On the analysis of indirect proofs: Contradiction and contraposition

Nicolas Jourdan & Oleksiy Yevdokimov University of Southern Queensland nicolas .jourdan@usq .edu .au oleksiy .yevdokimov@usq .edu .au

roof by contradiction is a very powerful mathematical technique. Indeed, Premarkable results such as the fundamental theorem of arithmetic can be proved by contradiction (e.g., Grossman, 2009, p. 248). This method of proof is also one of the oldest types of proof early Greek mathematicians developed. More than two millennia ago two of the most famous results in mathematics: The irrationality of 2 (Heath, 1921, p. 155), and the infinitude of prime numbers (Euclid, Elements IX, 20) were obtained through reasoning by contradiction. This method of proof was so well established in Greek mathematics that many of Euclid’s theorems and most of Archimedes’ important results were proved by contradiction. In the 17th century, proofs by contradiction became the focus of attention of philosophers and mathematicians, and the status and dignity of this method of proof came into question (Mancosu, 1991, p. 15). The debate centred around the traditional Aristotelian position that only causality could be the base of true scientific knowledge. In particular, according to this traditional position, a mathematical proof must proceed from the cause to the effect. Starting from false assumptions a proof by contradiction could not reveal the cause. As Mancosu (1996, p. 26) writes: “There was thus a consensus on

the part of these scholars that proofs by contradiction were inferior to direct Australian Senior Mathematics Journal vol. 30 no. 1 proofs on account of their lack of causality.” More recently, in the 20th century, the intuitionists went further and came to regard proof by contradiction as an invalid method of reasoning. In the classical approach the existence of an object can be proved by refuting its non-existence. To the intuitionist this is not acceptable. Led by the Dutch philosopher and mathematician L. E. J. Brouwer the intuitionists reject the law of excluded middle and only accept constructive proofs as valid proofs. Despite all its past criticism and even rejection by some members of the mathematical community, the majority of today’s mathematicians would agree with Rivaltus’ position: “Proofs by contradiction have the same dignity as direct proofs because, even if they do not give us the cause of why a certain

55 56 Australian Senior Mathematics Journal vol. 30 no. 1 Jourdan & Yevdokimov “If itisrainingthenIaminside”,and“Ioutside”. Fromthesetwofactswe 1. & deVilliers,2008,p.329).Morespecifically, HannaanddeVilliers are modus tollens.Accordingtothisrule,apropositionisprovedbyshowingthatits We nowinvestigatethetheoreticalfoundationsofproofbycontradiction.This 3. 2. Principles andStandardsdocumentseveralothermathematicscurricular For completenesswebriefly mentionhereanothercommonlyusedrule of write: “TherecentNationalCouncilofTeachers ofMathematics(NCTM) reasoning knownasmodusponens , whichcanbeformallystated as: not onlythatsomethingistrue,butalsowhyittrue.”(p.330).Theaim unattended whilststudyingandpractisingprooftechniques.We expectthe to understandtherolesofreasoningandprovinginmathematics”(Hanna be true.Goubault-LarrecqandMackie(1997,p. 4) giveasimpleexampleto Description ofageneralmodel by whichweknowthatastateofaffairsholds”(citedinMancosu,1991,p.34). by contrapositionareclarifiedthroughtheexaminationoftheirsimilarities can readilydeducethatitisnotraining.Moreformally, modustollenscanbe method of proof is an application of the rule of logical reasoning known as mathematics curriculumatalllevels.AsHannaanddeVilliers(2008,p.329) one firstassumesitsnegation¬ptobetrue.Onethengoesonshowthat misconceptions thatseemtoexistbetweenproofbycontradictionand means that“mathematicseducatorsfaceasignificanttaskingettingstudents educational jurisdictionsaroundtheworld.” of thepaperistodescribeageneralmodelproofbycontradiction.The challenging educators“tofostertheuseofproofasamethodtocertify illustrate modustollens.Supposeweknowthefollowing twofactstobetrue: implies documents have elevated the status of proof in school mathematics in several paper tobeofinterestundergraduatestudentsandtheiruniversityteachers false anditfollowslogicallythatitsnegation,the originalpropositionpmust falseness leadstounacceptableconsequences.Wishingproveastatementp, affection istobepredicatedonthesubject,theynonethelessgiveusareason as wellanyoneinterestedinindirectproofs. and peculiaritiesoftheindirectproofstructure,somewhichareoftenleft and differences.Moregenerallyweshedsomelightonthespecificdetails stated as: This renewedemphasisonproofandprovinginmathematicseducation Nowadays, proofhasbeenassignedamoreprominentplaceinthe ¬q ∴ ¬p p →q q, where qis already known to be false. By this argument, ¬pmust be (notq (ifp ) thenq ) On the analysis of indirect proofs Australian Senior Mathematics Journal vol. 30 no. 1 57 . exists there to the verbal 2 Contradiction = 2b 2 → a direct argument a direct argument Black box which proceeds from our (false) assumptions ) Figure 1. Model for proof by contradiction. Model for proof 1. Figure → for all, if-then—as well as the two quantifiers— found that none of the students new to the proof were or, is even” (p. 44). 2 2 then q (if p q → q opposite Assume the Such rules of reasoning are not part of mathematics itself. Rather, they itself. Rather, are not part of mathematics Such rules of reasoning In general, a mathematical statement may contain any number of logical In general, a mathematical statement To reiterate, the aim of a proof by contradiction is essentially to contradict by contradiction is essentially reiterate, the aim of a proof To The veryThe in many cases, be a opposite—can, the first step—assuming Their investigation into students’ difficulties in proving the irrationality Their investigation into students’ difficulties in proving the irrationality The ‘black box’ is truly the heart of the proof. It is usually a direct proof. It is usually heart of the truly the box’ is The ‘black stand above the subject matter to which they are applied. subject matter to which they are stand above the stumbling block for students. In his 1979 study, Williams (cited in Thompson, Williams his 1979 study, stumbling block for students. In several cognitive links to derive a new synthetic connection.” students. As Epp (1998, p. 711) states: “One of the most serious difficulties that students. As Epp (1998, p. 711) states: their own students have in actually constructing proofs by contradiction on as valid, owing to the fact that the initial assumption 1 = 0 is false. Obviously as valid, owing to the fact that the able to spontaneously link the algebraic statement a and algebra. In addition to carryingto addition In algebra. and each which in procedures sequential out of synthesis the requires often proof mathematical next, the cues action argument which proceeds from our (false) assumptions. What is in the box What (false) assumptions. our from which proceeds argument general model for proof by contradiction. general model for form of linkage different from the familiar routines of elementary arithmetic is utterly context dependent. Even for the simplest cases, the logical steps to is utterly context dependent. Even for the simplest cases, the logical one of our assumptions. If the aim is achieved then, as a consequence of modus modus of consequence a as then, achieved is aim the If assumptions. our of one is in supposing the wrong thing”. This first step is indeed crucial. Thompson crucial. is indeed first step This the wrong thing”. supposing is in difficulties at the onset”. connectives—and, effectively as many students appeared unwilling to argue from an hypothesis effectively as many students appeared of the number be taken may be far from obvious to the novice and require much insight. As be taken may be far from obvious to the novice and require much tollens, the original statement must be true. In Figure 1 below we propose a propose we below 1 Figure In true. be must statement original the tollens, they saw as untrue (p. 478). this difficulty considerably hindered students’ ability to use indirect proof use indirect ability to students’ hindered considerably difficulty this representation “a that, formulating the opposite of such statements is a difficult task for most that, formulating the opposite of Barnard and Tall (1997, p. 42) explain: “Mathematical proof introduces a (1997, p. 42) explain: “Mathematical proof introduces Barnard and Tall 2. p Many researchers (Epp, 1998; Lin et al., 2003, Thompson, 1996) report Thompson, 1996) al., 2003, Lin et 1998; researchers (Epp, Many 3. ∴ 0, many students did not view the argument 0, many students did not view the Asked to evaluate a proof that 1 ≠ 1. p 1996) presented students with a number of items dealing with indirect proof. 1996) presented students with a (1996, p. 476) adds, “If students have difficulty writing a negation, they have writing a difficulty have “If students p. 476) adds, (1996, 58 Australian Senior Mathematics Journal vol. 30 no. 1 Jourdan & Yevdokimov ¬p A conditionalstatementisaoftheform“ifp This partofthepaperexploresdifferencesandsimilaritiesthatexistbetween Italy. IntheUnitedStates,manytextbooksfailtoclearlydistinguishbetween Proof bycontrapositionrestsonthefactthatan implicationp two propositionsp these twotypesofproof.Epp’s Discrete MathematicswithApplications (2011)is teachers call‘proofbycontradiction’…bothproofcontrapositionand respectively. Inlogicconditionalstatementsareknownasimplicationsand both methodsasindirect methods ofproof.We willkeepwiththisterminology our assumption,thecontradictionmaytakeformofamathematical contradiction isnotalwaysthisapparent.Indeed,ratherthancontradicting contradicting one ofourassumptions will achievethedesiredaimbut closely relatedthattheyaresometimeconfusedandsotheirdifferences may bewrittenas“p contrapositive produced duringtheproof. our aimistoshow, usuallythrough adirectargument,thatthecontrapositive proof bycontradiction.”Thisconfusedstateofaffairsextendswellbeyond in thepaper. Bothmethodsarecloselyrelated.Infact,thetwoso proof bycontrapositionandcontradiction.Theliteraturerefersto implication isalsotrue. In aproofbycontrapositionboththestartingpoint fact weknowtobefalseoritmayevenemergeoutofourownconstruction Proof bycontraposition Proof bycontraposition bycontradictionversusproof Proof given set.“Allhumansaremortal”isanexampleofauniversalstatement. at acontradiction,butwhereisthatcontradictiontobefound?Certainly, an exception and she contends that proof by contraposition applies only to “a and Mariotti(2008,p.402)write:“InItaly, ingeneral,mathematiciansand and the aim are clear: assume that statements. Auniversalstatementisaaboutalltheelementsof specific classofstatements—thosethatareuniversalandconditional”(p. 204). similarities areworthwhileexploringandclarifying.Forexample,Antonini statement istrue.Bylogicalequivalencethisautomatically assuresusthatthe statements. Inthismethodofproof,thereisnocontradictiontobefound . Rather . To illustrate, belowisanexampleofaproofbycontraposition. To proceedfurtherweneedtoprovidemoredetailsonbothtypesof In aproofbycontradiction,theaimisnotalwaysclear. We dowanttoarrive ¬p

→ andq impliesq ¬q (notq referredtoasthehypothesisandconclusion ” (p impliesnot →q ¬q is true and show that it logically leads to insymbols). p ) aretwologicallyequivalent thenq

→ ”, withthe q andits On the analysis of indirect proofs Australian Senior Mathematics Journal vol. 30 no. 1 59 and we and ¢ ∈ k ). w a +1, l 1

n + w w ) o a i . l a k t

l = 2k = s

2 a and not q ' x n + n eg o a 2 (p i n t g k r a 2 e is even” respectively. Negating is even” respectively. c l o ( i 2 l b M ∧ ¬q p u = e o . m 1 i d d → ¬q → q + k 4 ∧ ¬q + ) q ∈ ¢ p 2 is even, then so is x is even, q 2 k ¬ ∈ ¢ ¬p ∨ ! 4 q ∧ is even” and “x ∀x p ∈ 2 = ¬ p ) and p ∀x ¬ 2 is odd” and “x is odd” respectively. Thus, we are odd” respectively. is odd” and “x is n ∧ ¬ ) ( 2 , 1 p 1 ¬ ¬ + + → q ≡ ≡ ≡ k is the conjunction p n ) 2 2 q ( = = → 2 2 → q x x p are now “x ( ¬ are defined as “x and ¬q and q ¬p p negation of p The universal statement becomes an existential statement. the The conditional statement becomes a conjunction. In particular, be an integer. Prove that if x Prove be an integer. We have proved that the square of an odd number is odd. In fact, by logical of an odd number is odd. In fact, have proved that the square We The logical steps shown below provide a justification for the logical The logical steps shown below provide a justification for the assume that statement to be false and we show that the logical consequences assume that statement to be false a proof by contradiction does not simply assume the conclusion to be false. a proof by contradiction does not simply assume the conclusion Proof by contradiction difference between the two methods of proof. In the context of implications difference between the two methods of proof. In the context of implications proceed this way: of our assumptions lead to a contradiction or an impossibility. For example, of our assumptions lead to a contradiction or an impossibility. equivalence between ¬(p of an odd number our starting point is therefore is point starting our number odd an of equivalence we have also proved Example 1. We note that Proof 1 can be Example 1. We equivalence we have also proved the two propositions, the statement we want to prove has the form the statement we want to prove the two propositions, two consequences: to prove that there is an infinite number of primes we assume from the onset the from assume we primes of number infinite an is there that prove to we are in a position to examine the main that number to be finite. Now, trying to show that, given any odd integer, its square is also odd. By definition also odd. square is its tryingany odd integer, given that, to show two odd numbers is odd. regarded as a special case of the more general statement that the product of product the that statement general more the of case special a as regarded where where Proof 1 Proof Example 1 Example Formally the statement can be written as Formally the statement Let x . Negating the entire statement has statement to be false Rather it assumes the entire 2.

We now turn our attention to the other method, the proof by contradiction. now turn our attention to the We As previously stated, in general, to prove a statement by contradiction, we by contradiction, statement to prove a general, in previously stated, As 1. 60 Australian Senior Mathematics Journal vol. 30 no. 1 Jourdan & Yevdokimov (2011), “…anystatementthatcanbeprovedbycontraposition As explainedpreviously, Example1hastheform A proofbycontradictionassumesthefollowingthree propositionstobetrue.

Let Essentially we are trying toshowthatwecannotfindanoddintegerwhose Essentially wearetrying here. We provebycontradictionthatforanyintegerx Proof 2 Example 2 we assumeboth“x with theassumption“x where we provecanbeseenas need to assume both p than oneproposition.Considerforinstancethefollowingexample. the directmode,whichisnotconsideredinpaper. AccordingtoEpp by contradiction. But the converse is not true. Statements such as ‘ contraposition. Thisisespeciallythecasewhenhypothesiscontainsmore completes theproof.We note,whilstitmaybepossibletostarttheproof divide a irrational’ can beprovedby contradiction but not bycontraposition”(p.204). proof bycontraposition. Some conditionalstatementsmaybeprovedbycontradictionbutnot Schematically, theproofcanbeseenas Symbolically, thestatement canbewrittenas a conditionalstatementsincesuchstatementsrequireatleasttwopropositions. assumptions. Choosing“x square isevenandourstartingpointcannowbeanyoftheabovetwo The result“x To illustrate the difference between the two methods we revisit Example 1 Indeed, “ a , p b c andc andq thenc |a p +b 1 2 aredefinedas“x ∃ bethreeintegers.Provethatifc doesnotdivideb ( is irrational” is a single proposition which cannotbe written as 2 a isodd”contradictstheotherassumption“x , x x b isodd”and“x , 2 2 c ∀ ∃ x = = ) ( and ¬q ∈ ( 2 a and 2 n ! , 2 ∈ :p k b iseven”,thatapproachwillchangetheproofto 3 + , + c isodd”weproceedalongthesamelinesas : 1 1 ) p , ∧¬q ) ∈

1 2

to be true as our starting point.This means that n ∧ ! = ∀ x 2 ∈ 3 p iseven”and“x . 4 leadtoacontradiction.

2 2

! k iseven”tobetrue.Formally, thestatement ( ∧ 2 ∈ ¢p c Figure 2 + ¬ c | a p 4  a q 2 + k leadtoacontradiction. b + →q ) 1 ∧ = ( c 2 ( / | 2 iseven”respectively. Nowwe a dividesa k ) 2 and → + 2 c ifx k / | ) b + 2 1 +c iseventhensox butc 2 iseven”.This c ¬q |b doesnot 2 is . On the analysis of indirect proofs Australian Senior Mathematics Journal vol. 30 no. 1 61

. ) (x = f crossing divides b ! ∈ . The proof is 2 x , and we have ∀

, ∈ 0 0

3 = > 2 2 0 , no further progress is r + − − 2 2 that produces a negative 1 0 k r x − 6 ) by c = (–1) = 3⋅(–1)5 + 2⋅(–1) + 1 1 ) + k is a continuous function with continuous a is 2 4 f r = + b ( . It follows by substitution ∈ . It follows x 2 2 3 + 1 has exactly one root. f 5 r k 1 − − , k e 1 = ) 1 c r 1 (or a ) ) r er + 2x 2 1 3 ( h k f + w −

x = c 1 c with k 2 3 1 ) k + 2x c k k 5 ( c 2 = –1, we get f + ( ' = = = 3 tells us ) exist, then the mean value theorem f 2 x c a a = k 2 2 : -axis. Given that Given -axis. k ) x + ) = 3x 2 + r 5 , a x 1 and r r 3 1 ( ) = 0. Figure 3(a) below confirms the graph of f ( , contradicting the second proposition p , contradicting the ∈ (x d c dx

). Guessing x ∃ = , the third b , the third . However, ) c (x 1 x ( = c f divides a = k -axis between –1 and 0. d dx , a few steps lead to a contradiction. Indeed, the first proposition the first proposition Indeed, lead to a contradiction. , a few steps ∈ (–1, 0) : f

+ b c (0) = 1, we need to find another value of x function crosses the crosses function more roots (say r the x value of f The uniqueness of the root is proved by contradiction. Assuming two or The uniqueness of the root is proved by contradiction. Assuming To establish the existence of a root it is sufficient to show that the To ∃ f ensures that = –3 – 2 – 2 – 1 = –6. Thus, the intermediate value theorem In fact, an attempt at a proof by contraposition quickly leads to an impasse at a proof by contraposition quickly In fact, an attempt The difference of two integers is an integer thus k of two integers is an integer thus The difference and ¬q 1 since we only have a single assumption to work with—namely c a single assumption to work since we only have shown that c arise from our own construction produced during the proof. The calculus arise from our own construction at the point x function cannot be both positive and equal to zero. function cannot be both positive possible. Further examples of proof by contradiction including geometry, linear algebra and calculus. As previously mentioned, the mentioned, previously As calculus. and algebra linear geometry, including complete. contradiction we are looking for may be on our original assumption or it may it or assumption original our on be may for are looking we contradiction where the constructed derivative category, example below falls into the latter means a Proof 3 Proof Example 3 Knowing nothing of the divisibility of a Knowing nothing of the divisibility Prove that the function f(x Besides number theory areas proof by contradiction may be applied in many 2.

In other words, there should be a horizontal tangent to the graph of y In other words, there should be a horizontal tangent to the graph We now show that, working from the first and third propositions, namely namely and third propositions, from the first show that, working now We p The proof is in two parts: (1) existence and (2) uniqueness. 1. 62 Australian Senior Mathematics Journal vol. 30 no. 1 Jourdan & Yevdokimov Figure 3(a) Figure 3(b) We alsohaveAC=CAI We haveAB=BAI

Pre-multiplying byC Let A Example 5 Proof 5 therefore AB=BC.Contradiction. is contradicted. Suppose thatA In thelinearalgebraexamplebelow, itisalsotheoriginalassumptionthat beaninvertiblematrix.ProvethatA Assume Given a triangle Proof 4 Example 4 which meansanytangenttothegraphhasapositivegradient to AC. original assumptioniscontradicted. itisdifferentiablesoits one root.Itisincreasingeverywhere, equal and equal. Itfollowsthatthetwotrianglesmustbecongruent, derivative isalwaysgreaterthanzero. get acontradiction.Thisfunctioncannothavemorethan and thereforecannotbehorizontal(Figure3(b)).Thus,we angled) triangles hastwodistinctinverses,namelyB The nextexampleisfromgeometry. Inthisexamplethe Now givesC , wheretheidentitymatrixis BD isperpendiculartoAC,that∠ADB=BDC90°. BD isamediansoAD=DC.Thereforethetwo(right- , soitfollowsthatAB=AC . BD is a median, prove that I (AB) =C = B ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ ABC such that the two sides ABD and 0 0 0 1 " 0 0 1 (AC). A 0 0 1 BCD have two corresponding sides # ! –1 Figure 4 isunique. 0 0 0 1 ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ andC D BD is not perpendicular (B AB and ≠C C ). BC are not On the analysis of indirect proofs Australian Senior Mathematics Journal vol. 30 no. 1 63 n m ! ! ! = is unique. ∈ ∉ ∉ . ∈ 1 x x 1 1 x x : ) ∧ → → 0 ! . That is, IB = IC. . That is, ! ≠ ! ! ∉ ∈ ∈ n ∉ x (

1 x 1 x x

:

! , , ! ∈ ! ! ∈ n ∈ ∈ = (CA)C , ∈ . x x x

m

∃ ∀ ∀ . ∃ ! ∈ 1 x , which means x . Contradiction. The inverse of A The inverse . Contradiction. ∈  contradicts the assumption x n m = = C x The introduction of mathematical proof in the classroom remains a The introduction of mathematical proof in the classroom remains a consensus that learners do find indirect types of proof quite difficult and a consensus that learners do find indirect types of proof quite difficult and the next line of the derivation is identical to the proof by contraposition. and the next line of the derivation is identical to the proof by contraposition. Concluding remarks formidable challenge to students given that, at this stage of their schooling, formidable challenge to students given that, at this stage of their focussed on the concept of contradiction mainly and a general model for this focussed on the concept of contradiction mainly and a general model do struggle with the conceptual and technical aspects of indirect proofs. As do struggle with the conceptual and technical aspects of indirect method of proof was offered. by contradiction as well as by contraposition. by contradiction between proof by contradiction and proof by contraposition. The paper also between proof by contradiction and proof by contraposition. The they are used to manipulating symbols through sequential steps. There is they are used to manipulating symbols through sequential steps. Proof 6 by contradiction Proof Proof 6 by contraposition Proof Example 6 Prove that the reciprocal of any irrational number is irrational. of any irrational number Prove that the reciprocal By associativity, we have (CA)B we By associativity,

Hence, B Our final example shows how the given conditional statement may be proved statement may be how the given conditional example shows Our final Our starting point is The result x This means that, by definition of the rational numbers, This means that, by definition of The negation of the original statement is The negation of the original statement That is, The contrapositive statement is The statement may be written symbolically as The statement may The paper explored and clarified the similarities and differences that exist The paper explored and clarified the similarities and differences Again, our starting point is 64 Australian Senior Mathematics Journal vol. 30 no. 1 Jourdan & Yevdokimov The authors wish to thank an anonymous reviewer for the useful comments Antonini, S.&MariottiM.A.(2008).Indirectproof:Whatisspecific tothiswayofproving? Acknowledgements Thompson, D.R.(1996).Learningandteachingindirectproof. Epp (1998,p.711)states,“Studentsfindproofbycontradictionconsiderably Williams, E.R.(1979).Aninvestigationofseniorhighstudents’understandingthenature of harder tomasterthandirectproof”.Indeed,learnersmaystrugglewith Heath, T. ofGreek mathematics:From ThalestoEuclid(Vol. L.(1921).Ahistory 1).NewYork: Hanna, G.&deVilliers,M.(2008).ICMIStudy19:Proofandproving inmathematics Proof theory andautomateddeduction . Dordrecht,The Goubault-Larrecq, J.&Mackie,I.(1997).Proof theory Grossman, P. (2009).Discrete mathematicsforcomputing(3rded.).NewYork: Palgrave Barnard, T. &Tall, D.(1997).Cognitiveunits,connectionsandmathematicalproof.In Mancosu, P. (Ed.).(1996).Philosophyofmathematicalpractice.Oxford,UK:OxfordUniversity Mancosu, P. (1991).Onthestatusofproofsbycontradictioninseventeenthcentury. The thirteen booksofEuclid’s elements(translationandcommentariesbyT.Euclid. (1956).Thethirteen L. Epp, S.(2011).Discrete mathematicswithapplications (4thed.).Boston,MA:Brooks/Cole& Epp, S.(1998).Aunifiedframeworkforproofanddisproof.TheMathematicsTeacher, 91(8), Lin, F., Lee,Y. &Wu Yu, J.(2003).Students’understandingofproofbycontradiction.In understanding theconceptofindirectproofsingeneralandproofby References other methods and didactic tools, e.g., counterexamples or the pigeon-hole contradiction inparticular. To addressthisissuefurther, andforlearning principle. But,thisisatopicforanotherinvestigation. purposes, proofbycontradictionmaybeconsideredinconjunctionwith and suggestionswhichhelpedtoimprovethepaper. ZDM – The International Journal ofMathematicsEducation,40(3),401–412. Journal ZDM –TheInternational Mathematics Education,(Vol. 2,pp.41–48).Lahti,Finland:UniversityofHelsinki. PMENA, (Vol. 4,pp.443–449).Hawaii: PME. mathematical proof. PhDdissertation,UniversityofAlberta,Canada. 474–482. Synthese, 88,15–41. 708–713. Heath). NewYork: DoverPublications. Cengage Learning. Macmillan. Dover Publications. E. Pehkonen(Ed.),Proceedings Conference ofthe21stInternational forthePsychologyof Press. ZDM – The International Journal ofMathematicsEducation,40(2),329–336. Journal education. ZDM–TheInternational Netherlands: KluwerAcademicPublishers. N. Pateman,B.Dougherty, &J.Zilliox(Eds),Proceedings oftheJointMeeting ofPMEand The MathematicsTeacher, 89 (6),