Barry Mazur, Visions, Dreams, and Mathematics
August 1, 2008
If someone asks us What is X? where X is some mathematical concept, we boldly answer, for we have been well trained in the art of definitions. All the fine articulations of logical structure are at our fingertips. If, however, someone asks us What does X mean? we respond as any human must respond when explaining the meaning of something: we are thrust into the whirlwind of interpretations, intentions, aims, expectations, desires, and shades of significance that, in effect, depend largely upon the story we have woven around the concept. Consider, for example, the innocuous question:
What does it mean to find X in the polynomial equation X² = 2?
We frame a narrative the minute we open our mouths to answer this question.
Continued here.
