Another quote from Bill Thurston, this with an arresting image of mathematical communication:
We have an inexorable instinct to convey through speech content that is not easily spoken. Because of this tendency, mathematics takes a highly symbolic, algebraic, and technical form. Few people listening to a technical discourse are hearing a story. Most readers of mathematics (if they happen not to be totally baffled) register only technical details – which are essentially different from the original thoughts we put into mathematical discourse. The meaning, the poetry, the music, and the beauty of mathematics are generally lost. It’s as if an audience were to attend a concert where the musicians, unable to perform in a way the audience could appreciate, just handed out copies of the score. In mathematics, it happens frequently that both the performers and the audience are oblivious to what went wrong, even though the failure of communication is obvious to all.” (Thurston 2011, page xi)
William P. Thurston : Foreword. The Best Writing on Mathematics: 2010. Edited by Mircea Pitici. Princeton, NJ, USA: Princeton University Press.