About the title

About the title

I changed the title of the blog on March 20, 2013 (it used to have the title “Notes of an owl”). This was my immediate reaction to the news the T. Gowers was presenting to the public the works of P. Deligne on the occasion of the award of the Abel prize to Deligne in 2013 (by his own admission, T. Gowers is not qualified to do this).

The issue at hand is not just the lack of qualification; the real issue is that the award to P. Deligne is, unfortunately, the best compensation to the mathematical community for the 2012 award of Abel prize to Szemerédi. I predicted Deligne before the announcement on these grounds alone. I would prefer if the prize to P. Deligne would be awarded out of pure appreciation of his work.



I believe that mathematicians urgently need to stop the growth of Gowers's influence, and, first of all, his initiatives in mathematical publishing. I wrote extensively about the first one; now there is another: to take over the arXiv overlay electronic journals. The same arguments apply.



Now it looks like this title is very good, contrary to my initial opinion. And there is no way back.
Showing posts with label William P. Thurston. Show all posts
Showing posts with label William P. Thurston. Show all posts

Thursday, August 23, 2012

William Thurston about the humanity and mathematics

Previous post: William P. Thurston, 1946-2012

To my regret, I found the included below short essay by Thurston at Mathoverflow only today. It is quite remarkable, presenting some not quite conventional ideas in a brilliant and succinct form. I do not think that many mathematicians will (be able to) argue that these ideas are wrong; they usually do not think about such matters at all. A young mathematician is usually so concentrated on technical work, proving theorems, publishing them, etc. that there is no time to think about if all this is valuable for the humanity, and if it is valuable, then why. Later on she or he is either still preoccupied with writing papers, despite the creative power had significantly diminished (there is at least a couple of ways to deal with this problem), or turns to a semi-administrative or purely administrative career and seeks the political influence and power.

I started to think about these questions long ago (my interest in semi-philosophical questions goes back to high school and owes a lot to a remarkable high school teacher) and eventually have come to essentially the same conclusions as Thurston. But I was unable to put them in writing with such clarity as Thurston did.

Bill Thurston's answer to the question "What can one contribute to mathematics?" October 30, 2010 at 2:55.

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.

(Note that this short essay is quite relevant to the discussion of T. Gowers in this blog.)

Next post: To appear

Wednesday, August 22, 2012

William P. Thurston, 1946-2012

Previous post: The twist ending. 4


William P. Thurston passed away yesterday (August 21) at 8:00 p.m. at a hospital in Rochester, NY, surrounded by the members of his family.

From the announcement of the Cornell Department of Mathematics (there is no permalink for this):

"All those who knew Bill, especially his many students and collaborators, know that nothing can replace his insight and personality. We are all terribly saddened by this loss."

American Mathematical Society posted a short obituary: William P. Thurston, 1946-2012.


William Thurston was the greatest geometer of the last century. The word "geometry" is very fashionable since about fifty years ago, and this phrase now calls for a clarification. William P. Thurston was able to see unexpected, remarkable, beautiful pictures hidden from all other mathematicians. After he showed them to other mathematicians, they saw and appreciated them also. His thinking was predominantly visual. Perhaps, in this respect he was the greatest geometer of all times.

It is extremely difficult to convey any visual concept by the means of a conventional mathematical text, even with a lot of illustrations. Some visual concepts are too complicated or too many dimensional (here the usual 3 dimensions are often already too many) to be adequately explained by a 2-dimensional picture. Apparently, mathematics lacks a proper language to efficiently describe visions of Thurston's level of complexity and originality. Perhaps, this is the main reason why Thurston did published only sketches or partial expositions of his results (his own published explanation is different, but compatible with this one). Some of his ideas were successfully translated into the conventional language and written down by other mathematicians. But some others are not, and the results themselves are reproved using different means. Of course, Thurston's visions are more important than his theorems, and I am afraid that some were lost completely already in the last century. I hope that his students and collaborators will write down and publish everything they learned from Thurston.


William P. Thurston was looking for alternative ways to convey visual concepts; his prize-winning book ''Three-Dimensional Geometry and Topology'' is an attempt to deal with this problem, but covers mostly pre-Thurston ideas.

Creating a proper language to describe visual mathematical concepts is, may be, the most important problem for the future generations. The layman language has the same deficiency; every description of a tree or a landscape relies on the previous experience of the readers with trees and landscapes in the real world. This recourse is not available to mathematicians creating new visual concepts, and by this reason the problem is so difficult that it is extremely rarely even acknowledged as a problem.


The passing away of William Thurston created a hole in the mathematical community which cannot be filled. Everybody who was happy enough to talk with him at least ten minutes knows how remarkable person he was. We lost not only an exceptional mathematician; we lost also an exceptional person. This deepens the feelings of loss, emptiness, sadness, and sorrow.


Next post: William Thurston about the humanity and mathematics.