I' t '~:r STRUCTURAL LEARNING n. Issues and Approaches JOSEPH M. SCANDURA, EDITOR GORDON AND BREACH, SCIENCE PUBLISHERS NEW YORK . LONDON. PARIS Copyright ~ 1976 by Gordon and Breach Science Publishers Inc., One Park Avenue, New York, N. Y. 10016, U. S. A. Editorial office for the United Kingdom Gordon and Breach Science Publishers Ltd., 42 William IV Street, London W. C. 2, England Editorial office for France Gordon & Breach, 7-9 rue Emile Dubois, 75014 Paris, France Library of Congress catalog card number 75-34846 ISBN 0-677-15110-1. All rights reserved. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher. Printed in Great Britain by Bell & Baln Ltd., Glasgow tv 0_ )-ttV'~~\I-d,..\L€.Q.Vl'" li t~~il~~ Aer' ~~ J05 <;c.~Juv'~ (€~) 4ov,u.o ...\ D-J tSre",~ ~Cl'l2~ P0 ~L~~~) lJ t:'v..;. '1ov ~ lov...k~Pc." I~ t q 7" THE NATURE OF A CORRECT THEORY OF PROOF AND ITS VALUE JOHN CORCORAN There are few employments of life in which it is not sometimes advantageous to pause for a short time, and reflect upon the nature of the end proposed. --Boole This is the second of a series of three articles dealing with application of linguistics and logic to the study of mathematical reasoning, especially in the setting of a concern for improvement of mathematical education. The present article presupposes the previous one. Herein we develop our ideas of the purposes of a theory of proof and the criterion of success to be applied to such theories. In addition we speculate at length concerning the specific kinds of uses to which a successful theory of proof may be put vis-a-vis improvement of various aspects of mathematical education. The final article will deal with the construction of such a theory. 1. PROOFS AND RULES OF INFERENCE As we have been using the word above, a proof is an articulation of deductive reasoning from premises to conclusion. Thus, when a mathematician writes a proof he is primarily interested in communicating his reasoning to others. He is explaining to others his reasoning that if the premises are true, the conclusion must also be true. Secondarily, he is recording a mental process/event--viz., the particular process of reasoning from those particular premises to that particular conclusion during a particular time interval. Regularity in Proofs. If we consider proofs that we have written or if we survey the proofs found in the literature of mathematics we find many repetitions of simple patterns. This is a cluI6to the fact that thewriting of proofs is a rule-governed activity. However, if we recall our experiences we will notice that in writing proofs we do not think of ourselves as following rules. It i.sonly after the fact that we see the patterns and postulate the existence of the rules to account for the regularity. This situation is analogous to the situation involving writing of sentences. After seeing many examples of sentences, we notice repeating patterns and postulate the existence of rules to account for 16The nature of rule-governed activity is treated in several articles in this book. 195 196 the regularity. not conl~ious of proofs. Corcoran Sentences are constructed according to rules but we are following rules in writing sentences. The same with .; :1 When you write a proof you are generally doing (or redoing) the reasoning that you are expressing in the proof. Moreover, when you are reasoning in a particular branch of mathematics (e.g., geometry or arithmetic) you are generally thinking about the subject matter of that branch--although, d as Hilbert, Eoole and others point out, if your reasoning is correct, the subject matter is irrelevant Snd the reasoning would apply equally well to any other subject matter.1 The point that I am makillg is that when you are wri ting a proof you are too busy to think of any rules even if you kne" which ones to think 0(. This is exactly analogous to speech: when you utter a sentence you are generally thinking about what the.sentence is abou t and thus are too busy to bo ther wi th rules. Indeed, for example, as you begin to learn a foreign language in a classroom sicuation, as long as you have to think of the rules you generally make rather dull conversation because you are too busy to give much thought to ,,'hatyou are talking about. Thus, carrying this over to reasoning, if you knew the rules explicitly and actually thought of them while you reasoned you would likely not get very far in your mathematics. Rules of Inference. Let us use the term "rule of inference" to refer to the rules according to which proofs are constructed. The rules of inference are rules for constructing proofs in the same way that the rules in a sentential grammar are rules for constructing sentences. Because of our hypothesis that the discourse level, which includes the proofs, must have kernel/compound structure there will be two types of rilles: initial string rules asserting that certain strings are proofs ab initio and production rules which build up compound proofs from simpler ones. As a result of my own experience in formulation of rules of inference it seems that each production rule can be written in the following form: if suchand-such is a proof then the result of adding so-and-so to the end of it 17rhe question of the reality of rules of either sort is in many respects analogous to the question of the reality of language structure briefl~ mentioned above in Section 3 of the first article in this series. l~he formal nature of reasoning was clearly presupposed if not explicitly recognized even by Aristotle. This is shown in my as yet unpublished article "A Mathematical Model of Aristotle's Syllogistic." It was explicitly recognized probably as early as 1851 by Boole (pp. 235ff). Hilbert's remarks quoted by Reid (pp. 57ff) show that he also was well aware of this fact very early in his career. However, despite the long history of this idea and despite widely published warnings by prominent mathematicians concerning misconstruals (e.g.,Poincare, Pl'.5f£) it has nevertheless been taken to imply that reasoning itself consists in a mindless application of computational techniques. The important point to realize in connection with present purposes is that,although subject m.,tter or content is irrelevant to soundnes, of reasoning in the sense that sound reasoning about one subject when reinterpreted correctly is equally sound when applied to another,it is still the case that reasoning divorced from all subject matter rarely, if ever, occurs in practice. Even ;Ulbert's heralded form-elltreatment of f;cometrywas, by Hilbert's own admission (p. 3), a codification of the fundalrental facts of our spatial intuition. Indeed, were Hilbert's proofs not understood in this way they would scarcely be understandable. The Nature of a Correct Theory of Proof and Its Value 197 is also a proof.19 This implies that each production-type rule of inference has the effect of lengthening an already existent proof. Since proofs frequently begin with assumptions laid down without proof, we may suppose that one initial string rule says that any finite list of sentences may be written down to start a proof provided that each such sentence is clearly marked as an assumption. Thus we might state the premise rule as follows: any finite list of sentences of the form 'Assume p'(for p a sentence)is a proof. Examples of production-type rules of inference are easy to think of. The rule of detachment (or modus ponens) can be stated: Any proof containing both p and 'if p then q' may be lengthened by adding q onto the end. Many other rules will corne to mind. Knowledge of Rules of Inference. It is important to distinguish a stronger and a weaker sense in which one may know a rule of inference. Let us say that a person has weak knowledge of a rule of inference if he reasons in accord with that rule. Thus weak knowledge of a rule of inference is a non-self-conscious kind of knowledge. All mathematicians and most people, I imagine, have weak knowledge of quite a few rules of inference although few people are self-conscious about the rules according to which they reason. On the other hand, let us say that a person has strong knowledge of a rule of inference if he can explain the details of the rule, pointout places where it is used, etc. Strong knowledge of a rule of inference is a very self-conscious kind of knowledge. Mathematicians generally have weak knowledge of many rules of inference and strong knowledge of very few. A logician who is poor at reasoning may have strong knowledge of many rules of inference and weak knowledge of very few, although most logicians, it seems, have weak knowledge and strong knowledge of many rules of inference. The same distinction carries over to knowledge of rules of sentence construction. All speakers of English have weak knowledge of many sentential rules whereas only linguists can be expected to have strong knowledge of more than a few such rules. Linguists make it their business to have strong knrn.ledge of rules of sentence construction whereas other speakers are content to be able to use the rules, i.e., to have weak knowledge of the rules. llaturally, it is not to be expected that everyone has even wcak k;wwledge of all rules of inference. Certainly the high school freshman could not be expectecl to know all of the rules of inference used by the professional mathcr.Jatician. In a sense, knowing a rule of inference involves an understanding of a type of logical connection. Of course, as people acquainted '.-lithmathercklticalcducation, we have all had the discourasing experience of seein8 a student mimic a teacher's pattern of reasoning without understanding it. In such cases, I believc, we will always be able to oscert£lin thnt the student has not le;lrncd the rule, but only the su?erficial .:lS;)Ccts of R few .1pplicatio\1s of it. Nevertheless, I must acknowlcrlge the t;}coretici11 possibj.lity of .:l student ',.;ho knu,ols how to use .::in i~pres5il,ly large class of rulc~ withõt IJt)dcrstandiñ any of thcQ. Such 3 ~t\.:dcnt could veri+:y d c~rrect proof of H c.o71clusioll fro:ii SOme ,-='!s'sunptions l(lor purely heuris tic rp.35~ns ....JC il:!..r.~ 1J~,i~r. the t(~l-m r':.roof" in such a ,.)oythat a partial proof (i:1itialseg;uent) is counted as a proof. Thus, a finished proof will be a "proof" which satisfies certair,additional conditions. This issue "ill be dealt with ;.nthe third article. 198 without believing that the he would not be willing to assumptions were true,then (CL fn.20below.) Corcoran conclusion actually followed from them--i.e., risk anything to defend the thesis that,if the the conclusion would necessarily also be true. ':i' '.•..7.•.. .. ,. ,., .• 1Ill, :j Even though a given person may not know all of the rules of inference (as the skills of mathematical reasoning evolve new rules may corne into use), it is most likely the case that most normal high school freshmen know several of the simpler rules. ~loreover, it is my view that 2.Q!!!£of the '. more complex rules are learned by developing skill in the use of the simp- ','.f ler rules and, then, seeing how steps may be skipped. This is certainly , not to suggest the obviously wrong conclusion that "quant'-'mjumps" do not occur. For example, it was probably not until the late 19th century that mathematicians began using the choice rule ~infer (Ef)(x)Rxf(x) from (x)(Ey)Rxy~ and it is difficult to see how this rule could be broken dõn into a deduction using significantly simpler rules. Indeed, '~uantum jumps" must have occurred--othenJise we would have no rules at all. The opinion concerning acquisition of knowledge of some of the more complex rules means that after a student has gone through a certain fixed pattern of detailed reasoning several times he may develop a feel for the upshot of the pattern and begin to omit the details in future proofs-thus, in effeet, gaining weak knO\~ledge of a more complex rule. liemay imagine that the professional mathematician, after years of experience in deductive reasoning, has developed >leak kno.~ledgeof very complex rules well beyond the comprehension of beginning students. From this point of view, it is natural to expect that as mathematical reasoning becomes increasingly sophisticated, more and more complex r~lcs of inference "ill evolve. If we wish we may even speculate that the mathematics student has two kinds of "vocabularies" of rules--an active vocabulary that he can actually use in doing proofs and a passive vocabulary of rules which he can "follow" but not use. This sort of hypothesis may partially account for inability of students to recrea te reasoning that they have followed in class. Correctness of Rules of Inference. ~e may wonder about correctness and incorrectness of rules of inference--is it conceivable that a small group of persons or even a whole society writes proofs according to incorrect rules? Indeed, suppose that everyone wrote proofs according to a certain rule, would not the universal acceptance of a rule ,,,,,keit correct? On a certain level, these are very easy questions once we recall that a proof is designed to show that a certain conclusion follo\Js from certain premises. If a conclusion follo.,sfrom some premises then it is impossible that the premises are true and the conclusion false. Thus if a system of rules could be used to prove a false sentence from a set of tr~e sentences then certainly at least one of the rules is incorrect or, in the terminology of logic, unsound. TilUS, it i.s possible that a s~all group or even a whole society writes proofs according to incorrect rule". (I;:is possihle uut I l,avene'Jerseen it happcn--although 1 bave see;lpecple make mistakes in proofs.) ~reover, COl.cerning this second question we can say that ~~ universal acceptance of a rule of inference loould not make it sound. 0 20It is instruc tive as well as awusing to imagine a "country" in which the system of reasoning devised by Copi (1954) were adopted as Incidentally, it follows from what has been said above that if a certain society writes proofs incorrectly then possibly someone could discover that fact--however, if a society writes proofs correctly then there seems to be no way of finding out for sure that it does. Parenthetically, I might add here that if 1 were an Intuitionist, I would have said that I had seen examples of the use of unsound rules. The Intuitionist, e.g.,Heyting (1956), would say that most mathematicians use unsound rules and that much of the literature of mathematics contains incorrect proofs. In particular, Intuitionists regard one of the forms of indirect proof as unsound. Let US consider this in a little more detail. The kind of indirect (or reductio ad absurdum) reasoning involved in the standard proof of the irrationality of 12 from the axioms of arithmetic pr~ceeds, after the (tacit) assumption of the axioms, by assumiñfhat J2 = nlm for some integers nand m and deducing a contradiction. This sort of reasoning is regarded as sound by the Intuitionists because what the Intuitionist ~ by "not p" is that the assumption of p leads to a contradiction. HOI"ever the Intuitionist does not regard as sound the other reductio rule which allows one to prove p from some assumptions by assuming "not p" and deriving a contradic tion. For him this would only prove "not-not-p" from original assumptions. ''Not-not-p''means that it is absurd to assume that p is absurd and, for the Intuitionist, this does not in turn mean that p itself is true. This view leads to the rejection of one rule of double negation (any proof containing "not-not-p" may be lengthened by adding p),and to the rejection of the rule of excluded middle (any proof may be lengthened by adding "p or not-p"). I . i 199 THEORIES OF PROOF2. The Nature of a Correct 11leory of Proof and Its Value -i .~ ".j U ~ ~_ BY:l theory of proof for English, say, I mean a discourse gramIMr (1) ;- ~ which is intended to describe some or all of the proofs expressible in ; ~ "English and (2) whose rules are intended to be rules of inferenc" known l • ~y persons who express their reasoning in English. If we are given such , ...J; t a theory, we may want to inquire concerning its correctness and its com- ~ -: ~prehensiveness. It would be natural to call it correct if each of its ~->rules were used by some speakers of English. (There are, of course, ~ rother possibilities but this one will suffice in this context.) Furthern.J; more, it would be natural to call it comprehensive if every rule used by ~ ....•Jany speaker of English I"ere included among its rules. Of course, thet ~ ~correctness and the comprehensiveness of a given theory of proof would r ~ be relative to a given time in order to leave open both the possibility i Sl-of"old" rules being abandoned and also the possibility of "new" rules ~ ~Jbeing "devised." I'~ 'i }The hope of ever getting a correct and comprehensive theory of proof is 5 .5 ~dim. Blltit ~s certainly possible to contribute toward such a theory. ~ -.J jThiS would be done first by considering one's own reasoning and try~nf, to •• :3 "for,nul"te the rules implicit therein. rhe next step would be to survey ~ ~he rna thematica1 Iiterature in an attempt to find correc t proofs that ~ -~ \f\are:: not COI"1structible by means of one's <J1Nnrules .'lnd which, therefore, \;. l' Jmay ;)epr"s'Je,edto I)econstructed according to "new" ~ules. After some7~ -of these "ere formulated the continuation of the project would involve , J~-----~ . "official reasoning." Parry (1965) has discovered several invalid '3 --J;~"~rgument~" whose :espec~~ve conclusions are deducible from their respec- ~ ~t~ve prem~se sets ~n COp1 s system. ~ ~ 2lcauman (1966) gives an interesting discussion of this proof. ~ ..:9 I r~d\ ~ 200 Corcoran getting other workers to formulate their own rules and to help in the survey of the literature. It is hard to imagine how one could ever determine whether a particular theory were comprehensive and, of course, if a theory were comprehensive relative to a fixed time it may very well not be comprehensive relative to a later time. To many readers, the above will sound at least utopian if not far-fetched. It may very well be utopian but, given the Chomsky-Harris idea of trying to develop a sentential grammar of English, the above can easily be seen as an application of the same core idea to a part of the totality of English discourses. Thus, the idea of a comprehensive discourse grãar for all of English is even more utopian. Now, as for being far-fetched, I would simply reply that it is no more far-fetched than the ideal of a comprehensive sentential grammar of English.and a considerable hody of researchers are developing this today. As soon as one seriously considers the project of working toward a correct and comprehensive theory of proof in English, he is quickly faced with a crucial consideration. Since a discourse grammar takes as a starting point a sentential grammar, and since a sentential grammar for English does not exist in anything like a complete form, it becomes clear that the project cannot be begun in a systematic fashion. This objection is well-taken but fortunately a reasonable substitute for a sentential grammar is available at least for the part of English used in mathe~atical proofs. As a result of centuries of logical analysis of mathematical discourse we now have formally defined symbolic languages which are sufficiently rich so that all of mathematical discourse can be syt~bolically stated.22 Thus, we may choose a formal language into which to translate proofs and use the grammar of this formal language as the sentential grammar needed for the theory of proof. Taking this path our resultant theory of proof will necessarily be an idealization of an actual theory of proof in the same sense that, say, a formal language for ãithoctic is an idealization of the part of English used in discourse about arithmetic. If it so happened that a group of mathematicians actually used a formal language in their investigations and they>"rote their proofs in the formal language then we could investigate the body of proofs as such without translating and without regarding ourselves as developing an idealization. (Cf. Church (1956), pp, 2, 3, £.7, fn. 108). 22Current symbolic languages can express all mathematical state"lents only in the sense that to each mathematical statement there corresponds a symbolic sentence having the same truth conditions. "his is not to say that for every mather.>aticalstatement there corresponds an equivalent symbolic 'lentencewhich olakes the same statement in the 'lameway. For exa",p!e, "No even number i,sodd" would be glossed as '-':;txCEz&O::)' Lec.c1use in none of the ~llrr~n:: innguages do :.:c find a I'nothing quant,i,fier." :-lorcover, the ~hrase "a, 0, and c are distinct 0bje,:ts;' ~.Jhir.:h Cl.ccurs ~-epenredly ill mathcr:l..'itics, must ~e glossed i:-I current lflng~i,q~~es by /~ tortured ~onstructiofj involving a c:oilj'..Jnction of thre~ inequ.:'oli.ti~s. Pro!;lc;.;is of this s0rt"., 0:"':ce n,")ticcd, arc easily solved. Indeed, Le.,....is ,111J 1.~:~':~f(:rd (pp. 306ff, 3~i5ff) have solved the above t"'O problf?:ms. ;~m.jcvcrJ 0.11 s~ch p::-oh lCflIs mus t be sol vt2d be f Oi:e a compreht2ns i ve theory of ijroo f can be ~:onstrt!cted. The reason is tildt the variety of regular reas0~ing pcs~il)lc in a la••guage depends on the linguistic devices ':lVailable, The Nature of a Correct T\:1eoryof Proof and Its Value 201 Noreover, the use of the symbol ic language may in the end be seen as a distinct advantage as it may enable the theory to tran2send English andprovide a theory of proof for other languages as ,,,ell. However, one should not overlook the possibility that the idiosyncrasies of the various languages will also make themselves known on the discourse level and, in particular, in the proofs expressible in the various languages. This is not to suggest that a conclusion may be provable from certain premises in one language ~ut not in another, though th is may be true. Our suggestion was that even if exactly the "same" conclusions are provable from the "same" premises in two different languages it may turn out that there are ~ of doing it in one language not ava ilable to the 0 ther. Both of these hypotheses are likely--and perhaps interesting to investigate. J. TIlE VALUE OF A THEORYOF PROOF Before we can consider the possible value of a theory of proof, we should try to determine specifications for a theory which could actually be developed. Otherwise, our speculations would be too hypothetical to be very interesting. In the first place we postulate the existence of a managably small set' of simple rules of inference which must be knO\m in order. for c,xample, to be able to prove the main theorems of plane geometry and arithmetic. It is immaterial whether these rules, which we ",i 11 call the bas ic rules, arc redundant. [A set of, say. three rules is redundant if everything thilt can be proved using all three can also be proved using only t\ ..•o.l \,'e can easily i:nagine that the basic rules can be discovered. It is my opinion that the basic rules could be discovered and formulated within a short time by several logicians working with several high school math- "matics teachers--provided that the =thematics teachers (1) had been in the habit of making up nel" proofs and encouraging their students to:> ,nal,e up new proofs and (2) had "een developing geometry in different 'yays from year to year. In a ther words. the :nathema tics teachers wor:<ing on the project must have some wide experience to refer to in these matters. ,;hat I i,ave in r:lind as a ",odel is the situation wh',rein several linguists ;J(lrk I'lith several native informnnts in dev~loping a sentential ,:rammar of an cxo t ic langu.:tge. In order to discuss the value (utility) of a theory of proof i~cn let us ilTk~gine that we h:lVe the basic rules neatly formulated. Now, when we ilre asking about the value of this theory of proof wh,~t we are really ::oncerned ",iti, is the pes:;ible answers to the fo11O'.•ing question: hOI... could a o;3thematical educator usc this theory to ir,'prove r:lathcmatical cC!UC:1 tion? 2.3I:~stC:1C of rcgal.-dir.g syr.;bolic lar.t;u~lge.s :-1.': ide<.i~i;~:-!tiO!1::; 01 ll::llllral li.lngu,:i~C~ t.;Ci.lC; li!1~;uists and lcsiciar:s prefer. to dL;tin(:: ...i~sh '\~:e iO;:,i- (:11 fer-nil of <1 scntC:occ fre-m lLi ".~;~'alillllat:icnl [arm'l ~1.nd to !'""c'..:ard ~':::::_ ;,olizc1t)oJj of :1 serlLer:c(: i!S .qn attl~: ..~pt co L:~/.?rcs:..; its l':,,::ic.:ll (Õ-i;~. Fl'~-)r.l __l~is point of vic\.' i1 di:':;t.:c:.~r'sf: t;r;:l;:1c;~r i:-d.r:"~d on a ~Jy";".:bolic la:l;_::"!.J.~\: ".'Di.!id zcncri1te the lõ~icill fUr:l1S of ci;,sca.1rS(~S (or dise'.;ur,;c .j"~(~r :;t.rilCturcs). Gr.J[)c:J.tic.:ul form:. or surface SU:l .•ctures of ser:lences a:id discourses nrc thought cf .:1S obtained !'rOt71 ti~eir logical [orrr:s 'Jr (ieep structures by means of encoding functions called t~3ñf0r~ations (~£ Kccrlan, 1%9). Il A theory of proof which included the basic rules would provide strong (self-conscious) knowledge of the rules of inference commonly used in elementary mathematics. It seems to me that there are four areas within mathematical education in which such knowledge would be of use, viz.,in teaching, in testing and guidance counseling, in curriculum design,and in attempts t6 understand the psychology of mathematical learning. Teachiñ. One important part of a mathematical education is learning to reason deductively and developing skill at it. There may be much more to learning to reason than merely acc;uiring knowledge and skill in the use of the rules--but certainly these are part of it. Imagine a teacher who has kn0\4led~e of the rules in both the weak and the strong senses, i.e., he not only kneH h0\4 to use them, but he also could refer to them explicitly, formulate them, etc. Such a teacher would be in a very advantageous pos i tion OJ is -a -"is trying to teach ma thema tica 1 reasonine. Firs t1y, he would be ber.ter able to detect ignorance of specific rules. Now, l-Jhen a teacher sees a student having difficulty with a proof he is left to his own ad hoc devices concerninr; diagnosis of the difficulty. Secondly, he woul~b~ble to be much more clear in his 0\4n writing of proofs because he could be self-consciously critical of his own proofs. Thirdly, he would have a guide in choosing exercises and examples. When the class is havine difficulty seeing a proof which involved a complicated application of a rule, the teacher would be able to choose another theorem which involves a simpler <~pplication of the same rule, and then, in presentin2, it to the class he could point out that the reasoning in the complicated case is similar to the reasonine in the simple case. All three of these points hinge on the advantage that an articulate teacher has over one who is merely expert in the subject matter. Consider, for example, Lile excellent tennis plnyer who is not articulate about what is involved in playing tennis. In trying to teach a beginner to play tennis, the e"pert player is reduced to showing. If he sees the student doing soelething \~ron/'.he cannot say exactly what is wrOl~g. Even in sh()l,lingthe student what the motions are like, the tencller \vill not knOt'" what to eX~:3ccrD.tc and he "ill not be able to distin[';uish his mm idiosyncrasies fror:J\4hat is essential about tennis. Finally, he will be poor at developinG drills, etc. 202 Corcoran Testiñ and GUld.:lnce Counselin;~. It ""ems to :"e that a student's ability in dcouc ti ve rC.:.lsoning is an iIilpor tan t inde:< of his m.:lthe;;la tical apti tude) his ability to learn mathematics. This reeans that a student \'Iho is skilled in underst<1ndinij and producin,~ r:klthemGticGl proofs ,,<illDe r.:ucl,r:Jore likely to bcncfi.~ from !TIathem.:ltic~i courses than one \.;ho docs not have sucb skills. l~ i~ c:)vious that .:l Polan h'l~o has a cnar.:lc teri7..1tion of '.;h.:.lt he \Jants to test is in a better position to c.!esi~n a tcr.t L~1an .1 r:~Cln\.;llo docs not have such a chGracteriz.:ttion. A theory of proof is G charGcterization of the abstrGct structure underlying reasoning ability Gnd it should provide ~ very useful framcw0rk [or designii12 tests ()f reasoning illJility. At tile very 1.Cilsta theory of proof '~Otild provide a better knowlcc!2c Gf what if. bcin2 2ca:;urcci in tests of reasoniñ ~lbility and, tilercfore, also in r.1::ltr,cni.:ltiC:l1 <:If.'ti rude tests. 1:1 ordf.'r to "-;2t ,1r:. 1.(!C.1 ()f h.}"..J s'...:cb tcst~ :1.:1)' b0. h,~lpiul in ;~uidailce COt':~"'f>cling \iC ñ].st [;rcct11at~ conccrrd,ng t!-.c ki.nds of t~licgs ti:.C1t r!li;.~ht be di~covere(l i)y use of t:}le :.c:;ts. For eXãi}le, aIle ~light be nble to show c:<p~~rir:1ejLt.:J.lly tl~.:lt unless a studer,t had ~1.cquired t.Jcak knO\..:lcd~-;e of the basic rules by :l certain age d~e chances of his ever hcill)3 COlilpctent in !iklthe:7l.:1ti.cs :ire very clim. rilis \.;ou1d cp.ahlc counselors to advise students concerning careers in mathci:lc:ltics and rC'lntp.c1 nrcas. l'1oreover, it The Nature of a Correct Theory of Proof and Its Value 203 is not unreasonable to suppose that normal mathematical development could be characterized in terms of the number and kind of rules learned at various ages (or at various testable stages). This would permit objective identification of unusually able and unusually backward students,again leading to nare efficient and more scientific counseling. The professional mathe~'tical educator can certainly conceive of other applications inthis vein. Curriculum Design. One of the aims of curriculum design is to trace a sequence of topics in mathematics which parallels the optimal development of the student's interests and abilities. The reason for this is the desire to give the student the maximum benefit from his formal educational experience. The idea is that the student is best educated by presenting to him at each stage in his education those concepts and proofs which he is best able to respond to. It is absurd either to present things which are too trivial or to present things that are beyond the student's ability. It seems to me then that a characterization of the development of rnather.uticalskill in terms of the number and kind of rules acquired at various ages would provide a valuable framework for use in the design of an efficient curriculum. It would at least permit the knowledge of what would be very difficult and what would be very easy, as far as reasoning is concerned, and this, in turn, would permit more rational choices among alternative theorems to be presented or between alternative developmentsof a particular topic. In addieion,one can easily imagine a battery of specific remedial programs each designed to teach a specific rule or cluster of rules. Such remedial programs used in conjunction with the diagnostic tests mentioned above might very well form a formidable weapon in trying to overcome in-adeqaate preparation. In ehe discussion of knowledge of rules of inference we suggested that complex rules are sometimes learned through experience with simpler ones. If this turns out to be true then the details of the interrelation of knowledge of complex and simple rules will be very important in the choice of alternative developments of a subject as well as in the designof drills and so on. ~inally, we return to the hypothesis of active and passive vocabularies of rules. ,he truth of this hypothesis would lend additional justification to the suggestions of Professor J. J. LeTourneau (personal communication) to the effect that there should be two separat~ but parallel mathematics programs--one aimed at developing skill and concrete experiencep in creating theorems and proofs, the other aimed at acquainting the I: student ,,,iththe bOdy of existent mathematical knowledge. Naturally, a 'I theory of the active vocabulary would be applied in the former, whereas./ the latter would use the passive theory. Pcyc!lOlogy. It is already clear enough that a theory of proof would provide a fruitful source of ideas for hypotheses and experiments in the jJ"ychologyof mathematical learning. Moreover, one might wish to con- .ider a more cooprehensive theory of proof as an idealized description of the r;;ore-or-Ie"sbehavioral aspects of the l'sychological processes of reasoning. ~e have already pointed out that the written (or spoken) proof is our only access to another person's reasoning processes. The ~ritten proof is a permanent record of the reasoning and, moreover, it is a "trace" of the behavioral aspec t of the reasoning. The rules of inference in ãcordance with which the proofs are written are thus moreI, Corcoran or-less behavioral "norms," Given all this, it is easy to speculate that a theory of proof could lead to a psychological theory of deductive reasoning--perhaps analogous to the way that Kepler's Laws describing the orbits of planets lead to a kinetic theory explaining the orbits in terms of the effects of forces. 204 Finally. on the subject of applications of a theory of proof, I would like to suggest that the quality of writing of mathematics texts could be 8reatly improved if the writers would take the trouble to learn the rules of inference used by their prospective audiences. A mature mathematician must learn how to reason in a fashion understandable to a freshman if he wants freshmen to learn the mathematics (and not just memorize). Frequently, the ñture mathematician encounters (in teaching) theorems which he sees "immediately" and he finds himself at a loss as to what to say to prove them. If he knew the rules of inference used by his class then he would know exactly what to say. If mathematics texts (and mathematics teaching) are improved in this way then one can expect that capable but non-genius students will be more able both to appreciate the beauty of mathematics and also to keep from "getting turned-off by the chicken scratching." Quite possibly all this could lead to the kind of improvement in the field of mathematics that we have seen after the rediscovery of the axiomatic method. In the axiomatic method we find the ideal of the deductive/definitional organization of branches of mathematics: a theory of proof provides a partial answer to the question of what deduction is. Following all of these hopeful speculations I want to emphasize tl.'O negative points. In the first place, none of the above applications will be easily or mechanically achieved despite the fact that much of the ~roundwork is done, A tremendous amount of very detailed creative thoufht, d ia logue and e:<perimen ta tion is needed. There is even cause to "Ionder whether there is a natural place to begin. And, there are pitfalls, one of which is the gap between the precision and simplicity of the symbolic languages, on the one hand, and the vagueness, ambiguity ane complexity of natural language on the other. Anyone seriously desiring to pursue any of the above applications must become e:<tre:nely sensitive to the nuances of normal English--and very few mathematicians have the patience for this. A pilot experiment in deductive reasoning recently conducted in a Philadelphia school ended distressingly because the s~bjects were diverted by too many linguistic red herrings in the test questions. Something can be ;:>erfectly clear Ln t".e symbol ic language ane perfectly confusing when translated mechanically into English. Paradoxically, the second negative pL'il~t issues f:-om the c:-:hiLJr-ating feeling of power and self-confide:1ce that a mathecaticaily cOIl'pet'.:nt person derives lrOii' learning to be articulilte about what he is ;~l)od at, i.c., fro1:l learning r1 clearly prese;)tcc and apparently CO:1ipre:-,en;,j'.ve t:H~ory of proof. Such a person naturally \vañs to teach the thcocy to his stucJer:ts--hllL if the s~udents are not yet good at reas()nitl~.~ t~C"y callnol appreci2te the si~;nifi.c3nce 0:': what they are leClrni":1~. (heY;:I.1Y l(:a:-ri t::c r'Jlc:3 .Jlid rJH:.Y may learn by,; [0 :..olluw L:1e rL;lcs. i,'h~) di.sas tcr is tnilt r~ey co::;(' to ::>,,=1 i~'ie that r:~:-lti:(::;~,Jt~calreasoni.nG i~ :'.oLhin;; :Lit folLy,.;i:-;.~~ rult';s. Ã; we pointed out in U,e b;..!ginr,ing of this :3.r~icle) if a p2rson has his minel Occul,ieu witt1 lhE' rl!les the~ the chances are slim that he will have any attention lefe for ehe subject ~aLter or for the deeper parts of reaso'ling. If a pers"t: learns the rules as e;:ternal rules (as prescriptions) and not as descriptions of what he already rioes (or would do naturally),the result is stultifying. If pressure is put on a student to accept ~ rule self-consciously before he knows the rule non-self-consciously (i.e., if a rule is imposed on a studenV. he will either rebel or lose his intellectual integrity,or adopt the view that it's all a silly game. Another equally undesirable but less disastrous effect of teaching an uncomprehensive theory of proof even to students who can appreciate it derives from the fact that they may reason according to rules not in the theory. In this case. the students will tend not to use the rules absent from this theory thus weakening their powers of reasoning. The upshot is that they will be poorer ~t reasoning after learning the theory than they were before learning it.2 ) . Ie "j i 'I The Nature of a Correct Theory of Proof and Its Value 205 2 i +Dr. l.lbert iiar,ltr.ond,late prof",'sor of philosophy at Jonns Hopkins l:niversit.y,.:"ported to the author in a personal communication the results of t2StH administered to logic students before and after his course. The tests involved making elementary inferences from material presented in the form of imaginary newspaper articles and narrations of fictional events. His report was to the effect that almost every subject was significantly ~ at elementary reasoning after the course. I I '; I , :