hatcher, foundations as a branch of mathematics

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Foundations as a Branch of MathematicsAuthor(s): William S. HatcherReviewed work(s):Source: Journal of Philosophical Logic, Vol. 1, No. 3/4 (Aug., 1972), pp. 349-358Published by: SpringerStable URL: http://www.jstor.org/stable/30226048 .

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FOUNDATIONS AS A BRANCH OFMATHEMATICS

WILLIAM S. HATCHER

I. PROLOGUE ON MATHEMATICAL LOGICIt is quiteclear the wayin which we can, andperhaps hould,considermathematicalogicas a branchof mathematics,n fact a partof appliedmathematics. he maintechniqueused in appliedmathematicss that ofstudyinga phenomenonby buildinga mathematicalmodel of it. Sucha model is an idealizationwhich approximateshe phenomenonbyconcentrating n a few relevant eaturesandignoringwhatseemsto belesssignificant.f themodel urnsout to havehighpredictive ndexplan-atoryvalue,at leastin some usefulsituations,t is subjectedo extensivetheoretical development and becomes a 'theory' (i.e. probability theory,heat theory, etc.).It is precisely in this way that mathematical logic can be viewed asapplied mathematics. The phenomenon which underlies mathematicallogic is common-sense logical inference. This phenomenon has severalaspects, in particular, language, 'reality', meaning, and deduction. Themathematical model of this situationconsists of thefollowing abstractions:formal languages for language, mathematical structures for reality,interpretation in structures for meaning, and formal deduction fordeduction. Though this model neglects many features of the actualphenomenon (most notably the psychology of the reasoner), it has beenjudged useful enough to be studied extensively.

Now, if one accepts the above as a reasonable summary of whatmathematicallogic is, then mathematicallogic is no more 'philosophical'than any otherbranch of appliedmathematics. The only reason one mightbe tempted o argueotherwise s because n logicit is, in part, hehumanthought process itself which is being modelled, thus apparentlyrenderingthe model more 'subjective'. Yet, if modern philosophical analysis hasshown anything, it has shown just how much subjectivityenters into ourmodels even of physical phenomena. Thus, logic would not seem to haveany special status on this account. One could even arguethat logic is moreJournalfPhilosophicalogic1 (1972)349-358.AllRightsReservedCopyright 1972byD.Reidel ublishingompany,ordrecht-Holland


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350 WILLIAM S. HATCHERobjectivethan physicssince it is more explicitabout the point wheresubjectivitynters n.Moreover, f one takes a thoroughgoing ragmatic iewpoint,here snot even a necessity o tryto distinguish bsolutelybetween hatpartofourworldview whichcomesfrom theviewerand thatpartwhichderivesfrom thethingviewed.We haveonlyto evaluate, hrough xperience ndusage,whetheror not a givenmodel of a givenphenomenonhasexplan-atoryvalue.Of course,there arephilosophicalproblemsconcerninghe natureofour knowledgeand the ultimateustificationor using any givenmathe-maticalmodel.But theseproblemsobtainequally or allphenomena ndtheir models and not peculiarly or logicalphenomena.Thus, it is notlogic which is philosophicalbut ratherthe whole questionof how weknow.

II. MATHEMATICS AND KNOWLEDGEIt is certainly rue that epistemologicalquestionshave been influencedbymathematics. he tradition f this influence oesback n a strong ormat least as far as Plato,andeveryone s awareof the centralroleplayedby mathematicsn theanalyticphilosophyof thetwentieth entury.But fmathematics has influenced epistemology, it is also fundamentally clearthat mathematics, ecause t claimsto know,mustitselfdependon thephilosophically prior question of how we know anything at all. It makesno sense to elaborate an epistemology of mathematics apart from theelaboration of epistemology tout court. The basic problem, then, is howto situate mathematics with respect to knowledge in general.Of course, we should beware of supposing a priori that 'mathematics'designates a clearly delineated part of our knowledge and that we haveonly to find the right objective criteria for characterizing t. This attitudebegs the question and departs from a pragmatic approach. Let us posethe problemrather n these terms: "Is it useful,and to whatdegree,toclassify a part of our knowledge as 'mathematical' and what are 'good'criteria for doing so?"

I feel that themostfruitfulwayto regardmathematicss to consider tto be the exact part of our thinking, our thinking about anything.' When-ever we objectify, abstract, and make precise our thinking we are doingmathematics n some level. From this view, it follows that there is no


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FOUNDATIONS AS A BRANCH OF MATHEMATICS 351specificallymathematical'ntuition.Anything s potentiallya source ofourintuition.

This view is certainly upportedby moderndevelopmentsn mathe-maticsn whichnon-numerical athematicaltructures rewidelyappliedto previouslyunmathematized'reas;andno one seriouslydoubts thattherewill be newand mportant pplicationsn thefuture.Theprocessof precising,abstracting ndobjectifyingnvolves heuseof variousconceptual,inguistic,andlogicaltools. We should,however,bewareof tryingto explainthe relationshipbetween hese tools in anytoo-simpleway.In all of ourthought, here s a giveand take betweenheconceptual ndtheformal, he intuitiveandthelogical.A concomitant fthe view of mathematics hat I am urging s that this duality s not adivorceandthatneither spect sprior o the otheror dominateshe other.I feel that t is wrong o consider hat mathematics asanyclearly-definedstartingpointorthatmathematicss basicallyoundedeitheron intuitionor on a formalismof some sort. Rather,both of these tools combineto createmathematics.A similarview is expressedn a recentpaperofLawverecf. Lawvere,1969,p. 281).The classicalmodernapproacheso foundations,howevermuchtheymaydiffer n theirconceptions r their ormulations, ll sharea commondesireof givinga 'onceandfor all'foundation or mathematics. ogicismsoughtto foundmathematics n our intuitionof purely ogicalrelation-ships.Intuitionismoughtto foundmathematics n our intuitionof theconcrete one is tempted o say our abstract ntuitionof the concrete).Formalismsoughtto found mathematics n a certain kind of formalsystem.Each one of these approacheshas subsequentlybeen seen to haveimportantdrawbacksdue to various kinds of inadequacieswhich itexhibits.Some of theseinadequacies reheightenedby the well-knownincompletenessndundecidabilityesultsas well as certainogicalantino-mies. Of the threebasicapproaches,he onewhichwasmostphilosophi-callyattractive n its initialformulationwas logicism.For if, in fact, itcould be shown thatour mathematicalntuition lows from ourlogicalintuition,considerable larification f theepistemologicalrocesswouldresult.Thisundoubtedly ccounts orthe tremendous ttractionogicismheld for strongthinkerssuch as Carnap,even when evidenceagainstlogicismcontinued o pile up.


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352 WILLIAM S. HATCHERSince ogicism s perhaps he strongesthesis n supportof a 'once and

for all' foundation for mathematics,I want here to reconsider it briefly.III. LOGICISM REVISITED

The first formulation of logicism was perhaps also the most precise sinceFrege not only formulated his philosophical ideas clearly but was highlyadept at technical manipulation and formal rigor. If we assume predicatelogic as given (and part of mathematics), then the two further axioms ofFrege's system are extensionality and abstraction. The underlyingphilos-ophy of the system could be described as follows: "When we formulateclearly the purely logical relationships between predicates and theirextensions, then we have, de facto, mathematics." The essence of thisintuition was the abstraction principle which says that every predicatehas an extension which is an element of the universe.

Now, Russell's paradox, which showed that the abstraction principlewas false in general, did not in itself destroy this philosophy. Russellshowed only that Frege's formulation was not correct. Philosophicallyspeaking, he showed that Frege's intuition of the logical relationshipbetweenpredicatesand extensions was innacurate. Russell's reformulationusing type theory was intended to recapture, correctly this time, theintuition which Frege had badly formulated. Although the axioms ofchoice and infinity were independent with respect to type theory, onecould argue, as Carnap did, that these should simply be taken as explicithypotheses in theorems depending on them, but that the abstractionprinciple in type theory)was still the basic intuitionon which to foundmathematics (cf. Carnap, 1970, pp. 345-6).

Indeed, in much of the current iterature on foundations, one still sensesa longing for Frege's system. Witness, for example, A. P. Morse whoevaluates his own system as one which captures"... the intuitive simplicityof Frege's beautiful but inconsistent system." (Morse, 1965, p. xxvii). Asis well known, Morse's system has a particularly strong form of theabstraction scheme as a basic principle of class generation.

What is it in the abstractionprinciplewhich leads authorsagainandagainto feel that it is so basic?Undoubtedlyt is, in largemeasure,heastonishing use which Frege and Dedekind were able to make of thisapparently simple principle. They showed, by explicit constructions, how


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FOUNDATIONS AS A BRANCH OF MATHEMATICS 353all current mathematical notions were to be defined as extensions ofpredicates of a purely logical sort. Thus, even though other mathemati-cians might feel that their spontaneous intuition of, for example, thenatural numbers was quite different from the logical one urged by Fregeand Dedekind, they could not but admit that the Frege-Dedekindconstructions ad succeeded n generating he naturalnumberson thebasis of theabstraction rinciple.Still, therewas a basic tension betweenthe spontaneousor 'natural'intuitionof mathematicalbjects uch as the naturalnumbers r the realnumbers,and the constructionsbased on one global logicalintuition.Thistension stillprevails oday.For example, he natural ntuitionof afunction is that of an operationwhereas the usual logical definitionconstructs unctionsas sets of orderedpairs.Philosophically,hen, thereal questionposed by logicism s: does our natural ntuition of thosestructuresneeded for mathematicson the one hand and our logicalintuition of the abstractionprincipleon the other lead us in the samedirection?More simply,is this tensionreal or imagined,essential ormarginal?There s one currentnotion of mathematicswhich cannot be definedin any evidentway from the abstractionprinciple,namelythat of achoice set. No attemptwas ever madeby Fregeor Dedekind o obtainthe choiceprinciple rom the abstractionprinciple.2Of course,by thetime that Zermelo had explicitlyformulated he choice principleandfocussed attention on it, Russell's paradox was already known. Theindependence of the choice principle in type theory left the questionunanswered.

However, one could argue that type theory is unnecessarilyrestrictivebecause there is not a complete ambiguityof types. This can be seen fromthe fact that there are sentences S in type theory such that the type lift S+of S (obtained from S by increasingthe type of each variable one unit) isprovable while S is not provable. Complete typical ambiguity can beobtained by adding the axiom scheme S--S+, where S is any sentence oftype theory. Such a system approximateseven closer our intuition of theabstraction principle.How does the axiom of choice fare in this system?Speckerhas answeredthis question by showing that the axiom of choice is false in type theorywith complete typical ambiguity (Specker, 1962).Thus, by approximating


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354 WILLIAM S. HATCHERmore closelyour logical intuitionof the abstractionprinciplewe haveactuallycontradictedour natural ntuition of the choice principle.Inotherwords,the tension betweenour two sorts of intuition s seen to beessentialand not marginal.The two intuitions ead us in differentdirec-tions.

Let us note in passingthat the axiom of infinity s provable n typetheorywithcomplete ypicalambiguity.The aboveexamplewould seem to saythat logicism s mistakenevenas a philosophicalideal, apart from the other technical objectionsresulting rom the various ncompleteness esults of GSdeland Tarski.Indeed, hetheoryof typeswithcomplete ypicalambiguitys equivalentto Quine'sNew Foundations cf. Specker,1962).This lattersystemwasoriginallyposedas a purely ormalgeneralizationf type theory,withoutany model or interpretation in mind. Though it originally found favorwith logicianssuch as Rosser,it has morerecentlycome to appearassomething of an oddity.

It is interestingto contrast all of this with Zermelo's system which wasoriginally an ad hoc attempt simply to list, without any philosophicaljustification, those principles actually used or needed by mathematicians.In other words, Zermelo's approachwas more in the line of a formulationof spontaneous mathematical intuition rather than that of derivingnotions from one global logical intuition. It may be fairly said that almostevery system of set theory seriously studied and used by mathematiciansin recent years has been a natural extension or modification of Zermelo's.

IV. FOUNDATIONS AS A BRANCH OF MATHEMATICS

The failure of logicism and the various inadequacies which continue toappear in connection with each of the traditional approaches to founda-tions strongly suggest that we should renounce the attempt at giving a'once and for all', absolute foundation to mathematics. It is my viewthat we should, in fact, accept this suggestion. I feel that the relativeapproach to foundations allows us to put mathematics into perspective.

Philosophically, we return to a healthy pragmatic openness in whichwe view mathematics ndlogicin less of a specialand exaltedway.Theirimportance till remains,but theirphilosophicaldependence n generalepistemologys frankly ecognized ndaccepted.


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FOUNDATIONS AS A BRANCH OF MATHEMATICS 355

Mathematics,he exactpartof ourthinking,becomesa phenomenonto be studied.We have an intuitivenotionof theprocessof abstracting,precisingand objectifying.Moreover,this processuses the deductivetechniques nd semanticnterpretations hich areusedgenerallyn ourthinking.These lattertechniquesalreadyhave a mathematicalmodelinmathematicalogic.Sincewe knowalreadyrom hevariousncomplete-ness theoremsof logicthatwe cannotformallyandconsistently apturethe total ntuitionof mathematics, emustcontentourselveswithvariouslanguageswhichcapture, n a naturalandconsistentway,an importantpartof ourintuition.Such a language callafoundationalystem.

A foundationalystem s a mathematicalbject.Thecomparativetudyof foundational ystem s clearlya branchof mathematics,moreparticu-larlya branchof mathematicalogic. I call this branchof mathematicsfoundations. uch a branchof mathematics oes notneedmore(or less)philosophical ustificationhanany other branchof knowledgeor thanmathematicstself. But, to repeat,the whole processderives ts philo-sophical ustificationromourwholeepistemology.The idea of naturalnessmentionedabove is an important ntuitivecriterion n elaborating oundationalsystems.A foundationalsystemshouldnotjust 'passively' apturea largeportionof mathematics, ut itshouldalsolead andstimulateourintuition. t shouldgiveus new ideasabouthowto objectifyandabstract.Therecentlydeveloped heoryof categoriesllustrates his pointverywell. Set theorywas originally ormulatedas a tool to clarifycertainnotions n analysis.Analysiswasessentiallyhe studyof certainnumbersystemswhichwere themselvespowerfulabstractions.As set theorywasdevelopednits ownright, hestructuralpproacho mathematicsecomemoreand more mportant.But as our ntuitionof structure rewstrongersettheoryappearedo be lessand essnatural ince heimportanteatureof a givenstructure re often irrelevanto its set-theoretical ropertieswhich ollow fromthe membershipelation.Category heoryhas shownitself to be extremelyusefulin clarifyinghe notion of structure nd insymplifyingndabstractinghe relevant eatures f structure iauniversalmappingproperties.This clarity shows the naturalnessof categorytheory.To takean example: he category-theoreticefinitionof the setof naturalnumbersas a universalobject (essentially he free Peanoalgebraon one generator)s significantly ifferent rom the set theoretic


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356 WILLIAM S. HATCHERintuitionof the numbersas countingsets.Lawvere's haracterizationfset exponentiation s rightadjoint o product s anotherexample.More-over, since both of these definitionsare specialcases of the category-theoreticnotionof universality, e are ed to thinkabout hese raditionalnotions n newways.This eads o new kindsof theorems nd theintuitingof new kindsof relationships.In fact, Lawverehas shown how to developa foundationalsystembased on category heory.The most recent form of this system,calleda Topos, is consideredby many to be the most fruitful foundationalsystem o be developedn the lastyears.Considerableimplificationndclarificationof logic, set theory, and algebraicgeometryhave alreadybeenachieved.

In sum,foundations an be considered branchof mathematicsnverymuch he samewaythatgroup heoryortopologyareconsidered ranchesof mathematics.A branchof mathematicss characterizedy the kindsof structures tudied and the kinds of questions reated.In the case offoundations, he structurestudiedare oundationalystems.Thekindsofquestions reatedare, orexample,herelative trength f various ystems,independencef axioms,consistency, ariouskindsof adequacy, tc.Traditionally, working'mathematicians uch as group theorists ortopologistshaveassumed hat a once andfor all foundation or mathe-maticshad alreadybeengivensomewhereby some logician.For them,foundationalquestionswererelegated o some nether and to be con-venientlyorgotten. t is thereforenterestingo observe hatfoundationalquestionsarenowbeingmore requently aised n all branches f mathe-matics. This is due in partto the fact that new techniqueshaveforcedresearcherso considerquestionswhich have differentanswerswithindifferentfoundationalsystems. (For example,many questionsin thetheory of abeliangroupsnow dependon whetheror nor one assumesstrongaxiomsof infinityn set theory.) t is alsodueto increasing ccep-tance of the relativeapproachto foundationswherebyfoundationalsystemsand their modelsare studied in the same way as groupsandtopologicalspaceshave been studied.

Philosophically,he viewof foundations amurginghas some of theflavorofMehlberg'spluralisticogicism' cf.Mehlberg, 962).Perhapshemain differences the importance accordto the naturalness rincipleas a basic ntuitive riterion.


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FOUNDATIONS AS A BRANCH OF MATHEMATICS 357Onemightmakethe followingsummaryof the situation:For formal-

ism, consistency s the primecriterion.Feelingthat consistency s nota sufficientguaranteeor justification or the foundationalnature of asystem,even a comprehensive ne, intuitionismseeks to build on anintuitive riterionof concreteness r constructibility. nemightviewmyfoundational ystemsas beingproperly ontained n the formerandnotcomparablewith thelatter.I would nsist on consistency utwouldrejectsomeconsistent ystemsasunnatural. hough regardmanyconstructivenotionsas natural, do not regarda notionas naturalsimplybecause,accordingo somemetaphorof concreteness,t is moreconcrete han analternative otion.The term 'metaphor' s particularly ppropriatehere, I feel. Sinceeverythingn mathematicss abstract,andhighlyso, it does not seemtomethatthere s any immediatelyvidentor apriorinotionof constructi-bility whichimposesitself upon us. Certainly t takes as great,if notgreater,powers of abstraction o visualizemany of the complicatedconstructivehierarchies urrentlyn voguethan it does to visualizeaninfiniteset such as that of the naturalnumbersor the real numbers. nfact, I feel that we may considerthat one of the main philosophicalproblemsn the epistemology f mathematicss that of finding he mostnaturalnotions of constructibility nd of justifyingthe naturalness fcertaincurrent constructive' otions.I am thinkingparticularly f vari-ous versionsof constructive nalysisandof constructive ierarchies.Universitdaval,Quebec,Canada

NOTESx This view wouldseem to bear someresemblanceo ideas whichHeytingattributes oBrouwer cf. Kleene, 1952, p. 51). However,I do not draw the sameconclusions asBrouwer.2 Interestingly nough,both Fregeand Dedekind did attempt o deducethe axiom ofinfinityfrom the abstractionprinciple.A theorem of infinity s provableon the basisof abstractionand extensionalityalone providedthat the language s sufficiently ich(cf. Hatcher,1968,pp. 103-9 and pp. 249-53).

BIBLIOGRAPHYCarnap,R., 1970,'The LogicistFoundationsof Mathematics',n Essayson BertrandRussell ed. by E. D. Klemke),Universityof IllinoisPress, Urbana,Ill.


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358 WILLIAM S. HATCHERHatcher, W., 1968,Foundationsf Mathematics, aunders& Co., Philadelphia.Kleene, S., 1952, Introduction o Metamathematics,N. J. Van Nostrand, Princeton.Lawvere,F., 1969,'Adjointnessn Foundations',Dialectica23, 281-96.Mehlberg,S., 1962, 'The Present Situation in the Philosophy of Mathematics', nLogic andLanguage,D. Reidel,Dordrecht.Morse, A., 1965,A Theory f Sets, AcademicPress,New York.Specker,E., 1962, TypicalAmbiguity',Logic,Methodology, ndPhilosophy f Science(Proc. Int. Cong., 1960),Stanford,pp. 116-24.

hatcher, foundations as a branch of mathematics

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