Excerpt from a Foreword by William Thurston

In Mirrored

Mirrored from http://matrixeditions.com/Thurstonforeword.html. Emphasis in foreword below not mine.

Excerpt from Foreword by William Thurston to
Teichmüller Theory and Applications to Geometry, Topology, and Dynamics
Volume I: Teichmüller Theory

by John Hubbard

I have long held a great admiration and appreciation for John Hamal Hubbard and his passionate engagement with mathematics. Hubbard has inspired me and many others. Passionate engagement is contagious. It shows through in his writing. This book develops a rich and interesting, interconnected body of mathematics that is also connected to many outside subjects. I commend it to you.

That’s the short version. Here’s a longer version:

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.