There are many snippets that can be found. I like the following bit of the foreword by Thurston to J. H. Hubbard's Teichmüller Theory. I share the remarks because I think you simply can't have enough of Bill Thurston's insights:
"Mathematics is a paradoxical, elusive subject, with the habit of
appearing clear and straightforward, then zooming away and leaving us
stranded in a blank haze.
Why?
It is easy to forget that mathematics is primarily a tool for human
thought. Mathematical thought is far better defined and far more
logical than everyday thought, and people can be fooled into thinking
of mathematics as logical, formal, symbolic reasoning. But this is far
from reality. Logic, formalization, and symbols can be very powerful
tools for humans to use, but we are actually very poor at purely
formal reasoning; computers are far better at formal computation and
formal reasoning, but humans are far better mathematicians.
The most important thing about mathematics is how it resides in the
human brain. Mathematics is not something we sense directly: it lives
in our imagination and we sense it only indirectly. The choices of how
it flows in our brains are not standard and automatic, and can be very
sensitive to cues and context. Our minds depend on many interconnected
special-purpose but powerful modules. We allocate everyday tasks to
these various modules instinctively and subconsciously.
The term `geometry', for instance, 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. One illustration
of this is the concept of correlation between two measurements on a
set, which is formally nearly identical with the concept of cosine of
the angle between two vectors. The content is almost the same (for
correlation, you first project to a hyperplane before measuring the
cosine of the angle), but the human psychology is very different. Each
mode of thinking has its own power, and ideally, people harness both
modes of thought to work together. However, in formalized expositions,
this psychological > difference vanishes.
In the same way, any idea in mathematics can be thought about in many
different ways, with competing advantages. When mathematics is
explained, formalized and written down, there is a strong tendency to
favor symbolic modes of thought at the expense of everything else,
because symbols are easier to write and more standardized than other
modes of reasoning. But when mathematics loses its connection to our
minds, it dissolves into a haze.
I've loved to read all my life. I went to New College of Sarasota,
Florida, a small college that was just starting up with a strong
emphasis on independent study, so I ended up learning a good deal of
mathematics by reading mathematics books. At that time, I prided
myself in reading quickly. I was really amazed by my first encounters
with serious mathematics textbooks. I was very interested and
impressed by the quality of the reasoning, but it was quite hard to
stay alert and focused. After a few experiences of reading a few pages
only to discover that I really had no idea what I'd just read, I
learned to drink lots of coffee, slow way down, and accept that I
needed to read these books at 1/10th or 1/50th standard reading speed,
pay attention to every single word and backtrack to look up all the
obscure numbers of equations and theorems in order to follow the
arguments. Even so, when something was ``left to the reader'', I
generally left it as well. At the time, 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.
John Hubbard approaches mathematics with his whole mind.
If you page through the current book, you will see many intriguing
figures. That is a first sign: figures are one of the most important
ways to keep our thought processes going in our whole brains, rather
than settling down into the linguistic, symbol-handling areas. Of
course, the figures in your imagination are even more important.
Geometric ideas can be conveyed with words and with symbols, sometimes
more effectively than with pictures, but a lack of figures is a good
indication of a lack of geometry.
Another important part of human thinking is the emotional aspect. In
mathematics, what is intriguing, puzzling, interesting, surprising,
boring, tedious, exciting is crucial; they are not incidental, they
shape how we think. Personally, my thinking was shaped by boredom: I
develop intense urges to come up with `easy' methods in order to avoid
tedious computations that are opaque to me. Hubbard, a principal
participant in the mathematics he is discussing, has done an excellent
job in conveying the drama."
