
The Existential Risk of Mathematical Error - gwern
http://www.gwern.net/The%20Existential%20Risk%20of%20Mathematical%20Error
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ttctciyf
Reminds me of the late lamented Pertti Lounesto who made himself quite
unpopular by looking for counterexamples to some accepted proofs and posting
about them repeatedly on Usenet. See
[http://users.tkk.fi/ppuska/mirror/Lounesto/counterexamples.h...](http://users.tkk.fi/ppuska/mirror/Lounesto/counterexamples.htm)
for example.

Lounesto was a controversial and sometimes reviled figure, but I (purely as a
non-mathematician and sometime reader of his posts) was very sorry to hear of
his death in a swimming incident about 10 years ago now. Nice that his website
is still available though.

I tend to agree with his idea that it's valuable to search for counterexamples
to newer proofs - a sort of application of Popperian principles in the
mathematical domain I suppose. I've no idea on the validity of his claims
about specific counterexamples, though.

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onosendai
This reminds me of an excellent short story by Ted Chiang, "Division by Zero",
which deals with a mathematician's deteriorating mental state and attempted
suicide as a consequence of proving the inconsistency of mathematics.

The whole thing
([http://www.fantasticmetropolis.com/i/division/full](http://www.fantasticmetropolis.com/i/division/full))
seems to be temporarily offline, but here's a pretty good summary:
[http://kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumbe...](http://kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumber=mf194)

~~~
MasterScrat
Fiction! it's a fiction. I didn't realize until after I finished. For some
reason I feel better now.

~~~
nitrogen
On the subject of mathematical fiction, the story of the secret number bleem
is interesting:
[http://www.strangehorizons.com/2000/20001120/secret_number.s...](http://www.strangehorizons.com/2000/20001120/secret_number.shtml)

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Bakkot
No mention of machine proof? Surely those raise the upper bound on our
confidence in a new result significantly.

~~~
betterunix
What machine proofs do is concentrate the risk of errors on the proof checker.
That is great if your proof checker is reliable, and _catastrophic_ if it is
unreliable. Fortunately, proof checkers need not be complicated; you only need
a small kernel, from which you can build the rest of the theory.

The other issue is usability. Coq is awesome, but proving something more
complicated than basic arithmetic results is _hard_ to do. Likewise with ACL2,
PVS, and similar systems. It is getting better, but we are still not quite
there.

~~~
kvb
It's hard to do, but it _is_ being done. See e.g. Gonthier's celebrated work
formalizing the Feit-Thompson theorem[1].

[1] [http://www.msr-inria.fr/news/feit-thomson-proved-in-coq/](http://www.msr-
inria.fr/news/feit-thomson-proved-in-coq/)

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AlexDanger
Great article, and I highly recommend the rest of the articles on gwern.net if
you're looking for thought provoking and well-researched writing on a range of
topics.

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elliotlai
Agda!

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yOutely
What is this blog and why is it getting popular? The articles are very long
and dense with no clear point.

"In some respects, there is nothing to be said; in other respects, there is
much to be said" Please edit your blog posts to remove phrases like this which
damage reading comprehension. Your opening sentence "Mathematical error has
been rarely examined except as a possibility and motivating reason for
research into formal methods" is long winded, vague, boring, and tells me
nothing about what this post is about.

I think you can write much shorter, clearer articles that are not dense,
rambling and inaccessible if you put some time into organization and structure
and read your blog critically from a layman's perspective.

~~~
gbog
You know, sometime some thought do actually require sentences with more than
five words, and some people might actually appreciate rich and flourishing
vocabularies. See Proust.

There is a limit, were the author is hiding emptiness and sterility behind
artificial obscurity and complexity, but I'd say this article is far from
that.

~~~
adrianN
Yes, some arguments require complicated reasoning. However, even in such cases
the author still is responsible for keeping the reader's interest. For this it
is for example helpful to start with an introduction to the topic and end with
a summary. Writing is also better if it doesn't contain phrases that add
little to nothing to the content.

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consider_this
"This sentence is false".

~~~
mjcohen
This sentence is not.

