Section (vi) The Unsettling Character of the Logical Conjunctions in the Definitions of SÈ T and SÇ T and the Complexity of the Notion of Supremum: the Varying Persuasion of Mathematical Arguments and the Importance of Semantic and Linguistic Clarity

Context: See Extract 6.6i and ii

Structure: In Extract 6.6i the tutor talks to Alan about what is wrong in his proof and then presents parts of CD2.5. In Extract 6.6ii, at Connie's request, the tutor presents parts of CD2.5.

The Episode: A Factual Account. See Extract 6.6i (Alan) and Extract 6.6ii (Connie)

An Interpretive Account: The Analysis

Alan's Problematic Proof. In fig.6b I present a reconstruction of Alan's proof and note that I find the third line of his proof hard to understand (why does aÏSÈ T imply that a<supS?). Alan then chooses a particular x in SÈ T and, taking advantage of the given options (that then x can be in either S or T), he assumes that x is in S. Then he claims he reached contradiction by observing that, by the definition of supS, x must be less than supS and, by the definition of a, x must also be less than a. From the third line of his proof a is less than supS, therefore Alan claims that he found a number, a, which is less than supS and for xÎS x<a, which contradicts the definition of supS as the lowest upper bound for S. I think that, contrary to the tutor's contention that Alan is making a claim about a specific x and then illegitimately twists his claim to a general x, Alan's claim is general about xÎS. I also think that the contradiction works if one assumes that a is not the maximum of supS and supT, that is if one assumes that supS>supT and chooses a to be supT, that is the smaller of the two. What is not clear is Alan's assumptions on which he tries to build the contradiction; also the third line of his proof blurs the presentation of his assumptions even more. In sum, I think that the tutor, confused by Alan's presentation, may have given a precipitate dismissal of Alan's idea. It is also possible that the fourth line of Alan's proof evokes the impression to the reader or the listener that the student switches illegitimately from a general element of SÈ T to those elements of SÈ T that are also elements of S.

The possibility that Alan, the tutor and myself do not seem to agree on whether Alan's approach is legitimate, highlights the intricate character of foundational analysis and by implication of axiomatic mathematics. From a semantic point of view, the lack of clarity in Alan's presentation (third line) as well as his inclination towards potentially misunderstood statements (fourth line) emphasise the imperative need that the novices acquire the skill to manipulate the very powerful linguistic and notational means of expression that formal mathematics can offer. If Alan's idea for the contradiction in CD2.5i is correct, then it has been sadly let down by his presentation; if it is not, then the ambiguity in his presentation has prevented the tutor — and possibly me — from pinpointing successfully its problematic aspects.

Connie's Problematic Conceptualisation and Manipulation of SÈT. One of the possible problems with Alan's proof above was the lack of clarity in selecting an element in SÈ T and maintaining the arbitrariness of this selection throughout the proof. Connie in Extract 6.6ii sounds also concerned (C1-C3) with selecting an element in SÈ T. She also sounds as if she is concerned with losing the arbitrariness of this selection if x belongs either to S or T. As a result she seems to suggest that x must be chosen as an element of SÈ T which belongs to both S and T. What Connie is missing then is that exactly by choosing x as an element of both S and T, she is definitely losing the arbitrariness of her choice.

In sum, in both occasions, the definition of SÈ T seems to engender an unease in the students' expressions which relate to CD2.5i. The unease seems to originate in the bi-lateral form of the definition (xÎSÈ T iff xÎS or xÎT) and the students' uncertainty about where x belongs (S, T, both S and T?) when chosen as xÎSÈ T. This multiplicity of options in the definition is perhaps unsettling as especially Connie's responses in C1-C3 illustrate. This unsettling emotion possibly results in the students' unsure handling of the related logical conjunctions and interferes with the formality of their proof.

The Non-Equivalent Psychological Power of Two Counterexamples. To prove that a set has a supremum, one has to prove that the set is non-empty and that it is bounded above. If any of these two conditions does not hold then the set has no supremum. In CD2.5ii to prove that sup(SÇT) is not necessarily equal to the min{supS, supT}, one can either show that SÇT does not necessarily have a supremum or that even if it does, this does not have to be min{supS, supT}. In the first case one can prove that SÇT has no supremum by illustrating a case in which for instance SÇT is the empty set; in the second case by pointing at two sets such as {1,2} and {1,3}. Both are logically equivalent and satisfactory counterexamples. They nevertheless seem to differ in persuasiveness: in the tutor's words (end of Extract 6.6i) the latter is 'slightly more convincing'. Similarly, students in other tutorials, that I observed but are not reported in this chapter, claimed that they did not stop looking for a counterexample for CD2.5ii until they finally thought of a pair of sets, such as {1,2} and {1,3}, even though they kept coming across more examples of the first case, namely pairs of sets with an empty intersection.

A possible interpretation for this reluctance towards counterexamples of the first kind, that is pairs of sets of an empty intersection, is

• first, that the counterexamples of the second kind refute sup(SÇT)=min{supS, supT} as opposed to the more formalistic refutation achieved by the counterexamples of the first kind. In other words, in answering a question such as 'is it true that a=b?', students and tutor here have expressed a psychological preference for

showing a case where a¹b (second kind)

instead of

showing a case where a does not exist (first kind).

• Secondly, in the case of CD2.5ii the counterexamples of the first kind did not question the upper-boundedness of the intersection but its non-emptiness. When discussing however the existence of the supremum it seems that, even though the two conditions, upper-boundedness and non-emptiness, are logically equivalent, in the mathematician's mind the former carries more weight than the latter.

So, in the above, I have tried to illustrate the difference between the epistemological and the psychological grounds of the tutor's and the students' preference. Epistemologically the counterexamples of the first kind are equivalent to the counterexamples of the second kind; psychologically however they do not seem to be equivalent. The significance of this distinction lies in the fact that the students, even after finding counterexamples of the first kind, continued to search for a psychologically satisfactory answer, that is counterexamples of the second kind. That is the priority of their own sense of conviction overshadowed the execution of the strictly mathematical task (to find a formally acceptable counterexample). In other words, the personal and subjective took over the priority from the impersonal and objective.

Connie's 'that's it?': a Reduced Concept Image of the Supremum of a Set. In order to prove that a number is the supremum of a set one has to show that this number is an upper bound of the set and also that it is its least upper bound. Evidence from Extracts 6.4, 6.6 and 6.7 illustrates that the novices quite commonly ignore the second condition. In Extracts 6.6 some intrinsic characteristics of the concept of supremum have been identified (semantic and conceptual) which, along with a general tendency of the novices to be careless about checking out all the prerequisite conditions for an implication, seem to offer a possible interpretation for C4, Connie's reduced conception of what is involved in proving that a number is the supremum of a set.

Conclusion: In the above, discussions of a problem on supSÇT and supSÈT gave rise to some idiosyncratic perceptions related to the definition of SÇT and SÈT (the possibly unsettling effect of the bi-lateral form of the definition of SÈT) and the definition of supremum (ignoring the second condition of the definition). At a metamathematical level the varying degree of conviction that different counterexamples seem to carry was related to the deeply subjective character of mathematical cognition. Finally the uncertainty generated by one of the students' presentation highlighted the necessity for the novices to acquire a clear and minimally-ambiguous semantic and linguistic command of their writing or presenting style.

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