Section (iv) The Problem of Clarifying What Knowledge Can Be Assumed in the Context of an Application of the Completeness Axiom

An Interpretive Account: The Analysis

The Impact of the Tutor's Attitude Towards Kelle's Written or Oral Suggestions. The tutor's approach here has such a tremendous impact on the student's approach to the tutorial that the shift from his intended to the eventually demonstrated style is visible. The tutor sets out with criticisms on his written proof for CD2.3 and elicits from him, with closed questioning, the definition and properties of the infimum of a set. Kelle complies but soon returns to his proof which he seems to wish to discuss. The tutor ignores his suggestion and outlines hers. During her exposition he interrupts her in order to highlight what he thinks went wrong with his proof, namely that he didn't know he could assume Completeness. The tutor then continues her exposition, this time with a tighter focus on his misunderstanding on assumed knowledge. Again in the end of her exposition he takes the initiative to recapitulate the syllogism for the proof in part ii (K5).

In the above, I think there is an underlying conflict of preferred dynamics. The student implicitly expresses a need for a stronger dialectic tutoring style (that is a tutorial in which the discussion of his own individual oral and written suggestions is given priority). The tutor on the other hand only slightly adapts to the student's implicitly expressed needs and her exposition indicates that she prefers a straightforward presentation of correct answers instead of discussing problematic ones.

Kelle's Misunderstood Approach. Kelle did not know that the Completeness of the Real Numbers (and in particular the version he knew: every bounded above set has a supremum) can be assumed and consequently re-formulated for sets that are bounded below. Hence he decided to re-write the statement-to-be-proved in a way that involves the supremum of a set which reminded him of Completeness. He then set out to prove Completeness. So he did not exactly, as the tutor contends in the beginning of the Episode, 'state what the question asked him to prove'.

So Kelle, despite having 'the right idea' of the proof,

As a result Kelle's proof has not taken off from its primary, intuitive grounds and has remained incomplete.

Kelle in general (see Context) seems to be a very reluctant formalist: he prefers the formulaic and concrete tasks of the Probability course and he is achieving a much better performance there than the 'mind-blowingly' rigorous course on Continuity and Differentiability. His handling of proof also suffers: as the tutor comments (see Context): he has proved the triangle inequality for n=2 and assumed that by simply saying 'by Mathematical Induction it is true for n=k' his proof was complete. Similarly to Extract 6.3, the proof that the lecturer used (via a geometric representation of the triangle inequality for n=2) acted as a strongly convincing argument for the novice who did not feel the need to strengthen his belief. As a result, Kelle added 'by Mathematical Induction it is true for n=k' as a formality only because he knew he was expected to cover all natural numbers. In sum Kelle does not behave as if he is convinced of the necessity for proof: once he is convinced, for instance by an illustrative picture, he seems to want to abstain from further action related to proof.