argument in favour of axiomatic mathematics in schools

8/6/2019 Argument in Favour of Axiomatic Mathematics in Schools

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THE AXIOMATIC METHOD IN HIGH-SCHOOL MATHEMATICSby

Patrick Suppes

TECHNICAL HEPORT NO. 95April 12, 1965

PSYCHOLOGY SERIES

Reproduction in Whole or in Part is Permitted forany Purpose of the United States Government

INSTITUTE FOR MATHEMATICAL STUDIES IN THE SOCIAL SCIENCESSTANFORD UNIVERSITYSTANFORD J CALIFORNIA


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*he Axiomatic Method in High-School MathematicsPatrick Suppes

Stanford University

Introduction. The importance of th e axiomatic method in modernmathematics scarcely needs a general defense. I t s widespread use inmany parts of mathematics and i t s l o ~ g history of importance in themathematics of ear l ier centuries provide clear evidence that it wil lcontinue to be of importance in the for eseeab le futur e of mathematics.On the other hand, the role of the axiomatic method in high-schoolmathematics is not as universally accepted, even though i t has had aplace in the teaching of school geometry throughout the history ofwestern culture since the time of Eudoxus. In the l as t decade therelevance of the axiomatic method even to the te aching of high-schoolgeometry has been challenged in some quarters.

The point of view that I want to presen t here is to make asvigorous a defense as I can of the importance of teaching t he axioma ticmethod in high-school mathematics. I real ize fu l l well that it is notsuff icient as an argument to point to the importance of t he axioma ti cmethod already in th e te ach ing o f uni ve rs it y mathematics and inmathematical research.

*This art icle was prepared at the invita t ion of the United StatesCommission on Mathematics Instruction and th e Conference Board of th eMathematical Sciences.


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There are th re e g en era l arguments I would l ike to advance foremphasizing the axiomatic approach to mathematics in the h igh-schoolcurriculum. In the f i r s t place t he axiomat ic approach provides animportant method of making the mathematics taught more elementary andthe subject more rest r icted from the standpoint of what th e studenthas to learn and encompass. An excellent example is provided by thereal numbers. We now expect students to know a good deal about th ereal number system by th e time they haVe finished high-school mathematics,at leas t the students who are college bound and who take a fu l l highschool mathematics curriculum. I f the properties of the real numbersystem are taught from an axiomatic standpoint , the student is given arest r icted l i s t of properties that are fundamental and from which a llothers can be derived. Moreover most of the axioms that are usedexpress elementary properties that generalize in a natural way whatthe student has learned in earl ier years. The axiomatic approach alsoassures th e student that the properties he must know are restr ic tedto those expressed by the axioms and the theorems that follow fromthe axioms. I t is my own bel ie f that i t is a l l too easy for usconf ident ly t o assume that students are clear about the p ro pe rt ie s o fthe real numbers and have a good intui t ive feeling for what the realnumbers are. This assumption, which I feel is unwarranted, is relat ivelyc ru cia l to those approaches to secondary-school geometry that leanheavily on the p rope rt ie s o f the real numbers.

Perhaps the best persepctive in which to conside r the axiomaticapproach to the real number system is to ask what are the al ternat ives.I t i s doubtful that many people feel it would be wiser to go through

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the ful l mathematical const ruct ion o f the real numbers via Dedekindcuts or Cauchy sequences of the rational numbers, and moreover it isdoubtful that they would want to construct the ra t ional numbersthemselves as certa in equivalence classes of ordered pairs of integers.An histor ical ly important alternative approach has been through thegeometrical theory of rea l magnitudes, but i t is doubtful that anyonewill want th e rea l numbers to be constructed out of geometricalent i t ies . In any case either of these prospective approaches i scer tainly less e lementary than the axiomatic algebraic approach. Isuppose that a fourth alternative is simply to leave the whole matterup in th e a ir and to develop in a higgledy-pigg ledy fashion thepropert ies needed, but I am skeptical that students wil l have th e r ightsort of confidence and clari ty about the properties of the real numberswhen this approach is taken seriously.

My second argument centers around the importance of developingin tui t ions for finding and giving mathematical proofs. I t is a Commoncomplaint about beginning graduate students in mathematics in the UnitedStates that one of their worst defects is t he ir i nab il it y to write acoherent mathematical proof. A for t ior i , th is is eve.n t ruer of undergraduate students of mathematics. And th e reasons for th is deficiencyare not hard to find. Both at the school and un ive rs it y l eve l expl ici ttraining in the writin g o f mathematical proofs and the expl ici t consideration of heurist ic methods for finding pro ofs a re woefully lackingin the curriculum. Systematic pursui t of the axi omatic method inhigh-school mathematics provides perhaps the best opportunity fort ra ining students at an early stage in th e finding and writ ing of

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mathematical pr oof s. I t is a ll to o easy to assume t h a t as s tu d e n tsdevelop an i n t u i t i o n fo r g e o m e tr ic a l f a c t s , fo r example, they w i l lalmost aut omat i cal l y be ab l e to produce coherent pr oof s. Pu t an o t h erway, what I am say i n g is t h a t I consi der i t jus t ap necessary to t rainth e i n t u i t i o n fo r f in d in g and writing mathematical proofs as to te a c hi n t u i t i v e knowledge o f geometry o r th e r e a l number system. I amcontending t h a t t r a i n i n g in th e f i n d i n g and w r i t i n g of p roo fs mustbe given as much e x p l i c i t a t t e n t i o n and should b eg i n as e a r l y aso t h e r p a r t s o f th e mathematical t r a i n i n g o f s t u d e n t s ; i t is i n th ec o n te x t o f th e elementary mathematics t a u g h t i n h ig h s ch ool t h a t th es t u d e n t ca n f i r s t l e a r n t o work i n a n a t u r a l and easy way with axiomsand th e p r o o f s o f theorems t h a t follow rigorously from the axioms.

My t h i r d argument fo r the use o f the axiomatic method i n h i g h -school mathematics cent er s around th e i n c r e a s i n g importance o f learninghow to think i n a mathematical f ashi on as the to ta l body o f mathematicsi t se l f i n c r e a s e s so r a p i d l y . I t is becoming c l e a r t h a t i t is reasonablyh o p el ess to ex p ect st udent s a t an y st age to master an y s u b s t a n t i a lp o rtio n o f e x ta n t mathematics. We can, o f c o u r s e , agree on th eg r e a t e r importance o f c e r t a i n p ar t s o f mathematics, b u t s t i l l a goodcase ca n be made t h a t perhaps th e b e s t thing we ca n do f o r ou r s t u d e n t sis to b e g in to te a c h them to think mathematically as e f f e c t i v e l yas p o s s i b l e . I t ha s been my own experience t h a t mathematiciansdi scussi ng curriculum a re uneasy i n an y a tte m p t t o c h a r a c t e r i z e whatthey c o n s id e r to be th e e s s e n t i a l n a t u r e o f mathematical t h i n k i n g .As mathematicians they are much more accustomed to t hi nki ng aboutmathematical o b j e c t s and proving f a c t s about th e s e o b j e c t s . I n th e

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same way, it i s much e a s i e r to g e t a c l e a r and s o p h i s t i c a t e d statementfrom a m a th em a ti ci an a b ou t why a given p r o o f is c o r r e c t or an o t h erp ro of c on ta in s an e r r o r than it is to g e t a su btle o r r e l a t i v e l ye la bo ra te e va lu at io n o f th e w or t h o f one kind of h e u r i s t i c r easo n i n gv er su s a n o th e r . Using an axiomatic approach i n th e t each i n g o f mathematics p r o v id e s a superb o p p o r tu n ity f o r more explici t emphasis onhow p ro o f s a r e found, and what h eu ri st i. c i de as are c e n t r a l to t h e i rdiscovery.

I would a l s o l i k e t o ur ge th e v ie wp oi nt t h a t t her e is no fundamentalc o n f l i c t between th e axi.omatic method as pursued i n pure mathematicsand the development o f ski l ls f o r s o lv in g problems in ap p l i ed mathematics.I t has become a l l to o f a s h io n a b le a t the p r e s e n t time to emphasize ac o n f l i c t between pure and a p p lie d mathematics, and t o be f o r th e oneand a g a i n s t the o t h e r i n terms o f what i.s t o b e e mp ha si ze d i n th emathematical t r a i n i n g o f s t u d e n t s . From a p s yc h ol o gi c al s t an d po i ntt her e is a v er y c lo se aff inity between th e c o r r e c t and completest at em en t o f th e mathematical c o n d itio n s t h a t c h a r a c t e r i z e a problemi n ap p l i ed mathematics and the s ta te m e n t o f axioms i n pure mathematics.I n b o th cases the ai m is from a mathematical s ta n d p o in t t o make th eproblem a t hand a tu b on i t s own bottom, so t o speak, w i t h o u t dependencei n i m p l i c i t and i l l - u n d e r s t o o d ways on o t h e r p a r t s o f mathematics o r

on o t h e r p h y s i c a l s id e c o nd it io ns . An ex p er i en ce t h a t I would claimis psychol ogi cal l y i d e n t i c a l to i s o l a t i n g th e m a t he m a ti c al f e at u .r e so f an a p p lie d problem is t h a t o f f in d in g axioms fo r some p a r t o fpure mathematics. A weakness o f our te a c h in g o f t h e axi.omatic methodis t h a t we to o seldom c o n f r o n t our st udent s w ith the problem o f

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formulating axioms, as opposed to deriving c o n s e ~ u e n c e s from aclearly stated, teacher-provided se t of axioms. In my own judgmentth e present teaching of axiomatic mathematics a t the school or universi ty level is more defic ient on th is point than any other.

To give these general remarks a greater sense of definiteness, Inow turn to some more constructive and part icular ideas about th eaxiomatic method under the headings of logic , algebra, geometry andcalculus.

Logic. As part of training in the axiomatic method in schoolmathematics, I would not advocate an excessive emphasis on logicas a self-contained discipl ine . For example, I do not really agreewith those mathematicians who feel that logic should be studied inthe form of Boolean algebra as an autonomous discipline early in themathematical t ra ining o f s tu dents . What I do feel is important isthat students be taught in an expl ic i t fashion class ica l rules oflogical in fe rence, l ea rn how to use these rules in deriving theoremsfrom given axioms, and to come to feel as much at home with simpleprinciples of inference l ike modus ponendo ponens as they do withelementary algorithms of arithmetic. I hasten to add that theseclassical and u b i ~ u i t o u s rules o f in fe rence need not be taught insymbolic form, nor do students need to be trained to write formalproofs in the sense of mathematical logic. What I have in mind isthat the student should be able to recognize without second thoughtthe cor re ct ne ss o f the inference:

I f th is figure is a s ~ u a r e , then th is figure is a ~ u a d r i l a t e r a l .This figure is a s ~ u a r e .

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Therefore, this figure is a quadri la teral .

And also to r ecognize the fallacious character of th e inference:

I f this figure i s a s qu are , th en th is figure is a quadrilateThis figure is a quadri la teral .Therefore, this figure i s a square. (Fallacious)

The classical forms of sententia l inference present no problem, andar e a lr ea dy covered in many of the modern textbooks on high-schoolgeometry. The real pedagogical problem centers around the making ofvalid inferences involving quant i f iers . In this respect it seems tome that the best approach i s the classical one of divide and conquer.What I mean by th is is that students should f i r s t be introduced tosubst i tut ion f or ind iv idual va ri ab le s where the only quantifiersimplicitly understood are universal quantifiers standing at th ebeginning of a sentence and whose scopes are the remainder of thesentence. No o the r un ive rsa l quantifiers and no existent ial quantifiersof any sor t should be considered a t this stage. For this restr ic teduse of quan t if i er s , e s sen t ia l ly only a simple rule of substitution ofterms f or va ri ab le s and a correspondingly simple rule of generalizat ioni s required. In the paragraphs below I t ry to indicate how far thissort of logic can carry us in th e elementary treatment of the algebraof real numbers and of vector geometry, without requir ing the introduction of exis ten t ia l quantif iers , and th e subtle problems ofinference tha t accompany these quant i f iers , Only a t a late stagein high-school mathematics would I recommend that inferences involving

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exis ten t ia l q u a n t i f i e r s be explic i t ly in tr o d u c e d , an d th en o nly s p a r i n g l y .The r o l e I see f o r logi 'cin te a c h in g o f th e axiomatic method in

h i g h - sch o o l mathematics should be c l e a r . Without t r a i n i n g in th epr ovi ng o f theorems th e development o f the a xi om a t i c method is as ter i le e n t e r p r i s e . I t i s i m por t a nt an d e ss en ti al t ha t st u d en t sl e a r n how to make i n f e r e n c e s from axioms in o r d er t o comprehend thepower o f the ax i o m at i c method. To be ab l e to make such i n f e r e n c e s ,they should be given t r a i n i n g in th e s ta n d a r d forms o f i nf er en ce t h a tthey may use i n l e a r n i n g to t hi nk out an d wr i t e down an accep t ab l emathematical p r o o f . From y e a r s o f g r ad i n g mathematical p r o o f s givenOn examinations a t th e u n i v e r s i t y level , I am f ir m ly convinced, as Ihave al r ead y i n d i c a t e d , t h a t th e abi l i ty to w r i t e a c ohe r e nt mathematicalp r o o f does n o t develop n a t u r a l l y even a t th e most elementary l e v e l s an dmust be a subj e c t o f expli.ci t t r a i n i n g .

A l ge br a . The in i t i a l framework o f l o g ic d e s c r ib e d above, i t issu g g est ed , s houl d be d e l i b e r a t e l y res t r ic ted t o q u a n t i f i e r - f r e esen t en ces i n o r d er t o avoid th e troublesome an d s u b t l e m at t er o fh an d l i n g existent ial q u a n t i f i e r s o r u n i v e r s a l q u a n t i f i e r s w i t hrestr icted scope. The s tu d e n t is a lr e a d y f a m i l i a r with this l o g i c ,f o r i t c or r e s ponds r a t h e r c lo se ly t o t he e le m en ta ry a r i t h m e t i c an dal g eb r a he h as ha d p r i o r t o e nte ri ng high s c h o o l. The b u l k o f th e

al g eb r a he h as l e a r n e d ca n now be c o d i f i e d i n an e le m en ta ry a x io m at icf ash i o n by d e r i v i n g th e consequences o f th e axioms f o r a Euclideanf ie ld , t h a t i s , an ordered f ield in which e v er y n o n- ne g at iv e elementis a sq u ar e. We may avoid a l l existent ial q u a n t i f i e r s by r e p l a c i n gth e t hr ee existent ial axioms.

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( \ Ix)(3 y)(x + y 0)(Vx) (Vy) [ i f Y t- 0 then (3 z)(x y,z)] ,(Vx) ( i f 0 < x then Cl y)(x y.y)

by the following three axioms which introduce the operations of sub-t rac t ion, division and ta kin g the square root of a positive number.

x - y z i f and only i f x Y + zI f y t- O then x y z i f and only if x y.zIf 0 < x then I[X y i f and only i f x y.y

In these terms then the elementary a lgebra taugh t in h igh schoo l shouldmainly center around the consequences of the axioms that define aEuclidean f ield . In view of the three axioms jus t sta ted, which ar eintroduced t o e li minate exis ten t ia l quant i f iers , the axioms as statedhere for a Euclidean field use the operation symbols f or add iti on ,mul tipl ica ti on , sub tr ac tion , d iv is ion , and taking the square root,the re la t ion symbol"


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(5) x.(y+z) = (x.z) + (x.z)(6) x +O = x(7) x . l = x(8) x - y = z i f and only i f x = y + z(9) If y 0 then x ; ' y = z i f and only i f x = y. z(10) If x < y then i t is ' not t h e ~ case that------------ y < x .(11) If x < y and y < z x < z .(12) If x y then x < y 9I. y < x(13) I f y < z then x + y < x + z (14) I f 0 < x and y < z then x.y < x.z(15) If 0 < x then VX = y i f and only i f x(16) 0 1 .

y.y .

At a la ter stage and toward the end of high-school mathematicsfor those who are taking a ful l program, i t wil l be appropriate to go onto the concept of a real closed f ield , that i s , a . f ie ld that is Euclideanand is such that every pOlynomial of an odd degree with coeff icientsin th e field has a zero in the f ield . (The choice of real closedfields as a terminal algebraic concept rests on the fact that everyreal closed field is elementari ly equivalent with the field of realnumbers; by th is I mean that every f irs t-order sentence which holdsin one of these two fields holds in th e other, and by ' f i r s t -o rde r 'is meant sentences whose variables range only over elements of thefield and not over sets of elements.)

Geometry. The appropriate axiomatic approach to elementarygeometry i s , as everyone knows, a much more controversial subject .

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I f a vector-space approach is used, then i t i s possible to use thequantifier-free methods just described for algebra, and this has theimportant advantage of continuing to keep th e structure of proofs simple.I t also means that proofs can be a natural extension of the techniquesa lr eady learned in algebra. Moreover, in l ine with the ear l ier remarkson algebra, I would propose vector space s ove r Euclidean fields as theproper elementary objects of the theory. The axiomatic approach can takeproper advantage of the fact that the vectors form an abelian group underaddition just as the real numbers or elements of a Euclidean f ie ld do,and therefore a l l elementary properties are shared. Elementary theoremsabout addition of vectors wil l have already been proved as elementarytheorems about addit ion of numbers. To this vector-space structure maybe added th e concept of the inner product of two v ectors to permit theintroduction of concepts of distance and perpendiculari ty. The quantifierfree axioms on the inner product are just t he f oll owing three, wherea and are real numbers, and x, y and z are vectors.

I f x f a then xx > a ,

Unfortunately, for the treatment of many geometrical f igures andthe i r properties, which we expect our students to know, the purely vectorspace approach does not provide a natural framework. For these developments, it i s my own conviction that an in t r ins ic axiomatization thatemphasizes the role of geometrical constructions i s the most appealing.I r e a l i z e ~ h o w e v e r , that there i s wide disagreement on th i s viewpoint.

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There are also pedagogical dif f icul t ies in providing a s t r ic t ly axiomaticapproach in terms of geometrical constructions. I wil l not pursue th e pointfurther here. I t should be mentioned that s t i l l a third approach toelementary geometry i s in terms of introducing geometr ica l t ransformationsas well as vectors. I t is thoroughly clear from recent discussions thatit wil l be some time before the pedagogically most s uit ab le s et ofaxioms wil l be hit upon in terms of any of th e approaches I have mentioned,but I would l ike to emphasize the importance of quantif ier-free methodsi f we expect our students to become adept at finding and writing correctproofs. The logical complexities of most axiomatic approaches togeometry a t the high-school level make i t dif f icul t for students toacquire a clear and sure-footed understanding of what mathematicalarguments are a l l about.

Calculus. Space does not permit many comments on how th e axiomaticapproach may be applied to the te ach ing of calculus in high school, butthe main thrust of what I want to say can be easi ly conjectured fromwhat I have a lre ady said about algebra and geometry. An axiomaticapproach in terms of ' epsi lon-delta ' concepts and proofs does not seemappropriate. What does seem pract ical i s an axiomatic algebraic approachto the calc ulu s o f elementary functions combined with conside rable stresson the intui t ive geometric and physical meaning of the derivative andintegral of an elementary function. Moreover, the student can explictlycheck th e axioms by computing the areas of rectangles and t r iangles orthe properties of rec t i l inear motion.

Again, I emphasize that a quantifier-free approach permits an easybut rigorous development of the elementary parts of the calculus, and

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th e stud ent can ~ u i c k l y be led to have a feel for the power of thecalculus in solving empirically meaningful problems.*

*For those interested in ~ u a n t i f i e r - f r e e arithmetic, an excellentsurvey is to be found in J . C. Shepherdson, "Non-standard models forfragments of number theory," The Theory of Models, edited by J . W.Addison, L. Henkin and A. Tarski, North-Holland Publishing Company,Amsterdam, 1965. One b ea utiful re sult is due to J . R. Shoenfield,"Open sentences and the induction axiom," Journal of Symbolic Logic,voL 23 (1958), pp. 7-12. He proves that a system of nine axiomsbased on the successor " predecessor P and addition + operations, andconstant 0 is not augmented in deductive power by the add it io n o f th einductive axiom for sentences without quantifiers. The nine axiomsare just these.

(1) x' f 0(2 ) PQ 0(3) Px' x(4) x + 0 x(5 ) x + y' (x + y)'(6) I f x f 0 then x (Px) ,(7) x + y y+ x(8) (x + y) + z x + (y + z)(9 ) If x + y x + z then y z

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argument in favour of axiomatic mathematics in schools

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