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.)

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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.

The twist ending. 4

Previous post: The twist ending. 3. R. Kirby.


Finally, a few thoughts about what I see as the main problem with Gowers's new journals projects: the intended competition with “Annals of Mathematics” (Princeton UP), “Inventiones Mathematicae” (Springer) and “J. of the AMS” (AMS). These three journals are widely recognized as the main and the most prestigious journals in mathematics. As I mentioned already, only one of them, “Inventiones”, is expensive.

In fact, its real price is unknown, and in a sense does not exist. Nobody subscribes to this journal alone; it is way too expensive for individual researchers, and libraries nowadays subscribe to huge packages of Springer journals and electronic books in all sciences and mathematics. As is well known the price of such a package is substantially lower (may be by an order of magnitude) than the sum of list prices of subscribed journals. Of course, these package deals are one of the main problems with big publishers: most of journals in these packages are of very limited interest or just a plain junk. My point here is that this practice makes the list price of a journal irrelevant. But I do consider “Inventiones” as a very expensive journal.

The Gowers-Tao-Cambridge UP project is planned as a competitor not only to “Inventiones”, but to all top mathematics journals, both the general ones and specialized. If the project succeeds, the main and the most influential journal will be not “Annals”, but the new one. This would be very much like a corporate hostile takeover. The power will be shifted from the mathematicians at the Princeton University and the Institute for Advanced Studies (both of which hire just the best mathematicians in the world available, without any regard to country of origin, citizenship, and all other irrelevant for mathematics qualities) to a much more narrow circle of T. Gowers’s friends and admirers.

The choice of the managing editor is, probably, the best for achieving such a goal. R. Kirby is the only mathematician who attempted something similar and succeeded. This story is told in the previous post. The choice of R. Kirby as the managing editors raises strong suspicion that the Gowers’s goal is the same as Kirby’s one. Only Kirby’s ambitions at the time were much more moderate: to control the main journal in one branch of mathematics. Gowers aims higher: to control the main journal in whole (or may be only pure?) mathematics.

I do realize that Kirby will deny my explanations of his motives, and so will Gowers. Both will claim that their goal was and is to ease access to the mathematical literature. Neither me, nor anybody else has a way to know what was and is going on in their minds. This can be judged only by their actions and the results of their actions. The result of Kirby’s project is that he controls the main journal in his area, and nothing is cheaper than it was. I expect that the result of Gowers's initiative will be the same.

So, this is the sad twist in the story: the only thing done by T. Gowers in the last 10-15 years (after his work on Banach spaces) which I wholeheartedly approved only two months ago, now seems (to me) to be a supporting campaign for his attempt to get even more power and influence in mathematics. The attention he got by inspiring the boycott of Elsevier and the accompanying attention to the problems of scientific publishing allowed T. Gowers to present his new journals as a solution of these problems.

And one should never forget that one of his goals is the elimination of mathematics as we know it, and turning mathematicians into service personnel for computers.


Next post: William P. Thurston, 1946-2012.

Conclusion of the series about Timothy Gowers: To be written.

The twist ending. 3. R. Kirby

Previous post: The twist ending. 2. A Cambridge don


R. Kirby (UC at Berkeley) is the Managing Editor of Gowers's journals. This justifies the following digression into Kirby's past achievements in scientific publishing.

In the 90ies he declared a war on the main journal in his field, namely “Topology”. Originally published by “Pergamon Press”, it was sold in early 90ies to Elsevier by late Robert Maxwell when his financial empire started to face serious problems. Of course, this wasn’t a good development, but it remained an excellent journal due, most likely, to its excellent and small editorial board. It was moderately expensive. I still fail to see any reason to single it out (as I don’t see any convincing justification for singling out the Elsevier in the recent boycott).

R. Kirby launched a new journal “Geometry&Topology” specifically intended to compete with “Topology” (and to eventually bring it down). It was published both online and in paper version. Online version was free; the paper version was very cheap initially. In contrast with “Topology”, the editorial board of “Geometry&Topology” was big and growing with time. The journal was also growing, and with the number of pages the price of the paper version was growing (the libraries were encouraged to subscribe to it; technically, for libraries the electronic access never was free). “Geometry&Topology” succeeded in diverting a lot of papers from “Topology”, and the editorial board of “Topology” was constantly pressured to attempt to lower the price (even when the individual subscription price to the paper version of “Topology&Geometry” surpassed that of “Topology”). Elsevier argued that the list price of a subscription is not relevant anymore (by the reasons I explained above using the example of “Inventiones”). The purpose of a relatively high list price, I believe, was to encourage participation in “package deals”. Eventually, the editorial board and Elsevier made a quite reasonable deal substantially lowering the price, but it was too late (and the list price already was not relevant).

On August 10, 2006 the whole editorial board of “Topology” resigned. Elsevier continued to publish the already accepted papers and managed to fill by them the 2007 volume. The subscription to 2007 volume was free for subscribers to the 2006 one. But the journal was, of course, dead.

Within a month (if I remember correctly, already in August 2006) “Geometry&Topology” closed free access to its electronic version or at least announced the imminent closing. Since then the access to the electronic version is by subscription only. Well, this is how much one can trust promises to be freely accessible in perpetuity. At the same time, “Geometry&Topology” doubled the subscription price, and invented some convoluted reason for quadrupling the subscription price for year 2007 (for any form of subscription, electronic or paper, individual or library). Being a member of our Library Committee, I attempted to understand their reasoning, but failed.

Nobody saved any money as a result of success of Kirby’s project. A slightly modified editorial board of “Topology” launched a replacement, “Journal of Topology” (apparently, Elsevier own the rights to the trademark “Topology”). “Geometry&Topology” is not a part of any package deal. I don’t know if the Oxford UP, the publisher of “Journal of Topology”, offers package deals, but our library had to subscribe to it as a standalone journal. So, the cost of subscription to specialized journals in the field of topology for our library substantially increased. If anybody was subscribing to “Geometry&Topology” or “Topology” as an individual, she or he, most likely, lost these subscriptions because of much higher prices.

During this struggle with “Topology” mathematicians gradually started to consider “Geometry&Topology” as the journal of choice for paper in topology and related fields.

So, the main result of Kirby’s ten-year effort is the fact that he now controls the main journal in his field (topology, of course). It seems that he had no chances to get into the editorial board of “Topology”. The co-author of his most famous papers, L. Siebenmann, was a member of the editorial board for decades, first as a regular member, then as a honorary one.


Next post: The twist ending. 4.

The twist ending. 2. A Cambridge don

Previous post: The twist ending. 1.

Mel Nathanson made a right on the target comment "Mel Nathanson Says, July 8, 2012, 9:14 a.m." in Gowers's blog about ethical issues stemming from the fact that Timothy Gowers is a professor at Cambridge University, of which the publishing house Cambridge University Press, the publisher of his new journals, is a for-profit branch. The university as a whole is non-profit, i.e. cannot distribute profits to people not employed by it.

Next post: The twist ending. 3. R. Kirby.


Behind the jump break I posted the complete text of Mel Nathanson's comment as an insurance against the disappearance of the original. Nothing on the web is really permanent, and I hope that Professor Mel Nathanson will not object to this and will not consider this to be a copyright infringement (I am relying on the "fair use" doctrine, but will remove the text at his request immediately).

The twist ending. 1

Previous post: T. Gowers about replacing mathematicians by computers. 2.



I thought that I more or less exhausted the topic of T. Gowers's mathematics and politics. I turned out to be wrong. The only aspect of Gowers's (quasi-)political activity which I supported was the initiated by him and supported by him boycott of Elsevier, the most predatory scientific publisher; namely the "Cost of Knowledge boycott". I had some reservations about the tactics (why Elsevier only, for example?), but felt that they are concerned with secondary issues and that the motives of Gowers are pure.

Well, in early July T. Tao published in his blog post "Forum of Mathematics, Pi and Forum of Mathematics, Sigma", which shed a lot of light on this political campaign. Further details were provided by T. Gowers himself in "A new open-access venture from Cambridge University Press".

It turned out that Gowers is also behind a project to establish a new electronic mathematical journal, or rather a system of new electronic journals, which will directly compete with the best existing journals, for example, with "Annals of Mathematics", which is usually regarded as simply the best one. In the words of T. Gowers:

"Thus, Pi papers will be at the level of leading general mathematics journals and will be an open-access alternative to them. Discussion is still going on about what precisely this means, but it looks as though the aim will probably be for Pi to be a serious competitor for Annals, Inventiones, the Journal of the AMS and the like."

Out of mentioned three journals, only the "Inventiones Mathematicae" (published by the second biggest scientific publisher after Reed-Elsevier, namely, Springer) is expensive. "Annals of Mathematics" is very cheap by any standards, and at the same time the most prestigious. One may suspect that it is subsidized by Princeton University, but I don't know. Why does it need any competition?


There is a buzz-word here: open access. Even the "Gold Open Access", which sounds great (this is what the buzz-words are for). Indeed, these journals are planned to be open for the readers, everybody will be able to download papers. But somebody is needed to pay at the very least for running a website, databases, for the servers. The "Gold" means that the authors pay. It is suggested that publishing an article in these "open" journals will cost the author $750.00 in current dollars, and the amount will be adjusted for inflation later. In order to attract authors, during the first three years this charge will be waived. Note that any new journal initially publishes mostly articles by the personal friends of the members of the editorial board; they will get a free ride. Gowers considers these three years free ride being really good news; I disagree and consider it to be a cheap trick to help launching his new journal(s).

I believe that it completely wrong to charge authors for publication. In the real world it is the authors who are paid if they done something good, be it a novel, a movie, or a painting. And what they will be paying for in this internet age? Not for the distribution of their papers, as before. Posting a paper at the ArXiv does this more efficiently than any journal. They will be paying for the prestige of the journal, i.e. for a line in CV which may increase their chances to get a good job, a salary raise, etc. This will introduce a new type of corruption into the mathematical community.

The idea of "gold open access" is very popular in the bio-medical sciences. If you work in a bio-med area, you need a big grant paying for your lab, equipment, lab technicians, etc. Adding to these huge costs only $750.00 per article is hardly noticeable (in fact, standard price for gold open access there is between two and three thousands depending on publisher). But mathematics is different. It is a cheap science. A lot of good mathematicians do not have any grants (about two thirds by an NSF estimate). In the current financial and political climate one cannot expect that their employers (the universities, except, perhaps, for a dozen of truly exceptional researches) will pay for publications. And $750.00 is not a negligible amount for a university professor, not to say about a graduate student.

I must mention that the idea of charging the author for the publication was realized in the past by some journals in the form of "page charges". The amount was proportional to the number of pages, since the typesetting costs were proportional; nowadays typesetting is done by the authors (which is, in fact, a hidden cost of publishing a paper), and only final touches are done by the publisher. Such journals existed about 30-something years ago. The author was never responsible for the payment, and if there was nobody to pay (no grant, the university has no such line in the budget) the paper was published anyhow. Still, the idea was abandoned in favor of the traditional publishing model: the one who wants to read a journal, pays for it. Exactly like in a grocery store: if you want an apple, then you pay for it, and not the farmer growing apple trees.


I believe that this idea of charging the authors for publications is much more morally reprehensible than anything done by Elsevier and is a sufficient ground for boycotting this Tao-Gowers initiative.

But this is not all...


Next post: The twist ending 2. A Cambridge don.