I first read the article by Thurston titled On Proof and Progress in Mathematics sometime before the summer of 2011 when I used that article as inspiration to teach three undergraduates some analysis through the vehicle of derivatives. This in turn eventually motivated me to write an undergraduate analysis text, In and Around Geometric Analysis – An Invitation. Here is an excerpt from the preface of that text …

I decided to write this text after teaching parts of the contents of this
text to three undergraduates during the summer of 2011. We met 2-3
hours a day, 2 days a week. The next year I expanded it a bit and
taught it to about 10 students – I believe it was 8 graduate students
and 2 undergraduate students. I did not assume a course in analysis
as a prerequisite, though students with and without that background
took the course.

[…]

Part of the inspiration for the course is Bill Thurston’s 1994 Bulletin
article, where he shows a few entries from a list of the different
ways in which the derivative can be understood. He made the state-
ment that “The list continues; there is no reason for it ever to stop”.
That stuck with me. The course notes that inspired this book were
based on the idea that derivatives can lead you (almost) anywhere in
analysis. And to prove it, I did just that, throwing in integration and
high dimensional geometry as needed.

Since then I have regularly assigned Bill Thurston’s article to both graduate and undergraduate students as either extra credit or required reading. I believe this article, with its informal authenticity and even vulnerability here and there, is a good example of the kind of writing that becomes influential because it penetrates more deeply than mere technical discussion. This is not unrelated to the fact that very often, the things we say and do that positively impact others often seem minor to us, certainly not big or even worth remembering. It is directly related to the fact that while Fred Almgren wrote his monster paper of about 1700 mimeographed pages (and is respected for the fact that in this paper there were three truly new ideas — from a conversation I had with Bill Allard), it was his expository papers that were widely read and very influential in many mathematician’s paths of exploration. An example is his wonderful Questions and Answers about Area Minimizing Surfaces and Geometric Measure Theory.

In On Proof and Progress in Mathematics, Bill Thurston describes and contrasts his experiences, first, in his essentially solo work on foliations (for which he won the Fields Medal), and then in his later work on the geometry of 3-manifolds, where he instead created a large community of mathematicians working together, with him concentrating on explaining his ideas and supporting the efforts of others to get involved, to understand the problems. While this entire paper is very much worth reading, it is the last section with the personal stories that is, to me, the most memorable. For stories integrate and ground everything. Reading that story again renews my determination to focus on helping students gain the wisdom necessary to find and maintain the right balance between getting results and explaining things deeply, generously.

Coming back to that course I taught those three undergraduates in the summer of 2011, the piece of the paper that inspired me was a minor point, almost a footnote, in an article focused on what it means to make progress in mathematics, to prove things, to contribute to the advancement of mathematical knowledge (however you define that).

Regarding minor things, former students and acquaintances have sometimes told me how some small thing I have said or done had a significant impact on them. Discussing this with others, I find this is fairly common. Things we might not even remember, that are simply part of who are are or what we do, are often what others find most helpful. Small, comparatively insignificant things that inspire larger significant things. Dwelling a little bit more on this experience, I arrive at a new, embodied sense for significance. Measuring significance (as I now do) by the impact something has on the happiness, the creativity, the ability to thrive for the other human beings in my smallest circles, and then aiming my intentions and attentions at maximizing this kind of significance, the usual measures of significance simply fade away.

The focus on those that you can literally reach out and touch, who need and can use your attention, feeds back into a sense of something accomplished, even when what is being accomplished is slow to emerge. The very act of directing attention, of exercising faith in those around you, in shedding the light that you have collected in their direction, carries in it its own reward.

Planting seeds is a very minor part of a gardener’s work of creating and maintaining a thriving garden. And good gardeners never worry about exactly which seeds grow … they plant the right seeds and then focus on keeping the soil right, supplying water, keeping the weeds out … in other words, they focus on getting and keeping the environment (i.e. the culture) right.

As we inhabit light filled gardens of small (and big), living things, we learn to embody principles making us collaborators in the creation of environments tuned for significance.

Moving back to cultures for, and optimal experiences in, mathematics, when we direct an embodied awareness towards learning to create and helping others create with us, we gain a deeper awareness how everything we do effects culture. And it is this living culture, created by our actions, that inspires (or inhibits) the rich, creative productivity (mathematical) humans need to thrive.

Set free, the human being integrates the mathematical universe.