Chapter 6: Foundational Analysis
PART I A Guide to the Paradigmatical Cases (Episodes) Presented in this Chapter
PART II Data and Analysis
In the following I present the factual and interpretive accounts and conclusions for the 8 Episodes of the table in Part I. In Part III then I synthesise the findings of Part II related to Foundational Analysis and briefly discuss the wider cognitive issues that are presented in the overall synthesis of the data analysis in Chapter 10.
(i) First Steps of Initiation Into Mathematical Formalism: Meaning and Proof of the Archimedean Property
(ii) The Problem of Clarifying What Knowledge Can Be Assumed in a Proof and the Role of Quantifiers in Establishing the Generality of a Proof
(iii) Mathematical Induction and the Triangle Inequality: Cultivating More Fruitful Uses of Intuition and Hindsight as Features of the Shift to More Expert Mathematical Practices
(iv) The Problem of Clarifying What Knowledge Can Be Assumed in the Context of an Application of the Completeness Axiom
(v) Preliminary Conceptions of Limit and Infinite Largeness. The Two-Step Battle Between Intuition and Formalisation: Conceptualising and Materialising the Necessity for Formal Proof
(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
(vii) The Overwhelming Linguistic and Conceptual Complexity of the Notions of Sup and Inf
(viii) The Difficulty of Realising and Justifying the Steps in a Proof and an Application of the Archimedean Property
PART III A Synthesis of the Findings in the Area of Foundational Analysis. Indications for the Cross-Topical Synthesis in Chapter 10