New comments to the post "What is mathematics?"

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There is a new thread of comments to the post "What is mathematics?" started by Sandro Magi. The post is dated April 3; this thread started on May 31. The thread is concerned with only one claim in that post: proofs are not needed at all for applications of mathematics.

Unfortunately, the very first phrase of Sandro Magi set the tone for the rest of the discussion: "This is blatantly false". I do not like to discuss things in such a manner: with a total lack of cooperation. The combinatorialists at Gowers's blog are much more friendly even after a direct attack on their field. But, I believe that the reason is not any kind of malice of either party. This dialog is a good illustration of the near impossiblity of people thinking linearly and verbally to understand people thinking visually. In this case the dialog of a mathematician (every mathematician thinks at least partially visually) and a software engineer turned out to be impossible. I encountered the same sort of difficulties while discussing essentially any other subject, from the movies to the current affairs. I see also this lack of understanding of visual and "the big picture" issues in the design and functionality of almost all the software.

Still, it seems to me that there are some important ideas in that discussion. Of course, it would be better to give a coherent exposition. But an attempt to write it would take a lot of time, and who knows when it would be ready.

If somebody wants to comment on any issue there, I suggest to post comments here; this will result in a more clear structure of comments. As an additional benefit for the next 30 days the comments here are not moderated; they are moderated at that post. This rule is subject to change without notice. :-) I would like to ask Sandro Magi to continue our discussion in comments to "What is mathematics?" and not here (of course, he is under not obligation to continue); then the whole discussion will be at the same place.


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William P. Thurston, 1946-2012

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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.