Mathematics is not about numbers, equations, computations, or
algorithms: it is about understanding.
Is this from Thurston? If yes, where and when it has been said. I've checked "ON PROOF AND PROGRESS IN MATHEMATICS" and it is not there.
[Math] William Thurston’s quote
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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."
There are also many very good interviews that can be found, such as this one with Connes, as well as the advice to young mathematicians in the Princeton Companion to Mathematics.
Most likely, this is just misremembering (or misattribution).
In his math writing Thurston did not make any predictions regarding what approach to the Geometrization Conjecture (GC) will be successful. I did not hear him making such predictions in his math lectures (but, of course, I heard only few), and I would be surprised if he did. What Thurston might have said to somebody privately, I do not know.
However, Thurston was well aware of the power of the Ricci Flow (RF) and in the early 1980s Thurston did write few pages with a sketch of the proof of the Orbifold Theorem which is the geometrization theorem for orientable irreducible orbifolds with nonempty orbifold locus:
W. P. Thurston, Three-manifolds with symmetry, preprint, Princeton University, 1982, 5 pages.
One of the steps of the proof was an application of Hamilton's 1982 result on manifolds of positive curvature (proven via RF, of course). A proof of the Orbifold Theorem was given later on in a work by Boileau, Leeb and Porti (broadly speaking, their proof follows Thurston's outline), shortly before appearance of Perelman's preprints. A bit different proof was later given in a work of Cooper, Hodgson and Kerckhoff. Both proofs use Hamilton's theorem at some point, as Thurston suggested.
- In the mid 1990s Thurston was developing his own approach to the part of the GC known as the Hyperbolization Conjecture, based on analyzing laminations/foliations in 3-manifolds:
- Three-manifolds, Foliations and Circles, I. arXiv:math/9712268.
- In the last chapter of my 2001 book
Kapovich, Michael, Hyperbolic manifolds and discrete groups, Progress in Mathematics (Boston, Mass.). 183. Boston, MA: Birkhäuser. xxv, 467 p. (2001). ZBL0958.57001.
I summarized several known approaches to the GC. Two approaches were differential-geometric (RF and the Mike Anderson's approach), one was via min-volume (a combination of differential and Alexandrov geometry) and one was coarse geometric (to the "hyperbolic part" of the conjecture). At the time, it was quite possible that the GC will be proven in a piece-meal fashion with one proof of the hyperbolization conjecture, another for the Poincare Conjecture and yet another for Spherical Space Forms Conjecture.
- Regarding Ryan's comment: There was nothing even close to a consensus on the most promising approach to the GC. Hamilton and Yau believed it to be the RF, Mike Anderson had his own approach, Besson, Courtois and Gallot were making progress via the min-volume approach, some people in (or close to) Geometric Group Theory (like myself) believed it was via Cannon's Conjecture (still open) and Coarse Hyperbolization, etc. And there was no such thing as "Thurston's program" for proving the GC. The closest thing to it was the plan for proving the Orbifold Theorem, but by the 1990s it was clear that this approach will not prove the full GC. In this list of successes one should also add the work of Gabai who in 1990s was proving directly some consequences of the Hyperbolization Conjecture, as well as the work of Tukia, Mess, Gabai, Casson and Jungreis, proving Seifert Conjecture.
Best Answer
This quote is from the book "Mathematicians: An Outer View of the Inner World" (Mariana Cook and Robert Clifford Gunning, Princeton University Press, 2009).
https://www.jstor.org/stable/j.ctt2jc8h2
"Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding. I’ve loved mathematics all my life, although I often doubted that mathematics would turn out to be my life’s..."