How many false theorems are there?LYBRHED@delphi.com
Date: Fri, 02 Jul 1993 02:04:14 -0400 (EDT)
Subject: How many false theorems are there?
Content-Type: TEXT/PLAIN; CHARSET=US-ASCII
Objection 9. QED is bound to fail in its own terms, because
mathematicians are already constructing accurate proofs.
The Manifesto says (Reply to Objection 8) "The standard of success or
failure [will be] whether it permits us to construct proofs
substantially more accurately than we can with current hand methods."
It follows that if current hand methods are already accurate, then QED
is doomed to fail by its own standard.
Therefore, it is necessary to demonstrate at the outset that current
hand methods leave substantial room for improvement. This has not been
In my letter of March 14 I said:
> After all, most theorems are true anyway. Correctness is
> not usually an issue in mathematics. Out of the thousands of
> theorems you have checked, how many have been discovered
> to be false? In the 18th century some mistakes were made
> concerning the convergence of series, but that was settled in
> the 19th century, and at this point the important areas of
> mathematics are usable as is. One does not have to worry
> about the soundness of Poincare's mathematics, or Hilbert's,
> or R.L. Moore's, or von Neumann's... etc.
You replied that 95% of the theorems you have checked turned out to be
> It turns out to be as difficult to sit down with a formal system
> and type in definitions and theorems correctly, as it is to write
> a program correctly. People always make mistakes that Nqthm
> catches. Generally, these are tiny little stupid mistakes.
> But mistakes!
All right, but that's not what I meant. We seem to be talking about two
different things. I was not asking, "How often are mistakes made in
entering a theorem into a proof checker?" No doubt many mistakes are
made in this process. What I meant was, "Once the proof has been
correctly entered into the proof checker, how often does it turn out
that the original theorem was false?" I am talking about theorems from
mathematics proper, not logic or computer science. When you said you
had checked thousands of theorems, I thought you meant theorems from
analysis, topology, algebra, geometry, and number theory.
Can you give an example of a mathematical theorem that was published in
a standard textbook or major journal, accepted by the mathematicians who
read it, and used by them in further work, and then found to be false?
By "found to be false" I don't mean a trivial lacuna was found in the
proof. I mean the theorem itself alleges something to be true which is
simply false, so that everything based on it falls down like a house of
cards. For example, it might turn out that a contraction mapping
doesn't have a fixed point after all, which would destroy most of
analysis. Has anything like this ever happened? I don't think so.
At the level of fixed point theorems, and the basic theorems about
vector fields, linear transformations, quotient groups, covering spaces,
and so forth -- the thousand or so theorems that are used as the
ingredients for all further work -- at that level there are no mistakes.
If the "building" metaphor is valid at all (which is not to be taken for
granted), what mathematicians build is more like a coral reef than a
house of cards. There may be some brittleness around the edges, but the
center of mathematics is as solid as a rock. (I'm saying the center is
solid, not the "foundation." Set theory is not the center; set theory
is as far out on the fringe in one direction as the four color theorem
is in another direction. (Incidentally, it isn't even necessary to know
something is true to "build on" it. Mathematicians have been
investigating what follows from the Riemann Hypothesis for more than a
hundred years, and this is a perfectly valid activity whether the R.H.
is ever proved or not.))
Objection 9 could be overcome in at least four different ways.
2. By meeting it head on. You could demonstrate that current hand
methods are not accurate. This could be done by answering my question
(Can you give an example of a theorem...) in the affirmative.
A few years ago, someone investigated how many papers in experimental
science contained errors. It turned out that a shocking number of
papers contained meaningless experiments, non sequiturs, statistical
fallacies, impossible data, etc. Maybe the same thing happens in
mathematics. But I'm saying it doesn't, and I think the burden of proof
is on the one who asserts that it does. The way to settle this question
is to ask for comments from the editors of ten major journals. How
often do they have to print corrections and retractions?
3. By changing the domain of application of QED to an area where hand
methods really are inadequate, such as programming. Programmers already
use compilers and debuggers, and still make mistakes; QED could be
conceived as a kind of super-debugger that would permit programmers to
write an entire operating system (or engineers to design a chip) and be
absolutely certain that it contains no bugs anywhere.
4. By changing the criterion of success or failure. You could go back
to Objection 8 and say that a certain region of mathematics is perfect,
and the purpose of QED is to extend that region. "The standard of
success or failure of the QED project will be whether it helps us to
extend the kingdom of perfection." That sounds more like a manifesto!
As you put it, what you are working toward is "the extremely wonderful
goal of having all mathematics in a single computing system." QED is
pyramid-building on a grand scale. The idea is to create a
mathematical jewel that sparkles like nothing ever has, because for the
first time each atom is in its exact place. Nothing less!