Thurston's Paean

I recently came across a beautiful and thoughtful answer on MathOverflow by the late mathematician William Thurston. A brief background about him from the Wikipedia article about him:

William Paul Thurston (October 30, 1946 – August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds.

Thurston was a professor of mathematics at Princeton University, University of California, Davis, and Cornell University. He was also a director of the Mathematical Sciences Research Institute.

MathOverflow makes all answers posted to the website available under a Creative Commons license. In particular, all answers posted before 08 Apr 2011 (UTC) are available under the terms of the Creative Commons Attribution-ShareAlike 2.5 Generic (CC BY-SA 2.5) license. Thurston wrote the answer I am about to share on 30 Oct 2010. Due to the license terms, this post too is available under the terms of the same license.

Thurston posted his answer while replying to a MathOverflow question: What's a mathematician to do?. The question enquires about how an ordinary mathematician can contribute to mathematics. Thurston's answer from mathoverflow.net/a/44213 is reproduced below:

It's not mathematics that you need to contribute to. It's deeper than that: how might you contribute to humanity, and even deeper, to the well-being of the world, by pursuing mathematics? Such a question is not possible to answer in a purely intellectual way, because the effects of our actions go far beyond our understanding. We are deeply social and deeply instinctual animals, so much that our well-being depends on many things we do that are hard to explain in an intellectual way. That is why you do well to follow your heart and your passion. Bare reason is likely to lead you astray. None of us are smart and wise enough to figure it out intellectually.

The product of mathematics is clarity and understanding. Not theorems, by themselves. Is there, for example any real reason that even such famous results as Fermat's Last Theorem, or the Poincaré conjecture, really matter? Their real importance is not in their specific statements, but their role in challenging our understanding, presenting challenges that led to mathematical developments that increased our understanding.

The world does not suffer from an oversupply of clarity and understanding (to put it mildly). How and whether specific mathematics might lead to improving the world (whatever that means) is usually impossible to tease out, but mathematics collectively is extremely important.

I think of mathematics as having a large component of psychology, because of its strong dependence on human minds. Dehumanized mathematics would be more like computer code, which is very different. Mathematical ideas, even simple ideas, are often hard to transplant from mind to mind. There are many ideas in mathematics that may be hard to get, but are easy once you get them. Because of this, mathematical understanding does not expand in a monotone direction. Our understanding frequently deteriorates as well. There are several obvious mechanisms of decay. The experts in a subject retire and die, or simply move on to other subjects and forget. Mathematics is commonly explained and recorded in symbolic and concrete forms that are easy to communicate, rather than in conceptual forms that are easy to understand once communicated. Translation in the direction conceptual -> concrete and symbolic is much easier than translation in the reverse direction, and symbolic forms often replaces the conceptual forms of understanding. And mathematical conventions and taken-for-granted knowledge change, so older texts may become hard to understand.

In short, mathematics only exists in a living community of mathematicians that spreads understanding and breaths life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others. All of us have clear understanding of a few things and murky concepts of many more. There is no way to run out of ideas in need of clarification. The question of who is the first person to ever set foot on some square meter of land is really secondary. Revolutionary change does matter, but revolutions are few, and they are not self-sustaining --- they depend very heavily on the community of mathematicians.

In the comments to the answer, one of the commenters was Suresh Venkatasubramanian who was a professor in the School of Computing at the University of Utah back then. He is now a professor of Computer Science and Data Science at Brown University. In his comment, Suresh proposed that this answer be called Thurston's Paean. Here is his complete comment:

This seems like an ideal counterpoint to Hardy's Lament. I'm calling it Thurston's Paean :). Seems poignant now that he has passed.

Thurston's answer does appear to be a perfect complement to Hardy's lament in the 1940 essay A Mathematician's Apology. While Hardy's lament is remarkably beautiful and introspective, it may also feel a little depressing at places. Thurston's post on the other hand is full of hope and purpose that goes beyond the actual work of doing mathematics. Indeed Thurston's Paean is a befitting title for his answer.