an edificeof a reduction of numbertheory o set theory.Thus, we should not conclude thatthese are newtruths nvolving henumbers alone. They are rather truths bout positions n an occur-rence of the natural number sequence in an occurrence of the settheoretichierarchy nd thus ultimately ruths bout sets.Untilnow I have spokenrather traightforwardlybout carryingout a semanticsfor numbertheory nce an occurrenceofthenaturalnumber sequence has been fixed.But no matterhow thoroughoureffortsofix n occurrenceof a pattern, here s, ngeneral,no factofthematter oncerningwhether he occurrence upposedlyfixed sor isnot the same as someother occurrence. fwe fix n occurrenceof thenaturalnumber sequence byidentifyingt as a certainsubpattern fsome more inclusivepattern, henwe willbe able to distinguish hisoccurrencefrommanyotherswithin he larger pattern.That is whygivinga formalsemanticsfor number theorywithin set theoreticframework ppears to make the numeralsreferabsolutely.We mustnot forget hatthis s relativeto takinga fixed occurrenceof the settheoretichierarchy or granted.We attemptto fixan occurrenceof a patternbyusing a repre-sentational cheme-a diagram,description, r theory-to pick tout.We mayeven pointto instancesof thepattern s well.But referentialrelativityntersat thispoint,for even our best efforts o not forestallquestionsconcerningwhether he occurrencefixed sthesame as otheroccurrence in a perhaps moreinclusivepattern).2'Thus reference srelative o akingn occurrencef patternsfixedThis simplymeans thatoursemantics or theory f a patternspredicatedupon taking t facevalue the representational chemethroughwhichwe have attemptedto fixan occurrenceof the pattern.22 he demand thatwe achieve


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546 NO(USabsolute referenceby fixing ccurrencesof patternsmore fullymissesmypoint thatthere s no more to do. For there s no factof the matterwhether an occurrence of a pattern s or is not the same as anotherexcept when theyare both subpatterns f the same pattern.23Since part of the "theory"of reference for a language involvesfixing he range of itsquantifiers, corollary f referential elativitysontological relativity.n my case this comes to the doctrinethat onlyrelative o taking n occurrenceof a pattern s fixeddoes it make senseto speak of the ontology positions) of a theoryof a pattern. WhereQuine proclaims hatwhatmakes sense are statements oncerninghowone theorymaybe interpretednanother have the parallel claim thatwhat makessense are statements hatone pattern ccurs nanother.24It may seem I have done away with he world or with he viewthattruth s a matterof the way in the world is. But I do not deny theexistence of patterns,nor do I claim that how a pattern s fails todeterminewhat s true n its theories.My claim s thatthere s enoughslippage between our theoriesof patterns nd the patterns hemselvesto affect eference.But it does not affect ruth: he truths f a theory fa pattern are invariant under all reinterpretationsn patterns con-gruent n it.

5. CONCLUSIONWhen Frege-and Dedekind too-asked whatnumberswere,mathe-maticshad neither n adequate characterization f thenaturalnumbersequence nor an axiomaticbasis fornumber heory. heirquestionwasinfact demand thatmathematics evelop an adequate conceptionofthe numbers.One wayto do this s toclarifyhe foundations f numbertheory tself, nd that clarificationwas a direct result of their work.Anotherway s to identify umberswith ntities orwhich we alreadyhave an adequate conception. Frege triedthistooby dentifyinghemwith ets,butenough doubt has been castupon our conceptionofsetsthat uch an identifications notregardedas conceptually larifyingnitself.This paper has takenyet nother pproach to thequestionof whatmathematical bjectsare. The problem s no longerone ofclarifyingour mathematicalconceptions.Rather it is a problem of findingaphilosophical nterpretationf them.My suggestion s thatmathemat-ical objectsare positions n a pattern s not intended as an ontologicalreduction. It could lead toone, ofcourse,but fwe reduce patterns osetsitwould be the old one witha detour added.) Myintentionwasinstead to offer notherwayofviewingnumbersand numbertheorywhichwould putthephenomena ofmultiple eductions nd ontologi-cal and referential elativityna clearer ight.My hope is thatwhenthey


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MATHEMATICS AS A SCIENCE OF PATTERNS 547are seen to arise in a fairly bvious waywithrespect to patterns, henthey will seem more comprehensiblewith regard to mathematicalstructures.25

REFERENCES[1] Benacerraf, P., "What Numbers Could Not Be", Philosophical eview,74(1965):47-73.[2] ,"MathematicalTruth",Journal fPhilosophy,0(1973): 661-79.[3] Corcoran,J., "On Definitional quivalence and Related Topics", Historynd Phi-losophy fLogic, 1(1980): 231-34.[4] Field, H., "Quine and the Correspondence Theory", PhilosophicalReview,84(1975): 23-50.[5] Jubien, M., "Ontology and MathematicalTruth", Noius,11(1977): 133-50.[6] Kitcher,P., "The Plightof the Platonist",Noius,12(1978): 119-36.[7] Parsons, C., "Ontology nd Mathematics", hilosophical eview, 0(1971): 151-76.[8] Quine, W.V., "OntologicalRelativity"nOntological elativitynd OtherEssaysNewYork: Columbia University ress, 1969).[9] Resnik, M., "More on Skolem's Paradox", Noius, (1969): 185-96.[10] "Mathematical Knowledge and PatternCognition",CanadianJournal fPhilosophy,(1975): 25-39.[1-1] ,Frege nd thePhilosophyfMathematicsIthaca: Cornell University ress,1980).[12] Steiner, M., Mathematical nowledgeIthaca: Cornell University res, 1975).[13] White, N., "WhatNumbers Are", Synthese,7(1974): 111-24.[14] Wilson,M., "The Double Standard nOntology",Philosophicaltudies, orthcoming

in 1981.NOTES

'I am usingthe term platonist" s it scommonlyused today n thephilosophy fmathematics o stand forrealismwithrespectto abstractmathematical bjects;I do notmean to commit myself hereby o Plato's theoryof ideas. My reasons forespousingplatonism re varied and complicated; some of them are foundin Resnik 9], [10], and[11]. In [2] Paul Benacerraf presents strong rgumentforgiving uniform emanticsformathematics nd the rest f science: fmathematicalermshaveone kindof semanticswhileother cientific ermshave anotherthen specialsemanticswill till e requiredforsentencescontainingboth kinds of term. This argumentcan be strengthened, s Iobserved ndiscussionwithBenacerraf and others t theGreensboroconference.For auniform emanticsformathematics nd science willbe requiredto obtain an account ofinferences nvolvingboth mathematical nd other scientific entences. n [11] p. 62 Itrace thisto Frege. Once the need fora uniform emantics nd a face-value ccount oflogical form is granted, the weightof the Tarski approach quickly carries one toplatonism.2Benacerraf[2] and [1] contain statementsof the first nd second problems,respectively. teiner discusses the former t length n [ 12] whileJubien 5] and Kitcher[6] elaborateon the latter.See also Maddy's paper in this ssue.3Cf.Jubien 5], Kitcher 6], Parsons [7], Benacerraf[1], Steiner 12].4Afuller nd perhaps differing ccountwillappear in a paper I am writing ora1982 AmericanPhilosophicalAssociationWesternDivision ymposium n mathematicalknowledge. t will ppear inNous XV, 1 (March 1982). A previousattempt ppeared inmy 10].5Notall patterns re knownthrough xposure totheir nstances-thus thequalifi-cations.6One of these was given byGilbertHarman in his commentson theGreensboroConference version of thispaper.


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5487I have added the parenthetical lause to parallel the model theoretic efinition fstructure. trictlypeaking,distinguished ositions, perations nd characteristics hichcannot be picked out using the relationsof the pattern tselfmust be viewed as distin-guished by seeing the pattern s embedded in another pattern.Thus (N, S, 0) containsdistinguishedposition,viz. 0, which can be picked out using the successor relation S.However, distinguished0 in (Rat,


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MATHEMATICS AS A SCIENCE OF PATTERNS 549geometry o notattempt odescribe single tructure ut deal with lasses of structures.An "if .. then" or deductivist pproach is appropriate forthese theories.17Non-categorical heories present a number of problems. First order numbertheory eemstobe a casewherewe had a clear dea of thepattern nd came torealize thatthe theorydid not completelydescribe t. We then used other devices to rule out theunintendedpatterns.n thecase ofsettheorywealso know hatwelack a fulldescriptionof the set theoretichierarchybut remain unsure of whichof the various structuresdescribableby set theory s the "intended"one. Notethatboth ntended and unintendedmodels of number theory nd set theorycontain congruentcores. I viewour initialversion fnumber heory nd ourcurrent ettheories s attempts o describe hesecores.That theyweresatisfied ydiverging xtensions f thecores nitially scaped our notice.18Both Field's and White's pproach are criticized n Kitcher 6].

191have putscare quotes around "theory" ecause on myview there re no facts fthe matterconcerningthe references of mathematical erms. I will not address theepistemological uestionsabout reference-such as,"How is tpossibleforus torefer opositions n patterns." hope to deal withthese in the paper mentioned n note 4.20MarkWilson n[ 14] counts ettheorywith henaturalnumbers s ur-elements ndpure set theory as ontologically quivalent. My view is that these are examples ofequivalentbutnoncongruentpatterns.As maybe evident, raditional uestions about atheory's ntologydo not finda place in my account.ilIn Quinean terms, ven a complete and categoricaltheory s reinterpretable nothertheories.22CfQuine: "In practiceof course we end the regressof coordinatesystemsbysomethingikepointing.And inpracticeweend theregress fbackground anguages, ndiscussions freference, y cquiescing nour mother ongueand taking tswords t facevalue" [8] p. 49.23Hartry ield offers n objection to Quine's referential elativity hich s predi-cated upon reading Quine's formulation reference s relative to some backgroundlanguage" as meaningthatreference nsteadofbeinga dyadicrelation x refers oy" satriadic relation"x refers to y relativeto BL". (ActuallyField reads Quine as sayingreferencetakes the form x refersto y relativeto translationmanual M". See [4] pp.206-208.) The objection s based upon a misunderstanding, believe.Quine and I arearguingthatreferencefails o make sensewhenan occurrenceof a patternbackgroundlanguage) is not accepted as fixed taken at facevalue). This is likepointingout thatapersonwho s not ofa certain ge cannotbe legally oundby contract.n either aseonemight ay"relative o . . . ", forexample, "relative o your being of age you are legallybound to pay Jones $100" or "relative to taking English at face value the Frenchexpression deux' refersto two". In neither case is an increase in the degree of therelationship ntended. Making explicit suppresseddegree in a predicatesuch as " ispresident" can be clarifying, ut it would be pointlessto demand thatQuine revise"'deux' refers o two" to read "'deux' refers o two n English".If you did not alreadyunderstand the expression"'deux' refersto two"as an Englishsentence,adding the -words"in English"would not clarifytforyou.24Whenone recalls thatforQuine a theory s alreadyinterpreted, he differencebetweenour views s notas greatas my alkofpatternsmightead one to believe. ndeedin conversation nd commentson an earlier draftof thispaper, Quine has noted thatsimilarityf my positions n patterns o hisobjectsof a theory. t is possibleto adduceconsiderations o show thatnon-mathematicalbjects chievetheir dentitynly hroughtheir elationships o otherobjects nd then rguethat hesetooare positionsnpatterns.This would lead to an extensionofmyviewsto include a completetheory freferentialrelativity,lthoughthe route to this onclusion wouldbe ratherdifferent hanQuine's.For worriesabout references o rabbitsvs. references o rabbitstagesare not worriesaboutmultiple ccurrences f a pattern.Rabbitparts,rabbit tagesand rabbitsmight epositionsnnon-congruent atterns,which ccur within ach other r areequivalent ndtherebyproduce relativity henomena.25Thispaperhasdeveloped inspurts ver a number fyears;thus, owe a largedebtofthanks. n oral comments n Benacerraf [2] Oswaldo Chateaubriand remarked hatwecan perceivecertain bstract ntities hrough heir nstances.Although was familiar


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550 NOUSwiththe view that mathematicss about structure hrougha number of sources, theseremarks uggested to me a footholdfor n epistemology orplatonism.Benacerrafhadhinted himself hat mathematicsmight e about structuren discussing he problem ofmultiplereductions. believethat was the first o propose a structuralistpproach toboth the epistemological nd ontological difficulties fplatonism n [10]. Since thenIhave benefitedfrom omments, ncouragement nd discussionwithmany ndividuals:James Anderson, CatherineAnderson, Charles Chihara, James Fetzer,Jane Fleener,Nicholas Goodman, Richard E. Grandy, Gilbert Harman, Penelope Maddy, VannMcGee, Richard Nunan,W. V. 0. Quine, JayF. Rosenberg,Frederick chmitt, aVerneShelton, Brian Skyrms,Mark Steiner,Mark Wilson,Paul Ziff nd the membersof myNationalEndowmentfor heHumanitiesSummer Seminar. want ogive special thankstoJohnCorcoran and Philip Kitcherwhose many etters nd conversations ave beenofcrucial ssistance. shouldadd that, omyknowledge, ew f those isted hink hat havegotten hings traightr right. arlier versions fthese deas havebeen presented o thefollowing: he BuffaloLogic Colloquium, Davidson College, the NorthCarolina Philo-sophical Society,the Universities f North Carolina at Greensboro and Chapel HillConference in the Philosophyof Mathematics, nd the philosophy colloquia of theUniversities f NorthCarolina at Chapel Hill and California at San Diego.

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Mathematics as a Science of Patterns Ontology and Reference (2)