Reasoning with Paradoxes in a Community of Mathematical Inquiry: An Exploration Toward Multidimensional Reasoning

Abstract

Introduction:  In this paper, reasoning is understood as «integrated reasoning» – that is, reasoning in a four-dimensional modality, manifested as formal, informal, interpersonal and philosophical. The first two modes are practiced in school to a greater or lesser degree, whatever the subject matter, but the last two are typically ignored by teachers except episodically. Formal reasoning is here understood as reasoning which is limited to obtaining definite results by applying explicit rules to clearly defined concepts and statements. In this sense, formal reasoning is only possible under special conditions within isolated and limited systems, such as formal logic or mathematics. But whenever we use a mathematical model to approximate, study or predict a real-life situation, we are necessarily bound to resort to the use of informal reasoning as well. Furthermore, the necessary hermeneutical process which is involved in the business of transitioning from the real world to formal systems and visa versa is far too complex to be reduced to formal and informal reasoning. It necessarily contains a metacognitive component, which is responsible for sustaining consistency as well as for coordinating between a real world situation and its formalized mathematical model. As such, it is a necessary component of creative thinking, problem solving, and «critical consciousness» in general. This dimension, which is referred to as philosophical reasoning by Cannon and Weinstein (1993) makes critical and evaluative thinking both possible and welcome in the mathematics classroom. It has the capacity to sustain ongoing doubt and inquiry through keeping various alternatives open. Finally, since most of the time the hermeneutical process is a social enterprise which involves reasoning with others who start from different points of view, it necessarily requires an ability for interpersonal reasoning, which is characterized by an emphasis on respecting others’ opinions and arguments, careful listening, and on committing oneself to a search for meaning through the negotiation of multiple perspectives.