Table Of ContentLogic, Epistemology, and the Unity of Science 34
Jacques Dubucs
Michel Bourdeau E ditors
Constructivity and
Computability
in Historical and
Philosophical
Perspective
Constructivity and Computability in Historical
and Philosophical Perspective
LOGIC, EPISTEMOLOGY, ANDTHE UNITY OFSCIENCE
VOLUME34
Editors
ShahidRahman,UniversityofLilleIII,France
JohnSymons,UniversityofTexasatElPaso,U.S.A.
ManagingEditor
AliAbasnezhad,UniversityofLilleIII,France
EditorialBoard
JeanPaulvanBendegem,FreeUniversityofBrussels,Belgium
JohanvanBenthem,UniversityofAmsterdam,theNetherlands
JacquesDubucs,UniversityofParisI-Sorbonne,France
AnneFagot-Largeault,CollègedeFrance,France
GöranSundholm,UniversiteitLeiden,TheNetherlands
BasvanFraassen,PrincetonUniversity,U.S.A.
DovGabbay,King’sCollegeLondon,U.K.
JaakkoHintikka,BostonUniversity,U.S.A.
KarelLambert,UniversityofCalifornia,Irvine,U.S.A.
GrahamPriest,UniversityofMelbourne,Australia
GabrielSandu,UniversityofHelsinki,Finland
HeinrichWansing,Ruhr-UniversityBochum,Germany
TimothyWilliamson,OxfordUniversity,U.K.
Logic, Epistemology, and the Unity of Science aims to reconsider the question of the
unityofscienceinlightofrecent developments inlogic.Atpresent, nosinglelogical,
semanticalormethodologicalframeworkdominatesthephilosophyofscience.However,
theeditorsofthisseriesbelievethatformaltechniqueslike,forexample,independence
friendlylogic,dialogicallogics,multimodallogics,gametheoreticsemanticsandlinear
logics,havethepotentialtocastnewlightonbasicissuesinthediscussionoftheunity
ofscience.
This series provides a venue where philosophers and logicians can apply specific
technicalinsightstofundamentalphilosophical problems.Whiletheseriesisopentoa
widevarietyofperspectives,includingthestudyandanalysisofargumentationandthe
criticaldiscussionoftherelationshipbetweenlogicandthephilosophy ofscience,the
aimistoprovideanintegratedpictureofthescientificenterpriseinallitsdiversity.
Moreinformationaboutthisseriesathttp://www.springer.com/series/6936
Jacques Dubucs (cid:129) Michel Bourdeau
Editors
Constructivity and
Computability in Historical
and Philosophical
Perspective
123
Editors
JacquesDubucs
MichelBourdeau
Institutd’histoireetdephilosophie
dessciencesetdestechniques
(IHPST;CNRS-Paris1-ENS)
Paris,France
ISSN2214-9775 ISSN2214-9783(electronic)
ISBN978-94-017-9216-5 ISBN978-94-017-9217-2(eBook)
DOI10.1007/978-94-017-9217-2
SpringerDordrechtHeidelbergNewYorkLondon
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Preface
This book coversthe almost 80 years from Turing’sseminal paper of 1936 to the
presentandfocusesontwodevelopmentsduringthatperiod.1
Ontheonehand,thedevelopmentofcomputabilitytheoryandcomplexitytheory
calledforgeneralizations,restrictions,andothermodificationsofTuring’soriginal
machine.On the otherhand,it became clear thatrecursiontheorycannotserveas
a foundationalanalysis of the notion of computable function(al),which has to be
acceptedasprimitive.
In different ways, both of these developments contributed to a unified view of
logic. The first development helped bringing various phenomena together into a
1In her recent book Logic and Philosophy of Mathematics in the Early Husserl (Dordrecht:
Springer 2010), Stefania Centrone has drawn attention to the fact that the first to call for a
theoreticalstudyofcomputabilityprobablywasHusserl.InhisPhilosophyofArithmeticof1891
(CollectedWorksvol.X,trl.D.Willard,Dordrecht:Springer2003),heformulatedthefollowing
‘generalpostulateofarithmetic’:‘thesymbolicformationsthataredifferentfromthesystematic
numbersmust,wherevertheyturnup,bereducedtothesystematicnumbersequivalenttothem,
astheirnormativeforms.Accordinglytherearises,asthefirsttaskofArithmetic,toseparateall
conceivablesymbolicmodesofformationofnumbersintotheirdistincttypes,andtodiscoverfor
eachtypethemethodsthatarereliableandassimpleaspossibleforcarryingoutthatreduction’
(p.277).Heconsideredthearithmeticaloperationstobebutthemethodstocarryoutthatreduction
(p.277),andunderstoodcomputation,inarithmeticbutalsomoregenerally,as‘anyrule-governed
mode of derivation of signs from signs within any algorithmic sign-system according to the
“laws”—orbetter:theconventions—for combination,separation, andtransformation peculiarto
thatsystem’(p.273).Healsoraisedthequestionofthecomputabilityofnumbersthataredefined
byasystemofequations(pp.296–298).Husserldidnot,however,attempttodevelopsuchatheory
ofcomputability,andhissuggestionseemsnottohavebeenpickeduponbyhiscontemporaries
either. Moreover, it seems that none of those who came to play a role in the development of
computabilitytheorywasawareofHusserl’ssuggestion.Theyhad,ofcourse,ampleindependent
motivation.Butitwouldbeinterestingtoknow,forexample,whetherSkolemknewthischapter
byHusserl.SkolemvisitedGöttingenduringthewinterof1915–1916;thatwasHusserl’slastterm
asaprofessorthere,beforemovingtoFreiburg.
v
vi Preface
framework in which the original Turing machine turned out to be a special, and
in some contexts ideal(ized), case. The second development dispelled a serious
misunderstanding according to which recursion theory should ideally replace
informal reflection on the notion of computable function. Instead, there is room
andindeedneedforboth,dependingonone’spurpose.2
Thetwodevelopmentsarediscussedinsevenchapters,asfollows.
GöranSundholm’sopeningchapter,ConstructiveRecursiveFunctions,Church’s
Thesis,andBrouwer’sTheoryoftheCreatingSubject,firstdiscussesrecursivever-
susconstructivefunctionsand,followingHeyting,stressesthatfromaconstructive
pointtheformercannotreplacethelatter.Thesecondhalfofthepapertreatsofthe
Kreisel-MyhilltheoryCSforBrouwer’sCreatingSubject,anditsrelationtoBHK
meaning-explanationsandKripke’sSchema.Kripke’sSchemaisreformulatedasa
principle and shown to be classically valid. Assuming existence of a verification-
object for this principle, a modification of a proof of conservativeness of Van
Dalen’s,isshowntogivearelativeBHKmeaningexplanationfortheKreisel-Myhill
connective.The resultoffersan explanationof why Kripke’sSchema can be used
asa replacementofthe TheoryofCreatingSubjectwhenformulatingBrouwerian
counter-examples.It also shows that the Theoryof Creating Subject is classically
valid.
Therelationbetweencomputationandmachineisasoldastheabacus,butonly
with Turing’s pioneering work to this relation became central to computability
theory. Chapter 2, Jean Mosconi’s The Developments of the Concept of Machine
Computabilityfrom1936tothe1960s,dealswiththisaspect,whichiscrucialfrom
atechnologicalpointofview.MosconiexplainshowTuring’sideasweregradually
adopted, developed and modified, leading to something much closer to the actual
computer.Thedevelopmentcanbedividedintothreestages.First,thereisastrong
contrastbetweenaquickacceptanceoftheconceptualanalysisofcomputationand
a scarce use of the technical potentialities of Turing’s contribution: While Gödel
quickly saw the philosophicalrelevance of Turing’s work (it made him overcome
hisobjectionstoChurch’sThesis),Postwasperhapsthefirsttoseeitsmathematical
fruitfulness. The Turing Machine enters in a crucial way in Post’s 1947 proof
of the algorithmic unsolvability of Thue’s problem. Then follows the rise of the
theory of Turing Machines in the 1950s. Here the Turing Machine was not used
to solve problemsor gain philosophicalinsights, but was taken as a properobject
of mathematical investigation. The main idea was to include the Turing Machine
in the ‘general and logical theory of automata’ that Von Neumann suggested to
develop in the late 1940s. From this point of view, it became possible to study
a larger class of automata: not only finite ones, as it was initially the case with
2Thatleavesopenthequestionwhetherthenotionsorrecursivefunctionandcomputablefunction
nevertheless have the same extension. Church’s Thesis asserts that they do; but within the
intuitionistictheory of the creating subject, Kripke has constructed a computable function that
isnotrecursive.ThiscounterexampleisdiscussedinthepapersbySundholmandVanAtteninthe
presentvolume.
Preface vii
ShannonorMcCulloch’sautomata,butalsoinfiniteones.Startingfromthegeneral
structure of the original Turing Machine, various restrictions and generalizations
were defined,leadingto a hierarchyof automata,and to a betterunderstandingof
what Turing Machines can achieve and how. Finally, the progress of technology
made it necessary to rework Turing’s model so as to establish the link between
computabilitytheoryandcomputerpractice.HaoWang’smachineB(1957),which
usesthenotionofinstructioninsteadofthenotionofstate,wasthefirstattemptto
bridgethegap,butacompletelyexplicitmodelofaprogramandregistermachine
wasarrivedatonly1963,byStephensonandSturgis.
With the development of computer science and the increasing demand for
feasibility,computabilitytheorygavebirthtocomplexitytheory.Chapters3and4,
by Serge Grigorieff and Marie Ferbus-Zanda, respectively, are dedicated to Kol-
mogorovcomplexity,which may be seen as an offspring of computabilitytheory.
Chapter 3, Grigorieff’sInformationand randomness,is the more technicaloneof
the two. Kolmogorovintroduceda radically new approach to the measurementof
information. In the combinatorial approach, the information content of an object
x was defined as the length of the shortest binary word which ‘encodes’ x.
Kolmogorov’sidea was to bringin the resourcesof computabilityby definingthe
informationcontentofx as thelengthof theshortestprogramwhichcomputesx.
The chapter givesthe main concepts and results, together with their proofs;some
relatednotionsofcomplexity,likeLevinmonotonecomplexityorSchnorrprocess
complexity are also presented. A new step was taken when Kolmogorov, and
independently Chaitin, noticed that the algorithmic theory of information could
also be used to give a definition of randomness, which was still lacking after the
axiomatizationof probabilitytheory. The basic idea is that a word is random if it
is incompressible,that is, if there is no shorter way to describe it. Martin-Löfhas
shownthatthenecessaryconditionwasalsoasufficientone.
Chapter 4, Application to Classification Theory, takes a more philosophical
stance and shows the relevance of Kolmogorov complexity to computer science,
whereit has alreadyfounda numberof usefulapplications.With the World Wide
Webanditshugenetworkofmachines,theanalysisofinformationprocessinghas
becomeevenmorechallengingandtheneedforaclassificationevenmoreurgent.
Up to now the two main approacheshave been classification by compression and
the so-called Google classification. But they still lack a good formalisation. As
theybothusecomplexity,Ferbus-ZandaproposestotakeKolmogorovalgorithmic
information theory as the mathematical foundation of information classification
we are looking for. The two main approaches can now be formulated in terms
of two types of definition of mathematical objects, namely iterative definitions,
based onset theoreticalunion,andinductiveorrecursivedefinitions,based onset
theoreticalintersection;theycanalsobeseenasbottom-upandtop-downversions
of the same underlyingtheory, of which Ferbus-Zanda gives a short presentation.
Furthermore,she shows how these two dual modes are also found in information
systems,particularlytherelationaldatabasemodelintroducedbyCoddinthe1970s.
Chapter5,ProofTheoreticalSemanticsandFeasibility,byJeanFichot,returns
toconstructivity.Accordingto oneofthebestknownjustificationsofconstructive
viii Preface
reasoning,themeaningoflogicalconstantsis givenby theirintroductionrulesor,
whatamountstothesame,bywhatcountsasacanonicalproofofit.Butcanonical
proofs are idealisations of the ones we normally use; depending on what types
of knowledge one admits, proof theoretical semantics may therefore be liable to
be invalidatedby the non-feasibilityof the canonicalproofs.Fichot’scontribution
reviewstwoattemptstoovercomethisdifficultyandtogoonestepfurthertowards
feasiblecanonicalproofs.Thefirstisthenewrecursion-theoreticcharacterisationof
polytimefunctionsgivenin1997byBellantoniandCook.Besidesnaturalnumbers,
they use feasible numbers and succeed in giving a meaningfulexplanation in the
style of Dummett for those numbers. Unfortunately,there remains a gap between
feasiblearithmeticandthefeasibletheoryofproofswearelookingfor.Thesecond
attempt is light affine logic, a system introduced by Asperti in 1998 and further
studiedby Baillot and Terrui.The mainidea comesfromlightlinear logic, where
Girardaddedanewmodaloperator,‘§’,totheofcourseoperator,‘!’,oflinearlogic.
Just as ‘!’ allows for contraction, ‘§’ allows for weakening. Besides the formulas
that express perennial propositions and can be contracted and reused as often as
wanted,and the formulasthat are simply true,we mustnow admita third kindof
formulas.But thissplitting of ‘!’into two modaloperatorsgivesveryusefultools
forcontrollingthecomputationalcomplexityofthecuteliminationprocedure.Jean
Fichot explainshow the justification for the logical rules of such a system can be
given.
The sixth chapter, Recursive Functions and Constructive Mathematics, by
Thierry Coquand, addresses one of the most fundamental questions concerning
the relations between constructivity and computability: is the theory of recursive
functions needed for a rigorous development of constructive mathematics? The
answer is negative in both the theoretical and the practical sense. The argument
proceeds in two steps. The first one shows how the success of recursion theory
fostered a lack of sense for constructivity.As was noted early on by Heyting and
Skolem,fromaconstructivepointofview,thetheoryofrecursivefunctionscannot
giveusaformaldefinitionoftheintuitivenotionofcomputablefunction.Kleene’s
definitionof(cid:2),forinstance,usesexistentialquantification:ifR.x;y/isarecursive
relation and if .x/EyR.x;y/ holds, then (cid:2)yR.x;y/ is a recursive function of x.
But we are left with a dilemma: if the quantifieris interpretednon-constructively,
the relation between computability and constructivity is lost; if it is interpreted
constructively,thenthedefinitioninfactpresupposessomenotionofacomputable
function.ThesecondstepinCoquand’sargumentbeginswithBishop’sFoundations
ofConstructiveAnalysis(1967).Thebookwasa breakthrough:Ittaughtusthatit
wasnotonlytheoreticallypossiblebutalsopracticallymoresatisfactorytointroduce
functionsinconstructivemathematicswithoutmentioningrecursivity.Muchcurrent
work in constructive mathematics strongly relies on Bishop’s ideas. It is also
noteworthy that one popular definition of constructive mathematics, according to
whichitismathematicsdevelopedusingintuitionisticlogic,isindependentofany
notionofalgorithm.
Thefinalchapter,GödelandIntuitionism,byMarkvanAtten,startswithabrief
survey of Gödel’s personal contacts with Brouwer and Heyting. Some examples
Preface ix
arediscussedwhereintuitionisticideashadadirectinfluenceonGödel’stechnical
work.Thenit is arguedthatthe closest rapprochementofGödelto intuitionismis
seen in the developmentof the Dialectica Interpretation,duringwhichhe came to
acceptthenotionofcomputablefunctionaloffinitetypeasprimitive.Itisshownthat
Gödel already thoughtof that possibility in the Princeton lectures on intuitionism
of Spring 1941, and evidence is presented that he adopted it in the same year or
thenext,longbeforethepublicationof1958.Draftmaterialfortherevisionofthe
DialecticapaperisdiscussedinwhichGödeldescribestheDialecticaInterpretation
asbeingbasedonanewintuitionisticinsightobtainedbyapplyingphenomenology,
and also notes thatrelate the new notionof reductiveproofto phenomenology.In
anappendix,attentionisdrawntonotesfromthearchiveaccordingtowhichGödel
anticipated autonomous transfinite progressions when writing his incompleteness
paper.
This book has grown out of the meeting ‘Computability and Constructivity in
Historical and Philosophical Perspective’, which took place at the École normale
supérieure in Paris, December 17–18,2006. The organiserswere Jacques Dubucs
(IHPST),MichelBourdeau(IHPST),Jean-PaulDelahaye(UniversitédesSciences
et Technologiesde Lille), and Gerhard Heinzmann (Université de Nancy II). The
meeting was a Joint Session of the two divisions of the International Union of
History and Philosophyof Science (IUHPS):the Division of Logic,Methodology
and Philosophy of Science (DLMPS) and the Division of History of Science and
Technology(DHST).
Paris,France JacquesDubucs
June2013 MichelBourdeau
