Clear Thinking, vague thinking and paradoxes

Abstract: Many undergraduate students of engineering and the exact sciences have difficulty with their mathematics courses due to insufficient proficiency in what we in this paper have termed clear thinking. We believe that this lack of proficiency is one of the primary causes underlying the common difficulties students face, leading to mistakes like the improper use of definitions and the improper phrasing of definitions, claimes and proofs. We further argue that clear thinking is not a skill that is acquired easily and naturally - it must be consciously learned and developed. The paper describes, using concrete examples, how the examination and analysis of classical paradoxes can be a fine tool for developing students' clear thinking. It also looks closely at the paradoxes themselves, and at the various solutions that have been proposed for them. We believe that the extensive literature on paradoxes has not always given clear thinking its due emphasis as an analytical tool. We therefore suggest that other discipkunes could also benefit from drawing upon the strategies employed by mathematicians to describe and examine the foundations of the problems they encounter.