

Major screwups in mathematics (why are there none) - nickb
http://blog.plover.com/math/major-screwups.html

======
cperciva
I had an incorrect proof in a paper I wrote in Mathematics of Computation; it
wasn't noticed until three years later, at which point I wrote another paper
in the same journal with a (correct) proof of the result -- with the slight
difference that the correct proof was five pages long where the incorrect
proof was five lines long.

The reason the result I claimed was not significantly wrong (the correct
theorem required the additional assumption that nobody does floating-point
arithmetic with less than five bits of precision) was because I discovered the
result experimentally: When I was writing the first paper, I needed an error
bound and wasn't sure what it should be, so I had my computer do millions of
trials on random inputs and tell me the worst rounding error which it
encountered. Only after experimentally convincing myself that I had the right
result did I try to come up with a proof.

EDIT: I should add that the result in question concerned the maximum rounding
error which could result from multiplying complex floating-point values -- so
it's not exactly an esoteric problem which nobody would care about.

~~~
danteembermage
Would you mind sharing a link to your papers? From scholar.google I'm guessing
this is the corrected version but it didn't list anything older.
(<http://www.daemonology.net/papers/fft.pdf>)

~~~
cperciva
_Would you mind sharing a link to your papers?_

You almost got there: <http://www.daemonology.net/papers/>

The paper you linked to contained the original (incorrect) proof at the start
of the proof to Theorem 5.1; the correct proof is given in
<http://www.daemonology.net/papers/complexmultiply.pdf> (Theorem 1).

------
mechanical_fish
I'd say it's because it's the field with (a) the strictest formal definition
of what "correct" means, and (b) no dependence on experiment, and hence no
data and no statistics.

Everything you need to determine whether or not a given proof is correct is
either in the paper in front of you, in the literature, or implied by the
literature. If you can't follow the implications, you ask the author of the
proof to clarify it.

Compare this to research in, say, medicine, where there are always millions of
variables and progress requires careful controls, a statistician, an enormous
budget that supports many trials of the same thing, and hope.

My hypothesis is that this is also why so many mathematicians do great work
when they're young: It's one of the few fields where you don't have to do
everything N times, because the error does _not_ depend inversely on the
square root of N. Once you've figured something out in math, it _stays_
figured out. So all you need to become a mathematician -- besides your own
mind -- is a library, the ability to read, and patience. (Although, in
practice, nobody becomes a great mathematician without a teacher or two to
guide them through the library. Even Ramanujan only found a couple of the
books on his own.)

~~~
eru
No, no, it's because Mathematicians are so smart!

------
bayareaguy
Well, in general it's hard to find errors unless you have something to look
for.

I don't know if these should count as "screwups" since the resolution of the
flaws in each led the way to better systems, but I'd offer the following:

\- Infinitesimals

\- Hilbert's program

~~~
hhm
Hilbert's program doesn't count, it wasn't a proof that was found to be false.

------
jorgeortiz85
Read: "Proofs and Refutations" by Imre Lakatos

