When mathematics is explained, formalized and written down, there is a — William Thurston

"The most important thing about mathematics is how it resides in the human brain."

"Mathematics is primarily a tool for human thought."

"People can be fooled into thinking of mathematics as logical, formal, symbolic reasoning. But this is far from reality...computers are far better at formal computation and formal reasoning, but humans are far better mathematicians."

"The term `geometry...refers to a pattern of processing within our brains related to our spatial and visual senses, more than it refers to a separate content area of mathematics."

"I was really amazed by my first encounters with serious mathematics textbooks...I could appreciate that the mathematics was an impressive intellectual edifice, and I could follow the steps of proofs. I assumed that such an elaborate buildup must be leading to a fantastic denouement, which I eagerly awaited -- and waited, and waited. It was only much later, after much of the mathematics I had studied had come alive for me that I came to appreciate how ineffective and denatured the standard ((definition theorem proof)n remark)m style is for communicating mathematics. When I reread some of these early texts, I was stunned by how well their formalism and indirection hid the motivation, the intuition and the multiple ways to think about their subjects: they were unwelcoming to the full human mind."