Four Basic Proof Techniques Used in Mathematics
Proofs and Refutations: The Logic of Mathematical Discovery By Imre Lakatos
Proofs and Refutations is a book by philosopher Imre Lakatos expounding his view of the progress of mathematics. The book is written as a series of Socratic dialogues involving a group of students who debate the proof of the Euler characteristic defined for the polyhedron. A central theme is that definitions are not carved in stone, but often have to be patched up in the light of later insights, in particular failed proofs. This gives mathematics a somewhat experimental flavour. At the end of the Introduction, Lakatos explains that his purpose is to challenge formalism in mathematics , and to show that informal mathematics grows by a logic of "proofs and refutations".
Machine Learning pp Cite as. To learn, a learner needs to formulate plans, monitor the plan execution to detect violated expectations, and then diagnose and rectify errors which the dis-confirming data reveal. In this paper, five heuristic methods are presented for repairing flawed beliefs. These beliefs are considered as theories that predict effects of actions. Theories presuppose particular structural characteristics. Each proposed theory fix produces as a by-product new domain concepts that capture environmental characteristics of instrumental value to the learner. The techniques proposed here provide the first analytical methods for constructing new knowledge.