[MD-sorular] Scientific American'dan Thurston hakkinda iki makale

[Ali Nesin]

http://blogs.scientificamerican.com/observations/2012/08/23/the-mathematical-legacy-of-william-thurston-1946-2012/ adresinden

The Mathematical Legacy of William Thurston (1946-2012)

By Evelyn Lamb | August 23, 2012

William Thurston. Photo courtesy of Cornell University Department of Mathematics

William Thurston, whose geometrization conjecture changed the fields of geometry and topology and whose approach to mathematics and mathematics education has reverberated throughout the mathematical world, died on August 21 following a battle with cancer. He has appeared in the pages of Scientific American in the article The Mathematics of Three-Dimensional Manifolds, which he co-wrote with Jeffrey Weeks, and as a figure in many of our other articles about mathematics.

When Thurston comes up in mathematical conversations, so do the words ¡°visionary¡± and ¡°intuitive.¡± ¡°When he knew something, he knew it intimately. His intuition knew where things were going,¡± says Bill Goldman, a mathematician at the University of Maryland and one of Thurston¡¯s students. ¡°He saw how to put everything together and saw much further than anyone else did at the time.¡±

Thurston earned his bachelor¡¯s degree at New College in Florida and his Ph.D. at the University of California-Berkeley in 1972. His early work revolutionized the study of geometric objects called foliations in the early 1970¡äs, to the extent that mathematicians started advising their students not to go into the field because Thurston seemed to be singlehandedly proving all the possible theorems. Starting in the mid-1970¡äs, he also revolutionized the field of low-dimensional geometry and topology. One of Thurston¡¯s most important contributions was the geometrization conjecture about the properties of three-manifolds. A three-manifold is an object in which every point looks locally like three-dimensional Euclidean space. (For a good example, look around you: objects in our world, and in fact the world itself, are three-manifolds.)

The geometrization conjecture states that three-manifolds that are closed and bounded can be decomposed into pieces, each of which has one of eight well understood geometric structures. The Poincar¨¦ conjecture, posed in 1904, was considered one of the most important unsolved problems in mathematics. It is just one case of Thurston¡¯s geometrization conjecture. At the time he made the conjecture, the related questions in dimensions one and two and in dimensions five and greater were understood, but little was known about dimensions three and four, where he worked.

In 1982, Thurston received a Fields Medal, one of the highest honors in mathematics, in part for his proof of the geometrization conjecture for a large class of manifolds called Haken manifolds. Reclusive Russian mathematician Grigori Perelman completed the proof of the geometrization conjecture in 2003, more than 20 years after it was made. For this achievement, Perelman was also awarded a Fields Medal, which he famously refused.

Thurston¡¯s instincts about what was true in mathematics were remarkable. ¡°He kind of had a truth filter. I¡¯d tell him my ideas, which were always wrong, and he¡¯d just sort of stare off in space and then come back and turn it into something correct. His mind rejected false mathematics,¡± says Goldman.

Thurston¡¯s intuitive approach to mathematics was in part a reaction to the formal style prevalent in the early 1970¡äs, which emphasized rigorous proofs at the expense of exposition. ¡°His intuitive style was pretty unconventional at the time, and a lot of the established mathematicians didn¡¯t appreciate him. That changed pretty quickly,¡± says Goldman, when Thurston¡¯s ideas started to transform entire fields of mathematics.

Thurston embraced efforts to make mathematics more accessible and enjoyable for students and the general public, especially in later years. In a 1994 article for the Bulletin of the American Mathematical Society, he wrote that the fundamental question for mathematicians should not be, ¡°How do mathematicians prove theorems?¡± but, ¡°How do mathematicians advance human understanding of mathematics?¡± He believed that this human understanding was what gave mathematics not only its utility but its beauty, and that mathematicians needed to improve their ability to communicate mathematical ideas rather than just the details of formal proofs.

He worked on projects to increase public understanding of mathematics and saw the mathematical sides of art and design. He co-developed a course called ¡°Geometry and the Imagination¡± designed to introduce deep geometric concepts to people who did not necessarily have an advanced background in math.  In 2010, he collaborated with designer Dai Fujiwara on mathematically-inspired fashion. About that project, he wrote, ¡°[Fujiwara] observed that we are both trying to understand the best 3-dimensional forms of 2-dimensional surfaces, and he noted that we each, independently, had come around to asking our students to peel oranges to explore these relationships. This resonated strongly with me, for I have long been fascinated (from a distance) by the art of clothing design and its connections to mathematics.¡±

Thurston worked at Princeton, MIT, University of California-Berkeley and Cornell University, among other places. He advised 33 students and has 157 mathematical descendants, including this author. In addition to his mathematical family, he is survived by his mother Margaret Thurston; his first wife Rachel Findley, from whom he was divorced; three children from his first marriage, Nathaniel Thurston, Dylan Thurston and Emily Thurston; his wife Julian Thurston; his two children from his second marriage Hannah Jade Thurston and Liam Thurston; his siblings Robert Thurston, Jean Baker and George Thurston; and two grandchildren.

Benson Farb, a mathematician at the University of Chicago and a student of Thurston, said in an email, ¡°in my opinion Thurston is underrated: his influence goes far beyond the (enormous) content of his mathematics. He changed the way geometers/topologists think about mathematics. He changed our idea of what it means to ¡®encounter¡¯ and ¡®interact with¡¯ a geometric object. The geometry that came before almost looks like pure symbol pushing in comparison.¡± His fundamental contributions to the field have influenced much of the mathematical world, and he will be greatly missed.
About the Author: Evelyn has not yet achieved her childhood dream of discovering the cure for AIDS in snake venom in the Amazon. She likes math. Follow on Twitter @evelynjlamb.

The views expressed are those of the author and are not necessarily those of Scientific American.



http://blogs.scientificamerican.com/cross-check/2012/08/24/how-william-thurston-rip-helped-bring-about-the-death-of-proof/
adresinden


How William Thurston (RIP) Helped Bring About ¡°The Death of Proof¡±

By John Horgan | August 24, 2012

William Thurston, who died on August 21 at the age of 65, would have hated this post¡¯s headline. Let me tell you why it¡¯s justified.

In 1993, when I was a full-time staff writer for Scientific American, my boss, Jonathan Piel, asked, or rather, commanded me to write an in-depth feature on something, anything, mathematical. Fercrissake, I was an English major! I whined. I could fake math knowledge for little news stories about the Mandelbrot set or Fermat¡¯s last theorem but a major article would be too hard! I urged Jonathan to assign the piece to my math-whiz colleague Paul Wallich. Piel was adamant. He wanted me, the ignoramus, to do it.

So after lots of bitching and moaning I started picking brains of Sci Am contributors¨Cincluding Wallich and two columnists, Ian Stewart and Key Dewdney¡ªfor ideas. I also began reading articles and popular books on math and interviewing big shots such as Andrew Wiles (who had just solved Fermat¡¯s last theorem), John Conway, Ronald Graham, David Mumford, Phillip Griffiths, John Milnor, Stephen Smale, Pierre Deligne and Thurston.

Mathematics, I soon realized, was undergoing an upheaval. Mathematicians were arguing heatedly about whether traditional proofs¡ªthe gold standard, since before Euclid, for demonstrating truth¡ªwere becoming obsolete. This debate resulted, in part, from the increasing complexity of modern mathematics, which seemed to be bumping up against the limits of human understanding. A case in point was Wiles¡¯s 200-page proof of Fermat¡¯s last theorem, which was too dense for most mathematicians to evaluate.

Some practitioners were relying on computers to test conjectures, graphically represent mathematical objects and construct proofs. Mathematicians were also being pressured to work on applications, such as cryptography and artificial vision, where the fundamental question shifts from ¡°Is it true?¡± to ¡°Does it work?¡±

Traditionalists lamented these shifts¡ªarguing, for example, that computer proofs produced answers without intellectual illumination¨Cbut others embraced them. Perhaps the most prominent advocate of change was Thurston, who had won a Fields Medal¡ªthe mathematical equivalent of a Nobel Prize¡ªin 1982 for delineating deep connections between topology and geometry. Thurston was advocating a more free-form, ¡°intuitive¡± style of mathematical discourse, with less emphasis on conventional proofs.

I chatted with Thurston over the phone and then flew to California to hang out with him for a couple days in Berkeley, where he ran a math center. We talked for hours about mathematical versus scientific truth, social-cultural influences on mathematics, the role of visualization in math and lots of other stuff. I was fascinated by the degree to which Thurston¡ªin some respects the consummate authority and insider¡ªwas challenging his field¡¯s axiomatic assumptions.

¡°The Death of Proof¡°¨Cillustrated by a ¡°video proof,¡± produced under Thurston¡¯s guidance, of one of his theorems on topology and geometry (see image)¡ªwas the cover story of the October 1993 Scientific American. I declared in the introduction:

¡°For millennia, mathematicians have measured progress in terms of what they could demonstrate through proofs¡ªthat is, a series of logical steps leading from a set of axioms to an irrefutable conclusion. Now the doubts riddling modern human thought have finally infected mathematics. Mathematicians may at last be forced to accept what many scientists and philosophers already have admitted: their assertions are, at best, only provisionally true, true until proved false.¡±

I cited Thurston as a major force driving this trend, noting that when talking about proofs Thurston ¡°sounds less like a disciple of Plato than of Thomas S. Kuhn, the philosopher who argued in his 1962 book, The Structure of Scientific Revolutions, that scientific theories are accepted for social reasons rather than because they are in any objective sense ¡®true.¡¯¡± I continued:

¡°¡®That mathematics reduces in principle to formal proofs is a shaky idea¡¯ peculiar to this century, Thurston asserts. ¡®In practice, mathematicians prove theorems in a social context,¡¯ he says. ¡®It is a socially conditioned body of knowledge and techniques.¡¯ The logician Kurt Godel demonstrated more than 60 years ago through his incompleteness theorem that ¡®it is impossible to codify mathematics,¡¯ Thurston notes. Any set of axioms yields statements that are self-evidently true but cannot be demonstrated with those axioms. Bertrand Russell pointed out even earlier that set theory, which is the basis of much of mathematics, is rife with logical contradictions related to the problem of self-reference¡­ ¡®Set theory is based on polite lies, things we agree on even though we know they¡¯re not true,¡¯ Thurston says. ¡®In some ways, the foundation of mathematics has an air of unreality.¡¯¡±

I let Thurston read a draft before publication. He made minor corrections and quibbled with some of my language but said he liked the overall thrust. After the article came out, the backlash¡ªin the form of letters charging me with sensationalism¨Cwas as intense as anything I¡¯ve encountered in my career. As a contributor to my Wikipedia page mentions, the article generated ¡°torrents of howls and complaints¡± from mathematicians, who were especially incensed by the article¡¯s title.

I expected, even welcomed, this criticism (and fortunately Piel, my boss, loved the article and had my back all the way). What I didn¡¯t expect was that Thurston would be one of the critics. He wrote a letter to Scientific American declaring that proofs are alive and well. ¡°The true drama of mathematics is more exciting than the melodrama suggested by the title, for this is a golden age for mathematics and for proof. A more appropriate title would have been ¡®The Life of Proof.¡¯¡±

I called Thurston and said, in effect, ¡°What the hell, man!¡± He told me that, once my article was published, he realized that it could harm his efforts to reform math, and so he had to distance himself from it. I understood. Scientific American published Thurston¡¯s letter and a few others in a later issue.

I have no regrets about ¡°The Death of Proof.¡± In fact, I¡¯m proud of it. After all, the mathematical trends I wrote about have continued, in no small part because of the leadership of Thurston. As Evelyn Lamb notes in a wonderful obituary for Scientific American, Thurston believed that ¡°human understanding was what gave mathematics not only its utility but its beauty, and that mathematicians needed to improve their ability to communicate mathematical ideas rather than just the details of formal proofs.¡± Thurston set forth his philosophy in a 1994 essay titled ¡°On Proof and Progress in Mathematics.¡± It¡¯s a fine piece, but I still prefer my article, title and all.
About the Author: Every week, John Horgan takes a puckish, provocative look at breaking science. A former staff writer at Scientific American, he is the author of four books, including The End of Science (Addison Wesley, 1996) and The End of War (McSweeney's Books, January 2012). Follow on Twitter @Horganism.

The views expressed are those of the author and are not necessarily those of Scientific American.