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## Different degrees of formality

• In this thesis we introduce the concept of the degree of formality. It is directed against a dualistic point of view, which only distinguishes between formal and informal proofs. This dualistic attitude does not respect the differences between the argumentations classified as informal and it is unproductive because the individual potential of the respective argumentation styles cannot be appreciated and remains untapped. This thesis has two parts. In the first of them we analyse the concept of the degree of formality (including a discussion about the respective benefits for each degree) while in the second we demonstrate its usefulness in three case studies. In the first case study we will repair Haskell B. Curry's view of mathematics, which incidentally is of great importance in the first part of this thesis, in light of the different degrees of formality. In the second case study we delineate how awareness of the different degrees of formality can be used to help students to learn how to prove. Third, we will show how theIn this thesis we introduce the concept of the degree of formality. It is directed against a dualistic point of view, which only distinguishes between formal and informal proofs. This dualistic attitude does not respect the differences between the argumentations classified as informal and it is unproductive because the individual potential of the respective argumentation styles cannot be appreciated and remains untapped. This thesis has two parts. In the first of them we analyse the concept of the degree of formality (including a discussion about the respective benefits for each degree) while in the second we demonstrate its usefulness in three case studies. In the first case study we will repair Haskell B. Curry's view of mathematics, which incidentally is of great importance in the first part of this thesis, in light of the different degrees of formality. In the second case study we delineate how awareness of the different degrees of formality can be used to help students to learn how to prove. Third, we will show how the advantages of proofs of different degrees of formality can be combined by the development of so called tactics having a medium degree of formality. Together the three case studies show that the degrees of formality provide a convincing solution to the problem of untapped potential.

Author: Sebastian BöhneORCiD urn:nbn:de:kobv:517-opus4-423795 https://doi.org/10.25932/publishup-42379 an introduction to the concept and a demonstration of its usefulness Vorstellung des Konzepts und Nachweis seiner Nützlichkeit Christoph Kreitz, Christoph BenzmüllerORCiD, Cezar IonescuORCiD Christoph Kreitz Doctoral Thesis English 2019 Universität Potsdam Universität Potsdam 2019/01/17 2019/02/12 Argumentation; Beweis; Beweisassistent; Beweisumgebung; Coq; Curry; Formalismus; Formalitätsgrad; Logik; Mathematikdidaktik; Mathematikphilosophie; TaktikCoq; Curry; argumentation; degree of formality; formalism; logic; mathematics education; philosophy of mathematics; proof; proof assistant; proof environment; tactic VI, 167 ```CCS -> Applied computing -> Education -> Interactive learning environments CCS -> Theory of computation -> Logic CCS -> Computing methodologies -> Symbolic and algebraic manipulation -> Symbolic and algebraic algorithms -> Theorem proving algorithms``` ST 130, SK 130 Mathematisch-Naturwissenschaftliche Fakultät / Institut für Informatik und Computational Science 0 Informatik, Informationswissenschaft, allgemeine Werke / 00 Informatik, Wissen, Systeme / 000 Informatik, Informationswissenschaft, allgemeine Werke 1 Philosophie und Psychologie / 10 Philosophie / 100 Philosophie und Psychologie 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik 03-XX MATHEMATICAL LOGIC AND FOUNDATIONS / 03Axx Philosophical aspects of logic and foundations 03-XX MATHEMATICAL LOGIC AND FOUNDATIONS / 03Fxx Proof theory and constructive mathematics / 03F07 Structure of proofs 97-XX MATHEMATICS EDUCATION / 97Dxx Education and instruction in mathematics 97-XX MATHEMATICS EDUCATION / 97Exx Foundations of mathematics / 97E50 Reasoning and proving in the mathematics classroom Keine Nutzungslizenz vergeben - es gilt das deutsche Urheberrecht