Explanation and Proof in Mathematics: Philosophical and Educational Perspectives

How many mathematicians does it take to prove whether a lightbulb is screwed in? These essays are from a workshop in Essen in 2006. The ways that the practice of mathematics have developed in the last three decades due to computerized visualization and experimentation may hint at its future. This book has three parts for seventeen chapters by eighteen contributors on proofs including their nature, aspects of teaching, cognitive development, experiments and diagrams. Proofs of the correctness by writing texts of algorithms were what ancient mathematicians used. Solutions are often based on common intuitions which can be further explored. Proofs are often discovered by mathematical experimentation, rather than deduction, which involves intuitive, inductive, or analogical reasoning through conjecture, verification, global or heuristic refutation, and understanding. Tools may be a proof of a mathematical form, or a way to explore a domain, via the notion of the semiotic potential of an artifact. Three different worlds of mathematics can be distinguished, the conceptual-embodied such as gears, perceptual-symbolic such as conics, and axiomatic-formal such as deductions. Individuals have warrants for truth that compensate for uncertainties in their mathematical proofs and that become more sophisticated over time. Situations that reinforce theoretical proofs over pragmatic are required for this type of research to result. Methods of proof can be used in other mathematical contexts. Though the history of mathematics encourages perseverance, each step of a proof stands without historical context since changes in language use may become a source of fallibility. The philosophy of mathematics shows the evolution of proofs and how they support empirical science and other symbolic endeavors. Much mathematical theorizing also occurs prior to the formulation of the axioms used as contextual definitions. The types of thesis as to why the Greeks invented proof include the socio-political, the internalist and the philosophical influence. Descartes’ arithmetization of geometry and the calculation of magnitudes was refined by Arnauld and Lamy. Can compare Frege and Russell, Peirce and Dewey, or Wittgenstein on how proof as picture shows what was proved and should get the same result, while proof as experiment shows procedure which can remain static and get different results. Lakoff and Núñez considered mathematics to be a cognitive system of conceptual metaphors based upon the sensory motor system.