Major screwups in mathematics
I don't remember how I got thinking about this, but for the past week
or so I've been trying to think of a major screwup in mathematics.
Specifically, I want a statement S such that:
I cannot think of an example.
- A purported (but erroneous) proof of S was published in the
mathematical literature, so that
- S was generally accepted as true for a significant period of time,
say at least two years, but
- S is actually false
There are many examples of statements that were believed without
proof that turned out to be false, such as any number of decidability
and completeness (non-)theorems. If it turns out that P=NP, this will
be one of those type, but as yet there is no generally accepted proof
to the contrary, so it is not an example. Similarly, if would be
quite surprising to learn that the Goldbach conjecture was false, but
at present mathematicians do not generally believe that it has been
proved to be true, so the Goldbach conjecture is not an example of
this type, and is unlikely ever to be.
There are a lot of results that could have gone one way or another,
such as the three-dimensional kissing number problem. In this case
some people believing they could go one way and some the other, and
then they found that it was one way, but no proof to the contrary was
ever widely accepted.
Then we have results like the independence of the parallel postulate,
where people thought for a long time that it should be implied by the
rest of Euclidean geometry, and tried to prove it, but couldn't, and
eventually it was determined to be independent. But again, there was
no generally accepted proof that it was implied by the other
postulates. So mathematics got the right answer in this case: the
mathematicians tried to prove a false statement, and failed, and then
eventually figured it out.
Alfred Kempe is famous for producing an erroneous proof of the
four-color map theorem, which was accepted for eleven years before the
error was detected. But the four-color map theorem is true. I
want an example of a false statement that was believed for
years because of an erroneous proof.
If there isn't one, that is an astonishing declaration of success for
all of mathematics and for its deductive methods. 2300 years without
one major screwup!
It seems too good to be true. Is it?
Glossary for non-mathematicians
[ Addendum 20080205: Readers suggested some examples, and I happened
upon one myself. For a summary, see this month's addenda. I also
wrote a detailed article
about a mistake of Kurt Gödel's. ]
- The "decidability and completeness" results I allude to include
the fact that the only systems of mathematical axioms strong enough
to prove all true statements of arithmetic, are those that are so
strong that they also prove all the false statements of
arithmetic. A number of results of this type were big surprises in
the early part of the 20th century.
- If "P=NP" were true, then it would be possible to efficiently find
solutions to any problem whose solutions could
be efficiently checked for correctness. For example, it is relatively
easy to check to see if a proposed conference schedule puts two
speakers in the same room at the same time, if it allots the right
amount of time for each talk, if it uses no more than the available
number of rooms, and so forth. But to generate such schedules seems
to be a difficult matter in general. "P=NP" would imply that this
problem, and many others that seem equally difficult, was actually easy.
- The Goldbach conjecture says that every even number is the
sum of two prime numbers.
- The kissing number problem takes a red ping-pong ball
and asks how many white ping-pong balls can simultaneously
touch it. It is easy to see that there is room for 12 white balls.
There is a lot of space left over, and for some time it was an open
question whether there was a way to fit in a 13th. The answer turns
out to be that there is not.
- The four-color map theorem asks whether any geographical map
(subject to certain restrictions) can be colored with only four colors
such that no two adjacent regions are the same color. It is quite
easy to see that at least four colors may be necessary (Belgium, France,
Germany, and Luxembourg, for example), and not hard to show that five
colors are sufficient.
- Classical Greek geometry contained a number of "postulates", such
as "any line can be extended to infinity" and "a circle can be drawn
with any radius around any center", but the fifth one, the notorious
"parallel postulate", was a complicated and obscure technical matter,
which turns out to be equivalent to the statement that, for any line L
and point P not on L, there is exactly one line
L' through P parallel to L. This in turn is
equivalent to the fact that classical geometry is done on a plane, and
not on a curved surface.
[ Addendum 20080206: Another article in this
series, asking readers for examples of a different type of screwup. ]
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