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.