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Date: Fri, 02 Jul 1993 02:13:20 -0400 (EDT) From: LYBRHED@delphi.com Subject: Motivation #1 To: qed@mcs.anl.gov Message-id: <01H01TNJWG0I8ZFOWH@delphi.com> X-Vms-To: INTERNET"qed@mcs.anl.gov" Mime-Version: 1.0 Content-Type: TEXT/PLAIN; CHARSET=US-ASCII Content-Transfer-Encoding: 7BIT Sender: qed-owner Precedence: bulk

Objection 10. The Manifesto begins by promising to solve a problem which either does not exist, or can't be solved. 1. The problem doesn't exist. > First, the increase of mathematical knowledge during the last > two hundred years has made the knowledge, let alone > understanding, of even the most important mathematical results > something beyond the capacity of any human. Poincare understood all the important mathematics of his time. That was one hundred years ago. Hilbert and Klein also understood all the important mathematics of the 1890s. How has the situation changed since then? Did it get harder and harder to survey all the important areas of mathematics, until at some point it became impossible? On the contrary. Forty years ago, Rene Thom could go to just about any seminar and jump right in, and so could Hermann Weyl and John von Neumann. And many others: participants in the Bourbaki seminars had to be prepared to discuss any part of mathematics. There were more mathematicians who understood all the important areas of math in the 1950s than there had been at the turn of the century. There are probably even more mathematicians in the present generation who have a grasp of mathematics as a whole. For example, there are probably at least a hundred mathematicians who have an attention span wide enough to comprehend Andrew Wiles's proof in its entirety, including the steps leading up to it, and who are also capable of seeing how similar methods could be applied to other problems. I'm not one of them and I don't even know most of their names, but I'm sure they exist. Their vision spans a larger mathematical horizon than what was visible to Poincare or Gauss. Of course no one can know everything in detail. That's not the point. In mathematics, as in anything else, you have to think on a strategic level, and put systems together out of black boxes. Some people will always be able to do that. How much of 20th century mathematics is really important? Years ago I overheard a conversation between a young assistant professor and some graduate students. The professor was describing his plan to launch a new journal. He had invented some obscure new branch of mathematics, and was going to publish a journal that no one but he and his clique would read; therefore no one would know that it was bogus; therefore he could get government grants, promote his career, etc. He was inviting the graduate students to join him in this project. His journal is, alas, not the only one of its kind. If you don't read this stuff, you're not missing anything. (Incidentally, a manifesto is exactly the place to draw a firm line between what is important and what isn't.) Even if you leave aside junk written by cynical opportunists, and only consider mathematics written by sincere mathematicians, most of it is ephemeral. Of course no one could read all of it, but so what? Let's go on to the next sentence of the Manifesto: > For example, few mathematicians, if any, will ever understand > the entirety of the recently settled structure of simple finite > groups or the proof of the four color theorem. This is supposed to be an "example" of what was just stated. Why is the proof of the Four Color Theorem one of the "most important results?" It is certainly famous. It is one of those things high school teachers love to talk about. They think it is An Important Theorem in Advanced Mathematics. But is it? If someone had found an elegant proof that showed how it relates to a larger body of knowledge, then it would have been important. As a conjecture, it seemed to be important; as a theorem, it turned out to be disappointing. In what sense is it beyond my capacity to know or to understand? I know what the theorem says, and I know that it was finally proved by brute force, using a computer-aided enumeration of cases. What else do I need to know about it? It may be true that no one will ever understand this proof in its entirety, but this does not imply or illustrate that it is impossible for today's mathematicians to have the same strategic understanding of mathematics that Poincare had. In mathematics or any other science, understanding is not the same as omniscience. It would be impossible for a biologist to know the unique biochemical characteristics of all the insects in the world, but it is possible to understand biology. (Ask Cairns-Smith and Kaufmann!) It is also possible to understand mathematics. It is easier now than it ever was. The "problem" QED is supposed to solve is an illusion. 2. The problem exists but can't be solved. There is a sense in which there is a problem of "too much mathematics." There is an enormous amount of mathematics which is undeniably important, and which is inaccessible to me, because it would take a thousand years to read it. But this is an ineluctable fact. It's not a problem that could be solved. QED could not possibly change the situation. Nothing can change it. One man can't absorb all the mathematics produced by hundreds. If you take the collected works of Abel, Riemann, Poincare, Goedel, Thom, and two or three hundred other mathematicians whose work is at least sometimes on more or less the same level, and enter all that mathematics into a proof checker, it would still take many lifetimes to read it all. In fact it would take longer to read mathematics through a QED terminal than it would take to read it in printed form. (I envision a QED terminal as something resembling an ORION terminal.) 3. The "problem" can be mitigated, but not with QED. To the extent that there is a problem of "too much math," the only way to mitigate it is to increase one's reading speed. QED goes in the wrong direction. If the target is to increase my reading speed, you don't want to force me to resolve informal statements into all their nitty gritty logical and set theoretic details. That would be like forcing me to read program listings in assembly language. It would slow me down. To increase my reading speed, what you want to do is present mathematics on a computer screen in such a way that I can take in more of it at a glance, and see relationships quickly that I would otherwise have to discover with great labor. Better yet, you want to present mathematics on the screen in such a way that it trains my imagination, so that eventually I can learn to see vector fields the way Riemann saw them, without the aid of a computer, and maybe even learn to see everything from the center, as he did. Therefore, if Motivation #1 is the main reason for pursuing the QED project, we should think less about how mathematics is represented in the computer, and more about how it is presented on the screen. Lyle Burkhead