Previous post: T. Gowers about replacing mathematicians by computers. 1.
As we do know too well by now, not all scientific or technological progress is unqualifiedly beneficial for the humanity. As one of the results of scientific research the humanity now has the ability to exterminate not only all humans, but also all the life on Earth. Dealing with this problem determined to a big extent the direction of development of western countries since shortly after WWII. There are not so dramatic examples also; a scientific research about humans may damage only minor part of the population, or even just the subjects of this research (during the last decades, such a research is carefully monitored in order to avoid any harm to the subjects).
Gowers’s project is an experiment on humans. I believe that replacing mathematicians by computers will do a lot of harm at least to the people who could find their joy and the meaning of life in doing mathematics. But the results, if the project succeeds, are not predictable. If we agree, together with André Weil, that mathematics is an indispensable part of our culture, then it hardly possible to predict what will happen without it.
There is also question if Gowers’s goal is achievable at all. He limited it in at least two significant respects. First, he would be satisfied even if computer will not surpass humans (as opposed to the designers of “Deep Blue”, who wanted and managed to surpass the best chess players). Second, he always speaks about proving theorems, and never about discovering analogies, introducing new definitions, etc. These aspects are the most important part of mathematics, not the theorems (compare the already quoted maxim by Manin). But only theorems matter in the Hungarian-style mathematics. Perhaps, this is the reason why Gowers never mentions these aspects of mathematics. It is hard to tell if this limited goal can be achieved. Given a statement, a computer definitely able sometimes to find a proof of it (or disprove it) by a sufficiently exhaustive search. If it is not able to give an answer, the problem remains open, exactly as in human mathematics. What kind of statements a computer will be able to deal with, is another question.
Some of the best problems are not a true-false type of questions. For example, the problem of defining a “good” cohomology theory for algebraic varieties over finite fields (to a big extent solved by Grothendieck), or the problem of defining higher algebraic K-functors (solved by Quillen). It is impossible for me to imagine a computer capable to invent new definitions or suggest problems based on vague analogies like these two problems, responsible for perhaps a half of really good mathematics after 1950.
It seems that I could feel safe: even in the gloomy Gowers’s future, there will be place for human mathematicians. In fact, the future theorems, stated as conjectures, always served as one of the main, or simply the main stimulus to invention of new definitions. In addition, the success of Gowers’s project will mean the end of mathematics as a profession. There will be no new mathematicians, of Serre’s level, or any other, simply because there will be no way to earn a living by doing human mathematics.
Next post: The twist ending. 1
Showing posts with label parallel processing. Show all posts
Showing posts with label parallel processing. Show all posts
Monday, June 4, 2012
T. Gowers about replacing mathematicians by computers. 1
Previous post: The Politics of Timothy Gowers. 3.
Starting with his “GAFA Visions” essay, T. Gowers promotes the idea that it is possible and desirable to design computers capable of proving theorems at a very high level, although he will be satisfied if such computers still will be not able to perform at the level of the very best mathematician, for example, at the level of Serre or Milnor. I attempted to discuss this topic with him in the comments to his post about this year Abel prize.
I had no plans for such a discussion, and the topic wasn’t selected by me. I made a spontaneous comment in another blog, which was a reaction to a reaction to a post about E. Szemerédi being awarded this year Abel prize. But I stated my position with many details in Gowers’s blog. T. Gowers replied to only three of my comments, and only partially. It seems that for many people it is hard to believe that a mathematician of the stature of T. Gowers may be interested in eliminating mathematics as a human activity, and this is why my comments in that blog made their way to Gowers’s one (one can find links in the latter).
For Gowers, the goal of designing computers capable of replacing mathematicians is fascinating by itself. Adding some details to his motivation, he claims that such computers cannot be designed without deep understanding of how humans prove theorems. He will not consider his goal achieved if the theorem-proving computer will operate in the manner of “Deep Blue” chess-playing computer, namely, by a huge and a massively parallel (like “Deep Blue”) search. Without any explanation, even after directly asked about this, he claims that in fact a computer operating in the manner of “Deep Blue” cannot be successful in proving theorems. In his opinion, such a computer should closely imitate humans (whence we will learn something about humans by designing such a computer), and that it is much simpler to imitate humans doing mathematics than other tasks.
In addition, Gowers holds the opinion that elimination of mathematics would be not a big loss, comparing it to losing many old professions to the technology.
Gowers’s position contradicts to the all the experience of the humanity. None of successful technologies imitates the way the humans act. No means of transportation imitates walking or running, for example. On the other end and closer to mathematics, no computer playing chess imitates human chess players.
Note that parallel processing (on which “Deep Blue” had heavily relied) is exactly that Gowers attempts to do with mathematics in his Polymath project. It seems that this project approaches the problem from the other end: it is an attempt to make humans to act like computers. This will definitely simplify the goal of imitating them by computers. Will they be humans after this?
Gowers’s position is a position of a scientist interested in learning how something functions and not caring about the cost; in his case not caring about the very survival of mathematics. In my opinion, this means that he is not a mathematician anymore. Of course, he proves theorems, relies on his mathematical experience in his destructive project, but these facts are uninteresting trivialities. I expect from mathematician affection toward mathematics and a desire of its continuing flourishing. (How many nominal mathematicians such a requirement will disqualify?)
Next post: T. Gowers about replacing mathematicians by computers. 2.
Starting with his “GAFA Visions” essay, T. Gowers promotes the idea that it is possible and desirable to design computers capable of proving theorems at a very high level, although he will be satisfied if such computers still will be not able to perform at the level of the very best mathematician, for example, at the level of Serre or Milnor. I attempted to discuss this topic with him in the comments to his post about this year Abel prize.
I had no plans for such a discussion, and the topic wasn’t selected by me. I made a spontaneous comment in another blog, which was a reaction to a reaction to a post about E. Szemerédi being awarded this year Abel prize. But I stated my position with many details in Gowers’s blog. T. Gowers replied to only three of my comments, and only partially. It seems that for many people it is hard to believe that a mathematician of the stature of T. Gowers may be interested in eliminating mathematics as a human activity, and this is why my comments in that blog made their way to Gowers’s one (one can find links in the latter).
For Gowers, the goal of designing computers capable of replacing mathematicians is fascinating by itself. Adding some details to his motivation, he claims that such computers cannot be designed without deep understanding of how humans prove theorems. He will not consider his goal achieved if the theorem-proving computer will operate in the manner of “Deep Blue” chess-playing computer, namely, by a huge and a massively parallel (like “Deep Blue”) search. Without any explanation, even after directly asked about this, he claims that in fact a computer operating in the manner of “Deep Blue” cannot be successful in proving theorems. In his opinion, such a computer should closely imitate humans (whence we will learn something about humans by designing such a computer), and that it is much simpler to imitate humans doing mathematics than other tasks.
In addition, Gowers holds the opinion that elimination of mathematics would be not a big loss, comparing it to losing many old professions to the technology.
Gowers’s position contradicts to the all the experience of the humanity. None of successful technologies imitates the way the humans act. No means of transportation imitates walking or running, for example. On the other end and closer to mathematics, no computer playing chess imitates human chess players.
Note that parallel processing (on which “Deep Blue” had heavily relied) is exactly that Gowers attempts to do with mathematics in his Polymath project. It seems that this project approaches the problem from the other end: it is an attempt to make humans to act like computers. This will definitely simplify the goal of imitating them by computers. Will they be humans after this?
Gowers’s position is a position of a scientist interested in learning how something functions and not caring about the cost; in his case not caring about the very survival of mathematics. In my opinion, this means that he is not a mathematician anymore. Of course, he proves theorems, relies on his mathematical experience in his destructive project, but these facts are uninteresting trivialities. I expect from mathematician affection toward mathematics and a desire of its continuing flourishing. (How many nominal mathematicians such a requirement will disqualify?)
Next post: T. Gowers about replacing mathematicians by computers. 2.
Wednesday, May 23, 2012
The Politics of Timothy Gowers. 3
Previous post: The Politics of Timothy Gowers. 2.
The preparations of Gowers to the elimination of mathematics are not limited to the elevation of the status of the most amenable to the computerization part of mathematics. T. Gowers uses other means also. His web page "Mathematical discussions” aims at developing some “more natural” ways to discover key mathematical ideas. By “more natural” Gowers apparently means “not requiring a sudden insight”. Some titles of his mini-articles are very telling. A good example is “How to solve basic analysis exercises without thinking.” To do mathematics without thinking is exactly what is needed for replacing mathematicians by computers. I consider this project as a failed one: no real way to discover key ideas without insight, not to say without thinking, is even hinted at in these notes. One of approaches used by Gowers is to reverse the history and shows how to use more recent ideas to discover the older ones, like in his note about the zeta-function. Euler’s and Riemann’s work on the zeta-function stimulated a lot of developments in analysis, and to use these developments to rediscover the main result of Euler looks like cheating. In other cases, like in his note about cubic equations, Gowers more or less rediscovers the original approaches. His approach to the cubic equations is very close to the one presented in every book about Galois theory paying some attention to the history.
My favorite part of this page is entitled "Topology”. It consists of only one phrase: "Watch this space”. This did not change at least since April 18, 2001 (according to the web archive). So, I am watching this space for more than a decade. Topology is the quintessential “first culture” mainstream mathematics, mathematics of Serre and Milnor. Completely missing, even without such a phrase, is algebraic geometry. It seems that the two most important developments in the twenties century mathematics are not amenable even to an attempt to eliminate or at least reduce the roles of insights and thinking.
The posts under the tag ‘Demystifying proofs’ in Gowers blog have the same goal and overlap with his “Mathematical discussions”.
Another project Gowers is actively promoting is called the “Polymath”. See posts in his blog under tags like “polymath”, “polymath1”, etc. Perhaps, the best place to start is the posts “Is massively collaborative mathematics possible?” and “Background to a polymath project”. The idea is, apparently, to prove theorems not by the usual process of an individual discovery or close interaction of few mathematicians, but by a massively parallel working of many mathematicians interacting on a special web site. This immediately brings to the memory famous computer “Deep Blue”, who won (at the second attempt) a chess match with Garry Kasparov, perhaps the best chess player of all times. “Deep Blue” relied on massively parallel computation, combined with the chess players’ insights (it turned out that without substantial help from human chess players the computer cannot beat Kasparov). Gowers attempt to arrange something similar but using only humans. This is, clearly, could be a good step toward replacing human mathematicians by computers, if successful.
To the best of my knowledge, the first attempt was somewhat successful, in the sense that it resulted in a published paper. But the result proved was not surprising at all, and the main contributions to the proof were made by very few mathematicians (perhaps, no more than three). The result was certainly accessible to a good mathematician working alone.
Much more can be found at “The polymath blog” and the Polymath1wiki (a Wiki-like site). Amazingly, 1/3 of the described there nine “Polymath projects” are devoted to solving specific problems from International Mathematical Olympiads. As is well known, a sufficiently bright and trained high school student can solve such problem in one-two hours.
So, it seems that the idea failed.
(It is worth to note that the domain name michaelnielsen.org, a subdomain of which is the Polymath1wiki, belongs to Michael Nielsen, who presents himself as “a writer, scientist, and programmer”. Perhaps, he is neither a writer, nor a scientist, nor a programmer, if these notions are understood in a sufficiently narrow sense. But we are not living at the times of André Weil, and nowadays he is definitely all of the above. The point is that even nowadays he is not classified as a mathematician.)
I think that all this gives a good idea of what I understand by the politics of Gowers.
He is also actively involved in a battle with big publishers over the prices of scientific journals. In this case his goals are quite close to my heart (in contrast with the already discussed activities), and I even signed an inspired by Gowers declaration of non-collaboration with the infamous publisher Reed-Elsevier. Still, I believe that his approach is misguided. Elsevier may be the most evil scientific publisher, but not the only one evil, and even the university presses and learned societies act in an evil manner more often than one may expect. From my point of view, the root of the problem is in the scientific community itself, and the solution can be found also only within this community. Everything depends on the transfer of the copyright from authors to publishers. Scientists need to refuse to transfer the copyright. But this is another topic.
Next post: T. Gowers about replacing mathematicians by computers. 1
The preparations of Gowers to the elimination of mathematics are not limited to the elevation of the status of the most amenable to the computerization part of mathematics. T. Gowers uses other means also. His web page "Mathematical discussions” aims at developing some “more natural” ways to discover key mathematical ideas. By “more natural” Gowers apparently means “not requiring a sudden insight”. Some titles of his mini-articles are very telling. A good example is “How to solve basic analysis exercises without thinking.” To do mathematics without thinking is exactly what is needed for replacing mathematicians by computers. I consider this project as a failed one: no real way to discover key ideas without insight, not to say without thinking, is even hinted at in these notes. One of approaches used by Gowers is to reverse the history and shows how to use more recent ideas to discover the older ones, like in his note about the zeta-function. Euler’s and Riemann’s work on the zeta-function stimulated a lot of developments in analysis, and to use these developments to rediscover the main result of Euler looks like cheating. In other cases, like in his note about cubic equations, Gowers more or less rediscovers the original approaches. His approach to the cubic equations is very close to the one presented in every book about Galois theory paying some attention to the history.
My favorite part of this page is entitled "Topology”. It consists of only one phrase: "Watch this space”. This did not change at least since April 18, 2001 (according to the web archive). So, I am watching this space for more than a decade. Topology is the quintessential “first culture” mainstream mathematics, mathematics of Serre and Milnor. Completely missing, even without such a phrase, is algebraic geometry. It seems that the two most important developments in the twenties century mathematics are not amenable even to an attempt to eliminate or at least reduce the roles of insights and thinking.
The posts under the tag ‘Demystifying proofs’ in Gowers blog have the same goal and overlap with his “Mathematical discussions”.
Another project Gowers is actively promoting is called the “Polymath”. See posts in his blog under tags like “polymath”, “polymath1”, etc. Perhaps, the best place to start is the posts “Is massively collaborative mathematics possible?” and “Background to a polymath project”. The idea is, apparently, to prove theorems not by the usual process of an individual discovery or close interaction of few mathematicians, but by a massively parallel working of many mathematicians interacting on a special web site. This immediately brings to the memory famous computer “Deep Blue”, who won (at the second attempt) a chess match with Garry Kasparov, perhaps the best chess player of all times. “Deep Blue” relied on massively parallel computation, combined with the chess players’ insights (it turned out that without substantial help from human chess players the computer cannot beat Kasparov). Gowers attempt to arrange something similar but using only humans. This is, clearly, could be a good step toward replacing human mathematicians by computers, if successful.
To the best of my knowledge, the first attempt was somewhat successful, in the sense that it resulted in a published paper. But the result proved was not surprising at all, and the main contributions to the proof were made by very few mathematicians (perhaps, no more than three). The result was certainly accessible to a good mathematician working alone.
Much more can be found at “The polymath blog” and the Polymath1wiki (a Wiki-like site). Amazingly, 1/3 of the described there nine “Polymath projects” are devoted to solving specific problems from International Mathematical Olympiads. As is well known, a sufficiently bright and trained high school student can solve such problem in one-two hours.
So, it seems that the idea failed.
(It is worth to note that the domain name michaelnielsen.org, a subdomain of which is the Polymath1wiki, belongs to Michael Nielsen, who presents himself as “a writer, scientist, and programmer”. Perhaps, he is neither a writer, nor a scientist, nor a programmer, if these notions are understood in a sufficiently narrow sense. But we are not living at the times of André Weil, and nowadays he is definitely all of the above. The point is that even nowadays he is not classified as a mathematician.)
I think that all this gives a good idea of what I understand by the politics of Gowers.
He is also actively involved in a battle with big publishers over the prices of scientific journals. In this case his goals are quite close to my heart (in contrast with the already discussed activities), and I even signed an inspired by Gowers declaration of non-collaboration with the infamous publisher Reed-Elsevier. Still, I believe that his approach is misguided. Elsevier may be the most evil scientific publisher, but not the only one evil, and even the university presses and learned societies act in an evil manner more often than one may expect. From my point of view, the root of the problem is in the scientific community itself, and the solution can be found also only within this community. Everything depends on the transfer of the copyright from authors to publishers. Scientists need to refuse to transfer the copyright. But this is another topic.
Next post: T. Gowers about replacing mathematicians by computers. 1
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