Date: Fri, 02 Jul 1993 02:13:20 -0400 (EDT)
Subject: Motivation #1
Content-Type: TEXT/PLAIN; CHARSET=US-ASCII
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
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.