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History of Induction and Recursion B

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History of Induction and Recursion B

  1. 1. 2012/2013 History of Mathematical Induction & Recursion [B]. [Anaccount of the historyof mathematical inductionandmathematicalrecursionwithan expansiontotransfinite inductionandtransfiniterecursion] Damien MacFarland Alexandre Borovik Math40000
  2. 2. History of Mathematical Induction & Recursion [B]. 1
  3. 3. History of Mathematical Induction & Recursion [B]. 2 Introduction: On the 3rd March 1845, GeorgFerdinandLudwigPhilippCantorwasborninSt Petersburg, Russia.Cantorwouldsoonmove toGermany where he wouldspendmostof the remainderof his life.Havingstudiedtobecome anengineerat hisfathers’ request,Cantorwouldturn to mathematics.Cantorwassaidto have shownexceptional skill inmathematicswhilstatschool, particularlyintrigonometry.However,whilstatuniversityinBerlin,Cantorcame underthe influence of three greatmathematicians –Karl Weierstrass,LeopardKroneckerandErnstKummer – and it was undertheirinfluence thatCantorshowedaninterestinnumbertheory. Havingfinisheduniversity,Cantorwouldeventuallyfindapositionatthe Universityof Halle where he workedalongside HeinrichHalle,whochallengedCantortoprove the problem of the uniquenessof representationof afunctionasa trigonometricseries.WiththisCantorturnedhis attentionfromnumbertheorytoanalysis.Cantorsolvedthe problemwithinayearof takingup the challenge.Trigonometricseriesrepresentationwoulddraw Cantor’sinterestawayfromanalysisand back towardsnumbertheory. In 1873, Cantor producedthree ground-breakingresults.The firsttwoshowedthatthe rational numbersandthe algebraicnumberswere bothcountable,i.e.couldbe putintoaone-to- one correspondence withthe natural numbers.The thirdresultwastoshow that the real numbers couldnot be put intoa one-to-one correspondencewiththe natural numbers,thusmakingthem uncountable.The resultwas verycontroversial atthe time andwouldleadtoa lot of toughyears for Cantor as he triedto justifyhisresulttothose whodoubtedit. One reasonfor the ridicule whichCantor’sworkreceivedwasbecause of itsintroductionof newconcepts suchas his transfinite numbers intomathematics.Anotherreasonwasitslevel of abstractionand use of philosophical arguments.WithCantorintroductionof new conceptshe often arguedextensivelyfromaphilosophical pointof view,distancinghisreasoningfrommathematics. Howeverthe maincause of confrontationtowardsCantor’sworkcame fromhisuse of the infinite. We knowthatboth the natural numbersandthe real numbershave nogreatestnumber. Both are infinite totalities.Cantor’sresultthatthe real numberswere notdenumerableessentially saidthat the real numbershad a differentinfinite entitythanthatof the natural numbers. What Cantor impliedisthattheyare more thanone type of infinity. Inordertojustifythis,Cantor embarkedona long,enduringjourneytodevelophistheoryof transfinite numbersandsolve his ContinuumHypothesis.
  4. 4. History of Mathematical Induction & Recursion [B]. 3 The concept of infinityhasalwaysbeenadifficultone.Fromthe time of the earlyGreek mathematiciansupuntil the time 18th Century,infinitywasalwaysseenasthe potential infinite; everywhere finite butstillwithoutend.Thiswasdue toAristotle whohadbannedthe actual infinite inthe 4th CenturyB.C.due to itscontradictive characteristics.However,inthe 19th Century, philosopherssuchasBolzanobeganto workon the infinite andinvestigatedthe paradoxeswhichit ledto withinmathematics.The maindigressfromthe use of infinityinmathematicscame fromits paradoxes.Dedekindturnedthisnegativeintoapositive andwasone of the firstmathematiciansto challenge the Aristotle stance onthe infinite. The paradoxesthatwere producedas a resultof infinityoftenstemmedfromtheir contradictionsof lawsanddefinitionsthathadbeenconstructedonlytoincorporate the finite. Dedekindtookone suchcontradiction andusedittodefine aninfiniteset.Fromthishe thendefined the natural numbersas one such infiniteset. However,before the time of CantorandDedekind, mathematicianshadconstructedtheirownprocedure of dealingwithinfinitequantities. Withinmathematicsthere are avaried collectionof mathematical proceduresandconcepts whichwe frequentlyuse.Some we use tocreate anddevelopnew results,otherswe use toverify and prove onesinwhichwe alreadyhave.Asmathematicshasdevelopedwithtime,ithasseenthe introductionof newconceptswhichhave servedinhelpingwithsuchdevelopment. These new conceptsare notalwaysperfectintheirbeginningbuthave beenmoulded intomore desirable forms. Withregardsto handlinginfinite quantities,the principlesof inductionandrecursionmade it possible tomanage the infinite withrespecttofinite terms. In thispiece of workI wishtoinvestigate the originsof the principlesof inductionand recursionandhowthe twoconfine the natural numberstoa couple of finite steps.Ishall give account forhow mathematical recursionwasnaturallyintroducedtomathematicsandhow itwas usedto produce one of the most importantresultsof the philosophyof mathematicsbyGödel. I shall thenworkmy way back, fromPascal whogave structure to the conceptof induction, throughthose whowere the firstto realize the benefitsandusesof sucha principle andshowed traces of the conceptthroughouttheirwork,usingtheirownsimilarversionstoobtainresults,and mostlyproof to results,inshort,finite,structuredmanners.ThenIshall addressthe name of mathematical inductionandhowthe principle becamepopularinthe 18th and 19th Century’s,paying particularattentiontohowbeneficial the conceptwastoPeanoand Dedekind. The principlesof inductionandrecursionare of amajor benefitintacklingthe totalityof the natural numbers,however,theycannotbe appliedtoCantor’stransfinite numbers.Thus,Ishall give an account of the originsCantor’stheoryof transfinitenumbers,the reasonshe felttheywere
  5. 5. History of Mathematical Induction & Recursion [B]. 4 needed,the struggle he hadfortheiracceptance,the differentpathshe tookintheirdevelopment and the resultingtheoryhe leftbehind.AfterCantor’slastmajorcontributiontothe theory inwhich he produced,itwas takenupby otherswhotriedto solve anyproblemswhichCantorleftbehind. The subjectwouldcome underthe influence of ErnstZermelowhowouldaddaxiomstoCantor’s naïveset theory.Throughoutthe developmentof the theoryof transfinite numberswe shall produce the conceptsof transfinite inductionandrecursionandgive anaccountof themtowards the end. I expectthe readertobe familiarwithabasicunderstandingof settheoryandhave a universitylevel of mathematical understanding.
  6. 6. History of Mathematical Induction & Recursion [B]. 5 Contents: Introduction:............................................................................................................................ 2 [1] Induction & Recursion:............................................................................................................. 6 [1.1] The Natural Numbers:.......................................................................................................6 [1.2] Mathematical recursion:....................................................................................................8 [1.3] Gödel’s Theorems:.......................................................................................................... 12 [1.4] Mathematical Induction: ................................................................................................. 16 [1.5] The Origin of Mathematical Induction:............................................................................. 20 [1.6] Maurolycus & Gersonides:............................................................................................... 25 [1.7] The Name – Mathematical Induction:............................................................................... 30 [1.8] Bibliography:................................................................................................................... 33 [2] Set Theory and Transfinite Numbers:...................................................................................... 34 [2.1] Giuseppe Peano:............................................................................................................. 35 [2.2] Richard Dedekind:........................................................................................................... 38 [2.3] Arithmetices principia and Was Sollen:............................................................................. 43 [2.4] Dedekind’s Interest in Infinity:......................................................................................... 46 [2.5] Georg Cantor:................................................................................................................. 49 [2.6] Cantor’s Quest for the Transfinite Numbers:..................................................................... 54 [2.7] Cantor’s Struggle:............................................................................................................ 61 [2.8] Post-Grundlagen:............................................................................................................ 65 [2.9] Contributions to the Founding Theory of Transfinite Numbers:.......................................... 72 [2.2.1] Part I:....................................................................................................................... 72 [2.2.2] Part II:...................................................................................................................... 82 [2.10] Aleph-One and Transfinite Induction:............................................................................. 88 [2.11] Post-Beiträge:............................................................................................................... 92 [2.12]: Axiomatic Set Theory and Transfinite Recursion:............................................................ 94 [2.13] Summary:..................................................................................................................... 99 [2.14] Bibliography:................................................................................................................103
  7. 7. History of Mathematical Induction & Recursion [B]. 6 [1] Induction& Recursion: [1.1] The Natural Numbers: In mathematics,there are numerousthingsthatwe take forgranted.Asmathematicshas progressedthroughthe ages,ithasgrown andbecome more abstract. Whenwe say ‘5’ whatdo we actuallymeanby‘5’; 5 apples,5 minutes,5metres,5 grainsof sand? Whenwe say that ‘ten equals twofives’we knowthatwhenwe have 2 sets/groups/clusters,eachof size/quantity/length5, and thenour total is 10, regardlessof any 10 possible concrete examples. Backinthe agesof the ancient Greeks,where mathematicshad essentiallygrownfrom, philosopherslike Plato,Socrates,the Pythagoreansetal, beganto ignore concrete numbersof ‘things’andtooktocounting‘things’ philosophically,withnoconsiderationof what actuallycountinganything,andthatliveson today. Buildinguponthis,there ismuchmore we take forgranted withregardsour number systems e.g.the definitionsof odd,even,prime,complex,irrational,integer.Anditdoesn’tstopat definitions,we cancontinue toinclude basic,essential propositions,(1 + 1 = 2; 4 isaneven number;Pythagoras’Theorem).There are formulasandsummationsthatwe expectanyotherpier, withsimilarlevelsof mathematical knowledge asourselves,to know andto believetobe true. One thingwe alwaysuse,withoutanyconsiderationastowhyit istrue,is the notionthat,as an example, 2 < 3.Of course thisdoesseemobvious;if one has2 applesbut3 orangesthenof course one has more orangesthan apples,whichistrue syntactically.Butwhydowe considerthe number2 to be strictlylessthan3? Andwhyin turnis 3 strictlylessthan4, 4 lessthan5? We overlookthe factthat the natural numbers are in a naturalorderingi .Startingfrom1, 1 is succeeded by 2, whichissucceededby3, and that by4 and that by5, then6, 7, 8… We obtainanynatural number 𝑛 from1. The Greeksthoughtof 1 not as a number,butas ‘unity’oras the ‘One’.Thusany numberisa collectionof units.We obtainone numberbytakingits predecessorandadding 1 tothat. But how do we obtainsaidpredecessor?Well,fromits predecessor;andthatpredecessorfromitsownpredecessor;thisinturnhasits ownpredecessor. Eventuallywe arrive at 1,which,thinkingasthe Greeksdo,iswhere we can go no furtherwiththe natural numbers. The natural numbersare a simple concept,beginningfrom 1,we add 1 to it 𝑛 − 1 timesto obtainthe natural number 𝑛. Thenadding1 to 𝑛 we obtain 𝑛’s successor, 𝑛 + 1.Thus everynatural i In fact they are ‘a well ordering’ and we shall cometo that later.
  8. 8. History of Mathematical Induction & Recursion [B]. 7 numberhasa successor;eachdependsonthe one definedbeforeit. Thislatternotionof defining somethingintermsof howwe define anotherbringsustomathematical recursion. Accordingto the Oxforddictionary,recursionisthe processof ‘repeatedapplicationof a rule,definition,orprocedure tosuccessive results’.We see resemblancesof recursionineveryday life,e.g.the Russian MatryoshkaDolls,The Droste Effect;howeverwe mainlyuse recursionin mathematics inorderto define suchthingsas sequences,series,relationsandfunctions.We canuse recursiontodefine sets,ortodefine unionsandintersectionsof sets. Andthere are manymore usesof recursionindifferentareasof mathematics - logic,statistics,graphtheory…
  9. 9. History of Mathematical Induction & Recursion [B]. 8 [1.2] Mathematical recursion: Mathematical recursion isthe processof definingamathematical process byrepetition;a functionorprocedure definedintermsof itself. We define the natural numbersbyrecursive definition:startingfrom 1,add1 to obtainthe nextnatural number.Fromthisnew number,namely 2, add 1 to it to obtainthe nextnatural number, 3.From3, we add 1 to obtain 4. Etc. Before we startto look furtherintomathematical recursion Iwouldlike tolookat a well-knownandcommon example,the Fibonacci Numbers, attributed toLeonardoPisano Bigollo(LeonardoFibonacci) [ca.1170- ca.1250] of Pisa. The son of a wealthybusinessman,Fibonacci hadanextendedinterestinthe mathematicsof the East and the Arabs.In 1202, after returningto Italyhavingvisited Egypt,Sicily,Greece andSyria,Fibonacci published Liberabaci,a textmainlyfocusingonthe base 10 arithmeticof al-KhwārizmīandAbūKāmil.In Liber abaci appearsthe Fibonacci Numbers. The Fibonacci Numbers are definedas:  Define 𝐹1 = 𝐹2 = 1;  For the natural number 𝑛 > 2,define 𝐹𝑛 = 𝐹𝑛−1 + 𝐹( 𝑛−2). Thus the 1st fewtermsof thissequence are 1,1, 2,3, 5,8,13, 21,34,55, … . Thisprocess for definingastructure throughan arbitrary numberiswhatisknownas Recursive Definition inmathematics. Notice thatwe have essentially2parts to the definitionof the Fibonacci Numbers: (1) A startingpoint,indexedbythe natural numbers 1 &2; (2) A rule forthe formulationof greaterFibonacci Numbers,indexed bythe naturalsgreater than 2. The rule correspondsto the 2 previouslydefinedFibonacci numbers. Generally,todefineamathematical procedurerecursively,we: (1) assigna Base Case; (2) setup a Recursive Step. The Base Caseservesas ourstartingpoint. 1 beingthe base case of the natural umbers; 𝐹1 and 𝐹2 serve as the base case of the Fibonacci Numbers.The base case isour reference pointfrom where we can continue, toformulate the remainderof mathematical procedure.Andwe dothis Figure 1: Leonardo Fibonacci.
  10. 10. History of Mathematical Induction & Recursion [B]. 9 fromour Recursive Step.The recursive stepallowsustocontinue to formulate more examplesof a procedure;itextendsourdefinition towardsa possible infinite numberof termsforour procedure.The recursive stepallowsustodescribe aninfinite numberof instances ina finite quantityandthe natural number 𝑛 holdsthe infinite factor. We can thinkof a recursive definitionasa sequence, 𝑢1, 𝑢2, 𝑢3, 𝑢4,… , 𝑢 𝑛,… indexedbythe natural numbers.Thisallows us to visualizethe sequence inanorderparallel tothe naturals and see the sequence asbeing acountable setof terms.We can draw up results,formulas,ratiosetc.betweenthe natural number 𝑛 andthe term 𝑢 𝑛 of our sequence orevendefine the our 𝑢 𝑛 intermsof the natural number 𝑛. Anexample of the latteristhe factorial function, whichwe know to be 𝑛! = ∏ 𝑘𝑛 1 . Howeverwe candefine the factorial functionrecursivelyasfollows:  Base Case:let1! = 1;  Recursive Step:define,for> 1, 𝑛! = 𝑛 ∙ ( 𝑛 − 1)! . Nowif we were to rename ourfactorial functioninthe structure of a sequence,thenwe couldlookat the previousdefinitionas:  Base Case:let 𝑢1 = 1,  Recursive Step:define,for 𝑛 > 1, 𝑢 𝑛 = 𝑛 ∙ 𝑢 𝑛−1 Thus the 1st fewtermsof the sequence are: 𝑢1 = 1, 𝑢2 = 2, 𝑢3 = 6, 𝑢4 = 24, 𝑢5 = 120... The Degree of recursion issaid tobe the numberof predecessorsthatare usedindefining any termby the recursive stepi.e.itisthe numberof termsdefinedinthe base case. Lookingback at examplespreviouslydefined,the Fibonacci numbersare of degree 2 whereasthe factorial functionisof degree 1. Beginningwiththe 1st 3 Fibonacci numbersasa new base case,we can define the Tribonacci Numbersbytakingthe sumof the previous3 defined numbersas opposed to2. Thus the Tribonacci Numbersare of degree 3.Similarlywe candefine the Fibonacci 𝑛-stepNumberSequence whichwillbe of degree 𝑛. Nowthat we have lookedatthe degree of recursionIgive a standard procedure for definitionof recursivedefinition: Figure 2: The Sierpinski Triangle – showing iterations of the Recursive Step.
  11. 11. History of Mathematical Induction & Recursion [B]. 10 1. Base Case:let 𝑢1 = 𝑛1, 𝑢2 = 𝑛2,…, 𝑢 𝑚 = 𝑛 𝑚 where 𝑚 isthe degree of recursionfor this definition and 𝑛1, 𝑛2,…, 𝑛 𝑚 are 𝑚 pre-definedterms. 2. Recursive Step:let 𝑢 𝑚+1 = 𝑓( 𝑢1, 𝑢2,… 𝑢 𝑚)where f is an 𝑚-dimensionalfunctionorrelation. Setsare anotherimportantentityinmathematicsthatcan be definedbyrecursion;we can define ℕ,the setof natural numbers,recursively inthe followingmanner:  Base case:1 ∈ ℕ;  Recursive Step:if 𝑛 ∈ ℕ, then 𝑛 + 1 ∈ ℕ;  ExtremalClause:ℕ is the smallestsetsatisfyingtheseconditions. There are otherpropertiesthataccompanyourset ℕ of the natural numbers,namelythe Peano-DedekindAxioms;howeverif we looksolelyat ℕ asa setof elements,withoutanyregardsto the ordering,thenwe can define ℕrecursivelyasabove. Otherexamplesof setsthatwe can define by recursionare the positive evenintegers,the integers, the setof triangularnumbers, andthe set of squaredintegers. Insettheoryall of these exampleswouldbe accompaniedwithextracriteria withregardstheirordering,butfornow I choose to overlookthis aswe are still justlookingatsets withspecificelements. Notice thatI have addedanotherpiece of criteriatothe definition of ℕ,the ExtremalClause. We dothisto distinguishbetween 2differentsetswhichmay satisfy boththe Base Case andthe Recursive Step butyetmay still be 2 differentsets.Forexample,the set {1,1.5, 2,2.5,3, 3.5,4, …} satisfiesthe Base Case andRecursive Stepafore mentionedyetitisnotthe setof natural numbers as it includessome rational numberswhich are not‘whole’numbers. Takinga (possibly infinite) collectionof sets,we candefine recursivelytheirunions, intersectionsandCartesiancrossproducts. Thisispossible becauseof the associativityof these actions. For example,suppose we have the collectionof sets 𝑆1, 𝑆2,𝑆3,… 𝑆 𝑛,… where the indices are randomlyassignedtothe setsof the collection,then a) The union can be definedas:  ⋃ 𝑆𝑖 = 𝑆𝑖 1 𝑖=1 ;  ⋃ 𝑆𝑖 = (⋃ 𝑆𝑖 𝑛 𝑖=1 )⋃ 𝑆 𝑛+1 𝑛+1 𝑖=1 . b) The intersectioncanbe definedas:  ⋂ 𝑆𝑖 = 𝑆1 1 𝑖=1 ;  ⋂ 𝑆𝑖 = (⋂ 𝑆𝑖 𝑛 𝑖=1 )⋂ 𝑆 𝑛+1 𝑛+1 𝑖=1 . c) The Cartesiancross productcan be definedas:  ∏ 𝑆𝑖 = 𝑆1 1 𝑖=1 ;
  12. 12. History of Mathematical Induction & Recursion [B]. 11  ∏ 𝑆𝑖 = (∏ 𝑆𝑖)𝑛 𝑖=1 𝑛+1 𝑖=1 × 𝑆 𝑛+1.
  13. 13. History of Mathematical Induction & Recursion [B]. 12 [1.3] Gödel’s Theorems: Mathematical recursionisa tool usedthroughoutmathematics;afew exampleswe have seen.Itcan be usedineveryareaof mathematics, fromlogicto geometry,fromstatisticstograph theory.Recursionseemstobe anatural processof mathematics,one withoutanyorigin. Itappears to be a tool forbuildingstructuressuchthatwe can produce theorems,propositions,oralgorithms baseduponthese structures. Recursive definitionseemstofollow fromthe natural numbers since we use the naturalsto structure thisform of definition.Aswithanydefinitioninmathematics,we canjustmake any assumptionorassigna certain criteriaor a specificprocedure; the truthof whichwe justassume to hold. We can do thisbecause we are more interestedinthe resultsthatwe canproduce from these assumptionsthanthe assumptionsthemselves.We use the definition asourpointof reference from whichwe aimto buildupon, toexplore andinvestigate. There isnorequirement toverify thatour definitionistrue,the truthvalue followsfromthe natural numbersandthe axiomsexhibitedthere. The originof recursiontherefore cannotbe pinned downto one specificpointintime.The Fibonacci numbers were developedbyFibonacci inthe 12th centurybutthey were saidto be knownto the Indians before that. Fibonacci wasalso knownto have readand studied alotof IndianandArabictext inhis time,tracesof whichcan be foundin Liber abaci.Each row of Pascal’sTriangle, usedtodefine binomial coefficients (whichcan themselvesbe recursivelydefined),canbe recursivelydefinedbythe row above it.However,althoughit isnamedafterBlaise Pascal wholivedduringthe 17th century, ‘his’triangle wasinvestigatedbythe Geeks,Chinese,Hindu and Arabicmathematiciansbefore him. The originof mathematical recursioncanbe hard to trace but itspopularityandimportance iswell known.One such importantresulttocome fromrecursion isthat of Kurt Gödel. Born 1906 inBrno, nowof the CzechRepublic, Gödel wasaphilosopheraswell asa mathematicianwhofocusedmainlyonthe logical aspectsof mathematics. Withthe Czechoslovak Republicdeclaringitsindependencefromthe Austro-HungarianEmpire in1918, Gödel movedto Figure 3 - A young Kurt Gödel ca. 1922.
  14. 14. History of Mathematical Induction & Recursion [B]. 13 Viennain1924 to studyat university,havingalwaysconsideredhimselftobe an Austrianlivingina Czechoslovakmajority. In 1931, Gödel had publishedthe paper‘ÜberformalunentscheidbareSätzederPrincipia Mathematica und verwandterSystemeI’whichcanbe translatedas ‘On Formally Undecidable Propositionsof Principa Mathematica and Related SystemsI’.The paper,originally appearingin ‘MonatsheftederMathematikund Physik’ Vol.38,wasof significantimportance tomathematical logicand philosophyasitcontainedGödel’s highlyimportantIncompletenessTheorems. The theorems were usedtoanswerthe 2nd problemof DavidHilbert’s “bootstrapping” Program whichasked“to provethatthey (the axiomsof arithmetic) arenotcontradictory,thatis, thata finite numberof logical stepsbased upon themcan never lead to contradictory results.” The axiomsof arithmeticthatHilbertisreferringtoare the axiomsof the PeanoArithmetic for the setof natural numbers, ℕ,whichappearedinPeano’s 1889 book, The Principles of Arithmetic by a New Method:i 1. 1 ∈ ℕ; 2. 𝑎 ∈ ℕ,∃𝑎′ ∈ ℕ 𝑠. 𝑡. 𝑎′ = 𝑎 + 1; 3. ∄𝑎 ∈ ℕ 𝑠. 𝑡. 𝑎 + 1 = 1; 4. 𝑎, 𝑏 ∈ ℕ, 𝑎 = 𝑏 ⇔ 𝑎 + 1 = 𝑏 + 1; 5. 𝐹𝑜𝑟 𝑀 ⊂ ℕ, if  1 ∈ ℕ;  𝑎 ∈ ℕ ⇒ 𝑎 + 1 ∈ ℕ; then 𝑀 = ℕ. In effect,the problemisasking –isthere anyproof that thisarithmeticis consistenti.e.yield no contradictions?Gödel usedhistheoremstoprove thatthe answertothisquestionwasinfact, no. Hilbert’sProgramwanted tosecure the foundationsof mathematicsandthendevelopits foundationsfurther.Hilbertwantedtofindoutwhatlayin store forthe future generationsof mathematicians,whatpossiblenewtechniquestheywoulduse, andmostimportantly,whatresults wouldtheyyield. Butsome thought Gödel’sanswersoughttodestroysucha possibility,whereas Gödel himself sawitas a route to developfurtherHilbert’swork. i I only give 5 of the original 9 axioms as they are sufficientto describethe Peano Arithmetic. The remaining4 axioms deal with the transitive,reflexiveand symmetric properties of equality in the natural numbers.
  15. 15. History of Mathematical Induction & Recursion [B]. 14 The IncompletenessTheorems: 1. Anyeffectivelygeneratedtheorycapableof expressingelementaryarithmeticcannotbe bothconsistentandcomplete.Inparticular,foranyconsistent,effectivelygenerated formal theory thatprovescertainbasicarithmetictruths,there isanarithmetical statement that istrue,but not provable inthe theory. 2. Anyformal effectivelygeneratedtheorythatincludesbasicarithmetical truthsandalso, certaintruthsabout formal provability,if the theoryincludesastatementof itsown consistency,then the theoryisinconsistent. Gödel'spaperconsistedof 11 propositions.PropositionsVIandXIare now what are called the IncompletenessTheoremsanditispropositionXI,the 2nd of the Incompletenesstheorems,that answersHilbert’sproblem.Once Gödel hasestablishedthe 1st of the 2 IncompletenessTheoremshe proceedstobuilduponitto produce the 2nd . FirstI state PropositionV: “Every recursiverelation is definablein the systemP (interpreted asto content), regardlessof whatinterpretation is given to the formulaeof P.” AfterprovingPropositionV,Gödel statesPropositionVI.He doesso inthisorderas he requires the recursive elementof PropositionV toprove VI. The 1962 Englishtranslation,byB. Meltzer,of Gödel’spaperisintroducedbyRichardB Braithwaite,whohighlightsthe importance of recursioninmathematicsandthe importance ithad for Gödel inhispiece of work: “Recursive definition enablesevery numberin a recursively defined infinitesequenceto be constructed according to a rule, so thata remarkabouttheinfinite sequencecan be constructed asa remarkabouttherule of construction and notasa remarkabouta given infinite totality.” “For the proof of Gödel’s‘Unprovability’theoremtheimportanceof recursivenesslies in the fact(Proposition V) thatevery statementof a recursive relationship holding between given numbers 𝑥1, 𝑥2,…, 𝑥 𝑛 is expressibleby a formula 𝑓 of theformalsystemP which is ‘provable’within Pif the statementis true and ‘disprovable’within P...if the statement is false.” Recursionremainsanessentialandvaluable assetthroughoutmathematics.The abilityto representthe infiniteinthe termsof the finite allowsfor anefficientand more abstractpractice of mathematics.Its quickandeffectivemethodhasbeenknownthroughoutthe historyof mathematics,fromthe time of the ancientGreeks,throughtothe Renaissance andonwardsto
  16. 16. History of Mathematical Induction & Recursion [B]. 15 today,where itremainsincommonuse,frommathematicsusedinprimaryschool,tothe highest and mostintellectual of levelsof mathematics.The mainpurpose of mathematical recursionisfor statingdefinitionsordefiningfunctions,sequences,series,etc.However,asmathematicsbecame more abstract and everclosertologic,the needformathematical proof became greater.Through this,a close relative of mathematical recursiondeveloped;thatof mathematical induction.
  17. 17. History of Mathematical Induction & Recursion [B]. 16 [1.4] Mathematical Induction: “Even in mathematicalsciences,ourprincipal instrumentsto discoverthe truth are induction and analogy.” – Pierre-SimonLaplace, EssaiPhilosophiquesurlesProbabilités. Mathematical induction,oftenreferredtoas The Principle of Induction, hasclose tiesto mathematical recursion; if recursionisthe processof buildinginmathematics,theninductionisthe processof checkingthat build.Essentiallyinductionisaformof proof formathematical procedures that are definedrecursivelyinmathematics.Alternatively,once we have definedaprocedure recursively,we cancheckits validitybyinduction. Mathematical inductionisnota methodof discovery,butamethodof provingthat whichhasalreadybeendiscovered. Inarithmeticinduction provesthatsome propertyholdsforall positive integers.Inlogicitprovesthatthata propertyholds for a language baseduponthe lengthof a sentence.We candefine setsrecursively,andprove propertiesaboutthese recursivelydefinedsetsbyinduction. Mathematical inductionbearsaclose resemblance tothe inductionusedinthe other sciences.The inductivemethodsof otherscienceslookatgeneralitiesfromspecificexamplesin orderto formulate ageneral andcommonconjecture thatcan be putforwardand usedinother casesto testits strength. Similarlyinmathematicswe canusuallysee that if propertyholdsordoes not hold forarbitrary cases. Butthis isnot proof that that propertydoesinfactholdfor everycase but itis an indicatorthatit mightjustdo so. Where mathematical inductiondiffersisin the factthat one case dependsuponanother due to the fact that we have recursivelydefinedourprocedure,andthatiswhere we formthe ‘proof’ aspect. If we findacertainpropertyassociatedtoa certaincase thenwe lookto the nextcase that followstosee if the same propertyholdsinthatcase as well. Howeverwe usuallyneedastarting point,sowe prove that a propertyholdsforthe most trivial of casesfirst.Fromthat we lookat the nextcase afterit. If the property holdsforthat case as well,thenwe lookatthe succeedingcase and see if the propertystill holds.Andsoon. Butcan we testthat a propertyholdsfora structure of a vast size,ora set of infinite cardinality?Itwouldbe verytime consumingtocheckif it istrue that 1 + 2 + 3 + 4 + ⋯+ 𝑛 = 𝑛( 𝑛+1) 2 for 𝑛 = 12 or 𝑛 = 12222. In mostcaseswe use the truth whichwe install inthe axiomsforthe natural numbers. This allowsusto prove thata propertyholdsforall the natural numbers 𝑛 withouthavingtoinspect each case one byone. But in othercases,as withsetsor logic,we use our recursive definitions.These
  18. 18. History of Mathematical Induction & Recursion [B]. 17 recursive definitionsdohave a startingpoint,namelytheirbase cases,whichwe have previously seen.Andwe knowthe natural numbers have the base case 1i . Mathematical inductionhasthe same structure;a base case and the stepthat passes fromone case to another,althoughwe referto thisas the InductiveStep as it isslightlydifferentfromourrecursive step. ItisthisInductive Stepthat provesthata propertyholdsforeverycase inthe structure. We can state The Principle of Induction forthe natural numbers inthe followingaxiom: Supposethat 𝑃( 𝑛) is a statementinvolving a generalnaturalnumber 𝑛.Then 𝑃(𝑛) is true forall forthe naturalnumbersif; 1. 𝑃(1)is true, and 2. 𝑃( 𝑘) ⇒ 𝑃( 𝑘 + 1) forall naturalnumbers 𝑘. Notice the comparisonof thisaxiomandthe 5th Peanoaxiomstatedinsection[1.3].They are,albeitslightlywordeddifferently,the same. The Principle of Inductionisanextensionof the algebraicandorderii axiomsthat we have forthe natural numbers.Inductionismerelyanextension of somethingwe oftenoverlookaboutthe natural numbers; startingfrom1,we can reach anyother natural numberbysimplyadding 1. The statement 𝑃(1) isthe Base Caseof The principle of induction,whereasoursecond statementisthe InductiveStep.Notice thatthere isan implicationinthe inductivestep.So,if 𝑃( 𝑘) is true,then 𝑃( 𝑘 + 1) isalso true.We may thinkthat thismeans thatwe have to prove that 𝑃( 𝑘) is true and thus itfollowsthat 𝑃( 𝑘 + 1) istrue also,but whatwe have to do isshow that – if it wasthe case that 𝑃( 𝑘) was true,thenfromthis 𝑃( 𝑘 + 1) is alsotrue. Thisis where we getourInductive Hypothesis;we assume,asa hypothesis,that 𝑃( 𝑘) istrue andunderthat assumptionwe prove that 𝑃( 𝑘 + 1) is true. The concept of mathematical inductionis – fora general property 𝑃( 𝑛) of anatural number 𝑛, whateveritmaybe,the base case checksthat 𝑃(1) is true and then,bythe inductive hypothesis, 𝑃(1) ⇒ 𝑃(2), and againby the inductive hypothesis 𝑃(2) ⇒ 𝑃(3) andthus 𝑃(3) ⇒ 𝑃(4) ⇒ 𝑃(5) ⇒ ⋯ ⇒ 𝑃( 𝑛) ⇒ 𝑃( 𝑛 + 1) ⇒ ⋯ NowI wouldlike togive anexample of proof byinduction toillustratehow we use the inductive hypothesistoprove the inductivestep.I shall prove thatthe followingstatementholds: For all natural numbers 𝑛, the number 𝑛2 + 𝑛 is even. i In some cases when usingmathematical induction,we can assumethat 0 ∈ 𝑵 as a property may hold for the casewhen 𝑛 = 0. ii I have not mentioned the orderingaxioms atthis pointbut they will appear in a later section.
  19. 19. History of Mathematical Induction & Recursion [B]. 18 So 𝑃( 𝑛) isthe statement‘𝑛2 + 𝑛 is even’.Thuswe are lookingfora natural number 𝑚 such that 2𝑚 = 𝑛2 + 𝑛. We proceednowwiththe structure laidout inthe inductionaxiombefore. Base case:When 𝑛 = 1, 𝑛2 + 𝑛 = 1 + 1 = 2 thus we have verifiedthat 𝑃(1)istrue. InductiveStep:Assume nowforan InductiveHypothesis that,foran arbitrary 𝑘, ‘𝑘2 + 𝑘 iseven’, thenwe seektoshowthat ( 𝑘 + 1)2 + ( 𝑘 + 1) = 2𝑝 for some natural number 𝑝. From our inductive hypothesiswe canassume furtherthatthere exists 𝑞 belongingto the natural numberssuchthat, 2𝑞 = 𝑘2 + 𝑘. However( 𝑘 + 1)2 + ( 𝑘 + 1) = 𝑘2 + 2𝑘 + 1 + 𝑘 + 1 = ( 𝑘2 + 𝑘) + 2𝑘 + 2 = 2𝑞 + 2𝑘 + 2 = 2(𝑞 + 𝑘 + 1). So if we take 𝑝 = 𝑞 + 𝑘 + 1 thenwe have a natural number 𝑝 suchthat 2𝑝 = ( 𝑘 + 1)2 + ( 𝑘 + 1) as required. ∎ Notice that,fromthe assumptionthat 𝑃(𝑘) is true for an arbitrary 𝑘, we were able to prove that 𝑃(𝑘 + 1) was alsotrue by manipulatingwhatwe knew about 𝑃( 𝑘 + 1) toreacha trivial conclusionbasedon 𝑃( 𝑘)’sassumption andgeneral arithmetic. We donot necessarilyhave tostartwith 𝑛 = 1 for the base case,there are certainfunctions or theoremsthatonlysatisfy‘𝑓𝑜𝑟 𝑛 > 𝑛0’forsome certainnatural number 𝑛0 i.e. 𝑃( 𝑛) istrue for all natural numbers 𝑛 ≥ 𝑛0.If thisis the case we justneedto make 2 slightadjustments toourusual procedure – the base case ischangedto 𝑃( 𝑛0) and the inductive step nolongerappliestoall natural numbers 𝑘,but onlyfor 𝑘 ≥ 𝑛0. Suchan example of thiscanbe seenwhencomparing 𝑛2 and2 𝑛. When 𝑛 = 1,2,3 we have 𝑛2 > 2 𝑛. However,for 𝑛 ≥ 4,we have 𝑛2 ≤ 2 𝑛. So inprovingthat the latterholds,we use inductionwiththe base case for 𝑛 = 4. Thenforhe inductive stepwe prove 𝑘2 ≤ 2 𝑘 ⇒ ( 𝑘 + 1)2 ≤ 2 𝑘+1 for all 𝑘 ≥ 4. Anothervariationof mathematical inductionisthe methodof Strong induction. We require thismethodwhenwe have the case that 𝑃( 𝑘) alone isnot enoughtoimplythat 𝑃( 𝑘 + 1) is holds but we actuallyrequire some, ormaybe evenall,of 𝑃(1), 𝑃(2),… , 𝑃( 𝑘 − 1) also. The axiomof Strong Induction isgivenas: Supposethat 𝑃( 𝑛) is a statementforsomenaturalnumber 𝑛.Then 𝑃(𝑛)is true for all naturalnumbers 𝑛 if: 1. 𝑃(1) is true,and 2. "𝑃(𝑛) ℎ𝑜𝑙𝑑𝑠 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛 ≤ 𝑘" ⇒ 𝑃(𝑘 + 1) holds for all such 𝑘. If anythingthisisjusta strongerversionof the axiomof induction,asall statementsproven by our standardinductioncanalsobe provenby stronginduction.We maywantto use strong inductiontoprove that 𝑛! = 𝑛 ∙ ( 𝑛 − 1)! .
  20. 20. History of Mathematical Induction & Recursion [B]. 19 In PeterEccles An Introduction to MathematicalReasoning,he givesthe followinganalogyof The Principle of Induction whichreallysumsupitsstrength: Supposethatwethinkof theintegers lined up like dominos.Theinductivestep tells us thatthey areclose enough foreach domino to knockoverthe nextone,the basecase tells usthatthe first domino fallsover,the conclusion is thatthey all fall over. Once it isknownthat the dominosare so close togetherinordertoknockanotherover,then there isno needtopusheach one over, andwe justneedtoknockover the first.Inductionwith the natural numbersisthe same,once we have seenthat 𝑃( 𝑛)shows that 𝑃( 𝑛 + 1) holds,thenall that isrequiredistoshowthat the base case is true. Mathematical inductiondevelopedaroundthe natural numbersandtheirordering.Then, inductionwasappliedtootherareasof mathematicswhere objectscanbe put intoan ordering,to prove propertiesandtheirtheorems,andtotackle the problemof representingthe infinite inthese areas – we have seenthat we can define aninfinite collectionof sets,sowe mayuse inductionto showthe propertyof inclusionamongstthese sets,basedontheircardinality.Inlogicwe canprove a theoremfora language byproof byinductiononthe lengththe sentencesbelongingtothis language. Mathematical inductiontodayisanessential tool withineveryareaof mathematics. Induction allowsustorepresentthe infinitewiththe finite.Itremovesall the complexityand longevity associatedwith proofsconcerningthe infinite.Itisa mathematical tool thathasgrown in importance until today.Inprovingtheorems,propositions,lemmas, whateveritmaybe,we have a certaingroupof techniquesinmathematicsthatwe can use, andinduction isone of these;its importance is unrivalled.
  21. 21. History of Mathematical Induction & Recursion [B]. 20 [1.5] The Origin of Mathematical Induction: As I have said,mathematical inductionisof great importance withinmathematics,but where didthe principle originate from?Whowasthe 1st persontodevelopthe principle?Whowas the firstto realize itsimportance?Whoeven,wasthe firsttoname the principle mathematical induction?What we alreadyknowatthispointis that inductionwasdevelopedfromthe natural numbersasthat is the basisof its structure;the orderinginwhichthe natural numbersexhibits. The exactoriginof the principle of mathematical inductionisunclearanditcouldbe debatedtobe attributedtomanydifferentgreatmathematiciansortovariousdifferentperiodswithinthe history of mathematics. We knowthatthe methodbearsa close resemblance tothe inductive processesof other sciences whichtendtobe basedon observationsandtoa degree,thatisone wayinwhich mathematical inductionwasdeveloped.Inductioninmathematicsisgenerallyamethodof proof;we do notjust developanyoldprocedure andsetouttoprove its validity.Insteadwe lookatpatterns fromexamplesandtryto developanatural property.Once we have thisproperty,we use induction to see if itis true.Thiscan be seenin Mathematicsand PlausibleReasoning,Vol.1– Induction and Analogy in Mathematics (1954) by George Pólyai : Wantingto developaformulaforthe sumof the squaresof the 1st 𝑛 natural numbers,Pólyacomparesthe ratioof ∑ 𝑖𝑛 1 and ∑ 𝑖2𝑛 1 . Here I wishto define ∑ 𝑖𝑛 1 = 𝑆 𝑛 and ∑ 𝑖𝑛 1 2 = 𝑆 𝑛 2. In doingthis,Pólyastates,withoutproof,that 𝑆 𝑛 = 1 + 2 + 3 + 4 + ⋯+ 𝑛 = 𝑛( 𝑛+1) 2 , whichis true andcan be proven,byinductionof all things. Lookingat the 1st 6 cases forpossible 𝑛,Pólyaobservesthatthe ratiosare givenby: 𝑛 1 2 3 4 5 6 𝑆 𝑛 𝑆 𝑛 2 3 3 5 3 7 3 9 3 11 3 13 3 From thisPólyaclaimsthat 𝑆 𝑛/(𝑆 𝑛 2)= (2𝑛 + 1)/3 whichiscertainlytrue for 𝑛 = 1,2, …,6. Takingthisintoconsiderationandmultiplyingbothsidesthruby 𝑆 𝑛 = 𝑛( 𝑛+1) 2 , it isobtainedthat 𝑆 𝑛 2 = 1 6 𝑛( 𝑛 + 1)( 𝑛 + 2). Thisremainstrue for 𝑛 = 1, 2,… ,6. i George Pólya was an Hungarian mathematician [1887-1985].
  22. 22. History of Mathematical Induction & Recursion [B]. 21 What Pólyahasdone is establisha‘conjecture’ashe callsit,all fromtakingat a few examplesandlookingforapatternin orderto presenthisequationfor 𝑆 𝑛 2.Now that he has seen that itworks fromthe 1st six cases,he goeson to prove thatit holdsfor 𝑛 = 7 by simplypluggingin the numbersof his equation.Notwantingtoprove thatthat hisequationistrue for 𝑛 = 8, 9,10, … Pólyaproceedstothe inductive step;assuminghisequationholdsforanarbitrarynatural number 𝑛, he proceedsinshowingthatitdoesalsoholdfor 𝑛 + 1. Pólyaconcludes the proof of hisequationfor the sum of the squaresof the first 𝑛 natural numbersbysaying: “If ourconjectureis true fora certain integer 𝑛, it remainsnecessarily truefor the next integer 𝑛 + 1. Yet we knowthatthe conjectureis true for 𝑛 = 1,2, 3,4,5, 6,7. Being true for7, it mustbe true also forthe nextinteger 8; being truefor 8, it mustbe true for 9; since true for9, also true for10, and so also for11, and so on.The conjectureis true forall integers;we succeeded in proving it in full generality.” The procedure inwhichhe developedhisequationissimilartothat of inductioninother sciences,bymakingobservations.Howeverinprovingthathisequationdoesinfactholdforall the natural numbersPólyarequirestoshowthe inductivestep.Makingobservationsandshowingthat theyare true for certaincases,doesnotprovide afull proof;thatis whyPólyarequiresthe inductive stepand that iswhere mathematical inductiondiffersfromthe inductionof othersciences. This inductionobviouslyinfluencedmathematicianstolookforrecurringpatternsinorder to formulate ‘conjectures’like Pólyaandthentryto prove theirvalidity. Like Pólyasays: “Has mathematicalinduction anything to do with induction (inscience)?Yes,ithas.” Howeverthiswouldonlybe the inspirationforthe base case;the inductive stepwouldnot have followedfrom the observationof examples. There are tracesof recurrentandinductive processesinthe worksof some GreekandHindu mathematicians. However,ratherthantryingto generalize aparticularmathematical method,they soughtto findone particularsolutionfromanother.Here theywere showingsigns of aninductive step,althoughfromone specificcase tothe next,asopposedtoan abstract case to the following abstract case. Examplesof suchare foundinthe work of both Theonof Smyra (fl.100 A.D.) and Proclus(412-485 A.D.) ontheirworkon findingnumbersrepresentingsidesanddiagonalsof squares. Bhāskara’s(1114-1185) cyclicmethodforsolvingindeterminate equationsof the form 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 𝑦 alsoshows traces. Furthertracescan be foundinEuclid’sElementsIXin PropositionXXwhere Euclidshowsthatthe numberof primesisinfinite.Howeveranytrace of mathematical inductionthatdidappearinthe work of the Greeksand Hinduswouldnothave been inthe same modernforminwhichwe see ittoday.
  23. 23. History of Mathematical Induction & Recursion [B]. 22 Duringthe 17th century,the FrenchmathematicianPierre de Fermatwas knownto have focusedalot of hiswork on infinitesimal calculusanditwasin a lettertoChristiaanHuygens inwhichFermatclaimed touse a methodcalled la descente infinite ou indefinite.The letterwastitled‘Relation des découvertesen la science desnombres’ andwasnot discovered until 1879, amongsta groupof papersthat had belongedto Huygens.Inthe paper,Fermatclaims he had foundthat hisnew methodwasapplicable in provingthe impossibilityof certain mathematical statements,before realizing thatitwas also applicable inprovingthe assertionof statementstoo. Fermat’s methodof infinitedescentdidcontainrecurrentmodesof inference,however,itwasnotmathematical inductioninits purityas Fermatwouldoftenjumpfrom acertaincase 𝑛 to 𝑛 − 𝑖, for some 𝑖 < 𝑛,skippingoverseveral cases ata time.Insteadof lookingtoprove aspecificbase case and an inductive step,Fermatchoosestoprove astatementforone certaincase andthenmake a connectiontoanotherinorderto prove thissecondcase.Here Fermat doesshow a trace of an inductive step. Fermatwas knownforpublishingonlyhisresultsandexcludinghismethods;itwasnot until 1995 that Fermat'sfamousLast Theoremwasproveni ;Fermathavingdiedin1665. Thiscouldbe the cause for, bypure chance, notseeing Fermat’smethodof infinite descentuntil 1879.The only occasionson whichFermatmade hismethodsknownwere inresponse tocritiquesof his work,who didnot believe hisresultstobe true or whosought a clearerexplanationof hisresults.One of these reasonscouldbe whyHuygenspossessedacopyof Fermat'smethodof infinite descent,andthe reasonthat we nowknowthat a recurrentmode of proof wasknownto Fermat.There isfurther evidence forthisinthe factthat Adrien-Marie Legendre,GustavLejeuneDirichletandLeonhard Eulerall usedsimilarmethodsof Fermat'smethodof infinitedescenttoprove alot of Fermat’s propositionsinthe space of time afterhisdeathandbefore the discoveryof Fermat’sletterto Huygens. There are alsotracesof mathematical inductionthatcan be foundinthe workof Blaise Pascal. Alsoa Frenchmathematicianof the 17th century,Pascal was workedcloselywith Fermaton i Andrews Wiles proof of Fermat’s Last Theorem was published in 1995.The famous theorem had been written in 1637 in the margin of Fermat's copy of Diophantus’ Arithmetica to which Fermat said themargin was not big enough to contain the proof as well. Figure 4 - Pierre de Fermat.
  24. 24. History of Mathematical Induction & Recursion [B]. 23 the foundationsof modernprobabilitytheory. Pascal isfamoustodayforPascal’sTrianglewhichwas mentionedearlier,howeveritwasknowntomanyin the daysof Pascal and Fermatas the arithmetical triangle.In1665, Pascal's Traité du trianglearithmétiquei waspublished.Pascal usedthe treatise toshowthe applicationsof the arithmetical triangle inthe theoryof combinations,the theoryof probabilitiesandtothe calculationsof the powersof the binomial coefficients.Pascal showedthatthe binomial coefficient 𝐶 𝑘 𝑛 canbe foundbythe equation 𝑛 ∙ ( 𝑛 − 1) ∙ ( 𝑛 − 2) ∙ …∙ ( 𝑛 − 𝑘 + 1) 𝑘! which,if we multiplythruby ( 𝑛−𝑘)! ( 𝑛−𝑘)! we obtain the formthat we use today 𝑛! ( 𝑘!)( 𝑛−𝑘)! . Pascal usedthe arithmetical triangle tocalculate the share of a total stake in a game of dice between2players whichhasbeen stopped prematurely.Letthe firstof the 2 playersbe person A and the secondperson B. Withperson A needing 𝑛 pointstowinand B needing 𝑚 points,Pascal usedthe arithmeticaltriangletocalculate that the ratio of A to B shouldbe givenas the sum of the 1st 𝑛 numbersof the 𝑗 𝑡ℎ row of the arithmetical triangletothe sumof the remaining 𝑚 numbersof the same row,where 𝑗 = 𝑚 + 𝑛. Alternatively,letnumbersof the 𝑗 𝑡ℎ rowof the arithmetical triangle be givenbythe sequence 𝑎1, 𝑎2,… , 𝑎 𝑛,𝑏1, 𝑏2,… , 𝑏 𝑚.Then the ratio of A to B is givenby ∑𝑎𝑖 ∶ ∑𝑏𝑗 . Here 𝑗 = 𝑚 + 𝑛 is the numberof throwsof the dice remaininginthe game whenitis stoppedearly.Pascal proved inhistreatise,throughalemma,thathisratioiscorrect for 𝑗 = 1 and thenproved inthe nextlemmathat,if itis alsocorrect for some natural number,thenitisalso correct for the nextnatural numbergreaterthanit.ThenPascal concluded thathisratiois correct for everynatural numberthat 𝑗 can be.This ispreciselymathematical inductionof today.Pascal did not refertoit as mathematicalinduction oruse a 2 stepprocedure;insteadhe hadprovena base case inone lemmaandthenproven an inductive stepinthe nextlemmabefore he made aseparate conclusion –that hisratio istrue foreverynatural number. Due to the popularityof Pascal'streatise onthe arithmetical triangle,he couldbe deemedas one of the reasonsas to howthe principle of mathematical inductionbecame knownat the time of and inthe time afterthe 17th Century. Pascal couldhave alsobroughtthe principle of mathematical inductiontothe attentionof Fermatthroughtheirworktogetheronprobabilitytheory. Pascal was i Translated into English as - A treatiseon the Arithmetical Triangle;it appeared firstin 1654. Figure 5 - Blaise Pascal.
  25. 25. History of Mathematical Induction & Recursion [B]. 24 certainlyone of the firstmathematicianstouse the principleof inductionsimilartoitscurrentform, ina systematicway andto realise the implicationof the inductivestep. However,toPascal was knownthe workof an ItalianmathematiciancalledMaurolycus. InaletterfromPascal to Pierre de Carcavii ,Pascal refersto Maurolycusfor the proof that twice the 𝑛 𝑡ℎ triangularnumberminus 𝑛 equals 𝑛2 . Although Pascal doesnotmentionMaurolycusinhis Traité du triangle arithmétique, he was well aware of Maurolycuswhose workcouldhave beenthe inspirationof Pascal'smethodof induction. i In the letter Lettre de Dettonville á Carcavi.
  26. 26. History of Mathematical Induction & Recursion [B]. 25 [1.6] Maurolycus & Gersonides: FranciscusMaurolycus(1494-1575) was an Italian mathematicianwhoworkedontranslationsfromsome of the most famousGreekmathematicians –Euclid,Archimedes, Theodosius.His workwas of great importance forthe transitof Greekworkto Europe. In 1575, Maurolycus publishedatreatise onarithmetictitled Arithmeticorumlibri duo found inhis book D. Francisci Maurolyci Opuscula Mathematica.Here,Maurolycususesamode of inference in a systematicway,buildingupfromthe firstcase to the next,to demonstrate simple propositionsbefore movingontoprove harder, more complicatedonesinasimilarfashion.Proposition XIof this treatise isthe propositiontowhichPascal credited Maurolycus inthe lettertoCarcavi.Maurolycus provedthis propositionthrough2previouslystatedpropositionsand variousdefinitions. Therefore Pascal didnotgethisideaforhisinductive proof fromthe Maurolycus propositionthathe hadreferencedtoCarcavi.Howeverinthe same treatise,2otherpropositions may have caughtthe eye of Pascal. PropositionsXIIIandXV of the treatise asfollows: (13)Every squarenumberplusthefollowing odd numberequalsthefollowing squarenumber. (15)The sumof thefirst 𝑛 odd integers is equal to the 𝑛 𝑡ℎ squarenumber. In modern notationthiswouldprecede asfollows: (13) ( 𝑛 + 1)2 = 𝑛2 + 𝑂 𝑛+1 for 𝑂 𝑛 = 2𝑛 − 1. (15) 𝑂1 + 𝑂2 + ⋯+ 𝑂 𝑛 = 𝑛2. In the modernnotationitisclearto see that there isa connectionbetween the 2propositions. Maurolycus stateshisproof of PropositionXV asfollows “By a previousa previousproposition (namelyPropositionXIII) thefirstsquare number(unity) added to thefollowing odd number(3) makesthefollowing square number(4);and this second squarenumber(4) added to the 3rd odd number(5) makes the 3rd squarenumber(9);and likewise the3rd squarenumber(9) added to the 4th odd number(7) makesthe 4th squarenumber(16);and so successively to infinity the proposition isdemonstrated by therepeated application of Proposition XIII.”i i This is the translation given by W.H. Bussey in American Mathematical Monthly, No.5, Vol.14, May 1917. Figure 6 - Franciscus Maurolycus.
  27. 27. History of Mathematical Induction & Recursion [B]. 26 Proposition XV isachievedby repeateduse of proposition XIIIwhichactsas an inductive step. ClearlyMaurolycusproof isan example of mathematical induction.Againitisnotinthe current structure as we use today butit isin a systematic,step-by-stepprogression fromthe firstcase tothe next,andthento the next,andto the next,andso on towardsinfinity.One difference between Pascal'smode of inferenceandthatof Maurolycus,isMaurolycusdoesnotmake an inductive assumption/hypothesiswhereasPascal does,makinghismethodof proof more abstract. Now I come to Levi BenGershon,(Gersonidesin Latin).A Rabbi bornin 1288 inLanguedocinwhat would now be the southerncoastof moderndayFrance, Gersonideshadabroaderbackgroundto his mathematical careerthanMaurolycus.Although Gersonidesdid tooworkonEuclid’sElements, he also worked ona lot of the oldArabicand Hindutexts,suchas Bhāskara,who I mentionedbefore,who isknowntohave usedhiscyclicmethod to solve indeterminateequations of the form 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 𝑦. Gersonides’1321 work – MaaseiHoshevi , couldbe calleda piece of work that wasaheadof itstime; Gersonides usesletterstorepresentarbitrarynumbers, onlyJordanusNemorariuswasknowntohave alsohave done this at that time.Anotherreasonthat MaaseiHoshev couldbe consideredasaheadof its time is because of Gersonidesuse of the methodof whathe calledrising step-by-step whichhassimilaritiestoMaurolycus’workonrepresentingthe infinite. Thisstep-by-stepmethodcanbe foundin Propositions9to 12 inclusive,whichInow give (inmodern notation): (9) 𝑎( 𝑏𝑐) = 𝑏( 𝑎𝑐) = 𝑐(𝑎𝑏). (10) 𝑎( 𝑏𝑐𝑑) = 𝑏( 𝑎𝑐𝑑) = 𝑐( 𝑎𝑏𝑑) = 𝑑( 𝑎𝑏𝑐). Gersonidesused Proposition9to prove Proposition10before makingthe followingstatement: In thismannerof rising step-by-step,itis proved to infinity.Thus,… the result of multiplying onenumberby a productof other numberscontainsany oneof these numbersasmany timesas the productof all the others.” i Title is taken from Exodus 26:1 to roughly mean The Work of the Calculator. Figure 7 - A stamp of Isreal showing Gersonides' invention, Jacob's Staff, which was used for measuring nautical and astronomical measurements.
  28. 28. History of Mathematical Induction & Recursion [B]. 27 WhichI interpretas 𝑎𝑗( 𝑎1 𝑎2… 𝑎 𝑖 … 𝑎 𝑛) = 𝑎 𝑖( 𝑎1 𝑎2… 𝑎𝑗 … 𝑎 𝑛). (11) 𝑎( 𝑏𝑐𝑑) = ( 𝑎𝑏)( 𝑐𝑑) = ( 𝑎𝑐)( 𝑏𝑑) = ⋯ AgainGersonideswrote towardsanextensiontowardsinfinity: Similarly, it is shown to infinityby thesame kind of demonstration.Therefore,any numbercontainstheproductof any two of its factorasmany times asthe productof the remaining factors. (12) In modernlanguage,Proposition12statesthat multiplicationisbothassociative and commutative.Gersonidesprovesthisusingthe previous3propositionstoshow how factors can be grouped intodifferentstringsof differentlengths,whichisnotreallyafull proof. Althoughnone of these propositionsare provedby induction,theydoshow whatGersonides meantby rising step-by-step anditdoesbare some resemblance toaninductive step. Gersonides continued tostate and prove a longseriesof propositions;IskiptoPropositions 63– 65 on permutations: (63) 𝑃 𝑛+1 = ( 𝑛 + 1) 𝑃𝑛. (64)𝑃2 𝑛 = 𝑛( 𝑛 − 1). (65)𝑃𝑗 +1 𝑛 = ( 𝑛 − 𝑗) 𝑃𝑗 𝑛 . Gersonidesprovedeachone of these propositionsinturn,usingProposition63to prove Proposition65.Once thisis done,Gersonided concludesbysaying: Thusit hasbeen proven thatthe permutationsof ordera given numberfroma second given numberof elements equalsthe numberswhosefactorsareasmany asthefirst given numberand they are the integersin their naturalorder,thelast being thesecond given number. Thus Gersonideshad proventhat 𝑃𝑗 𝑛 = ∏ 𝑖𝑛 𝑛−𝑗+1 .One can see that, whenwe take 𝑗 = 2 we getProposition64,whichhad beenprovenpreviouslybyGersonides.Gersonidesclaimedthatthese 3 propositionsare enoughtoprove thisgeneral result,andtheyare.FromProposition64,we use Proposition65to prove the case for 𝑗 = 3 and fromthat,we applyitagainto prove that the result holdsfor 𝑗 = 4 andapplyit againfor 𝑗 = 5, andso on. Proposition 65acts as an inductive step,from any arbitrary 𝑗 to 𝑗 + 1. Thus Gersonideshad usedinductiontoprove hisresultonthe numberof permutationsof order 𝑗 of 𝑛 elements. Againwe cannotsaythatthisishow we use inductiontoday,butGersonides methoddoescontainthe essence of modern induction havingprovenaparticularcase as well asa
  29. 29. History of Mathematical Induction & Recursion [B]. 28 recursive step. The proof of proposition42inthiswork wasalsoprovenina similarfashion. However,there wasalack of an assumptioninorderto make the inductive step,similartothatof Maurolycusand again,Gersonidesworkwasconstructive,buildinguptothe result,whereaswith inductiontoday,we seektoprove whatwe have alreadystated. The source as to Gersonidesinfluencetoinvestigate anduse arecursive mode of inference inorder to prove mathematical procedures liesinthe Hebrew communityof Gersonidestime where the subjectwasinvestigatedatan earlystage. Aswell asBhāskara,Gersonidescouldhave been influencedbythe worksof SeferYetsirahandhis Bookof Creation,whichinvolvesarecursive mode of lookingatthe permutationsof the twenty-twolettersof the Hebrew alphabetandisbelievedto be from the secondcentury.Anotherpossible influence is Rabbi ShabbetaiBenAbrahamDonnolo, (913-970), whoprovedthat 𝑛 letterscanbe arranged 𝑛! waysin a similarfashiontoarecursive method. In MasseiHoshev,Gersonidesmentioned thatthe readerof histextshouldbe aware,and capable of understanding, the 7th ,8th and 9th booksof Euclid’s Elementswhere we have already seen a small trace of a recursive mode of inference. However,itwouldbe possibletoassume that Gersonidesfoundmodesof recursive definitionandinduction,throughouthismathematical career, inthe worksof othersand we couldevenassume furtherthat GersonideshadsetoutinMassei Hoshevto investigate the applicationof recursivelydefinedstructuresandthe applicationof recursioninmathematical proof inastructured,systematicway. Andit seemstome that Gersonideswasthe firsttodo this.I feel thatGersonideswasthe firstto realize the significanceof the orderingof the natural numbersandhow he couldapplyit to proofsinorder torepresentthe a propertyof the infiniteintermsof the finite. There isonlyone furthermathematicianthatIfeel we couldconsiderforthe invention of mathematical inductionand thatisCampanus. AnItalianmathematicianof the 13th century, Campanusof Novara workedontranslating Euclid’sElementsintoLatin.Indoingthishe includedhisownversionof the proof thatthe goldenratioi isirrational. The methodusedby Campanusinhisproof was similartothat of Fermat.Campanus,like Fermat,useda descendingmethodof progression,jumping i (1+√5) 2 is the Golden Ratio, which is said to be found to occur naturally throughoutall of life. Figure 8 - The Golden Ratio.
  30. 30. History of Mathematical Induction & Recursion [B]. 29 sporadicallyovercertaincases,toprove othercertaincases. However,Idonot feel thatthis methodisfullyrepresentativeof the mathematical induction of today.Modern inductionrepresents everypossiblecase whereasthe methodsof Campanusand Fermatappeardisjoint andfull of gaps.Althoughtheirmethodsdoshow tracesof induction, they lack thatcontinuous,connected,one afteranothersequence.Ifeel thatthe methodsof Gersonides, Maurolycusand Pascal are strongerand closertothe methodin whichwe now use today,not because theyprecede fromacertain,finite,base case towardsinfinity,butforthe reasonthatthese methodswouldnotskipoveranyparticularcase;theyrepresentthe continuousprogressionfrom one discrete case to another. In addition,Ibelieve thatthe same three men,Gersonides,MaurolycusandPascal were more aware of the significance intheirmethod;Fermatcouldalsobe includedinthisregard but since he neverreleasedhisarguments,we will neverknow if he wasaware of such significance.As for Campanus,Iwouldhave toregard himas one of the menwhoinfluence the formerfour.Iregard himin the same groupas Bhāskara, Theon,Proclus, Euclidandothers,whoshowedtracesof the mode of inference andrecursion, whousedamethodsimilartoinductionasa one-off proof, significantonlyatthat one particulartime.MoreoverI would considerGersonidestobe the first mathematiciantounderstandthe meaningandsignificance of arecursive mode of inference andto give ita stepby stepstructure. Furthermore IregardMaurolycusas the firstmanto see the significance of applyingthisrecursive mode of inference toproofs whereasPascal broughtittothe attentionof manyothers.
  31. 31. History of Mathematical Induction & Recursion [B]. 30 [1.7] The Name – Mathematical Induction: For the originof the term MathematicalInduction,agroupof othermathematicians who were aware of the methodare to be credited. Fermatreferredtohisnew methodas la descente infinite ou indefinite,Gersonidesreferredtohismethodof rising step-by-step.HoweverMaurolycus and Pascal didnot assignanyparticularname to theirmode of inference. Itisevidentthatthe term mathematicalinduction isderivedinitiallyfrom the observational induction of sciences asmanyseen it as an adaptationof that concept. In JohnWallis’1656 work, Arithmetica Infinitorum, Wallisi usedthe methodof inductionusedinscience and simplyreferstothismethodas induction.InProposition16of thiswork,Wallislooked tofindthe ratioof the 1st 𝑛 squared numberstothe product ( 𝑛 + 1) 𝑛2. Wallisproceedsto observe that,inthe 1st 6 cases,the ratioturns outto be 1 3 + 𝑥ii where 𝑥 < 1 decreasesasthe size of 𝑛 increases.Wallis thenconcludedthat lim 𝑛→∞ 𝑥 = 0. Thismethod wasreferredto by Wallisas per moduminductionis.iii AsWallisproceeded throughthe remainderof thispiece of work,he relied freely on thismethodof inductionsimilartothatof natural science. Wallisdidfeel stronglyaboutthe scientificinductionandthatitcouldeasilybe appliedto mathematics.Inhis1685 treatise onAlgebra,Wallisstated: “ThosePropositions...demonstratedby way of Induction:which isplain,obvious,and easy;and wherethingsproceed in a clear regularorder (ashere they do),very satisfactory.” “I look upon Induction asa very good method of investigation;asthatwhich doth very lead usto the easy discovery of a General Rule.” However,in1686, JacobBernoulliiv recommendedinhis Acta Eruditorum thatWalliscould improve hismethodof inductionbyintroducingthe argumentfromanarbitrary 𝑛 to 𝑛 + 1. This appearsto be the 1st appearance of the InductiveHypothesis andthe beginningof the modernform i John Walliswas an English mathematician,(1616 –1703). ii Just likePolya noted in section [1.5]. iiiLatin for - by way of induction. iv Jacob Bernoulli (1655 – 1705) was a Swiss mathematician and partof a largemathematical family. Figure 9 - John Wallis.
  32. 32. History of Mathematical Induction & Recursion [B]. 31 of Mathematical Induction. Bernoulli usedthisnew 𝑛 to 𝑛 + 1 argumentto prove the binomial theoreminhis ArsConjectandii . FlorianCajoriii ,ahistorianinmathematics, referstothe methodusedbyWallisasincomplete and referstois as IncompleteInduction,whichgivesrise tothe CompleteInduction thatCajori defined asthe methodbyBernoulli. Formore thana century afterBernoulli’srecommendationto Wallis, Induction wasbeingusedasthe name forboththe methodsof WallisandBernoulli. The two methods were seeminglyunpopularinthistime anditwasin fact Bernoulli'smethodthatwasless knownat the time Most whousedeithermethodactuallyusedthe methodwithoutassigninga specificname. Howeverinthe 1830s, thischanged.George Peacock(1791 – 1858) wasan Englishmathematicianwho publishedhisTreatise onAlgebrain1830. In thisTreatise, Peacocktalked of a “law of formation extended by induction to any number”. Inexplainingthe argument from 𝑛 to 𝑛 + 1, Peacockreferred tohismethodas DemonstrativeInduction. AugustusDe Morgan (1806 – 1871) wasa British mathematicianwhose name ismainlyassociatedtothe lawsof negationonthe conjunctionanddisjunctionof sets.In1838, De Morgan publishedinthe Penny Cyclopaediahis Induction(Mathematics) inwhichhe described clearlymathematicalinductionand itssimilarities/differencestoinductioninphysics.De Morganshowed how mathematical induction shouldbe appliedthroughtwo clear,well-describedexampleswhereapropositionisstatedand thenprovenviaan inductive step(usinganinductive hypothesisonlyinthe 1st example),before referringback to a base case.De Morgan referred toinductionas successiveinduction at the beginningof thispiece of work, howeverhe laterreferstothe method asMathematical Induction; the firstpublishedoccasiononwhichthe termhadbeenused. Both the terms DemonstrativeInduction andMathematicalInductionbecame popularinthe time afterbutthe formertermfell intodisuse asmostmathematiciansbeganto adoptthe latter. The term VollständigeInduktion wasusedbyGermanmathematiciansinthe 19th century,most notablybyRichard Dedekindinhis1887 Was Sind und Was Sollen die Zahlen.It wasthisusage by i Published in 1713,after Jacob Bernoulli’s death. ii Origin of the Name Mathematical Induction,The American Mathematical Monthly, vol.25, number 5, 1918. Figure 10 - Augustus De Morgan.
  33. 33. History of Mathematical Induction & Recursion [B]. 32 Dedekindthatpopularizedthe methodinGermany,althoughthe methodwasslightlydifferentto that of Peacockand De Morgan. In 1863, Isaac Todhunter(1820 – 1884), an Englishmathematician, popularizedDe Morgan’s mathematicalinduction inhisAlgebra forBeginners usingthe methodto prove variousexamples.One suchexample thatTodhunterspoke of was: “The sumof (the first) 𝑛 termsof the series 1, 3,5,7, …is 𝑛2. This assertion wecan see to be truein somecases...wewish to howeverto provethistheoremuniversally”. Usinginduction,inthe same manneras De Morgan, Todhunterproved the above universally, for all possible casesof 𝑛.Realisingthe full benefitof mathematicalinduction, Todhunterthen stated: “The method of mathematicalinduction may bethusdescribed:we provethatif a theoremis true in one case,whateverthatcasemay be, it is true in anothercasewhich may be the nextcase;hence it is true in the nextcase,and hence in the nextto that, and so on;henceit mustbe true in every case afterthat which it began..... Themethod of mathematicalinduction is asrigid as any otherprocessin mathematics.” Todhunterreferred tothe method directlyas mathematicalinduction –the title thatDe Morgan assignedtoit and the title whichwe use today. Inthe centurythat followed Todhunter’s Algebra,mathematical inductionhasbecome even more abstract and has been acceptedacrossthe mathematical worldas an essential tool formathematical proof. Todayits structure has become more like thatof a procedure thatis followed inastepby step manneras we have seenearlier. Howeverthe name andthe basic concepthave remainedintact; fromGersonides,toPascal,throughtoDe Morgan and onwards until itsmodernformtoday.Figure 11 - Isaac Todhunter
  34. 34. History of Mathematical Induction & Recursion [B]. 33 [1.8] Bibliography: 1. N L Briggs, Discrete Mathematics,Revised Edition,1989, OxfordUniversityPress,P8-10. 2. J L Hein, Discrete Mathematics,2nd Edition,2003, JonesandBartlettPublishers,P145-146. 3. H Eves, An Introduction to theHistory of Mathematics,4th Edition, 1976, Holt,Rinehart& Winston,P209-212. 4. R C Penner, Discrete Mathematics:Proof Techniquesand MathematicalStructures,1999, WorldScientificPub.Co.Inc.,P141. 5. K Gödel,on Formally UndecidablePropositionsof Principa Mathematica and Related Systems, EnglishTranslationbyB.Meltzer,1962, Oliver&Boyd LTD. IntroductionbyRB Braithwaite FBA. 6. S C Kleene,MathematicalLogic,1967, JohnWiley&Sons,Inc.,P250. 7. J W DawsonJr., Logical Dilemmas,the Life and Work of Kurt Gödel, 1997, A.K. Peters,P3-21, P53-79. 8. DavidHilbert,MathematicalProblems, Bulletin of theMathematicalSociety, 1902, Vol.8, Number10, P437-479, translatedbyM WinstonNewson. 9. I Grattan-Guinness, Search forMathematicalRoots1870-1940, 2000, PrincetonUniversity Press,P227. 10. I Grattan-Guinness, Search forMathematicalRoots1870-1940, 2000, PrincetonUniversity Press,P227. 11. P Eccles, An Introduction to MathematicalReasoning, 2007,Cambridge UniversityPress, P39-51. 12. G Polya, Mathematicsand PlausableReasoning,Vol.1:Induction and Analogy in Mathematics,1954, OxfordUniversityPress,P108-111 13. F Cajori, ÜberdasWesen der Mathematik, Bulletin of American MathematicalSociety,1909, Vol.15, Number8, P407. 14. F Cajori, History of Mathematics,5th Edition, 1991, Vol.2, ChelseaPublishingCompany, P142. Also,whole textusedforbirth/deathdatesof Mathematiciansandtheirwork. 15. G Vacca,Maurolycus, TheFirst Discoverer of the Principle of MathematicalInduction,Bulletin of the American MathematicalSociety,1909, Vol.16, Number2,P70-73. 16. N L Rabinovitch, RabbiLeviBen Gershon and the Originsof MathematicalInduction,Archive forHistory of Exact Sciences, 1970, Vol.6, Issue 3, communicatedbyCTruesdell. 17. F Cajori, Origion of the Name‘MathematicalInduction’,TheAmerican Mathematical Monthly,1918, Vol.25, Number5, P197-201. 18. A De Morgan, Induction (Mathematics),Penny Encyclopaedia,1838,Vol.12, London. 19. I Todhunter, Algebra forBeginners,4th Edition,1866, Macmillan& Co.,P281-284.
  35. 35. History of Mathematical Induction & Recursion [B]. 34 [2] Set Theory and Transfinite Numbers: The historyof howmathematical inductionandrecursioncame tobe is an importantone. The journeythatboth have takento progressthroughthe agesto theirmodernformshasbeena labouringone.Bothmathematical applicationshave showntracesof theirimportance asfarback as the time of Pythagorasand Plato,partlydue to the resultsthattheywere able toproduce,partly due to the struggle withthe infinite atthe time. Mostof thishas beenaddressedinthe firstpartof thispiece of work.One thingI didnot investigate fullywasthe significancethatinductionhadon mathematicsinthe late 19th and early20th Century’sandinparticular,itseffectonnumbertheory. Duringthistime,numbertheorybecame animportantinteresttoa lotof mathematicians, the most importantbeingGiuseppe Peanoof Italy,RichardDedekindandGeorgCantor,bothof Germany.The importance numbertheory hadat the time wasnot just froma mathematical pointof viewbutalsofroma philosophical one.The mainaimwas to establish foundationsforthe theoryof numbersanditsarithmeticsuchthat it shouldbe soundandfree of contradiction. The workof PeanoandDedekindhadthe mostsignificance inthisarea. Mathematical inductionwasanessential tool forGiuseppe Peanoinestablishinghisnow famous,anduniversallyaccepted,axioms whichIhave previouslymentioned.The use of induction by Peanohelpedhimconvert mathematicsintoasymbolic,logical form;infactthe simplicity of induction shinesthroughinPeano’ssymbolicnotations. ThereforeIwouldlike tostartbylookingat the work of Peano,mostnotablyhis Arithmetices principia,nova method exposita,asitcontinueson fromthe workdone inthe previoussection.Thiswill thenleadme toRichardDedekind. In hisday,Dedekindpublishedtwofamouspiecesof work – Stetigkeitund irrationalle Zahlen and Was Sind und Was Sollen die Zahlen? The latterof whichhasa real significance onthis piece of work,althoughIshall mentionthe formerbrieflyinvariousareasbeyondthispoint. Although Dedekind’s WasSollen waspublishedbefore Peano’s Arithmetices principia,Iwishtolook at both inthisorderas the workof Dedekind,baringhuge similarityandimportance tothe workof Peano,alsohasan overlappingconnectiontothe workof Cantor. The work of Georg Cantor isthe maininterestof thissection.The developmentof the transfinite ordinalsandcardinalsbyCantorinthe late 18th century as a new subjectinmathematics was notan easyone but it doesremaintothisday and itbecame one of the mostimportantareasof mathematicsof the 20th century, bringingmathematicsinto alogical formandbringingmathematics closerto itsphilosophical roots.Towardsthe endof thiswork,havingintroducedCantor’s naïveset theory, we shall have come across transfinite inductionandrecursion, whichare slightlydifferent formsof the inductionandrecursionalreadyinvestigated.
  36. 36. History of Mathematical Induction & Recursion [B]. 35 [2.1] Giuseppe Peano: Giuseppe PeanowasanItalianmathematicianwhowasborninCuneoinnorth-westItaly, nearthe borderof Austria,on27th August1858. Havingmovedto Turinin1871 withhisuncle to furtherhisstudies,Peanoenrolledin agraduate programin mathematicsat the Universityof Turin in1876. Upon graduatingin1880, Peanowasofferedaposition of workat the universitywhichhe acceptedandwhere he remaineduntil hisdeathin1932. Peanosoonbecame a professorof the infinitesimal calculusatthe university,althoughhe didhave otherinterestsoutside of thisarea, mostnotably,the foundationsof mathematicsand,tofill the time,Peanohadakeeninterestin linguisticstudies. Peanopublishedinexcessof 200 papersduringhislife andthe firstof these came in1884, entitled Calcolo differenzialee principii di calcolo integrale. Peanodedicatedthiswork toaformer teacher,AngeloGenocchi,bypublishingitunder Genocchi’s name andassigninghisownname asa subtitle of the work.Itwas inthispiece of work,and hisworkas a lecturer,thatthe needfora higherstandardof rigour in mathematicsbecame cleartoPeano. Whenwritingtowardsa publication,Peanolikedtokeeptohisownhighstandardof rigourwhile atthe same time making hisworksimple andeasyto understandandfollow,soitwasusual forPeano,justlike inancient Greekscripts,to showa demonstrative formof writing. All of thisis evidentinPeano’s1889 Arithmetices principia,nova method exposita,whichcan be translatedas The Principles of Arithmetic,Presented by a New Method.The needforsuch work was apparenttoPeano,to worktowardsthat more rigorousand simple foundationof mathematics that he thoughtwas necessary,andtoachieve this,Peano introduced symbolstorepresent full Figure 12 - Giuseppe Peano (1858-1932)
  37. 37. History of Mathematical Induction & Recursion [B]. 36 mathematical structures andalsoused letterstorepresentwhole propositionsandpropositional functions;he wasthe firstto do so,there wasno needforPeanoto write full labouringsentences explainingwhathe wantedtoachieve. Whenpublished,hissymbolicworkprobablyappearedasa code to its readers,butthe simplicityandeasyflow of readingwouldhave beeneasytopickup. Othersymbolsthatwere alsointroducedwhichwe stilluse todaywere ∈ forthe inclusionof an elementinaset, ⊂ for the containmentof asubsetina largerparentset, ∪ and ∩ for the unionand intersection,respectively,of two sets. Peanopaid specialattentiontothe distinctionbetween ∈and ⊂ as to leadto no ambiguity. For Peanoto establishhisnew foundationshe hadtostart at the most essential andbasic area of mathematics,the natural numbers.Peanowaslookingtoestablishanaxiomaticsystemfor lookingatthe natural numbersandtheirproperties, presented throughhissymbols.Withthe natural numbersaxiomatized,anypropositionortheoreminmathematicscouldbe verifiedby the truth heldinthe axioms. Toquote Peano: “with thesenotations,every proposition assumestheformand theprecision thatequationshavein algebra”. Arithmeticesprincipia was publishedasa 36 page pamphlet.The openingintroductionwas 16 pagesdevotedtodefiningandexplaining,rigorouslyof course,the new symbolsthatPeano wishedtointroduce. The remainderbeganwithfour definitionsandthese were followed bythe axiomsof whichtheywere nine intotal.Thus, Arithmeticesprincipia beganas follows: “The sign 𝑁 meansnumber(positiveinteger). The sign 1 meansunity. The sign 𝑎 + 1 meansthesuccessorof 𝑎, or 𝑎 plus1. The sign = meansis equalto. 1. 1 ∈ 𝑁. 2. 𝑎 ∈ 𝑁, 𝑎 = 𝑎. 3. 𝑎, 𝑏 ∈ 𝑁, 𝑖𝑓 𝑎 = 𝑏 𝑡ℎ𝑒𝑛 𝑏 = 𝑎. 4. 𝑎, 𝑏, 𝑐 ∈ 𝑁, 𝑖𝑓 𝑎 = 𝑏 𝑎𝑛𝑑 𝑏 = 𝑐, 𝑡ℎ𝑒𝑛 𝑎 = 𝑐. 5. 𝑖𝑓 𝑎 = 𝑏, 𝑎𝑛𝑑 𝑏 ∈ 𝑁, 𝑡ℎ𝑒𝑛 𝑎 ∈ 𝑁. 6. 𝑎 ∈ 𝑁, 𝑎 + 1 ∈ 𝑁. 7. 𝑎, 𝑏 ∈ 𝑁, 𝑖𝑓 𝑎 = 𝑏, 𝑡ℎ𝑒𝑛 𝑎 + 1 = 𝑏 + 1. 8. 𝑎 ∈ 𝑁, 𝑎 + 1 ≠ 1. 9. 𝐾 ⊂ 𝑁, 𝑖𝑓 ( 𝑎) 1 ∈ 𝐾 𝑎𝑛𝑑 ( 𝑏) 𝑖𝑓, 𝑓𝑜𝑟 𝑎 ∈ 𝑁, 𝑡ℎ𝑒𝑛 𝑎 + 1 ∈ 𝑁; 𝑡ℎ𝑒𝑛 𝐾 = 𝑁.”
  38. 38. History of Mathematical Induction & Recursion [B]. 37 Axioms2,3, 4 & 5, are the axiomsthatwe have not seenbefore.Todaythese canbe considered as trivial oras part of the underlyinglogicof equality. Notice thataxiom9,the axiomof induction,is statedslightlydifferent fromthatof before.i Withthe inductionaxiombeingstatedlast,Peano followshisaxiomswithdefiningthe natural numbers,byinduction: 2 = 1 + 1; 3 = 2 + 1; 4 = 3 + 1; and so forth. Withthis,Peanointroducesaddition,subtraction,maximumandminimumnumbers, multiplication,powers,division andthenproceedstomove onto theoremsonnumbertheory, rational andirrational numbers,theoremsonopenandclosedintervals.Mostof the work in Arithmeticesprincipia is provenbyinductionorconstructedfrom 𝑁 and the successorfunctionii undera mode of recursion.The detailsof these howeverIdonot wishtoinvestigate,aswe shall see all of these froma set-theoreticpointof view whenlookingatthe work of Dedekind. From the axiomof induction, Peanodoesnothave todeduce the validityof the propertiesof his natural numbers.ThisiscontrastingwithDedekindaswe shall see.We know the axiomcarries propertiesoverthe whole of the natural numberswhenthe propertyholdsforthe basecaseand, underthe inductivehypothesis,itholdsforthe inductivestep.This axiom, alongwiththe,whatwe may call logical, precedingaxioms,isall Peanorequirestodefinestructuresof,andpropertieson, the natural numbers. Today,Peano’saxiomsare consideredareference pointforthe natural numbers.However,they are notalwaysreferredtoas exclusivelybelongingtoPeano;some refertothe axiomsasthe Peano- DedekindAxioms. i This is due to wanting to keep to the original translation of Peano’s own statement of the axiom, found in Jean van Heijenhoort’s Frege to Gödel. ii 𝑓( 𝑎) = 𝑎 + 1. This is simply the third of the four terms defined by Peano at the beginningand how I shall refer to it from here.
  39. 39. History of Mathematical Induction & Recursion [B]. 38 [2.2] Richard Dedekind: Figure 13 - Richard Dedekind (1831-1916) “One of the wholly great in the history of mathematics, now and in the past” – EdmundLandauon Dedekind. JuliusWilhelm RichardDedekind wasbornthe 6th of October1831 in Brunswicki ,Lower Saxony,Germany. Whenhe wasyounger,Dedekindfoundhimself more interestedinphysicsand chemistrythanmathematics,however,atthe age of 17 and discontentedbythe lackof reasoninghe foundinphysics,Dedekindturnedhisattentionstowardsmathematicsinsearchformore logically soundreasoning.Itwasin1850 thatDedekind begantoattend the Universityof Göttingen.Here, havingbeenborninthe same town andattendedthe same college,Dedekindwouldstudyunderthe influenceof Carl FriedrichGauss. Aftertwoyears,Dedekindgraduatedwithhisdoctorate before takingupa lecturingrole at the universityin1854. It is believedthatDedekindwasthe first,in1857, to give a course on Galois Theoryat a universitylevel;he gave the course onlytotwo students.Inthe same year,Dedekind movedonto take up a positionata polytechnicinZurich.1862 saw Dedekindreturnto Brunswick to take up a positionasa professorat a technical highschool.Thisseemslike twostepsbackwardsfor Dedekind,butitappearsthatDedekindlivedanobscure,secludedlife,awayfromthe demandand attentionthatother,lesscapable,mathematicianswere receiving. In1904, it appearedinTeubner’s i This is an Anglicization of Braunschweig.
  40. 40. History of Mathematical Induction & Recursion [B]. 39 CalendarforMathematicians thatDedekindhaddiedonthe 4th of September,1899, although Dedekinddidinformthe editorthathe wasin fact “in perfecthealth” onthat day havingspentit withhis“honoured friend Georg Cantorof Halle”. In 1888, a yearbefore hisfictional death,Dedekindpublishedhisfamous WasSind und Was Sollen Die Zahlen? The secondeditionof thiswaspublishedin1893 and thiswas translatedin1901 by WoosterW. BemanintoEnglishandgivingthe title theNatureand Meaning of Numbers, althougha directtranslationwouldbe WhatAreNumbersand WhatShould They Be?From this work,like thatof Peano,Dedekindwaslookingtoestablishahigherrigourof mathematics,butin doingso,he tacklesthe age oldelephantinthe room – infinity.Infinityplaysamajorrole inwhatis to come in laterparts of this workand we have touchedon the infinitebeforewithmathematical inductionandrecursion.Ishall sayno more on the complexitiesof the infiniteatpresentasI would like togive a brief summaryof WasSollen, highlightingthe keypointsof interest. Was Sollen differsinapproachfromPeano’s Arithmeticesprincipia as Dedekinddoesnottake the axiomaticapproach of Peano,insteadinsistingonaset-theoreticapproachtodefiningthe natural numbers.Indoingso,Dedekindbeginswiththe infinite anddefinesafiniteseti asa set contrastingfromthat of an infinite one;thispartof Dedekind’sworkiswhathe shareswithCantor. Dedekindusesthismethodof workingbackfromthe infinitetodefinethe natural numbers. Was Sollen consistsof 172 paragraphs;each assignedanumberinunisonbyDedekind.These paragraphsare thengroupedtogetherin14 differentsections.The secondeditionhasa new preface attachedwhere Dedekindwantedtoaddressessomeof hiscritiques;howeverhe couldfindno justificationforanycriticismthatthe firsteditionmetandthe new preface ismerelyan acknowledgementthatDedekindhadtriedtoaddressanyissues. In the preface to the firsteditionof WasSollen,Dedekind addresseshisdesire foramore rigorousfoundationtomathematics: “In science nothing capableof proof oughtto beaccepted withoutproof.Though this demand seemsso reasonableyetI cannotregard it as having been meteven in the mostrecent methodsof laying the foundationsof thesimplestscienceii … numbersare free creationsof the human mind…It is only the purely logical processof building up the science of numbersand by thusacquiring the continuousnumber-domain thatweare prepared accurately to investigateour notionsof spaceand time by bringing theminto relation withthis number-domain created in ourmind.” i Dedekind referred to a set as a system. ii Here Dedekind singles out,in a footnote, the work at the time of Kronecker, Schröder, and von Helmholtz.
  41. 41. History of Mathematical Induction & Recursion [B]. 40 In WasSollen Dedekindreferred tosetsas systems consistingof things(Dinge) thatcouldbe “considered froma common pointof view,associated in the mind”.Although,inlatersectionsof Was Sollen,Dedekindreferredtothese thingssimplyaselementsand tookthe same conceptof Peanobyusinglowercase letterstorepresenttheseelements.Dedekindalsoreferred tofunctions as transformations. As alreadynoted,DedekindandPeanotackle theirworkfromdifferentangles.Peano’s axiomaticapproachallowsPeanotogive hisaxioms,alongwithhis4new definitions,andexpand fromthere to produce desiredresults.ThusPeanoestablishesthe arithmeticof the natural numbers fromhis axioms.Incontrast,Dedekind’sconstructionof the natural numbershasto be more progressive.The foundationsof the workbyDedekindconsistof aseriesof definitionsand correspondingtheorems.Thusthe earlysectionsof Dedekind’s WasSollen addressthese requirements. In the thirdsection, Similarityof Transformations.SimilarSystems,Dedekindsaysthattwo setsare similar if theycan be putin to a one-to-one correspondenceundera similar function,one that isbijective.Thisisfollowedbythe definitionof a class, by whichall setsthatare similartoone anothercan be groupedtogetherandany one of the setsin the class can be the representativeof the class as a meansof identifyingthe classfromanyotherclassof similarsets.The conceptof similarity betweensetsishowDedekinddefinesinfinitesetsandfromthis,finite sets. ¶64. “Definition. A system 𝑆 is said to be infinite when it is similar to a properpartof itself; in the contrary case, 𝑆 is said to be finite.” Thus,a set 𝑆 is an infinitesetif ithasa one-to-one correspondence toaproperi subsetof itself.Onthe otherhand,if thisisnot the case,then 𝑆 issaidto be a finite set.Dedekindhasnow definedwhatitmeansfora setto be infinite andhighlighteditsdifference fromafinite set. ¶66. “Theorem. There existsinfinite systems.” Althoughin1872’s Stetigkeitund Irrationalle Zahlen,Dedekind hadalreadyusedinfinite collectionswhencreatinghisidealsii ,the needtojustifythe existence of infinite sets inWasSollen was because hisdefinitionof the natural numbersdependedupontheirexistence.Thus,according to Dedekind,if infinitesets exist,thenthe natural numbersexist;asomewhatpressure toaccept theirexistence.The proof of thistheoremiscompletelyphilosophical andnon-mathematical and shall be returnedtoindue course.Dedekindcontinuedbydistinguishingbetweeninfinite andfinite i 𝑈 is a proper subsetof 𝑆 if itis not equal to 𝑆, i.e. 𝑈 is strictly contained in 𝑆; 𝑈 ⊂ 𝑆. ii Dedekind defined an ideal as an infinitecollection of algebraic real numbers;numbers that are the root of a non-zero polynomial in 1 variable,with rational coefficients.
  42. 42. History of Mathematical Induction & Recursion [B]. 41 sets,howeverbeforeDedekindcould define the natural numbersthere wasthe needtodefinea simply infinite set. A set 𝑆 is simply infinite whenthere isa similarfunction 𝜙: 𝑆 → 𝑆, suchthat 𝑆 can be obtainedfromthe unionof repeatedapplicationof 𝜙 onanelementof 𝑆 thatisnot in 𝜙( 𝑆)i , i.e. 𝑆 is generatedbyanelementthatisnotin the image of 𝜙. Thisdefinitionis thenreferredtoa particularset, 𝑁, andan elementthatgeneratesthisset; the Base-element“which we shall denoteby the symbol1”. The definitionof asimplyinfinite set, 𝑁, was simplifiedbyDedekindtothe existence of asimilarfunction 𝜙of 𝑁 and an element1 suchthat the followingconditionsare satisfied: 𝛼: Define 𝑁′ = 𝜙( 𝑁),then 𝑁′ ⊂ 𝑁. 𝛽: 𝑁 = 𝜙(1), that is, 1 isthe base-elementof 𝑁. 𝛾:1 ∉ 𝑁′. 𝛿: 𝜙 issimilar,i.e.aone-to-onefunction. It wouldbe possible totake the fourrequirements 𝛼, 𝛽, 𝛾, 𝛿, andassume theirexistence as axiomsfora set 𝑁, howeverDedekind didnotfeel therewasaneedforaxiomsanddefined the natural numbersas follows: ¶73. “Definition. If in theconsideration of a simply infinite system 𝑁 set in order by a transformation 𝜙weentirely neglect thespecial character of the elements;simply retaining their distinguishabilityand taken into accountonly the relationsto one anotherin which they are placed by the order-setting transformation 𝜙,then are these elementscalled naturalnumbers orordinalnumbers orsimply numbers,and thebase- element 1 is called the base-numberof thenumber-series 𝑁.With referenceto this freeing the elements fromevery othercontent(abstraction) wearejustified in calling numbersa free creation of thehuman mind.The relationsor lawswhich are derived entirely fromthe conditions 𝛼, 𝛽, 𝛾, 𝛿, in (71) and thereforearealwaysthe samein all ordered simply infinite systems,whatevernamesmay happen to begiven to the individualelements,fromthe first objectof the science of numbersorarithmeticii .” The remainderof thisdefinitionfocused onhow the elements andsubsetsof 𝑁 are closed under 𝜙. It wasalso noted byDedekind that“thetransform 𝑛′of a number 𝑛 is also called the numberfollowing 𝑛”. i Thus 𝑆 =∪ 𝜙 𝑛( 𝑠), 𝑠 ∈ 𝑆, where 𝜙 𝑛 is the 𝑛th iterate of 𝜙. ii From here the elements of the set 𝑁 arereferred to as numbers.
  43. 43. History of Mathematical Induction & Recursion [B]. 42 Thiswas howDedekinddefined the natural numbers;it wassufficienttohave the simply infinite set 𝑁,awhole,singleentity,orderedbythe repeateduse of the function 𝜙 suchthat,for 𝑛 ∈ 𝑁, 𝜙( 𝑛) ∈ 𝑁. The definitionof the natural numberwasfollowedbyaseriesof theoremsthatcoincide with thisnewdefinition of the natural numbers.Thesetheoremscan be seenasPeano’saxioms.For example,Peano’s8th axiom, 𝑎 ∈ 𝑁, 𝑎 + 1 ≠ 1,can be comparedto paragraph 79 of WasSollen whichsays “every numberotherthan thebase-number1 isan element of 𝑁′.” The followingparagraph,paragraph80, iswhenDedekinddefinedinductionuponthe natural numbers.Dedekindhadpreviouslydefinedinductiononthe notionsof chainsinan early sectionbutstatedthat that formation, the “theoremof induction”,served only asabasisfor inductionuponthe natural numbers.Aswasstatedinthe previoussectionof thiswork,Dedekind referredtoinductionas VöllstandigeInduktioni andfromthispointin WasSollen, itwas usedto prove the majorityof the theoremsthatare to follow.Inparagraph126, the “theoremof the definition by induction”is introduced;this wasthenusedtodefine addition,multiplicationand exponentiationwiththeirsubsequenttheoremsandpropertiesprovenbyinduction. i Beman translates this as complete induction.
  44. 44. History of Mathematical Induction & Recursion [B]. 43 [2.3] Arithmetices principia and Was Sollen: The authors of Arithmetices principia and Was Sollen bothhad the same objective inwriting theirpublications;toestablishthatmore rigorousfoundationtomathematicsthatbothDedekind and Peanothought mathematicsrequired atthe time.AlthoughDedekindandPeano attackedtheir problemfromdifferentanglesandtakingdifferentapproaches,bothwere successfulinobtaining theirobjective and theirideas were sostrongthatthey are still beingusedas a basisfor mathematics today. Dedekind’sopposite approachfromPeano wasinhow the natural numbers were defined; havingfirstdefinedwhatitmeanstobe an infinitesetandsayingthatthe natural numbersare such a set undera particularone-to-onefunction,whereas Peanobuiltupfromunity,usinghisaxioms.As we have seenPeano’suse of induction enabledhimtoworkwiththe infinitefromafinite perspective;itallowed Peanotorepresent the potentialinfinity,being finite ateachpointandnever actuallyreachingan infinite totality.Incontrast, Dedekind’sdefinitionof the natural numbers isnot free of the existence of the actual infinite. Atthe time thiswasa boldand unusual approachand Peano’saxiomswouldhave beenmore readilyacceptedbythe mathematical community. For Dedekind,the infinitewaseasierthanitscounterpart;the existence of aninfinite setandthe definitionof the natural numbers,paragraphs 66 and72 respectively, were addressedbefore the specificfinitecase of theorem81; 𝑛 isnot equal toits successor 𝑛′. To acceptthe existence of an actual infinite setwaswhateveryothermathematician atthe time wasavoidingorwhose existence was absurd to thembutit led to a newwayof lookingatthe natural numbers. Thisnewoutlookonthe most simplestof all mathematical areashada huge bearingonthe worksof others(those whodidn’trejectit) andthe proof of the existenceof aninfinitesethelpedin mathematiciansacceptingthe workof GeorgCantorwhoat thistime hadalreadybegun topublish hisworkson transfinite numbers. We have seenthatDedekind’sdefinitionof the natural numbers,paragraph73, wasvery wordyand itcame froma philosophical viewpoint,justlikethe ‘proof’of the existence of simple infinite sets. Dedekindcouldhave definedthe natural numbersinamore mathematical structure but thisapproachallowedDedekindtoabstractnumberfromanyneedof justificationof thought, much like inthe same veinasPeano’suse of the inductionaxiom.Thismethodof abstractionwas fundamental tobothPeanoandDedekindtocreate theirfoundations;itallowedbothmento distance theirideasfromanyconcrete thing (Dinge) orfrom any justificationinnature andtheir abstractionleadthemtogenerality. Theirrigorousfoundationswere laidoutandtodaytheystill exist,togetherinfact,as the Dedekind-Peano Axioms.
  45. 45. History of Mathematical Induction & Recursion [B]. 44 Dedekind’sacceptance of anactual infinitesetisworthconsideration.Before the theoremof paragraph 66, thereexist infinite sets,any theoremthathad beenstatedwasalwaysfollowedbya proof that stuck to mathematical reasoning.Howeverwiththistheorem,Dedekindbeganhisproof with“My own realm of thoughts”whichimmediatelysuggeststhatthere were tobe no mathematical reasoninginvolvedwiththisproof,andindeedthatwasthe case.Dedekindinstead argueshiscase froma philosophical pointof view,butwhy? Thiscouldbe because of the statusthat the infinite hadwithinmathematicsatthe time – bannedbyAristotle inthe age of the great Greekmathematicians,noone daredtoacceptits existence fromthenon,whetheritbe potentialoractual.An example of afamousmathematician whowouldneveracceptthe infiniteisGalileoi whobelievedinthe axiomsof Euclid’s Elementsand inparticular– the wholeis greater than thepart.Therefore,toGalileo,tohave abijective correspondence betweenasetanda propersubsetof that setwouldmeandenyingthe truthof Euclid’saxiom.Dedekindtriedaddressingthisbytakingthis paradox andusingitasa definitionfor the infinite,essentiallysaying,thatthe infiniteisthatwhichisnotinfinite.Thus,the infinite would have differentpropertiestoitscomplementandvice-versa. However,anyconsiderationof the infinite atthe time wouldhave beenmetwithsternobjection by many.Most of the ‘justifications’thatwere usedtodothisusuallycame froma philosophical pointof view.Thuswasthiswhy Dedekindfounditnecessarytoargue hispointinsucha manner? Another,simplerreasonforthiscouldbe that Dedekindsimplybelievedittobe true,but couldnot formulize amathematical proof tojustifyit. Howeverthe questionarises,shouldtheorem66 be a mathematicaltheoremorshouldit come underthe postulate/axiomtitleandjustbe acceptedastrue,in orderfor Dedekindto continue hisworkmaintainingacontinuousmathematicaltruth?;foreverytheoremthatisbased uponthat of 66 wouldbe basedona philosophicaljustification. The structure of Dedekind’s WasSollen isdifferentfromthatof Peano’s.Dedekinddidnot feel the needtodefine the usual arithmeticdefinitionsof additionandmultiplicationimmediately afterdefiningthe natural numbers,insteaddefiningthe structure of the natural numbersandthe orderingof itselements.One reasonforthiscouldhave beenbecause Dedekind’sdefinitionof the natural numbersdependedonthe existence of infinite sets,soaneedtoinvestigatethe simple infinite set 𝑁furthermayhave stoodinthe way.Anotherreasoncouldbe because Dedekinddidnot feel the needtodoso as the needto define the structure of 𝑁 andthe orderof itselementswhere more important.ThisallowedDedekindtodistinguishfurtherbetweenfinite andinfinitesets,use i Galileo Galilei (1564-1642),Italy.
  46. 46. History of Mathematical Induction & Recursion [B]. 45 definitionbyinduction,andtalkof the classesof infinitesets;all of thisbefore additionof the natural numbers.Maybe there wasa hiddenagendaof tacklingthe actual infinite that Dedekind wantedtodisguise behindthe natural numbers. Whateverreasonthere wasforthe order of Dedekind’swork,itremainsthatDedekindhad more material thathe wantedtopublishandshow toothers than simplydefiningthe natural numbers.Thisextramaterial wasmainlyfocusedonthe actual infinite anditwasparallel tothe workbeingdone byGeorg Cantorat the time.
  47. 47. History of Mathematical Induction & Recursion [B]. 46 [2.4] Dedekind’s Interest in Infinity: We have seenthatDedekindfoundsignificanceinthe infinite;fromthe infinite idealsof Stetigkeit to the infinite systems of WasSollen. In hiswork,Dedekindtriedtojustifythe use of infinity;byusinginfinite setsasabasisfor hisdefinitionof the natural numbers,Dedekindtriedto make it difficultandcontradictorytodenythe existenceof suchsets.We know that Dedekindwas seekingtoestablishsounderfoundationsformathematics,so toinclude infinitesetsinhis foundations,Dedekindmusthave alsothoughtthatinfinite setswereanimportantpartof mathematics. Once the natural numberswere definedandthose theoremssimilartothe axiomsof Peano stated,Dedekindproceededtodefinethe structure of the natural numbersunderthe greaterthan and lessthan relations.AfterthisDedekindstated animportantresult,one whichCantorwouldfind importantinhisownwork onwell-orderedsets: ¶96. “Theorem. In every part 𝑇of 𝑁 there existsone and only one leastnumber 𝑘, i.e. a number 𝑘 which is less than every othernumbercontained in 𝑇.” Thus,in anysubsetof the natural numbers,there isaminimal element,one smallerthan everyotherelementinthe subset,accordingtothe orderon the natural numbers.Thiscan be extendedtoanysetthatis similartothe natural numbersor a subsetof the natural numbers;soit appliestoall simplyinfinite sets.Thiscoincideswiththe theoreminparagraph72, whichsaysthat everysimplyinfinite sethasasubsetthatis similartothe setof natural numbers.Eventually Dedekind proved thatall simplyinfinite setsare similartothe setof natural numbersand these simplyinfinitesetscanbe groupedintodifferentclassescorrespondingtowhichsubsetof the natural numberstheyare similarto. Today,we wouldsaythat thismeansthat any simplyinfinitesetisone thatiscountable and therefore amodel of the natural numbers.Inthe time of Dedekind,the natural numberswere knownto be countable (theycountthemselves essentially),butothernumbersystemswere abitof a grey areaat the time,especiallythe real numbers.Cantorbelievedthatthe real numberswere of a differentsizethan the natural numbers,butthisfacedstiff oppositionfromKronecker,whosaidthat the ideaof twodifferent(potentially) infinitesizeswasridiculous.However,tosaythat all simply infinite setscanbe classedtogether,one couldworkoutthatthat wouldmean,since theydonot have a leastelement,the setof real numberscouldnotbe putinto the same classas the natural numbers.Elaboratingonthis,noteverysubsetof the real numberswouldhave aleastmember,take the openinterval (0,1) andcompare it with [0, 1].
  48. 48. History of Mathematical Induction & Recursion [B]. 47 The needforDedekindtodistinguishbetweenthe finite andthe infiniteisanessential partof Was Sollen.The differencesbetweenthe twoare laidoutinsimplisticformsothat theycan be easily followedandmore readilyaccepted.WhatDedekindtriedtoestablish,wastostopmathematicians fromtacklingthe infinite fromafinite perspective.Asmathematicianswereapplyingpropertiesand theoremsdefinedonthe finite,tothe infinite,theywere notgettingconsistencyor producingthe resultsthatwere expected.Thisisevidentinthe case of Galileo andEuclid’saxiomabove andthis treatmentof the infinite still happenedinthe 18th century.Thus,there were some whowere unwillingtoinvestigate andworkwiththe infinite.Butthe workof Dedekindtriedtocurbthisway of thinkingandattract more mathematicians toacceptthe conceptof the actual infinite. Anotherimportantaspectof WasSollen isDedekind’sfinal section.Itisentitled TheNumber of Elements in a Finite System and itaddressesthe size of a finite set.The mostnotable paragraphin thissectionis161 whichI shall give initsentirety: ¶161. “Definition.If a set 𝛴 is a finite system,then thereexists oneand only onesingle number 𝑛 to which 𝑍 𝑛~ 𝛴, fora system 𝑍 𝑛.This number 𝑛 is called the number(Anzahl) of elements in 𝛴; it showshowmany elementsarecontained in 𝛴. If numbersareused to expressaccurately thisdeterminateproperty of finite systemsthey are called cardinal numbers.If thereis a similar transformation 𝜓: 𝑍 𝑛 → 𝛴, then wecan say thatthe elementsof 𝛴 are counted and setin order by 𝜓 in determinatemanner,and call 𝑎 𝑚 the 𝑚th element of 𝛴; if 𝑚 < 𝑛, then 𝑎 𝑚 is called the element following 𝑎 𝑚 i,and 𝑎 𝑛 is called the last element.In this counting of theelements thereforethenumbers 𝑚 appearagain asordinalnumbers.” Thus,the elementsof the finitesetcanbe indexedbythe natural numbersandif the order of the elementsof the finite setare arrangedsuchthat theyfollow the orderof theirindex,thenwe can say that theyfollowthe orderof the natural numbersandcan thus be calledordinal numbers,as statedinDedekind’sdefinitionof the natural numbers.Conversely,the definitionsaysthat, if we abstract fromthe orderof the elementsinaset,thenwe obtainthe cardinal numberof the set. Dedekinddoesnotstate whenanumberisan ordinal number,butthe implicationisthatthisis whenthe numberfollows the orderingof the numbersthatprecede itii .Bothcardinal andordinal numberswere the keyinterestsof cantorat the time,whowastryingto introduce bothtohis transfinite settheory.The backingof Dedekindwouldhave beenamajorboostto theiracceptance as the oppositionagainst Cantorandhistheorywasmounting. i I think this should havesaid 𝑎 𝑛 is thenumber following 𝑎 𝑚 . ii Thankfully Cantor gives a better definition of ordinal number.
  49. 49. History of Mathematical Induction & Recursion [B]. 48 Cantor beganto workon infinitesetswhenthe needtoinvestigate furtherthe cardinalityof infinite setsandthe numberof pointsinthe continuumi became clear.ItappearsthatDedekindsaw cardinalityasan importantaspectof everyset;fromthe remark at the endof WasSollen, Dedekind statesthat the cardinalityof a setdoesnotchange undersimilarfunctions,thusthere canbe many setsthat share the same cardinalityandbe collectedtogetherandtreatedasone andthe same. However,Dedekinddoesnotinvestigate thisasit“doesnotlie in the line of this memoirto go furtherinto their discussion”,noristhere any considerationof the cardinalityof infinitesets,surely theyjusthave infinite cardinality anyway. The newfoundationsof the infinite,andthe finite,wouldhave servedasabasisfor the work of GeorgCantor,not for Cantorto buildhisworkuponit as he had alreadybeguntopublishworkon histransfinite numbers,butforothermathematicianstoreference anydoubtsinthe workof Cantor. i The real numbers, or equivalently,the open interval (0, 1) of the real numbers.

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