FormalMathematicsand MathematicalPractice PeterKoepke,MathematicalInstitute,UniversityofBonn,Germany 14thCongressofLogic,Methodology,andPhilosophyofSc

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1 FormalMathematicsand MathematicalPractice PeterKoepke,MathematicalInstitute,UniversityofBonn,Germany 14thCongressofLogic,Methodology,andPhilosophyofScience SpecialSymposium:MathematicsandtheNewTechnologies Nancy,July22,2011

2 Formalmathematics Formalmathematicshasbecomeanestablishedresearcharea Formalmathematicsisalreadycoveringsubstantialmathematicalresults. 3. Formalmathematicsisbeginningtointeractwithresearchmathematics. 4. Formalmathematicscouldbecomepartofmathematicalpractice. Theacceptanceof formal mathematicsinmathematical practicewill dependon 5. thenaturalnessofitsuse. 6. Thenaturalnessofformalmathematicscanbeincreasedconsiderably. 7. Formalmathematicswillbecomeaneverydaytoolinmathematicalpractice(?)

3 Foundationsofformalmathematics :Uber formalunentscheidbaresatze derprincipia KurtGodel MathematicaundverwandterSystemeI. Exaktheithat DieEntwicklungderMathematikinderRichtungzugro erer,da weitegebietevonihrformalisiertwurden,in bekanntlichdazugefuhrt derart,da dasbeweisennacheinigenwenigenmechanischenregeln vollzogenwerdenkann.dieumfassendstenderzeitaufgestelltenformalen SystemesinddasSystemderPrincipiaMathematica(PM)einerseits,das Zermelo-Fraenkelsche(vonJ.v.Neumannweiterausgebildete)AxiomensystemderMengenlehreandererseits.DiesebeidenSystemesindsoweit, da alleheuteindermathematikangewendetenbeweismethodeninihnen formalisiert,d.h.aufeinigewenigeaxiomeundschlu regelnzuruckgefuhrt sind.

4 Foundationsofformalmathematics :Uber formalunentscheidbaresatze derprincipia KurtGodel MathematicaundverwandterSystemeI. Asiswell-knownthedevelopmentofmathematicstowardsgreaterexactnesshasleadtothepointthatlargeareaswereformalisedinawaythat proofscanbecarriedoutaccordingtoasmallnumberofmechanicalrules. [...]Thesetwosystemsaresufficientlydevelopedsothatallproofmethods currentlyappliedinmathematicscanbeformalisedinthem,[...].

5 Foundationsofformalmathematics :Uber formalunentscheidbaresatze derprincipia KurtGodel MathematicaundverwandterSystemeI. Asiswell-knownthedevelopmentofmathematicstowardsgreaterexactnesshasleadtothepointthatlargeareaswereformalisedinawaythat proofscanbecarriedoutaccordingtoasmallnumberofmechanicalrules. [...]Thesetwosystemsaresufficientlydevelopedsothatallproofmethods currentlyappliedinmathematicscanbeformalisedinthem,[...].

6 formal mathematics: modelingmathematicsbymathematicallogic Foundationsof Mathematics Mathematicallogic basicnotions symbols -Syntax Languageformulas statements proofs formalderivations( ) structures sets,relations,etc -Settheoretic Ontologysatisfactionrelation semantics truth implication logicalimplication( )...

7 formal mathematics: modelingmathematicsbyformallogic Foundationsof completenesstheorem: = tion formalmathematics:toactuallyproduceformalderivationsfrominformalproofs excellentagreementbetweenontologyandsemantics: agroupisasetsuchthat... completeagreementbetweensyntaxandsemantics: Godel hence:everyproofcanbereplacedbyaformalderiva-

8 Onthefeasabilityofformalmathematics N.Bourbaki: [...]suchaprojectisabsolutelyunrealizable:thetiniestproofat thebeginningsofthetheoryofsetswouldalreadyrequire severalhundredsofsignsforitscompleteformalization.[...] formalizedmathematicscannotinpracticebewrittendownin full,[...] J.McCarthy: Proofstobecheckedbycomputermaybebrieferandeasierto writethantheinformalproofsacceptabletomathematicians. Thisisbecausethecomputercanbeaskedtodomuchmore worktocheckeachstepthanahumaniswillingtodo,andthis permitslongerandfewersteps.

9 Thedevelopmentofformalmathematicssystems Automath,deBruijn,~1967 Mizar,Trybulec,~1973 Isabelle/Isar,Paulson,Nipkow,Wenzel,~2002 Coq HOLLight,Harrison manyothersystems

10 SubstantialmathematicsinAutomath vanbenthemjutting,1977 CheckingEdmundLandau,GrundlagenderAnalysis

11 SubstantialmathematicsinMizar BanachFixedPointTheoremforcompactspaces,theBrouwer Theorem, thebirkhoff VarietyTheoremfor FixedPoint manysortedalgebras,fermat'slittletheorem,thefundamental TheoremofAlgebra,theFundamental Theoremof thegodel CompletenessTheorem, thehahn- Arithmetic, BanachTheoremforcomplexandrealspaces,theJordan CurveTheoremforspecialpolygons,...

12 SubstantialmathematicsinIsabelle/Isar 'sproofoftherelative largest...istheformalizationofgodel consistencyoftheaxiomofchoice

13 SubstantialmathematicsinCoq FundamentalTheoremofAlgebrabyHermanGeuvers,Freek Wiedijk,JanZwanenburg,RandyPollack,HenkBarendregt ProofofBuchberger'salgorithmbyLaurentThery,Henrik Persson...

14 SubstantialmathematicsinHOLLight fundamentaltheoremof Fundamentaltheoremofcalculus, algebra,inversefunctiontheorem,brouwer'sfixpointtheorem, -Skolem Jordancurvetheorem,compactnessandLowenheim 'sfirstincompletenesstheorem,... theorem,godel

15 BigtheoremsinIsabelle/Isar PrimeNumberTheorem,elementaryproof,byJ.Avigadetal.

16 BigtheoremsinHOLLight PrimeNumberTheorem,analyticalproof,byJ.Harrison

17 BigtheoremsinCoq FourColourTheorem,byG.Gonthier

18 Theflyspeckproject FormalproofoftheKeplerconjecture(T.Halesetal.),using varioussystems.

19 puterprogramsourcecode. Sofar,formalmathematicsdoesnotfitcommonmathematicalpractice(s). Whyisformalmathematicsnotyetwidelyused? Wiedijk:Theotherreasonthattherehasnotbeenmuch progressonthevisionfromtheqedmanifestoisthat currentlyformalizedmathematicsdoesnotresemble realmathematicsatall.formalproofslooklikecom-

20 Naturalnessinformalmathematics inputandoutputlanguagesandformats userinterfaces logics backgroundtheories automation...

21 proofs? Naturallanguage mathematicallanguage=naturallanguage+formulas usecomputationallinguisticstoparseinputlanguage formalgrammarsdefinecontrollednaturallanguagesfor mathematics discourserepresentationtheorycanhandlemathemat- icaltextsasdiscourses(likestories) usenaturallanguagetodescribe/steercoqorholstyle

22 Userinterfaces A TEX-stylelanguageformathematicaltypesetting usel mergeexistingformal mathematicsinterfaceswith L A TEXeditors usel A TEXqualitywysiwygeditorslikeTEX MACS

23 Logics classicallogic rich(dependent)typesystem declarativeproofstyle manyfiguresofproof/argumentation

24 andsubstitutions,beforeinvokingautomatictheorem andtextstructure Automation usestronggeneralorspecialpurposeautomaticthe- oremproverstoderivestatementsinproofsfrompre- cedingpremises use``reasoner''modulesforstraightforwardinferences provers premiseselectionaccordingtonaturallanguagecues

25 TheSADproject:SystemforAutomatedDeduction A.Lyaletski,A.Paskevich,K.Verchinine ForTheL:FormulaTheoryLanguage,acombinationof controlledenglishwithmathematicalnotation Reasonerperforming`òbvious''inferences genericinterfacetofirst-orderprovers

26 TheSADproject:SystemforAutomatedDeduction TheoremMain. Forallnonzeronaturalnumbersn,m,pifp*(m*m)=(n*n) thenpiscompound. Proofbyinduction.Letn,m,pbenonzeronaturalnumbers. Assumethatp*(m*m)=(n*n).Assumethatpisprime. Hencepdividesn*nandpdividesn.Takeq=n/p. Thenm*m=p*(q*q).Indeedp*(m*m)=p*(p*(q*q)). m<n.indeedn<=m=>n*n<=m*m. Hencepiscompound. qed.

27 matics modelsnatural languageproofsusingcomputer-supportedmethodsofformallinguistics(naturallanguage andformal logic (automatic theorem processing) provers) TheNaprocheproject:Naturallanguageproofchecking studiesthesyntaxandsemanticsofthelanguageof mathematicalproofs,emphasizingnaturallanguageand naturalargumentation,relatingthemtoformalmathe-

28 ? Theorem30:Forallx,y,z,x (y+z)=(x E.Landau,GrundlagenderAnalysis,1930 y)+(x z). Proof:Fixx, y.x (y +1)=x y =x y + x=(x y)+(x 1). Nowsupposex (y + z)= (x y)+ (x z). Thenx (y + z )=x ((y + z) ) = (x (y + z)) + x = ((x y) + (x z)) + x = (x y) + ((x z)+x)=(x y)+(x z ). Thusbyinduction,forallz,x (y+z)=(x y)+(x z).qed.

29 CombiningSADwithnaturallanguage Theorem 1. (Fuerstenberg) Let S = {Arseq(0, r) r is prime}. Then S is infinite. Proof. We have S ={1,1}, indeed n belongs to S iff n has a prime divisor. Assume that S is finite. Then S is closed and S is open. Take p such that ArSeq(1,p) S. Claim. ArSeq(1,p) has an element that does not belong to {1,1}. Proof. 1 + p and 1p are elements of ArSeq(1, p). 1 + p 1 1 p p 1 1p 1. qed. Contradiction.

30 vincedthatformalverificationofmathematicswilleven- tuallybecomecommonplace. philosophyofmathematics? Outlook mathe- Combinebestpracticesfromvariousformal maticssystemstoobtainnaturalnessandpower willthisachieveacceptancebymathematicalpractioneers JeremyAvigad:Onapersonalnote,Iamentirelycon- Whatwouldbetheimplicationsofawidespreaduseof formalmathematicsforthemethodology,practice,and

31 ThankYou!

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